html/ 0000755 0001750 0001750 00000000000 11301733652 011673 5 ustar cpaulin cpaulin html/index_global_H.html 0000644 0001750 0001750 00000022476 11301735615 015473 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_axiom_R.html 0000644 0001750 0001750 00000022476 11301735615 015362 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_variable_O.html 0000644 0001750 0001750 00000022476 11301735615 016027 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_section_S.html 0000644 0001750 0001750 00000022676 11301735615 015714 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
S (section)
sets [in
Sets ]
html/index_global_F.html 0000644 0001750 0001750 00000070024 11301735615 015461 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
F
fcomp [definition, in
Ccpo ]
fcomp2 [definition, in
Ccpo ]
fcomp2_simpl [lemma, in
Ccpo ]
fcomp_simpl [lemma, in
Ccpo ]
fcont [record, in
Ccpo ]
fcontinuous [projection, in
Ccpo ]
fcontm [projection, in
Ccpo ]
Fcontm [definition, in
Ccpo ]
Fcontm_continuous [instance, in
Ccpo ]
fcontm_fcont_comp_simpl [lemma, in
Ccpo ]
fcontm_monotonic [instance, in
Ccpo ]
fcont0 [definition, in
Ccpo ]
fcont2_comp [definition, in
Ccpo ]
fcont2_COMP [definition, in
Ccpo ]
fcont2_comp_simpl [lemma, in
Ccpo ]
fcont_app [definition, in
Ccpo ]
fcont_app_continuous [lemma, in
Ccpo ]
fcont_app_simpl [lemma, in
Ccpo ]
fcont_COMP [definition, in
Ccpo ]
fcont_comp [definition, in
Ccpo ]
fcont_Comp [definition, in
Ccpo ]
fcont_comp_continuous [instance, in
Ccpo ]
fcont_Comp_continuous2 [instance, in
Ccpo ]
fcont_comp_le_compat [lemma, in
Ccpo ]
fcont_COMP_simpl [lemma, in
Ccpo ]
fcont_Comp_simpl [lemma, in
Ccpo ]
fcont_comp_simpl [lemma, in
Ccpo ]
fcont_continuous [lemma, in
Ccpo ]
fcont_continuous2 [instance, in
Ccpo ]
fcont_cpo [instance, in
Ccpo ]
fcont_eq [lemma, in
Ccpo ]
fcont_eq_compat2 [lemma, in
Ccpo ]
fcont_eq_elim [lemma, in
Ccpo ]
fcont_eq_intro [lemma, in
Ccpo ]
fcont_fun [definition, in
Ccpo ]
fcont_ishift [definition, in
Ccpo ]
fcont_le [lemma, in
Ccpo ]
fcont_le_compat2 [lemma, in
Ccpo ]
fcont_le_elim [lemma, in
Ccpo ]
fcont_le_intro [lemma, in
Ccpo ]
fcont_lub [definition, in
Ccpo ]
fcont_lub_continuous [instance, in
Ccpo ]
fcont_lub_simpl [lemma, in
Ccpo ]
fcont_mshift [definition, in
Ccpo ]
fcont_ord [instance, in
Ccpo ]
fcont_SEQ [definition, in
Ccpo ]
fcont_SEQ_simpl [lemma, in
Ccpo ]
fcpo [instance, in
Ccpo ]
fcpo_lub_simpl [lemma, in
Ccpo ]
feq [definition, in
Misc ]
feq_refl [lemma, in
Misc ]
feq_sym [lemma, in
Misc ]
feq_trans [lemma, in
Misc ]
finite [inductive, in
Sets ]
finite_dec [lemma, in
Sets ]
finite_full_dec [lemma, in
Sets ]
finite_incl [lemma, in
Sets ]
finite_inter [lemma, in
Sets ]
finite_rem [lemma, in
Sets ]
finite_union [lemma, in
Sets ]
fin_add [lemma, in
Sets ]
fin_add_in [lemma, in
Sets ]
fin_empty [lemma, in
Sets ]
fin_eqset [lemma, in
Sets ]
fin_eq_add [constructor, in
Sets ]
fin_eq_empty [constructor, in
Sets ]
fixp [definition, in
Ccpo ]
Fixp [definition, in
Ccpo ]
fixp_continuous [lemma, in
Ccpo ]
fixp_continuous_eq [lemma, in
Ccpo ]
fixp_cte [definition, in
Ccpo ]
fixp_eq [lemma, in
Ccpo ]
fixp_ind [lemma, in
Ccpo ]
fixp_inv [lemma, in
Ccpo ]
fixp_le [lemma, in
Ccpo ]
fixp_le_compat [lemma, in
Ccpo ]
fixp_monotonic [instance, in
Ccpo ]
Fixp_simpl [lemma, in
Ccpo ]
fmon [record, in
Ccpo ]
fmono [instance, in
Ccpo ]
fmonotonic [projection, in
Ccpo ]
fmonotonic [instance, in
Ccpo ]
fmonotonic2 [instance, in
Ccpo ]
fmont [projection, in
Ccpo ]
fmon2_mon [instance, in
Ccpo ]
fmon_cpo [instance, in
Ccpo ]
fmon_cte_comp [lemma, in
Ccpo ]
fmon_diag_simpl [lemma, in
Ccpo ]
fmon_eq [lemma, in
Ccpo ]
fmon_le [lemma, in
Ccpo ]
fmon_le_compat2 [lemma, in
Ccpo ]
fmon_lub_simpl [lemma, in
Ccpo ]
fnatO_elim [lemma, in
Ccpo ]
fnatO_intro [definition, in
Ccpo ]
ford [instance, in
Ccpo ]
ford_eq_elim [lemma, in
Ccpo ]
ford_eq_intro [lemma, in
Ccpo ]
ford_le_elim [lemma, in
Ccpo ]
ford_le_intro [lemma, in
Ccpo ]
fstable [instance, in
Ccpo ]
fstable2 [instance, in
Ccpo ]
full [definition, in
Sets ]
fun2 [definition, in
Ccpo ]
fun2_monotonic [instance, in
Ccpo ]
fun2_mon2 [lemma, in
Ccpo ]
fun_ext [definition, in
Ccpo ]
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_projection_C.html 0000644 0001750 0001750 00000023063 11301735615 016373 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
C (projection)
continuous2_intro [in
Ccpo ]
cont_intro [in
Ccpo ]
html/index_section_J.html 0000644 0001750 0001750 00000022476 11301735615 015701 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_projection__.html 0000644 0001750 0001750 00000022476 11301735615 016436 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_instance_R.html 0000644 0001750 0001750 00000023244 11301735615 016043 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
R (instance)
Rules.Mchoice_mon2 [in
Prog ]
Rules.Mif_mon2 [in
Prog ]
Rules.prod_distr_mon2 [in
Prog ]
html/index_library_W.html 0000644 0001750 0001750 00000022476 11301735615 015716 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_module_B.html 0000644 0001750 0001750 00000022725 11301735615 015507 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
B (module)
Bernoulli [in
Bernoulli ]
html/index_variable_L.html 0000644 0001750 0001750 00000022476 11301735615 016024 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_global_J.html 0000644 0001750 0001750 00000022476 11301735615 015475 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_definition_A.html 0000644 0001750 0001750 00000023140 11301735615 016341 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
A (definition)
add [in
Sets ]
admissible [in
Ccpo ]
app2 [in
Ccpo ]
html/toc.html 0000644 0001750 0001750 00000037635 11301735615 013365 0 ustar cpaulin cpaulin
Table of contents
html/index_record_E.html 0000644 0001750 0001750 00000022476 11301735615 015506 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_projection_I.html 0000644 0001750 0001750 00000022476 11301735615 016410 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_instance_B.html 0000644 0001750 0001750 00000022755 11301735615 016031 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
B (instance)
Bernoulli.Fbern_mon [in
Bernoulli ]
html/index_axiom_I.html 0000644 0001750 0001750 00000022476 11301735615 015351 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_axiom_C.html 0000644 0001750 0001750 00000022476 11301735615 015343 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_instance_N.html 0000644 0001750 0001750 00000022700 11301735615 016033 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
N (instance)
natO [in
Ccpo ]
html/index_instance_K.html 0000644 0001750 0001750 00000022476 11301735615 016042 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_inductive_M.html 0000644 0001750 0001750 00000023041 11301735615 016217 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
M (inductive)
monotonic [in
Ccpo ]
monotonic2 [in
Ccpo ]
html/index_global_D.html 0000644 0001750 0001750 00000024250 11301735615 015457 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
D
Dbot [projection, in
Ccpo ]
dec [definition, in
Sets ]
diag [definition, in
Ccpo ]
diag_le_compat [lemma, in
Ccpo ]
diag_shift [lemma, in
Ccpo ]
double_app [definition, in
Ccpo ]
double_lub_diag [lemma, in
Ccpo ]
double_lub_shift [lemma, in
Ccpo ]
D0 [projection, in
Ccpo ]
html/index_definition_Q.html 0000644 0001750 0001750 00000022476 11301735615 016374 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_axiom_G.html 0000644 0001750 0001750 00000022476 11301735615 015347 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_global_X.html 0000644 0001750 0001750 00000022476 11301735615 015513 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_lemma_Y.html 0000644 0001750 0001750 00000025037 11301735615 015343 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
Y (lemma)
Ycart.FYcart_lt_simpl [in
Ycart ]
Ycart.FYcart_simpl [in
Ycart ]
Ycart.FYcart_val [in
Ycart ]
Ycart.p_diff [in
Ycart ]
Ycart.p_equation [in
Ycart ]
Ycart.p_lub [in
Ycart ]
Ycart.P_prod [in
Ycart ]
Ycart.Ycart_eqx [in
Ycart ]
Ycart.Ycart_ltx [in
Ycart ]
Ycart.Ycart_one_simpl [in
Ycart ]
Ycart.Ycart_term1 [in
Ycart ]
Ycart.Ycart_term2 [in
Ycart ]
html/index_projection_X.html 0000644 0001750 0001750 00000022476 11301735615 016427 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_inductive_J.html 0000644 0001750 0001750 00000022476 11301735615 016227 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_lemma_J.html 0000644 0001750 0001750 00000022476 11301735615 015330 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_library_K.html 0000644 0001750 0001750 00000022476 11301735615 015702 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_module_E.html 0000644 0001750 0001750 00000022476 11301735615 015515 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_projection_E.html 0000644 0001750 0001750 00000022476 11301735615 016404 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_constructor_H.html 0000644 0001750 0001750 00000022476 11301735615 016620 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_module_F.html 0000644 0001750 0001750 00000022476 11301735615 015516 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_constructor_Z.html 0000644 0001750 0001750 00000022476 11301735615 016642 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_inductive_K.html 0000644 0001750 0001750 00000022476 11301735615 016230 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_constructor_S.html 0000644 0001750 0001750 00000023055 11301735615 016625 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
S (constructor)
stable2_intro [in
Ccpo ]
stable_def [in
Ccpo ]
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_
(1807 entries)
Instance Index
A
B C D
E
F G
H
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K
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X
Y Z
_
(109 entries)
Projection Index
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B
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_
(32 entries)
Record Index
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E
F G
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J
K
L
M N
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R
S T
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X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
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R S T
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W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
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(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
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W
X
Y
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_
(12 entries)
Inductive Index
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B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
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R S T
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Y Z
_
(277 entries)
Module Index
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E
F
G
H
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L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
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R
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W
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Y Z
_
(13 entries)
html/Misc.html 0000644 0001750 0001750 00000070116 11301735614 013461 0 ustar cpaulin cpaulin
Misc
Library Misc
Set Implicit Arguments .
Require Export Setoid .
Definition of iterator compn
compn f u n x is defined as
Fixpoint compn (
A :Type)(f:A ->
A ->
A ) (
x :A) (
u :nat ->
A ) (
n :nat) {
struct n }:
A :=
match n with O =>
x | (
S p ) =>
f (
u p ) (
compn f x u p )
end .
Lemma comp0 :
forall (
A :Type) (
f :A ->
A ->
A ) (
x :A) (
u :nat ->
A ),
compn f x u 0 =
x .
trivial .
Save .
Lemma compS :
forall (
A :Type) (
f :A ->
A ->
A ) (
x :A) (
u :nat ->
A ) (
n :nat),
compn f x u (
S n ) =
f (
u n ) (
compn f x u n ).
trivial .
Save .
Lemma if_then :
forall (
P :Prop) (
b :{
P }+{ ~
P })(
A :Type)(p
q :A),
P -> (
if b then p else q ) =
p .
destruct b ;
simpl ;
intuition .
Save .
Lemma if_else :
forall (
P :Prop) (
b :{
P }+{ ~
P })(
A :Type)(p
q :A),
~
P -> (
if b then p else q ) =
q .
destruct b ;
simpl ;
intuition .
Save .
Definition class (
A :Prop) := ~ ~
A ->
A .
Lemma class_neg :
forall A :Prop,
class ( ~
A ).
unfold class ;
intuition .
Save .
Lemma class_false :
class False .
unfold class ;
intuition .
Save .
Hint Resolve class_neg class_false .
Definition orc (
A B :Prop) :=
forall C :Prop,
class C -> (
A ->
C ) -> (
B ->
C ) ->
C .
Lemma orc_left :
forall A B :Prop,
A ->
orc A B .
red ;
intuition .
Save .
Lemma orc_right :
forall A B :Prop,
B ->
orc A B .
red ;
intuition .
Save .
Hint Resolve orc_left orc_right .
Lemma class_orc :
forall A B ,
class (
orc A B ).
repeat red ;
intros .
apply H0 ;
red ;
intro .
apply H ;
red ;
intro .
apply H3 ;
apply H4 ;
auto .
Save .
Implicit Arguments class_orc [].
Lemma orc_intro :
forall A B , ( ~
A -> ~
B ->
False ) ->
orc A B .
intros ;
apply class_orc ;
red ;
intros .
apply H ;
red ;
auto .
Save .
Lemma class_and :
forall A B ,
class A ->
class B ->
class (
A /\
B ).
unfold class ;
intuition .
Save .
Lemma excluded_middle :
forall A ,
orc A ( ~
A ).
red ;
intros .
apply H ;
red ;
intro .
intuition .
Save .
Definition exc (
A :Type)(P:A ->
Prop ) :=
forall C :Prop,
class C -> (
forall x :A,
P x ->
C ) ->
C .
Lemma exc_intro :
forall (
A :Type)(P:A ->
Prop ) (
x :A),
P x ->
exc P .
red ;
firstorder .
Save .
Lemma class_exc :
forall (
A :Type)(P:A ->
Prop ),
class (
exc P ).
repeat red ;
intros .
apply H0 ;
clear H0 ;
red ;
intro .
apply H ;
clear H ;
red ;
intro H2 .
apply H2 ;
intros ;
auto .
apply H0 ;
apply (
H1 x );
auto .
Save .
Lemma exc_intro_class :
forall (
A :Type) (
P :A ->
Prop ), ((
forall x , ~
P x ) ->
False ) ->
exc P .
intros ;
apply class_exc ;
red ;
intros .
apply H ;
red ;
intros ;
auto .
apply H0 ;
apply exc_intro with (
x :=x);
auto .
Save .
Lemma not_and_elim_left :
forall A B , ~ (
A /\
B ) ->
A -> ~B.
intuition .
Save .
Lemma not_and_elim_right :
forall A B , ~ (
A /\
B ) ->
B -> ~A.
intuition .
Save .
Hint Resolve class_orc class_and class_exc excluded_middle .
Lemma class_double_neg :
forall P Q :
Prop ,
class Q -> (
P ->
Q ) -> ~ ~
P ->
Q .
intros .
apply (
excluded_middle (
A :=P));
auto .
Save .
Definition feq A B (
f g :
A ->
B ) :=
forall x ,
f x =
g x .
Lemma feq_refl :
forall A B (
f :A->B),
feq f f .
red ;
trivial .
Save .
Lemma feq_sym :
forall A B (
f g :
A ->
B ),
feq f g ->
feq g f .
unfold feq ;
auto .
Save .
Lemma feq_trans :
forall A B (
f g h :
A ->
B ),
feq f g ->
feq g h ->
feq f h .
unfold feq ;
intros .
transitivity (
g x );
auto .
Save .
Hint Resolve feq_refl .
Hint Immediate feq_sym .
Hint Unfold feq .
Add Parametric Relation (
A B :
Type ) : (
A ->
B ) (
feq (
A :=A) (
B :=B))
reflexivity proved by (
feq_refl (
A :=A) (
B :=B))
symmetry proved by (
feq_sym (
A :=A) (
B :=B))
transitivity proved by (
feq_trans (
A :=A) (
B :=B))
as feq_rel .
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Global Index
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Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
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W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
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Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/Choice.html 0000644 0001750 0001750 00000041147 11301735615 013763 0 ustar cpaulin cpaulin
Choice
Library Choice
Choice.v: An example of probabilistic choice
Require Export Prog .
Set Implicit Arguments .
Definition of a probabilistic choice
We interpret the probabilistic program which executes
two probabilistic programs and and then make a choice
between the two computed results.
let rec p () = let x = p1 () in let y = p2 () in choice x y
Section CHOICE .
Variable A :
Type .
Variables p1 p2 :
distr A .
Variable choice :
A ->
A ->
A .
Definition p :
distr A :=
Mlet p1 (
fun x =>
Mlet p2 (
fun y =>
Munit (
choice x y ))).
Main result
We estimate the probability for to satisfy given estimations for
both and .
Assumptions
We need extra properties on , and .
and terminate with probability
value on is not less than the sum of values of on separate elements.
If is a boolean function it means than if one of or satisfies
then will also satisfy
Hypothesis p1_terminates : (
mu p1 (
fone A ))==1.
Hypothesis p2_terminates : (
mu p2 (
fone A ))==1.
Variable Q :
MF A .
Hypothesis choiceok :
forall x y ,
Q x +
Q y <=
Q (
choice x y ).
Lemma choicerule :
forall k1 k2 ,
k1 <=
mu p1 Q ->
k2 <=
mu p2 Q -> (
k1 * ([1-]
k2 ) +
k2 ) <=
mu p Q .
unfold p ,Mlet,star;
simpl ;
unfold unit ;
intros .
apply Ole_trans with
(
mu p1 (
fun x :
A =>
mu p2 (
fun y :
A =>
Q x * ([1-] (
Q y )) +
Q y ))).
apply Ole_trans with
(
mu p1 (
fun x :
A =>
((
mu p2 (
fun y :
A =>
Q x * ([1-] (
Q y ))))
+(mu
p2 Q )))).
change
(
k1 * [1-]
k2 +
k2 <=
mu p1 (
fplus (
fun x :
A =>
mu p2 (
fun y :
A =>
Q x * [1-] (
Q y )))
(
fun x =>
mu p2 Q ))).
assert (
H1 :
fplusok (
fun x :
A =>
mu p2 (
fun y :
A =>
Q x * [1-] (
Q y )))
(
fun x =>
mu p2 Q )).
repeat red ;
unfold finv ;
intros .
apply Ole_trans with (
mu p2 (
fun y :
A => [1-] (
Q y ))).
apply (
mu_monotonic p2 );
auto .
apply (
mu_stable_inv p2 Q ).
setoid_rewrite (
mu_stable_plus p1 H1 ).
assert (
Heq1 :mu
p1 (
fun _ :
A =>
mu p2 Q ) ==
mu p2 Q ).
exact (
mu_cte_eq p1 (
mu p2 Q )
p1_terminates ).
rewrite Heq1 .
apply Ole_trans with (
k1 * [1-] (
mu p2 Q ) + (
mu p2 Q ));
auto .
Usimpl .
apply Ole_trans with (
mu p1 (
fun x :
A => (
Q x ) *
mu p2 (
finv Q ))).
apply Ole_trans with ((
mu p1 Q ) * (
mu p2 (
finv Q ))).
rewrite (
mu_one_inv p2 p2_terminates Q );
auto .
rewrite (
mu_stable_mult_right p1 (
mu p2 (
finv Q ))
Q );
auto .
apply (
mu_monotonic p1 );
intro x .
rewrite <- (
mu_stable_mult p2 (
Q x ) (
finv Q ));
auto .
apply (
mu_monotonic p1 );
intro x .
assert (
fplusok (
fun y :
A =>
Q x * [1-] (
Q y ))
Q ).
intro y ;
unfold finv ;
auto .
rewrite <- (
mu_stable_plus p2 H1 );
auto .
apply (
mu_monotonic p1 );
intro x .
apply (
mu_monotonic p2 );
intro y .
apply Ole_trans with ((
Q x ) + (
Q y ));
auto .
Save .
End CHOICE .
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Global Index
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K
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_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
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R S T U V
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Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_axiom_P.html 0000644 0001750 0001750 00000022476 11301735615 015360 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_global_W.html 0000644 0001750 0001750 00000022476 11301735615 015512 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_instance_X.html 0000644 0001750 0001750 00000022476 11301735615 016057 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_definition_U.html 0000644 0001750 0001750 00000065113 11301735615 016373 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
U (definition)
union [in
Sets ]
Universe.U1 [in
Utheory ]
Univ_prop.B2U [in
Uprop ]
Univ_prop.cauchy [in
Uprop ]
Univ_prop.diff [in
Uprop ]
Univ_prop.exp2 [in
Uprop ]
Univ_prop.fconj [in
Uprop ]
Univ_prop.Fconj [in
Uprop ]
Univ_prop.Fcte [in
Uprop ]
Univ_prop.fcte [in
Uprop ]
Univ_prop.fdiv [in
Uprop ]
Univ_prop.fesp [in
Uprop ]
Univ_prop.Finf [in
Uprop ]
Univ_prop.finf [in
Uprop ]
Univ_prop.finv [in
Uprop ]
Univ_prop.flub [in
Uprop ]
Univ_prop.fminus [in
Uprop ]
Univ_prop.Fmult [in
Uprop ]
Univ_prop.fmult [in
Uprop ]
Univ_prop.fone [in
Uprop ]
Univ_prop.fplus [in
Uprop ]
Univ_prop.fplusok [in
Uprop ]
Univ_prop.Fsup [in
Uprop ]
Univ_prop.fsup [in
Uprop ]
Univ_prop.full [in
Uprop ]
Univ_prop.fzero [in
Uprop ]
Univ_prop.G [in
Uprop ]
Univ_prop.glb [in
Uprop ]
Univ_prop.has_lim [in
Uprop ]
Univ_prop.Ieq [in
Uprop ]
Univ_prop.Iin [in
Uprop ]
Univ_prop.Iincl [in
Uprop ]
Univ_prop.Ilim [in
Uprop ]
Univ_prop.Imult [in
Uprop ]
Univ_prop.Imultk [in
Uprop ]
Univ_prop.inf [in
Uprop ]
Univ_prop.Iplus [in
Uprop ]
Univ_prop.is_limit [in
Uprop ]
Univ_prop.liminf [in
Uprop ]
Univ_prop.limsup [in
Uprop ]
Univ_prop.Low [in
Uprop ]
Univ_prop.Max [in
Uprop ]
Univ_prop.max [in
Uprop ]
Univ_prop.mean [in
Uprop ]
Univ_prop.MF [in
Uprop ]
Univ_prop.MFopp [in
Uprop ]
Univ_prop.min [in
Uprop ]
Univ_prop.Min [in
Uprop ]
Univ_prop.mufix [in
Uprop ]
Univ_prop.NB2U [in
Uprop ]
Univ_prop.Nmult [in
Uprop ]
Univ_prop.NMult [in
Uprop ]
Univ_prop.Nmult_def [in
Uprop ]
Univ_prop.nufix [in
Uprop ]
Univ_prop.Pmin [in
Uprop ]
Univ_prop.pmin [in
Uprop ]
Univ_prop.Prod [in
Uprop ]
Univ_prop.prod [in
Uprop ]
Univ_prop.prod_inf [in
Uprop ]
Univ_prop.retract [in
Uprop ]
Univ_prop.seq_max [in
Uprop ]
Univ_prop.serie [in
Uprop ]
Univ_prop.Serie [in
Uprop ]
Univ_prop.Sigma [in
Uprop ]
Univ_prop.sigma [in
Uprop ]
Univ_prop.singl [in
Uprop ]
Univ_prop.sMax [in
Uprop ]
Univ_prop.sup [in
Uprop ]
Univ_prop.UDiv [in
Uprop ]
Univ_prop.UEsp [in
Uprop ]
Univ_prop.Uesp [in
Uprop ]
Univ_prop.Uexp [in
Uprop ]
Univ_prop.UExp [in
Uprop ]
Univ_prop.Uglb [in
Uprop ]
Univ_prop.UInv [in
Uprop ]
Univ_prop.UInvopp [in
Uprop ]
Univ_prop.UInv_simpl [in
Uprop ]
Univ_prop.Ult [in
Uprop ]
Univ_prop.Ulub [in
Uprop ]
Univ_prop.UMinus [in
Uprop ]
Univ_prop.Uminus [in
Uprop ]
Univ_prop.UMult [in
Uprop ]
Univ_prop.UMult_simpl [in
Uprop ]
Univ_prop.Uopp [in
Uprop ]
Univ_prop.Up [in
Uprop ]
Univ_prop.UPlus [in
Uprop ]
Univ_prop.UPlus_simpl [in
Uprop ]
Univ_prop.U1min [in
Uprop ]
Univ_prop.wretract [in
Uprop ]
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_section_M.html 0000644 0001750 0001750 00000022476 11301735615 015704 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_instance_D.html 0000644 0001750 0001750 00000022476 11301735615 016033 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_constructor__.html 0000644 0001750 0001750 00000022476 11301735615 016647 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_axiom_Y.html 0000644 0001750 0001750 00000022476 11301735615 015371 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_global_C.html 0000644 0001750 0001750 00000106624 11301735615 015464 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
C
Ccpo [library]
Choice [module, in
Choice ]
Choice [library]
Choice.CHOICE [section, in
Choice ]
Choice.choicerule [lemma, in
Choice ]
Choice.CHOICE.A [variable, in
Choice ]
Choice.CHOICE.choice [variable, in
Choice ]
Choice.CHOICE.choiceok [variable, in
Choice ]
Choice.CHOICE.p1 [variable, in
Choice ]
Choice.CHOICE.p1_terminates [variable, in
Choice ]
Choice.CHOICE.p2 [variable, in
Choice ]
Choice.CHOICE.p2_terminates [variable, in
Choice ]
Choice.CHOICE.Q [variable, in
Choice ]
Choice.p [definition, in
Choice ]
class [definition, in
Misc ]
class_and [lemma, in
Misc ]
class_double_neg [lemma, in
Misc ]
class_exc [lemma, in
Misc ]
class_false [lemma, in
Misc ]
class_neg [lemma, in
Misc ]
class_orc [lemma, in
Misc ]
comp [definition, in
Ccpo ]
compl [definition, in
Sets ]
compn [definition, in
Misc ]
compS [lemma, in
Misc ]
comp0 [lemma, in
Misc ]
comp_monotonic2 [instance, in
Ccpo ]
comp_monotonic_left [lemma, in
Ccpo ]
comp_monotonic_right [lemma, in
Ccpo ]
comp_simpl [lemma, in
Ccpo ]
cont [constructor, in
Ccpo ]
continuous [record, in
Ccpo ]
continuous [inductive, in
Ccpo ]
continuous2 [record, in
Ccpo ]
continuous2 [inductive, in
Ccpo ]
continuous2_app [lemma, in
Ccpo ]
continuous2_comp [lemma, in
Ccpo ]
continuous2_comp2 [lemma, in
Ccpo ]
continuous2_continuous [lemma, in
Ccpo ]
continuous2_eq_compat [lemma, in
Ccpo ]
continuous2_intro [projection, in
Ccpo ]
continuous2_intro [constructor, in
Ccpo ]
continuous2_left [lemma, in
Ccpo ]
continuous2_right [lemma, in
Ccpo ]
continuous2_sym [lemma, in
Ccpo ]
continuous_comp [lemma, in
Ccpo ]
continuous_continuous2 [lemma, in
Ccpo ]
continuous_eq_compat [lemma, in
Ccpo ]
continuous_sym [lemma, in
Ccpo ]
cont0 [instance, in
Ccpo ]
cont2 [definition, in
Ccpo ]
cont2_continuous [instance, in
Ccpo ]
cont_app [definition, in
Ccpo ]
cont_app_monotonic [instance, in
Ccpo ]
cont_app_simpl [lemma, in
Ccpo ]
cont_intro [projection, in
Ccpo ]
cont_intro [constructor, in
Ccpo ]
Cover [library]
CoverFun [module, in
Cover ]
CoverFun.carac [definition, in
Cover ]
CoverFun.carac_eq_compat [lemma, in
Cover ]
CoverFun.carac_ge [definition, in
Cover ]
CoverFun.carac_gt [definition, in
Cover ]
CoverFun.carac_incl [lemma, in
Cover ]
CoverFun.carac_le [definition, in
Cover ]
CoverFun.carac_lt [definition, in
Cover ]
CoverFun.carac_monotonic [lemma, in
Cover ]
CoverFun.carac_one [lemma, in
Cover ]
CoverFun.carac_zero [lemma, in
Cover ]
CoverFun.cover [definition, in
Cover ]
CoverFun.cover_compl [lemma, in
Cover ]
CoverFun.cover_dec [lemma, in
Cover ]
CoverFun.cover_elim [lemma, in
Cover ]
CoverFun.cover_empty [lemma, in
Cover ]
CoverFun.cover_eqset_eq_stable [lemma, in
Cover ]
CoverFun.cover_eqset_stable [lemma, in
Cover ]
CoverFun.cover_eq_one [lemma, in
Cover ]
CoverFun.cover_eq_one_elim [lemma, in
Cover ]
CoverFun.cover_eq_one_elim_class [lemma, in
Cover ]
CoverFun.cover_eq_stable [lemma, in
Cover ]
CoverFun.cover_eq_zero [lemma, in
Cover ]
CoverFun.cover_eq_zero_elim [lemma, in
Cover ]
CoverFun.cover_esp_fun [lemma, in
Cover ]
CoverFun.cover_esp_fun_le [lemma, in
Cover ]
CoverFun.cover_esp_mult_left [lemma, in
Cover ]
CoverFun.cover_esp_mult_right [lemma, in
Cover ]
CoverFun.cover_incl_eq [lemma, in
Cover ]
CoverFun.cover_incl_fle [lemma, in
Cover ]
CoverFun.cover_inter_esp [lemma, in
Cover ]
CoverFun.cover_inter_mult [lemma, in
Cover ]
CoverFun.cover_let [lemma, in
Cover ]
CoverFun.cover_mult_fun [lemma, in
Cover ]
CoverFun.cover_ok [lemma, in
Cover ]
CoverFun.cover_orc_0_1 [lemma, in
Cover ]
CoverFun.cover_union [lemma, in
Cover ]
CoverFun.cover_unit [lemma, in
Cover ]
CoverFun.cover_zero_one [lemma, in
Cover ]
CoverFun.distr_restr [definition, in
Cover ]
CoverFun.distr_restr_simpl [lemma, in
Cover ]
CoverFun.fconj_zero_one [lemma, in
Cover ]
CoverFun.fesp_conj_zero_one [lemma, in
Cover ]
CoverFun.fesp_zero_one [definition, in
Cover ]
CoverFun.fesp_zero_one_mult_left [lemma, in
Cover ]
CoverFun.fesp_zero_one_mult_right [lemma, in
Cover ]
CoverFun.finv_zero_one [lemma, in
Cover ]
CoverFun.fplus_zero_one [lemma, in
Cover ]
CoverFun.ge_dec [definition, in
Cover ]
CoverFun.gt_dec [definition, in
Cover ]
CoverFun.is_ge [definition, in
Cover ]
CoverFun.is_gt [definition, in
Cover ]
CoverFun.is_le [definition, in
Cover ]
CoverFun.is_lt [definition, in
Cover ]
CoverFun.le_dec [definition, in
Cover ]
CoverFun.lt_dec [definition, in
Cover ]
CoverFun.Mrestr [definition, in
Cover ]
CoverFun.Mrestr0 [lemma, in
Cover ]
CoverFun.Mrestr1 [lemma, in
Cover ]
CoverFun.Mrestr_simpl [lemma, in
Cover ]
CoverFun.mu_cut [lemma, in
Cover ]
CoverFun.nb_elts [definition, in
Cover ]
CoverFun.nb_elts_lt_ge [lemma, in
Cover ]
CoverFun.nb_elts_lt_le [lemma, in
Cover ]
CoverFun.nb_elts_true [lemma, in
Cover ]
CoverFun.Random_carac [lemma, in
Cover ]
CoverFun.Random_le [lemma, in
Cover ]
CoverFun.Random_lt [lemma, in
Cover ]
CoverFun.range_cover [lemma, in
Cover ]
CoverFun.retract_fin [definition, in
Cover ]
CoverFun.retract_fin_incl [lemma, in
Cover ]
CoverFun.retract_fin_inv [lemma, in
Cover ]
CoverFun.retract_fin_le [lemma, in
Cover ]
CoverFun.SigmaFinite [section, in
Cover ]
CoverFun.SigmaFinite.A [variable, in
Cover ]
CoverFun.SigmaFinite.decA [variable, in
Cover ]
CoverFun.SigmaFinite.RandomFinite [section, in
Cover ]
CoverFun.sigma_fin [definition, in
Cover ]
CoverFun.sigma_fin_cte [lemma, in
Cover ]
CoverFun.sigma_fin_eqset [lemma, in
Cover ]
CoverFun.sigma_fin_eq_compat [lemma, in
Cover ]
CoverFun.sigma_fin_incl [lemma, in
Cover ]
CoverFun.sigma_fin_inv [lemma, in
Cover ]
CoverFun.sigma_fin_monotonic [lemma, in
Cover ]
CoverFun.sigma_fin_mult [lemma, in
Cover ]
CoverFun.sigma_fin_plus [lemma, in
Cover ]
CoverFun.sigma_fin_prod_ge [lemma, in
Cover ]
CoverFun.sigma_fin_prod_le [lemma, in
Cover ]
CoverFun.sigma_fin_prod_maj [lemma, in
Cover ]
CoverFun.sigma_fin_rem [lemma, in
Cover ]
CoverFun.sigma_fin_unique [lemma, in
Cover ]
CoverFun.zero_one [definition, in
Cover ]
CoverFun.zero_one_cover [lemma, in
Cover ]
CoverFun.zero_one_not_one [lemma, in
Cover ]
CoverFun.zero_one_not_zero [lemma, in
Cover ]
CP [module, in
Ycart ]
cpo [record, in
Ccpo ]
cpo_ord_equiv [definition, in
Ccpo ]
cshift [definition, in
Ccpo ]
cshift_continuous2 [instance, in
Ccpo ]
cshift_simpl [lemma, in
Ccpo ]
cte [definition, in
Ccpo ]
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_library_L.html 0000644 0001750 0001750 00000022476 11301735615 015703 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_definition_W.html 0000644 0001750 0001750 00000022476 11301735615 016402 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_projection_K.html 0000644 0001750 0001750 00000022476 11301735615 016412 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_record_A.html 0000644 0001750 0001750 00000022476 11301735615 015502 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_projection_A.html 0000644 0001750 0001750 00000022476 11301735615 016400 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_lemma_O.html 0000644 0001750 0001750 00000025420 11301735615 015325 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
O (lemma)
Oeq_le [in
Ccpo ]
Oeq_le_sym [in
Ccpo ]
Oeq_refl [in
Ccpo ]
Oeq_refl_eq [in
Ccpo ]
Oeq_sym [in
Ccpo ]
Oeq_trans [in
Ccpo ]
Ole_antisym [in
Ccpo ]
Ole_eq_compat [in
Ccpo ]
Ole_eq_left [in
Ccpo ]
Ole_eq_right [in
Ccpo ]
Ole_refl [in
Ccpo ]
Ole_refl_eq [in
Ccpo ]
Ole_refl_eq_inv [in
Ccpo ]
Ole_trans [in
Ccpo ]
orc_intro [in
Misc ]
orc_left [in
Misc ]
orc_right [in
Misc ]
html/index_variable_Z.html 0000644 0001750 0001750 00000022476 11301735615 016042 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_definition_N.html 0000644 0001750 0001750 00000023041 11301735615 016356 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
N (definition)
notempty [in
Sets ]
nth_finite [in
Sets ]
html/index_definition_Z.html 0000644 0001750 0001750 00000022476 11301735615 016405 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_variable_Y.html 0000644 0001750 0001750 00000023117 11301735615 016032 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
Y (variable)
Ycart.YcartExample.K [in
Ycart ]
Ycart.YcartExample.N [in
Ycart ]
html/index_definition_M.html 0000644 0001750 0001750 00000025602 11301735615 016362 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
M (definition)
mfun2 [in
Ccpo ]
Monad.le_esp [in
Monads ]
Monad.le_plus [in
Monads ]
Monad.le_plus_cte [in
Monads ]
Monad.M [in
Monads ]
Monad.stable_inv [in
Monads ]
Monad.stable_mult [in
Monads ]
Monad.stable_plus [in
Monads ]
Monad.star [in
Monads ]
Monad.unit [in
Monads ]
mon2 [in
Ccpo ]
mon_comp [in
Ccpo ]
mon_ord_equiv [in
Ccpo ]
mseq_cte [in
Ccpo ]
mseq_lift_left [in
Ccpo ]
mseq_lift_right [in
Ccpo ]
mshift [in
Ccpo ]
html/index_inductive_G.html 0000644 0001750 0001750 00000022476 11301735615 016224 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_inductive_C.html 0000644 0001750 0001750 00000023045 11301735615 016211 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
C (inductive)
continuous [in
Ccpo ]
continuous2 [in
Ccpo ]
html/index_constructor_A.html 0000644 0001750 0001750 00000022476 11301735615 016611 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_section_K.html 0000644 0001750 0001750 00000022476 11301735615 015702 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_library_F.html 0000644 0001750 0001750 00000022476 11301735615 015675 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_section_N.html 0000644 0001750 0001750 00000022476 11301735615 015705 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_module__.html 0000644 0001750 0001750 00000022476 11301735615 015547 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
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Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
D (projection)
Dbot [in
Ccpo ]
D0 [in
Ccpo ]
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Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
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Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
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Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/all-gal.html 0000644 0001750 0001750 00005244735 11301733653 014117 0 ustar cpaulin cpaulin
Library Misc
Definition of iterator compn
compn f u n x is defined as
Fixpoint compn (
A :Type)(f:A ->
A ->
A ) (
x :A) (
u :nat ->
A ) (
n :nat) {
struct n }:
A :=
match n with O =>
x | (
S p ) =>
f (
u p ) (
compn f x u p )
end .
Lemma comp0 :
forall (
A :Type) (
f :A ->
A ->
A ) (
x :A) (
u :nat ->
A ),
compn f x u 0 =
x .
Lemma compS :
forall (
A :Type) (
f :A ->
A ->
A ) (
x :A) (
u :nat ->
A ) (
n :nat),
compn f x u (
S n ) =
f (
u n ) (
compn f x u n ).
Lemma if_then :
forall (
P :Prop) (
b :{
P }+{ ~
P })(
A :Type)(p
q :A),
P -> (
if b then p else q ) =
p .
Lemma if_else :
forall (
P :Prop) (
b :{
P }+{ ~
P })(
A :Type)(p
q :A),
~
P -> (
if b then p else q ) =
q .
Definition class (
A :Prop) := ~ ~
A ->
A .
Lemma class_neg :
forall A :Prop,
class ( ~
A ).
Lemma class_false :
class False .
Hint Resolve class_neg class_false .
Definition orc (
A B :Prop) :=
forall C :Prop,
class C -> (
A ->
C ) -> (
B ->
C ) ->
C .
Lemma orc_left :
forall A B :Prop,
A ->
orc A B .
Lemma orc_right :
forall A B :Prop,
B ->
orc A B .
Hint Resolve orc_left orc_right .
Lemma class_orc :
forall A B ,
class (
orc A B ).
Implicit Arguments class_orc [].
Lemma orc_intro :
forall A B , ( ~
A -> ~
B ->
False ) ->
orc A B .
Lemma class_and :
forall A B ,
class A ->
class B ->
class (
A /\
B ).
Lemma excluded_middle :
forall A ,
orc A ( ~
A ).
Definition exc (
A :Type)(P:A ->
Prop ) :=
forall C :Prop,
class C -> (
forall x :A,
P x ->
C ) ->
C .
Lemma exc_intro :
forall (
A :Type)(P:A ->
Prop ) (
x :A),
P x ->
exc P .
Lemma class_exc :
forall (
A :Type)(P:A ->
Prop ),
class (
exc P ).
Lemma exc_intro_class :
forall (
A :Type) (
P :A ->
Prop ), ((
forall x , ~
P x ) ->
False ) ->
exc P .
Lemma not_and_elim_left :
forall A B , ~ (
A /\
B ) ->
A -> ~B.
Lemma not_and_elim_right :
forall A B , ~ (
A /\
B ) ->
B -> ~A.
Hint Resolve class_orc class_and class_exc excluded_middle .
Lemma class_double_neg :
forall P Q :
Prop ,
class Q -> (
P ->
Q ) -> ~ ~
P ->
Q .
Definition feq A B (
f g :
A ->
B ) :=
forall x ,
f x =
g x .
Lemma feq_refl :
forall A B (
f :A->B),
feq f f .
Lemma feq_sym :
forall A B (
f g :
A ->
B ),
feq f g ->
feq g f .
Lemma feq_trans :
forall A B (
f g h :
A ->
B ),
feq f g ->
feq g h ->
feq f h .
Hint Resolve feq_refl .
Hint Immediate feq_sym .
Hint Unfold feq .
Add Parametric Relation (
A B :
Type ) : (
A ->
B ) (
feq (
A :=A) (
B :=B))
reflexivity proved by (
feq_refl (
A :=A) (
B :=B))
symmetry proved by (
feq_sym (
A :=A) (
B :=B))
transitivity proved by (
feq_trans (
A :=A) (
B :=B))
as feq_rel .
Library Ccpo
Ccpo.v: Specification and properties of a cpo
Definition eq_rel {
A } (
E1 E2 :relation
A ) :=
forall x y ,
E1 x y <->
E2 x y .
Class Order {
A } (
E :relation
A ) (
R :relation
A ) :=
{
reflexive :>
Reflexive R ;
order_eq :
forall x y ,
R x y /\
R y x <->
E x y ;
transitive :>
Transitive R }.
Instance OrderEqRefl `{
Order A E R } :
Reflexive E .
Save .
Instance OrderEqSym `{
Order A E R } :
Symmetric E .
Save .
Instance OrderEqTrans `{
Order A E R } :
Transitive E .
Save .
Instance OrderEquiv `{
Order A E R } :
Equivalence E .
Save .
Class ord A :=
{
Oeq :
relation A ;
Ole :
relation A ;
order_rel :>
Order Oeq Ole }.
Instance OrdSetoid `(
o :ord
A ) :
Setoid A .
Defined .
Add Parametric Relation {
A } {
o :ord
A } :
A (@
Oeq _ o )
reflexivity proved by OrderEqRefl
symmetry proved by OrderEqSym
transitivity proved by OrderEqTrans
as Oeq_setoid .
Infix "<=" :=
Ole .
Infix "==" :=
Oeq :
type_scope .
Lemma Ole_refl_eq :
forall {
O } {
o :ord
O } (
x y :O),
x ==
y ->
x <=
y .
Hint Immediate @
Ole_refl_eq .
Lemma Ole_refl_eq_inv :
forall {
O } {
o :ord
O } (
x y :O),
x ==
y ->
y <=
x .
Hint Immediate @
Ole_refl_eq_inv .
Lemma Ole_trans :
forall {
O } {
o :ord
O } (
x y z :O),
x <=
y ->
y <=
z ->
x <=
z .
Lemma Ole_refl :
forall {
O } {
o :ord
O } (
x :O),
x <=
x .
Hint Resolve @
Ole_refl .
Add Parametric Relation {
A } {
o :ord
A } :
A (@
Ole _ o )
reflexivity proved by Ole_refl
transitivity proved by Ole_trans
as Ole_setoid .
Lemma Ole_antisym :
forall {
O } {
o :ord
O } (
x y :O),
x <=
y ->
y <=
x ->
x ==y.
Hint Immediate @
Ole_antisym .
Lemma Oeq_refl :
forall {
O } {
o :ord
O } (
x :O),
x ==
x .
Hint Resolve @
Oeq_refl .
Lemma Oeq_refl_eq :
forall {
O } {
o :ord
O } (
x y :O),
x =y ->
x ==
y .
Hint Resolve @
Oeq_refl_eq .
Lemma Oeq_sym :
forall {
O } {
o :ord
O } (
x y :O),
x ==
y ->
y ==
x .
Lemma Oeq_le :
forall {
O } {
o :ord
O } (
x y :O),
x ==
y ->
x <=
y .
Lemma Oeq_le_sym :
forall {
O } {
o :ord
O } (
x y :O),
x ==
y ->
y <=
x .
Hint Resolve @
Oeq_le .
Hint Immediate @
Oeq_sym @
Oeq_le_sym .
Lemma Oeq_trans
:
forall {
O } {
o :ord
O } (
x y z :O),
x ==
y ->
y ==
z ->
x ==
z .
Hint Resolve @
Oeq_trans .
Instance Ole_eq_compat_iff `(
o :ord
A ) :
Morphism (
Oeq (
A :=A) ==>
Oeq (
A :=A) ==>
iff ) (
Ole (
ord :=o)).
Save .
Equivalence of orders
Definition eq_ord {
O } (
o1 o2 :ord
O ) :=
eq_rel (
Ole (
ord :=o1)) (
Ole (
ord :=o2)).
Lemma eq_ord_equiv :
forall {
O } (
o1 o2 :ord
O ),
eq_ord o1 o2 ->
eq_rel (
Oeq (
ord :=o1)) (
Oeq (
ord :=o2)).
Lemma Ole_eq_compat :
forall {
O } {
o :ord
O } (
x1 x2 :
O ),
x1 ==
x2 ->
forall x3 x4 :
O ,
x3 ==
x4 ->
x1 <=
x3 ->
x2 <=
x4 .
Lemma Ole_eq_right :
forall {
O } {
o :ord
O } (
x y z :
O ),
x <=
y ->
y ==
z ->
x <=
z .
Lemma Ole_eq_left :
forall {
O } {
o :ord
O } (
x y z :
O ),
x ==
y ->
y <=
z ->
x <=
z .
Add Parametric Morphism `{
o :ord
A } : (
Oeq (
A :=A))
with signature Oeq &
Oeq &
iff as Oeq_iff_morphism .
Qed .
Add Parametric Morphism `{
o :ord
A } : (
Ole (
A :=A))
with signature Oeq &
Oeq &
iff as Ole_iff_morphism .
Qed .
Add Parametric Morphism `{
o :ord
A } : (
Ole (
A :=A))
with signature Ole -->
Ole &
Basics.impl as Ole_impl_morphism .
Qed .
Definition Iord : forall O {o :ord O }, ord O .Defined .Implicit Arguments Iord [[o ]].
Definition fun_ext A B (R :relation B ) : relation (A -> B ) := fun f g => forall x , R (f x ) (g x ).Implicit Arguments fun_ext [B ].
ford f g := forall x, f x <= g x
Instance ford A O {
o :ord
O } :
ord (
A ->
O ) :=
{
Oeq :=fun_ext
A (
Oeq (
A :=O));
Ole :=fun_ext
A (
Ole (
A :=O))}.
Defined .
Lemma ford_le_elim :
forall A O (
o :ord
O ) (
f g :A ->
O ),
f <=
g ->
forall n ,
f n <=
g n .
Hint Immediate ford_le_elim .
Lemma ford_le_intro :
forall A O (
o :ord
O ) (
f g :A ->
O ), (
forall n ,
f n <=
g n ) ->
f <=
g .
Hint Resolve ford_le_intro .
Lemma ford_eq_elim :
forall A O (
o :ord
O ) (
f g :A ->
O ),
f ==
g ->
forall n ,
f n ==
g n .
Hint Immediate ford_eq_elim .
Lemma ford_eq_intro :
forall A O (
o :ord
O ) (
f g :A ->
O ), (
forall n ,
f n ==
g n ) ->
f ==
g .
Hint Resolve ford_eq_intro .
Definition and properties
Class monotonic `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f :
O1 ->
O2 ) :=
monotonic_def :
forall x y ,
x <=
y ->
f x <=
f y .
Lemma monotonic_intro :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f :
O1 ->
O2 ),
(
forall x y ,
x <=
y ->
f x <=
f y ) ->
monotonic f .
Hint Resolve @
monotonic_intro .
Instance monotonic_morphism `{
o1 :ord
O1 } `{
o2 :ord
O2 }
(
f :
O1 ->
O2 ) {
m :monotonic
f }
:
Morphism (
Ole (
A :=O1) &
Ole (
A :=O2))
f .
Save .
Class stable `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f :
O1 ->
O2 ) :=
stable_def :
forall x y ,
x ==
y ->
f x ==
f y .
Hint Unfold stable .
Lemma stable_intro :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f :
O1 ->
O2 ),
(
forall x y ,
x ==
y ->
f x ==
f y ) ->
stable f .
Hint Resolve @
stable_intro .
Instance stable_morphism `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f :
O1 ->
O2 ) {
s :stable
f }
:
Morphism (
Oeq (
A :=O1) &
Oeq (
A :=O2))
f .
Save .
Type classes Opaque monotonic stable .
Instance monotonic_stable `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f :
O1 ->
O2 ) {
m :monotonic
f }
:
stable f .
Save .
Type of monotonic functions
Record fmon `{
o1 :ord
O1 } `{
o2 :ord
O2 }:=
mon
{
fmont :>
O1 ->
O2 ;
fmonotonic :
monotonic fmont }.
Implicit Arguments mon [[
O1 ] [
o1 ] [
O2 ] [
o2 ] [
fmonotonic ]].
Implicit Arguments fmon [[
o1 ] [
o2 ]].
Hint Resolve @
fmonotonic .
Notation "O1 -m> O2" := (
fmon O1 O2 )
(
right associativity ,
at level 30) :
O_scope .
Notation "O1 --m> O2" := (
fmon O1 (
o1 :=Iord
O1 )
O2 )
(
right associativity ,
at level 30) :
O_scope .
Notation "O1 --m-> O2" := (
fmon O1 (
o1 :=Iord
O1 )
O2 (
o2 :=Iord
O2 ))
(
right associativity ,
at level 30) :
O_scope .
Notation "O1 -m-> O2" := (
fmon O1 O2 (
o2 :=Iord
O2 ))
(
right associativity ,
at level 30) :
O_scope .
Open Scope O_scope .
Lemma mon_simpl :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f :O1 ->
O2 ){
mf :
monotonic f }
x ,
mon f x =
f x .
Hint Resolve @
mon_simpl .
Instance fstable `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f :O1 -m>
O2 ) :
stable f .
Save .
Hint Resolve @
fstable .
Lemma fmon_le :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f :O1 -m>
O2 )
x y ,
x <=
y ->
f x <=
f y .
Hint Resolve @
fmon_le .
Lemma fmon_eq :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f :O1 -m>
O2 )
x y ,
x ==
y ->
f x ==
f y .
Hint Resolve @
fmon_eq .
Instance fmono O1 O2 {
o1 :ord
O1 } {
o2 :ord
O2 } :
ord (
O1 -m>
O2 )
:= {
Oeq :=
fun (
f g :
O1 -m>
O2 )=>
forall x ,
f x ==
g x ;
Ole :=
fun (
f g :
O1 -m>
O2 )=>
forall x ,
f x <=
g x }.
Defined .
Lemma mon_le_compat :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f g :O1 ->
O2 )
{
mf :monotonic
f } {
mg :monotonic
g },
f <=
g ->
mon f <=
mon g .
Hint Resolve @
mon_le_compat .
Lemma mon_eq_compat :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f g :O1->
O2 )
{
mf :monotonic
f } {
mg :monotonic
g },
f ==
g ->
mon f ==
mon g .
Hint Resolve @
mon_eq_compat .
Add Parametric Morphism `{
o1 :ord
O1 } `{
o2 :ord
O2 }
: (
fmont (
O1 :=O1) (
O2 :=O2))
with signature Oeq &
Oeq &
Oeq as fmont_eq_morphism .
Qed .
Monotonicity and dual order
Lemma Imonotonic `{o1 :ord O1 } `{o2 :ord O2 } (f :O1 -> O2 ) {m :monotonic f } monotonic (o1 :=Iord O1 ) (o2 :=Iord O2 ) f .Hint Extern 2 (@monotonic _ (Iord _ ) _ (Iord _ ) _ ) => apply @Imonotonic typeclass_instances .Definition imon `{o1 :ord O1 } `{o2 :ord O2 } (f :O1 -> O2 ) {m :monotonic f } : O1 --m-> O2 := mon (o1 :=Iord O1 ) (o2 :=Iord O2 ) f .Lemma imon_simpl : forall `{o1 :ord O1 } `{o2 :ord O2 } (f :O1 -> O2 ) {m :monotonic f } (x :O1),imon f x = f x .
Iord (A -> U )A -> Iord U
Lemma Iord_app {A } `{o1 :ord O1 } (x : A ) : ((A -> O1 ) --m-> O1 ).
Imon f
Definition Imon :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 }, (
O1 -m>
O2 ) -> (
O1 --m->
O2 ).
Defined .
Lemma Imon_simpl :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f :O1 -m>
O2 )(
x :O1),
Imon f x =
f x .
Monotonic functions with 2 arguments
Class monotonic2 `{
o1 :
ord O1 } `{
o2 :
ord O2 } `{
o3 :ord
O3 } (
f :O1 ->
O2 ->
O3 ) :=
monotonic2_intro :
forall (
x y :O1) (
z t :O2),
x <=
y ->
z <=
t ->
f x z <=
f y t .
Instance mon2_intro `{
o1 :
ord O1 } `{
o2 :
ord O2 } `{
o3 :ord
O3 } (
f :O1 ->
O2 ->
O3 )
{
m1 :monotonic
f } {
m2 :
forall x ,
monotonic (
f x )} :
monotonic2 f | 10.
Save .
Lemma mon2_elim1 `{
o1 :
ord O1 } `{
o2 :
ord O2 } `{
o3 :ord
O3 } (
f :O1 ->
O2 ->
O3 )
{
m :monotonic2
f } :
monotonic f .
Lemma mon2_elim2 `{
o1 :
ord O1 } `{
o2 :
ord O2 } `{
o3 :ord
O3 } (
f :O1 ->
O2 ->
O3 )
{
m :monotonic2
f } :
forall x ,
monotonic (
f x ).
Hint Immediate @
mon2_elim1 @
mon2_elim2 :
typeclass_instances .
Definition mon_comp {
A } `{
o1 :
ord O1 } `{
o2 :
ord O2 } (
f :A ->
O1 ->
O2 ) {
mf :forall
x ,
monotonic (
f x )}
:
A ->
O1 -m>
O2 :=
fun x =>
mon (
f x ).
Instance mon_fun_mon `{
o1 :
ord O1 } `{
o2 :
ord O2 } `{
o3 :ord
O3 } (
f :O1 ->
O2 ->
O3 )
{
m :monotonic2
f } :
monotonic (
fun x =>
mon (
f x )).
Save .
Class stable2 `{
o1 :
ord O1 } `{
o2 :
ord O2 } `{
o3 :ord
O3 } (
f :O1 ->
O2 ->
O3 ) :=
stable2_intro :
forall (
x y :O1) (
z t :O2),
x ==y ->
z ==
t ->
f x z ==
f y t .
Instance monotonic2_stable2 `{
o1 :
ord O1 } `{
o2 :
ord O2 } `{
o3 :ord
O3 }
(
f :O1 ->
O2 ->
O3 ) {
m :monotonic2
f } :
stable2 f .
Save .
Type classes Opaque monotonic2 stable2 .
Definition mon2 `{
o1 :
ord O1 } `{
o2 :
ord O2 } `{
o3 :ord
O3 } (
f :O1 ->
O2 ->
O3 )
{
mf :monotonic2
f } :
O1 -m>
O2 -m>
O3 :=
mon (
fun x =>
mon (
f x )).
Lemma mon2_simpl :
forall `{
o1 :
ord O1 } `{
o2 :
ord O2 } `{
o3 :ord
O3 } (
f :O1 ->
O2 ->
O3 )
{
mf :monotonic2
f }
x y ,
mon2 f x y =
f x y .
Hint Resolve @
mon2_simpl .
Lemma mon2_le_compat :
forall `{
o1 :
ord O1 } `{
o2 :
ord O2 } `{
o3 :ord
O3 }
(
f g :O1 ->
O2 ->
O3 ) {
mf :
monotonic2 f } {
mg :monotonic2
g },
f <=
g ->
mon2 f <=
mon2 g .
Definition fun2 `{
o1 :
ord O1 } `{
o2 :
ord O2 } `{
o3 :ord
O3 } (
f :O1 ->
O2 -m>
O3 )
:
O1 ->
O2 ->
O3 :=
fun x =>
f x .
Instance fmon2_mon `{
o1 :
ord O1 } `{
o2 :
ord O2 } `{
o3 :ord
O3 } (
f :O1 ->
O2 -m>
O3 ) :
forall x :O1,
monotonic (
fun2 f x ).
Save .
Instance fun2_monotonic `{
o1 :
ord O1 } `{
o2 :
ord O2 } `{
o3 :ord
O3 }
(
f :O1 ->
O2 -m>
O3 ) {
mf :monotonic
f } :
monotonic (
fun2 f ).
Save .
Hint Resolve @
fun2_monotonic .
Instance fmonotonic2 `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 } (
f :O1 -m>
O2 -m>
O3 )
:
monotonic2 (
fun2 f ).
Save .
Hint Resolve @
fmonotonic2 .
Definition mfun2 `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 } (
f :O1 -m>
O2 -m>
O3 )
:
O1 -m> (
O2 ->
O3 ) :=
mon (
fun2 f ).
Lemma mfun2_simpl :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 } (
f :O1 -m>
O2 -m>
O3 )
x y ,
mfun2 f x y =
f x y .
Instance mfun2_mon `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 }
(
f :O1 -m>
O2 -m>
O3 )
x :
monotonic (
mfun2 f x ).
Save .
Lemma mon2_fun2 :
forall `{
o1 :
ord O1 } `{
o2 :
ord O2 } `{
o3 :ord
O3 }
(
f :O1 -m>
O2 -m>
O3 ),
mon2 (
fun2 f ) ==
f .
Lemma fun2_mon2 :
forall `{
o1 :
ord O1 } `{
o2 :
ord O2 } `{
o3 :ord
O3 }
(
f :O1 ->
O2 ->
O3 ) {
mf :monotonic2
f } ,
fun2 (
mon2 f ) ==
f .
Hint Resolve @
mon2_fun2 @
fun2_mon2 .
Instance fstable2 `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 } (
f :O1 -m>
O2 -m>
O3 )
:
stable2 (
fun2 f ).
Save .
Hint Resolve @
fstable2 .
Definition Imon2 :
forall `{
o1 :
ord O1 } `{
o2 :
ord O2 } `{
o3 :ord
O3 },
(
O1 -m>
O2 -m>
O3 ) -> (
O1 --m>
O2 --m->
O3 ).
Defined .
Lemma Imon2_simpl :
forall `{
o1 :
ord O1 } `{
o2 :
ord O2 } `{
o3 :ord
O3 }
(
f :O1 -m>
O2 -m>
O3 ) (
x :O1) (
y :
O2 ),
Imon2 f x y =
f x y .
Lemma Imonotonic2 `{
o1 :
ord O1 } `{
o2 :
ord O2 } `{
o3 :ord
O3 }
(
f :O1 ->
O2 ->
O3 ){
mf :
monotonic2 f }
:
monotonic2 (
o1 :=Iord
O1 ) (
o2 :=Iord
O2 ) (
o3 :=Iord
O3 )
f .
Hint Extern 2 (@
monotonic2 _ (
Iord _ )
_ (
Iord _ )
_ (
Iord _ )
_ ) =>
apply @
Imonotonic2
:
typeclass_instances .
Definition imon2 `{
o1 :
ord O1 } `{
o2 :
ord O2 } `{
o3 :ord
O3 }
(
f :O1 ->
O2 ->
O3 ){
mf :
monotonic2 f } :
O1 --m>
O2 --m->
O3 :=
mon2 (
o1 :=Iord
O1 ) (
o2 :=Iord
O2 ) (
o3 :=Iord
O3 )
f .
Lemma imon2_simpl :
forall `{
o1 :
ord O1 } `{
o2 :
ord O2 } `{
o3 :ord
O3 }
(
f :O1 ->
O2 ->
O3 ){
mf :
monotonic2 f } (
x :O1) (
y :O2),
imon2 f x y =
f x y .
Instance natO :
ord nat :=
{
Oeq :=
fun n m :
nat =>
n =
m ;
Ole :=
fun n m :
nat => (
n <=
m )%nat}.
Defined .
Lemma le_Ole :
forall n m , ((
n <=
m )%nat)->
n <=
m .
Hint Resolve le_Ole .
Lemma nat_monotonic :
forall {
O } {
o :ord
O }
(
f :nat ->
O ), (
forall n ,
f n <=
f (
S n )) ->
monotonic f .
Hint Resolve @
nat_monotonic .
Definition fnatO_intro :
forall {
O } {
o :ord
O } (
f :nat ->
O ), (
forall n ,
f n <=
f (
S n )) ->
nat -m>
O .
Defined .
Lemma fnatO_elim :
forall {
O } {
o :ord
O } (
f :nat -m>
O ) (
n :nat),
f n <=
f (
S n ).
Hint Resolve @
fnatO_elim .
(mseq_lift_left f n) k = f (n+k)
Definition seq_lift_left {
O } (
f :nat ->
O )
n :=
fun k =>
f (
n +k)%nat.
Instance mon_seq_lift_left
:
forall {
O } {
o :ord
O } (
f :nat ->
O ) {
m :monotonic
f },
monotonic (
seq_lift_left f n ).
Save .
Definition mseq_lift_left :
forall {
O } {
o :ord
O } (
f :nat -m>
O ) (
n :nat),
nat -m>
O .
Defined .
Lemma mseq_lift_left_simpl :
forall {
O } {
o :ord
O } (
f :nat -m>
O ) (
n k :nat),
mseq_lift_left f n k =
f (
n +k)%nat.
Lemma mseq_lift_left_le_compat :
forall {
O } {
o :ord
O } (
f g :nat -m>
O ) (
n :nat),
f <=
g ->
mseq_lift_left f n <=
mseq_lift_left g n .
Hint Resolve @
mseq_lift_left_le_compat .
Add Parametric Morphism {
O } {
o :ord
O } : (@
mseq_lift_left _ o )
with signature Oeq &
eq &
Oeq
as mseq_lift_left_eq_compat .
Save .
Hint Resolve @
mseq_lift_left_eq_compat .
Add Parametric Morphism {
O } {
o :ord
O }: (@
seq_lift_left O )
with signature Oeq &
eq &
Oeq
as seq_lift_left_eq_compat .
Save .
Hint Resolve @
seq_lift_left_eq_compat .
(mseq_lift_right f n) k = f (k+n)
Definition seq_lift_right {
O } (
f :nat ->
O )
n :=
fun k =>
f (
k +n)%nat.
Instance mon_seq_lift_right
:
forall {
O } {
o :ord
O } (
f :nat ->
O ) {
m :monotonic
f },
monotonic (
seq_lift_right f n ).
Save .
Definition mseq_lift_right :
forall {
O } {
o :ord
O } (
f :nat -m>
O ) (
n :nat),
nat -m>
O .
Defined .
Lemma mseq_lift_right_simpl :
forall {
O } {
o :ord
O } (
f :nat -m>
O ) (
n k :nat),
mseq_lift_right f n k =
f (
k +n)%nat.
Lemma mseq_lift_right_le_compat :
forall {
O } {
o :ord
O } (
f g :nat -m>
O ) (
n :nat),
f <=
g ->
mseq_lift_right f n <=
mseq_lift_right g n .
Hint Resolve @
mseq_lift_right_le_compat .
Add Parametric Morphism {
O } {
o :ord
O } : (
mseq_lift_right (
o :=o))
with signature Oeq &
eq &
Oeq
as mseq_lift_right_eq_compat .
Save .
Add Parametric Morphism {
O } {
o :ord
O }: (@
seq_lift_right O )
with signature Oeq &
eq &
Oeq
as seq_lift_right_eq_compat .
Save .
Hint Resolve @
seq_lift_right_eq_compat .
Lemma mseq_lift_right_left :
forall {
O } {
o :ord
O } (
f :nat -m>
O )
n ,
mseq_lift_left f n ==
mseq_lift_right f n .
Monotonicity and functions
Instance shift_mon_fun {
A } `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f :O1 -m> (
A ->
O2 )) :
forall x :A,
monotonic (
fun (
y :O1) =>
f y x ).
Save .
Definition shift {
A } `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f :O1 -m> (
A ->
O2 )) :
A ->
O1 -m>
O2
:=
fun x => (
mon (
fun y =>
f y x )).
Infix "<o>" :=
shift (
at level 30,
no associativity ) :
O_scope .
Lemma shift_simpl :
forall {
A } `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f :O1 -m> (
A ->
O2 ))
x y ,
(
f <o>
x )
y =
f y x .
Lemma shift_le_compat :
forall {
A } `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f g :O1 -m> (
A ->
O2 )),
f <=
g ->
shift f <=
shift g .
Hint Resolve @
shift_le_compat .
Add Parametric Morphism {
A } `{
o1 :ord
O1 } `{
o2 :ord
O2 }
: (
shift (
A :=A) (
O1 :=O1) (
O2 :=O2))
with signature Oeq &
eq &
Oeq
as shift_eq_compat .
Save .
Instance ishift_mon {
A } `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f :A -> (
O1 -m>
O2 )) :
monotonic (
fun (
y :O1) (
x :A) =>
f x y ).
Save .
Definition ishift {
A } `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f :A -> (
O1 -m>
O2 )) :
O1 -m> (
A ->
O2 )
:=
mon (
fun (
y :O1) (
x :A) =>
f x y ) (
fmonotonic :=ishift_mon
f ).
Lemma ishift_simpl :
forall {
A } `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f :A -> (
O1 -m>
O2 ))
x y ,
ishift f x y =
f y x .
Lemma ishift_le_compat :
forall {
A } `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f g :A -> (
O1 -m>
O2 )),
f <=
g ->
ishift f <=
ishift g .
Hint Resolve @
ishift_le_compat .
Add Parametric Morphism {
A } `{
o1 :ord
O1 } `{
o2 :ord
O2 }
: (
ishift (
A :=A) (
O1 :=O1) (
O2 :=O2))
with signature Oeq &
eq &
Oeq
as ishift_eq_compat .
Save .
Instance shift_fun_mon `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 } (
f :O1 -m> (
O2 ->
O3 ))
{
m :forall
x ,
monotonic (
f x )} :
monotonic (
shift f ).
Save .
Instance shift_mon2 `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 } (
f :O1 -m>
O2 -m>
O3 )
:
monotonic2 (
fun x y =>
f y x ).
Save .
Hint Resolve @
shift_mon_fun @
shift_fun_mon @
shift_mon2 .
Definition mshift `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 } (
f :O1 -m>
O2 -m>
O3 )
:
O2 -m>
O1 -m>
O3 :=
mon2 (
fun x y =>
f y x ).
Definition id O {
o :ord
O } :
O ->
O :=
fun x =>
x .
Instance mon_id :
forall {
O :Type} {
o :ord
O },
monotonic (
id O ).
Save .
Definition cte A `{
o1 :ord
O1 } (
c :O1) :
A ->
O1 :=
fun x =>
c .
Instance mon_cte :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
c :O2),
monotonic (
cte O1 c ).
Save .
Definition mseq_cte {
O } {
o :ord
O } (
c :O) :
nat -m>
O :=
mon (
cte nat c ).
Add Parametric Morphism `{
o1 :ord
O1 } `{
o2 :ord
O2 } : (@
cte O1 O2 _ )
with signature Ole &
Ole as cte_le_compat .
Save .
Add Parametric Morphism `{
o1 :ord
O1 } `{
o2 :ord
O2 } : (@
cte O1 O2 _ )
with signature Oeq &
Oeq as cte_eq_compat .
Save .
Instance mon_diag `{
o1 :ord
O1 } `{
o2 :ord
O2 }(
f :O1 -m> (
O1 -m>
O2 ))
:
monotonic (
fun x =>
f x x ).
Save .
Hint Resolve @
mon_diag .
Definition diag `{
o1 :ord
O1 } `{
o2 :ord
O2 }(
f :O1 -m> (
O1 -m>
O2 )) :
O1 -m>
O2
:=
mon (
fun x =>
f x x ).
Lemma fmon_diag_simpl :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f :O1 -m> (
O1 -m>
O2 )) (
x :O1),
diag f x =
f x x .
Lemma diag_le_compat :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f g :O1 -m> (
O1 -m>
O2 )),
f <=
g ->
diag f <=
diag g .
Hint Resolve @
diag_le_compat .
Add Parametric Morphism `{
o1 :ord
O1 } `{
o2 :ord
O2 } : (
diag (
O1 :=O1) (
O2 :=O2))
with signature Oeq &
Oeq as diag_eq_compat .
Save .
Lemma diag_shift :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f :
O1 -m>
O1 -m>
O2 ),
diag f ==
diag (
mshift f ).
Hint Resolve @
diag_shift .
Lemma mshift_simpl :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 }
(
h :O1 -m>
O2 -m>
O3 ) (
x :
O2 ) (
y :O1),
mshift h x y =
h y x .
Lemma mshift_le_compat :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 }
(
f g :O1 -m>
O2 -m>
O3 ),
f <=
g ->
mshift f <=
mshift g .
Hint Resolve @
mshift_le_compat .
Add Parametric Morphism `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 } : (@
mshift O1 _ O2 _ O3 _ )
with signature Oeq &
Oeq as mshift_eq_compat .
Save .
Lemma mshift2_eq :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 } (
h :
O1 -m>
O2 -m>
O3 ),
mshift (
mshift h ) ==
h .
Instance monotonic_comp `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 }
(
f :O2 ->
O3 ){
mf :
monotonic f } (
g :O1 ->
O2 ){
mg :monotonic
g } :
monotonic (
fun x =>
f (
g x )).
Save .
Hint Resolve @
monotonic_comp .
Instance monotonic_comp_mon `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 }
(
f :O2 -m>
O3 )(
g :O1 -m>
O2 ) :
monotonic (
fun x =>
f (
g x )).
Save .
Hint Resolve @
monotonic_comp_mon .
Definition comp `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 } (
f :O2 -m>
O3 ) (
g :O1 -m>
O2 )
:
O1 -m>
O3 :=
mon (
fun x =>
f (
g x )).
Infix "@" :=
comp (
at level 35) :
O_scope .
Lemma comp_simpl :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 }
(
f :O2 -m>
O3 ) (
g :O1 -m>
O2 ) (
x :O1), (
f@g )
x =
f (
g x ).
Add Parametric Morphism `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 }: (@
comp O1 _ O2 _ O3 _ )
with signature Ole ++>
Ole ++>
Ole
as comp_le_compat .
Save .
Hint Immediate @
comp_le_compat .
Add Parametric Morphism `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 } : (@
comp O1 _ O2 _ O3 _ )
with signature Oeq &
Oeq &
Oeq
as comp_eq_compat .
Save .
Hint Immediate @
comp_eq_compat .
(f@2 g) h x = f (g x) (h x)
Instance mon_app2 `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 } `{
o4 :ord
O4 }
(
f :O2 ->
O3 ->
O4 ) (
g :O1 ->
O2 ) (
h :O1 ->
O3 )
{
mf :monotonic2
f }{
mg :monotonic
g } {
mh :monotonic
h }
:
monotonic (
fun x =>
f (
g x ) (
h x )).
Save .
Instance mon_app2_mon `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 } `{
o4 :ord
O4 }
(
f :O2 -m>
O3 -m>
O4 ) (
g :O1 -m>
O2 ) (
h :O1 -m>
O3 )
:
monotonic (
fun x =>
f (
g x ) (
h x )).
Save .
Definition app2 `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 } `{
o4 :ord
O4 }
(
f :O2 -m>
O3 -m>
O4 ) (
g :O1 -m>
O2 ) (
h :O1 -m>
O3 ) :
O1 -m>
O4
:=
mon (
fun x =>
f (
g x ) (
h x )).
Infix "@2" :=
app2 (
at level 70) :
O_scope .
Add Parametric Morphism `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 } `{
o4 :ord
O4 }:
(@
app2 O1 _ O2 _ O3 _ O4 _ )
with signature Ole ++>
Ole ++>
Ole ++>
Ole
as app2_le_compat .
Save .
Hint Immediate @
app2_le_compat .
Add Parametric Morphism `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 } `{
o4 :ord
O4 }:
(@
app2 O1 _ O2 _ O3 _ O4 _ )
with signature Oeq &
Oeq &
Oeq &
Oeq
as app2_eq_compat .
Save .
Hint Immediate @
app2_eq_compat .
Lemma app2_simpl :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 } `{
o4 :ord
O4 }
(
f :O2 -m>
O3 -m>
O4 ) (
g :O1 -m>
O2 ) (
h :O1 -m>
O3 ) (
x :O1),
(
f@2 g )
h x =
f (
g x ) (
h x ).
Lemma comp_monotonic_right :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 } (
f :
O2 -m>
O3 ) (
g1 g2 :O1 -m>
O2 ),
g1 <=
g2 ->
f @
g1 <=
f @
g2 .
Hint Resolve @
comp_monotonic_right .
Lemma comp_monotonic_left :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 } (
f1 f2 :
O2 -m>
O3 ) (
g :O1 -m>
O2 ),
f1 <=
f2 ->
f1 @
g <=
f2 @
g .
Hint Resolve @
comp_monotonic_left .
Instance comp_monotonic2 :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 },
monotonic2 (@
comp O1 _ O2 _ O3 _ ).
Save .
Hint Resolve @
comp_monotonic2 .
Definition fcomp `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 } :
(
O2 -m>
O3 ) -m> (
O1 -m>
O2 ) -m> (
O1 -m>
O3 ) :=
mon2 (@
comp O1 _ O2 _ O3 _ ).
Implicit Arguments fcomp [[
o1 ] [
o2 ] [
o3 ]].
Lemma fcomp_simpl :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 }
(
f :O2 -m>
O3 ) (
g :O1 -m>
O2 ),
fcomp _ _ _ f g =
f@g .
Definition fcomp2 `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 } `{
o4 :ord
O4 } :
(
O3 -m>
O4 ) -m> (
O1 -m>
O2 -m>
O3 ) -m> (
O1 -m>
O2 -m>
O4 ):=
(
fcomp O1 (
O2 -m>
O3 ) (
O2 -m>
O4 ))@(
fcomp O2 O3 O4 ).
Implicit Arguments fcomp2 [[
o1 ] [
o2 ] [
o3 ] [
o4 ]].
Lemma fcomp2_simpl :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 } `{
o4 :ord
O4 }
(
f :O3 -m>
O4 ) (
g :O1 -m>
O2 -m>
O3 ) (
x :O1)(
y :O2),
fcomp2 _ _ _ _ f g x y =
f (
g x y ).
Lemma fmon_le_compat2 :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 }
(
f :
O1 -m>
O2 -m>
O3 ) (
x y :O1) (
z t :O2),
x <=y ->
z <=t ->
f x z <=
f y t .
Hint Resolve fmon_le_compat2 .
Lemma fmon_cte_comp :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 }
(
c :O3)(
f :O1 -m>
O2 ), (
mon (
cte O2 c )) @
f ==
mon (
cte O1 c ).
Basic operators of omega-cpos
Constant :
lub : limit of monotonic sequences
Class cpo `{
o :ord
D } :
Type :=
mk_cpo
{
D0 :
D ;
lub :
forall (
f :nat -m>
D ),
D ;
Dbot :
forall x :D,
D0 <=
x ;
le_lub :
forall (
f :
nat -m>
D ) (
n :nat),
f n <=
lub f ;
lub_le :
forall (
f :
nat -m>
D ) (
x :D), (
forall n ,
f n <=
x ) ->
lub f <=
x }.
Implicit Arguments cpo [[
o ]].
Notation "0" :=
D0 :
O_scope .
Hint Resolve @
Dbot @
le_lub @
lub_le .
Definition mon_ord_equiv :
forall `{
o :ord
D1 } `{
o1 :ord
D2 } {
o2 :ord
D2 },
eq_ord o1 o2 ->
fmon D1 D2 (
o2 :=o2) ->
fmon D1 D2 (
o2 :=o1).
Defined .
Lemma mon_ord_equiv_simpl :
forall `{
o :ord
D1 } `{
o1 :ord
D2 } {
o2 :ord
D2 }
(
H :eq_ord
o1 o2 ) (
f :fmon
D1 D2 (
o2 :=o2)) (
x :D1),
mon_ord_equiv H f x =
f x .
Definition cpo_ord_equiv `{
o1 :ord
D } (
o2 :ord
D )
:
eq_ord o1 o2 ->
cpo (
o :=o1)
D ->
cpo (
o :=o2)
D .
Defined .
Add Parametric Morphism `{
c :cpo
D } : (
lub (
cpo :=c))
with signature Ole ++>
Ole as lub_le_compat .
Save .
Hint Resolve @
lub_le_compat .
Add Parametric Morphism `{
c :cpo
D }: (
lub (
cpo :=c))
with signature Oeq &
Oeq as lub_eq_compat .
Save .
Hint Resolve @
lub_eq_compat .
Notation "'mlub' f" := (
lub (
mon f )) (
at level 60) :
O_scope .
Lemma mlub_le_compat :
forall `{
c :cpo
D } (
f g :nat ->
D ) {
mf :monotonic
f } {
mg :monotonic
g },
f <=
g ->
mlub f <=
mlub g .
Hint Resolve @
mlub_le_compat .
Lemma mlub_eq_compat :
forall `{
c :cpo
D } (
f g :nat ->
D ) {
mf :monotonic
f } {
mg :monotonic
g },
f ==
g ->
mlub f ==
mlub g .
Hint Resolve @
mlub_eq_compat .
Lemma le_mlub :
forall `{
c :cpo
D } (
f :nat ->
D ) {
m :monotonic
f } (
n :nat),
f n <=
mlub f .
Lemma mlub_le :
forall `{
c :cpo
D }(
f :nat ->
D ) {
m :monotonic
f }(
x :D), (
forall n ,
f n <=
x ) ->
mlub f <=
x .
Hint Resolve @
le_mlub @
mlub_le .
Instance lub_mon `{
c :cpo
D } :
monotonic lub .
Save .
Definition Lub `{
c :cpo
D } : (
nat -m>
D ) -m>
D :=
mon lub .
Instance monotonic_lub_comp {
O } {
o :ord
O } `{
c :cpo
D } (
f :O ->
nat ->
D ){
mf :monotonic2
f }:
monotonic (
fun x =>
mlub (
f x )).
Save .
Lemma lub_cte :
forall `{
c :cpo
D } (
d :D),
mlub (
cte nat d ) ==
d .
Hint Resolve @
lub_cte .
Lemma mlub_lift_right :
forall `{
c :cpo
D } (
f :nat -m>
D )
n ,
lub f ==
mlub (
seq_lift_right f n ).
Hint Resolve @
mlub_lift_right .
Lemma mlub_lift_left :
forall `{
c :cpo
D } (
f :nat -m>
D )
n ,
lub f ==
mlub (
seq_lift_left f n ).
Hint Resolve @
mlub_lift_left .
Lemma lub_lift_right :
forall `{
c :cpo
D } (
f :nat -m>
D )
n ,
lub f ==
lub (
mseq_lift_right f n ).
Hint Resolve @
lub_lift_right .
Lemma lub_lift_left :
forall `{
c :cpo
D } (
f :nat -m>
D )
n ,
lub f ==
lub (
mseq_lift_left f n ).
Hint Resolve @
lub_lift_left .
Lemma lub_le_lift :
forall `{
c :cpo
D } (
f g :nat -m>
D )
(
n :nat), (
forall k ,
n <=
k ->
f k <=
g k ) ->
lub f <=
lub g .
Lemma lub_eq_lift :
forall `{
c :cpo
D } (
f g :nat -m>
D ) {
m :monotonic
f } {
m' :monotonic
g }
(
n :nat), (
forall k ,
n <=
k ->
f k ==
g k ) ->
lub f ==
lub g .
(lub_fun h) x = lub_n (h n x)
Definition lub_fun {
A } `{
c :cpo
D } (
h :
nat -m> (
A ->
D )) :
A ->
D
:=
fun x =>
mlub (
h <o>
x ).
Instance lub_shift_mon {
O } {
o :ord
O } `{
c :cpo
D } (
h :
nat -m> (
O -m>
D ))
:
monotonic (
fun (
x :O) =>
lub (
mshift h x )).
Save .
Hint Resolve @
lub_shift_mon .
Instance fcpo {
A :
Type } `(
c :cpo
D ) :
cpo (
A ->
D ) :=
{
D0 :=
fun x :A => (0:D);
lub :=
fun f =>
lub_fun f }.
Defined .
Lemma fcpo_lub_simpl :
forall {
A } `{
c :cpo
D } (
h :nat -m> (
A ->
D ))(
x :A),
(
lub h )
x =
mlub (
h <o>
x ).
Lemma lub_ishift :
forall {
A } `{
c :cpo
D } (
h :A -> (
nat -m>
D )),
lub (
ishift h ) ==
fun x =>
lub (
h x ).
Cpo of monotonic functions
Instance fmon_cpo {
O } {
o :ord
O } `{
c :cpo
D } :
cpo (
O -m>
D ) :=
{
D0 :=
mon (
cte O (0:D));
lub :=
fun h :nat -m> (
O -m>
D ) =>
mon (
fun (
x :O) =>
lub (
cpo :=c) (
mshift h x ))}.
Defined .
Lemma fmon_lub_simpl :
forall {
O } {
o :ord
O } `{
c :cpo
D }
(
h :nat -m> (
O -m>
D ))(
x :O), (
lub h )
x =
lub (
mshift h x ).
Hint Resolve @
fmon_lub_simpl .
Instance mon_fun_lub :
forall {
O } {
o :ord
O } `{
c :cpo
D }
(
h :nat -m> (
O ->
D )) {
mh :forall
n ,
monotonic (
h n )},
monotonic (
lub h ).
Save .
Link between lubs on ordinary functions and monotonic functions
Lemma lub_mon_fcpo :
forall {
O } {
o :ord
O } `{
c :cpo
D } (
h :nat -m> (
O -m>
D )),
lub h ==
mon (
lub (
mfun2 h )).
Lemma lub_fcpo_mon :
forall {
O } {
o :ord
O } `{
c :cpo
D } (
h :nat -m> (
O ->
D ))
{
mh :forall
x ,
monotonic (
h x )},
lub h ==
lub (
mon2 h ).
Lemma double_lub_diag :
forall `{
c :cpo
D } (
h :
nat -m>
nat -m>
D ),
lub (
lub h ) ==
lub (
diag h ).
Hint Resolve @
double_lub_diag .
Lemma double_lub_shift :
forall `{
c :cpo
D } (
h :
nat -m>
nat -m>
D ),
lub (
lub h ) ==
lub (
lub (
mshift h )).
Hint Resolve @
double_lub_shift .
Lemma lub_comp_le :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } (
f :D1 -m>
D2 ) (
h :
nat -m>
D1 ),
lub (
f @
h ) <=
f (
lub h ).
Hint Resolve @
lub_comp_le .
Lemma lub_app2_le :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }
(
F :D1 -m>
D2 -m>
D3 ) (
f :
nat -m>
D1 ) (
g :
nat -m>
D2 ),
lub ((
F @2
f )
g ) <=
F (
lub f ) (
lub g ).
Hint Resolve @
lub_app2_le .
Class continuous `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } (
f :D1 -m>
D2 ) :=
cont_intro :
forall (
h :
nat -m>
D1 ),
f (
lub h ) <=
lub (
f @
h ).
Type classes Opaque continuous .
Lemma continuous_eq_compat :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 }(
f g :D1 -m>
D2 ),
f ==
g ->
continuous f ->
continuous g .
Add Parametric Morphism `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } : (@
continuous D1 _ _ D2 _ _ )
with signature Oeq &
iff
as continuous_eq_compat_iff .
Save .
Lemma lub_comp_eq :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } (
f :D1 -m>
D2 ) (
h :
nat -m>
D1 ),
continuous f ->
f (
lub h ) ==
lub (
f @
h ).
Hint Resolve @
lub_comp_eq .
Instance cont0 `{c1 :cpo D1 } `{c2 :cpo D2 } : continuous (mon (cte D1 (0:D2))).Save .Implicit Arguments cont0 [].
double_app f g n m = f m (g n)
Definition double_app `{o1 :ord O1 } `{o2 :ord O2 } `{o3 :ord O3 } `{o4 : ord O4 } f :O1 -m> O3 -m> O4 ) (g :O2 -m> O3 ) O2 -m> (O1 -m> O4 ) := mon ((mshift f ) @ g ).
Class continuous2 `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 } (
F :D1 -m>
D2 -m>
D3 ) :=
continuous2_intro :
forall (
f :
nat -m>
D1 ) (
g :nat -m>
D2 ),
F (
lub f ) (
lub g ) <=
lub ((
F @2
f )
g ).
Lemma continuous2_app :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }
(
F :
D1 -m>
D2 -m>
D3 ) {
cF :continuous2
F } (
k :D1),
continuous (
F k ).
Type classes Opaque continuous2 .
Lemma continuous2_eq_compat :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 } (
f g :
D1 -m>
D2 -m>
D3 ),
f ==
g ->
continuous2 f ->
continuous2 g .
Lemma continuous2_continuous :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }
(
F :
D1 -m>
D2 -m>
D3 ),
continuous2 F ->
continuous F .
Hint Immediate @
continuous2_continuous .
Lemma continuous2_left :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }
(
F :
D1 -m>
D2 -m>
D3 ) (
h :nat -m>
D1 ) (
x :D2),
continuous F ->
F (
lub h )
x <=
lub (
mshift (
F @
h )
x ).
Lemma continuous2_right :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }
(
F :
D1 -m>
D2 -m>
D3 ) (
x :D1)(
h :nat -m>
D2 ),
continuous2 F ->
F x (
lub h ) <=
lub (
F x @
h ).
Lemma continuous_continuous2 :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }
(
F :
D1 -m>
D2 -m>
D3 ) (
cFr :
forall k :D1,
continuous (
F k )) (
cF :
continuous F ),
continuous2 F .
Hint Resolve @
continuous2_app @
continuous2_continuous @
continuous_continuous2 .
Lemma lub_app2_eq :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }
(
F :
D1 -m>
D2 -m>
D3 ) {
cFr :forall
k :D1,
continuous (
F k )} {
cF :
continuous F },
forall (
f :nat -m>
D1 ) (
g :nat -m>
D2 ),
F (
lub f ) (
lub g ) ==
lub ((
F@2 f )
g ).
Lemma lub_cont2_app2_eq :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }
(
F :
D1 -m>
D2 -m>
D3 ){
cF :
continuous2 F },
forall (
f :nat -m>
D1 ) (
g :nat -m>
D2 ),
F (
lub f ) (
lub g ) ==
lub ((
F@2 f )
g ).
Lemma mshift_continuous2 :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }
(
F :
D1 -m>
D2 -m>
D3 ),
continuous2 F ->
continuous2 (
mshift F ).
Hint Resolve @
mshift_continuous2 .
Lemma monotonic_sym :
forall `{
o1 :ord
D1 } `{
o2 :ord
D2 } (
F :
D1 ->
D1 ->
D2 ),
(
forall x y ,
F x y ==
F y x ) -> (
forall k :D1,
monotonic (
F k )) ->
monotonic F .
Hint Immediate @
monotonic_sym .
Lemma monotonic2_sym :
forall `{
o1 :ord
D1 } `{
o2 :ord
D2 } (
F :
D1 ->
D1 ->
D2 ),
(
forall x y ,
F x y ==
F y x ) -> (
forall k :D1,
monotonic (
F k )) ->
monotonic2 F .
Hint Immediate @
monotonic2_sym .
Lemma continuous_sym :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } (
F :
D1 -m>
D1 -m>
D2 ),
(
forall x y ,
F x y ==
F y x ) -> (
forall k :D1,
continuous (
F k )) ->
continuous F .
Lemma continuous2_sym :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } (
F :
D1 -m>D1 -m>D2),
(
forall x y ,
F x y ==
F y x ) -> (
forall k ,
continuous (
F k )) ->
continuous2 F .
Hint Resolve @
continuous2_sym .
continuity is preserved by composition
Lemma continuous_comp :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }
(
f :D2 -m>
D3 )(
g :D1 -m>
D2 ),
continuous f ->
continuous g ->
continuous (
mon (
f@g )).
Hint Resolve @
continuous_comp .
Lemma continuous2_comp :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 } `{
c4 :cpo
D4 }
(
f :D1 -m>
D2 )(
g :D2 -m>
D3 -m>
D4 ),
continuous f ->
continuous2 g ->
continuous2 (
g @
f ).
Hint Resolve @
continuous2_comp .
Lemma continuous2_comp2 :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 } `{
c4 :cpo
D4 }
(
f :D3 -m>
D4 )(
g :D1 -m>
D2 -m>
D3 ),
continuous f ->
continuous2 g ->
continuous2 (
fcomp2 D1 D2 D3 D4 f g ).
Hint Resolve @
continuous2_comp2 .
Cpo of continuous functions
Instance lub_continuous `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }
(
f :nat -m> (
D1 -m>
D2 )) {
cf :forall
n ,
continuous (
f n )}
:
continuous (
lub f ).
Save .
Record fcont `{
c1 :cpo
D1 } `{
c2 :cpo
D2 }:
Type
:=
cont {
fcontm :>
D1 -m>
D2 ;
fcontinuous :
continuous fcontm }.
Hint Resolve @
fcontinuous .
Implicit Arguments fcont [[
o ][
c1 ] [
o0 ][
c2 ]].
Implicit Arguments cont [[
D1 ][
o ][
c1 ] [
D2 ][
o0 ][
c2 ] [
fcontinuous ]].
Infix "-c>" :=
fcont (
at level 30,
right associativity ) :
O_scope .
Definition fcont_fun `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } (
f :D1 -c>
D2 ) :
D1 ->
D2 :=
fun x =>
f x .
Instance fcont_ord `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } :
ord (
D1 -c>D2)
:= {
Oeq :=
fun f g =>
forall x ,
f x ==
g x ;
Ole :=
fun f g =>
forall x ,
f x <=
g x }.
Defined .
Lemma fcont_le_intro :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } (
f g :
D1 -c>
D2 ),
(
forall x ,
f x <=
g x ) ->
f <=
g .
Lemma fcont_le_elim :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } (
f g :
D1 -c>
D2 ),
f <=
g ->
forall x ,
f x <=
g x .
Lemma fcont_eq_intro :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } (
f g :
D1 -c>
D2 ),
(
forall x ,
f x ==
g x ) ->
f ==
g .
Lemma fcont_eq_elim :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } (
f g :
D1 -c>
D2 ),
f ==
g ->
forall x ,
f x ==
g x .
Lemma fcont_le :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } (
f :
D1 -c>
D2 ) (
x y :
D1 ),
x <=
y ->
f x <=
f y .
Hint Resolve @
fcont_le .
Lemma fcont_eq :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } (
f :
D1 -c>
D2 ) (
x y :
D1 ),
x ==
y ->
f x ==
f y .
Hint Resolve @
fcont_eq .
Definition fcont0 D1 `{
c1 :cpo
D1 }
D2 `{
c2 :cpo
D2 } :
D1 -c>
D2 :=
cont (
mon (
cte D1 (0:D2))).
Instance fcontm_monotonic :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 },
monotonic (
fcontm (
D1 :=D1) (
D2 :=D2)).
Save .
Definition Fcontm D1 `{
c1 :cpo
D1 }
D2 `{
c2 :cpo
D2 } : (
D1 -c>
D2 ) -m> (
D1 -m>
D2 ) :=
mon (
fcontm (
D1 :=D1) (
D2 :=D2)).
Instance fcont_lub_continuous :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } (
f :nat -m> (
D1 -c>
D2 )),
continuous (
lub (
D :=D1 -m>
D2 ) (
Fcontm D1 D2 @
f )).
Save .
Definition fcont_lub `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } : (
nat -m> (
D1 -c>
D2 )) ->
D1 -c>
D2 :=
fun f =>
cont (
lub (
D :=D1 -m>
D2 ) (
Fcontm D1 D2 @
f )).
Instance fcont_cpo `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } :
cpo (
D1 -c>
D2 ) :=
{
D0 :=fcont0
D1 D2 ;
lub :=fcont_lub (
D1 :=D1) (
D2 :=D2)}.
Defined .
Definition fcont_app {
O } {
o :ord
O } `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } (
f :
O -m>
D1 -c>
D2 ) (
x :D1) :
O -m>
D2
:=
mshift (
Fcontm D1 D2 @
f )
x .
Infix "<__>" :=
fcont_app (
at level 70) :
O_scope .
Lemma fcont_app_simpl :
forall {
O } {
o :ord
O } `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } (
f :
O -m>
D1 -c>
D2 )(
x :D1)(
y :O),
(
f <__>
x )
y =
f y x .
Instance ishift_continuous :
forall {
A :Type} `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } (
f :
A -> (
D1 -c>
D2 )),
continuous (
ishift f ).
Qed .
Definition fcont_ishift {
A :Type} `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } (
f :
A -> (
D1 -c>
D2 ))
:
D1 -c> (
A ->
D2 ) :=
cont _ (
fcontinuous :=ishift_continuous
f ).
Instance mshift_continuous :
forall {
O } {
o :ord
O } `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } (
f :
O -m> (
D1 -c>
D2 )),
continuous (
mshift (
Fcontm D1 D2 @
f )).
Save .
Definition fcont_mshift {
O } {
o :ord
O } `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } (
f :
O -m> (
D1 -c>
D2 ))
:
D1 -c>
O -m>
D2 :=
cont (
mshift (
Fcontm D1 D2 @
f )).
Lemma fcont_app_continuous :
forall {
O } {
o :ord
O } `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } (
f :
O -m>
D1 -c>
D2 ) (
h :nat -m>
D1 ),
f <__> (
lub h ) <=
lub (
D :=O -m>
D2 ) ((
fcont_mshift f ) @
h ).
Lemma fcont_lub_simpl :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } (
h :nat -m>
D1 -c>
D2 )(
x :D1),
lub h x =
lub (
h <__>
x ).
Instance cont_app_monotonic :
forall `{
o1 :ord
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 } (
f :D1 -m>
D2 -m>
D3 )
(
p :forall
k ,
continuous (
f k )),
monotonic (
O2 :=D2 -c>
D3 ) (
fun (
k :D1) =>
cont _ (
fcontinuous :=p
k )).
Qed .
Definition cont_app `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 } (
f :D1 -m>
D2 -m>
D3 )
(
p :forall
k ,
continuous (
f k )) :
D1 -m> (
D2 -c>
D3 )
:=
mon (
fun k =>
cont (
f k ) (
fcontinuous :=p
k )).
Lemma cont_app_simpl :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }(
f :D1 -m>
D2 -m>
D3 )(
p :forall
k ,
continuous (
f k ))
(
k :D1),
cont_app f p k =
cont (
f k ).
Instance cont2_continuous `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 } (
f :D1 -m>
D2 -m>
D3 )
(
p :continuous2
f ) :
continuous (
cont_app f (
continuous2_app f )).
Qed .
Definition cont2 `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 } (
f :D1 -m>
D2 -m>
D3 )
{
p :continuous2
f } :
D1 -c> (
D2 -c>
D3 )
:=
cont (
cont_app f (
continuous2_app f )).
Instance Fcontm_continuous `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } :
continuous (
Fcontm D1 D2 ).
Save .
Hint Resolve @
Fcontm_continuous .
Instance fcont_comp_continuous :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }
(
f :D2 -c>
D3 ) (
g :D1 -c>
D2 ),
continuous (
f @
g ).
Save .
Definition fcont_comp `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 } (
f :D2 -c>
D3 ) (
g :D1 -c>
D2 )
:
D1 -c>
D3 :=
cont (
f @
g ).
Infix "@_" :=
fcont_comp (
at level 35) :
O_scope .
Lemma fcont_comp_simpl :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }
(
f :D2 -c>
D3 )(
g :D1 -c>
D2 ) (
x :D1), (
f @
_ g )
x =
f (
g x ).
Lemma fcontm_fcont_comp_simpl :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }
(
f :D2 -c>
D3 )(
g :D1 -c>
D2 ),
fcontm (
f @
_ g ) =
f @
g .
Lemma fcont_comp_le_compat :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }
(
f g :
D2 -c>
D3 ) (
k l :D1 -c>
D2 ),
f <=
g ->
k <=
l ->
f @
_ k <=
g @
_ l .
Hint Resolve @
fcont_comp_le_compat .
Add Parametric Morphism `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }
: (@
fcont_comp _ _ c1 _ _ c2 _ _ c3 )
with signature Ole ++>
Ole ++>
Ole as fcont_comp_le_morph .
Save .
Add Parametric Morphism `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }
: (@
fcont_comp _ _ c1 _ _ c2 _ _ c3 )
with signature Oeq &
Oeq &
Oeq as fcont_comp_eq_compat .
Save .
Definition fcont_Comp D1 `{
c1 :cpo
D1 }
D2 `{
c2 :cpo
D2 }
D3 `{
c3 :cpo
D3 }
: (
D2 -c>
D3 ) -m> (
D1 -c>
D2 ) -m>
D1 -c>
D3
:=
mon2 _ (
mf :=fcont_comp_le_compat (
D1 :=D1) (
D2 :=D2) (
D3 :=D3)).
Lemma fcont_Comp_simpl :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }
(
f :D2 -c>
D3 ) (
g :D1 -c>
D2 ),
fcont_Comp D1 D2 D3 f g =
f @
_ g .
Instance fcont_Comp_continuous2
:
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 },
continuous2 (
fcont_Comp D1 D2 D3 ).
Save .
Definition fcont_COMP D1 `{
c1 :cpo
D1 }
D2 `{
c2 :cpo
D2 }
D3 `{
c3 :cpo
D3 }
: (
D2 -c>
D3 ) -c> (
D1 -c>
D2 ) -c>
D1 -c>
D3
:=
cont2 (
fcont_Comp D1 D2 D3 ).
Lemma fcont_COMP_simpl :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }
(
f :
D2 -c>
D3 ) (
g :D1 -c>
D2 ),
fcont_COMP D1 D2 D3 f g =
f @
_ g .
Definition fcont2_COMP D1 `{
c1 :cpo
D1 }
D2 `{
c2 :cpo
D2 }
D3 `{
c3 :cpo
D3 }
D4 `{
c4 :cpo
D4 }
: (
D3 -c>
D4 ) -c> (
D1 -c>
D2 -c>
D3 ) -c>
D1 -c>
D2 -c>
D4 :=
(
fcont_COMP D1 (
D2 -c>
D3 ) (
D2 -c>
D4 )) @
_ (
fcont_COMP D2 D3 D4 ).
Definition fcont2_comp `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 } `{
c4 :cpo
D4 }
(
f :D3 -c>
D4 )(
F :D1 -c>
D2 -c>
D3 ) :=
fcont2_COMP D1 D2 D3 D4 f F .
Infix "@@_" :=
fcont2_comp (
at level 35) :
O_scope .
Lemma fcont2_comp_simpl :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 } `{
c4 :cpo
D4 }
(
f :D3 -c>
D4 )(
F :D1 -c>
D2 -c>
D3 )(
x :D1)(
y :D2), (
f @@
_ F )
x y =
f (
F x y ).
Lemma fcont_le_compat2 :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 } (
f :
D1 -c>
D2 -c>
D3 )
(
x y :
D1 ) (
z t :
D2 ),
x <=
y ->
z <=
t ->
f x z <=
f y t .
Hint Resolve @
fcont_le_compat2 .
Lemma fcont_eq_compat2 :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 } (
f :
D1 -c>
D2 -c>
D3 )
(
x y :
D1 ) (
z t :
D2 ),
x ==
y ->
z ==
t ->
f x z ==
f y t .
Hint Resolve @
fcont_eq_compat2 .
Lemma fcont_continuous :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } (
f :D1 -c>
D2 )(
h :nat-m>
D1 ),
f (
lub h ) <=
lub (
f @
h ).
Hint Resolve @
fcont_continuous .
Instance fcont_continuous2 :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }
(
f :D1 -c>
D2 -c>
D3 ),
continuous2 (
Fcontm D2 D3 @
f ).
Save .
Hint Resolve @
fcont_continuous2 .
Instance cshift_continuous2 :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }
(
f :D1 -c>
D2 -c>
D3 ),
continuous2 (
mshift (
Fcontm D2 D3 @
f )).
Save .
Hint Resolve @
cshift_continuous2 .
Definition cshift `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 } (
f :D1 -c>
D2 -c>
D3 )
:
D2 -c>
D1 -c>
D3 :=
cont2 (
mshift (
Fcontm D2 D3 @
f )).
Lemma cshift_simpl :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }
(
f :D1 -c>
D2 -c>
D3 ) (
x :D2) (
y :D1),
cshift f x y =
f y x .
Definition fcont_SEQ D1 `{
c1 :cpo
D1 }
D2 `{
c2 :cpo
D2 }
D3 `{
c3 :cpo
D3 }
: (
D1 -c>
D2 ) -c> (
D2 -c>
D3 ) -c>
D1 -c>
D3 :=
cshift (
fcont_COMP D1 D2 D3 ).
Lemma fcont_SEQ_simpl :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }
(
f :
D1 -c>
D2 ) (
g :D2 -c>
D3 ),
fcont_SEQ D1 D2 D3 f g =
g @
_ f .
Instance Id_mon :
forall `{
o1 :ord
O1 },
monotonic (
fun x :O1 =>
x ).
Save .
Definition Id O1 {
o1 :ord
O1 } :
O1 -m>
O1 :=
mon (
fun x =>
x ).
Lemma Id_simpl :
forall `{
o1 :ord
O1 } (
x :O1),
Id O1 x =
x .
Fixpoint iter_ `{
c :
cpo D } (
f :
D -m>
D )
n {
struct n } :
D
:=
match n with O => 0 |
S m =>
f (
iter_ f m )
end .
Lemma iter_incr :
forall `{
c :
cpo D } (
f :
D -m>
D )
n ,
iter_ f n <=
f (
iter_ f n ).
Hint Resolve @
iter_incr .
Instance iter_mon :
forall `{
c :
cpo D } (
f :
D -m>
D ),
monotonic (
iter_ f ).
Save .
Definition iter `{
c :
cpo D } (
f :
D -m>
D ) :
nat -m>
D :=
mon (
iter_ f ).
Definition fixp `{
c :
cpo D } (
f :
D -m>
D ) :
D :=
mlub (
iter_ f ).
Lemma fixp_le :
forall `{
c :
cpo D } (
f :
D -m>
D ),
fixp f <=
f (
fixp f ).
Hint Resolve @
fixp_le .
Lemma fixp_eq :
forall `{
c :
cpo D } (
f :
D -m>
D ) {
mf :continuous
f },
fixp f ==
f (
fixp f ).
Lemma fixp_inv :
forall `{
c :
cpo D } (
f :
D -m>
D )
g ,
f g <=
g ->
fixp f <=
g .
Definition fixp_cte :
forall `{
c :cpo
D } (
d :D),
fixp (
mon (
cte D d )) ==
d .
Save .
Hint Resolve @
fixp_cte .
Lemma fixp_le_compat :
forall `{
c :cpo
D } (
f g :
D -m>
D ),
f <=
g ->
fixp f <=
fixp g .
Hint Resolve @
fixp_le_compat .
Instance fixp_monotonic `{
c :cpo
D } :
monotonic fixp .
Save .
Add Parametric Morphism `{
c :cpo
D } : (
fixp (
c :=c))
with signature Oeq &
Oeq as fixp_eq_compat .
Save .
Hint Resolve @
fixp_eq_compat .
Definition Fixp D `{
c :cpo
D } : (
D -m>
D ) -m>
D :=
mon fixp .
Lemma Fixp_simpl :
forall `{
c :cpo
D } (
f :D-m>D),
Fixp D f =
fixp f .
Instance iter_monotonic `{
c :cpo
D } :
monotonic iter .
Save .
Definition Iter D `{
c :cpo
D } : (
D -m>
D ) -m> (
nat -m>
D ) :=
mon iter .
Lemma IterS_simpl :
forall `{
c :cpo
D }
f n ,
Iter D f (
S n ) =
f (
Iter D f n ).
Lemma iterO_simpl :
forall `{
c :cpo
D } (
f :
D -m>
D ),
iter f O = (0:D).
Lemma iterS_simpl :
forall `{
c :cpo
D }
f n ,
iter f (
S n ) =
f (
iter f n ).
Lemma iter_continuous :
forall `{
c :cpo
D } (
h :
nat -m> (
D -m>
D )),
(
forall n ,
continuous (
h n )) ->
iter (
lub h ) <=
lub (
mon iter @
h ).
Hint Resolve @
iter_continuous .
Lemma iter_continuous_eq :
forall `{
c :cpo
D } (
h :
nat -m> (
D -m>
D )),
(
forall n ,
continuous (
h n )) ->
iter (
lub h ) ==
lub (
mon iter @
h ).
Lemma fixp_continuous :
forall `{
c :cpo
D } (
h :
nat -m> (
D -m>
D )),
(
forall n ,
continuous (
h n )) ->
fixp (
lub h ) <=
lub (
mon fixp @
h ).
Hint Resolve @
fixp_continuous .
Lemma fixp_continuous_eq :
forall `{
c :cpo
D } (
h :
nat -m> (
D -m>
D )),
(
forall n ,
continuous (
h n )) ->
fixp (
lub h ) ==
lub (
mon fixp @
h ).
Definition admissible `{
c :cpo
D }(
P :D->Type) :=
forall f :
nat -m>
D , (
forall n ,
P (
f n )) ->
P (
lub f ).
Lemma fixp_ind :
forall `{
c :cpo
D }(
F :D -m>
D )(
P :D->Type),
admissible P ->
P 0 -> (
forall x ,
P x ->
P (
F x )) ->
P (
fixp F ).
Ltac continuity cont Cont Hcont :=
match goal with
| |- (
Ole ?x1 (
lub (
mon (
fun (
n :nat) =>
cont (@?g
n ))))) =>
let f :=
fresh "f"
in (
pose (
f :=g);
assert (
monotonic f ) ;
[
auto | (
transitivity (
lub (
Cont@ (
mon f ))); [
rewrite <-
Hcont |
auto ])]
)
end .
Ltac gen_monotonic :=
match goal with |-
context [(@
mon _ _ _ _ ?f ?mf)] =>
generalize (
mf :monotonic
f )
end .
Ltac gen_monotonic1 f :=
match goal with |-
context [(@
mon _ _ _ _ f ?mf)] =>
generalize (
mf :monotonic
f )
end .
Library Utheory
Utheory.v: Specification of , interval
Require Export Misc .
Require Export Ccpo .
Open Local Scope O_scope .
Constants : and
Constructor : Unth
Operations : , , inv
Relations : ,
Module Type Universe .
Parameter U :
Type .
Instance ordU :
ord U .
Instance cpoU :
cpo U .
Delimit Scope U_scope with U .
Parameter s Uplus Umult Udiv :
U ->
U ->
U .
Parameter Uinv :
U ->
U .
Parameter Unth :
nat ->
U .
Infix "+" :=
Uplus :
U_scope .
Infix "*" :=
Umult :
U_scope .
Infix "/" :=
Udiv :
U_scope .
Notation "[1-] x" := (
Uinv x ) (
at level 35,
right associativity ) :
U_scope .
Notation "[1/]1+ n" := (
Unth n ) (
at level 35,
right associativity ) :
U_scope .
Open Local Scope U_scope .
Definition U1 :
U := [1-] 0.
Notation "1" :=
U1 :
U_scope .
Hypothesis Udiff_0_1 : ~0 == 1.
Hypothesis Uplus_sym :
forall x y :U,
x +
y ==
y +
x .
Hypothesis Uplus_assoc :
forall x y z :U,
x + (
y +
z ) ==
x +
y +
z .
Hypothesis Uplus_zero_left :
forall x :U, 0 +
x ==
x .
Hypothesis Umult_sym :
forall x y :U,
x *
y ==
y *
x .
Hypothesis Umult_assoc :
forall x y z :U,
x * (
y *
z ) ==
x *
y *
z .
Hypothesis Umult_one_left :
forall x :U, 1 *
x ==
x .
Hypothesis Uinv_one : [1-] 1 == 0.
Hypothesis Umult_div :
forall x y , ~0 ==
y ->
x <=
y ->
y * (
x /y) ==
x .
Hypothesis Udiv_le_one :
forall x y , ~0 ==
y ->
y <=
x -> (
x /y) == 1.
Hypothesis Udiv_by_zero :
forall x y , 0 ==
y -> (
x /y) == 0.
Property : holds when does not overflow
Hypothesis Uinv_plus_left :
forall x y ,
y <= [1-]
x -> [1-] (
x +
y ) +
x == [1-]
y .
Property :
holds when does not overflow
Hypothesis Udistr_plus_right :
forall x y z ,
x <= [1-]
y -> (
x +
y ) *
z ==
x *
z +
y *
z .
Hypothesis Udistr_inv_right :
forall x y :U, [1-] (
x *
y ) == ([1-]
x ) *
y + [1-]
y .
The relation is compatible with operators
Properties of simplification in case there is no overflow
Hypothesis archimedian :
forall x , ~0 ==
x ->
exc (
fun n => [1/]1+n <=
x ).
Stability properties of lubs with respect to + and *
Hypothesis Uplus_right_continuous :
forall k ,
continuous (
mon (
Uplus k )).
Hypothesis Umult_right_continuous :
forall k ,
continuous (
mon (
Umult k )).
End Universe .
Declare Module Univ :Universe.
Export Univ .
Hint Resolve Udiff_0_1 Unth_prop .
Hint Resolve Uplus_sym Uplus_assoc Umult_sym Umult_assoc .
Hint Resolve Uinv_one Uinv_plus_left Umult_div Udiv_le_one Udiv_by_zero .
Hint Resolve Uplus_zero_left Umult_one_left Udistr_plus_right Udistr_inv_right .
Hint Resolve Uplus_mon_right Umult_mon_right Uinv_le_compat .
Hint Resolve lub_le le_lub Uplus_right_continuous Umult_right_continuous .
Hint Resolve Ule_total Ule_class .
Library Uprop
Uprop.v : Properties of operators on [0,1]
Require Export Utheory .
Require Export Arith .
Require Export Omega .
Open Local Scope U_scope .
Direct consequences of axioms
Lemma Uplus_le_compat_right :
forall x y z :U,
y <=
z ->
x +
y <=
x +
z .
Hint Resolve Uplus_le_compat_right .
Instance Uplus_mon2 :
monotonic2 Uplus .
Save .
Hint Resolve Uplus_mon2 .
Lemma Uplus_le_compat_left :
forall x y z :U,
x <=
y ->
x +
z <=
y +
z .
Hint Resolve Uplus_le_compat_left .
Lemma Uplus_le_compat :
forall x y z t ,
x <=
y ->
z <=
t ->
x +
z <=
y +
t .
Hint Immediate Uplus_le_compat .
Lemma Uplus_eq_compat_left :
forall x y z :U,
x ==
y ->
x +
z ==
y +
z .
Hint Resolve Uplus_eq_compat_left .
Lemma Uplus_eq_compat_right :
forall x y z :U,
x ==
y -> (
z +
x ) == (
z +
y ).
Hint Resolve Uplus_eq_compat_left Uplus_eq_compat_right .
Add Morphism Uplus with signature Oeq &
Oeq &
Oeq as Uplus_eq_compat .
Qed .
Hint Immediate Uplus_eq_compat .
Add Morphism Uinv with signature Oeq &
Oeq as Uinv_eq_compat .
Qed .
Hint Resolve Uinv_eq_compat .
Lemma Uplus_zero_right :
forall x :U,
x + 0 ==
x .
Hint Resolve Uplus_zero_right .
Lemma Uinv_opp_left :
forall x , [1-]
x +
x == 1.
Hint Resolve Uinv_opp_left .
Lemma Uinv_opp_right :
forall x ,
x + [1-]
x == 1.
Hint Resolve Uinv_opp_right .
Lemma Uinv_inv :
forall x :
U , [1-] [1-]
x ==
x .
Hint Resolve Uinv_inv .
Lemma Unit :
forall x :U,
x <= 1.
Hint Resolve Unit .
Lemma Uinv_zero : [1-] 0 = 1.
Lemma Ueq_class :
forall x y :U,
class (
x ==y).
Lemma Ueq_double_neg :
forall x y :
U , ~ ~ (
x ==
y ) ->
x ==
y .
Hint Resolve Ueq_class .
Hint Immediate Ueq_double_neg .
Lemma Ule_orc :
forall x y :
U ,
orc (
x <=y) (~
x <=y).
Implicit Arguments Ule_orc [].
Lemma Ueq_orc :
forall x y :U,
orc (
x ==y) (~
x ==y).
Implicit Arguments Ueq_orc [].
Lemma Upos :
forall x :U, 0 <=
x .
Lemma Ule_0_1 : 0 <= 1.
Hint Resolve Upos Ule_0_1 .
Properties of == derived from properties of
Definition UPlus :
U -m>
U -m>
U :=
mon2 Uplus .
Definition UPlus_simpl :
forall x y ,
UPlus x y =
x +y.
Save .
Instance Uplus_continuous2 :
continuous2 (
mon2 Uplus ).
Save .
Hint Resolve Uplus_continuous2 .
Lemma Umult_le_compat_right :
forall x y z :U,
y <=
z ->
x *
y <=
x *
z .
Hint Resolve Umult_le_compat_right .
Instance Umult_mon2 :
monotonic2 Umult .
Save .
Lemma Umult_le_compat_left :
forall x y z :U,
x <=
y ->
x *
z <=
y *
z .
Hint Resolve Umult_le_compat_left .
Lemma Umult_le_compat :
forall x y z t ,
x <=
y ->
z <=
t ->
x *
z <=
y *
t .
Hint Immediate Umult_le_compat .
Definition UMult :
U -m>
U -m>
U :=
mon2 Umult .
Lemma Umult_eq_compat_left :
forall x y z :U,
x ==
y -> (
x *
z ) == (
y *
z ).
Hint Resolve Umult_eq_compat_left .
Lemma Umult_eq_compat_right :
forall x y z :U,
x ==
y -> (
z *
x ) == (
z *
y ).
Hint Resolve Umult_eq_compat_left Umult_eq_compat_right .
Definition UMult_simpl :
forall x y ,
UMult x y =
x *y.
Save .
Instance Umult_continuous2 :
continuous2 (
mon2 Umult ).
Save .
Hint Resolve Umult_continuous2 .
Add Morphism Umult with signature Oeq &
Oeq &
Oeq
as Umult_eq_compat .
Qed .
Hint Immediate Umult_eq_compat .
Instance Uinv_mon :
monotonic (
o1 :=Iord
U )
Uinv .
Save .
Definition UInv :
U --m>
U :=
mon (
o1 :=Iord
U )
Uinv .
Definition UInv_simpl :
forall x ,
UInv x = [1-]
x .
Save .
Lemma Ule_eq_compat :
forall x1 x2 :
U ,
x1 ==
x2 ->
forall x3 x4 :
U ,
x3 ==
x4 ->
x1 <=
x3 ->
x2 <=
x4 .
Definition and properties of
Definition Ult (
r1 r2 :U) :
Prop := ~ (
r2 <=
r1 ).
Infix "<" :=
Ult :
U_scope .
Hint Unfold Ult .
Add Morphism Ult with signature Oeq &
Oeq &
iff as Ult_eq_compat_iff .
Save .
Lemma Ult_eq_compat :
forall x1 x2 :
U ,
x1 ==
x2 ->
forall x3 x4 :
U ,
x3 ==
x4 ->
x1 <
x3 ->
x2 <
x4 .
Lemma Ult_class :
forall x y ,
class (
x <y).
Hint Resolve Ult_class .
Lemma Ule_zero_eq : forall x :U, x <= 0 -> x == 0.Lemma Uge_one_eq : forall x :U, 1 <= x -> x == 1.Hint Immediate Ule_zero_eq Uge_one_eq .
Lemma Ult_neq :
forall x y :U,
x <
y -> ~x ==
y .
Lemma Ult_neq_rev :
forall x y :U,
x <
y -> ~y ==
x .
Lemma Ult_trans :
forall x y z ,
x <y ->
y <z ->
x <z.
Lemma Ult_le :
forall x y :U,
x <
y ->
x <=
y .
Lemma Ule_diff_lt :
forall x y :
U ,
x <=
y -> ~x==y ->
x <
y .
Hint Immediate Ult_neq Ult_neq_rev Ult_le .
Hint Resolve Ule_diff_lt .
Lemma Ult_neq_zero :
forall x , ~0 ==
x -> 0 <
x .
Hint Resolve Ule_total Ult_neq_zero .
Lemma Udistr_plus_left :
forall x y z ,
y <= [1-]
z ->
x * (
y +
z ) ==
x *
y +
x *
z .
Lemma Udistr_inv_left :
forall x y , [1-](
x *
y ) == (
x * ([1-]
y )) + [1-]
x .
Hint Resolve Uinv_eq_compat Udistr_plus_left Udistr_inv_left .
Lemma Uplus_perm2 :
forall x y z :U,
x + (
y +
z ) ==
y + (
x +
z ).
Lemma Umult_perm2 :
forall x y z :U,
x * (
y *
z ) ==
y * (
x *
z ).
Lemma Uplus_perm3 :
forall x y z :
U , (
x + (
y +
z )) ==
z + (
x +
y ).
Lemma Umult_perm3 :
forall x y z :
U , (
x * (
y *
z )) ==
z * (
x *
y ).
Hint Resolve Uplus_perm2 Umult_perm2 Uplus_perm3 Umult_perm3 .
Lemma Uinv_simpl :
forall x y :
U , [1-]
x == [1-]
y ->
x ==
y .
Hint Immediate Uinv_simpl .
Lemma Umult_decomp :
forall x y ,
x ==
x *
y +
x * [1-]
y .
Hint Resolve Umult_decomp .
More properties on + and * and Uinv
Lemma Umult_one_right :
forall x :U,
x * 1 ==
x .
Hint Resolve Umult_one_right .
Lemma Umult_one_right_eq :
forall x y :U,
y == 1 ->
x *
y ==
x .
Hint Resolve Umult_one_right_eq .
Lemma Umult_one_left_eq :
forall x y :U,
x == 1 ->
x *
y ==
y .
Hint Resolve Umult_one_left_eq .
Lemma Udistr_plus_left_le :
forall x y z :
U ,
x * (
y +
z ) <=
x *
y +
x *
z .
Lemma Uplus_eq_simpl_right :
forall x y z :U,
z <= [1-]
x ->
z <= [1-]
y -> (
x +
z ) == (
y +
z ) ->
x ==
y .
Lemma Ule_plus_right :
forall x y ,
x <=
x +
y .
Lemma Ule_plus_left :
forall x y ,
y <=
x +
y .
Hint Resolve Ule_plus_right Ule_plus_left .
Lemma Ule_mult_right :
forall x y ,
x *
y <=
x .
Lemma Ule_mult_left :
forall x y ,
x *
y <=
y .
Hint Resolve Ule_mult_right Ule_mult_left .
Lemma Uinv_le_perm_right :
forall x y :U,
x <= [1-]
y ->
y <= [1-]
x .
Hint Immediate Uinv_le_perm_right .
Lemma Uinv_le_perm_left :
forall x y :U, [1-]
x <=
y -> [1-]
y <=
x .
Hint Immediate Uinv_le_perm_left .
Lemma Uinv_le_simpl :
forall x y :U, [1-]
x <= [1-]
y ->
y <=
x .
Hint Immediate Uinv_le_simpl .
Lemma Uinv_double_le_simpl_right :
forall x y ,
x <=y ->
x <= [1-][1-]
y .
Hint Resolve Uinv_double_le_simpl_right .
Lemma Uinv_double_le_simpl_left :
forall x y ,
x <=y -> [1-][1-]
x <=
y .
Hint Resolve Uinv_double_le_simpl_left .
Lemma Uinv_eq_perm_left :
forall x y :U,
x == [1-]
y -> [1-]
x ==
y .
Hint Immediate Uinv_eq_perm_left .
Lemma Uinv_eq_perm_right :
forall x y :U, [1-]
x ==
y ->
x == [1-]
y .
Hint Immediate Uinv_eq_perm_right .
Lemma Uinv_eq_simpl :
forall x y :U, [1-]
x == [1-]
y ->
x ==
y .
Hint Immediate Uinv_eq_simpl .
Lemma Uinv_double_eq_simpl_right :
forall x y ,
x ==y ->
x == [1-][1-]
y .
Hint Resolve Uinv_double_eq_simpl_right .
Lemma Uinv_double_eq_simpl_left :
forall x y ,
x ==y -> [1-][1-]
x ==
y .
Hint Resolve Uinv_double_eq_simpl_left .
Lemma Uinv_plus_right :
forall x y ,
y <= [1-]
x -> [1-] (
x +
y ) +
y == [1-]
x .
Hint Resolve Uinv_plus_right .
Lemma Uplus_eq_simpl_left :
forall x y z :U,
x <= [1-]
y ->
x <= [1-]
z -> (
x +
y ) == (
x +
z ) ->
y ==
z .
Lemma Uplus_eq_zero_left :
forall x y :U,
x <= [1-]
y -> (
x +
y ) ==
y ->
x == 0.
Lemma Uinv_le_trans :
forall x y z t ,
x <= [1-]
y ->
z <=x ->
t <=y ->
z <= [1-]
t .
Lemma Uinv_plus_left_le :
forall x y , [1-]
y <= [1-](
x +y) +x.
Lemma Uinv_plus_right_le :
forall x y , [1-]
x <= [1-](
x +y) +y.
Hint Resolve Uinv_plus_left_le Uinv_plus_right_le .
Lemma neq_sym :
forall x y :U, ~x==y -> ~y==x.
Hint Immediate neq_sym .
Lemma Uinv_neq_compat :
forall x y , ~x ==
y -> ~ [1-]
x == [1-]
y .
Lemma Uinv_neq_simpl :
forall x y , ~ [1-]
x == [1-]
y -> ~x ==
y .
Hint Resolve Uinv_neq_compat .
Hint Immediate Uinv_neq_simpl .
Lemma Uinv_neq_left :
forall x y , ~x == [1-]
y -> ~ [1-]
x ==
y .
Lemma Uinv_neq_right :
forall x y , ~ [1-]
x ==
y -> ~x == [1-]
y .
Lemma Ult_antirefl :
forall x :U, ~x <
x .
Lemma Ult_0_1 : (0 < 1).
Lemma Ule_lt_trans :
forall x y z :U,
x <=
y ->
y <
z ->
x <
z .
Lemma Ult_le_trans :
forall x y z :U,
x <
y ->
y <=
z ->
x <
z .
Hint Resolve Ult_0_1 Ult_antirefl .
Lemma Ule_neq_zero :
forall (
x y :U), ~0 ==
x ->
x <=
y -> ~ 0 ==
y .
Lemma Uplus_neq_zero_left :
forall x y , ~0 ==
x -> ~0 ==
x +y.
Lemma Uplus_neq_zero_right :
forall x y , ~0 ==
y -> ~0 ==
x +y.
Lemma not_Ult_le :
forall x y , ~x <
y ->
y <=
x .
Lemma Ule_not_lt :
forall x y ,
x <=
y -> ~y <
x .
Hint Immediate not_Ult_le Ule_not_lt .
Theorem Uplus_le_simpl_left :
forall x y z :
U ,
z <= [1-]
x ->
z +
x <=
z +
y ->
x <=
y .
Lemma Uplus_lt_compat_left :
forall x y z :U,
z <= [1-]
y ->
x <
y -> (
x +
z ) < (
y +
z ).
Lemma Uplus_lt_compat_right :
forall x y z :U,
z <= [1-]
y ->
x <
y -> (
z +
x ) < (
z +
y ).
Hint Resolve Uplus_lt_compat_right Uplus_lt_compat_left .
Lemma Uplus_lt_compat :
forall x y z t :U,
z <= [1-]
x ->
t <= [1-]
y ->
x <
y ->
z <
t -> (
x +
z ) < (
y +
t ).
Hint Immediate Uplus_lt_compat .
Lemma Uplus_lt_simpl_left :
forall x y z :U,
z <= [1-]
y -> (
z +
x ) < (
z +
y ) ->
x <
y .
Lemma Uplus_lt_simpl_right :
forall x y z :U,
z <= [1-]
y -> (
x +
z ) < (
y +
z ) ->
x <
y .
Lemma Uplus_one_le :
forall x y ,
x +
y == 1 -> [1-]
y <=
x .
Hint Immediate Uplus_one_le .
Theorem Uplus_eq_zero :
forall x ,
x <= [1-]
x -> (
x +
x ) ==
x ->
x == 0.
Lemma Umult_zero_left :
forall x , 0 *
x == 0.
Hint Resolve Umult_zero_left .
Lemma Umult_zero_right :
forall x , (
x * 0) == 0.
Hint Resolve Uplus_eq_zero Umult_zero_right .
Lemma Umult_zero_left_eq :
forall x y ,
x == 0 ->
x *
y == 0.
Lemma Umult_zero_right_eq :
forall x y ,
y == 0 ->
x *
y == 0.
Lemma Umult_zero_eq :
forall x y z ,
x == 0 ->
x *
y ==
x *
z .
Compatibility of operations with respect to order.
Lemma Umult_le_simpl_right :
forall x y z , ~0 ==
z -> (
x *
z ) <= (
y *
z ) ->
x <=
y .
Hint Resolve Umult_le_simpl_right .
Lemma Umult_simpl_right :
forall x y z , ~0 ==
z -> (
x *
z ) == (
y *
z ) ->
x ==
y .
Lemma Umult_simpl_left :
forall x y z , ~0 ==
x -> (
x *
y ) == (
x *
z ) ->
y ==
z .
Lemma Umult_lt_compat_left :
forall x y z , ~0 ==
z ->
x <
y -> (
x *
z ) < (
y *
z ).
Lemma Umult_lt_compat_right :
forall x y z , ~0 ==
z ->
x <
y -> (
z *
x ) < (
z *
y ).
Lemma Umult_lt_simpl_right :
forall x y z , ~0 ==
z -> (
x *
z ) < (
y *
z ) ->
x <
y .
Lemma Umult_lt_simpl_left :
forall x y z , ~0 ==
z -> (
z *
x ) < (
z *
y ) ->
x <
y .
Hint Resolve Umult_lt_compat_left Umult_lt_compat_right .
Lemma Umult_zero_simpl_right :
forall x y , 0 ==
x *y -> ~0 ==
x -> 0 ==
y .
Lemma Umult_zero_simpl_left :
forall x y , 0 ==
x *y -> ~0 ==
y -> 0 ==
x .
Lemma Umult_neq_zero :
forall x y , ~0 ==
x -> ~0 ==
y -> ~0 ==
x *y.
Hint Resolve Umult_neq_zero .
Lemma Umult_lt_zero :
forall x y , 0 <
x -> 0 <
y -> 0 <
x *y.
Hint Resolve Umult_lt_zero .
Lemma Umult_lt_compat :
forall x y z t ,
x <
y ->
z <
t ->
x *
z <
y *
t .
Lemma Uplus_one :
forall x y , [1-]
x <=
y ->
x +
y == 1.
Hint Resolve Uplus_one .
Lemma Uplus_one_right :
forall x ,
x + 1 == 1.
Lemma Uplus_one_left :
forall x :U, 1 +
x == 1.
Hint Resolve Uplus_one_right Uplus_one_left .
Lemma Uinv_mult_simpl :
forall x y z t ,
x <= [1-]
y -> (
x *
z ) <= [1-] (
y *
t ).
Hint Resolve Uinv_mult_simpl .
Lemma Umult_inv_plus :
forall x y ,
x * [1-]
y +
y ==
x +
y * [1-]
x .
Hint Resolve Umult_inv_plus .
Lemma Umult_inv_plus_le :
forall x y z ,
y <=
z ->
x * [1-]
y +
y <=
x * [1-]
z +
z .
Hint Resolve Umult_inv_plus_le .
Lemma Uplus_lt_Uinv :
forall x y ,
x +y < 1 ->
x <= [1-]
y .
Lemma Uinv_lt_perm_left :
forall x y :
U , [1-]
x <
y -> [1-]
y <
x .
Lemma Uinv_lt_perm_right :
forall x y :
U ,
x < [1-]
y ->
y < [1-]
x .
Hint Immediate Uinv_lt_perm_left Uinv_lt_perm_right .
Lemma Uinv_lt_one :
forall x , 0 <
x -> [1-]
x < 1.
Lemma Uinv_lt_zero :
forall x ,
x < 1 -> 0 < [1-]
x .
Hint Resolve Uinv_lt_one Uinv_lt_zero .
Lemma orc_inv_plus_one :
forall x y ,
orc (
x <=[1-]
y ) (
x +y==1).
Lemma Umult_lt_right :
forall p q ,
p <1 -> 0 <
q ->
p *
q <
q .
Lemma Umult_lt_left :
forall p q , 0 <
p ->
q < 1 ->
p *
q <
p .
Hint Resolve Umult_lt_right Umult_lt_left .
Fixpoint Uexp (
x :U) (
n :nat) {
struct n } :
U :=
match n with 0 => 1 | (
S p ) =>
x *
Uexp x p end .
Infix "^" :=
Uexp :
U_scope .
Lemma Uexp_1 :
forall x ,
x ^1==x.
Lemma Uexp_0 :
forall x ,
x ^0==1.
Lemma Uexp_zero :
forall n , (0<n)%nat -> 0^n==0.
Lemma Uexp_one :
forall n , 1^n==1.
Lemma Uexp_le_compat_right :
forall x n m , (
n <=m)%nat ->
x ^m <=
x ^n.
Lemma Uexp_le_compat_left :
forall x y n ,
x <=y ->
x ^n <=
y ^n.
Hint Resolve Uexp_le_compat_left Uexp_le_compat_right .
Lemma Uexp_le_compat :
forall x y (
n m :nat),
x <=y ->
n <=m ->
x ^m <=
y ^n.
Instance Uexp_mon2 :
monotonic2 (
o1 :=Iord
U ) (
o3 :=Iord
U )
Uexp .
Save .
Definition UExp :
U --m> (
nat -m->
U ) :=
mon2 Uexp .
Add Morphism Uexp with signature Oeq &
eq &
Oeq as Uexp_eq_compat .
Save .
Lemma Uexp_inv_S :
forall x n , ([1-]
x ^(S
n ))==x*([1-]
x ^n)+[1-]
x .
Lemma Uexp_lt_compat :
forall p q n , (
O <n)%nat->(p<q)->(p^n<q^n).
Hint Resolve Uexp_lt_compat .
Lemma Uexp_lt_zero :
forall p n , (0<p)->(0<p^n).
Hint Resolve Uexp_lt_zero .
Lemma Uexp_lt_one :
forall p n , (0<n)%nat->(p<1)->(p^n<1).
Hint Resolve Uexp_lt_one .
Lemma Uexp_lt_antimon :
forall p n m , (
n <m)%nat-> 0<p ->
p < 1 ->
p ^m <
p ^n.
Hint Resolve Uexp_lt_antimon .
Lemma Udiv_mult :
forall x y , ~0==y ->
x <=
y -> (
x /y)*y ==
x .
Hint Resolve Udiv_mult .
Lemma Umult_div_le :
forall x y ,
y * (
x /
y ) <=
x .
Hint Resolve Umult_div_le .
Lemma Udiv_mult_le :
forall x y , (
x /y)*y <=
x .
Hint Resolve Udiv_mult_le .
Lemma Udiv_le_compat_left :
forall x y z ,
x <=
y ->
x /z <=
y /z.
Hint Resolve Udiv_le_compat_left .
Lemma Udiv_eq_compat_left :
forall x y z ,
x ==
y ->
x /z ==
y /z.
Hint Resolve Udiv_eq_compat_left .
Lemma Umult_div_le_left :
forall x y z , ~0==y ->
x *y<=z ->
x <=
z /y.
Lemma Udiv_le_compat_right :
forall x y z , ~0==y ->
y <=
z ->
x /z <=
x /y.
Hint Resolve Udiv_le_compat_right .
Lemma Udiv_eq_compat_right :
forall x y z ,
y ==
z ->
x /z ==
x /y.
Hint Resolve Udiv_eq_compat_right .
Add Morphism Udiv with signature Oeq &
Oeq &
Oeq as Udiv_eq_compat .
Save .
Add Morphism Udiv with signature Ole ++>
Oeq &
Ole as Udiv_le_compat .
Save .
Lemma Umult_div_eq :
forall x y z , ~0 ==
y ->
x *y ==
z ->
x ==
z /y.
Lemma Umult_div_le_right :
forall x y z ,
x <=
y *z ->
x /z <=
y .
Lemma Udiv_le :
forall x y , ~0==y ->
x <=
x /y.
Lemma Udiv_zero :
forall x , 0/x==0.
Hint Resolve Udiv_zero .
Lemma Udiv_zero_eq :
forall x y , 0==x ->
x /y==0.
Hint Resolve Udiv_zero_eq .
Lemma Udiv_one :
forall x , ~0==x ->
x /1==x.
Hint Resolve Udiv_one .
Lemma Udiv_refl :
forall x , ~0==x ->
x /x==1.
Hint Resolve Udiv_refl .
Lemma Umult_div_assoc :
forall x y z ,
y <=
z -> (
x *y) /
z ==
x * (
y /z).
Lemma Udiv_mult_assoc :
forall x y z ,
x <=
y *z ->
x /(y*z) == (
x /y)/z.
Lemma Udiv_inv :
forall x y , ~0==y -> [1-](
x /y) <= ([1-]
x )/y.
Lemma Uplus_div_inv :
forall x y z ,
x +y <=
z ->
x <=[1-]
y ->
x /z <= [1-](
y /z).
Hint Resolve Uplus_div_inv .
Lemma Udiv_plus_le :
forall x y z ,
x /z +
y /z <= (
x +y)/z.
Hint Resolve Udiv_plus_le .
Lemma Udiv_plus :
forall x y z , (
x +y)/z ==
x /z +
y /z.
Hint Resolve Udiv_plus .
Instance Udiv_mon :
forall k ,
monotonic (
fun x => (
x /k)).
Save .
Definition UDiv (
k :U) :
U -m>
U :=
mon (
fun x => (
x /k)).
Lemma UDiv_simpl :
forall (
k :U)
x ,
UDiv k x =
x /k.
Definition and properties of
A conjonction operation which coincides with min and mult
on 0 and 1, see Morgan & McIver
Definition Uesp (
x y :U) := [1-] ([1-]
x + [1-]
y ).
Infix "&" :=
Uesp (
left associativity ,
at level 40) :
U_scope .
Lemma Uinv_plus_esp :
forall x y , [1-] (
x +
y ) == [1-]
x & [1-]
y .
Hint Resolve Uinv_plus_esp .
Lemma Uinv_esp_plus :
forall x y , [1-] (
x &
y ) == [1-]
x + [1-]
y .
Hint Resolve Uinv_esp_plus .
Lemma Uesp_sym :
forall x y :
U ,
x &
y ==
y &
x .
Lemma Uesp_one_right :
forall x :
U ,
x & 1 ==
x .
Lemma Uesp_one_left :
forall x :
U , 1 &
x ==
x .
Lemma Uesp_zero :
forall x y ,
x <= [1-]
y ->
x &
y == 0.
Hint Resolve Uesp_sym Uesp_one_right Uesp_one_left Uesp_zero .
Lemma Uesp_zero_right :
forall x :
U ,
x & 0 == 0.
Lemma Uesp_zero_left :
forall x :
U , 0 &
x == 0.
Hint Resolve Uesp_zero_right Uesp_zero_left .
Add Morphism Uesp with signature Oeq &
Oeq &
Oeq as Uesp_eq_compat .
Save .
Lemma Uesp_le_compat :
forall x y z t ,
x <=y ->
z <=t ->
x &z <=
y &t.
Hint Immediate Uesp_le_compat Uesp_eq_compat .
Lemma Uesp_zero_one_mult_left :
forall x y ,
orc (
x == 0) (
x == 1) ->
x &
y ==
x *
y .
Lemma Uesp_zero_one_mult_right :
forall x y ,
orc (
y == 0) (
y == 1) ->
x &
y ==
x *
y .
Hint Immediate Uesp_zero_one_mult_left Uesp_zero_one_mult_right .
Instance Uesp_mon :
monotonic2 Uesp .
Save .
Definition UEsp :
U -m>
U -m>
U :=
mon2 Uesp .
Lemma UEsp_simpl :
forall x y ,
UEsp x y =
x &
y .
Lemma Uesp_le_left :
forall x y ,
x &
y <=
x .
Lemma Uesp_le_right :
forall x y ,
x &
y <=
y .
Hint Resolve Uesp_le_left Uesp_le_right .
Lemma Uesp_plus_inv :
forall x y , [1-]
y <=
x ->
x ==
x &
y + [1-]
y .
Hint Resolve Uesp_plus_inv .
Lemma Uesp_le_plus_inv :
forall x y ,
x <=
x &
y + [1-]
y .
Hint Resolve Uesp_le_plus_inv .
Lemma Uplus_inv_le_esp :
forall x y z ,
x <=
y + ([1-]
z ) ->
x &
z <=
y .
Hint Immediate Uplus_inv_le_esp .
Definition and properties of
Definition Uminus (
x y :U) := [1-] ([1-]
x +
y ).
Infix "-" :=
Uminus :
U_scope .
Lemma Uminus_le_compat_left :
forall x y z ,
x <=
y ->
x -
z <=
y -
z .
Lemma Uminus_le_compat_right :
forall x y z ,
y <=
z ->
x -
z <=
x -
y .
Hint Resolve Uminus_le_compat_left Uminus_le_compat_right .
Lemma Uminus_le_compat :
forall x y z t ,
x <=
y ->
t <=
z ->
x -
z <=
y -
t .
Hint Immediate Uminus_le_compat .
Add Morphism Uminus with signature Oeq &
Oeq &
Oeq as Uminus_eq_compat .
Save .
Hint Immediate Uminus_eq_compat .
Lemma Uminus_zero_right :
forall x ,
x - 0 ==
x .
Lemma Uminus_one_left :
forall x , 1 -
x == [1-]
x .
Lemma Uminus_le_zero :
forall x y ,
x <=
y ->
x -
y == 0.
Hint Resolve Uminus_zero_right Uminus_one_left Uminus_le_zero .
Lemma Uminus_zero_left :
forall x , 0 -
x == 0.
Hint Resolve Uminus_zero_left .
Lemma Uminus_eq :
forall x ,
x -
x == 0.
Hint Resolve Uminus_eq .
Lemma Uminus_le_left :
forall x y ,
x -
y <=
x .
Hint Resolve Uminus_le_left .
Lemma Uminus_le_inv :
forall x y ,
x -
y <= [1-]
y .
Hint Resolve Uminus_le_inv .
Lemma Uminus_plus_simpl :
forall x y ,
y <=
x -> (
x -
y ) +
y ==
x .
Lemma Uminus_plus_zero :
forall x y ,
x <=
y -> (
x -
y ) +
y ==
y .
Hint Resolve Uminus_plus_simpl Uminus_plus_zero .
Lemma Uminus_plus_le :
forall x y ,
x <= (
x -
y ) +
y .
Hint Resolve Uminus_plus_le .
Lemma Uesp_minus_distr_left :
forall x y z , (
x &
y ) -
z == (
x -
z ) &
y .
Lemma Uesp_minus_distr_right :
forall x y z , (
x &
y ) -
z ==
x & (
y -
z ).
Hint Resolve Uesp_minus_distr_left Uesp_minus_distr_right .
Lemma Uesp_minus_distr :
forall x y z t , (
x &
y ) - (
z +
t ) == (
x -
z ) & (
y -
t ).
Hint Resolve Uesp_minus_distr .
Lemma Uminus_esp_simpl_left :
forall x y , [1-]
x <=
y ->
x - (
x &
y ) == [1-]
y .
Lemma Uplus_esp_simpl :
forall x y , (
x - (
x &
y ))+y ==
x +y.
Hint Resolve Uminus_esp_simpl_left Uplus_esp_simpl .
Lemma Uminus_esp_le_inv :
forall x y ,
x - (
x &
y ) <= [1-]
y .
Hint Resolve Uminus_esp_le_inv .
Lemma Uplus_esp_inv_simpl :
forall x y ,
x <= [1-]
y -> (
x +
y ) & [1-]
y ==
x .
Hint Resolve Uplus_esp_inv_simpl .
Lemma Uplus_inv_esp_simpl :
forall x y ,
x <=
y -> (
x + [1-]
y ) &
y ==
x .
Hint Resolve Uplus_inv_esp_simpl .
Definition and properties of max
Definition max (
x y :
U ) :
U := (
x -
y ) +
y .
Lemma max_eq_right :
forall x y :
U ,
y <=
x ->
max x y ==
x .
Lemma max_eq_left :
forall x y :
U ,
x <=
y ->
max x y ==
y .
Hint Resolve max_eq_right max_eq_left .
Lemma max_eq_case :
forall x y :
U ,
orc (
max x y ==
x ) (
max x y ==
y ).
Add Morphism max with signature Oeq &
Oeq &
Oeq as max_eq_compat .
Save .
Lemma max_le_right :
forall x y :
U ,
x <=
max x y .
Lemma max_le_left :
forall x y :
U ,
y <=
max x y .
Hint Resolve max_le_right max_le_left .
Lemma max_le :
forall x y z :
U ,
x <=
z ->
y <=
z ->
max x y <=
z .
Lemma max_le_compat :
forall x y z t :
U ,
x <=
y ->
z <=
t ->
max x z <=
max y t .
Hint Immediate max_le_compat .
Instance max_mon :
monotonic2 max .
Save .
Definition Max :
U -m>
U -m>
U :=
mon2 max .
Lemma max_eq_mult :
forall k x y ,
max (
k *x) (
k *y) ==
k *
max x y .
Lemma max_eq_plus_cte_right :
forall x y k ,
max (
x +k) (
y +k) == (
max x y ) +
k .
Hint Resolve max_eq_mult max_eq_plus_cte_right .
Definition and properties of min
Definition min (
x y :
U ) :
U := [1-] ((
y -
x ) + [1-]
y ).
Lemma min_eq_right :
forall x y :
U ,
x <=
y ->
min x y ==
x .
Lemma min_eq_left :
forall x y :
U ,
y <=
x ->
min x y ==
y .
Hint Resolve min_eq_right min_eq_left .
Lemma min_eq_case :
forall x y :
U ,
orc (
min x y ==
x ) (
min x y ==
y ).
Add Morphism min with signature Oeq &
Oeq &
Oeq as min_eq_compat .
Save .
Lemma min_le_right :
forall x y :
U ,
min x y <=x.
Lemma min_le_left :
forall x y :
U ,
min x y <=
y .
Hint Resolve min_le_right min_le_left .
Lemma min_le :
forall x y z :
U ,
z <=
x ->
z <=
y ->
z <=
min x y .
Lemma Uinv_min_max :
forall x y , [1-](
min x y )==max ([1-]
x ) ([1-]
y ).
Lemma Uinv_max_min :
forall x y , [1-](
max x y )==min ([1-]
x ) ([1-]
y ).
Lemma min_mult :
forall x y k ,
min (
k *
x ) (
k *
y ) ==
k * (
min x y ).
Hint Resolve min_mult .
Lemma min_plus :
forall x1 x2 y1 y2 ,
(
min x1 x2 ) + (
min y1 y2 ) <=
min (
x1 +y1) (
x2 +y2).
Hint Resolve min_plus .
Lemma min_plus_cte :
forall x y k ,
min (
x +
k ) (
y +
k ) == (
min x y ) +
k .
Hint Resolve min_plus_cte .
Lemma min_le_compat :
forall x1 y1 x2 y2 ,
x1 <=y1 ->
x2 <=y2 ->
min x1 x2 <=
min y1 y2 .
Hint Immediate min_le_compat .
Instance min_mon2 :
monotonic2 min .
Save .
Definition Min :
U -m>
U -m>
U :=
mon2 min .
Lemma min_sym :
forall x y ,
min x y ==
min y x .
Hint Resolve min_sym .
Lemma incr_decomp_aux :
forall f g :
nat -m>
U ,
forall n1 n2 , (
forall m , ~ ((
n1 <=m)%nat /\
f n1 <=
g m ))
-> (
forall m , ~((n2<=m)%nat /\
g n2 <=
f m )) -> (
n1 <=n2)%nat ->
False .
Lemma incr_decomp :
forall f g :
nat -m>
U ,
orc (
forall n ,
exc (
fun m => (
n <=m)%nat /\
f n <=
g m ))
(
forall n ,
exc (
fun m => (
n <=m)%nat /\
g n <=
f m )).
Lemma Uplus_minus_simpl_right :
forall x y ,
y <= [1-]
x -> (
x +
y ) -
y ==
x .
Hint Resolve Uplus_minus_simpl_right .
Lemma Uplus_minus_simpl_left :
forall x y ,
y <= [1-]
x -> (
x +
y ) -
x ==
y .
Lemma Uminus_assoc_left :
forall x y z , (
x -
y ) -
z ==
x - (
y +
z ).
Hint Resolve Uminus_assoc_left .
Lemma Uminus_perm :
forall x y z , (
x -
y ) -
z == (
x -
z ) -
y .
Hint Resolve Uminus_perm .
Lemma Uminus_le_perm_left :
forall x y z ,
y <=
x ->
x -
y <=
z ->
x <=
z +
y .
Lemma Uplus_le_perm_left :
forall x y z ,
x <=
y +
z ->
x -
y <=
z .
Lemma Uminus_eq_perm_left :
forall x y z ,
y <=
x ->
x -
y ==
z ->
x ==
z +
y .
Lemma Uplus_eq_perm_left :
forall x y z ,
y <= [1-]
z ->
x ==
y +
z ->
x -
y ==
z .
Hint Resolve Uminus_le_perm_left Uminus_eq_perm_left .
Hint Resolve Uplus_le_perm_left Uplus_eq_perm_left .
Lemma Uminus_le_perm_right :
forall x y z ,
z <=
y ->
x <=
y -
z ->
x +
z <=
y .
Lemma Uplus_le_perm_right :
forall x y z ,
z <= [1-]
x ->
x +
z <=
y ->
x <=
y -
z .
Hint Resolve Uminus_le_perm_right Uplus_le_perm_right .
Lemma Uminus_le_perm :
forall x y z ,
z <=
y ->
x <= [1-]
z ->
x <=
y -
z ->
z <=
y -
x .
Hint Resolve Uminus_le_perm .
Lemma Uminus_eq_perm_right :
forall x y z ,
z <=
y ->
x ==
y -
z ->
x +
z ==
y .
Hint Resolve Uminus_eq_perm_right .
Lemma Uminus_plus_perm :
forall x y z ,
y <=
x ->
z <= [1-]
x -> (
x -
y ) +
z == (
x +
z ) -
y .
Lemma Uminus_zero_le :
forall x y ,
x -
y == 0 ->
x <=
y .
Lemma Uminus_lt_non_zero :
forall x y ,
x <
y -> ~0 ==
y -
x .
Hint Immediate Uminus_zero_le Uminus_lt_non_zero .
Lemma Ult_le_nth_minus :
forall x y ,
x <
y ->
exc (
fun n =>
x <=
y - [1/]1+n).
Lemma Ult_le_nth_plus :
forall x y ,
x <
y ->
exc (
fun n :
nat =>
x + [1/]1+n <=
y ).
Lemma Uminus_distr_left :
forall x y z , (
x -
y ) *
z == (
x *
z ) - (
y *
z ).
Hint Resolve Uminus_distr_left .
Lemma Uminus_distr_right :
forall x y z ,
x * (
y -
z ) == (
x *
y ) - (
x *
z ).
Hint Resolve Uminus_distr_right .
Lemma Uminus_assoc_right :
forall x y z ,
y <=
x ->
z <=
y ->
x - (
y -
z ) == (
x -
y ) +
z .
Lemma Uplus_minus_assoc_right :
forall x y z ,
y <= [1-]
x ->
z <=
y ->
x + (
y -
z ) == (
x +
y ) -
z .
Lemma Udiv_minus :
forall x y z , ~0 ==
z ->
x <=
z -> (
x -
y ) /
z ==
x /z -
y /z.
Lemma Umult_inv_minus :
forall x y ,
x * [1-]
y ==
x -
x *
y .
Hint Resolve Umult_inv_minus .
Lemma Uinv_mult_minus :
forall x y , ([1-]
x ) *
y ==
y -
x *
y .
Hint Resolve Uinv_mult_minus .
Lemma Uminus_plus_perm_right :
forall x y z ,
y <=
x ->
y <=
z -> (
x -
y ) +
z ==
x + (
z -
y ).
Hint Resolve Uminus_plus_perm_right .
Lemma Uminus_triangular : forall x y z , x - y <= (x - z ) + (z - y ).Hint Resolve Uminus_triangular .
Definition and properties of generalized sums
Definition sigma : (
nat ->
U ) ->
nat -m>
U .
Defined .
Lemma sigma_0 :
forall (
f :
nat ->
U ),
sigma f O == 0.
Lemma sigma_S :
forall (
f :nat ->
U ) (
n :nat),
sigma f (
S n ) = (
f n ) + (
sigma f n ).
Lemma sigma_1 :
forall (
f :
nat ->
U ),
sigma f (
S 0) ==
f O .
Lemma sigma_incr :
forall (
f :
nat ->
U ) (
n m :nat), (
n <=
m )%nat ->
sigma f n <=
sigma f m .
Hint Resolve sigma_incr .
Lemma sigma_eq_compat :
forall (
f g :
nat ->
U ) (
n :nat),
(
forall k , (
k <
n )%nat ->
f k ==
g k ) ->
sigma f n ==
sigma g n .
Lemma sigma_le_compat :
forall (
f g :
nat ->
U ) (
n :nat),
(
forall k , (
k <
n )%nat ->
f k <=
g k ) ->
sigma f n <=
sigma g n .
Lemma sigma_S_lift :
forall (
f :nat ->
U ) (
n :nat),
sigma f (
S n ) == (
f O ) + (
sigma (
fun k =>
f (
S k ))
n ).
Lemma sigma_plus_lift :
forall (
f :nat ->
U ) (
n m :nat),
sigma f (
n +m)%nat ==
sigma f n +
sigma (
fun k =>
f (
n +k)%nat)
m .
Lemma sigma_zero :
forall f n , (
forall k , (
k <n)%nat ->
f k ==0)->
sigma f n ==0.
Lemma sigma_not_zero :
forall f n k , (
k <n)%nat -> 0 <
f k -> 0 <
sigma f n .
Lemma sigma_zero_elim :
forall f n , (
sigma f n )==0->forall
k , (
k <n)%nat ->
f k ==0.
Hint Resolve sigma_eq_compat sigma_le_compat sigma_zero .
Lemma sigma_le :
forall f n k , (
k <n)%nat ->
f k <=
sigma f n .
Hint Resolve sigma_le .
Lemma sigma_minus_decr :
forall f n , (
forall k ,
f (
S k ) <=
f k ) ->
sigma (
fun k =>
f k -
f (
S k ))
n ==
f O -
f n .
Lemma sigma_minus_incr :
forall f n , (
forall k ,
f k <=
f (
S k )) ->
sigma (
fun k =>
f (
S k ) -
f k )
n ==
f n -
f O .
Definition and properties of generalized products
Definition prod (
alpha :
nat ->
U ) (
n :nat) :=
compn Umult 1
alpha n .
Lemma prod_0 :
forall (
f :
nat ->
U ),
prod f 0 = 1.
Lemma prod_S :
forall (
f :nat ->
U ) (
n :nat),
prod f (
S n ) = (
f n ) * (
prod f n ).
Lemma prod_1 :
forall (
f :
nat ->
U ),
prod f (
S 0) ==
f O .
Lemma prod_S_lift :
forall (
f :nat ->
U ) (
n :nat),
prod f (
S n ) == (
f O ) * (
prod (
fun k =>
f (
S k ))
n ).
Lemma prod_decr :
forall (
f :
nat ->
U ) (
n m :nat), (
n <=
m )%nat ->
prod f m <=
prod f n .
Hint Resolve prod_decr .
Lemma prod_eq_compat :
forall (
f g :
nat ->
U ) (
n :nat),
(
forall k , (
k <
n )%nat ->
f k ==
g k ) -> (
prod f n ) == (
prod g n ).
Lemma prod_le_compat :
forall (
f g :
nat ->
U ) (
n :nat),
(
forall k , (
k <
n )%nat ->
f k <=
g k ) ->
prod f n <=
prod g n .
Lemma prod_zero :
forall f n k , (
k <n)%nat ->
f k ==0 ->
prod f n ==0.
Lemma prod_not_zero :
forall f n , (
forall k , (
k <n)%nat -> 0 <
f k )-> 0 <
prod f n .
Lemma prod_zero_elim :
forall f n ,
prod f n ==0 ->
exc (
fun k => (
k <n)%nat /\
f k ==0).
Hint Resolve prod_eq_compat prod_le_compat prod_not_zero .
Lemma prod_le :
forall f n k , (
k <n)%nat ->
prod f n <=
f k .
Lemma prod_minus :
forall f n ,
prod f n -
prod f (
S n ) == ([1-]
f n ) *
prod f n .
Definition Prod : (
nat ->
U ) ->
nat -m->
U .
Defined .
Lemma Prod_simpl :
forall f n ,
Prod f n =
prod f n .
Hint Resolve Prod_simpl .
Lemma Unth_zero : [1/]1+0 == 1.Notation "[1/2]" := (Unth 1).Lemma Unth_one : [1/2] == [1-] [1/2].Hint Resolve Unth_zero Unth_one .Lemma Unth_one_plus : [1/2] + [1/2] == 1.Hint Resolve Unth_one_plus .Lemma Unth_one_refl : forall t , [1/2] * t + [1/2] * t == t .Lemma Unth_not_null : forall n , ~ (0 == [1/]1+n).Hint Resolve Unth_not_null .Lemma Unth_lt_zero : forall n , 0 < [1/]1+n.Hint Resolve Unth_lt_zero .Lemma Unth_inv_lt_one : forall n , [1-][1/]1+n<1.Hint Resolve Unth_inv_lt_one .Lemma Unth_not_one : forall n , ~ (1 == [1-][1/]1+n).Hint Resolve Unth_not_one .Lemma Unth_prop_sigma : forall n , [1/]1+n == [1-] (sigma (fun k => [1/]1+n) n ).Hint Resolve Unth_prop_sigma .Lemma Unth_sigma_n : forall n : nat , ~ (1 == sigma (fun k => [1/]1+n) n ).Lemma Unth_sigma_Sn : forall n : nat , 1 == sigma (fun k => [1/]1+n) (S n ).Hint Resolve Unth_sigma_n Unth_sigma_Sn .Lemma Unth_decr : forall n , [1/]1+(S n ) < [1/]1+n.Hint Resolve Unth_decr .Lemma Unth_anti_mon :forall n m , (n <= m )%nat -> [1/]1+m <= [1/]1+n.Hint Resolve Unth_anti_mon .Lemma Unth_le_half : forall n , [1/]1+(S n ) <= [1/2].Hint Resolve Unth_le_half .
Definition mean (
x y :U) := [1/2] *
x + [1/2] *
y .
Lemma mean_eq :
forall x :U,
mean x x ==x.
Lemma mean_le_compat_right :
forall x y z ,
y <=
z ->
mean x y <=
mean x z .
Lemma mean_le_compat_left :
forall x y z ,
x <=
y ->
mean x z <=
mean y z .
Hint Resolve mean_eq mean_le_compat_left mean_le_compat_right .
Lemma mean_lt_compat_right :
forall x y z ,
y <
z ->
mean x y <
mean x z .
Lemma mean_lt_compat_left :
forall x y z ,
x <
y ->
mean x z <
mean y z .
Hint Resolve mean_eq mean_le_compat_left mean_le_compat_right .
Hint Resolve mean_lt_compat_left mean_lt_compat_right .
Lemma mean_le_up :
forall x y ,
x <=
y ->
mean x y <=
y .
Lemma mean_le_down :
forall x y ,
x <=
y ->
x <=
mean x y .
Lemma mean_lt_up :
forall x y ,
x <
y ->
mean x y <
y .
Lemma mean_lt_down :
forall x y ,
x <
y ->
x <
mean x y .
Hint Resolve mean_le_up mean_le_down mean_lt_up mean_lt_down .
Lemma le_half_inv :
forall x ,
x <= [1/2] ->
x <= [1-]
x .
Hint Immediate le_half_inv .
Lemma ge_half_inv :
forall x , [1/2] <=
x -> [1-]
x <=
x .
Hint Immediate ge_half_inv .
Lemma Uinv_le_half_left :
forall x ,
x <= [1/2] -> [1/2] <= [1-]
x .
Lemma Uinv_le_half_right :
forall x , [1/2] <=
x -> [1-]
x <= [1/2].
Hint Resolve Uinv_le_half_left Uinv_le_half_right .
Lemma half_twice :
forall x , (
x <= [1/2]) -> ([1/2]) * (
x +
x ) ==
x .
Lemma half_twice_le :
forall x , ([1/2]) * (
x +
x ) <=
x .
Lemma Uinv_half :
forall x , ([1/2]) * ([1-]
x ) + ([1/2]) == [1-] (([1/2]) *
x ).
Lemma half_esp :
forall x , ([1/2] <=
x ) -> ([1/2]) * (
x &
x ) + [1/2] ==
x .
Lemma half_esp_le :
forall x ,
x <= ([1/2]) * (
x &
x ) + [1/2].
Hint Resolve half_esp_le .
Lemma half_le :
forall x y ,
y <= [1-]
y ->
x <=
y +
y -> ([1/2]) *
x <=
y .
Lemma half_Unth :
forall n , ([1/2])*([1/]1+n) <= [1/]1+(S
n ).
Hint Resolve half_le half_Unth .
Lemma half_exp :
forall n , [1/2]^n == [1/2]^(S
n ) + [1/2]^(S
n ).
Definition diff (
x y :U) := (
x -
y ) + (
y -
x ).
Lemma diff_eq :
forall x ,
diff x x == 0.
Hint Resolve diff_eq .
Lemma diff_sym :
forall x y ,
diff x y ==
diff y x .
Hint Resolve diff_sym .
Lemma diff_zero :
forall x ,
diff x 0 ==
x .
Hint Resolve diff_zero .
Add Morphism diff with signature Oeq &
Oeq &
Oeq as diff_eq_compat .
Qed .
Hint Immediate diff_eq_compat .
Lemma diff_plus_ok :
forall x y ,
x -
y <= [1-](
y -
x ).
Hint Resolve diff_plus_ok .
Lemma diff_Uminus :
forall x y ,
x <=
y ->
diff x y ==
y -
x .
Lemma diff_Uplus_le :
forall x y ,
x <=
diff x y +
y .
Hint Resolve diff_Uplus_le .
Lemma diff_triangular :
forall x y z ,
diff x y <=
diff x z +
diff y z .
Hint Resolve diff_triangular .
Lemma Ule_lt_lim :
forall x y , (
forall t ,
t <
x ->
t <=
y ) ->
x <=
y .
Lemma Ule_nth_lim :
forall x y , (
forall p ,
x <=
y + [1/]1+p) ->
x <=
y .
Properties of least upper bounds
Lemma lub_un :
mlub (
cte nat 1) == 1.
Hint Resolve lub_un .
Lemma UPlusk_eq :
forall k ,
UPlus k ==
mon (
Uplus k ).
Lemma UMultk_eq :
forall k ,
UMult k ==
mon (
Umult k ).
Lemma UPlus_continuous_right :
forall k ,
continuous (
UPlus k ).
Hint Resolve UPlus_continuous_right .
Lemma UPlus_continuous_left :
continuous UPlus .
Hint Resolve UPlus_continuous_left .
Lemma UMult_continuous_right :
forall k ,
continuous (
UMult k ).
Hint Resolve UMult_continuous_right .
Lemma UMult_continuous_left :
continuous UMult .
Hint Resolve UMult_continuous_left .
Lemma lub_eq_plus_cte_left :
forall (
f :nat -m>
U ) (
k :U),
lub ((
UPlus k ) @
f ) ==
k +
lub f .
Hint Resolve lub_eq_plus_cte_left .
Lemma lub_eq_mult :
forall (
k :U) (
f :nat -m>
U ),
lub ((
UMult k ) @
f ) ==
k *
lub f .
Hint Resolve lub_eq_mult .
Lemma lub_eq_plus_cte_right :
forall (
f :
nat -m>
U ) (
k :U),
lub ((
mshift UPlus k ) @
f ) ==
lub f +
k .
Hint Resolve lub_eq_plus_cte_right .
Lemma min_lub_le :
forall f g :
nat -m>
U ,
lub ((
Min @2
f )
g ) <=
min (
lub f ) (
lub g ).
Lemma min_lub_le_incr_aux :
forall f g :
nat -m>
U ,
(
forall n ,
exc (
fun m => (
n <=m)%nat /\
f n <=
g m ))
->
min (
lub f ) (
lub g ) <=
lub ((
Min @2
f )
g ).
Lemma min_lub_le_incr :
forall f g :
nat -m>
U ,
min (
lub f ) (
lub g ) <=
lub ((
Min @2
f )
g ).
Lemma lub_eq_esp_right :
forall (
f :
nat -m>
U ) (
k :
U ),
lub ((
mshift UEsp k ) @
f ) ==
lub f &
k .
Hint Resolve lub_eq_esp_right .
Lemma Udiv_continuous :
forall (
k :U),
continuous (
UDiv k ).
Hint Resolve Udiv_continuous .
Definition glb (
f :nat -m->
U ) := [1-](
lub (
UInv @
f )).
Lemma glb_le :
forall (
f :
nat -m->
U ) (
n :
nat ),
glb f <= (
f n ).
Lemma le_glb :
forall (
f :
nat -m->
U ) (
x :U), (
forall n :
nat ,
x <=
f n )->
x <=
glb f .
Hint Resolve glb_le le_glb .
Definition Uopp :
cpo (
o :=Iord
U )
U .
Defined .
Lemma Uopp_lub_simpl
:
forall h :
nat -m->
U ,
lub (
cpo :=Uopp)
h =
glb h .
Lemma Uopp_mon_seq :
forall f :nat -m->
U ,
forall n m :nat, (
n <=
m )%nat ->
f m <=
f n .
Hint Resolve Uopp_mon_seq .
Infinite product
Definition prod_inf (f : nat -> U ) : U := glb (Prod f ).
Properties of glb
Lemma glb_le_compat :
forall f g :
nat -m->
U , (
forall x ,
f x <=
g x ) ->
glb f <=
glb g .
Hint Resolve glb_le_compat .
Lemma glb_eq_compat :
forall f g :
nat -m->
U ,
f ==
g ->
glb f ==
glb g .
Hint Resolve glb_eq_compat .
Lemma glb_cte :
forall c :
U ,
glb (
mon (
cte nat (
o1 :=(Iord
U ))
c )) ==
c .
Hint Resolve glb_cte .
Lemma glb_eq_plus_cte_right :
forall (
f :
nat -m->
U ) (
k :
U ),
glb (
Imon (
mshift UPlus k ) @
f ) ==
glb f +
k .
Hint Resolve glb_eq_plus_cte_right .
Lemma glb_eq_plus_cte_left :
forall (
f :
nat -m->
U ) (
k :
U ),
glb (
Imon (
UPlus k ) @
f ) ==
k +
glb f .
Hint Resolve glb_eq_plus_cte_left .
Lemma glb_eq_mult :
forall (
k :
U ) (
f :
nat -m->
U ),
glb (
Imon (
UMult k ) @
f ) ==
k *
glb f .
Lemma Imon2_plus_continuous
:
continuous2 (
c1 :=Uopp) (
c2 :=Uopp) (
c3 :=Uopp) (
imon2 Uplus ).
Hint Resolve Imon2_plus_continuous .
Lemma Uinv_continuous :
continuous (
c1 :=Uopp)
UInv .
Definition UInvopp :
U -m->
U .
Defined .
Lemma UInvopp_simpl :
forall x ,
UInvopp x = [1-]
x .
Lemma Uinvopp_continuous :
continuous (
c2 :=Uopp)
UInvopp .
Hint Resolve Uinv_continuous Uinvopp_continuous .
Instance Uminus_mon2 :
monotonic2 (
o2 :=Iord
U )
Uminus .
Save .
Definition UMinus :
U -m>
U --m>
U :=
mon2 Uminus .
Lemma UMinus_simpl :
forall x y ,
UMinus x y =
x -
y .
Lemma Uminus_continuous2 :
continuous2 (
c2 :=Uopp)
UMinus .
Hint Resolve Uminus_continuous2 .
Lemma glb_le_esp :
forall f g :nat -m->
U , (
glb f ) & (
glb g ) <=
glb ((
imon2 Uesp @2
f )
g ).
Hint Resolve glb_le_esp .
Lemma Uesp_min :
forall a1 a2 b1 b2 ,
min a1 b1 &
min a2 b2 <=
min (
a1 &
a2 ) (
b1 &
b2 ).
Defining lubs of arbitrary sequences
Fixpoint seq_max (
f :nat ->
U ) (
n :nat) :
U :=
match n with
O =>
f O |
S p =>
max (
seq_max f p ) (
f (
S p ))
end .
Lemma seq_max_incr :
forall f n ,
seq_max f n <=
seq_max f (
S n ).
Hint Resolve seq_max_incr .
Lemma seq_max_le :
forall f n ,
f n <=
seq_max f n .
Hint Resolve seq_max_le .
Instance seq_max_mon :
forall (
f :nat ->
U ),
monotonic (
seq_max f ).
Save .
Definition sMax (
f :nat ->
U ) :
nat -m>
U :=
mon (
seq_max f ).
Lemma sMax_mult :
forall k (
f :nat->U),
sMax (
fun n =>
k *
f n ) ==
UMult k @
sMax f .
Lemma sMax_plus_cte_right :
forall k (
f :nat->
U ),
sMax (
fun n =>
f n +
k ) ==
mshift UPlus k @
sMax f .
Definition Ulub (
f :nat ->
U ) :=
lub (
sMax f ).
Lemma le_Ulub :
forall f n ,
f n <=
Ulub f .
Lemma Ulub_le :
forall f x , (
forall n ,
f n <=
x ) ->
Ulub f <=
x .
Hint Resolve le_Ulub Ulub_le .
Lemma Ulub_le_compat :
forall f g :
nat ->U,
f <=
g ->
Ulub f <=
Ulub g .
Hint Resolve Ulub_le_compat .
Add Morphism Ulub with signature Oeq &
Oeq as Ulub_eq_compat .
Save .
Hint Resolve Ulub_eq_compat .
Lemma Ulub_eq_mult :
forall k (
f :nat->U),
Ulub (
fun n =>
k *
f n )==
k *
Ulub f .
Lemma Ulub_eq_plus_cte_right :
forall (
f :nat->U)
k ,
Ulub (
fun n =>
f n +
k )==
Ulub f +
k .
Hint Resolve Ulub_eq_mult Ulub_eq_plus_cte_right .
Lemma Ulub_eq_esp_right :
forall (
f :
nat ->
U ) (
k :
U ),
Ulub (
fun n =>
f n &
k ) ==
Ulub f &
k .
Hint Resolve lub_eq_esp_right .
Lemma Ulub_le_plus :
forall f g ,
Ulub (
fun n =>
f n +
g n ) <=
Ulub f +
Ulub g .
Hint Resolve Ulub_le_plus .
Definition Uglb (
f :nat ->
U ) :U := [1-]
Ulub (
fun n => [1-](
f n )).
Lemma Uglb_le :
forall (
f :
nat ->
U ) (
n :
nat ),
Uglb f <=
f n .
Lemma le_Uglb :
forall (
f :
nat ->
U ) (
x :U), (
forall n :
nat ,
x <=
f n )->
x <=
Uglb f .
Hint Resolve Uglb_le le_Uglb .
Lemma Uglb_le_compat :
forall f g :
nat ->U,
f <=
g ->
Uglb f <=
Uglb g .
Hint Resolve Uglb_le_compat .
Add Morphism Uglb with signature Oeq &
Oeq as Uglb_eq_compat .
Save .
Hint Resolve Uglb_eq_compat .
Lemma Uglb_eq_plus_cte_right :
forall (
f :
nat ->
U ) (
k :
U ),
Uglb (
fun n =>
f n +
k ) ==
Uglb f +
k .
Hint Resolve Uglb_eq_plus_cte_right .
Lemma Uglb_eq_mult :
forall (
k :
U ) (
f :
nat ->
U ),
Uglb (
fun n =>
k *
f n ) ==
k *
Uglb f .
Hint Resolve Uglb_eq_mult Uglb_eq_plus_cte_right .
Lemma Uglb_le_plus :
forall f g ,
Uglb f +
Uglb g <=
Uglb (
fun n =>
f n +
g n ).
Hint Resolve Uglb_le_plus .
Lemma Ulub_lub :
forall f :nat -m>
U ,
Ulub f ==
lub f .
Hint Resolve Ulub_lub .
Lemma Uglb_glb :
forall f :nat -m->
U ,
Uglb f ==
glb f .
Hint Resolve Uglb_glb .
Lemma lub_le_plus :
forall (
f g :
nat -m>
U ),
lub ((
UPlus @2
f )
g ) <=
lub f +
lub g .
Hint Resolve lub_le_plus .
Lemma glb_le_plus :
forall (
f g :nat -m->
U ) ,
glb f +
glb g <=
glb ((
Imon2 UPlus @2
f )
g ).
Hint Resolve glb_le_plus .
Lemma lub_eq_plus :
forall f g :
nat -m>
U ,
lub ((
UPlus @2
f )
g ) ==
lub f +
lub g .
Hint Resolve lub_eq_plus .
Lemma glb_mon :
forall f :
nat -m>
U ,
Uglb f ==
f O .
Lemma lub_inv :
forall (
f g :
nat -m>
U ), (
forall n ,
f n <= [1-]
g n ) ->
lub f <= [1-] (
lub g ).
Lemma glb_lift_left :
forall (
f :nat -m->
U )
n ,
glb f ==
glb (
mon (
seq_lift_left f n )).
Hint Resolve glb_lift_left .
Lemma Ulub_mon :
forall f :
nat -m->
U ,
Ulub f ==
f O .
Lemma lub_glb_le :
forall (
f :nat -m>
U ) (
g :nat -m->
U ),
(
forall n ,
f n <=
g n ) ->
lub f <=
glb g .
Lemma lub_lub_inv_le :
forall f g :nat -m>
U ,
(
forall n ,
f n <= [1-]
g n ) ->
lub f <= [1-]
lub g .
Lemma Uplus_opp_continuous_right :
forall k ,
continuous (
c1 :=Uopp) (
c2 :=Uopp) (
Imon (
UPlus k )).
Lemma Uplus_opp_continuous_left :
continuous (
c1 :=Uopp) (
c2 :=fmon_cpo (
o :=Iord
U ) (
c :=Uopp))(Imon2
UPlus ).
Hint Resolve Uplus_opp_continuous_right Uplus_opp_continuous_left .
Lemma glb_eq_plus :
forall (
f g :
nat -m->
U ),
glb ((
Imon2 UPlus @2
f )
g ) ==
glb f +
glb g .
Hint Resolve glb_eq_plus .
Instance sigma_mon :monotonic
sigma .
Save .
BUG V8.2-1
Definition Sigma : (nat -> U) -m> nat-m> U
:= mon sigma.
Anomaly: Non-functional construction. Please report.
Definition Sigma : (
nat ->
U ) -m>
nat -m>
U
:=
mon sigma (
fmonotonic :=sigma_mon).
Lemma Sigma_simpl :
forall f ,
Sigma f =
sigma f .
Lemma sigma_continuous1 :
continuous Sigma .
Lemma sigma_lub1 :
forall (
f :
nat -m> (
nat ->
U ))
n ,
sigma (
lub f )
n ==
lub ((
mshift Sigma n ) @
f ).
Definition MF (
A :Type) :
Type :=
A ->
U .
Instance MFcpo (
A :Type) :
cpo (
MF A ).
Defined .
Definition MFopp (
A :Type) :
cpo (
o :=Iord (
A ->
U )) (
MF A ).
Defined .
Lemma MFopp_lub_eq :
forall (
A :Type) (
h :nat-m->
MF A ),
lub (
cpo :=MFopp
A )
h ==
fun x =>
glb (
Iord_app x @
h ).
Lemma fle_intro :
forall (
A :Type) (
f g :
MF A ), (
forall x ,
f x <=
g x ) ->
f <=
g .
Hint Resolve fle_intro .
Lemma feq_intro :
forall (
A :Type) (
f g :
MF A ), (
forall x ,
f x ==
g x ) ->
f ==
g .
Hint Resolve feq_intro .
Definition fplus (
A :Type) (
f g :
MF A ) :
MF A :=
fun x =>
f x +
g x .
Definition fmult (
A :Type) (
k :U) (
f :
MF A ) :
MF A :=
fun x =>
k *
f x .
Definition finv (
A :Type) (
f :
MF A ) :
MF A :=
fun x => [1-]
f x .
Definition fzero (
A :Type) :
MF A :=
fun x => 0.
Definition fdiv (
A :Type) (
k :U) (
f :
MF A ) :
MF A :=
fun x => (
f x ) /
k .
Definition flub (
A :Type) (
f :
nat -m>
MF A ) :
MF A :=
lub f .
Lemma fplus_simpl :
forall (
A :Type)(f
g :
MF A ) (
x :
A ),
fplus f g x =
f x +
g x .
Lemma fmult_simpl :
forall (
A :Type)(k:U) (
f :
MF A ) (
x :
A ),
fmult k f x =
k *
f x .
Lemma fdiv_simpl :
forall (
A :Type)(k:U) (
f :
MF A ) (
x :
A ),
fdiv k f x =
f x /
k .
Implicit Arguments fzero [].
Lemma fzero_simpl :
forall (
A :Type)(x :
A ),
fzero A x = 0.
Lemma finv_simpl :
forall (
A :Type)(f :
MF A ) (
x :
A ),
finv f x = [1-]
f x .
Lemma flub_simpl :
forall (
A :Type)(f:nat -m>
MF A ) (
x :A),
(
flub f )
x =
lub (
f <o>
x ).
Hint Resolve fplus_simpl fmult_simpl fzero_simpl finv_simpl flub_simpl .
Definition fone (
A :Type) :
MF A :=
fun x => 1.
Implicit Arguments fone [].
Lemma fone_simpl :
forall (
A :Type) (
x :A),
fone A x = 1.
Definition fcte (
A :Type) (
k :U):
MF A :=
fun x =>
k .
Implicit Arguments fcte [].
Lemma fcte_simpl :
forall (
A :Type) (
k :U) (
x :A),
fcte A k x =
k .
Definition fminus (
A :Type) (
f g :MF
A ) :
MF A :=
fun x =>
f x -
g x .
Lemma fminus_simpl :
forall (
A :Type) (
f g :
MF A ) (
x :A),
fminus f g x =
f x -
g x .
Definition fesp (
A :Type) (
f g :MF
A ) :
MF A :=
fun x =>
f x &
g x .
Lemma fesp_simpl :
forall (
A :Type) (
f g :
MF A ) (
x :A),
fesp f g x =
f x &
g x .
Definition fconj (
A :Type)(f
g :MF
A ) :
MF A :=
fun x =>
f x *
g x .
Lemma fconj_simpl :
forall (
A :Type) (
f g :
MF A ) (
x :A),
fconj f g x =
f x *
g x .
Lemma MF_lub_simpl :
forall (
A :Type) (
f :
nat -m>
MF A ) (
x :A),
lub f x =
lub (
f <o>x).
Hint Resolve MF_lub_simpl .
Lemma fplus_eq_compat :
forall A (
f1 f2 g1 g2 :MF
A ),
f1 ==f2 ->
g1 ==g2 ->
fplus f1 g1 ==
fplus f2 g2 .
Add Parametric Morphism (
A :Type) : (@
fplus A )
with signature Oeq &
Oeq &
Oeq
as fplus_feq_compat_morph .
Save .
Instance fplus_mon2 :
forall A ,
monotonic2 (
fplus (
A :=A)).
Save .
Hint Resolve fplus_mon2 .
Lemma fplus_le_compat :
forall A (
f1 f2 g1 g2 :MF
A ),
f1 <=f2 ->
g1 <=g2 ->
fplus f1 g1 <=
fplus f2 g2 .
Add Parametric Morphism A : (@
fplus A )
with signature Ole ++>
Ole ++>
Ole
as fplus_fle_compat_morph .
Save .
Lemma finv_eq_compat :
forall A (
f g :MF
A ),
f ==g ->
finv f ==
finv g .
Add Parametric Morphism A : (@
finv A )
with signature Oeq &
Oeq
as finv_feq_compat_morph .
Save .
Instance finv_mon :
forall A ,
monotonic (
o2 :=Iord (
MF A )) (
finv (
A :=A)).
Save .
Hint Resolve finv_mon .
Lemma finv_le_compat :
forall A (
f g :MF
A ),
f <=
g ->
finv g <=
finv f .
Add Parametric Morphism A : (@
finv A )
with signature Ole -->
Ole as finv_fle_compat_morph .
Save .
Lemma fmult_eq_compat :
forall A k1 k2 (
f1 f2 :MF
A ),
k1 ==
k2 ->
f1 ==
f2 ->
fmult k1 f1 ==
fmult k2 f2 .
Add Parametric Morphism A : (@
fmult A )
with signature Oeq &
Oeq &
Oeq as fmult_feq_compat_morph .
Save .
Instance fmult_mon2 :
forall A ,
monotonic2 (
fmult (
A :=A)).
Save .
Hint Resolve fmult_mon2 .
Lemma fmult_le_compat :
forall A k1 k2 (
f1 f2 :MF
A ),
k1 <=
k2 ->
f1 <=
f2 ->
fmult k1 f1 <=
fmult k2 f2 .
Add Parametric Morphism A : (@
fmult A )
with signature Ole ++>
Ole ++>
Ole as fmult_fle_compat_morph .
Save .
Lemma fminus_eq_compat :
forall A (
f1 f2 g1 g2 :MF
A ),
f1 ==
f2 ->
g1 ==
g2 ->
fminus f1 g1 ==
fminus f2 g2 .
Add Parametric Morphism A : (@
fminus A )
with signature Oeq &
Oeq &
Oeq as fminus_feq_compat_morph .
Save .
Instance fminus_mon2 :
forall A ,
monotonic2 (
o2 :=Iord (
MF A )) (
fminus (
A :=A)).
Save .
Hint Resolve fminus_mon2 .
Lemma fminus_le_compat :
forall A (
f1 f2 g1 g2 :MF
A ),
f1 <=
f2 ->
g2 <=
g1 ->
fminus f1 g1 <=
fminus f2 g2 .
Add Parametric Morphism A : (@
fminus A )
with signature Ole ++>
Ole -->
Ole as fminus_fle_compat_morph .
Save .
Lemma fesp_eq_compat :
forall A (
f1 f2 g1 g2 :MF
A ),
f1 ==f2 ->
g1 ==g2 ->
fesp f1 g1 ==
fesp f2 g2 .
Add Parametric Morphism A : (@
fesp A )
with signature Oeq &
Oeq &
Oeq as fesp_feq_compat_morph .
Save .
Instance fesp_mon2 :
forall A ,
monotonic2 (
fesp (
A :=A)).
Save .
Hint Resolve fesp_mon2 .
Lemma fesp_le_compat :
forall A (
f1 f2 g1 g2 :MF
A ),
f1 <=f2 ->
g1 <=g2 ->
fesp f1 g1 <=
fesp f2 g2 .
Add Parametric Morphism A : (@
fesp A )
with signature Ole ++>
Ole ++>
Ole as fesp_fle_compat_morph .
Save .
Lemma fconj_eq_compat :
forall A (
f1 f2 g1 g2 :MF
A ),
f1 ==f2 ->
g1 ==g2 ->
fconj f1 g1 ==
fconj f2 g2 .
Add Parametric Morphism A : (@
fconj A )
with signature Oeq &
Oeq &
Oeq
as fconj_feq_compat_morph .
Save .
Instance fconj_mon2 :
forall A ,
monotonic2 (
fconj (
A :=A)).
Save .
Hint Resolve fconj_mon2 .
Lemma fconj_le_compat :
forall A (
f1 f2 g1 g2 :MF
A ),
f1 <=
f2 ->
g1 <=
g2 ->
fconj f1 g1 <=
fconj f2 g2 .
Add Parametric Morphism A : (@
fconj A )
with signature Ole ++>
Ole ++>
Ole
as fconj_fle_compat_morph .
Save .
Hint Immediate fplus_le_compat fplus_eq_compat fesp_le_compat fesp_eq_compat
fmult_le_compat fmult_eq_compat fminus_le_compat fminus_eq_compat
fconj_eq_compat .
Hint Resolve finv_eq_compat .
Lemma fle_fplus_left :
forall (
A :Type) (
f g :
MF A ),
f <=
fplus f g .
Lemma fle_fplus_right :
forall (
A :Type) (
f g :
MF A ),
g <=
fplus f g .
Lemma fle_fmult :
forall (
A :Type) (
k :U)(f :
MF A ),
fmult k f <=
f .
Lemma fle_zero :
forall (
A :Type) (
f :
MF A ),
fzero A <=
f .
Lemma fle_one :
forall (
A :Type) (
f :
MF A ),
f <=
fone A .
Lemma feq_finv_finv :
forall (
A :Type) (
f :
MF A ),
finv (
finv f ) ==
f .
Lemma fle_fesp_left :
forall (
A :Type) (
f g :
MF A ),
fesp f g <=
f .
Lemma fle_fesp_right :
forall (
A :Type) (
f g :
MF A ),
fesp f g <=
g .
Lemma fle_fconj_left :
forall (
A :Type) (
f g :
MF A ),
fconj f g <=
f .
Lemma fle_fconj_right :
forall (
A :Type) (
f g :
MF A ),
fconj f g <=
g .
Lemma fconj_decomp :
forall A (
f g :
MF A ),
f ==
fplus (
fconj f g ) (
fconj f (
finv g )).
Hint Resolve fconj_decomp .
Compatibility of addition of two functions
Definition fplusok (
A :Type) (
f g :
MF A ) :=
f <=
finv g .
Hint Unfold fplusok .
Lemma fplusok_sym :
forall (
A :Type) (
f g :
MF A ) ,
fplusok f g ->
fplusok g f .
Hint Immediate fplusok_sym .
Lemma fplusok_inv :
forall (
A :Type) (
f :
MF A ) ,
fplusok f (
finv f ).
Hint Resolve fplusok_inv .
Lemma fplusok_le_compat :
forall (
A :Type)(f1
f2 g1 g2 :MF
A ),
fplusok f2 g2 ->
f1 <=
f2 ->
g1 <=
g2 ->
fplusok f1 g1 .
Hint Resolve fle_fplus_left fle_fplus_right fle_zero fle_one feq_finv_finv finv_le_compat
fle_fmult fle_fesp_left fle_fesp_right fle_fconj_left fle_fconj_right .
Lemma fconj_fplusok :
forall (
A :Type)(f
g h :MF
A ),
fplusok g h ->
fplusok (
fconj f g ) (
fconj f h ).
Hint Resolve fconj_fplusok .
Definition Fconj A :
MF A -m>
MF A -m>
MF A :=
mon2 (
fconj (
A :=A)).
Lemma Fconj_simpl :
forall A f g ,
Fconj A f g =
fconj f g .
Lemma fconj_sym :
forall A (
f g :
MF A ),
fconj f g ==
fconj g f .
Lemma lub_MF_simpl :
forall A (
h :
nat -m>
MF A ) (
x :A),
lub h x =
lub (
h <o>
x ).
Instance fconj_continuous2 A :
continuous2 (
Fconj A ).
Save .
Definition Fmult A :
U -m>
MF A -m>
MF A :=
mon2 (
fmult (
A :=A)).
Lemma Fmult_simpl :
forall A k f ,
Fmult A k f =
fmult k f .
Lemma fmult_continuous2 :
forall A ,
continuous2 (
Fmult A ).
Fixpoints of functions of type
Section FixDef .
Variable A :Type.
Variable F :
MF A -m>
MF A .
Definition mufix :
MF A :=
fixp F .
Definition G :
MF A --m->
MF A :=
Imon F .
Definition nufix :
MF A :=
fixp (
c :=MFopp
A )
G .
Lemma mufix_inv :
forall f :
MF A ,
F f <=
f ->
mufix <=
f .
Hint Resolve mufix_inv .
Lemma nufix_inv :
forall f :MF
A ,
f <=
F f ->
f <=
nufix .
Hint Resolve nufix_inv .
Lemma mufix_le :
mufix <=
F mufix .
Hint Resolve mufix_le .
Lemma nufix_sup :
F nufix <=
nufix .
Hint Resolve nufix_sup .
Lemma mufix_eq :
continuous F ->
mufix ==
F mufix .
Hint Resolve mufix_eq .
Lemma nufix_eq :
continuous (
c1 :=MFopp
A ) (
c2 :=MFopp
A )
G ->
nufix ==
F nufix .
Hint Resolve nufix_eq .
End FixDef .
Hint Resolve mufix_le mufix_eq nufix_sup nufix_eq .
Definition Fcte (
A :Type) (
f :MF
A ) :
MF A -m>
MF A :=
mon (
cte (
MF A )
f ).
Lemma mufix_cte :
forall (
A :Type) (
f :MF
A ),
mufix (
Fcte f ) ==
f .
Lemma nufix_cte :
forall (
A :Type) (
f :MF
A ),
nufix (
Fcte f ) ==
f .
Hint Resolve mufix_cte nufix_cte .
Properties of (pseudo-)barycenter of two points
Lemma Uinv_bary :
forall a b x y :
U ,
a <= [1-]
b ->
[1-] (
a *
x +
b *
y ) ==
a * [1-]
x +
b * [1-]
y + [1-] (
a +
b ).
Hint Resolve Uinv_bary .
Lemma Uinv_bary_le :
forall a b x y :
U ,
a <= [1-]
b ->
a * [1-]
x +
b * [1-]
y <= [1-] (
a *
x +
b *
y ).
Hint Resolve Uinv_bary_le .
Lemma Uinv_bary_eq :
forall a b x y :
U ,
a == [1-]
b ->
[1-] (
a *
x +
b *
y ) ==
a * [1-]
x +
b * [1-]
y .
Hint Resolve Uinv_bary_eq .
Lemma bary_refl_eq :
forall a b x ,
a == [1-]
b ->
a *
x +
b *
x ==
x .
Hint Resolve bary_refl_eq .
Lemma bary_le_left :
forall a b x y , [1-]
b <=
a ->
x <=
y ->
x <=
a *
x +
b *
y .
Lemma bary_le_right :
forall a b x y ,
a <= [1-]
b ->
x <=
y ->
a *
x +
b *
y <=
y .
Hint Resolve bary_le_left bary_le_right .
Lemma bary_up_eq :
forall a b x y :
U ,
a == [1-]
b ->
x <=
y ->
a *
x +
b *
y ==
x +
b * (
y -
x ).
Lemma bary_up_le :
forall a b x y :
U ,
a <= [1-]
b ->
a *
x +
b *
y <=
x +
b * (
y -
x ).
Lemma bary_Uminus_left :
forall a b x y :
U ,
a <= [1-]
b -> (
a *
x +
b *
y ) -
x <=
b * (
y -
x ).
Lemma bary_Uminus_left_eq :
forall a b x y :
U ,
a == [1-]
b ->
x <=
y -> (
a *
x +
b *
y ) -
x ==
b * (
y -
x ).
Lemma Uminus_bary_left
:
forall a b x y :
U , [1-]
a <=
b ->
x - (
a *
x +
b *
y ) <=
b * (
x -
y ).
Lemma Uminus_bary_left_eq
:
forall a b x y :
U ,
a == [1-]
b ->
y <=
x ->
x - (
a *
x +
b *
y ) ==
b * (
x -
y ).
Hint Resolve bary_up_eq bary_up_le bary_Uminus_left Uminus_bary_left bary_Uminus_left_eq Uminus_bary_left_eq .
Lemma diff_bary_left_eq
:
forall a b x y :
U ,
a == [1-]
b ->
diff x (
a *
x +
b *
y ) ==
b *
diff x y .
Hint Resolve diff_bary_left_eq .
Lemma Uinv_half_bary :
forall x y :
U , [1-] ([1/2] *
x + [1/2] *
y ) == [1/2] * [1-]
x + [1/2] * [1-]
y .
Hint Resolve Uinv_half_bary .
Properties of generalized sums sigma
Lemma sigma_plus :
forall (
f g :
nat ->
U ) (
n :nat),
sigma (
fun k => (
f k ) + (
g k ))
n ==
sigma f n +
sigma g n .
Definition retract (
f :
nat ->
U ) (
n :
nat ) :=
forall k , (
k <
n )%nat ->
f k <= [1-] (
sigma f k ).
Lemma retract0 :
forall (
f :
nat ->
U ),
retract f 0.
Lemma retract_pred :
forall (
f :
nat ->
U ) (
n :
nat ),
retract f (
S n ) ->
retract f n .
Lemma retractS :
forall (
f :
nat ->
U ) (
n :
nat ),
retract f (
S n ) ->
f n <= [1-] (
sigma f n ).
Hint Immediate retract_pred retractS .
Lemma retractS_inv :
forall (
f :
nat ->
U ) (
n :
nat ),
retract f (
S n ) ->
sigma f n <= [1-]
f n .
Hint Immediate retractS_inv .
Lemma retractS_intro :
forall (
f :
nat ->
U ) (
n :
nat ),
retract f n ->
f n <= [1-] (
sigma f n ) ->
retract f (
S n ).
Hint Resolve retract0 retractS_intro .
Lemma retract_lt :
forall (
f :
nat ->
U ) (
n :
nat ),
sigma f n < 1 ->
retract f n .
Lemma retract_unif :
forall (
f :
nat ->
U ) (
n :
nat ),
(
forall k , (
k <=n)%nat ->
f k <= [1/]1+n) ->
retract f (
S n ).
Hint Resolve retract_unif .
Lemma sigma_mult :
forall (
f :
nat ->
U )
n c ,
retract f n ->
sigma (
fun k =>
c * (
f k ))
n ==
c * (
sigma f n ).
Hint Resolve sigma_mult .
Lemma sigma_prod_maj :
forall (
f g :
nat ->
U )
n ,
sigma (
fun k => (
f k ) * (
g k ))
n <=
sigma f n .
Hint Resolve sigma_prod_maj .
Lemma sigma_prod_le :
forall (
f g :
nat ->
U ) (
c :U), (
forall k , (
f k ) <=
c )
->
forall n ,
retract g n ->
sigma (
fun k => (
f k ) * (
g k ))
n <=
c * (
sigma g n ).
Lemma sigma_prod_ge :
forall (
f g :
nat ->
U ) (
c :U), (
forall k ,
c <= (
f k ))
->
forall n , (
retract g n ) ->
c * (
sigma g n ) <= (
sigma (
fun k => (
f k ) * (
g k ))
n ).
Hint Resolve sigma_prod_maj sigma_prod_le sigma_prod_ge .
Lemma sigma_inv :
forall (
f g :
nat ->
U ) (
n :nat), (
retract f n ) ->
[1-] (
sigma (
fun k =>
f k *
g k )
n ) == (
sigma (
fun k =>
f k * [1-] (
g k ))
n ) + [1-] (
sigma f n ).
Definition of Nmult n x n */ x
Fixpoint Nmult (
n :
nat ) (
x :
U ) {
struct n } :
U :=
match n with O => 0 | (
S O ) =>
x |
S p =>
x + (
Nmult p x )
end .
Condition for n */ x
Definition Nmult_def (
n :
nat ) (
x :
U ) :=
match n with O =>
True |
S p =>
x <= [1/]1+p
end .
Lemma Nmult_def_O :
forall x ,
Nmult_def O x .
Hint Resolve Nmult_def_O .
Lemma Nmult_def_1 :
forall x ,
Nmult_def (
S O )
x .
Hint Resolve Nmult_def_1 .
Lemma Nmult_def_intro :
forall n x ,
x <= [1/]1+n ->
Nmult_def (
S n )
x .
Hint Resolve Nmult_def_intro .
Lemma Nmult_def_Unth :
forall n ,
Nmult_def (
S n ) ([1/]1+n).
Hint Resolve Nmult_def_Unth .
Lemma Nmult_def_pred :
forall n x ,
Nmult_def (
S n )
x ->
Nmult_def n x .
Hint Immediate Nmult_def_pred .
Lemma Nmult_defS :
forall n x ,
Nmult_def (
S n )
x ->
x <= [1/]1+n.
Hint Immediate Nmult_defS .
Lemma Nmult_def_class :
forall n p ,
class (
Nmult_def n p ).
Hint Resolve Nmult_def_class .
Infix "*/" :=
Nmult (
at level 60) :
U_scope .
Add Morphism Nmult_def with signature eq &
Oeq &
iff as Nmult_def_eq_compat .
Save .
Lemma Nmult_def_zero :
forall n ,
Nmult_def n 0.
Hint Resolve Nmult_def_zero .
Lemma Nmult_0 :
forall (
x :U),
O */x = 0.
Lemma Nmult_1 :
forall (
x :U), (
S O )*/x =
x .
Lemma Nmult_zero :
forall n ,
n */ 0 == 0.
Lemma Nmult_SS :
forall (
n :nat) (
x :U),
S (
S n ) */x =
x + (
S n */
x ).
Lemma Nmult_2 :
forall (
x :U), 2*/x =
x +
x .
Lemma Nmult_S :
forall (
n :nat) (
x :U),
S n */
x ==
x + (
n */x).
Hint Resolve Nmult_1 Nmult_SS Nmult_2 Nmult_S .
Add Morphism Nmult with signature eq &
Oeq &
Oeq as Nmult_eq_compat .
Save .
Hint Immediate Nmult_eq_compat .
Lemma Nmult_eq_compat_left :
forall (
n :nat) (
x y :U),
x ==
y ->
n */
x ==
n */
y .
Lemma Nmult_eq_compat_right :
forall (
n m :nat) (
x :U), (
n =
m )%nat ->
n */
x ==
m */
x .
Hint Resolve Nmult_eq_compat_right .
Lemma Nmult_le_compat_right :
forall n x y ,
x <=
y ->
n */
x <=
n */
y .
Lemma Nmult_le_compat_left :
forall n m x , (
n <=
m )%nat ->
n */
x <=
m */
x .
Hint Resolve Nmult_eq_compat_right Nmult_le_compat_right Nmult_le_compat_left .
Lemma Nmult_le_compat :
forall (
n m :nat)
x y ,
n <=
m ->
x <=
y ->
n */
x <=
m */
y .
Hint Immediate Nmult_le_compat .
Instance Nmult_mon2 :
monotonic2 Nmult .
Save .
Definition NMult :
nat -m>
U -m>
U :=mon2
Nmult .
Lemma Nmult_sigma :
forall (
n :nat) (
x :U),
n */
x ==
sigma (
fun k =>
x )
n .
Hint Resolve Nmult_sigma .
Lemma Nmult_Unth_prop :
forall n :nat, [1/]1+n == [1-] (
n */ ([1/]1+n)).
Hint Resolve Nmult_Unth_prop .
Lemma Nmult_n_Unth :
forall n :nat,
n */ [1/]1+n == [1-] ([1/]1+n).
Lemma Nmult_Sn_Unth :
forall n :nat,
S n */ [1/]1+n == 1.
Hint Resolve Nmult_n_Unth Nmult_Sn_Unth .
Lemma Nmult_ge_Sn_Unth :
forall n k , (
S n <=
k )%nat ->
k */ [1/]1+n == 1.
Lemma Nmult_le_n_Unth :
forall n k , (
k <=
n )%nat ->
k */ [1/]1+n <= [1-] ([1/]1+n).
Hint Resolve Nmult_ge_Sn_Unth Nmult_le_n_Unth .
Lemma Nmult_Umult_assoc_left :
forall n x y ,
Nmult_def n x ->
n */(x*y) == (
n */x)*y.
Hint Resolve Nmult_Umult_assoc_left .
Lemma Nmult_Umult_assoc_right :
forall n x y ,
Nmult_def n y ->
n */(x*y) ==
x *(n*/y).
Hint Resolve Nmult_Umult_assoc_right .
Lemma plus_Nmult_distr :
forall n m x , (
n +
m ) */
x == (
n */
x ) + (
m */
x ).
Lemma Nmult_Uplus_distr :
forall n x y ,
n */ (
x +
y ) == (
n */
x ) + (
n */
y ).
Lemma Nmult_mult_assoc :
forall n m x , (
n *
m ) */
x ==
n */ (
m */
x ).
Lemma Nmult_Unth_simpl_left :
forall n x , (
S n ) */ ([1/]1+n *
x ) ==
x .
Lemma Nmult_Unth_simpl_right :
forall n x , (
S n ) */ (
x * [1/]1+n) ==
x .
Hint Resolve Nmult_Umult_assoc_right plus_Nmult_distr Nmult_Uplus_distr
Nmult_mult_assoc Nmult_Unth_simpl_left Nmult_Unth_simpl_right .
Lemma Uinv_Nmult :
forall k n , [1-] (
k */ [1/]1+n) == ((
S n ) -
k ) */ [1/]1+n.
Lemma Nmult_neq_zero :
forall n x , ~0==x -> ~0==S
n */
x .
Hint Resolve Nmult_neq_zero .
Lemma Nmult_le_simpl :
forall (
n :nat) (
x y :U),
Nmult_def (
S n )
x ->
Nmult_def (
S n )
y -> (
S n */
x ) <= (
S n */
y ) ->
x <=
y .
Lemma Nmult_Unth_le :
forall (
n1 n2 m1 m2 :nat),
(
n2 *
S n1 <=
m2 *
S m1 )%nat ->
n2 */ [1/]1+m1 <=
m2 */ [1/]1+n1.
Lemma Nmult_Unth_eq :
forall (
n1 n2 m1 m2 :nat),
(
n2 *
S n1 =
m2 *
S m1 )%nat ->
n2 */ [1/]1+m1 ==
m2 */ [1/]1+n1.
Hint Resolve Nmult_Unth_le Nmult_Unth_eq .
Lemma Nmult_Unth_factor :
forall (
n m1 m2 :nat),
(
n *
S m2 =
S m1 )%nat ->
n */ [1/]1+m1 == [1/]1+m2.
Hint Resolve Nmult_Unth_factor .
Lemma Nmult_def_lt :
forall n x ,
n */
x <1 ->
Nmult_def n x .
Hint Immediate Nmult_def_lt .
Conversion from booleans to U
Definition B2U :MF
bool :=
fun (
b :bool) =>
if b then 1
else 0.
Definition NB2U :MF
bool :=
fun (
b :bool) =>
if b then 0
else 1.
Lemma B2Uinv :
NB2U ==
finv B2U .
Lemma NB2Uinv :
B2U ==
finv NB2U .
Hint Resolve B2Uinv NB2Uinv .
Definition pmin (p :U) (n :nat) := p - ([1/2]^n).Add Morphism pmin with signature Oeq & eq & Oeq as pmin_eq_compat .Save .
Lemma pmin_esp_S :
forall p n ,
pmin (
p &
p )
n ==
pmin p (
S n ) &
pmin p (
S n ).
Lemma pmin_esp_le :
forall p n ,
pmin p (
S n ) <= [1/2] * (
pmin (
p &
p )
n ) + [1/2].
Lemma pmin_plus_eq :
forall p n ,
p <= [1/2] ->
pmin p (
S n ) == [1/2] * (
pmin (
p +
p )
n ).
Lemma pmin_0 :
forall p :U,
pmin p O == 0.
Lemma pmin_le :
forall (
p :U) (
n :nat),
p - ([1/]1+n) <=
pmin p n .
Hint Resolve pmin_0 pmin_le .
Lemma pmin_le_compat :
forall p (
n m :
nat ),
n <=
m ->
pmin p n <=
pmin p m .
Hint Resolve pmin_le_compat .
Instance pmin_mon :
forall p ,
monotonic (
pmin p ).
Save .
Definition Pmin (
p :U) :nat -m>
U :=
mon (
pmin p ).
Lemma le_p_lim_pmin :
forall p ,
p <=
lub (
Pmin p ).
Lemma le_lim_pmin_p :
forall p ,
lub (
Pmin p ) <=
p .
Hint Resolve le_p_lim_pmin le_lim_pmin_p .
Lemma eq_lim_pmin_p :
forall p ,
lub (
Pmin p ) ==
p .
Hint Resolve eq_lim_pmin_p .
Particular case where p = 1
Definition U1min := Pmin 1.Lemma eq_lim_U1min : lub U1min == 1.Lemma U1min_S : forall n , U1min (S n ) == [1/2]*(U1min n ) + [1/2].Lemma U1min_0 : U1min O == 0.Hint Resolve eq_lim_U1min U1min_S U1min_0 .Lemma glb_half_exp : glb (UExp [1/2]) == 0.Hint Resolve glb_half_exp .Lemma Ule_lt_half_exp : forall x y , (forall p , x <= y + [1/2]^p) -> x <= y .Lemma half_exp_le_half : forall p , [1/2]^(S p ) <= [1/2].Hint Resolve half_exp_le_half .Lemma twice_half_exp : forall p , [1/2]^(S p ) + [1/2]^(S p ) == [1/2]^p.Hint Resolve twice_half_exp .
Fixpoint exp2 (
n :nat) :
nat :=
match n with O => (1%nat) |
S p => (2 * (
exp2 p ))%nat
end .
Lemma exp2_pos :
forall n , (
O <
exp2 n )%nat.
Hint Resolve exp2_pos .
Lemma S_pred_exp2 :
forall n ,
S (
pred (
exp2 n ))=exp2
n .
Hint Resolve S_pred_exp2 .
Notation "k /2^ p" := (
k */ ([1/2])^p) (
at level 35,
no associativity ).
Lemma Unth_eq :
forall n p ,
n */
p == [1-]
p ->
p == [1/]1+n.
Lemma Unth_half :
forall n , (
O <n)%nat -> [1/]1+(pred (
n +n)) == [1/2] * [1/]1+pred
n .
Lemma Unth_exp2 :
forall p , [1/2]^p == [1/]1+pred (
exp2 p ).
Hint Resolve Unth_exp2 .
Lemma Nmult_exp2 :
forall p , (
exp2 p )/2^p == 1.
Hint Resolve Nmult_exp2 .
Tactic for simplification of goals
Ltac Usimpl :=
match goal with
|-
context [(
Uplus 0 ?x)] =>
setoid_rewrite (
Uplus_zero_left x )
| |-
context [(
Uplus ?x 0)] =>
setoid_rewrite (
Uplus_zero_right x )
| |-
context [(
Uplus 1 ?x)] =>
setoid_rewrite (
Uplus_one_left x )
| |-
context [(
Uplus ?x 1)] =>
setoid_rewrite (
Uplus_one_right x )
| |-
context [(
Umult 0 ?x)] =>
setoid_rewrite (
Umult_zero_left x )
| |-
context [(
Umult ?x 0)] =>
setoid_rewrite (
Umult_zero_right x )
| |-
context [(
Umult 1 ?x)] =>
setoid_rewrite (
Umult_one_left x )
| |-
context [(
Umult ?x 1)] =>
setoid_rewrite (
Umult_one_right x )
| |-
context [(
Uesp 0 ?x)] =>
setoid_rewrite (
Uesp_zero_left x )
| |-
context [(
Uesp ?x 0)] =>
setoid_rewrite (
Uesp_zero_right x )
| |-
context [(
Uesp 1 ?x)] =>
setoid_rewrite (
Uesp_one_left x )
| |-
context [(
Uesp ?x 1)] =>
setoid_rewrite (
Uesp_one_right x )
| |-
context [(
Uminus 0 ?x)] =>
setoid_rewrite (
Uminus_le_zero 0
x );
[
idtac |apply (
Upos x )]
| |-
context [(
Uminus ?x 0)] =>
setoid_rewrite (
Uminus_zero_right x )
| |-
context [(
Uminus ?x 1)] =>
setoid_rewrite (
Uminus_le_zero x 1);
[
idtac |apply (
Unit x )]
| |-
context [([1-] ([1-] ?x))] =>
setoid_rewrite (
Uinv_inv x )
| |-
context [([1-] 1)] =>
setoid_rewrite Uinv_one
| |-
context [([1-] 0)] =>
setoid_rewrite Uinv_zero
| |-
context [([1/]1+O)] =>
setoid_rewrite Unth_zero
| |-
context [?x^O] =>
setoid_rewrite (
Uexp_0 x )
| |-
context [?x^(S
O )] =>
setoid_rewrite (
Uexp_1 x )
| |-
context [0^(?n)] =>
setoid_rewrite Uexp_zero ; [
idtac |omega]
| |-
context [
U1 ^(?n)] =>
setoid_rewrite Uexp_one
| |-
context [(
Nmult 0 ?x)] =>
setoid_rewrite (
Nmult_0 x )
| |-
context [(
Nmult 1 ?x)] =>
setoid_rewrite (
Nmult_1 x )
| |-
context [(
Nmult ?n 0)] =>
setoid_rewrite (
Nmult_zero n )
| |-
context [(
sigma ?f
O )] =>
setoid_rewrite (
sigma_0 f )
| |-
context [(
sigma ?f (
S O ))] =>
setoid_rewrite (
sigma_1 f )
| |- (
Ole (
Uplus ?x ?y) (
Uplus ?x ?z)) =>
apply Uplus_le_compat_right
| |- (
Ole (
Uplus ?x ?z) (
Uplus ?y ?z)) =>
apply Uplus_le_compat_left
| |- (
Ole (
Uplus ?x ?z) (
Uplus ?z ?y)) =>
setoid_rewrite (
Uplus_sym z y );
apply Uplus_le_compat_left
| |- (
Ole (
Uplus ?x ?y) (
Uplus ?z ?x)) =>
setoid_rewrite (
Uplus_sym x y );
apply Uplus_le_compat_left
| |- (
Ole (
Uinv ?y) (
Uinv ?x)) =>
apply Uinv_le_compat
| |- (
Ole (
Uminus ?x ?y) (
Uplus ?x ?z)) =>
apply Uminus_le_compat_right
| |- (
Ole (
Uminus ?x ?z) (
Uplus ?y ?z)) =>
apply Uminus_le_compat_left
| |- ((
Uinv ?x) == (
Uinv ?y)) =>
apply Uinv_eq_compat
| |- ((
Uplus ?x ?y) == (
Uplus ?x ?z)) =>
apply Uplus_eq_compat_right
| |- ((
Uplus ?x ?z) == (
Uplus ?y ?z)) =>
apply Uplus_eq_compat_left
| |- ((
Uplus ?x ?z) == (
Uplus ?z ?y)) =>
setoid_rewrite (
Uplus_sym z y );
apply Uplus_eq_compat_left
| |- ((
Uplus ?x ?y) == (
Uplus ?z ?x)) =>
setoid_rewrite (
Uplus_sym x y );
apply Uplus_eq_compat_left
| |- ((
Uminus ?x ?y) == (
Uplus ?x ?z)) =>
apply Uminus_eq_compat ;[
apply Oeq_refl |idtac]
| |- ((
Uminus ?x ?z) == (
Uplus ?y ?z)) =>
apply Uminus_eq_compat ;[
idtac |apply
Oeq_refl ]
| |- (
Ole (
Umult ?x ?y) (
Umult ?x ?z)) =>
apply Umult_le_compat_right
| |- (
Ole (
Umult ?x ?z) (
Umult ?y ?z)) =>
apply Umult_le_compat_left
| |- (
Ole (
Umult ?x ?z) (
Umult ?z ?y)) =>
setoid_rewrite (
Umult_sym z y );
apply Umult_le_compat_left
| |- (
Ole (
Umult ?x ?y) (
Umult ?z ?x)) =>
setoid_rewrite (
Umult_sym x y );
apply Umult_le_compat_left
| |- ((
Umult ?x ?y) == (
Umult ?x ?z)) =>
apply Umult_eq_compat_right
| |- ((
Umult ?x ?z) == (
Umult ?y ?z)) =>
apply Umult_eq_compat_left
| |- ((
Umult ?x ?z) == (
Umult ?z ?y)) =>
setoid_rewrite (
Umult_sym z y );
apply Umult_eq_compat_left
| |- ((
Umult ?x ?y) == (
Umult ?z ?x)) =>
setoid_rewrite (
Umult_sym x y );
apply Umult_eq_compat_left
end .
Properties of current values
Notation "[1/3]" := (Unth 2%nat).Notation "[1/4]" := (Unth 3%nat).Notation "[1/8]" := (Unth 7).Notation "[3/4]" := (Uinv [1/4]).Lemma half_square : [1/2]*[1/2]==[1/4].Lemma half_cube : [1/2]*[1/2]*[1/2]==[1/8].Lemma three_quarter_decomp : [3/4]==[1/2]+[1/4].
Equality is not true, even for monotonic sequences fot instance n/m
Lemma Ulub_Uglb_exch_le :
forall f :
nat ->
nat ->
U ,
Ulub (
fun n =>
Uglb (
fun m =>
f n m )) <=
Uglb (
fun m =>
Ulub (
fun n =>
f n m )).
Record IU : Type := mk_IU {low :U; up :U; proper :low <= up }.Hint Resolve proper .
the all set : [0,1]
Definition full := mk_IU 0 1 (Upos 1).
singleton : [x ]
Definition singl (
x :U) :=
mk_IU x x (
Ole_refl x ).
down segment : [0,x]
Definition inf (x :U) := mk_IU 0 x (Upos x ).
up segment : [x ,1]
Definition sup (x :U) := mk_IU x 1 (Unit x ).
Definition Iin (
x :U) (
I :IU) :=
low I <=
x /\
x <=
up I .
Definition Iincl I J :=
low J <=
low I /\
up I <=
up J .
Definition Ieq I J :=
low I ==
low J /\
up I ==
up J .
Hint Unfold Iin Iincl Ieq .
Lemma Iin_low :
forall I ,
Iin (
low I )
I .
Lemma Iin_up :
forall I ,
Iin (
up I )
I .
Hint Resolve Iin_low Iin_up .
Lemma Iin_singl_elim :
forall x y ,
Iin x (
singl y ) ->
x ==
y .
Lemma Iin_inf_elim :
forall x y ,
Iin x (
inf y ) ->
x <=
y .
Lemma Iin_sup_elim :
forall x y ,
Iin x (
sup y ) ->
y <=
x .
Lemma Iin_singl_intro :
forall x y ,
x ==
y ->
Iin x (
singl y ).
Lemma Iin_inf_intro :
forall x y ,
x <=
y ->
Iin x (
inf y ).
Lemma Iin_sup_intro :
forall x y ,
y <=
x ->
Iin x (
sup y ).
Hint Immediate Iin_inf_elim Iin_sup_elim Iin_singl_elim .
Hint Resolve Iin_inf_intro Iin_sup_intro Iin_singl_intro .
Lemma Iin_class :
forall I x ,
class (
Iin x I ).
Lemma Iincl_class :
forall I J ,
class (
Iincl I J ).
Lemma Ieq_class :
forall I J ,
class (
Ieq I J ).
Hint Resolve Iin_class Iincl_class Ieq_class .
Lemma Iincl_in :
forall I J ,
Iincl I J ->
forall x ,
Iin x I ->
Iin x J .
Lemma Iincl_low :
forall I J ,
Iincl I J ->
low J <=
low I .
Lemma Iincl_up :
forall I J ,
Iincl I J ->
up I <=
up J .
Hint Immediate Iincl_low Iincl_up .
Lemma Iincl_refl :
forall I ,
Iincl I I .
Hint Resolve Iincl_refl .
Lemma Iincl_trans :
forall I J K ,
Iincl I J ->
Iincl J K ->
Iincl I K .
Instance IUord :
ord IU := {
Oeq :=
fun I J =>
Ieq I J ;
Ole :=
fun I J =>
Iincl J I }.
Defined .
Lemma low_le_compat :
forall I J :IU,
I <=
J ->
low I <=
low J .
Lemma up_le_compat :
forall I J :
IU ,
I <=
J ->
up J <=
up I .
Instance low_mon :
monotonic low .
Save .
Definition Low :
IU -m>
U :=
mon low .
Instance up_mon :
monotonic (
o2 :=Iord
U )
up .
Save .
Definition Up :
IU -m->
U :=
mon (
o2 :=Iord
U )
up .
Lemma Ieq_incl :
forall I J ,
Ieq I J ->
Iincl I J .
Lemma Ieq_incl_sym :
forall I J ,
Ieq I J ->
Iincl J I .
Hint Immediate Ieq_incl Ieq_incl_sym .
Lemma lincl_eq_compat :
forall I J K L ,
Ieq I J ->
Iincl J K ->
Ieq K L ->
Iincl I L .
Lemma lincl_eq_trans :
forall I J K ,
Iincl I J ->
Ieq J K ->
Iincl I K .
Lemma Ieq_incl_trans :
forall I J K ,
Ieq I J ->
Iincl J K ->
Iincl I K .
Lemma Iincl_antisym :
forall I J ,
Iincl I J ->
Iincl J I ->
Ieq I J .
Hint Immediate Iincl_antisym .
Lemma Ieq_refl :
forall I ,
Ieq I I .
Hint Resolve Ieq_refl .
Lemma Ieq_sym :
forall I J ,
Ieq I J ->
Ieq J I .
Hint Immediate Ieq_sym .
Lemma Ieq_trans :
forall I J K ,
Ieq I J ->
Ieq J K ->
Ieq I K .
Lemma Isingl_eq :
forall x y ,
Iincl (
singl x ) (
singl y ) ->
x ==y.
Hint Immediate Isingl_eq .
Lemma Iincl_full :
forall I ,
Iincl I full .
Hint Resolve Iincl_full .
Definition Iplus I J :=
mk_IU (
low I +
low J ) (
up I +
up J )
(
Uplus_le_compat _ _ _ _ (
proper I ) (
proper J )).
Lemma low_Iplus :
forall I J ,
low (
Iplus I J )=low
I +
low J .
Lemma up_Iplus :
forall I J ,
up (
Iplus I J )=up
I +
up J .
Lemma Iplus_in :
forall I J x y ,
Iin x I ->
Iin y J ->
Iin (
x +y) (
Iplus I J ).
Lemma lplus_in_elim :
forall I J z ,
low I <= [1-]
up J ->
Iin z (
Iplus I J )
->
exc (
fun x =>
Iin x I /\
exc (
fun y =>
Iin y J /\
z ==x+y)).
Definition Imult I J :=
mk_IU (
low I *
low J ) (
up I *
up J )
(
Umult_le_compat _ _ _ _ (
proper I ) (
proper J )).
Lemma low_Imult :
forall I J ,
low (
Imult I J ) =
low I *
low J .
Lemma up_Imult :
forall I J ,
up (
Imult I J ) =
up I *
up J .
Definition Imultk p I :=
mk_IU (
p *
low I ) (
p *
up I ) (
Umult_le_compat_right p _ _ (
proper I )).
Lemma low_Imultk :
forall p I ,
low (
Imultk p I ) =
p *
low I .
Lemma up_Imultk :
forall p I ,
up (
Imultk p I ) =
p *
up I .
Lemma Imult_in :
forall I J x y ,
Iin x I ->
Iin y J ->
Iin (
x *y) (
Imult I J ).
Lemma Imultk_in :
forall p I x ,
Iin x I ->
Iin (
p *x) (
Imultk p I ).
Definition Ilim :
forall I :
nat -m>
IU ,
IU .
Defined .
Lemma low_lim :
forall (
I :nat -m>
IU ),
low (
Ilim I ) =
lub (
Low @
I ).
Lemma up_lim :
forall (
I :nat -m>
IU ),
up (
Ilim I ) =
glb (
Up @
I ).
Lemma lim_Iincl :
forall (
I :nat -m>
IU )
n ,
Iincl (
Ilim I ) (
I n ).
Hint Resolve lim_Iincl .
Lemma Iincl_lim :
forall J (
I :nat -m>IU), (
forall n ,
Iincl J (
I n )) ->
Iincl J (
Ilim I ).
Lemma IIim_incl_stable :
forall (
I J :nat -m>
IU ), (
forall n ,
Iincl (
I n ) (
J n )) ->
Iincl (
Ilim I ) (
Ilim J ).
Hint Resolve IIim_incl_stable .
Instance IUcpo :
cpo IU := {
D0 :=full;
lub :=Ilim}.
Defined .
Definition fsup (
f :nat->U) (
n :nat) :=
Ulub (
fun k =>
f (
n +k)%nat).
Definition finf (
f :nat->U) (
n :nat) :=
Uglb (
fun k =>
f (
n +k)%nat).
Lemma fsup_incr :
forall (
f :nat->U)
n ,
fsup f (
S n ) <=
fsup f n .
Hint Resolve fsup_incr .
Lemma finf_incr :
forall (
f :nat->U)
n ,
finf f n <=
finf f (
S n ).
Hint Resolve finf_incr .
Instance fsup_mon :
forall f ,
monotonic (
o2 :=Iord
U ) (
fsup f ).
Save .
Instance finf_mon :
forall f ,
monotonic (
finf f ).
Save .
Definition Fsup (
f :nat->U) :
nat -m->
U :=
mon (
fsup f ).
Definition Finf (
f :nat->U) :
nat -m>
U :=
mon (
finf f ).
Lemma fn_fsup :
forall f n ,
f n <=
fsup f n .
Hint Resolve fn_fsup .
Lemma finf_fn :
forall f n ,
finf f n <=
f n .
Hint Resolve finf_fn .
Definition limsup f :=
glb (
Fsup f ).
Definition liminf f :=
lub (
Finf f ).
Lemma le_liminf_sup :
forall f ,
liminf f <=
limsup f .
Hint Resolve le_liminf_sup .
Definition has_lim f :=
limsup f <=
liminf f .
Lemma eq_liminf_sup :
forall f ,
has_lim f ->
liminf f ==
limsup f .
Definition cauchy f :=
forall (
p :nat),
exc (
fun M :nat =>
forall n m ,
(
M <=
n )%nat -> (
M <=
m )%nat ->
f n <=
f m + [1/2]^p).
Definition is_limit f (
l :U) :=
forall (
p :nat),
exc (
fun M :nat =>
forall n ,
(
M <=
n )%nat ->
f n <=
l + [1/2]^p /\
l <=
f n + [1/2]^p).
Lemma cauchy_lim :
forall f ,
cauchy f ->
is_limit f (
limsup f ).
Lemma has_limit_cauchy :
forall f l ,
is_limit f l ->
cauchy f .
Lemma limit_le_unique :
forall f l1 l2 ,
is_limit f l1 ->
is_limit f l2 ->
l1 <=
l2 .
Lemma limit_unique :
forall f l1 l2 ,
is_limit f l1 ->
is_limit f l2 ->
l1 ==
l2 .
Hint Resolve limit_unique .
Lemma has_limit_compute :
forall f l ,
is_limit f l ->
is_limit f (
limsup f ).
Lemma limsup_eq_mult :
forall k (
f :
nat ->
U ),
limsup (
fun n =>
k *
f n ) ==
k *
limsup f .
Lemma liminf_eq_mult :
forall k (
f :
nat ->
U ),
liminf (
fun n =>
k *
f n ) ==
k *
liminf f .
Lemma limsup_eq_plus_cte_right :
forall k (
f :
nat ->
U ),
limsup (
fun n => (
f n ) +
k ) ==
limsup f +
k .
Lemma liminf_eq_plus_cte_right :
forall k (
f :
nat ->
U ),
liminf (
fun n => (
f n ) +
k ) ==
liminf f +
k .
Lemma limsup_le_plus :
forall (
f g :
nat ->
U ),
limsup (
fun x =>
f x +
g x ) <=
limsup f +
limsup g .
Lemma liminf_le_plus :
forall (
f g :
nat ->
U ),
liminf f +
liminf g <=
liminf (
fun x =>
f x +
g x ).
Hint Resolve liminf_le_plus limsup_le_plus .
Lemma limsup_le_compat :
forall f g :
nat ->
U ,
f <=
g ->
limsup f <=
limsup g .
Lemma liminf_le_compat :
forall f g :
nat ->
U ,
f <=
g ->
liminf f <=
liminf g .
Hint Resolve limsup_le_compat liminf_le_compat .
Lemma limsup_eq_compat :
forall f g :
nat ->
U ,
f ==
g ->
limsup f ==
limsup g .
Lemma liminf_eq_compat :
forall f g :
nat ->
U ,
f ==
g ->
liminf f ==
liminf g .
Hint Resolve liminf_eq_compat limsup_eq_compat .
Lemma limsup_inv :
forall f :
nat ->
U ,
limsup (
fun x => [1-]
f x ) == [1-]
liminf f .
Lemma liminf_inv :
forall f :
nat ->
U ,
liminf (
fun x => [1-]
f x ) == [1-]
limsup f .
Hint Resolve limsup_inv liminf_inv .
Limits of arbitrary sequences
Lemma liminf_incr :
forall f :nat -m>
U ,
liminf f ==
lub f .
Lemma limsup_incr :
forall f :nat -m>
U ,
limsup f ==
lub f .
Lemma has_limit_incr :
forall f :nat -m>
U ,
has_lim f .
Lemma liminf_decr :
forall f :nat -m->
U ,
liminf f ==
glb f .
Lemma limsup_decr :
forall f :nat-m->U,
limsup f ==
glb f .
Lemma has_limit_decr :
forall f :nat -m->
U ,
has_lim f .
Lemma has_limit_sum :
forall f g :
nat ->
U ,
has_lim f ->
has_lim g ->
has_lim (
fun x =>
f x +
g x ).
Lemma has_limit_inv :
forall f :
nat ->
U ,
has_lim f ->
has_lim (
fun x => [1-]
f x ).
Lemma has_limit_cte :
forall c ,
has_lim (
fun n =>
c ).
Definition and properties of series : infinite sums
Definition serie (
f :
nat ->
U ) :
U :=
lub (
sigma f ).
Lemma serie_le_compat :
forall (
f g :
nat ->
U ),
(
forall k ,
f k <=
g k ) ->
serie f <=
serie g .
Lemma serie_eq_compat :
forall (
f g :
nat ->
U ),
(
forall k ,
f k ==
g k ) ->
serie f ==
serie g .
Lemma serie_sigma_lift :
forall (
f :nat ->
U ) (
n :nat),
serie f ==
sigma f n +
serie (
fun k =>
f (
n +
k )%nat).
Lemma serie_S_lift :
forall (
f :nat ->
U ),
serie f ==
f O +
serie (
fun k =>
f (
S k )).
Lemma serie_zero :
forall f , (
forall k ,
f k ==0) ->
serie f ==0.
Lemma serie_not_zero :
forall f k , 0 <
f k -> 0 <
serie f .
Lemma serie_zero_elim :
forall f ,
serie f == 0 ->
forall k ,
f k ==0.
Hint Resolve serie_eq_compat serie_le_compat serie_zero .
Lemma serie_le :
forall f k ,
f k <=
serie f .
Lemma serie_minus_incr :
forall f :nat -m>
U ,
serie (
fun k =>
f (
S k ) -
f k ) ==
lub f -
f O .
Lemma serie_minus_decr :
forall f :
nat -m->
U ,
serie (
fun k =>
f k -
f (
S k )) ==
f O -
glb f .
Lemma serie_plus :
forall (
f g :
nat ->
U ),
serie (
fun k => (
f k ) + (
g k )) ==
serie f +
serie g .
Definition wretract (
f :
nat ->
U ) :=
forall k ,
f k <= [1-] (
sigma f k ).
Lemma retract_wretract :
forall f , (
forall n ,
retract f n ) ->
wretract f .
Lemma wretract_retract :
forall f ,
wretract f ->
forall n ,
retract f n .
Hint Resolve wretract_retract .
Lemma wretract_lt :
forall (
f :
nat ->
U ), (
forall (
n :
nat ),
sigma f n < 1) ->
wretract f .
Lemma retract_zero_wretract :
forall f n ,
retract f n -> (
forall k , (
n <=
k )%nat ->
f k == 0) ->
wretract f .
Lemma wretract_le :
forall f g :
nat ->U,
f <=
g ->
wretract g ->
wretract f .
Lemma serie_mult :
forall (
f :
nat ->
U )
c ,
wretract f ->
serie (
fun k =>
c *
f k ) ==
c *
serie f .
Hint Resolve serie_mult .
Lemma serie_prod_maj :
forall (
f g :
nat ->
U ),
serie (
fun k =>
f k *
g k ) <=
serie f .
Hint Resolve serie_prod_maj .
Lemma serie_prod_le :
forall (
f g :
nat ->
U ) (
c :U), (
forall k ,
f k <=
c )
->
wretract g ->
serie (
fun k =>
f k *
g k ) <=
c *
serie g .
Lemma serie_prod_ge :
forall (
f g :
nat ->
U ) (
c :U), (
forall k ,
c <= (
f k ))
->
wretract g ->
c *
serie g <=
serie (
fun k =>
f k *
g k ).
Hint Resolve serie_prod_le serie_prod_ge .
Lemma serie_inv_le :
forall (
f g :
nat ->
U ),
wretract f ->
serie (
fun k =>
f k * [1-] (
g k )) <= [1-] (
serie (
fun k =>
f k *
g k )).
Definition Serie : (
nat ->
U ) -m>
U .
Defined .
Lemma Serie_simpl :
forall f ,
Serie f =
serie f .
Lemma serie_continuous :
continuous Serie .
Library Monads
Monads.v: Monads for randomized constructions
Definition of monadic operators as the cpo of monotonic oerators
Definition M (A :Type) := MF A -m> U .Instance app_mon (A :Type) (x :A) : monotonic (fun (f :MF A ) => f x ).Save .Definition unit (A :Type) (x :A) : M A := mon (fun (f :MF A ) => f x ).Definition star : forall (A B :Type), M A -> (A -> M B ) -> M B .Defined .Lemma star_simpl : forall (A B :Type) (a :M A ) (F :A -> M B )(f :MF B ),star a F f = a (fun x => F x f ).
Properties of monadic operators
Lemma law1 :
forall (
A B :Type) (
x :A) (
F :A ->
M B ) (
f :MF
B ),
star (
unit x )
F f ==
F x f .
Lemma law2 :
forall (
A :Type) (
a :M
A ) (
f :MF
A ),
star a (
fun x :A =>
unit x )
f ==
a (
fun x :A =>
f x ).
Lemma law3 :
forall (
A B C :Type) (
a :M
A ) (
F :A ->
M B ) (
G :B ->
M C )
(
f :MF
C ),
star (
star a F )
G f ==
star a (
fun x :A =>
star (
F x )
G )
f .
Properties of distributions
Expected properties of measures
Definition stable_inv (
A :Type) (
m :M
A ) :
Prop :=
forall f :MF
A ,
m (
finv f ) <= [1-] (
m f ).
Definition stable_plus (
A :Type) (
m :M
A ) :
Prop :=
forall f g :MF
A ,
fplusok f g ->
m (
fplus f g ) == (
m f ) + (
m g ).
Definition le_plus (
A :Type) (
m :M
A ) :
Prop :=
forall f g :MF
A ,
fplusok f g -> (
m f ) + (
m g ) <=
m (
fplus f g ).
Definition le_esp (
A :Type) (
m :M
A ) :
Prop :=
forall f g :
MF A , (
m f ) & (
m g ) <=
m (
fesp f g ).
Definition le_plus_cte (
A :Type) (
m :M
A ) :
Prop :=
forall (
f :
MF A ) (
k :U),
m (
fplus f (
fcte A k )) <=
m f +
k .
Definition stable_mult (
A :Type) (
m :M
A ) :
Prop :=
forall (
k :U) (
f :MF
A ),
m (
fmult k f ) ==
k * (
m f ).
Lemma stable_minus_distr :
forall (
A :Type) (
m :M
A ),
stable_plus m ->
stable_inv m ->
forall (
f g :
MF A ),
g <=
f ->
m (
fminus f g ) ==
m f -
m g .
Hint Resolve stable_minus_distr .
Lemma inv_minus_distr :
forall (
A :Type) (
m :M
A ),
stable_plus m ->
stable_inv m ->
forall (
f :
MF A ),
m (
finv f ) ==
m (
fone A ) -
m f .
Hint Resolve inv_minus_distr .
Lemma le_minus_distr :
forall (
A :
Type )(
m :M
A ),
forall (
f g :A ->
U ),
m (
fminus f g ) <=
m f .
Hint Resolve le_minus_distr .
Lemma le_plus_distr :
forall (
A :
Type )(
m :M
A ),
stable_plus m ->
stable_inv m ->
forall (
f g :MF
A ),
m (
fplus f g ) <=
m f +
m g .
Hint Resolve le_plus_distr .
Lemma le_esp_distr :
forall (
A :
Type ) (
m :M
A ),
stable_plus m ->
stable_inv m ->
le_esp m .
Lemma unit_stable_eq :
forall (
A :Type) (
x :A),
stable (
unit x ).
Lemma star_stable_eq :
forall (
A B :Type) (
m :M
A ) (
F :A ->
M B ),
stable (
star m F ).
Lemma unit_stable_inv :
forall (
A :Type) (
x :A),
stable_inv (
unit x ).
Lemma star_stable_inv :
forall (
A B :Type) (
m :M
A ) (
F :A ->
M B ),
stable_inv m -> (
forall a :A,
stable_inv (
F a )) ->
stable_inv (
star m F ).
Lemma unit_stable_plus : forall (A :Type) (x :A), stable_plus (unit x ).Lemma star_stable_plus : forall (A B :Type) (m :M A ) (F :A -> M B ),stable_plus m ->forall a :A, forall f g , fplusok f g -> (F a f ) <= Uinv (F a g ))forall a :A, stable_plus (F a )) -> stable_plus (star m F ).Lemma unit_le_plus : forall (A :Type) (x :A), le_plus (unit x ).Lemma star_le_plus : forall (A B :Type) (m :M A ) (F :A -> M B ),le_plus m ->forall a :A, forall f g , fplusok f g -> (F a f ) <= Uinv (F a g ))forall a :A, le_plus (F a )) -> le_plus (star m F ).
Lemma unit_stable_mult : forall (A :Type) (x :A), stable_mult (unit x ).Lemma star_stable_mult : forall (A B :Type) (m :M A ) (F :A -> M B ),stable_mult m -> (forall a :A, stable_mult (F a )) -> stable_mult (star m F ).
Lemma unit_continuous :
forall (
A :Type) (
x :A),
continuous (
unit x ).
Lemma star_continuous :
forall (
A B :
Type ) (
m :
M A )(
F :
A ->
M B ),
continuous m -> (
forall x ,
continuous (
F x )) ->
continuous (
star m F ).
Library Probas
Probas.v: The monad for distributions
Definition of distribution
Distributions are monotonic measure functions such that
Record distr (A :Type) : Type := mu : M A ;mu_stable_inv : stable_inv mu ; mu_stable_plus : stable_plus mu ;mu_stable_mult : stable_mult mu ;mu_continuous : continuous mu }.Hint Resolve mu_stable_plus mu_stable_inv mu_stable_mult mu_continuous .
Lemma mu_monotonic : forall (A : Type )(m : distr A ), monotonic (mu m ).Hint Resolve mu_monotonic .Implicit Arguments mu_monotonic [A ].Lemma mu_stable_eq : forall (A : Type )(m : distr A ), stable (mu m ).Hint Resolve mu_stable_eq .Implicit Arguments mu_stable_eq [A ].Lemma mu_zero : forall (A : Type )(m : distr A ), mu m (fzero A ) == 0.Hint Resolve mu_zero .Lemma mu_zero_eq : forall (A : Type )(m : distr A ) f , forall x , f x == 0) -> mu m f == 0.Lemma mu_one_inv : forall (A : Type )(m :distr A ),mu m (fone A ) == 1 -> forall f , mu m (finv f ) == [1-] (mu m f ).Hint Resolve mu_one_inv .Lemma mu_fplusok : forall (A : Type )(m :distr A ) f g , fplusok f g -> mu m f <= [1-] mu m g .Hint Resolve mu_fplusok .Lemma mu_le_minus : forall (A : Type )(m :distr A ) (f g :MF A ),mu m (fminus f g ) <= mu m f .Hint Resolve mu_le_minus .Lemma mu_le_plus : forall (A : Type )(m :distr A ) (f g :MF A ),mu m (fplus f g ) <= mu m f + mu m g .Hint Resolve mu_le_plus .Lemma mu_cte : forall (A : Type )(m :(distr A )) (c :U),mu m (fcte A c ) == c * mu m (fone A ).Hint Resolve mu_cte .Lemma mu_cte_le : forall (A : Type )(m :distr A ) (c :U), mu m (fcte A c ) <= c .Lemma mu_cte_eq : forall (A : Type )(m :distr A ) (c :U),mu m (fone A ) == 1 -> mu m (fcte A c ) == c .Hint Resolve mu_cte_le mu_cte_eq .Lemma mu_stable_mult_right : forall (A : Type )(m :distr A ) (c :U) (f : MF A ),mu m (fun x => (f x ) * c ) == (mu m f ) * c .Lemma mu_stable_minus : forall (A :Type) (m :distr A )(f g : MF A ),g <= f -> mu m (fun x => f x - g x ) == mu m f - mu m g .Lemma mu_inv_minus : forall (A :Type) (m :distr A )(f : MF A ), mu m (finv f ) == mu m (fone A ) - mu m f .Lemma mu_stable_le_minus : forall (A :Type) (m :distr A )(f g : MF A ),mu m f - mu m g <= mu m (fun x => f x - g x ).Lemma mu_inv_minus_inv : forall (A :Type) (m :distr A )(f : MF A ), mu m (finv f ) + [1-](mu m (fone A )) == [1-](mu m f ).Lemma mu_le_esp_inv : forall (A :Type) (m :distr A )(f g : MF A ),mu m (finv f )) & mu m g <= mu m (fesp f g ).Hint Resolve mu_le_esp_inv .Lemma mu_stable_inv_inv : forall (A :Type) (m :distr A )(f : MF A ),mu m f <= [1-] mu m (finv f ).Hint Resolve mu_stable_inv_inv .Lemma mu_stable_div : forall (A :Type) (m :distr A )(k :U)(f : MF A ),f <= fcte A k -> mu m (fdiv k f ) == mu m f / k .Lemma mu_stable_div_le : forall (A :Type) (m :distr A )(k :U)(f : MF A ),mu m (fdiv k f ) <= mu m f / k .Lemma mu_le_esp : forall (A :Type) (m :distr A )(f g : MF A ),mu m f & mu m g <= mu m (fesp f g ).Hint Resolve mu_le_esp .Lemma mu_esp_one : forall (A :Type)(m:distr A )(f g :MF A ),m f -> mu m g == mu m (fesp f g ).Lemma mu_esp_zero : forall (A :Type)(m:distr A )(f g :MF A ),mu m (finv f ) <= 0 -> mu m g == mu m (fesp f g ).Instance Odistr (A :Type) : ord (distr A ) := Ole := fun (f g : distr A ) => mu f <= mu g ;Oeq := fun (f g : distr A ) => mu f == mu g }.Defined .
Monadic operators for distributions
Definition Munit : forall A :Type, A -> distr A .Defined .Definition Mlet : forall A B :Type, distr A -> (A -> distr B ) -> distr B .Defined .Lemma Munit_simpl : forall (A :Type) (q :A -> U ) x , mu (Munit x ) q = q x .Lemma Mlet_simpl : forall (A B :Type) (m :distr A ) (M :A -> distr B ) (f :B -> U ),mu (Mlet m M ) f = mu m (fun x => (mu (M x ) f )).
Operations on distributions
Lemma Munit_eq_compat : forall A (x y : A ), x = y -> Munit x == Munit y .Lemma Mlet_le_compat : forall (A B : Type ) (m1 m2 :distr A ) (M1 M2 : A -> distr B ), m1 <= m2 -> M1 <= M2 -> Mlet m1 M1 <= Mlet m2 M2 .Hint Resolve Mlet_le_compat .Add Parametric Morphism (A B : Type ) : (Mlet (A :=A) (B :=B))with signature Ole & Ole & Ole as Mlet_le_morphism .Save .Instance Mlet_mon2 : forall (A B : Type ), monotonic2 (@Mlet A B ).Save .Definition MLet (A B : Type ) : distr A -m> (A -> distr B ) -m> distr B mon2 (@Mlet A B ).Lemma MLet_simpl : forall (A B :Type) (m :distr A ) (M :A -> distr B )(f :B -> U ),mu (MLet A B m M ) f = mu m (fun x => mu (M x ) f ).Lemma Mlet_eq_compat : forall (A B : Type ) (m1 m2 :distr A ) (M1 M2 : A -> distr B ), m1 == m2 -> M1 ==M2 -> Mlet m1 M1 == Mlet m2 M2 .Hint Resolve Mlet_eq_compat .Add Parametric Morphism (A B : Type ) : (Mlet (A :=A) (B :=B))with signature Oeq & Oeq & Oeq as Mlet_eq_morphism .Save .Lemma mu_le_compat : forall (A :Type) (m1 m2 :distr A ),m1 <= m2 -> forall f g : A -> U , f <= g -> mu m1 f <= mu m2 g .Lemma mu_eq_compat : forall (A :Type) (m1 m2 :distr A ),m1 == m2 -> forall f g : A -> U , f == g -> mu m1 f == mu m2 g .Hint Immediate mu_le_compat mu_eq_compat .Add Parametric Morphism (A : Type ) : (mu (A :=A))with signature Ole & Ole as mu_le_morphism .Save .Add Parametric Morphism (A : Type ) : (mu (A :=A))with signature Oeq & Oeq as mu_eq_morphism .Save .
Properties of monadic operators
Lemma Mlet_unit : forall (A B :Type) (x :A) (m :A -> distr B ), Mlet (Munit x ) m == m x .Lemma Mlet_ext : forall (A :Type) (m :distr A ), Mlet m (fun x => Munit x ) == m .Lemma Mlet_assoc : forall (A B C :Type) (m1 : distr A ) (m2 :A -> distr B ) (m3 :B -> distr C ),Mlet (Mlet m1 m2 ) m3 == Mlet m1 (fun x :A => Mlet (m2 x ) m3 ).Lemma let_indep : forall (A B :Type) (m1 :distr A ) (m2 : distr B ) (f :MF B ), mu m1 (fun _ => mu m2 f ) == mu m1 (fone A ) * (mu m2 f ).
Definition distr_null : forall A : Type , distr A .Defined .Lemma le_distr_null : forall (A :Type) (m : distr A ), distr_null A <= m .Hint Resolve le_distr_null .
Definition Mmult A (k :MF A ) (m :M A ) : M A .Defined .Lemma Mmult_simpl : forall A (k :MF A ) (m :M A ) f , Mmult k m f = m (fun x => k x * f x ).Lemma Mmult_stable_inv : forall A (k :MF A ) (d :distr A ), stable_inv (Mmult k (mu d )).Lemma Mmult_stable_plus : forall A (k :MF A ) (d :distr A ), stable_plus (Mmult k (mu d )).Lemma Mmult_stable_mult : forall A (k :MF A ) (d :distr A ), stable_mult (Mmult k (mu d )).Lemma Mmult_continuous : forall A (k :MF A ) (d :distr A ), continuous (Mmult k (mu d )).Definition distr_mult A (k :MF A ) (d :distr A ) : distr A .Defined .Definition Mdiv A (k :U) (m :M A ) : M A := UDiv k @ m .Lemma Mdiv_simpl : forall A k (m :M A ) f , Mdiv k m f = m f / k .Lemma Mdiv_stable_inv : forall A (k :U) (d :distr A )(dk : mu d (fone A ) <= k ),stable_inv (Mdiv k (mu d )).Lemma Mdiv_stable_plus : forall A (k :U)(d:distr A ), stable_plus (Mdiv k (mu d )).Lemma Mdiv_stable_mult : forall A (k :U)(d:distr A )(dk : mu d (fone A ) <= k ), stable_mult (Mdiv k (mu d )).Lemma Mdiv_continuous : forall A (k :U)(d:distr A ), continuous (Mdiv k (mu d )).Definition distr_div A (k :U) (d :distr A ) (dk : mu d (fone A ) <= k )distr A .Defined .
Conditional probabilities
Definition mcond A (m :M A ) (f :MF A ) : M A .Defined .Lemma mcond_simpl : forall A (m :M A ) (f g : MF A ),mcond m f g = m (fconj f g ) / m f .Lemma mcond_stable_plus : forall A (m :distr A ) (f : MF A ), stable_plus (mcond (mu m ) f ).Lemma mcond_stable_inv : forall A (m :distr A ) (f : MF A ), stable_inv (mcond (mu m ) f ).Lemma mcond_stable_mult : forall A (m :distr A ) (f : MF A ), stable_mult (mcond (mu m ) f ).Lemma mcond_continuous : forall A (m :distr A ) (f : MF A ), continuous (mcond (mu m ) f ).Definition Mcond A (m :distr A ) (f :MF A ) : distr A :=Build_distr (mcond_stable_inv m f ) (mcond_stable_plus m f )mcond_stable_mult m f ) (mcond_continuous m f ).Lemma Mcond_total : forall A (m :distr A ) (f :MF A ), mu m f -> mu (Mcond m f ) (fone A ) == 1.Lemma Mcond_simpl : forall A (m :distr A ) (f g :MF A ), mu (Mcond m f ) g = mu m (fconj f g ) / mu m f .Hint Resolve Mcond_simpl .Lemma Mcond_zero_stable : forall A (m :distr A ) (f g :MF A ), mu m g == 0 -> mu (Mcond m f ) g == 0.Lemma Mcond_null : forall A (m :distr A ) (f g :MF A ), mu m f == 0 -> mu (Mcond m f ) g == 0.Lemma Mcond_conj : forall A (m :distr A ) (f g :MF A ), mu m (fconj f g ) == mu (Mcond m f ) g * mu m f .Lemma Mcond_decomp : forall A (m :distr A ) (f g :MF A ), mu m g == mu (Mcond m f ) g * mu m f + mu (Mcond m (finv f )) g * mu m (finv f ).Lemma Mcond_bayes : forall A (m :distr A ) (f g :MF A ), mu (Mcond m f ) g == (mu (Mcond m g ) f * mu m g ) / (mu m f ).
Least upper bound of increasing sequences of distributions
Lemma M_lub_simpl : forall A (h : nat -m> M A ) (f :MF A ),lub h f = lub (mshift h f ).Section Lubs .Variable A : Type .Definition Mu : distr A -m> M A .Defined .Variable muf : nat -m> distr A .Definition mu_lub : distr A .Defined .Lemma mu_lub_le : forall n :nat, muf n <= mu_lub .Lemma mu_lub_sup : forall m : distr A , (forall n :nat, muf n <= m ) -> mu_lub <= m .End Lubs .Hint Resolve mu_lub_le mu_lub_sup .
distributions seen as a Ccpo
Instance cdistr : forall (A :Type), cpo (distr A ) :=D0 := distr_null A ; lub :=mu_lub (A :=A)}.Defined .Lemma distr_lub_simpl : forall A (h : nat -m> distr A ) (f :MF A ),mu (lub h ) f = lub (mshift (Mu A @ h ) f ).Hint Resolve distr_lub_simpl .
Definition Mfix (
A B :Type) (
F : (
A ->
distr B ) -m> (
A ->
distr B ))
:
A ->
distr B :=
fixp F .
Definition MFix (
A B :Type) : ((
A ->
distr B ) -m> (
A ->
distr B )) -m> (
A ->
distr B )
:=
Fixp (
A ->
distr B ).
Lemma Mfix_le :
forall (
A B :Type) (
F : (
A ->
distr B ) -m> (
A ->
distr B )) (
x :A),
Mfix F x <=
F (
Mfix F )
x .
Lemma Mfix_eq :
forall (
A B :Type) (
F : (
A ->
distr B ) -m> (
A ->
distr B )),
continuous F ->
forall (
x :A),
Mfix F x ==
F (
Mfix F )
x .
Hint Resolve Mfix_le Mfix_eq .
Lemma Mfix_le_compat :
forall (
A B :Type) (
F G : (
A ->
distr B )-m> (
A ->
distr B )),
F <=
G ->
Mfix F <=
Mfix G .
Definition Miter (
A B :Type) :=
Ccpo.iter (
D :=A ->
distr B ).
Lemma Mfix_le_iter :
forall (
A B :Type) (
F :(A ->
distr B ) -m> (
A ->
distr B )) (
n :nat),
Miter F n <=
Mfix F .
Section Continuity .Variables A B :Type.Instance Mlet_continuous_right forall a :distr A , continuous (D1 := A -> distr B ) (D2 :=distr B ) (MLet A B a ).Save .Lemma Mlet_continuous_left continuous (D1 :=distr A ) (D2 :=(A -> distr B ) -m> distr B ) (MLet A B ).Hint Resolve Mlet_continuous_right Mlet_continuous_left .Lemma Mlet_continuous2 : continuous2 (D1 :=distr A ) (D2 := A ->distr B ) (D3 :=distr B ) (MLet A B ).Hint Resolve Mlet_continuous2 .Lemma Mlet_lub_le : forall (mun :nat -m> distr A ) (Mn : nat -m> (A -> distr B )),Mlet (lub mun ) (lub Mn ) <= lub ((MLet A B @2 mun ) Mn ).Lemma Mfix_continuous : forall (Fn : nat -m> (A -> distr B ) -m> (A -> distr B )),forall n , continuous (Fn n )) -> Mfix (lub Fn ) <= lub (MFix A B @ Fn ).End Continuity .
distribution for flip
The distribution associated to
flip () is
Definition flip : M bool := mon (fun (f : bool -> U ) => [1/2] * (f true ) + [1/2] * (f false )).Lemma flip_stable_inv : stable_inv flip .Lemma flip_stable_plus : stable_plus flip .Lemma flip_stable_mult : stable_mult flip .Lemma flip_continuous : continuous flip .Definition ctrue : MF bool := fun (b :bool) => if b then 1 else 0.Definition cfalse : MF bool := fun (b :bool) => if b then 0 else 1.Lemma flip_ctrue : flip ctrue == [1/2].Lemma flip_cfalse : flip cfalse == [1/2].Hint Resolve flip_ctrue flip_cfalse .Definition Flip : distr bool .Defined .Lemma Flip_simpl : forall f , mu Flip f = [1/2] * (f true ) + [1/2] * (f false ).
Uniform distribution beween 0 and n
Require Arith .
Definition of fnth
fnth n k is defined as
Definition fnth (n :nat) : nat -> U := fun k => [1/]1+n.
Lemma Unth_eq : forall n , Unth n == [1-] (sigma (fnth n ) n ).Hint Resolve Unth_eq .Lemma sigma_fnth_one : forall n , sigma (fnth n ) (S n ) == 1.Hint Resolve sigma_fnth_one .Lemma Unth_inv_eq : forall n , [1-] ([1/]1+n) == sigma (fnth n ) n .Lemma sigma_fnth_sup : forall n m , (m > n ) -> sigma (fnth n ) m == sigma (fnth n ) (S n ).Lemma sigma_fnth_le : forall n m , (sigma (fnth n ) m ) <= (sigma (fnth n ) (S n )).Hint Resolve sigma_fnth_le .
fnth
Lemma fnth_retract : forall n :nat,(retract (fnth n ) (S n )).Implicit Arguments fnth_retract [].
distributions and general summations
Definition sigma_fun A (f :nat -> MF A ) (n :nat) : MF A := fun x => sigma (fun k => f k x ) n .Definition serie_fun A (f :nat -> MF A ) : MF A := fun x => serie (fun k => f k x ).Definition Sigma_fun A (f :nat -> MF A ) : nat -m> MF A := ishift (fun x => Sigma (fun k => f k x )).Lemma Sigma_fun_simpl : forall A (f :nat -> MF A ) (n :nat),Sigma_fun f n = sigma_fun f n .Lemma serie_fun_lub_sigma_fun : forall A (f :nat -> MF A ), serie_fun f == lub (Sigma_fun f ).Hint Resolve serie_fun_lub_sigma_fun .Lemma sigma_fun_0 : forall A (f :nat -> MF A ), sigma_fun f 0 == fzero A .Lemma sigma_fun_S : forall A (f :nat->MF A ) (n :nat), sigma_fun f (S n ) == fplus (f n ) (sigma_fun f n ).Lemma mu_sigma_le : forall A (d :distr A ) (f :nat -> MF A ) (n :nat), mu d (sigma_fun f n ) <= sigma (fun k => mu d (f k )) n .Lemma retract_fplusok : forall A (f :nat -> MF A ) (n :nat), forall x , retract (fun k => f k x ) n ) -> forall k , (k < n )%nat -> fplusok (f k ) (sigma_fun f k ).Lemma mu_sigma_eq : forall A (d :distr A ) (f :nat -> MF A ) (n :nat), forall x , retract (fun k => f k x ) n ) -> mu d (sigma_fun f n ) == sigma (fun k => mu d (f k )) n .Lemma mu_serie_le : forall A (d :distr A ) (f :nat -> MF A ), mu d (serie_fun f ) <= serie (fun k => mu d (f k )).Lemma mu_serie_eq : forall A (d :distr A ) (f :nat -> MF A ), forall x , wretract (fun k => f k x )) -> mu d (serie_fun f ) == serie (fun k => mu d (f k )).Lemma wretract_fplusok : forall A (f :nat -> MF A ), forall x , wretract (fun k => f k x )) -> forall k , fplusok (f k ) (sigma_fun f k ).
Instance discrete_mon : forall A (c : nat -> U ) (p : nat -> A ),monotonic (fun f : A ->U => serie (fun k => c k * f (p k ))).Save .Definition discrete A (c : nat -> U ) (p : nat -> A ) : M A := mon (fun f : A ->U => serie (fun k => c k * f (p k ))).Lemma discrete_simpl : forall A (c : nat -> U ) (p : nat -> A ) f , discrete c p f = serie (fun k => c k * f (p k )).Lemma discrete_stable_inv : forall A (c : nat -> U ) (p : nat -> A ), wretract c -> stable_inv (discrete c p ).Lemma discrete_stable_plus : forall A (c : nat -> U ) (p : nat -> A ), stable_plus (discrete c p ).Lemma discrete_stable_mult : forall A (c : nat -> U ) (p : nat -> A ), wretract c -> stable_mult (discrete c p ).Lemma discrete_continuous : forall A (c : nat -> U ) (p : nat -> A ), continuous (discrete c p ).Record discr (A :Type) : Type := coeff : nat -> U ; coeff_retr : wretract coeff ; points : nat -> A }.Hint Resolve coeff_retr .Definition Discrete : forall A , discr A -> distr A .Defined .Lemma Discrete_simpl : forall A (d :discr A ), mu (Discrete d ) = discrete (coeff d ) (points d ).Definition is_discrete (A :Type) (m : distr A ) := exists d : discr A , m == Discrete d .
distribution for random n
The distribution associated to
random n is
we cannot factorize because of possible overflow
Instance random_mon : forall n , monotonic (fun (f :MF nat ) => sigma (fun k => Unth n * f k ) (S n )).Save .Definition random (n :nat):M nat := mon (fun (f :MF nat ) => sigma (fun k => Unth n * f k ) (S n )).Lemma random_simpl : forall n (f : MF nat ), random n f = sigma (fun k => Unth n * f k ) (S n ).
Lemma random_stable_inv : forall n , stable_inv (random n ).Lemma random_stable_plus : forall n , stable_plus (random n ).Lemma random_stable_mult : forall n , stable_mult (random n ).Lemma random_continuous : forall n , continuous (random n ).Definition Random (n :nat) : distr nat .Defined .Lemma Random_simpl : forall (n :nat), mu (Random n ) = random n .Lemma Random_total : forall n : nat , mu (Random n ) (fone nat ) == 1.Hint Resolve Random_total .Lemma Random_inv : forall f n , mu (Random n ) (finv f ) == [1-] (mu (Random n ) f ).Hint Resolve Random_inv .
Ltac mu_plus d := match goal with context [fmont (mu d ) (fun x => (Uplus (@?f x ) (@?g x )))] => rewrite (mu_stable_plus d (f :=f) (g :=g))end .Ltac mu_mult d := match goal with context [fmont (mu d ) (fun x => (Umult ?k (@?f x )))] => rewrite (mu_stable_mult d k f )end .
Library Prog
Prog.v: Composition of distributions
Definition Mif (
A :Type) (
b :distr
bool ) (
m1 m2 :
distr A )
:=
Mlet b (
fun x :bool =>
if x then m1 else m2 ).
Lemma Mif_le_compat :
forall (
A :Type) (
b1 b2 :distr
bool ) (
m1 m2 n1 n2 :
distr A ),
b1 <=
b2 ->
m1 <=
m2 ->
n1 <=
n2 ->
Mif b1 m1 n1 <=
Mif b2 m2 n2 .
Hint Resolve Mif_le_compat .
Instance Mif_mon2 :
forall (
A :Type)
b ,
monotonic2 (
Mif (
A :=A)
b ).
Save .
Definition MIf :
forall (
A :Type),
distr bool -m>
distr A -m>
distr A -m>
distr A .
Defined .
The distribution associated to pchoice p m1 m2
Definition pchoice :
forall A ,
U ->
M A ->
M A ->
M A .
Defined .
Lemma pchoice_simpl :
forall A p (
m1 m2 :M
A )
f ,
pchoice p m1 m2 f =
p *
m1 f + [1-]
p *
m2 f .
Definition Mchoice (
A :Type) (
p :U) (
m1 m2 :
distr A ) :
distr A .
Defined .
Lemma Mchoice_simpl :
forall A p (
m1 m2 :distr
A )
f ,
mu (
Mchoice p m1 m2 )
f =
p *
mu m1 f + [1-]
p *
mu m2 f .
Lemma Mchoice_le_compat :
forall (
A :Type) (
p :U) (
m1 m2 n1 n2 :
distr A ),
m1 <=m2 ->
n1 <=n2 ->
Mchoice p m1 n1 <=
Mchoice p m2 n2 .
Hint Resolve Mchoice_le_compat .
Add Parametric Morphism (
A :Type) : (
Mchoice (
A :=A))
with signature Oeq &
Oeq &
Oeq &
Oeq
as Mchoice_eq_compat .
Save .
Hint Immediate Mchoice_eq_compat .
Instance Mchoice_mon2 :
forall (
A :Type) (
p :U),
monotonic2 (
Mchoice (
A :=A)
p ).
Save .
Definition MChoice A (
p :U) :
distr A -m>
distr A -m>
distr A :=
mon2 (
Mchoice (
A :=A)
p ).
Lemma MChoice_simpl :
forall A (
p :U) (
m1 m2 :
distr A ),
MChoice A p m1 m2 =
Mchoice p m1 m2 .
Lemma Mchoice_sym_le :
forall (
A :Type) (
p :U) (
m1 m2 :
distr A ),
Mchoice p m1 m2 <=
Mchoice ([1-]
p )
m2 m1 .
Hint Resolve Mchoice_sym_le .
Lemma Mchoice_sym :
forall (
A :Type) (
p :U) (
m1 m2 :
distr A ),
Mchoice p m1 m2 ==
Mchoice ([1-]
p )
m2 m1 .
Lemma Mchoice_continuous_right
:
forall (
A :Type) (
p :U) (
m :
distr A ),
continuous (
D1 :=distr
A ) (
D2 :=distr
A ) (
MChoice A p m ).
Hint Resolve Mchoice_continuous_right .
Lemma Mchoice_continuous_left :
forall (
A :Type) (
p :U),
continuous (
D1 :=distr
A ) (
D2 :=distr
A -m>
distr A ) (
MChoice A p ).
Lemma Mchoice_continuous :
forall (
A :Type) (
p :U),
continuous2 (
D1 :=distr
A ) (
D2 :=distr
A ) (
D3 :=distr
A ) (
MChoice A p ).
Definition im_distr (
A B :
Type ) (
f :A ->
B ) (
m :distr
A ) :
distr B :=
Mlet m (
fun a =>
Munit (
f a )).
Lemma im_distr_simpl :
forall A B (
f :A ->
B ) (
m :distr
A )(
h :B ->
U ),
mu (
im_distr f m )
h =
mu m (
fun a =>
h (
f a )).
Add Parametric Morphism (
A B :
Type ) : (
im_distr (
A :=A) (
B :=B))
with signature (
feq (
A :=A) (
B :=B)) &
Oeq &
Oeq
as im_distr_eq_compat .
Save .
Lemma im_distr_comp :
forall A B C (
f :A ->
B ) (
g :B ->
C ) (
m :distr
A ),
im_distr g (
im_distr f m ) ==
im_distr (
fun a =>
g (
f a ))
m .
Lemma im_distr_id :
forall A (
f :A ->
A ) (
m :distr
A ), (
forall x ,
f x =
x ) ->
im_distr f m ==
m .
Definition prod_distr (
A B :
Type )(
d1 :distr
A )(
d2 :distr
B ) :
distr (
A *B) :=
Mlet d1 (
fun x =>
Mlet d2 (
fun y =>
Munit (
x ,y))).
Add Parametric Morphism (
A B :
Type ) : (
prod_distr (
A :=A) (
B :=B))
with signature Ole ++>
Ole ++>
Ole
as prod_distr_le_compat .
Save .
Hint Resolve prod_distr_le_compat .
Add Parametric Morphism (
A B :
Type ) : (
prod_distr (
A :=A) (
B :=B))
with signature Oeq &
Oeq &
Oeq
as prod_distr_eq_compat .
Save .
Hint Immediate prod_distr_eq_compat .
Instance prod_distr_mon2 :
forall (
A B :Type),
monotonic2 (
prod_distr (
A :=A) (
B :=B)).
Save .
Definition Prod_distr (
A B :Type):
distr A -m>
distr B -m>
distr (
A *B) :=
mon2 (
prod_distr (
A :=A) (
B :=B)).
Lemma Prod_distr_simpl :
forall (
A B :Type)(d1:
distr A ) (
d2 :distr
B ),
Prod_distr A B d1 d2 =
prod_distr d1 d2 .
Lemma prod_distr_rect :
forall (
A B :
Type ) (
d1 :
distr A ) (
d2 :distr
B ) (
f :A ->
U )(
g :B ->
U ),
mu (
prod_distr d1 d2 ) (
fun xy =>
f (
fst xy ) *
g (
snd xy )) ==
mu d1 f *
mu d2 g .
Lemma prod_distr_fst :
forall (
A B :
Type ) (
d1 :
distr A ) (
d2 :distr
B ) (
f :A ->
U ),
mu (
prod_distr d1 d2 ) (
fun xy =>
f (
fst xy )) ==
mu d2 (
fone B ) *
mu d1 f .
Lemma prod_distr_snd :
forall (
A B :
Type ) (
d1 :
distr A ) (
d2 :distr
B ) (
g :B ->
U ),
mu (
prod_distr d1 d2 ) (
fun xy =>
g (
snd xy )) ==
mu d1 (
fone A ) *
mu d2 g .
Lemma prod_distr_fst_eq :
forall (
A B :
Type ) (
d1 :
distr A ) (
d2 :distr
B ),
mu d2 (
fone B ) == 1 ->
im_distr (
fst (
A :=A) (
B :=B)) (
prod_distr d1 d2 ) ==
d1 .
Lemma prod_distr_snd_eq :
forall (
A B :
Type ) (
d1 :
distr A ) (
d2 :distr
B ),
mu d1 (
fone A ) == 1 ->
im_distr (
snd (
A :=A) (
B :=B)) (
prod_distr d1 d2 ) ==
d2 .
Definition swap A B (
x :A*B) :
B *
A := (
snd x ,fst
x ).
Definition arg_swap A B (
f :
MF (
A *B)) :
MF (
B *A) :=
fun z =>
f (
swap z ).
Definition Arg_swap A B :
MF (
A *B) -m>
MF (
B *A).
Defined .
Lemma Arg_swap_simpl :
forall A B f ,
Arg_swap A B f =
arg_swap f .
Definition prod_distr_com A B (
d1 :
distr A ) (
d2 :distr
B ) (
f :
MF (
A *
B )) :=
mu (
prod_distr d1 d2 )
f ==
mu (
prod_distr d2 d1 ) (
arg_swap f ).
Lemma prod_distr_com_eq_compat :
forall A B (
d1 :
distr A ) (
d2 :distr
B ) (
f g :
MF (
A *
B )),
f ==
g ->
prod_distr_com d1 d2 f ->
prod_distr_com d1 d2 g .
Lemma prod_distr_com_rect :
forall (
A B :
Type ) (
d1 :
distr A ) (
d2 :distr
B ) (
f :A ->
U )(
g :B ->
U ),
prod_distr_com d1 d2 (
fun xy =>
f (
fst xy ) *
g (
snd xy )).
Lemma prod_distr_com_cte :
forall (
A B :
Type ) (
d1 :
distr A ) (
d2 :distr
B ) (
c :U),
prod_distr_com d1 d2 (
fcte (
A *B)
c ).
Lemma prod_distr_com_one :
forall (
A B :
Type ) (
d1 :
distr A ) (
d2 :distr
B ),
prod_distr_com d1 d2 (
fone (
A *B)).
Lemma prod_distr_com_plus :
forall (
A B :
Type ) (
d1 :
distr A ) (
d2 :distr
B ) (
f g :MF (
A *B)),
fplusok f g ->
prod_distr_com d1 d2 f ->
prod_distr_com d1 d2 g ->
prod_distr_com d1 d2 (
fplus f g ).
Lemma prod_distr_com_mult :
forall (
A B :
Type ) (
d1 :
distr A ) (
d2 :distr
B ) (
k :U)(f:MF (
A *B)),
prod_distr_com d1 d2 f ->
prod_distr_com d1 d2 (
fmult k f ).
Lemma prod_distr_com_inv :
forall (
A B :
Type ) (
d1 :
distr A ) (
d2 :distr
B ) (
f :MF (
A *B)),
prod_distr_com d1 d2 f ->
prod_distr_com d1 d2 (
finv f ).
Lemma prod_distr_com_lub :
forall (
A B :
Type ) (
d1 :
distr A ) (
d2 :distr
B ) (
f :nat -m>
MF (
A *B)),
(
forall n ,
prod_distr_com d1 d2 (
f n )) ->
prod_distr_com d1 d2 (
lub f ).
Lemma prod_distr_com_refl :
forall A B (
d1 :distr
A ) (
d2 :distr
B ) (
f :MF (
A *B)),
prod_distr_com d1 d2 f ->
prod_distr_com d2 d1 (
arg_swap f ).
Lemma discrete_commute :
forall A B (
d1 :distr
A ) (
d2 :distr
B ) (
f :MF (
A *B)),
is_discrete d1 ->
prod_distr_com d1 d2 f .
Definition fst_distr A B (
m :
distr (
A *B)) :
distr A :=
im_distr (
fst (
B :=B))
m .
Definition snd_distr A B (
m :
distr (
A *B)) :
distr B :=
im_distr (
snd (
B :=B))
m .
Definition range A (
P :A ->
Prop ) (
d :
distr A ) :=
forall f , (
forall x ,
P x -> 0 ==
f x ) -> 0 ==
mu d f .
Lemma range_le :
forall A (
P :A ->
Prop ) (
d :distr
A ),
range P d ->
forall f g , (
forall a ,
P a ->
f a <=
g a ) ->
mu d f <=
mu d g .
Lemma range_eq :
forall A (
P :A ->
Prop ) (
d :distr
A ),
range P d ->
forall f g , (
forall a ,
P a ->
f a ==
g a ) ->
mu d f ==
mu d g .
Prog.v: Axiomatic semantics
Definition of correctness judgements
is defined as
is defined as
Definition ok (A :Type) (p :U) (e :distr A ) (q :A -> U ) := p <= mu e q .Definition okfun (A B :Type)(p:A -> U )(e :A -> distr B )(q :A -> B -> U )forall x :A, ok (p x ) (e x ) (q x ).Definition okup (A :Type) (p :U) (e :distr A ) (q :A -> U ) := mu e q <= p .Definition upfun (A B :Type)(p:A -> U )(e :A -> distr B )(q :A -> B -> U )forall x :A, okup (p x ) (e x ) (q x ).
Lemma ok_le_compat :
forall (
A :Type) (
p p' :U) (
e :distr
A ) (
q q' :A ->
U ),
p' <=
p ->
q <=
q' ->
ok p e q ->
ok p' e q' .
Lemma ok_eq_compat :
forall (
A :Type) (
p p' :U) (
e e' :distr
A ) (
q q' :A ->
U ),
p' ==
p ->
q ==
q' ->
e ==
e' ->
ok p e q ->
ok p' e' q' .
Lemma okfun_le_compat :
forall (
A B :Type) (
p p' :A ->
U ) (
e :A ->
distr B ) (
q q' :A ->
B ->
U ),
p' <=
p ->
q <=
q' ->
okfun p e q ->
okfun p' e q' .
Lemma ok_mult :
forall (
A :Type)(k
p :U)(e:distr
A )(
f :
A ->
U ),
ok p e f ->
ok (
k *p)
e (
fmult k f ).
Lemma ok_inv :
forall (
A :Type)(p:U)(e:distr
A )(
f :
A ->
U ),
ok p e f ->
mu e (
finv f ) <= [1-]
p .
Lemma okup_le_compat :
forall (
A :Type) (
p p' :U) (
e :distr
A ) (
q q' :A ->
U ),
p <=
p' ->
q' <=
q ->
okup p e q ->
okup p' e q' .
Lemma okup_eq_compat :
forall (
A :Type) (
p p' :U) (
e e' :distr
A ) (
q q' :A ->
U ),
p ==
p' ->
q ==
q' ->
e ==
e' ->
okup p e q ->
okup p' e' q' .
Lemma upfun_le_compat :
forall (
A B :Type) (
p p' :A ->
U ) (
e :A ->
distr B )
(
q q' :A ->
B ->
U ),
p <=
p' ->
q' <=q ->
upfun p e q ->
upfun p' e q' .
Lemma okup_mult :
forall (
A :Type)(k
p :U)(e:distr
A )(
f :
A ->
U ),
okup p e f ->
okup (
k *p)
e (
fmult k f ).
Lemma apply_rule :
forall (
A B :Type)(a:(distr
A ))(
f :A ->
distr B )(
r :U)(p:A ->
U )(
q :B ->
U ),
ok r a p ->
okfun p f (
fun x =>
q ) ->
ok r (
Mlet a f )
q .
Lemma okup_apply_rule :
forall (
A B :Type)(a:distr
A )(
f :A ->
distr B )(
r :U)(p:A ->
U )(
q :B ->
U ),
okup r a p ->
upfun p f (
fun x =>
q ) ->
okup r (
Mlet a f )
q .
Lemma lambda_rule : forall (A B :Type)(f:A -> distr B )(p :A -> U )(q :A -> B -> U ),forall x :A, ok (p x ) (f x ) (q x )) -> okfun p f q .Lemma okup_lambda_rule : forall (A B :Type)(f:A -> distr B )(p :A -> U )(q :A -> B -> U ),forall x :A, okup (p x ) (f x ) (q x )) -> upfun p f q .
Lemma combiok :
forall (
A :Type)
p q (
f1 f2 :
A ->
U ),
p <= [1-]
q ->
fplusok (
fmult p f1 ) (
fmult q f2 ).
Hint Resolve combiok .
Lemma fmult_fplusok :
forall (
A :Type)
p q (
f1 f2 :
A ->
U ),
fplusok f1 f2 ->
fplusok (
fmult p f1 ) (
fmult q f2 ).
Hint Resolve fmult_fplusok .
Lemma ifok :
forall f1 f2 ,
fplusok (
fmult f1 ctrue ) (
fmult f2 cfalse ).
Hint Resolve ifok .
Lemma Mif_eq :
forall (
A :Type)(b:(distr
bool ))(
f1 f2 :distr
A )(
q :MF
A ),
mu (
Mif b f1 f2 )
q == (
mu f1 q ) * (
mu b ctrue ) + (
mu f2 q ) * (
mu b cfalse ).
Lemma ifrule :
forall (
A :Type)(b:(distr
bool ))(
f1 f2 :distr
A )(
p1 p2 :U)(q:A ->
U ),
ok p1 f1 q ->
ok p2 f2 q
->
ok (
p1 * (
mu b ctrue ) +
p2 * (
mu b cfalse )) (
Mif b f1 f2 )
q .
Lemma okup_ifrule :
forall (
A :Type)(b:(distr
bool ))(
f1 f2 :distr
A )(
p1 p2 :U)(q:A ->
U ),
okup p1 f1 q ->
okup p2 f2 q
->
okup (
p1 * (
mu b ctrue ) +
p2 * (
mu b cfalse )) (
Mif b f1 f2 )
q .
Rule for fixpoints
with , an increasing sequence of functions starting from
Section Fixrule .
Variables A B :
Type .
Variable F : (
A ->
distr B ) -m> (
A ->
distr B ).
Section Ruleseq .
Variable q :
A ->
B ->
U .
Lemma fixrule_Ulub :
forall (
p :
A ->
nat ->
U ),
(
forall x :A,
p x O == 0)->
(
forall (
i :nat) (
f :A ->
distr B ),
(
okfun (
fun x =>
p x i )
f q ) ->
okfun (
fun x =>
p x (
S i )) (
fun x =>
F f x )
q )
->
okfun (
fun x =>
Ulub (
p x )) (
Mfix F )
q .
Lemma fixrule :
forall (
p :
A ->
nat -m>
U ),
(
forall x :A,
p x O == 0)->
(
forall (
i :nat) (
f :A ->
distr B ),
(
okfun (
fun x =>
p x i )
f q ) ->
okfun (
fun x =>
p x (
S i )) (
fun x =>
F f x )
q )
->
okfun (
fun x =>
lub (
p x )) (
Mfix F )
q .
Lemma fixrule_up_Ulub :
forall (
p :
A ->
nat ->
U ),
(
forall (
i :nat) (
f :A ->
distr B ),
(
upfun (
fun x =>
p x i )
f q ) ->
upfun (
fun x =>
p x (
S i )) (
fun x =>
F f x )
q )
->
upfun (
fun x =>
Ulub (
p x )) (
Mfix F )
q .
Lemma fixrule_up_lub :
forall (
p :
A ->
nat -m>
U ),
(
forall (
i :nat) (
f :A ->
distr B ),
(
upfun (
fun x =>
p x i )
f q ) ->
upfun (
fun x =>
p x (
S i )) (
fun x =>
F f x )
q )
->
upfun (
fun x =>
lub (
p x )) (
Mfix F )
q .
Lemma okup_fixrule_glb :
forall p :
A ->
nat -m->
U ,
(
forall (
i :nat) (
f :A ->
distr B ),
(
upfun (
fun x =>
p x i )
f q ) ->
upfun (
fun x =>
p x (
S i )) (
fun x =>
F f x )
q )
->
upfun (
fun x =>
glb (
p x )) (
Mfix F )
q .
End Ruleseq .
Lemma okup_fixrule_inv :
forall (
p :
A ->
U ) (
q :
A ->
B ->
U ),
(
forall (
f :A ->
distr B ),
upfun p f q ->
upfun p (
fun x =>
F f x )
q )
->
upfun p (
Mfix F )
q .
Rules using commutation properties
Section TransformFix .
Section Fix_muF .
Variable q :
A ->
B ->
U .
Variable muF :
MF A -m>
MF A .
Definition admissible (
P :(A ->
distr B ) ->
Prop ) :=
P 0 /\
forall f ,
P f ->
P (
F f ).
Lemma admissible_true :
admissible (
fun f =>
True ).
Lemma admissible_le_fix :
continuous (
D1 :=A ->
distr B ) (
D2 :=A ->
distr B )
F ->
admissible (
fun f =>
f <=
Mfix F ).
Lemma muF_stable :
stable muF .
Definition mu_muF_commute_le :=
forall f x ,
f <=
Mfix F ->
mu (
F f x ) (
q x ) <=
muF (
fun y =>
mu (
f y ) (
q y ))
x .
Hint Unfold mu_muF_commute_le .
Section F_muF_results .
Hypothesis F_muF_le :
mu_muF_commute_le .
Lemma mu_mufix_le :
forall x ,
mu (
Mfix F x ) (
q x ) <=
mufix muF x .
Hint Resolve mu_mufix_le .
Lemma muF_le :
forall f ,
muF f <=
f
->
forall x ,
mu (
Mfix F x ) (
q x ) <=
f x .
Hypothesis muF_F_le :
forall f x ,
f <=
Mfix F ->
muF (
fun y =>
mu (
f y ) (
q y ))
x <=
mu (
F f x ) (
q x ).
Lemma mufix_mu_le :
forall x ,
mufix muF x <=
mu (
Mfix F x ) (
q x ).
End F_muF_results .
Hint Resolve mu_mufix_le mufix_mu_le .
Lemma mufix_mu :
(
forall f x ,
f <=
Mfix F ->
mu (
F f x ) (
q x ) ==
muF (
fun y =>
mu (
f y ) (
q y ))
x )
->
forall x ,
mufix muF x ==
mu (
Mfix F x ) (
q x ).
Hint Resolve mufix_mu .
End Fix_muF .
Section Fix_Term .
Definition pterm :
MF A :=
fun (
x :A) =>
mu (
Mfix F x ) (
fone B ).
Variable muFone :
MF A -m>
MF A .
Hypothesis F_muF_eq_one :
forall f x ,
f <=
Mfix F ->
mu (
F f x ) (
fone B ) ==
muFone (
fun y =>
mu (
f y ) (
fone B ))
x .
Hypothesis muF_cont :
continuous muFone .
Lemma muF_pterm :
pterm ==
muFone pterm .
Hint Resolve muF_pterm .
End Fix_Term .
Section Fix_muF_Term .
Variable q :
A ->
B ->
U .
Definition qinv x y := [1-]
q x y .
Variable muFqinv :
MF A -m>
MF A .
Hypothesis F_muF_le_inv :
mu_muF_commute_le qinv muFqinv .
Lemma muF_le_term :
forall f ,
muFqinv (
finv f ) <=
finv f ->
forall x ,
f x &
pterm x <=
mu (
Mfix F x ) (
q x ).
Lemma muF_le_term_minus :
forall f ,
f <=
pterm ->
muFqinv (
fminus pterm f ) <=
fminus pterm f ->
forall x ,
f x <=
mu (
Mfix F x ) (
q x ).
Variable muFq :
MF A -m>
MF A .
Hypothesis F_muF_le :
mu_muF_commute_le q muFq .
Lemma muF_eq :
forall f ,
muFq f <=
f ->
muFqinv (
finv f ) <=
finv f ->
forall x ,
pterm x == 1 ->
mu (
Mfix F x ) (
q x ) ==
f x .
End Fix_muF_Term .
End TransformFix .
Section LoopRule .
Variable q :
A ->
B ->
U .
Variable stop :
A ->
distr bool .
Variable step :
A ->
distr A .
Variable a :
U .
Definition Loop :
MF A -m>
MF A .
Defined .
Lemma Loop_eq :
forall f x ,
Loop f x =
mu (
stop x ) (
fun b =>
if b then a else mu (
step x )
f ).
Definition loop :=
mufix Loop .
Lemma Mfixvar :
(
forall (
f :A ->
distr B ),
okfun (
fun x =>
Loop (
fun y =>
mu (
f y ) (
q y ))
x ) (
fun x =>
F f x )
q )
->
okfun loop (
Mfix F )
q .
Definition up_loop :
MF A :=
nufix Loop .
Lemma Mfixvar_up :
(
forall (
f :A ->
distr B ),
upfun (
fun x =>
Loop (
fun y =>
mu (
f y ) (
q y ))
x ) (
fun x =>
F f x )
q )
->
upfun up_loop (
Mfix F )
q .
End LoopRule .
End Fixrule .
Distributions operates on intervals
Definition Imu :
forall A :Type,
distr A -> (
A ->
IU ) ->
IU .
Defined .
Lemma low_Imu :
forall (
A :Type) (
e :distr
A ) (
F :
A ->
IU ),
low (
Imu e F ) =
mu e (
fun x =>
low (
F x )).
Lemma up_Imu :
forall (
A :Type) (
e :distr
A ) (
F :
A ->
IU ),
up (
Imu e F ) =
mu e (
fun x =>
up (
F x )).
Lemma Imu_monotonic :
forall (
A :Type) (
e :distr
A ) (
F G :
A ->
IU ),
(
forall x ,
Iincl (
F x ) (
G x )) ->
Iincl (
Imu e F ) (
Imu e G ).
Lemma Imu_stable_eq :
forall (
A :Type) (
e :distr
A ) (
F G :
A ->
IU ),
(
forall x ,
Ieq (
F x ) (
G x )) ->
Ieq (
Imu e F ) (
Imu e G ).
Hint Resolve Imu_monotonic Imu_stable_eq .
Lemma Imu_singl :
forall (
A :Type) (
e :distr
A ) (
f :A ->
U ),
Ieq (
Imu e (
fun x =>
singl (
f x ))) (
singl (
mu e f )).
Lemma Imu_inf :
forall (
A :Type) (
e :distr
A ) (
f :A ->
U ),
Ieq (
Imu e (
fun x =>
inf (
f x ))) (
inf (
mu e f )).
Lemma Imu_sup :
forall (
A :Type) (
e :distr
A ) (
f :A ->
U ),
Iincl (
Imu e (
fun x =>
sup (
f x ))) (
sup (
mu e f )).
Lemma Iin_mu_Imu :
forall (
A :Type) (
e :distr
A ) (
F :A ->
IU ) (
f :A ->
U ),
(
forall x ,
Iin (
f x ) (
F x )) ->
Iin (
mu e f ) (
Imu e F ).
Hint Resolve Iin_mu_Imu .
Definition Iok (
A :Type) (
I :IU) (
e :distr
A ) (
F :A ->
IU ) :=
Iincl (
Imu e F )
I .
Definition Iokfun (
A B :Type)(I:A ->
IU ) (
e :A ->
distr B ) (
F :A ->
B ->
IU )
:=
forall x ,
Iok (
I x ) (
e x ) (
F x ).
Lemma Iin_mu_Iok :
forall (
A :Type) (
I :IU) (
e :distr
A ) (
F :A ->
IU ) (
f :A ->
U ),
(
forall x ,
Iin (
f x ) (
F x )) ->
Iok I e F ->
Iin (
mu e f )
I .
Lemma Iok_le_compat :
forall (
A :Type) (
I J :IU) (
e :distr
A ) (
F G :A ->
IU ),
Iincl I J -> (
forall x ,
Iincl (
G x ) (
F x )) ->
Iok I e F ->
Iok J e G .
Lemma Iokfun_le_compat :
forall (
A B :Type) (
I J :A ->
IU ) (
e :A ->
distr B ) (
F G :A ->
B ->
IU ),
(
forall x ,
Iincl (
I x ) (
J x )) -> (
forall x y ,
Iincl (
G x y ) (
F x y )) ->
Iokfun I e F ->
Iokfun J e G .
Lemma Iunit_eq : forall (A : Type ) (a :A) (F :A -> IU ), Ieq (Imu (Munit a ) F ) (F a ).
Lemma Ilet_eq :
forall (
A B :
Type ) (
a :distr
A ) (
f :A ->
distr B )(
I :IU)(G:B ->
IU ),
Ieq (
Imu (
Mlet a f )
G ) (
Imu a (
fun x =>
Imu (
f x )
G )).
Hint Resolve Ilet_eq .
Lemma Iapply_rule :
forall (
A B :
Type ) (
a :distr
A ) (
f :A ->
distr B )(
I :IU)(F:A ->
IU )(
G :B ->
IU ),
Iok I a F ->
Iokfun F f (
fun x =>
G ) ->
Iok I (
Mlet a f )
G .
Lemma Ilambda_rule : forall (A B :Type)(f:A -> distr B )(F :A -> IU )(G :A -> B -> IU ),forall x :A, Iok (F x ) (f x ) (G x )) -> Iokfun F f G .
Lemma Imu_Mif_eq :
forall (
A :Type)(b:distr
bool )(
f1 f2 :distr
A )(
F :A ->
IU ),
Ieq (
Imu (
Mif b f1 f2 )
F ) (
Iplus (
Imultk (
mu b ctrue ) (
Imu f1 F )) (
Imultk (
mu b cfalse ) (
Imu f2 F ))).
Lemma Iifrule :
forall (
A :Type)(b:(distr
bool ))(
f1 f2 :distr
A )(
I1 I2 :IU)(F:A ->
IU ),
Iok I1 f1 F ->
Iok I2 f2 F
->
Iok (
Iplus (
Imultk (
mu b ctrue )
I1 ) (
Imultk (
mu b cfalse )
I2 )) (
Mif b f1 f2 )
F .
Rule for fixpoints
with , an decreasing sequence of intervals functions () such that contains for all .
Section IFixrule .
Variables A B :
Type .
Variable F : (
A ->
distr B ) -m> (
A ->
distr B ).
Section IRuleseq .
Variable Q :
A ->
B ->
IU .
Variable I :
A ->
nat -m>
IU .
Lemma Ifixrule :
(
forall x :A,
Iin 0 (
I x O )) ->
(
forall (
i :nat) (
f :A ->
distr B ),
(
Iokfun (
fun x =>
I x i )
f Q ) ->
Iokfun (
fun x =>
I x (
S i )) (
fun x =>
F f x )
Q )
->
Iokfun (
fun x =>
Ilim (
I x )) (
Mfix F )
Q .
End IRuleseq .
Section ITransformFix .
Section IFix_muF .
Variable Q :
A ->
B ->
IU .
Variable ImuF : (
A ->
IU ) -m> (
A ->
IU ).
Lemma ImuF_stable :
forall I J ,
I ==J ->
ImuF I ==
ImuF J .
Section IF_muF_results .
Hypothesis Iincl_F_ImuF :
forall f x ,
f <=
Mfix F ->
Iincl (
Imu (
F f x ) (
Q x )) (
ImuF (
fun y =>
Imu (
f y ) (
Q y ))
x ).
Lemma Iincl_fix_ifix :
forall x ,
Iincl (
Imu (
Mfix F x ) (
Q x )) (
fixp (
D :=A ->
IU )
ImuF x ).
Hint Resolve Iincl_fix_ifix .
End IF_muF_results .
End IFix_muF .
End ITransformFix .
End IFixrule .
Lemma Flip_ctrue :
mu Flip ctrue == [1/2].
Lemma Flip_cfalse :
mu Flip cfalse == [1/2].
Lemma ok_Flip :
forall q :
bool ->
U ,
ok ([1/2] *
q true + [1/2] *
q false )
Flip q .
Lemma okup_Flip :
forall q :
bool ->
U ,
okup ([1/2] *
q true + [1/2] *
q false )
Flip q .
Hint Resolve ok_Flip okup_Flip Flip_ctrue Flip_cfalse .
Lemma Flip_eq :
forall q :
bool ->
U ,
mu Flip q == [1/2] *
q true + [1/2] *
q false .
Hint Resolve Flip_eq .
Lemma IFlip_eq :
forall Q :
bool ->
IU ,
Ieq (
Imu Flip Q ) (
Iplus (
Imultk [1/2] (
Q true )) (
Imultk [1/2] (
Q false ))).
Hint Resolve IFlip_eq .
Rules for total (well-founded) fixpoints
Section Wellfounded .
Variables A B :
Type .
Variable R :
A ->
A ->
Prop .
Hypothesis Rwf :
well_founded R .
Variable F :
forall x , (
forall y ,
R y x ->
distr B ) ->
distr B .
Definition WfFix :
A ->
distr B :=
Fix Rwf (
fun _ =>
distr B )
F .
Hypothesis Fext :
forall x f g , (
forall y (
p :R
y x ),
f y p ==
g y p ) ->
F f ==
F g .
Lemma Acc_iter_distr :
forall x ,
forall r s :
Acc R x ,
Acc_iter (
fun _ =>
distr B )
F r ==
Acc_iter (
fun _ =>
distr B )
F s .
Lemma WfFix_eq :
forall x ,
WfFix x ==
F (
fun (
y :A) (
p :R
y x ) =>
WfFix y ).
Variable P :
distr B ->
Prop .
Hypothesis Pext :
forall m1 m2 ,
m1 ==
m2 ->
P m1 ->
P m2 .
Lemma WfFix_ind :
(
forall x f , (
forall y (
p :R
y x ),
P (
f y p )) ->
P (
F f ))
->
forall x ,
P (
WfFix x ).
End Wellfounded .
Ltac distrsimpl :=
match goal with
| |- (
Ole (
fmont (
mu ?d1) ?f) (
fmont (
mu ?d2) ?g)) =>
apply (
mu_le_compat (
m1 :=d1) (
m2 :=d2) (
Ole_refl d1 ) (
f :=f) (
g :=g));
intro
| |- (
Oeq (
fmont (
mu ?d1) ?f) (
fmont (
mu ?d2) ?g)) =>
apply (
mu_eq_compat (
m1 :=d1) (
m2 :=d2) (
Oeq_refl d1 ) (
f :=f) (
g :=g));
intro
| |- (
Oeq (
Munit ?x) (
Munit ?y)) =>
apply (
Munit_eq_compat x y )
| |- (
Oeq (
Mlet ?x1 ?f) (
Mlet ?x2 ?g))
=>
apply (
Mlet_eq_compat (
m1 :=x1) (
m2 :=x2) (
M1 :=f) (
M2 :=g) (
Oeq_refl x1 ));
intro
| |- (
Ole (
Mlet ?x1 ?f) (
Mlet ?x2 ?g))
=>
apply (
Mlet_le_compat (
m1 :=x1) (
m2 :=x2) (
M1 :=f) (
M2 :=g) (
Ole_refl x1 ));
intro
| |-
context [(
fmont (
mu (
Mlet ?m ?M)) ?f)] =>
rewrite (
Mlet_simpl m M f )
| |-
context [(
fmont (
mu (
Munit ?x)) ?f)] =>
rewrite (
Munit_simpl f x )
| |-
context [(
Mlet (
Mlet ?m ?M) ?f)] =>
rewrite (
Mlet_assoc m M f )
| |-
context [(
Mlet (
Munit ?x) ?f)] =>
rewrite (
Mlet_unit x f )
| |-
context [(
fmont (
mu Flip ) ?f)] =>
rewrite (
Flip_simpl f )
| |-
context [(
fmont (
mu (
Discrete ?d)) ?f)] =>
rewrite (
Discrete_simpl d );
rewrite (
discrete_simpl (
coeff d ) (
points d )
f )
| |-
context [(
fmont (
mu (
Random ?n)) ?f)] =>
rewrite (
Random_simpl n );
rewrite (
random_simpl n f )
| |-
context [(
fmont (
mu (
Mif ?b ?f ?g)) ?h)] =>
unfold Mif
| |-
context [(
fmont (
mu (
Mchoice ?p ?m1 ?m2)) ?f)] =>
rewrite (
Mchoice_simpl p m1 m2 f )
| |-
context [(
fmont (
mu (
im_distr ?f ?m)) ?h)] =>
rewrite (
im_distr_simpl f m h )
| |-
context [(
fmont (
mu (
prod_distr ?m1 ?m2)) ?h)] =>
unfold prod_distr
| |-
context [((
mon ?f (
fmonotonic :=?mf)) ?x)] =>
rewrite (
mon_simpl f (
mf :=mf)
x )
end .
Library Sets
Sets.v: Definition of sets as predicates over a type A
Section sets .
Variable A :
Type .
Variable decA :
forall x y :A, {
x =y}+{x<>y}.
Definition set :=
A ->Prop.
Definition full :
set :=
fun (
x :A) =>
True .
Definition empty :
set :=
fun (
x :A) =>
False .
Definition add (
a :A) (
P :set) :
set :=
fun (
x :A) =>
x =a \/ (
P x ).
Definition singl (
a :A) :set :=
fun (
x :A) =>
x =a.
Definition union (
P Q :set) :set :=
fun (
x :A) => (
P x ) \/ (
Q x ).
Definition compl (
P :set) :set :=
fun (
x :A) => ~P
x .
Definition inter (
P Q :set) :set :=
fun (
x :A) => (
P x ) /\ (
Q x ).
Definition rem (
a :A) (
P :set) :set :=
fun (
x :A) =>
x <>a /\ (
P x ).
Finite sets given as an enumeration of elements
Emptyness is decidable for finite sets
Properties of operations on sets
Lemma incl_empty :
forall P ,
incl empty P .
Lemma incl_empty_false :
forall P a ,
incl P empty -> ~
P a .
Lemma incl_add_empty :
forall (
a :A) (
P :set), ~
incl (
add a P )
empty .
Lemma eqset_empty_false :
forall P a ,
eqset P empty ->
P a ->
False .
Hint Immediate incl_empty_false eqset_empty_false incl_add_empty .
Lemma incl_rem_stable :
forall a P Q ,
incl P Q ->
incl (
rem a P ) (
rem a Q ).
Lemma incl_add_stable :
forall a P Q ,
incl P Q ->
incl (
add a P ) (
add a Q ).
Lemma incl_rem_add_iff :
forall a P Q ,
incl (
rem a P )
Q <->
incl P (
add a Q ).
Lemma incl_rem_add :
forall (
a :A) (
P Q :set),
(
P a ) ->
incl Q (
rem a P ) ->
incl (
add a Q )
P .
Lemma incl_add_rem :
forall (
a :A) (
P Q :set),
~
Q a ->
incl (
add a Q )
P ->
incl Q (
rem a P ) .
Hint Immediate incl_rem_add incl_add_rem .
Lemma eqset_rem_add :
forall (
a :A) (
P Q :set),
(
P a ) ->
eqset Q (
rem a P ) ->
eqset (
add a Q )
P .
Lemma eqset_add_rem :
forall (
a :A) (
P Q :set),
~
Q a ->
eqset (
add a Q )
P ->
eqset Q (
rem a P ).
Hint Immediate eqset_rem_add eqset_add_rem .
Lemma add_rem_eq_eqset :
forall x (
P :set),
eqset (
add x (
rem x P )) (
add x P ).
Lemma add_rem_diff_eqset :
forall x y (
P :set),
x <>y ->
eqset (
add x (
rem y P )) (
rem y (
add x P )).
Lemma add_eqset_in :
forall x (
P :set),
P x ->
eqset (
add x P )
P .
Hint Resolve add_rem_eq_eqset add_rem_diff_eqset add_eqset_in .
Lemma add_rem_eqset_in :
forall x (
P :set),
P x ->
eqset (
add x (
rem x P ))
P .
Hint Resolve add_rem_eqset_in .
Lemma rem_add_eq_eqset :
forall x (
P :set),
eqset (
rem x (
add x P )) (
rem x P ).
Lemma rem_add_diff_eqset :
forall x y (
P :set),
x <>y ->
eqset (
rem x (
add y P )) (
add y (
rem x P )).
Lemma rem_eqset_notin :
forall x (
P :set), ~P
x ->
eqset (
rem x P )
P .
Hint Resolve rem_add_eq_eqset rem_add_diff_eqset rem_eqset_notin .
Lemma rem_add_eqset_notin :
forall x (
P :set), ~P
x ->
eqset (
rem x (
add x P ))
P .
Hint Resolve rem_add_eqset_notin .
Lemma rem_not_in :
forall x (
P :set), ~
rem x P x .
Lemma add_in :
forall x (
P :set),
add x P x .
Lemma add_in_eq :
forall x y P ,
x =y ->
add x P y .
Lemma add_intro :
forall x (
P :set)
y ,
P y ->
add x P y .
Lemma add_incl :
forall x (
P :set),
incl P (
add x P ).
Lemma add_incl_intro :
forall x (
P Q :set), (
Q x ) -> (
incl P Q ) -> (
incl (
add x P )
Q ).
Lemma rem_incl :
forall x (
P :set),
incl (
rem x P )
P .
Hint Resolve rem_not_in add_in rem_incl add_incl .
Lemma union_sym :
forall P Q :
set ,
eqset (
union P Q ) (
union Q P ).
Lemma union_empty_left :
forall P :
set ,
eqset P (
union P empty ).
Lemma union_empty_right :
forall P :
set ,
eqset P (
union empty P ).
Lemma union_add_left :
forall (
a :A) (
P Q :
set ),
eqset (
add a (
union P Q )) (
union P (
add a Q )).
Lemma union_add_right :
forall (
a :A) (
P Q :
set ),
eqset (
add a (
union P Q )) (
union (
add a P )
Q ).
Hint Resolve union_sym union_empty_left union_empty_right union_add_left union_add_right .
Lemma union_incl_left :
forall P Q ,
incl P (
union P Q ).
Lemma union_incl_right :
forall P Q ,
incl Q (
union P Q ).
Lemma union_incl_intro :
forall P Q R ,
incl P R ->
incl Q R ->
incl (
union P Q )
R .
Hint Resolve union_incl_left union_incl_right union_incl_intro .
Lemma incl_union_stable :
forall P1 P2 Q1 Q2 ,
incl P1 P2 ->
incl Q1 Q2 ->
incl (
union P1 Q1 ) (
union P2 Q2 ).
Hint Immediate incl_union_stable .
Lemma inter_sym :
forall P Q :
set ,
eqset (
inter P Q ) (
inter Q P ).
Lemma inter_empty_left :
forall P :
set ,
eqset empty (
inter P empty ).
Lemma inter_empty_right :
forall P :
set ,
eqset empty (
inter empty P ).
Lemma inter_add_left_in :
forall (
a :A) (
P Q :
set ),
(
P a ) ->
eqset (
add a (
inter P Q )) (
inter P (
add a Q )).
Lemma inter_add_left_out :
forall (
a :A) (
P Q :
set ),
~
P a ->
eqset (
inter P Q ) (
inter P (
add a Q )).
Lemma inter_add_right_in :
forall (
a :A) (
P Q :
set ),
Q a ->
eqset (
add a (
inter P Q )) (
inter (
add a P )
Q ).
Lemma inter_add_right_out :
forall (
a :A) (
P Q :
set ),
~
Q a ->
eqset (
inter P Q ) (
inter (
add a P )
Q ).
Hint Resolve inter_sym inter_empty_left inter_empty_right inter_add_left_in inter_add_left_out inter_add_right_in inter_add_right_out .
Removing an element from a finite set
Selecting elements in a finite set
A set with size > 1 contains at least 2 different elements
Library Cover
Cover.v: Characteristic functions
Properties of zero_one functions
Definition zero_one (
A :Type)(f:MF
A ) :=
forall x ,
orc (
f x == 0) (
f x == 1).
Hint Unfold zero_one .
Lemma zero_one_not_one :
forall (
A :Type)(f:MF
A )
x ,
zero_one f -> ~ 1 <=
f x ->
f x == 0.
Lemma zero_one_not_zero :
forall (
A :Type)(f:MF
A )
x ,
zero_one f -> ~
f x <= 0 ->
f x == 1.
Hint Resolve zero_one_not_one zero_one_not_zero .
Definition fesp_zero_one :
forall (
A :Type)(f
g :MF
A ),
zero_one f ->
zero_one g ->
zero_one (
fesp f g ).
Save .
Lemma fesp_conj_zero_one :
forall (
A :Type)(f
g :MF
A ),
zero_one f ->
fesp f g ==
fconj f g .
Lemma fconj_zero_one :
forall (
A :Type)(f
g :MF
A ),
zero_one f ->
zero_one g ->
zero_one (
fconj f g ).
Lemma fplus_zero_one :
forall (
A :Type)(f
g :MF
A ),
zero_one f ->
zero_one g ->
zero_one (
fplus f g ).
Lemma finv_zero_one :
forall (
A :Type)(f :MF
A ),
zero_one f ->
zero_one (
finv f ).
Lemma fesp_zero_one_mult_left :
forall (
A :Type)(f:MF
A )(
p :U),
zero_one f ->
forall x ,
f x &
p ==
f x *
p .
Lemma fesp_zero_one_mult_right :
forall (
A :Type)(p:U)(f:MF
A ),
zero_one f ->
forall x ,
p &
f x ==
p *
f x .
Hint Resolve fesp_zero_one_mult_left fesp_zero_one_mult_right .
Definition cover (
A :Type)(P:set
A )(
f :MF
A ) :=
forall x , (
P x -> 1 <=
f x ) /\ (~
P x ->
f x <= 0).
Lemma cover_eq_one :
forall (
A :Type)(P:set
A )(
f :MF
A ) (
z :A),
cover P f ->
P z ->
f z == 1.
Lemma cover_eq_zero :
forall (
A :Type)(P:set
A )(
f :MF
A ) (
z :A),
cover P f -> ~
P z ->
f z == 0.
Lemma cover_orc_0_1 :
forall (
A :Type)(P:set
A )(
f :MF
A ),
cover P f ->
forall x ,
orc (
f x == 0) (
f x == 1).
Lemma cover_zero_one :
forall (
A :Type)(P:set
A )(
f :MF
A ),
cover P f ->
zero_one f .
Lemma zero_one_cover :
forall (
A :Type)(f:MF
A ),
zero_one f ->
cover (
fun x => 1 <=
f x )
f .
Lemma cover_esp_mult_left :
forall (
A :Type)(P:set
A )(
f :MF
A )(
p :U),
cover P f ->
forall x ,
f x &
p ==
f x *
p .
Lemma cover_esp_mult_right :
forall (
A :Type)(P:set
A )(
p :U)(f:MF
A ),
cover P f ->
forall x ,
p &
f x ==
p *
f x .
Hint Immediate cover_esp_mult_left cover_esp_mult_right .
Lemma cover_elim :
forall (
A :Type)(P:set
A )(
f :MF
A ),
cover P f ->
forall x ,
orc (~P
x /\
f x == 0) (
P x /\
f x == 1).
Lemma cover_eq_one_elim_class :
forall (
A :Type)(P
Q :set
A )(
f :MF
A ),
cover P f ->
forall z ,
f z == 1 ->
class (
Q z ) ->
incl P Q ->
Q z .
Lemma cover_eq_one_elim :
forall (
A :Type)(P:set
A )(
f :MF
A ),
cover P f ->
forall z ,
f z == 1 -> ~ ~
P z .
Lemma cover_eq_zero_elim :
forall (
A :Type)(P:set
A )(
f :MF
A ) (
z :A),
cover P f ->
f z == 0 -> ~
P z .
Lemma cover_unit :
forall (
A :Type)(P:set
A )(
f :MF
A )(
a :A),
cover P f ->
P a -> 1 <=
mu (
Munit a )
f .
Lemma cover_let :
forall (
A B :Type)(m1:
distr A )(
m2 :
A ->distr
B ) (
P :set
A )(
cP :MF
A )(
f :MF
B )(
p :U),
cover P cP -> 1 <=
mu m1 cP -> (
forall x :A,
P x ->
p <=
mu (
m2 x )
f ) ->
p <=
mu (
Mlet m1 m2 )
f .
Lemma cover_incl_fle :
forall (
A :Type)(P
Q :set
A )(
f g :MF
A ),
cover P f ->
cover Q g ->
incl P Q ->
f <=
g .
Lemma cover_incl_eq :
forall (
A :Type)(P:set
A )(
f g :MF
A ),
cover P f ->
cover P g ->
f ==
g .
Lemma cover_eqset_stable :
forall (
A :Type)(P
Q :set
A )(
EQ :eqset
P Q )(
f :MF
A ),
cover P f ->
cover Q f .
Lemma cover_eq_stable :
forall (
A :Type)(P:set
A )(
f g :MF
A ),
cover P f ->
f ==
g ->
cover P g .
Lemma cover_eqset_eq_stable :
forall (
A :Type)(P
Q :set
A )(
f g :MF
A ),
cover P f ->
eqset P Q ->
f ==
g ->
cover Q g .
Add Parametric Morphism (
A :Type) : (
cover (
A :=A))
with signature eqset (
A :=A) &
Oeq &
iff as cover_eqset_compat .
Save .
Lemma cover_union :
forall (
A :Type)(P
Q :set
A )(
f g :
MF A ),
cover P f ->
cover Q g ->
cover (
union P Q ) (
fplus f g ).
Lemma cover_inter_esp :
forall (
A :Type)(P
Q :set
A )(
f g :
MF A ),
cover P f ->
cover Q g ->
cover (
inter P Q ) (
fesp f g ).
Lemma cover_inter_mult :
forall (
A :Type)(P
Q :set
A )(
f g :
MF A ),
cover P f ->
cover Q g ->
cover (
inter P Q ) (
fun x =>
f x *
g x ).
Lemma cover_compl :
forall (
A :Type)(P:set
A )(
f :MF
A ),
cover P f ->
cover (
compl P ) (
finv f ).
Lemma cover_empty :
forall (
A :Type),
cover (
empty A ) (
fzero A ).
Caracteristic functions for decidable predicates
Definition carac (
A :Type)(P:set
A )(
Pdec :
dec P ) :
MF A
:=
fun z =>
if Pdec z then 1
else 0.
Lemma carac_incl :
forall (
A :Type)(P
Q :A ->
Prop )(
Pdec :
dec P )(
Qdec :
dec Q ),
incl P Q ->
carac Pdec <=
carac Qdec .
Lemma carac_monotonic :
forall (
A B :Type)(P:A ->
Prop )(
Q :B->Prop)(Pdec:
dec P )(
Qdec :
dec Q )
x y ,
(
P x ->
Q y ) ->
carac Pdec x <=
carac Qdec y .
Hint Resolve carac_monotonic .
Lemma carac_eq_compat :
forall (
A B :Type)(P:A ->
Prop )(
Q :B->Prop)(Pdec:
dec P )(
Qdec :
dec Q )
x y ,
(
P x <->
Q y ) ->
carac Pdec x ==
carac Qdec y .
Hint Resolve carac_eq_compat .
Lemma carac_one :
forall (
A :Type)(P:A ->
Prop )(
Pdec :dec
P )(
z :A),
P z ->
carac Pdec z == 1.
Lemma carac_zero :
forall (
A :Type)(P:A ->
Prop )(
Pdec :dec
P )(
z :A),
~
P z ->
carac Pdec z == 0.
Lemma cover_dec :
forall (
A :Type)(P:set
A )(
Pdec :
dec P ),
cover P (
carac Pdec ).
Lemma cover_mult_fun :
forall (
A :Type)(P:set
A )(
cP :
MF A )(
f g :A->U),
(
forall x ,
P x ->
f x ==
g x ) ->
cover P cP ->
forall x ,
cP x *
f x ==
cP x *
g x .
Lemma cover_esp_fun :
forall (
A :Type)(P:set
A )(
cP :
MF A )(
f g :A->U),
(
forall x ,
P x ->
f x ==
g x ) ->
cover P cP ->
forall x ,
cP x &
f x ==
cP x &
g x .
Lemma cover_esp_fun_le :
forall (
A :Type)(P:set
A )(
cP :
MF A )(
f g :A->U),
(
forall x ,
P x ->
f x <=
g x ) ->
cover P cP ->
forall x ,
cP x &
f x <=
cP x &
g x .
Hint Resolve cover_esp_fun_le .
Lemma cover_ok :
forall (
A :Type)(P
Q :set
A )(
f g :
MF A ),
(
forall x ,
P x -> ~
Q x ) ->
cover P f ->
cover Q g ->
fplusok f g .
Hint Resolve cover_ok .
Assuming m is a distribution under assumption P and cP is 0 or 1, builds
a distribution which is m if cP is 1 and 0 otherwise
Definition Mrestr A (
cp :U) (
m :M
A ) :
M A :=
UMult cp @
m .
Lemma Mrestr_simpl :
forall A cp (
m :M
A )
f ,
Mrestr cp m f =
cp * (
m f ).
Lemma Mrestr0 :
forall A cP (
m :M
A ),
cP <= 0 ->
forall f ,
Mrestr cP m f == 0.
Lemma Mrestr1 :
forall A cP (
m :M
A ), 1 <=
cP ->
forall f ,
Mrestr cP m f ==
m f .
Definition distr_restr :
forall A (
P :Prop) (
cp :U) (
m :M
A ),
((
P -> 1 <=
cp ) /\ (~
P ->
cp <= 0)) -> (
P ->
stable_inv m ) ->
(
P ->
stable_plus m ) -> (
P ->
stable_mult m ) -> (
P ->
continuous m )
->
distr A .
Defined .
Lemma distr_restr_simpl :
forall A (
P :Prop) (
cp :U) (
m :M
A )
(
Hp : (
P -> 1 <=
cp ) /\ (~
P ->
cp <= 0)) (
Hinv :P ->
stable_inv m )
(
Hplus :P ->
stable_plus m )(
Hmult :P ->
stable_mult m )(
Hcont :P ->
continuous m )
f ,
mu (
distr_restr cp Hp Hinv Hplus Hmult Hcont )
f =
cp *
m f .
Modular reasoning on programs
Lemma range_cover :
forall A (
P :A ->
Prop )
d cP ,
range P d ->
cover P cP ->
forall f ,
mu d f ==
mu d (
fun x =>
cP x *
f x ).
Lemma mu_cut :
forall (
A :Type)(m:distr
A )(
P :set
A )(
cP f g :MF
A ),
cover P cP -> (
forall x ,
P x ->
f x ==
g x ) -> 1<=mu
m cP ->
mu m f ==
mu m g .
Uniform measure on finite sets
Section SigmaFinite .Variable A :Type.Variable decA : forall x y :A, {x =y}+{~x=y}.Section RandomFinite .
Distribution for random_fin P
The distribution associated to
random_fin P is
with the size of
we cannot factorize because of possible overflow
Fixpoint sigma_fin (
f :A->U)(P:A->Prop)(FP:finite
P ){
struct FP }:U :=
match FP with
| (
fin_eq_empty eq ) => 0
| (
fin_eq_add x Q nQx FQ eq ) => (
f x ) +
sigma_fin f FQ
end .
Definition retract_fin (
P :A->Prop) (
f :A->U) :=
forall Q (
FQ :
finite Q ),
incl Q P ->
forall x , ~(Q
x ) -> (
P x ) ->
f x <= [1-](
sigma_fin f FQ ).
Lemma retract_fin_inv :
forall (
P :A->Prop) (
f :A->U),
retract_fin P f ->
forall Q (
FQ :
finite Q ),
incl Q P ->
forall x , ~(Q
x ) -> (
P x ) ->
sigma_fin f FQ <=[1-]
f x .
Hint Immediate retract_fin_inv .
Lemma retract_fin_incl :
forall P Q f ,
retract_fin P f ->
incl Q P ->
retract_fin Q f .
Lemma sigma_fin_monotonic :
forall (
f g :
A ->
U )(
P :A->Prop)(FP:finite
P ),
(
forall x ,
P x -> (
f x )<=(g
x ))->
sigma_fin f FP <=
sigma_fin g FP .
Lemma sigma_fin_eq_compat :
forall (
f g :
A ->
U )(
P :A->Prop)(FP:finite
P ),
(
forall x ,
P x -> (
f x )==(g
x ))->
sigma_fin f FP ==
sigma_fin g FP .
Lemma retract_fin_le :
forall (
P :A->Prop) (
f g :A->U),
(
forall x ,
P x ->
f x <=
g x ) ->
retract_fin P g ->
retract_fin P f .
Lemma sigma_fin_mult :
forall (
f :
A ->
U )
c (
P :A->Prop)(FP:finite
P ),
retract_fin P f ->
sigma_fin (
fun k =>
c *
f k )
FP ==
c *
sigma_fin f FP .
Lemma sigma_fin_plus :
forall (
f g :
A ->
U ) (
P :A->Prop)(FP:finite
P ),
sigma_fin (
fun k =>
f k +
g k )
FP ==
sigma_fin f FP +
sigma_fin g FP .
Lemma sigma_fin_prod_maj :
forall (
f g :
A ->
U )(
P :A->Prop)(FP:finite
P ),
sigma_fin (
fun k =>
f k *
g k )
FP <=
sigma_fin f FP .
Lemma sigma_fin_prod_le :
forall (
f g :
A ->
U ) (
c :U) , (
forall k ,
f k <=
c ) ->
forall (
P :A->Prop)(FP:finite
P ),
retract_fin P g ->
sigma_fin (
fun k =>
f k *
g k )
FP <=
c *
sigma_fin g FP .
Lemma sigma_fin_prod_ge :
forall (
f g :
A ->
U ) (
c :U) , (
forall k ,
c <=
f k ) ->
forall (
P :A->Prop)(FP:finite
P ),
retract_fin P g ->
c *
sigma_fin g FP <=
sigma_fin (
fun k =>
f k *
g k )
FP .
Hint Resolve sigma_fin_prod_maj sigma_fin_prod_ge sigma_fin_prod_le .
Lemma sigma_fin_inv :
forall (
f g :
A ->
U )(
P :A->Prop)(FP:finite
P ),
retract_fin P f ->
[1-]
sigma_fin (
fun k =>
f k *
g k )
FP ==
sigma_fin (
fun k =>
f k * [1-]
g k )
FP + [1-]
sigma_fin f FP .
Lemma sigma_fin_eqset :
forall f P Q (
FP :finite
P ) (
e :eqset
P Q ),
(
sigma_fin f (
fin_eqset e FP )) = (
sigma_fin f FP ).
Lemma sigma_fin_rem :
forall f P (
FP :finite
P )
a ,
P a ->
sigma_fin f FP ==
f a +
sigma_fin f (
finite_rem decA a FP ).
Lemma sigma_fin_incl :
forall f P (
FP :finite
P )
Q (
FQ :finite
Q ),
(
incl P Q ) ->
sigma_fin f FP <=
sigma_fin f FQ .
Lemma sigma_fin_unique :
forall f P Q (
FP :finite
P ) (
FQ :finite
Q ), (
eqset P Q ) ->
sigma_fin f FP ==
sigma_fin f FQ .
Lemma sigma_fin_cte :
forall c P (
FP :finite
P ),
sigma_fin (
fun _ =>
c )
FP == (
size FP ) */
c .
End RandomFinite .
End SigmaFinite .
Properties of the Random distribution
Definition le_dec (
n :nat) :
dec (
fun x => (
x <=
n )%nat).
Defined .
Definition lt_dec (
n :nat) :
dec (
fun x => (
x <
n )%nat).
Defined .
Definition gt_dec :
forall x ,
dec (
lt x ).
Defined .
Definition ge_dec :
forall x ,
dec (
le x ).
Defined .
Definition carac_eq n :=
carac (
eq_nat_dec n ).
Definition carac_le n :=
carac (
le_dec n ).
Definition carac_lt n :=
carac (
lt_dec n ).
Definition carac_gt n :=
carac (
gt_dec n ).
Definition carac_ge n :=
carac (
ge_dec n ).
Definition is_eq (
n :nat) :
cover (
fun x =>
n =
x ) (
carac_eq n ) :=
cover_dec (
eq_nat_dec n ).
Definition is_le (
n :nat) :
cover (
fun x => (
x <=n)%nat) (
carac_le n ) :=
cover_dec (
le_dec n ).
Definition is_lt (
n :nat) :
cover (
fun x => (
x <
n )%nat) (
carac_lt n ) :=
cover_dec (
lt_dec n ).
Definition is_gt (
n :nat) :
cover (
fun x => (
n <
x )%nat) (
carac_gt n ):=
cover_dec (
gt_dec n ).
Definition is_ge (
n :nat) :
cover (
fun x => (
n <=
x )%nat) (
carac_ge n ) :=
cover_dec (
ge_dec n ).
Lemma carac_gt_S :
forall x y ,
carac_gt (
S y ) (
S x ) ==
carac_gt y x .
Lemma carac_lt_S :
forall x y ,
carac_lt (
S x ) (
S y ) ==
carac_lt x y .
Lemma carac_le_S :
forall x y ,
carac_le (
S x ) (
S y ) ==
carac_le x y .
Lemma carac_ge_S :
forall x y ,
carac_ge (
S x ) (
S y ) ==
carac_ge x y .
Lemma carac_eq_S :
forall x y ,
carac_eq (
S x ) (
S y ) ==
carac_eq x y .
Hint Resolve carac_le_S carac_eq_S carac_lt_S carac_ge_S carac_gt_S .
count the number of elements between 0 and n-1 which satisfy P
Fixpoint nb_elts (
P :nat ->
Prop )(
Pdec :
dec P )(
n :nat) {
struct n } :
nat :=
match n with
0 => 0%nat
|
S n =>
if Pdec n then (
S (
nb_elts Pdec n ))
else (
nb_elts Pdec n )
end .
Lemma nb_elts_true :
forall (
P :nat ->
Prop )(
Pdec :
dec P )(
n :nat),
(
forall k , (
k <
n )%nat ->
P k ) ->
nb_elts Pdec n =n.
Hint Resolve nb_elts_true .
the probability for a random number between 0 and n to satisfy P is equal
to the number of elements below n which satisfy P divided by n+1
Lemma Random_carac : forall (P :nat -> Prop )(Pdec : dec P )(n :nat),mu (Random n ) (carac Pdec ) == (nb_elts Pdec (S n )) */ [1/]1+n.Lemma nb_elts_lt_le : forall k n , (k <= n )%nat -> nb_elts (lt_dec k ) n = k .Lemma nb_elts_lt_ge : forall k n , (n <= k )%nat -> nb_elts (lt_dec k ) n = n .Hint Resolve nb_elts_lt_ge nb_elts_lt_le .Lemma Random_lt : forall n k , mu (Random n ) (carac_lt k ) == k */ [1/]1+n.Hint Resolve Random_lt .Lemma Random_le : forall n k , mu (Random n ) (carac_le k ) == (S k ) */ [1/]1+n.Hint Resolve Random_le .
Library IterFlip
IterFlip.v: An example of probabilistic termination
Definition of a random walk
We interpret the probabilistic program
let rec iter x = if flip() then iter (x+1) else x
Require Import ZArith .
Instance Fiter_mon :
monotonic (
fun (
f :Z ->
distr Z ) (
x :Z) =>
Mif Flip (
f (
Zsucc x )) (
Munit x )).
Save .
Definition Fiter : (
Z ->
distr Z ) -m> (
Z ->
distr Z )
:=
mon (
fun f (
x :Z) =>
Mif Flip (
f (
Zsucc x )) (
Munit x )).
Lemma Fiter_simpl :
forall f x ,
Fiter f x =
Mif Flip (
f (
Zsucc x )) (
Munit x ).
Definition iterflip :
Z ->
distr Z :=
Mfix Fiter .
Main result
Probability for
iter to terminate is
Auxiliary function
Definition
Fixpoint p_ (
n :
nat ) :
U :=
match n with O => 0 | (
S n ) => [1/2] *
p_ n + [1/2]
end .
Lemma p_incr :
forall n ,
p_ n <=
p_ (
S n ).
Hint Resolve p_incr .
Definition p :
nat -m>
U :=
fnatO_intro p_ p_incr .
Lemma pS_simpl :
forall n ,
p (
S n ) = [1/2] *
p n + [1/2].
Lemma p_eq :
forall n :nat,
p n == [1-]([1/2]^n).
Hint Resolve p_eq .
Lemma p_le :
forall n :nat, [1-]([1/]1+n) <=
p n .
Hint Resolve p_le .
Lemma lim_p_one : 1 <=
lub p .
Hint Resolve lim_p_one .
Proof of probabilistic termination
Definition q1 (
z1 z2 :Z) := 1.
Lemma iterflip_term :
okfun (
fun k => 1)
iterflip q1 .
Library Choice
Choice.v: An example of probabilistic choice
Definition of a probabilistic choice
We interpret the probabilistic program which executes
two probabilistic programs and and then make a choice
between the two computed results.
let rec p () = let x = p1 () in let y = p2 () in choice x y
Section CHOICE .
Variable A :
Type .
Variables p1 p2 :
distr A .
Variable choice :
A ->
A ->
A .
Definition p :
distr A :=
Mlet p1 (
fun x =>
Mlet p2 (
fun y =>
Munit (
choice x y ))).
Main result
We estimate the probability for to satisfy given estimations for
both and .
Assumptions
We need extra properties on , and .
and terminate with probability
value on is not less than the sum of values of on separate elements.
If is a boolean function it means than if one of or satisfies
then will also satisfy
Hypothesis p1_terminates : (
mu p1 (
fone A ))==1.
Hypothesis p2_terminates : (
mu p2 (
fone A ))==1.
Variable Q :
MF A .
Hypothesis choiceok :
forall x y ,
Q x +
Q y <=
Q (
choice x y ).
Lemma choicerule : forall k1 k2 , k1 <= mu p1 Q -> k2 <= mu p2 Q -> (k1 * ([1-] k2 ) + k2 ) <= mu p Q .End CHOICE .
Library Bernoulli
Bernoulli.v: Simulating Bernoulli and Binomial distributions
Require Export Cover .
Require Export Misc .
Program for computing a Bernoulli distribution
bernoulli p gives true with probability p
and false with probability (1-p)
let rec bernoulli x =
if flip
then (if x < 1/2 then false else bernoulli (2 p - 1))
else (if x < 1/2 then bernoulli (2 p) else true)
Hypothesis dec_demi :
forall x :
U , {
x < [1/2]}+{[1/2] <=
x }.
Instance Fbern_mon :
monotonic
(
fun (
f :U ->
distr bool )
p =>
Mif Flip
(
if dec_demi p then Munit false else f (
p &
p ))
(
if dec_demi p then f (
p +
p )
else Munit true )).
Save .
Definition Fbern : (
U ->
distr bool ) -m> (
U ->
distr bool )
:=
mon (
fun f p =>
Mif Flip
(
if dec_demi p then Munit false else f (
p &
p ))
(
if dec_demi p then f (
p +
p )
else Munit true )).
Lemma Fbern_simpl :
forall f p ,
Fbern f p =
Mif Flip
(
if dec_demi p then Munit false else f (
p &
p ))
(
if dec_demi p then f (
p +
p )
else Munit true ).
Definition bernoulli :
U ->
distr bool :=
Mfix Fbern .
Properties of the Bernoulli program
Proofs using fixpoint rules
Definition Mubern (
q :
bool ->
U ) :
MF U -m>
MF U .
Defined .
Lemma Mubern_simpl :
forall (
q :
bool ->
U )
f p ,
Mubern q f p =
if dec_demi p then [1/2]*(q
false )+[1/2]*(f (
p +p))
else [1/2]*(f (
p &p)) + [1/2]*(q
true ).
Lemma Mubern_eq :
forall (
q :
bool ->
U ) (
f :U ->
distr bool ) (
p :U),
mu (
Fbern f p )
q ==
Mubern q (
fun y =>
mu (
f y )
q )
p .
Hint Resolve Mubern_eq .
Lemma Bern_eq :
forall q :
bool ->
U ,
forall p ,
mu (
bernoulli p )
q ==
mufix (
Mubern q )
p .
Hint Resolve Bern_eq .
Lemma Bern_commute :
forall q :
bool ->
U ,
mu_muF_commute_le Fbern (
fun (
x :U)=>q) (
Mubern q ).
Hint Resolve Bern_commute .
Lemma Bern_term :
forall p ,
mu (
bernoulli p ) (
fone bool ) == 1.
Hint Resolve Bern_term .
p is an invariant of Mubern qtrue
Lemma MuBern_true :
forall p ,
Mubern B2U (
fun q =>
q )
p ==
p .
Hint Resolve MuBern_true .
Lemma MuBern_false :
forall p ,
Mubern (
finv B2U ) (
finv (
fun q =>
q ))
p == [1-]
p .
Hint Resolve MuBern_false .
Lemma Bern_true :
forall p ,
mu (
bernoulli p )
B2U ==
p .
Lemma Bern_false :
forall p ,
mu (
bernoulli p )
NB2U == [1-]
p .
Lemma Mubern_inv :
forall (
q :
bool ->
U ) (
f :U ->
U ) (
p :U),
Mubern (
finv q ) (
finv f )
p == [1-]
Mubern q f p .
Invariant pmin p
Property :
Definition qtrue (p :U) := B2U .Definition qfalse (p :U) := NB2U .Lemma bernoulli_true : okfun (fun p => p ) bernoulli qtrue .
Property :
Lemma bernoulli_false : okfun (fun p => [1-] p ) bernoulli qfalse .
Probability for the result of to be true is exactly
Lemma qtrue_qfalse_inv :
forall (
b :bool) (
x :U),
qtrue x b == [1-] (
qfalse x b ).
Lemma bernoulli_eq_true :
forall p ,
mu (
bernoulli p ) (
qtrue p ) ==
p .
Lemma bernoulli_eq_false :
forall p ,
mu (
bernoulli p ) (
qfalse p )== [1-]
p .
Lemma bernoulli_eq :
forall p f ,
mu (
bernoulli p )
f ==
p *
f true + ([1-]
p ) *
f false .
Lemma bernoulli_total :
forall p ,
mu (
bernoulli p ) (
fone bool )==1.
gives with probability
Definition and properties of binomial coefficients
Fixpoint comb (
k n :nat) {
struct n } :
nat :=
match n with O =>
match k with O => (1%nat) | (
S l ) =>
O end
| (
S m ) =>
match k with O => (1%nat)
| (
S l ) => ((
comb l m ) + (
comb k m ))%nat
end
end .
Lemma comb_0_n :
forall n ,
comb 0
n = 1%nat.
Lemma comb_not_le :
forall n k ,
le (
S n )
k ->
comb k n =0%nat.
Lemma comb_Sn_n :
forall n ,
comb (
S n )
n =0%nat.
Lemma comb_n_n :
forall n ,
comb n n = (1%nat).
Lemma comb_1_Sn :
forall n ,
comb 1 (
S n ) = (
S n ).
Lemma comb_inv :
forall n k , (
k <=n)%nat ->
comb k n =
comb (
n -k)
n .
Lemma comb_n_Sn :
forall n ,
comb n (
S n ) = (
S n ).
Definition fc (
p :U)(n
k :nat) := (
comb k n ) */ (
p ^k * ([1-]
p )^(n-k)).
Lemma fcp_0 :
forall p n ,
fc p n O == ([1-]
p )^n.
Lemma fcp_n :
forall p n ,
fc p n n ==
p ^n.
Lemma fcp_not_le :
forall p n k , (
S n <=
k )%nat ->
fc p n k == 0.
Lemma fc0 :
forall n k ,
fc 0
n (
S k ) == 0.
Hint Resolve fc0 .
Add Morphism fc with signature Oeq &
eq &
eq &
Oeq
as fc_eq_compat .
Save .
Hint Resolve fc_eq_compat .
Lemma sigma_fc0 :
forall n k ,
sigma (
fc 0
n ) (
S k ) ==1.
Lemma retract_class :
forall f n ,
class (
retract f n ).
Hint Resolve retract_class .
Lemma fc_retract :
forall p n , ([1-]
p ^n ==
sigma (
fc p n )
n ) ->
retract (
fc p n ) (
S n ).
Hint Resolve fc_retract .
Lemma fc_Nmult_def :
forall p n k , ([1-]
p ^n ==
sigma (
fc p n )
n ) ->
Nmult_def (
comb k n ) (
p ^k * ([1-]
p ) ^(n-k)).
Hint Resolve fc_Nmult_def .
Lemma fcp_S :
forall p n k , ([1-]
p ^n ==
sigma (
fc p n )
n ) ->
fc p (
S n ) (
S k ) ==
p * (
fc p n k ) + ([1-]
p ) * (
fc p n (
S k )).
Lemma sigma_fc_1 :
forall p n , ([1-]
p ^n ==
sigma (
fc p n )
n ) ->1==sigma (
fc p n ) (
S n ).
Hint Resolve sigma_fc_1 .
Lemma Uinv_exp :
forall p n , [1-](
p ^n)==sigma (
fc p n )
n .
Hint Resolve Uinv_exp .
Lemma Nmult_comb :
forall p n k ,
Nmult_def (
comb k n ) (
p ^
k * ([1-]
p ) ^ (
n -
k )).
Hint Resolve Nmult_comb .
Definition of binomial distribution
Fixpoint binomial (p :U)(n:nat) {struct n }: distr nat :=match n with O => (Munit O )S m => Mlet (binomial p m )fun x => Mif (bernoulli p ) (Munit (S x )) (Munit x ))end .
Properties of binomial distribution
Lemma binomial_eq_k :forall p n k , mu (binomial p n ) (carac_eq k ) == fc p n k .Lemma binomial_le_n :forall p n , 1 <= mu (binomial p n ) (carac_le n ).Lemma binomial_le_S : forall p n k , mu (binomial p (S n )) (carac_le (S k )) ==p * (mu (binomial p n ) (carac_le k )) + ([1-]p ) * (mu (binomial p n ) (carac_le (S k ))).Lemma binomial_lt_S : forall p n k , mu (binomial p (S n )) (carac_lt (S k )) ==p * (mu (binomial p n ) (carac_lt k )) + ([1-]p ) * (mu (binomial p n ) (carac_lt (S k ))).
Library Ycart
Ycart.v: An exemple of partial termination
Section YcartExample .
Program giving an example of partiality
Given a function K : nat ->nat, N:nat
let rec ycart p x = if random p < K x then x else ycart (x+1)
The probability of termination is
Variable K :
nat ->
nat .
Variable N :
nat .
Instance FYcart_mon :
monotonic
(
fun (
f :nat ->
distr nat )
n =>
Mlet (
Random N ) (
fun x =>
if lt_dec (
K n )
x then Munit n else f (
S n ))).
Save .
Definition FYcart : (
nat ->
distr nat ) -m> (
nat ->
distr nat )
:=
mon (
fun (
f :nat ->
distr nat )
n =>
Mlet (
Random N ) (
fun x =>
if lt_dec (
K n )
x then Munit n else f (
S n ))).
Lemma FYcart_simpl :
forall f n ,
FYcart f n =
Mlet (
Random N ) (
fun x =>
if lt_dec (
K n )
x then Munit n else f (
S n )).
Definition Ycart :
nat ->
distr nat :=
Mfix FYcart .
Definition F x := (
K x ) */ [1/]1+N.
Lemma FYcart_val :
forall (
q :
MF nat )
f x ,
mu (
FYcart f x )
q ==
F x *
q x + [1-](
F x ) *
mu (
f (
S x ))
q .
Definition P (
x :
nat ) :
nat -m->
U :=
Prod (
fun i => [1-]
F (
x +i)).
Definition p (
x :nat) :
nat -m>
U :=
sigma (
fun k =>
F (
x +k) *
P x k ).
Lemma P_prod :
forall x k ,
F (
x +k) *
P x k ==
P x k -
P x (
S k ).
Hint Resolve P_prod .
Lemma p_diff :
forall x n , (
p x n :
U ) == [1-]
P x n .
Hint Resolve p_diff .
Lemma p_lub :
forall x ,
lub (
p x ) == [1-]
prod_inf (
fun i => [1-]
F (
x +i)).
Hint Resolve p_lub .
Lemma p_equation :
forall x n ,
p x (
S n ) ==
F x + [1-](
F x ) *
p (
S x )
n .
Hint Resolve p_equation .
Lemma Ycart_term1 :
forall x ,
mu (
Ycart x ) (
fone nat ) == [1-]
prod_inf (
fun i => [1-]
F (
x +i)).
A shorter proof using mu (Ycart x) (f_one nat) = mu h. muYcart h x
Instance Ycart_one_mon
:
monotonic (
fun (
p :nat ->
U ) (
y :nat) =>
F y + [1-](
F y ) *
p (
S y )).
Save .
Definition Ycart_one : (
nat ->
U ) -m> (
nat ->
U ) :=
mon (
fun (
p :nat->U) (
y :nat) =>
F y + [1-](
F y ) *
p (
S y )).
Lemma Ycart_one_simpl :
forall p y ,
Ycart_one p y =
F y + [1-](
F y ) *
p (
S y ).
Lemma Ycart_term2 :
forall x ,
mu (
Ycart x ) (
fone nat ) == [1-]
prod_inf (
fun i => [1-]
F (
x +i)).
Instance FYcart_lt_mon :
monotonic (
fun (
p :
nat ->
U ) (
y :nat) => [1-](
F y ) *
p (
S y )).
Save .
Definition FYcart_lt : (
nat ->
U ) -m> (
nat ->
U ) :=
mon (
fun (
p :nat->U) (
y :nat) => [1-](
F y ) *
p (
S y )).
Lemma FYcart_lt_simpl :
forall (
p :nat ->
U ) (
y :nat),
FYcart_lt p y = [1-](
F y ) *
p (
S y ).
Lemma Ycart_ltx :
forall x ,
mu (
Ycart x ) (
carac_lt x ) <= 0.
Lemma Ycart_eqx :
forall x ,
mu (
Ycart x ) (
carac (
eq_nat_dec x )) ==
F x .
End YcartExample .
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H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
This page has been generated by
coqdoc
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Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_section_R.html 0000644 0001750 0001750 00000026065 11301735615 015707 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
R (section)
Rules.Fixrule [in
Prog ]
Rules.Fixrule.LoopRule [in
Prog ]
Rules.Fixrule.Ruleseq [in
Prog ]
Rules.Fixrule.TransformFix [in
Prog ]
Rules.Fixrule.TransformFix.Fix_muF [in
Prog ]
Rules.Fixrule.TransformFix.Fix_muF.F_muF_results [in
Prog ]
Rules.Fixrule.TransformFix.Fix_muF_Term [in
Prog ]
Rules.Fixrule.TransformFix.Fix_Term [in
Prog ]
Rules.IFixrule [in
Prog ]
Rules.IFixrule.IRuleseq [in
Prog ]
Rules.IFixrule.ITransformFix [in
Prog ]
Rules.IFixrule.ITransformFix.IFix_muF [in
Prog ]
Rules.IFixrule.ITransformFix.IFix_muF.IF_muF_results [in
Prog ]
Rules.Wellfounded [in
Prog ]
html/index_lemma_R.html 0000644 0001750 0001750 00000071066 11301735615 015337 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
R (lemma)
rem_add_diff_eqset [in
Sets ]
rem_add_eqset_notin [in
Sets ]
rem_add_eq_eqset [in
Sets ]
rem_eqset_notin [in
Sets ]
rem_incl [in
Sets ]
rem_not_in [in
Sets ]
Rules.Acc_iter_distr [in
Prog ]
Rules.admissible_le_fix [in
Prog ]
Rules.admissible_true [in
Prog ]
Rules.apply_rule [in
Prog ]
Rules.Arg_swap_simpl [in
Prog ]
Rules.combiok [in
Prog ]
Rules.discrete_commute [in
Prog ]
Rules.fixrule [in
Prog ]
Rules.fixrule_Ulub [in
Prog ]
Rules.fixrule_up_lub [in
Prog ]
Rules.fixrule_up_Ulub [in
Prog ]
Rules.Flip_cfalse [in
Prog ]
Rules.Flip_ctrue [in
Prog ]
Rules.Flip_eq [in
Prog ]
Rules.fmult_fplusok [in
Prog ]
Rules.Iapply_rule [in
Prog ]
Rules.Ifixrule [in
Prog ]
Rules.IFlip_eq [in
Prog ]
Rules.ifok [in
Prog ]
Rules.ifrule [in
Prog ]
Rules.Iifrule [in
Prog ]
Rules.Iincl_fix_ifix [in
Prog ]
Rules.Iin_mu_Imu [in
Prog ]
Rules.Iin_mu_Iok [in
Prog ]
Rules.Ilambda_rule [in
Prog ]
Rules.Ilet_eq [in
Prog ]
Rules.ImuF_stable [in
Prog ]
Rules.Imu_inf [in
Prog ]
Rules.Imu_Mif_eq [in
Prog ]
Rules.Imu_monotonic [in
Prog ]
Rules.Imu_singl [in
Prog ]
Rules.Imu_stable_eq [in
Prog ]
Rules.Imu_sup [in
Prog ]
Rules.im_distr_comp [in
Prog ]
Rules.im_distr_id [in
Prog ]
Rules.im_distr_simpl [in
Prog ]
Rules.Iokfun_le_compat [in
Prog ]
Rules.Iok_le_compat [in
Prog ]
Rules.Iunit_eq [in
Prog ]
Rules.lambda_rule [in
Prog ]
Rules.Loop_eq [in
Prog ]
Rules.low_Imu [in
Prog ]
Rules.Mchoice_continuous [in
Prog ]
Rules.Mchoice_continuous_left [in
Prog ]
Rules.Mchoice_continuous_right [in
Prog ]
Rules.Mchoice_le_compat [in
Prog ]
Rules.Mchoice_simpl [in
Prog ]
Rules.MChoice_simpl [in
Prog ]
Rules.Mchoice_sym [in
Prog ]
Rules.Mchoice_sym_le [in
Prog ]
Rules.Mfixvar [in
Prog ]
Rules.Mfixvar_up [in
Prog ]
Rules.Mif_eq [in
Prog ]
Rules.Mif_le_compat [in
Prog ]
Rules.mufix_mu [in
Prog ]
Rules.mufix_mu_le [in
Prog ]
Rules.muF_eq [in
Prog ]
Rules.muF_le [in
Prog ]
Rules.muF_le_term [in
Prog ]
Rules.muF_le_term_minus [in
Prog ]
Rules.muF_pterm [in
Prog ]
Rules.muF_stable [in
Prog ]
Rules.mu_mufix_le [in
Prog ]
Rules.okfun_le_compat [in
Prog ]
Rules.okup_apply_rule [in
Prog ]
Rules.okup_eq_compat [in
Prog ]
Rules.okup_fixrule_glb [in
Prog ]
Rules.okup_fixrule_inv [in
Prog ]
Rules.okup_Flip [in
Prog ]
Rules.okup_ifrule [in
Prog ]
Rules.okup_lambda_rule [in
Prog ]
Rules.okup_le_compat [in
Prog ]
Rules.okup_mult [in
Prog ]
Rules.ok_eq_compat [in
Prog ]
Rules.ok_Flip [in
Prog ]
Rules.ok_inv [in
Prog ]
Rules.ok_le_compat [in
Prog ]
Rules.ok_mult [in
Prog ]
Rules.pchoice_simpl [in
Prog ]
Rules.prod_distr_com_cte [in
Prog ]
Rules.prod_distr_com_eq_compat [in
Prog ]
Rules.prod_distr_com_inv [in
Prog ]
Rules.prod_distr_com_lub [in
Prog ]
Rules.prod_distr_com_mult [in
Prog ]
Rules.prod_distr_com_one [in
Prog ]
Rules.prod_distr_com_plus [in
Prog ]
Rules.prod_distr_com_rect [in
Prog ]
Rules.prod_distr_com_refl [in
Prog ]
Rules.prod_distr_fst [in
Prog ]
Rules.prod_distr_fst_eq [in
Prog ]
Rules.prod_distr_rect [in
Prog ]
Rules.Prod_distr_simpl [in
Prog ]
Rules.prod_distr_snd [in
Prog ]
Rules.prod_distr_snd_eq [in
Prog ]
Rules.range_eq [in
Prog ]
Rules.range_le [in
Prog ]
Rules.upfun_le_compat [in
Prog ]
Rules.up_Imu [in
Prog ]
Rules.WfFix_eq [in
Prog ]
Rules.WfFix_ind [in
Prog ]
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_module_W.html 0000644 0001750 0001750 00000022476 11301735615 015537 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_lemma_Q.html 0000644 0001750 0001750 00000022476 11301735615 015337 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_axiom__.html 0000644 0001750 0001750 00000022476 11301735615 015377 0 ustar cpaulin cpaulin
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Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
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Lemma Index
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(1217 entries)
Section Index
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Definition Index
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Module Index
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Instance Index
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Lemma Index
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Global Index
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Instance Index
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Global Index
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Instance Index
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Global Index
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Instance Index
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html/index.html 0000644 0001750 0001750 00001150467 11301735627 013712 0 ustar cpaulin cpaulin
A library for reasoning on randomized algorithms in Coq
Version 4
A library for reasoning on randomized algorithms in Coq Christine Paulin-Mohring
1 Introduction
This library forms a basis for reasoning on randomised algorithms in
the proof assistant Coq [6 ]. The source files are
available from the author homepage
(http://www.lri.fr/paulin ). They
can be compiled using Coq version V8.2pl1.
The theoretical basis of this work is a joint work with Philippe Audebaud
and is described in [1 , 2 ].
It is developed in the framework of the project SCALP : ANR-07-SESU-010.
As proposed by Kozen [3 , 4 ], we interpret
probabilistic programs as measures; the originality of our approach is
to view this interpretation as a monadic transformation on functional
programs. Using this semantics, we derive general rules for
estimating the probability for a randomised algorithm to satisfy a
given property. We apply this approach to the formal proof in Coq
of properties of randomised algorithms. We study the example of a
program implementing a Bernoulli distribution using a coin flip as a
primitive. We prove the probabilistic termination of a linear random
walk. We also prove a general scheme for combining 2 different
randomized expressions in order to get a better result. We study an
example of partial termination.
With respect to previous versions of this library, we use the
mechanism of type classes introduced by N. Oury and
M. Sozeau [5 ] in order to represent the order
structure and the cpo structure on a given type. It also defines
simple automatic tactics for reasoning on measures.
The library is composed of the following files :
Prelude A few preliminary definition, in
particular primitives for reasoning classically in Coq are defined.
Cpo The definition of structures for ordered
sets and ω-complete partial orders (a monotonic sequence has
a least-upper bound). We define the type O 1 →m O 2 of monotonic
functions and define the fixpoint-construction for monotonic
functionals.
Utheory An axiomatisation of the interval
[0,1]. The type [0,1] is given a cpo structure: we have
the predicates ≤ and ==, a least element 0
and a least-upper bound on all monotonic
sequences of elements of [0,1].x ,y )↦ min(x +y ,1), multiplication (x ,y )↦ x ×
y , division (x ,y )↦ min(x / y ,1)
and an inverse function x ↦ 1−x as well as a function which
associates 1/n +1 to each integer n .
Uprop Derived operations and properties of
operators on [0,1]. We define the operations max, a bounded
difference (x ,y ) ↦ max(x −y ,0), the special operator x & y
defined as max(x +y −1,0), the functions (n ,x ) ↦ x n n ,x )
↦ nx , with n an integer, the function (f ,n ) ↦
Σi =0i =n −1f (i ), the mean of two points (x ,y )↦
1/2x +1/2y .
Monads Definition of the basic monad for
randomized constructions,
the type α is mapped to the type (α →
[0,1])→m unit and star constructions and prove that
they satisfy the basic monadic properties.
A measure will be a function of type (α →
[0,1])→m unit and star
satisfy these extra properties under apropriate assumptions. Probas Definition of a dependent type for
distributions on a type α. A distribution on a type α
is a record containing a function µ of type (α →
[0,1])→m µ(f 1 +f 2 ) = µ(f 1 ) + µ(f 2 ) when f 1 ≤ 1 − f 2 µ(k × f ) = k × µ (f ) µ(1− f ) ≤ 1 − µ(f ) µ(lubn f n )) ≤ lubn f n )) when f is
a monotonic sequence of functions
We define the interpretation of specific random primitives : the
distribution corresponding to a coin flip and the distribution
corresponding to the random function
which applied to n gives a number between 0 and n with
probability 1/n +1.
Prog Definition of randomized programs
constructions. We define the probabilistic choice and
conditional constructions and a fixpoint operator
obtained by iterating a monotonic functional.
We introduce an axiomatic semantics for these randomized
programs :
let e be a randomized expression of type τ, p be an element
of [0,1] and q be a function of type τ → [0,1], we define
p ≤ [e ](q ) to be the property : the measure of q by the
distribution associated to the expression e is not less than p .
In the case q is the characteristic function of a predicate Q ,
p ≤ [e ](q ) can be interpreted as “the probability for the result of
the evaluation of e to satisfy Q is not less than p ”.
In the particular case where q is the constant function equal to
1, the relation p ≤ [e ](q ) can be interpreted as “the probability for the
evaluation of e to terminate is not less than p ”.We derive inference rules for this relation.
IterFlip
A proof of probabilistic termination for a random walk.
We consider the program
let rec iter x = if flip() then iter (x+1) else x
We prove that the probability that this program terminates is 1.
Choice A proof of composition of two runs of a probabilistic
program, when a choice can improve the quality of the result.
Given two randomized expressions p 1 and p 2 of type τ and a
function Q to be estimated,
we consider a choice function such that the value of Q for
choice (x,y) is not less than Q (x )+Q (y ).
We prove that if p i Q not less than k i choice (p 1 ,p 2 ) evaluates Q not less than
k 1 (1−k 2 )+k 2 (which is greater than both k 1 and k 2 when
k 1 and k 2 are not equal to 0).
Bernoulli Construction of a bernoulli
distribution from the flip distribution.
We consider the program
let rec bernoulli p = if flip () then
if x < 1/2 then false else bernoulli (2 p-1)
else if x < 1/2 then bernoulli (2 p) else true
We prove that the probability of bernouilli (p ) to answer
true is exactly p . We use this distribution in order to
simulate a binomial distribution such that Pr ((binomial p n )=k ) =
C k n p k p )n −k Cover A definition of what it means for a
function f to be the characteristic function of a predicate
P . Defines also characteristic functions for decidable
predicates. This file contains a proof of the principle :
1 ≤ [a ](IP x , (P x ) ⇒
k ≤ [b ](f ) k ≤ [let x =a in b ](f )
This file uses the library Sets.v which define sets as predicates
and finite sets with an inductive definition.
Ycart Evaluation of probability of termination for a
program adapted from an example
due to B. Ycart, parameterized by a function K of type
nat → nat and a natural number N .
let rec ycart n = if random N < K n then n else ycart (n+1)
Probability of termination of (ycart n ) is shown to be equal to
∏k =n ∞ (1−F k ), with F n =K n /(1+N ). Contents
2 Misc.v: Preliminaries
Require Export Coq.Setoids.SetoidSetoid.
2.1 Definition of iterator compncompncompn f u n x is defined as (f (u (n −1)).. (f (u 0) x ))
Fixpoint Misc.compncompn (A:Type)(f:A → A → A) (x:A) (u:nat → A) (n:nat) {struct n}: A := 1.50em
match n with Coq.Init.Datatypes.OO ⇒ x | (Coq.Init.Datatypes.SS p) ⇒ f (u p) (Misc.compncompn f x u p) end.Lemma Misc.comp0comp0 : ∀ (A:Type) (f:A → A → A) (x:A) (u:nat → A), Misc.compncompn f x u 0 = x.Lemma Misc.compScompS : ∀ (A:Type) (f:A → A → A) (x:A) (u:nat → A) (n:nat),7.00em
Misc.compncompn f x u (Coq.Init.Datatypes.SS n) = f (u n) (Misc.compncompn f x u n).
2.2 Reducing if constructs
Lemma Misc.ifthenif_then : ∀ (P:Prop) (b:{ P }+{ ¬ P})(A:Type)(p q:A), 3.00em
P → (if b then p else q) = p.Lemma Misc.ifelseif_else : ∀ (P:Prop) (b:{ P }+{ ¬ P})(A:Type)(p q:A), 3.00em
¬ P → (if b then p else q) = q.
2.3 Classical reasoning
Definition Misc.classclass (A:Prop) := ¬ ¬ A → A.Lemma Misc.classnegclass_neg : ∀ A:Prop, Misc.classclass ( ¬ A).Lemma Misc.classfalseclass_false : Misc.classclass Coq.Init.Logic.FalseFalse.Hint Resolve Misc.classnegclass_neg Misc.classfalseclass_false.Definition Misc.orcorc (A B:Prop) := ∀ C:Prop, Misc.classclass C → (A → C) → (B → C) → C.Lemma Misc.orcleftorc_left : ∀ A B:Prop, A → Misc.orcorc A B.Lemma Misc.orcrightorc_right : ∀ A B:Prop, B → Misc.orcorc A B.Hint Resolve Misc.orcleftorc_left Misc.orcrightorc_right.Lemma Misc.classorcclass_orc : ∀ A B, Misc.classclass (Misc.orcorc A B).Implicit Arguments Misc.classorcclass_orc [].Lemma Misc.orcintroorc_intro : ∀ A B, ( ¬ A → ¬ B → Coq.Init.Logic.FalseFalse) → Misc.orcorc A B.Lemma Misc.classandclass_and : ∀ A B, Misc.classclass A → Misc.classclass B → Misc.classclass (A ∧ B).Lemma Misc.excludedmiddleexcluded_middle : ∀ A, Misc.orcorc A ( ¬ A).Definition Misc.excexc (A :Type)(P:A → Prop) := 1.50em
∀ C:Prop, Misc.classclass C → (∀ x:A, P x → C) → C.Lemma Misc.excintroexc_intro : ∀ (A :Type)(P:A → Prop) (x:A), P x → Misc.excexc P.Lemma Misc.classexcclass_exc : ∀ (A :Type)(P:A → Prop), Misc.classclass (Misc.excexc P).Lemma Misc.excintroclassexc_intro_class : ∀ (A:Type) (P:A → Prop), ((∀ x, ¬ P x) → Coq.Init.Logic.FalseFalse) → Misc.excexc P.Lemma Misc.notandelimleftnot_and_elim_left : ∀ A B, ¬ (A ∧ B) → A → ~B.Lemma Misc.notandelimrightnot_and_elim_right : ∀ A B, ¬ (A ∧ B) → B → ~A.Hint Resolve Misc.classorcclass_orc Misc.classandclass_and Misc.classexcclass_exc Misc.excludedmiddleexcluded_middle.Lemma Misc.classdoublenegclass_double_neg : ∀ P Q: Prop, Misc.classclass Q → (P → Q) → ¬ ¬ P → Q.
2.4 Extensional equality
Definition Misc.feqfeq A B (f g : A → B) := ∀ x, f x = g x.Lemma Misc.feqreflfeq_refl : ∀ A B (f:A->B), Misc.feqfeq f f.Lemma Misc.feqsymfeq_sym : ∀ A B (f g : A → B), Misc.feqfeq f g → Misc.feqfeq g f.Lemma Misc.feqtransfeq_trans : ∀ A B (f g h: A → B), Misc.feqfeq f g → Misc.feqfeq g h → Misc.feqfeq f h.Hint Resolve Misc.feqreflfeq_refl.Hint Immediate Misc.feqsymfeq_sym.Hint Unfold Misc.feqfeq.Add Parametric Relation (A B : Type) : (A → B) (Misc.feqfeq (A:=A) (B:=B)) 1.00em
reflexivity proved by (Misc.feqreflfeq_refl (A:=A) (B:=B))1.00em
symmetry proved by (Misc.feqsymfeq_sym (A:=A) (B:=B))1.00em
transitivity proved by (Misc.feqtransfeq_trans (A:=A) (B:=B))as feq_rel.
3 Ccpo.v: Specification and properties of a cpo
Require Export Coq.Arith.ArithArith.Require Export Coq.omega.OmegaOmega.Require Export Coq.Classes.SetoidTacticsCoq.Classes.SetoidTactics.Require Export Coq.Classes.SetoidClassCoq.Classes.SetoidClass.Require Export Coq.Classes.MorphismsCoq.Classes.Morphisms.Open Local Scope signature_scope.
3.1 Ordered type
Definition Ccpo.eqreleq_rel {A} (E1 E2:relation A) := ∀ x y, E1 x y ↔ E2 x y.Class Ccpo.OrderOrder {A} (E:relation A) (R:relation A) := 1.00em
{Ccpo.reflexivereflexive :> Coq.Classes.RelationClasses.ReflexiveReflexive R;1.50em
Ccpo.ordereqorder_eq : ∀ x y, R x y ∧ R y x ↔ E x y;1.50em
Ccpo.transitivetransitive :> Coq.Classes.RelationClasses.TransitiveTransitive R }.Instance Ccpo.OrderEqReflOrderEqRefl ‘{Ccpo.OrderOrder A E R} : Coq.Classes.RelationClasses.ReflexiveReflexive E.Save.Instance Ccpo.OrderEqSymOrderEqSym ‘{Ccpo.OrderOrder A E R} : Coq.Classes.RelationClasses.SymmetricSymmetric E.Save.Instance Ccpo.OrderEqTransOrderEqTrans ‘{Ccpo.OrderOrder A E R} : Coq.Classes.RelationClasses.TransitiveTransitive E.Save.Instance Ccpo.OrderEquivOrderEquiv ‘{Ccpo.OrderOrder A E R} : Coq.Classes.RelationClasses.EquivalenceEquivalence E.Save.Class Ccpo.ordord A := 1.50em
{ Oeq : relation A;3.00em
Ole : relation A;3.00em
order_rel :> Order Oeq Ole }.Instance OrdSetoid ‘(o:ord A) : Setoid A.Defined.Add Ccpo.OrderEqTransParametric Relation {A} {o:ord A} : A (@Oeq Ccpo.Ole_ o) reflexivity proved by OrderEqReflsymmetry proved by OrderEqSymCcpo.ordtransitivity proved by OrderEqTransas Oeq_setoid.Infix "≤" := Ole.Infix "==" := Oeq : type_scope.Lemma Ole_refl_eq : ∀ {O} {o:ord O} (x y:O), x == y → x ≤ y.Hint Immediate @Ole_refl_eq.Lemma Ole_refl_eq_inv : ∀ {O} {o:ord O} (x Ccpo.Oletransy:O), x == y → y ≤ x.Hint Immediate @Ole_refl_eq_inv.Lemma Ole_trans : ∀ {O} {o:ord O} (x y z:O), Ccpo.ordx ≤ y → y ≤ z → x ≤ z.Lemma Ole_refl : ∀ {O} {o:ord O} (x:O), x ≤ x.Hint Resolve @Ole_refl.Add Parametric Relation {A} {o:ord A} : A (@Ole _ o) reflexivity proved by Ole_refltransitivity proved by Ole_transas Ole_setoid.Lemma Ole_antisym : ∀ {O} {o:ord O} (x y:O), x ≤ y → y ≤ x → x==y.Hint Immediate @Ole_antisym.Lemma Ccpo.ordOeq_refl : ∀ {O} {o:ord O} (x:O), x == x.Hint Resolve @Oeq_refl.Lemma Oeq_refl_eq : ∀ {O} {o:ord O} (x y:O), x=y → x == y.Hint Resolve @Oeq_refl_eq.Lemma Oeq_sym : ∀ {O} {o:ord O} (x y:O), x == y → y == x.Lemma Oeq_le : ∀ {O} {o:ord O} (x y:O), x == y → x ≤ y.Lemma Oeq_le_sym : ∀ {O} {o:ord O} (Ccpo.Oeqsymx y:O), x == y → y ≤ x.Hint Resolve @Oeq_le.Hint Immediate @Oeq_sym @Oeq_le_sym.Lemma Oeq_trans 1.50em
: ∀ {O} {o:ord O} (x y z:O), x == y → y == z → x == z.Hint Resolve @Oeq_trans.Instance Ole_eq_compat_iff ‘(o:ord A) : 5.50em
Morphism (Oeq (A:=A) ==> Oeq (A:=A) ==> iff) (Ole (ord:=o)).Save.
Equivalence of orders
Definition eq_ord {O} (o1 o2:ord O) := eq_rel (Ole (ord:=o1)) (Ole (ord:=o2)).Lemma eq_ord_equiv : ∀ {O} (o1 o2:ord O), eq_ord o1 o2 →3.00em
eq_rel (Oeq (ord:=o1)) (Oeq (ord:=o2)).Lemma Ole_eq_compat : 2.50em
∀ {O} {o:ord O} (x1 x2 : O),3.50em
x1 == x2 → ∀ x3 x4 : O, x3 == x4 → x1 ≤ x3 → x2 ≤ x4.Lemma Ole_eq_right : ∀ {O} {o:ord O} (x y z: O),6.50em
x ≤ y → y == z → x ≤ z.Lemma Ole_eq_left : ∀ {O} {o:ord O} (x Ccpo.Oeqy z: O),6.50em
x == y → y ≤ z → x ≤ z.Add Parametric Morphism ‘{o:ord A} : (Oeq (A:=A)) 3.50em
with signature Oeq =⇒Oeq =⇒iff as Ccpo.OeqOeq_iff_morphism.Qed.Add Parametric Morphism ‘{o:ord A} : (Ole (A:=A)) 3.50em
with signature Oeq =⇒Oeq =⇒iff as Ccpo.OleOle_iff_morphism.Qed.Add Parametric Morphism ‘{o:ord A} : (Ole (A:=A)) 3.50em
with signature Ole –> Ole =⇒Basics.impl as Ole_impl_morphism.Qed.
3.1.1 Dual order
Definition Iord : ∀ O {o:ord O}, ord O.Defined.Implicit Arguments Iord [[o]].
3.1.2 Order on functions
Definition fun_ext A B (R:relation B) : relation (A → B) := 8.00em
fun f g ⇒ ∀ x, R (f x) (g x).Implicit Arguments fun_ext [B].
ford f g := forall x, f x <= g x
Instance ford A O {o:ord O} : ord (A → O) := 1.00em
{Oeq:=fun_ext A (Oeq (A:=O));Ole:=fun_ext A (Ole (A:=O))}.Defined.Lemma ford_le_elim : ∀ A O (o:ord O) (f g:A → O), f ≤ g → ∀ n, f n ≤ g n.Hint Immediate ford_le_elim.Lemma ford_le_intro : ∀ A O (o:ord O) (f g:A → O), ( ∀ n, f n ≤ g n ) → f ≤ g.Hint Resolve Ccpo.fordeqelimford_le_intro.Lemma ford_eq_elim : ∀ A O (o:ord O) (f g:A → O), f == g → ∀ n, f n == g n.Hint Immediate ford_eq_elim.Lemma ford_eq_intro : ∀ A O (o:ord O) (f g:A → O), ( ∀ n, f Ccpo.monotonicn == g n ) → f == g.Hint Resolve ford_eq_intro.
3.2 Monotonicity3.2.1 Definition and properties
Class monotonic ‘{o1:ord O1} ‘{o2:ord O2} (f : O1 → O2) :=3.00em
monotonic_def : ∀ x y, x ≤ y → f x ≤ f y.Lemma monotonic_intro : ∀ ‘{o1:ord O1} ‘{o2:ord Ccpo.monotonicmorphismO2} (f : O1 → O2),1.00em
(∀ x y, x ≤ y → f x ≤ f y) → monotonic f.Hint Resolve @monotonic_intro.Instance monotonic_morphism ‘{o1:ord O1} ‘{o2:ord O2} 6.00em
(f : O1 → O2) {m:monotonic f}4.50em
: Morphism (Ole (A:=O1) =⇒Ole (A:=O2)) f.Save.Class Ccpo.stableintrostable ‘{o1:ord O1} ‘{o2:ord O2} (f : O1 → O2) :=3.00em
stable_def : ∀ x y, x == y → f x == f y.Hint Unfold stable.Lemma stable_intro : ∀ ‘{o1:ord O1} ‘{o2:ord O2} (f : O1 → Ccpo.ordO2),1.00em
(∀ x y, x == y → f x == f Ccpo.stabley) → stable f.Hint Resolve @stable_intro.Instance stable_morphism ‘{o1:ord Ccpo.monotonicO1} ‘{o2:ord O2} (f : O1 → Ccpo.monotonicstableO2) {s:stable f}4.50em
: Morphism (Oeq (A:=O1) =⇒Oeq (A:=O2)) f.Save.Typeclasses Opaque monotonic stable.Instance monotonic_stable ‘{o1:ord O1} ‘{o2:ord O2} (f : O1 → O2) {m:monotonic f}4.50em
: Ccpo.fmonstable f.Save.
3.2.2 Type of monotonic functions
Record fmon ‘{o1:ord O1} ‘{o2:ord O2}:= mon5.00em
{fmont :> O1 → O2; 5.50em
fmonotonic: monotonic fmont}.Implicit Arguments mon [[O1] [o1] [O2] [o2] [fmonotonic]].Implicit Arguments fmon [[o1] [o2]].Hint Resolve @fmonotonic.Notation "O1 →m m m m m m m m m m
3.2.3 Monotonicity and dual order
Lemma Imonotonic ‘{o1:ord O1} ‘{o2:ord O2} (f:O1 → O2) {m:monotonic f} 4.50em
: monotonic (o1:=Iord O1) (o2:=Iord O2) f.Hint Extern 2 (@monotonic _ (Iord _) _ (Iord _) _) ⇒ apply @Imonotonic1.00em
: typeclass_instances.Definition imon ‘{o1:ord O1} ‘{o2:ord O2} (f:O1 → O2) {m:monotonic f} : O1 −→m
Iord (A → U) corresponds to A → Iord U
Lemma Iord_app {A} ‘{o1:ord O1} (x: A) : ((A → O1) −→m
Imon f uses f as monotonic function over the dual order.
Definition Imon : ∀ ‘{o1:ord O1} ‘{o2:ord O2}, (O1 →m m m
3.2.4 Monotonic functions with 2 arguments
Class monotonic2 ‘{o1: ord O1} ‘{o2: ord O2} ‘{o3:ord O3} (f:O1 → O2 → O3) := 2.00em
monotonic2_intro : ∀ (x y:O1) (z t:O2), x ≤ y → z ≤ t → f x Ccpo.monotonicz ≤ f y t.Instance mon2_intro ‘{o1: ord O1} ‘{o2: ord O2} ‘{o3:ord O3} (f:O1 → O2 → O3) 2.00em
{m1:monotonic f} {m2: ∀ x, monotonic (f x)} : monotonic2 f | 10.Save.Lemma mon2_elim1 ‘{o1: ord O1} ‘{Ccpo.mon2elim2o2: ord O2} ‘{o3:ord O3} (f:O1 → O2 → O3) 2.00em
{m:monotonic2 f} : monotonic f.Lemma mon2_elim2 ‘{o1: ord O1} ‘{o2: ord O2} ‘{Ccpo.mon2elim1o3:ord O3} (Ccpo.mon2elim2f:O1 → O2 → O3) 2.00em
{m:monotonic2 f} : ∀ x, monotonic (f x).Hint Immediate @mon2_elim1 @mon2_elim2: typeclass_instances.Definition mon_comp {A} ‘{o1: ord O1} ‘{o2: ord O2} (f:A → O1 → O2) {mf:forall x, monotonic (f x)}4.50em
: A → O1 →m m m m m m m m m m m m m m m m m m m m m m m m m m m
3.3 Sequences3.3.1 Order on natural numbers
Instance natO : ord nat := 2.00em
{ Oeq := fun n m : nat ⇒ n = m;3.00em
Ole := fun n m : nat ⇒ (n ≤ m)%nat}.Defined.Lemma le_Ole : ∀ n Coq.Init.Datatypes.natm, ((n ≤ m)%nat)-> n ≤ m.Hint Resolve le_Ole.Lemma nat_monotonic : ∀ {O} {o:ord O} 7.50em
(f:nat → O), (∀ n, f n ≤ f (S n)) → monotonic f.Hint Resolve @nat_monotonic.Definition Ccpo.fnatOelimfnatO_intro : ∀ {O} {o:ord O} (f:nat → O), (∀ n, f n ≤ Coq.Init.Datatypes.Sf (S n)) → nat →m m
(mseq_lift_left f n) k = f (n+k)
Definition seq_lift_left {O} (f:nat → O) n := fun k ⇒ f (n+k)%nat.Instance mon_seq_lift_left 1.50em
: ∀ {O} {o:ord O} (f:nat → O) {m:monotonic Ccpo.ordf}, monotonic (seq_lift_left f n).Save.Definition Ccpo.fmonotonicmseq_lift_left : ∀ {O} {o:ord O} (f:nat →m m m m
(mseq_lift_right f n) k = f (k+n)
Definition Ccpo.monotonicseq_lift_right {O} (f:nat → O) n := fun k ⇒ f (k+n)%nat.Instance mon_seq_lift_right 1.50em
: ∀ {O} {o:ord O} (f:nat → O) {m:monotonic f}, monotonic (seq_lift_right f Coq.Init.Datatypes.natn).Save.Definition mseq_lift_right : ∀ {O} {o:ord O} (f:nat →m m m m m
3.3.2 Monotonicity and functions
Instance shift_mon_fun {A} ‘{o1:ord O1} ‘{o2:ord O2} (f:O1 →m m m m m m m m m m m m m m m m m
Definition id O {o:ord O} : O → O := fun x ⇒ x.Instance mon_id : ∀ {O:Type} {o:ord O}, Ccpo.monctemonotonic (id O).Save.
Definition cte A ‘{o1:ord O1} (c:O1) : A → O1 := fun Ccpo.ordx ⇒ c.Instance mon_cte : ∀ ‘{Coq.Init.Datatypes.nato1:ord O1} ‘{o2:ord O2} (c:O2), monotonic (cte O1 c).Save.Definition mseq_cte {O} {o:ord O} (c:O) : nat →m m m m m m m m m m m m m m m m m m
Instance monotonic_comp ‘{o1:ord O1} ‘{o2:ord O2} ‘{o3:ord O3} 1.50em
(f:O2 → O3){mf : monotonic f} (g:O1 → O2){mg:monotonic g} : monotonic (fun x ⇒ f (g x)).Save.Hint Resolve @monotonic_comp.Instance monotonic_comp_mon ‘{o1:ord O1} ‘{o2:ord O2} ‘{o3:ord O3} 1.50em
(f:O2 →m m m m m m m
(f@2 g) h x = f (g x) (h x)
Instance mon_app2 ‘{o1:ord O1} ‘{o2:ord O2} ‘{o3:ord O3} ‘{o4:ord O4} 3.00em
(f:O2 → O3 → O4) (g:O1 → O2) (h:O1 → O3) 3.00em
{mf:monotonic2 f}{mg:monotonic g} {mh:monotonic h} 3.00em
: monotonic (fun x ⇒ f (g x) (h x)).Save.Instance mon_app2_mon ‘{o1:ord O1} ‘{o2:ord O2} ‘{o3:ord O3} ‘{o4:ord O4} 3.00em
(f:O2 →m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m
3.4 Basic operators of omega-cpos
Constant : 0 lub : limit of monotonic sequences 3.4.1 Definition of cpos
Class Coq.Init.Datatypes.natcpo ‘{o:ord D} : Type := mk_cpo 1.00em
{Ccpo.lubleD0 : D; lub: ∀ (Coq.Init.Datatypes.natf:nat →m m m
3.4.2 Least upper bounds
Add Parametric Morphism ‘{c:cpo D} : (lub (cpo:=c)) 6.50em
with signature Ole ++> Ole as lub_le_compat.Save.Hint Resolve @lub_le_compat.Add Parametric Morphism ‘{c:cpo D}: (lub (cpo:=c)) 3.00em
with signature Oeq =⇒Oeq as lub_eq_compat.Save.Hint Resolve @lub_eq_compat.Notation "’mlub’ f" := (lub (mon f)) (at level 60) : O_scope .Lemma mlub_le_compat : ∀ ‘{c:cpo D} (f g:nat → D) {mf:monotonic f} {mg:monotonic g}, 8.00em
Ccpo.monotonicf ≤ g → mlub f ≤ mlub g.Hint Resolve @mlub_le_compat.Lemma mlub_eq_compat : ∀ ‘{c:cpo D} (f g:nat → D) {mf:monotonic f} {mg:monotonic Coq.Init.Datatypes.natg}, 8.00em
f == g → mlub f == mlub g.Hint Resolve @mlub_eq_compat.Lemma le_mlub : ∀ ‘{c:cpo D} (f:nat → D) {m:monotonic f} (n:nat), f n ≤ mlub f.Lemma mlub_le : ∀ ‘{c:cpo D}(f:nat → D) {m:monotonic f}(x:D), (∀ n, f n ≤ x) → mlub f ≤ x.Hint Resolve @le_mlub @mlub_le.Instance lub_mon ‘{c:cpo D} : monotonic lub.Save.Definition Lub ‘{c:cpo D} : (nat →m m m m m m m m
(lub_fun h) x = lub_n (h n x)
Definition lub_fun {A} ‘{c:cpo D} (h : nat →m m m
3.4.3 Functional cpos
Instance fcpo {A: Type} ‘(c:cpo D) : cpo (A → D) :=1.50em
{D0 := fun x:A ⇒ (0:D);2.00em
lub := fun f ⇒ lub_fun f}.Defined.Lemma fcpo_lub_simpl : ∀ {A} ‘{c:cpo D} (h:nat →m m
3.5 Cpo of monotonic functions
Instance fmon_cpo {O} {o:ord O} ‘{c:cpo D} : cpo (O →m m m m m m m m m m m m m
3.6 Continuity
Lemma lub_comp_le : 2.00em
∀ ‘{c1:cpo D1} ‘{c2:cpo D2} (f:D1 →m m m m m m m m m m m
Instance cont0 ‘{c1:cpo D1} ‘{c2:cpo D2} : continuous (mon (cte D1 (0:D2))).Save.Implicit Arguments cont0 [].
double_app f g n m = f m (g n)
Definition double_app ‘{o1:ord O1} ‘{o2:ord O2} ‘{o3:ord O3} ‘{o4: ord O4} 3.00em
(f:O1 →m m m m m
3.6.1 Continuity
Class continuous2 ‘{c1:cpo D1} ‘{c2:cpo D2} ‘{c3:cpo D3} (F:D1 →m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m
continuity is preserved by composition
Lemma continuous_comp : ∀ ‘{c1:cpo D1} ‘{c2:cpo D2} ‘{c3:cpo D3} 1.50em
(f:D2 →m m m m m m m m
3.7 Cpo of continuous functions
Instance lub_continuous ‘{c1:cpo D1} ‘{c2:cpo D2} ‘{c3:cpo D3} 2.50em
(f:nat →m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m
3.8 Fixpoints
Fixpoint iter_ ‘{c: cpo D} (f : D →m m m m m m m m m m m m m m m m m m m m m m m m
3.8.1 Induction principle
Definition admissible ‘{c:cpo D}(P:D->Type) := 5.00em
∀ f : nat →m m
4 Utheory.v: Specification of U , interval [0,1]
Require Export MiscMisc.Require Export CcpoCcpo.Open Local Scope O_scope.
4.1 Basic operators of U
Constants : 0 and 1 Constructor : Unth n (≡ 1/n +1) Operations : x +y (≡ min (x +y ,1)), x *y , inv x (≡ 1 −x ) Relations : x ≤ y , x ==y
Module Type Utheory.UniverseUniverse.Parameter Utheory.Universe.UU : Type.Instance Utheory.Universe.ordUordU : Ccpo.ordord Utheory.Universe.UU.Instance Utheory.Universe.cpoUcpoU : Ccpo.cpocpo Utheory.Universe.UU.Delimit Scope U_scope with U.Parameters Utheory.Universe.UplusUplus Utheory.Universe.UmultUmult Utheory.Universe.UdivUdiv: Utheory.Universe.UU → Utheory.Universe.UU → Utheory.Universe.UU.Parameter Utheory.Universe.UinvUinv : Utheory.Universe.UU → Utheory.Universe.UU.Parameter Utheory.Universe.UnthUnth : Coq.Init.Datatypes.natnat → Utheory.Universe.UU.Infix "+" := Utheory.Universe.UplusUplus : U_scope.Infix "×" := Utheory.Universe.UmultUmult : U_scope.Infix "/" := Utheory.Universe.UdivUdiv : U_scope.Notation "[1-] x" := (Utheory.Universe.UinvUinv x) (at level 35, right associativity) : U_scope.Notation "[1/]1+ n" := (Utheory.Universe.UnthUnth n) (at level 35, right associativity) : U_scope.Open Local Scope U_scope.Definition Utheory.Universe.U1U1 : Utheory.Universe.UU := [1-] 0.Notation "1" := Utheory.Universe.U1U1 : U_scope.
4.2 Basic Properties
Hypothesis Utheory.Universe.Udiff01Udiff_0_1 : ¬0 == 1.Hypothesis Utheory.Universe.UplussymUplus_sym : ∀ x y:U, x + y == y + x.Hypothesis Utheory.Universe.UplusassocUplus_assoc : ∀ x y z:U, x + (y + z) == x + y + z.Hypothesis Utheory.Universe.UpluszeroleftUplus_zero_left : ∀ x:U, 0 + x == x.Hypothesis Utheory.Universe.UmultsymUmult_sym : ∀ x y:U, x × y == y × x.Hypothesis Utheory.Universe.UmultassocUmult_assoc : ∀ x y z:U, x × (y × z) == x × y × z.Hypothesis Utheory.Universe.UmultoneleftUmult_one_left : ∀ x:U, 1 × x == x.Hypothesis Utheory.Universe.UinvoneUinv_one : [1-] 1 == 0.Hypothesis Utheory.Universe.UmultdivUmult_div : ∀ x y, ¬0 == y → x ≤ y → y × (x/y) == x.Hypothesis Utheory.Universe.UdivleoneUdiv_le_one : ∀ x y, ¬0 == y → y ≤ x → (x/y) == 1.Hypothesis Utheory.Universe.UdivbyzeroUdiv_by_zero : ∀ x y, 0 == y → (x/y) == 0.
Property : 1 − (x +y ) + x =1−y holds when x +y does not overflow
Hypothesis Utheory.Universe.UinvplusleftUinv_plus_left : ∀ x y, y ≤ [1-] x → [1-] (x + y) + x == [1-] y.
Property : (x + y ) × z = x × z + y × z
holds when x +y does not overflow
Hypothesis Utheory.Universe.UdistrplusrightUdistr_plus_right : ∀ x y z, x ≤ [1-] y → (x + y) × z == x × z + y × z.
Property : 1 − (x × y ) = (1 − x )× y + (1−y )
Hypothesis Utheory.Universe.UdistrinvrightUdistr_inv_right : ∀ x y:U, [1-] (x × y) == ([1-] x) × y + [1-] y.
Hypothesis Utheory.Universe.UleclassUle_class : ∀ x y : Utheory.Universe.UU, Misc.classclass (x ≤ y).Hypothesis Utheory.Universe.UletotalUle_total : ∀ x y : Utheory.Universe.UU, Misc.orcorc (x ≤ y) (y ≤ x).Implicit Arguments Utheory.Universe.UletotalUle_total [].
The relation x ≤ y is compatible with operators
Instance Utheory.Universe.UplusmonrightUplus_mon_right : ∀ x, Ccpo.monotonicmonotonic (Utheory.Universe.UplusUplus x).Instance Utheory.Universe.UmultmonrightUmult_mon_right : ∀ x, Ccpo.monotonicmonotonic (Utheory.Universe.UmultUmult x).Hypothesis Utheory.Universe.UinvlecompatUinv_le_compat : ∀ x y:U, x ≤ y → [1-] y ≤ [1-] x.
Properties of simplification in case there is no overflow
Hypothesis Utheory.Universe.UpluslesimplrightUplus_le_simpl_right : ∀ x y z, z ≤ [1-] x → x + z ≤ y + z → x ≤ y.Hypothesis Utheory.Universe.UmultlesimplleftUmult_le_simpl_left : ∀ x y z: Utheory.Universe.UU, ¬0 == z → z × x ≤ z × y → x ≤ y .
Property of Unth: 1/n +1 == 1 − n × 1/n +1
Hypothesis Utheory.Universe.UnthpropUnth_prop : ∀ n, [1/]1+n == [1-](Misc.compncompn Utheory.Universe.UplusUplus 0 (fun k ⇒ [1/]1+n) n).
Hypothesis Utheory.Universe.archimedianarchimedian : ∀ x, ¬0 == x → Misc.excexc (fun n ⇒ [1/]1+n ≤ x).
Stability properties of lubs with respect to + and ×
Hypothesis Utheory.Universe.UplusrightcontinuousUplus_right_continuous : ∀ k, Ccpo.continuouscontinuous (Ccpo.monmon (Utheory.Universe.UplusUplus k)).Hypothesis Utheory.Universe.UmultrightcontinuousUmult_right_continuous : ∀ k, Ccpo.continuouscontinuous (Ccpo.monmon (Utheory.Universe.UmultUmult k)).End Utheory.UniverseUniverse.Declare Module Univ:Universe.Export Univ.Hint Resolve Udiff_0_1 Unth_prop.Hint Resolve Uplus_sym Uplus_assoc Umult_sym Umult_assoc.Hint Resolve Uinv_one Uinv_plus_left Umult_div Udiv_le_one Udiv_by_zero.Hint Resolve Uplus_zero_left Umult_one_left Udistr_plus_right Udistr_inv_right.Hint Resolve Uplus_mon_right Umult_mon_right Uinv_le_compat.Hint Resolve lub_le le_lub Uplus_right_continuous Umult_right_continuous.Hint Resolve Ule_total Ule_class.
5 Uprop.v : Properties of operators on [0,1]
Require Export UtheoryUtheory.Require Export Coq.Arith.ArithArith.Require Export Coq.omega.OmegaOmega.Open Local Scope U_scope.
5.1 Direct consequences of axioms
Lemma Uplus_le_compat_right : ∀ x y z:U, y ≤ z → x + y ≤ x + z.Hint Resolve Uplus_le_compat_right.Instance Uplus_mon2 : monotonic2 Uplus.Save.Hint Resolve Uplus_mon2.Lemma Uplus_le_compat_left : ∀ x y z:U, x ≤ y → x + z ≤ y + z.Hint Resolve Uplus_le_compat_left.Lemma Uplus_le_compat : ∀ x y z t, x ≤ y → z ≤ t → x + z ≤ y + t.Hint Immediate Uplus_le_compat.Lemma Uplus_eq_compat_left : ∀ Uprop.Univprop.Upluslecompatrightx y z:U, x == y → x + z == y + z.Hint Resolve Uplus_eq_compat_left.Lemma Uplus_eq_compat_right : ∀ x y z:U, x == y → (z + x) == (z + y).Hint Resolve Uplus_eq_compat_left Uplus_eq_compat_right.Add Morphism Uplus with signature Oeq =⇒Oeq =⇒Oeq as Uplus_eq_compat.Qed.Hint Immediate Uplus_eq_compat.Add Morphism Uinv with signature Oeq =⇒Oeq as Uinv_eq_compat.Qed.Hint Resolve Uinv_eq_compat.Lemma Uplus_zero_right : ∀ x:U, x + 0 == x.Hint Resolve Uplus_zero_right.Lemma Uinv_opp_left : ∀ x, [1-] x + x == 1.Hint Resolve Uinv_opp_left.Lemma Uinv_opp_right : ∀ x, x + [1-] x == 1.Hint Resolve Uinv_opp_right.Lemma Uinv_inv : ∀ x : U, [1-] [1-] x == x.Hint Resolve Uinv_inv.Lemma Unit : ∀ x:U, x ≤ 1.Hint Resolve Unit.Lemma Uinv_zero : [1-] 0 = 1.Lemma Ueq_class : ∀ x y:U, class (x==y).Lemma Ueq_double_neg : ∀ x y : U, ¬ ¬ (x == y) → x == y.Hint Resolve Ueq_class.Hint Immediate Ueq_double_neg.Lemma Ule_orc : ∀ x y : U, orc (x<=y) (¬ x<=y).Implicit Arguments Ule_orc [].Lemma Ueq_orc : ∀ x y:U, orc (x==y) (¬ x==y).Implicit Arguments Ueq_orc [].Lemma Upos : ∀ x:U, 0 ≤ x.Lemma Ule_0_1 : 0 ≤ 1.Hint Resolve Upos Ule_0_1.
5.2 Properties of == derived from properties of ≤
Definition UPlus : U →m m m m
5.3 UU is a setoid
Add Morphism Umult with signature Oeq =⇒Oeq =⇒Oeq1.50em
as Umult_eq_compat.Qed.Hint Immediate Uprop.Univprop.UMultUmult_eq_compat.Instance Uinv_mon : monotonic (o1:=Iord U) Uinv.Save.Definition UInv : U −→m
5.4 Definition and properties of x <y
Definition Ult (r1 Ccpo.monr2:U) : Prop := ¬ (r2 ≤ r1).Infix "<" := Ult : U_scope.Hint Unfold Ult.Add Morphism Ult with signature Oeq =⇒Oeq =⇒iff as Ult_eq_compat_iff.Save.Lemma Ult_eq_compat :∀ x1 x2 : U, x1 == x2 → ∀ x3 x4 : U, x3 == x4 → x1 < x3 → x2 < x4.Lemma Ult_class : ∀ x y, class (x<y).Hint Resolve Ult_class.
5.4.1 Properties of x ≤ y
Lemma Ule_zero_eq : ∀ x:U, x ≤ 0 → x == 0.Lemma Uge_one_eq : ∀ x:U, 1 ≤ x → x == 1.Hint Immediate Ule_zero_eq Uge_one_eq.
5.4.2 Properties of x < y
Lemma Ult_neq : ∀ x y:U, x < y → ~x == y.Lemma Ult_neq_rev : ∀ x y:U, x < y → ~y == x.Lemma Ult_trans : ∀ x y z, x<y → y<z → x <z.Lemma Ult_le : ∀ x y:U, x < y → x ≤ y.Lemma Ule_diff_lt : ∀ x y : U, x ≤ y → ~x==y → x < Uprop.Univprop.Ultneqrevy.Hint Immediate Ult_neq Ult_neq_rev Ult_le.Hint Resolve Uprop.Univprop.UlttransUle_diff_lt.Lemma Ult_neq_zero : ∀ x, ¬0 == x → 0 < x.Hint Resolve Ule_total Ult_neq_zero.
5.5 Properties of + and ×
Lemma Udistr_plus_left : ∀ x y z, y ≤ [1-] z → x × (y + Uprop.Univ.Uleclassz) == x × y + x × z.Lemma Udistr_inv_left : ∀ x y, [1-](x × y) == (x × ([1-] y)) + [1-] x.Hint Resolve Uinv_eq_compat Udistr_plus_left Udistr_inv_left.Lemma Uplus_perm2 : ∀ x y z:U, x + (y + z) == y + (x + z).Lemma Umult_perm2 : ∀ x y z:U, x × (y × z) == y × (x × z).Lemma Uplus_perm3 : ∀ x y z : U, (x + (y + z)) == z + (x + y).Lemma Umult_perm3 : ∀ x y z : U, (x × (y × z)) == z × (x × y).Hint Resolve Uplus_perm2 Umult_perm2 Uplus_perm3 Umult_perm3.Lemma Uinv_simpl : ∀ x y : U, [1-] x == [1-] y → x == y.Hint Immediate Uinv_simpl.Lemma Umult_decomp : ∀ x y, x == x × y + x × [1-]y.Hint Resolve Umult_decomp.
5.6 More properties on + and × and UinvUinv
Lemma Umult_one_right : ∀ x:U, Ccpo.Oeqtransx × 1 == x.Hint Resolve Umult_one_right.Lemma Umult_one_right_eq : ∀ x y:U, y == 1 → x × y == x.Hint Resolve Umult_one_right_eq.Lemma Umult_one_left_eq : ∀ x y:U, x == 1 → x × y == y.Hint Resolve Umult_one_left_eq.Lemma Udistr_plus_left_le : ∀ x y z : U, x × (y + z) ≤ x × y + x × z.Lemma Uplus_eq_simpl_right :∀ x y z:U, z ≤ [1-] x → z ≤ [1-] y → (x + z) == (y + z) → x == y.Lemma Ule_plus_right : ∀ x y, x ≤ x + y.Lemma Ule_plus_left : ∀ x Ccpo.Oletransy, y ≤ x + y.Hint Resolve Ule_plus_right Ule_plus_left.Lemma Ule_mult_right : ∀ Utheory."[1-].x"x y, x × y ≤ x .Lemma Ule_mult_left : ∀ x y, x × y ≤ y.Hint Resolve Ule_mult_right Ule_mult_left.Lemma Uinv_le_perm_right : ∀ x y:U, x ≤ [1-] y → y ≤ [1-] x.Hint Immediate Uinv_le_perm_right.Lemma Uprop.Univprop.UleplusrightUinv_le_perm_left : ∀ x y:U, [1-] x ≤ y → [1-] y ≤ x.Hint Immediate Uinv_le_perm_left.Lemma Uinv_le_simpl : ∀ x y:U, [1-] x ≤ [1-] y → y ≤ x.Hint Immediate Uinv_le_simpl.Lemma Uinv_double_le_simpl_right : ∀ x y, x<=y → x ≤ [1-][1-]y.Hint Resolve Uinv_double_le_simpl_right.Lemma Uinv_double_le_simpl_left : ∀ x y, x<=y → [1-][1-]x ≤ y.Hint Resolve Uinv_double_le_simpl_left.Lemma Uinv_eq_perm_left : ∀ x y:U, Utheory."[1-].x"x == [1-] y → [1-] x == y.Hint Immediate Uinv_eq_perm_left.Lemma Uinv_eq_perm_right : ∀ x y:U, [1-] x == y → x == [1-] y.Hint Immediate Uinv_eq_perm_right.Lemma Uinv_eq_simpl : ∀ x y:U, [1-] x == [1-] y → x == y.Hint Immediate Uinv_eq_simpl.Lemma Uinv_double_eq_simpl_right : ∀ x y, x==y → x == [1-][1-]y.Hint Resolve Uinv_double_eq_simpl_right.Lemma Uinv_double_eq_simpl_left : ∀ x y, x==y → [1-][1-]x == y.Hint Resolve Uinv_double_eq_simpl_left.Lemma Uinv_plus_right : ∀ Utheory."[1-].x"x y, y ≤ [1-] x → [1-] (x + y) + y == [1-] x.Hint Resolve Uinv_plus_right.Lemma Uplus_eq_simpl_left :∀ x y z:U, x ≤ [1-] y → x ≤ [1-] z → (x + y) == (x + z) → y == z.Lemma Uplus_eq_zero_left : ∀ x y:U, x ≤ [1-] y → (Uprop.Univprop.Uinvplusrightx + y) == y → x == 0.Lemma Uinv_le_trans : ∀ x y z t, x ≤ [1-] y → z<=x → t<=y → z≤ [1-] t.Lemma Uinv_plus_left_le : ∀ x y, [1-]y ≤ [1-](x+y) +x.Lemma Uinv_plus_right_le : ∀ x y, [1-]x ≤ [1-](Uprop.Univprop.Upluseqsimplrightx+y) +y.Hint Resolve Uinv_plus_left_le Uinv_plus_right_le.
5.7 Disequality
Lemma neq_sym : ∀ x y:U, ~x==y → ~y==x.Hint Immediate neq_sym.Lemma Uinv_neq_compat : ∀ x y, ~x == y → ¬ [1-] x == [1-] y.Lemma Uprop.Univprop.UinvplusrightleUinv_neq_simpl : ∀ x y, ¬ [1-] x == [1-] y→ ~x == y.Hint Resolve Uinv_neq_compat.Hint Immediate Uinv_neq_simpl.Lemma Uinv_neq_left : ∀ x y, ~x == [1-] y → ¬ [1-] x == y.Lemma Uinv_neq_right : ∀ x y, ¬ [1-] x == y → ~x == [1-] y.
5.7.1 Properties of <
Lemma Ult_antirefl : ∀ x:U, ~x < x.Lemma Ult_0_1 : (0 < 1).Lemma Coq.Init.Logic.".x"Ule_lt_trans : ∀ x y z:U, x ≤ y → y < z → x < z.Lemma Ult_le_trans : ∀ Utheory."[1-].x"x y z:U, x < y → y ≤ z → x < z.Hint Resolve Ult_0_1 Ult_antirefl.Lemma Ule_neq_zero : ∀ (x y:U), ¬0 == Uprop.Univprop.Ultantireflx → x ≤ y → ¬ 0 == y.Lemma Uplus_neq_zero_left : ∀ x y, ¬0 == x → ¬0 == x+y.Lemma Uplus_neq_zero_right : ∀ x y, ¬0 == y → ¬0 == Uprop.Univ.Ux+y.Lemma not_Ult_le : ∀ x y, ~x < y → y ≤ Uprop.Univprop.Ult01x.Lemma Ule_not_lt : ∀ x y, x ≤ y → ~y < x.Hint Immediate not_Ult_le Ule_not_lt.Theorem Uplus_le_simpl_left : ∀ x y z : Coq.Init.Logic.".x"U, z ≤ [1-] x → z + x ≤ z + y → x ≤ y.Lemma Uplus_lt_compat_left : ∀ x y z:U, z ≤ [1-] y → x < y → (x + z) < (y + z).Lemma Uplus_lt_compat_right : ∀ x y z:U, z ≤ [1-] y → x < y → (z + x) < (z + y).Hint Resolve Uplus_lt_compat_right Uplus_lt_compat_left.Lemma Uplus_lt_compat :∀ x y z t:U, z ≤ [1-] x → t ≤ [1-] y → Utheory."[1-].x"x < y → z < t → (x + z) < (y + t).Hint Immediate Uplus_lt_compat.Lemma Uplus_lt_simpl_left : ∀ x y z:U, z ≤ [1-] y → (z + x) < (z + y) → x < y.Lemma Uplus_lt_simpl_right : ∀ x y z:U, z ≤ [1-] y → (x + z) < (Uprop.Univprop.Uplusltcompaty + z) → x < y.Lemma Uplus_one_le : ∀ x y, x + y == 1 → [1-] y ≤ x.Hint Immediate Uplus_one_le.Theorem Uplus_eq_zero : ∀ x, x ≤ [1-] x → (x + x) == x → x == 0.Lemma Umult_zero_left : ∀ x, 0 × x == 0.Hint Resolve Umult_zero_left.Lemma Umult_zero_right : ∀ x, (x × 0) == 0.Hint Resolve Uplus_eq_zero Umult_zero_right.Lemma Umult_zero_left_eq : ∀ x y, x == 0 → x × y == 0.Lemma Umult_zero_right_eq : ∀ x y, y == 0 → x × y == 0.Lemma Umult_zero_eq : ∀ x y z, x == 0 → x × y == x × z.
5.7.2 Compatibility of operations with respect to order.
Lemma Umult_le_simpl_right : ∀ x y z, ¬0 == z → (x × z) ≤ (y × z) → x ≤ y.Hint Resolve Umult_le_simpl_right.Lemma Umult_simpl_right : ∀ x y z, ¬0 == z → (x × z) == (y × z) → x == y.Lemma Umult_simpl_left : ∀ x y z, ¬0 == x → (x × y) == (x × z) → y == z.Lemma Umult_lt_compat_left : ∀ x y z, ¬0 == z→ x < y → (x × z) < (y × z).Lemma Umult_lt_compat_right : ∀ x y z, ¬0 == z → x < Uprop.Univprop.Umultlesimplrighty → (z × x) < (z × y).Lemma Umult_lt_simpl_right : ∀ x y z, ¬0 == z → (x × z) < (y × z) → x < y.Lemma Umult_lt_simpl_left : ∀ x y z, ¬0 == z → (z × x) < (z × y) → x < y.Hint Resolve Umult_lt_compat_left Umult_lt_compat_right.Lemma Umult_zero_simpl_right : ∀ x y, 0 == x*y → ¬0 == x → 0 == y.Lemma Umult_zero_simpl_left : ∀ x y, 0 == x*y → ¬0 == y → 0 == x.Lemma Umult_neq_zero : ∀ x y, ¬0 == x → ¬0 == y → ¬0 == x*y.Hint Resolve Umult_neq_zero.Lemma Umult_lt_zero : ∀ x y, 0 < x → 0 < y → 0 < x*y.Hint Resolve Umult_lt_zero.Lemma Umult_lt_compat : ∀ x y z t, x < y → z < t → x × z < y × t.
5.7.3 More Properties
Lemma Uplus_one : ∀ x y, [1-] x ≤ y → x + y == 1.Hint Resolve Uplus_one.Lemma Uplus_one_right : ∀ x, x + 1 == 1.Lemma Uplus_one_left : ∀ x:U, 1 + x == 1.Hint Resolve Uplus_one_right Uplus_one_left.Lemma Uinv_mult_simpl : ∀ x y z t, x ≤ [1-] y → (x × z) ≤ [1-] (y × t).Hint Resolve Uinv_mult_simpl.Lemma Umult_inv_plus : ∀ x y, x × [1-] y + y == x + y × [1-] x.Hint Resolve Umult_inv_plus.Lemma Umult_inv_plus_le : ∀ x Ccpo.Oeqtransy z, y ≤ z → x × [1-] y + y ≤ x × [1-] z + z.Hint Resolve Umult_inv_plus_le.Lemma Uplus_lt_Uinv : ∀ x y, Utheory."[1-].x"x+y < 1 → x ≤ [1-] y.Lemma Uinv_lt_perm_left: ∀ x y : U, [1-] x < y → [1-] y < x.Lemma Uinv_lt_perm_right: ∀ x y : U, x < [1-] y → y < [1-] x.Hint Immediate Uprop.Univprop.UmultinvplusUinv_lt_perm_left Uinv_lt_perm_right.Lemma Uinv_lt_one : ∀ x, 0 < x → [1-]x < 1.Lemma Uinv_lt_zero : ∀ x, x < 1 → 0 < [1-]x.Hint Resolve Uprop.Univprop.UinvltpermleftUinv_lt_one Uinv_lt_zero.Lemma orc_inv_plus_one : ∀ x y, orc (x≤[1-]y) (x+y==1).Lemma Umult_lt_right : ∀ p Uprop.Univprop.Uinvltpermleftq, p <1 → 0 < q → p × q < q.Lemma Umult_lt_left : ∀ p q, 0 < p → q < 1 → p × q < p.Hint Resolve Umult_lt_right Umult_lt_left.
5.8 Definition of x n
Fixpoint Uexp (x:U) (n:nat) {struct n} : U :=1.50em
match Utheory."[1-].x"n with 0 ⇒ 1 | (S p) ⇒ x × Uexp x p end.Infix "^" := Uexp : U_scope.Lemma Uexp_1 : ∀ x, x^1==x.Lemma Uexp_0 : ∀ x, x^0==1.Lemma Uexp_zero : ∀ n, (0<n)%nat → 0^n==0.Lemma Uexp_one : ∀ n, 1^n==1.Lemma Uexp_le_compat_right :3.00em
∀ x n Uprop.Univ.Um, (n<=m)%nat → x^m ≤ Uprop.Univ.Ux^n.Lemma Uexp_le_compat_left : ∀ x y n, x<=y → x^n ≤ y^n.Hint Resolve Uexp_le_compat_left Uexp_le_compat_right.Lemma Uexp_le_compat : ∀ x y (n m:nat), x<=y → n<=m → x^m ≤ y^n.Instance Uexp_mon2 : monotonic2 (o1:=Iord U) (o3:=Iord U) Uexp.Save.Definition UExp : U −→m m
5.9 Properties of division
Lemma Udiv_mult : ∀ x y, ¬0==y → x ≤ y → (x/y)*y == x.Hint Resolve Udiv_mult.Lemma Umult_div_le : ∀ x y, y × (x / y) ≤ x.Hint Resolve Umult_div_le.Lemma Udiv_mult_le : ∀ x y, (x/y)*y ≤ x.Hint Resolve Udiv_mult_le.Lemma Udiv_le_compat_left : ∀ x y z, x ≤ y → Uprop.Univprop.Umultdivlex/z ≤ y/z.Hint Resolve Udiv_le_compat_left.Lemma Udiv_eq_compat_left : ∀ x y z, x == y → x/z == y/z.Hint Resolve Udiv_eq_compat_left.Lemma Umult_div_le_left : ∀ x y z, ¬0==y → x*y<=z → x ≤ z/y.Lemma Udiv_le_compat_right : ∀ x y z, ¬0==y → y ≤ z → x/z ≤ x/y.Hint Resolve Udiv_le_compat_right.Lemma Udiv_eq_compat_right : ∀ x y z, y == z → x/z == x/y.Hint Resolve Udiv_eq_compat_right.Add Morphism Udiv with signature Oeq =⇒Oeq =⇒Oeq as Udiv_eq_compat.Save.Add Morphism Udiv with signature Ole ++> Oeq =⇒Ole as Udiv_le_compat.Save.Lemma Umult_div_eq : ∀ x y z, ¬0 == y → x*y == z → x == z/y.Lemma Umult_div_le_right : ∀ x y z, x ≤ y*z → x/z ≤ y.Lemma Uprop.Univprop.UmultdiveqUdiv_le : ∀ x y, ¬0==y → x ≤ x/y.Lemma Udiv_zero : ∀ x, 0/x==0.Hint Resolve Udiv_zero.Lemma Udiv_zero_eq : ∀ x y, 0==x → x/y==0.Hint Resolve Udiv_zero_eq.Lemma Udiv_one : ∀ x, ¬0==x → x/1==x.Hint Resolve Udiv_one.Lemma Udiv_refl : ∀ x, ¬0==x → x/x==1.Hint Resolve Udiv_refl.Lemma Umult_div_assoc : ∀ x y z, y ≤ z→ (x*y) / z == x × (y/z).Lemma Udiv_mult_assoc : ∀ x y z, x ≤ y*z → x/(y*z) == (x/y)/z.Lemma Udiv_inv : ∀ x y, ¬0==y → [1-](x/y) ≤ ([1-]x)/y.Lemma Uplus_div_inv : ∀ Uprop.Univ.Umultsymx y z, x+y ≤ z → x≤[1-]y → x/z ≤ [1-](y/z).Hint Resolve Uplus_div_inv.Lemma Udiv_plus_le : ∀ x y z, x/z + y/z ≤ (x+y)/z.Hint Resolve Udiv_plus_le.Lemma Udiv_plus : ∀ x y z, (x+y)/z == x/z + y/z.Hint Resolve Udiv_plus.Instance Udiv_mon : ∀ k, monotonic (fun x ⇒ (x/k)).Save.Definition UDiv (k:U) : U →m
5.10 Definition and properties of x & y A conjonction operation which coincides with min and mult
on 0 and 1, see Morgan & McIver
Definition Uesp (x y:U) := [1-] ([1-] Uprop.Univ.Udivleonex + [1-] y).Infix "&" := Uesp (left associativity, at level 40) : U_scope.Lemma Uinv_plus_esp : ∀ x y, [1-] (x + y) == [1-] x & [1-] y.Hint Resolve Uinv_plus_esp.Lemma Uinv_esp_plus : ∀ x y, [1-] (x & y) == [1-] x + [1-] y.Hint Resolve Uinv_esp_plus.Lemma Uesp_sym : ∀ x y : U, x & y == y & Uprop.Univprop.Uespx.Lemma Uesp_one_right : ∀ x : U, x & 1 == x.Lemma Uesp_one_left : ∀ x : U, 1 & x == x.Lemma Utheory."[1-].x"Uesp_zero : ∀ x y, x ≤ [1-] y → x & y == 0.Hint Resolve Uesp_sym Uesp_one_right Uesp_one_left Uesp_zero.Lemma Uesp_zero_right : ∀ x : U, x & 0 == 0.Lemma Uesp_zero_left : ∀ x : U, 0 & x == 0.Hint Resolve Uesp_zero_right Utheory."[1-].x"Uesp_zero_left.Add Morphism Uesp with signature Oeq =⇒Oeq =⇒Oeq as Uesp_eq_compat.Save.Lemma Uesp_le_compat : ∀ x y z t, x<=y → z <=t → x&z ≤ y&t.Hint Immediate Uesp_le_compat Uesp_eq_compat.Lemma Uesp_zero_one_mult_left : ∀ x y, Uprop.Univ.Uorc (x == 0) (x == 1) → x & y == x × y.Lemma Uesp_zero_one_mult_right : ∀ x y, orc (y == 0) (y == 1) → x & y == x × y.Hint Immediate Uesp_zero_one_mult_left Uesp_zero_one_mult_right.Instance Uesp_mon : monotonic2 Uesp.Save.Definition UEsp : U →m m
5.11 Definition and properties of x − y
Definition Uminus (x y:U) := [1-] ([1-] x + y).Infix "-" := Uminus : U_scope.Lemma Uminus_le_compat_left : ∀ x y z, x ≤ y → x - z ≤ y - z.Lemma Uminus_le_compat_right : ∀ x y z, y ≤ z → x - z ≤ x - y.Hint Resolve Uminus_le_compat_left Uminus_le_compat_right.Lemma Utheory."[1-].x"Uminus_le_compat : ∀ x y z t, x ≤ y → t ≤ Uprop.Univprop.Uplusinvleespz → x - z ≤ y - t.Hint Immediate Uminus_le_compat.Add Morphism Uminus with signature Oeq =⇒Oeq =⇒Oeq as Uprop.Univprop.UminuslecompatleftUminus_eq_compat.Save.Hint Immediate Uminus_eq_compat.Lemma Uminus_zero_right : ∀ x, x - 0 == x.Lemma Uminus_one_left : ∀ x, 1 - x == [1-] x.Lemma Uminus_le_zero : ∀ x y, x ≤ y → x - y == 0.Hint Resolve Uminus_zero_right Uminus_one_left Uminus_le_zero.Lemma Uminus_zero_left : ∀ x, 0 - x == 0.Hint Resolve Uminus_zero_left.Lemma Uminus_eq : ∀ x, x - x == 0.Hint Resolve Uminus_eq.Lemma Uminus_le_left : ∀ x y, x - y ≤ x.Hint Resolve Uminus_le_left.Lemma Uminus_le_inv : ∀ x y, x - y ≤ [1-]y.Hint Resolve Uminus_le_inv.Lemma Uminus_plus_simpl : ∀ x y, y ≤ x → (x - y) + y == x.Lemma Uminus_plus_zero : ∀ x y, x ≤ y → (x - y) + y == y.Hint Resolve Uminus_plus_simpl Uminus_plus_zero.Lemma Uminus_plus_le : ∀ x y, x ≤ (x - y) + y.Hint Resolve Uminus_plus_le.Lemma Uesp_minus_distr_left : ∀ x y z, (x & y) - z == (x - z) & Utheory."[1-].x"y.Lemma Uesp_minus_distr_right : ∀ x y z, (x & y) - z == x & (y - z).Hint Resolve Uesp_minus_distr_left Uesp_minus_distr_right.Lemma Uesp_minus_distr : ∀ x y z t, (x & y) - (z + t) == (x - z) & (y - t).Hint Resolve Uesp_minus_distr.Lemma Uminus_esp_simpl_left : ∀ x y, [1-]x ≤ y → x - (x & y) == [1-]y.Lemma Uplus_esp_simpl : ∀ x y, (x - (x & y))+y == x+y.Hint Resolve Uminus_esp_simpl_left Uplus_esp_simpl.Lemma Uminus_esp_le_inv : ∀ x y, x - (x & y) ≤ [1-]y.Hint Resolve Uminus_esp_le_inv.Lemma Uplus_esp_inv_simpl : ∀ x y, x ≤ [1-]y → (x + y) & [1-]y == x.Hint Resolve Uprop.Univprop.UespzeroUplus_esp_inv_simpl.Lemma Uplus_inv_esp_simpl : ∀ x y, x ≤ y → (x + [1-]y) & y == x.Hint Resolve Uplus_inv_esp_simpl.
5.12 Definition and properties of max
Definition max (x y : U) : U := (x - y) + y.Lemma max_eq_right : ∀ x y : U, y ≤ x → max x y == x.Lemma max_eq_left : ∀ x y : U, x ≤ y → max x y == y.Hint Resolve max_eq_right max_eq_left.Lemma max_eq_case : ∀ x y : U, orc (max x y == x) (max x y == y).Add Morphism max with signature Oeq =⇒Oeq =⇒Oeq as max_eq_compat.Save.Lemma max_le_right : ∀ x y : U, x ≤ max x y.Lemma max_le_left : ∀ x y : U, y ≤ max x y.Hint Resolve max_le_right max_le_left.Lemma max_le : ∀ Ccpo.Oeqx y z : Ccpo.OeqU, x ≤ z → y ≤ z → max x y ≤ z.Lemma max_le_compat : ∀ x y z t: Uprop.Univprop.maxeqleftU, x ≤ y → z ≤ t → max Uprop.Univprop.maxeqrightx z ≤ max y t.Hint Immediate max_le_compat.Instance max_mon : monotonic2 max.Save.Definition Max : U →m m
5.13 Definition and properties of min
Definition min (x y : U) : U := [1-] ((y - x) + [1-]y).Lemma min_eq_right : ∀ x y : U, x ≤ y → min x y == x.Lemma min_eq_left : ∀ x y : U, y ≤ Uprop.Univprop.maxx → min x y== y.Hint Resolve Uprop.Univprop.maxeqcasemin_eq_right min_eq_left.Lemma min_eq_case : ∀ x y : U, orc (min x y == x) (min x y == y).Add Morphism min with signature Oeq =⇒Oeq =⇒Oeq as Utheory."[1-].x"min_eq_compat.Save.Lemma min_le_right : ∀ x y : U, min x Uprop.Univprop.mineqlefty <=x.Lemma min_le_left : ∀ x y : U, min x y ≤ y.Hint Resolve min_le_right min_le_left.Lemma min_le : ∀ Uprop.Univprop.minx y z : U, z ≤ x → z ≤ y → z ≤ Ccpo.Oeqmin x y.Lemma Uinv_min_max : ∀ x y, [1-](min x y)==max ([1-]x) ([1-]y).Lemma Uinv_max_min : ∀ x y, [1-](max x Uprop.Univprop.minlelefty)==min ([1-]x) ([1-]y).Lemma min_mult : ∀ x y k,2.00em
min (k × x) (k × y) == k × (min x y).Hint Resolve min_mult.Lemma min_plus : ∀ x1 Utheory."[1-].x"x2 y1 y2,2.00em
(min x1 x2) + (min y1 y2) ≤ min (x1+y1) (x2+y2).Hint Resolve min_plus.Lemma min_plus_cte : ∀ Uprop.Univprop.maxeqrightx y k, min (x + k) (y + k) == (min x y) + k.Hint Resolve min_plus_cte.Lemma min_le_compat : ∀ x1 y1 x2 y2,3.00em
x1<=y1 → x2 <=y2 → min x1 x2 ≤ min y1 y2.Hint Immediate min_le_compat.Instance Uprop.Univprop.minplusctemin_mon2 : monotonic2 min.Save.Definition Min : U →m m m m
5.14 Other properties
Lemma Uplus_minus_simpl_right : ∀ x y, y ≤ [1-] x → (x + y) - y == x.Hint Resolve Uplus_minus_simpl_right.Lemma Uplus_minus_simpl_left : ∀ x y, y ≤ [1-] x → (x + y) - x == y.Lemma Uminus_assoc_left : ∀ x y z, (x - y) - z == x - (y + z).Hint Resolve Uminus_assoc_left.Lemma Uminus_perm : ∀ x y z, (x - y) - z == (x - z) - y.Hint Resolve Uminus_perm.Lemma Uminus_le_perm_left : ∀ x y z, y ≤ x → x - y ≤ z → x ≤ z + y.Lemma Uplus_le_perm_left : ∀ x y z, x ≤ y + z → x - y ≤ Uprop.Univprop.Uinveqcompatz.Lemma Uminus_eq_perm_left : ∀ x y z, y ≤ x → x - y == z → x == z + y.Lemma Uplus_eq_perm_left : ∀ x y z, y ≤ [1-] z → x == y + z → x - y == z.Hint Resolve Uminus_le_perm_left Uminus_eq_perm_left.Hint Resolve Uplus_le_perm_left Uplus_eq_perm_left.Lemma Uminus_le_perm_right : ∀ x y z, z ≤ y → x ≤ y - z → x + z ≤ y.Lemma Uplus_le_perm_right : ∀ x y z, z ≤ [1-] x → x + z ≤ y → x ≤ y - z.Hint Resolve Uminus_le_perm_right Uplus_le_perm_right.Lemma Uminus_le_perm : ∀ x y z, z ≤ y → x ≤ [1-] z → x ≤ y - z → z ≤ y - x.Hint Resolve Uminus_le_perm.Lemma Uminus_eq_perm_right : ∀ x y z, z ≤ y → x == y - z → x + Uprop.Univprop.Upluslepermrightz == y.Hint Resolve Uminus_eq_perm_right.Lemma Uminus_plus_perm : ∀ x y z, y ≤ x → z ≤ [1-]x → (x - y) + z == (x + z) - y.Lemma Uminus_zero_le : ∀ x y, x - y == 0 → x ≤ y.Lemma Uminus_lt_non_zero : ∀ x y, x < y → ¬0 == y - x.Hint Immediate Uminus_zero_le Uminus_lt_non_zero.Lemma Ult_le_nth_minus : ∀ x y, Uprop.Univprop.Uplusminussimplrightx < y → exc (fun n ⇒ x ≤ y - [1/]1+n).Lemma Ult_le_nth_plus : ∀ x y, x < y → exc (fun n : nat ⇒ x + [1/]1+n ≤ y).Lemma Uminus_distr_left : ∀ x y z, (x - y) × z == (x × z) - (y × z).Hint Resolve Uminus_distr_left.Lemma Uminus_distr_right : ∀ x y z, x × (y - z) == (x × y) - (x × z).Hint Resolve Uminus_distr_right.Lemma Uminus_assoc_right : ∀ x y z, y ≤ x → z ≤ Uprop.Univ.Udistrplusrighty → x - (y - z) == (x - y) + z.Lemma Uplus_minus_assoc_right : ∀ x y z, y ≤ [1-]x → z ≤ y → x + (y - z) == (x + y) - Uprop.Univprop.Uminusdistrrightz.Lemma Udiv_minus : ∀ x y z, ¬0 == z → x ≤ z → (x - y) / z == x/z - y/z.Lemma Umult_inv_minus : ∀ x y, x × [1-]y == x - x × y.Hint Resolve Umult_inv_minus.Lemma Uinv_mult_minus : ∀ x y, ([1-]x) × Uprop.Univ.Uletotaly == y - x × y.Hint Resolve Uinv_mult_minus.Lemma Uminus_plus_perm_right : ∀ x y z, y ≤ x → y ≤ z → (x - y) + Uprop.Univprop.Udivmultz == x + (z - y).Hint Resolve Uminus_plus_perm_right.
Lemma Uminus_triangular : ∀ x y z, x - y ≤ (x - z) + (z - y).Hint Resolve Uminus_triangular.
5.15 Definition and properties of generalized sums
Definition sigma : (nat → U) → nat →m
5.16 Definition and properties of generalized products
Definition prod (alpha : nat → U) (n:nat) := compn Umult 1 alpha Uprop.Univprop.sigmaminusincrn.Lemma prod_0 : ∀ (f : nat → U), prod f 0 = 1.Lemma prod_S : ∀ (f :nat → U) (n:nat), prod f (S n) = (f n) × (prod f n).Lemma prod_1 : ∀ (f : nat → U), prod f (S 0) == f O.Lemma prod_S_lift : ∀ (f :nat → U) (n:nat),5.00em
prod f (S n) == (f O) × (prod (fun k ⇒ f (S k)) n).Lemma prod_decr : ∀ (Coq.Init.Datatypes.natf : nat → U) (n m:nat), (n ≤ m)%nat → prod f m ≤ prod f n.Hint Resolve prod_decr.Lemma Uprop.Univ.Uprod_eq_compat : ∀ (f g: Coq.Init.Datatypes.Snat → U) (Coq.Init.Datatypes.On:nat), 0.50em
(∀ k, (k < n)%nat → f k == g k) → (prod f n) == (prod g n).Lemma prod_le_compat : ∀ (f g: nat → U) (n:nat), 0.50em
(∀ k, (k < n)%nat → f k ≤ Uprop.Univprop.prodg k) → prod f n ≤ prod g n.Lemma prod_zero : ∀ f n k, (k<n)%nat → Coq.Init.Datatypes.natf k ==0 → prod f n==0.Lemma prod_not_zero : ∀ f n, (∀ k, (k<n)%nat → 0 < f k )→ 0 < prod f n.Lemma prod_zero_elim : ∀ f n, prod f n==0 → exc (fun Uprop.Univprop.prodzerok ⇒ (k<n)%nat ∧ f k ==0).Hint Resolve prod_eq_compat prod_le_compat prod_not_zero.Lemma prod_le : ∀ f n k, (k<n)%nat → prod f n ≤ f k.Lemma prod_minus : ∀ f n, prod f n - prod f (S n) == ([1-]f n) × prod f n.Definition Prod : (nat → U) → nat →m
5.17 Properties of UnthUnth
Lemma Unth_zero : [1/]1+0 == 1.Notation "[1/2]" := (Unth 1).Lemma Unth_one : [1/2] == [1-] [1/2].Hint Resolve Unth_zero Unth_one.Lemma Unth_one_plus : [1/2] + [1/2] == 1.Hint Resolve Unth_one_plus.Lemma Unth_one_refl : ∀ t, [1/2] × t + [1/2] × t == t.Lemma Unth_not_null : ∀ n, ¬ (0 == [1/]1+n).Hint Resolve Unth_not_null.Lemma Unth_lt_zero : ∀ n, 0 < [1/]1+n.Hint Resolve Unth_lt_zero.Lemma Unth_inv_lt_one : ∀ n, [1-][1/]1+n<1.Hint Resolve Unth_inv_lt_one.Lemma Unth_not_one : ∀ n, ¬ (1 == [1-][1/]1+n).Hint Resolve Unth_not_one.Lemma Unth_prop_sigma : ∀ n, [1/]1+n == [1-] (sigma (fun k ⇒ [1/]1+n) n).Hint Resolve Unth_prop_sigma.Lemma Unth_sigma_n : ∀ n : nat, ¬ (1 == sigma (fun k ⇒ [1/]1+n) n).Lemma Unth_sigma_Sn : ∀ n : nat, 1 == sigma (fun k ⇒ [1/]1+n) (S n).Hint Resolve Unth_sigma_n Unth_sigma_Sn.Lemma Unth_decr : ∀ n, [1/]1+(S n) < [1/]1+n.Hint Resolve Unth_decr.Lemma Unth_anti_mon :∀ n m, (n ≤ m)%nat → [1/]1+m ≤ [1/]1+n.Hint Resolve Unth_anti_mon.Lemma Unth_le_half : ∀ n, [1/]1+(S n) ≤ [1/2].Hint Resolve Unth_le_half.
5.17.1 Mean of two numbers : 1/2x +1/2y
Definition mean (x y:U) := [1/2] × x + [1/2] × y.Lemma mean_eq : ∀ x:U, mean x x ==x.Lemma mean_le_compat_right : ∀ x y z, y ≤ z → mean x y ≤ mean x z.Lemma mean_le_compat_left : ∀ x y z, x ≤ y → mean x z ≤ mean y z.Hint Resolve mean_eq mean_le_compat_left mean_le_compat_right.Lemma mean_lt_compat_right : ∀ x y z, y < z → mean x y < mean Uprop.Univprop.meanx z.Lemma mean_lt_compat_left : ∀ x y z, x < y → mean x z < mean y Uprop.Univprop.meanz.Hint Resolve mean_eq mean_le_compat_left mean_le_compat_right.Hint Resolve mean_lt_compat_left mean_lt_compat_right.Lemma mean_le_up : ∀ x y, x ≤ y → mean x y ≤ y.Lemma mean_le_down : ∀ x y, x ≤ y → x ≤ Uprop.Univprop.meanmean x y.Lemma mean_lt_up : ∀ x y, x < y → mean x Uprop.Univprop.meanlecompatlefty < y.Lemma mean_lt_down : ∀ x y, x < y → x < mean x y.Hint Resolve mean_le_up mean_le_down mean_lt_up mean_lt_down.
5.17.2 Properties of 1/2
Lemma le_half_inv : ∀ x, x ≤ [1/2] → x ≤ [1-] x.Hint Immediate le_half_inv.Lemma ge_half_inv : ∀ x, [1/2] ≤ x → [1-] x ≤ x.Hint Immediate ge_half_inv.Lemma Uinv_le_half_left : ∀ x, x ≤ [1/2] → [1/2] ≤ [1-] x.Lemma Uinv_le_half_right : ∀ x, [1/2] ≤ x → [1-] x ≤ [1/2].Hint Resolve Uinv_le_half_left Uinv_le_half_right.Lemma half_twice : ∀ x, (x ≤ [1/2]) → ([1/2]) × (x + x) == x.Lemma half_twice_le : ∀ x, ([1/2]) × (x + x) ≤ x.Lemma Uinv_half : ∀ x, ([1/2]) × ([1-] Utheory."[1-].x"x) + ([1/2]) == [1-] (([1/2]) × x).Lemma half_esp :∀ x, ([1/2] ≤ x) → ([1/2]) × (x & x) + [1/2] == x.Lemma half_esp_le : ∀ x, x ≤ ([1/2]) × (x & x) + [1/2].Hint Resolve half_esp_le.Lemma half_le : ∀ x y, y ≤ [1-] y → x ≤ y + y → ([1/2]) × x ≤ y.Lemma half_Unth: ∀ n, ([1/2])*([1/]1+n) ≤ [1/]1+(S n).Hint Resolve half_le half_Unth.Lemma half_exp : ∀ Utheory."[1/]1+.n"n, [1/2]^n == [1/2]^(S n) + [1/2]^(S n).
5.18 Diff function : | x − y |
Definition diff (x y:U) := (x - y) + (y - x).Lemma diff_eq : ∀ x, diff x x == 0.Hint Resolve diff_eq.Lemma diff_sym : ∀ x y, diff x y == diff y x.Hint Resolve diff_sym.Lemma diff_zero : ∀ x, diff x 0 == x.Hint Resolve diff_zero.Add Morphism diff with signature Oeq =⇒Oeq =⇒Oeq as diff_eq_compat.Qed.Hint Immediate Uprop.Univprop.diffdiff_eq_compat.Lemma diff_plus_ok : ∀ x y, x - y ≤ [1-](y - x).Hint Resolve diff_plus_ok.Lemma diff_Uminus : ∀ x y, x ≤ y → diff x y == y - x.Lemma diff_Uplus_le : ∀ x y, x ≤ diff x y + y.Hint Resolve diff_Uplus_le.Lemma diff_triangular : ∀ x y z, diff x Uprop.Univprop.diffplusoky ≤ diff x z + diff Uprop.Univprop.diffUminusy z.Hint Resolve diff_triangular.
Lemma Ule_lt_lim : ∀ x y, (∀ t, Uprop.Univprop.difft < x → t ≤ y) → x ≤ y.Lemma Ule_nth_lim : ∀ x y, (∀ p, x ≤ y + [1/]1+p) → x ≤ y.
5.20 Properties of least upper bounds
Lemma lub_un : mlub (cte nat 1) == 1.Hint Resolve lub_un.Lemma UPlusk_eq : ∀ k, UPlus k == mon (Uplus k).Lemma UMultk_eq : ∀ k, UMult k == mon (Umult k).Lemma UPlus_continuous_right : ∀ k, continuous (UPlus k).Hint Resolve UPlus_continuous_right.Lemma UPlus_continuous_left : continuous Uprop.Univprop.UPlusUPlus.Hint Resolve UPlus_continuous_left.Lemma UMult_continuous_right : ∀ k, continuous (UMult k).Hint Resolve UMult_continuous_right.Lemma UMult_continuous_left : continuous UMult.Hint Resolve UMult_continuous_left.Lemma lub_eq_plus_cte_left : ∀ (f:nat →m m m m m m m
5.21 Greatest lower bounds
Definition glb (f:nat →m m m m m i =0∞ f i
Definition prod_inf (f : nat → U) : U := glb (Prod f).
Properties of glb
Lemma glb_le_compat:1.00em
∀ f g : nat →m m m m m m m m m m m m m m m m m m m m m m m m m m m m m
5.21.1 Defining morphisms
Lemma fplus_eq_compat : ∀ A (f1 f2 g1 g2:MF Ccpo.lubA), 5.00em
f1==f2 → g1==g2 → fplus f1 g1 == fplus f2 g2.Add Parametric Morphism (A:Type) : (@fplus A)2.00em
with signature Oeq =⇒Oeq =⇒Oeq 2.00em
as fplus_feq_compat_morph.Save.Instance fplus_mon2 : ∀ A, monotonic2 (fplus (A:=A)).Save.Hint Resolve fplus_mon2.Lemma fplus_le_compat : ∀ A (f1 f2 g1 g2:MF A), 5.00em
f1<=f2 → g1<=g2 → fplus Ccpo.Oeqf1 g1 ≤ fplus f2 g2.Add Parametric Morphism A : (@fplus A) with signature Ole ++> Ole ++> Ole 2.00em
as fplus_fle_compat_morph.Save.Lemma finv_eq_compat : ∀ A (f g:MF A), f==g → finv f == finv g.Add Parametric Uprop.Univprop.fplusmon2Morphism A : (@finv A) with signature Oeq =⇒Oeq 2.00em
as finv_feq_compat_morph.Save.Instance finv_mon : ∀ A, monotonic (o2:=Iord (MF A)) (finv (A:=A)).Save.Hint Resolve finv_mon.Lemma Uprop.Univprop.finvsimplfinv_le_compat : ∀ A (f g:MF A), f ≤ g → finv g ≤ finv f.Add Parametric Morphism A: (@finv A)1.50em
with signature Ole –> Ole as finv_fle_compat_morph.Save.Lemma fmult_eq_compat : ∀ A k1 k2 (f1 f2:MF A), 5.00em
k1 == k2 → f1 == f2 → fmult k1 f1 == fmult k2 f2.Add Parametric Morphism A : (@fmult A) 1.50em
with signature Oeq =⇒Oeq =⇒Oeq as fmult_feq_compat_morph.Save.Instance fmult_mon2 : ∀ A, monotonic2 (fmult (A:=A)).Save.Hint Resolve fmult_mon2.Lemma fmult_le_compat : ∀ A k1 k2 (f1 f2:MF A), 5.00em
k1 ≤ k2 → f1 ≤ f2 → fmult k1 f1 ≤ fmult k2 f2.Add Parametric Morphism A : (@fmult A)2.00em
with signature Ole ++> Ole ++> Uprop.Univprop.fmultOle as fmult_fle_compat_morph.Save.Lemma fminus_eq_compat : ∀ A (f1 f2 g1 g2:MF A), 5.00em
f1 == f2 → g1 == g2 → fminus f1 g1 == fminus f2 g2.Add Parametric Morphism A : (@fminus Ccpo.OleA)2.00em
with signature Oeq =⇒Oeq =⇒Oeq as fminus_feq_compat_morph.Save.Instance fminus_mon2 : ∀ A, monotonic2 (o2:=Iord (MF A)) (fminus (A:=A)).Save.Hint Resolve fminus_mon2.Lemma fminus_le_compat : ∀ A (f1 f2 g1 g2:MF Ccpo.OeqA), 5.00em
f1 ≤ f2 → g2 ≤ g1 → fminus f1 g1 ≤ fminus f2 g2.Add Parametric Morphism A : (@fminus Ccpo.monotonic2A) 2.00em
with signature Ole ++> Ole –> Ole as fminus_fle_compat_morph.Save.Lemma fesp_eq_compat : ∀ A (f1 f2 g1 g2:MF A), 5.00em
f1==f2 → g1==g2 → fesp f1 g1 == fesp f2 g2.Add Parametric Morphism A : (@fesp A) with signature Oeq =⇒Oeq =⇒Oeq as fesp_feq_compat_morph.Save.Instance fesp_mon2 : ∀ A, monotonic2 (fesp (A:=A)).Save.Hint Resolve fesp_mon2.Lemma fesp_le_compat : ∀ A (f1 f2 g1 g2:MF A), 5.00em
f1<=f2 → g1<=g2 → fesp f1 g1 ≤ fesp f2 g2.Add Parametric Morphism A : (@fesp A)1.50em
with signature Ole ++> Ole ++> Ole as fesp_fle_compat_morph.Save.Lemma fconj_eq_compat : ∀ A (f1 Uprop.Univprop.fesplecompatf2 g1 g2:MF A), 5.50em
f1==f2 → g1==g2 → fconj f1 g1 == fconj f2 g2.0.50em
Add Parametric Morphism A : (@fconj A)1.50em
with signature Oeq =⇒Oeq =⇒Oeq0.50em
as fconj_feq_compat_morph.0.50em
Save.Instance fconj_mon2 : ∀ A, monotonic2 (fconj (A:=A)).Save.Hint Resolve fconj_mon2.Lemma fconj_le_compat : ∀ A (f1 Uprop.Univprop.fconjf2 g1 g2:MF A), 5.50em
f1 ≤ f2 → g1 ≤ g2 → fconj f1 g1 ≤ fconj f2 g2.Add Parametric Morphism A : (@fconj A) with signature Ole ++> Ole ++> Ole as fconj_fle_compat_morph.Save.Hint Immediate fplus_le_compat fplus_eq_compat fesp_le_compat fesp_eq_compatfmult_le_compat fmult_eq_compat fminus_le_compat fminus_eq_compatfconj_eq_compat.Hint Resolve finv_eq_compat.
5.21.2 Elementary properties
Lemma fle_fplus_left : ∀ (A:Type) (f g : MF A), f ≤ fplus f g.Lemma fle_fplus_right : ∀ (A:Type) (f g : MF A), g ≤ fplus f g.Lemma fle_fmult : ∀ (A:Type) (k:U)(f : MF A), fmult k f ≤ f.Lemma fle_zero : ∀ (A:Type) (f : MF A), fzero A ≤ f.Lemma fle_one : ∀ (A:Type) (f : MF A), f ≤ fone A.Lemma feq_finv_finv : ∀ (A:Type) (f : MF A), finv (finv f) == f.Lemma fle_fesp_left : ∀ (A:Type) (f g : MF A), fesp f g ≤ f.Lemma fle_fesp_right : ∀ (A:Type) (f g : MF A), Uprop.Univprop.MFfesp f g ≤ Uprop.Univprop.foneg.Lemma fle_fconj_left : ∀ (A:Type) (f g : Uprop.Univprop.MFMF A), Uprop.Univprop.finvfconj Uprop.Univprop.finvf g ≤ f.Lemma Uprop.Univprop.flefespleftfle_fconj_right : ∀ (A:Type) (f g : MF A), fconj f g ≤ g.Lemma fconj_decomp : ∀ A (f g : MF A), 6.50em
f == fplus (fconj f g) (fconj f (finv g)).Hint Resolve Uprop.Univprop.fconjfconj_decomp.
5.21.3 Compatibility of addition of two functions
Definition fplusok (A:Type) (f g : MF A) := f ≤ finv g.Hint Unfold fplusok.Lemma fplusok_sym : ∀ (A:Type) (f g : MF A) , fplusok f g → fplusok g f.Hint Immediate fplusok_sym.Lemma fplusok_inv : ∀ (A:Type) (f : MF A) , fplusok f (finv f).Hint Resolve fplusok_inv.Lemma fplusok_le_compat : ∀ (Uprop.Univprop.fplusokA:Type)(f1 f2 g1 g2:MF A), 3.00em
fplusok f2 g2 → f1 ≤ f2 → g1 ≤ g2 → fplusok f1 g1.Hint Resolve fle_fplus_left fle_fplus_right fle_zero fle_one feq_finv_finv finv_le_compatfle_fmult fle_fesp_left fle_fesp_right fle_fconj_left fle_fconj_right.Lemma fconj_fplusok : ∀ (A:Type)(f g h:MF A), 6.00em
fplusok g h → fplusok (fconj f g) (fconj f h).Hint Resolve fconj_fplusok.Definition Fconj A : Uprop.Univprop.flefplusrightMF A →m m m m m
5.22 Fixpoints of functions of type A →[0,1]
Section FixDef.Variable A :Type.Variable F : MF A →m m m
5.23 Properties of (pseudo-)barycenter of two points
Lemma Uinv_bary :1.50em
∀ a b x y : U, a ≤ [1-]b → 3.00em
[1-] (a × x + b × y) == a × [1-] x + b × [1-] y + [1-] (a + b).Hint Resolve Uinv_bary.Lemma Uinv_bary_le : 1.50em
∀ a b x y : U, a ≤ [1-]b → a × [1-] x + b × [1-] y ≤ [1-] (a × x + b × y).Hint Resolve Uinv_bary_le.Lemma Uinv_bary_eq : ∀ a b x y : U, a == [1-]b → 3.00em
[1-] (a × x + b × y) == a × [1-] x + Utheory."[1-].x"b × [1-] y.Hint Resolve Uinv_bary_eq.Lemma bary_refl_eq : ∀ Utheory."[1-].x"a b x, a == [1-]b → a × x + b × x == x.Hint Resolve bary_refl_eq.Lemma bary_le_left : ∀ a b x Utheory."[1-].x"y, [1-]b ≤ a → x ≤ y → x ≤ a × x + b × y.Lemma bary_le_right : ∀ a b x y, a ≤ [1-]b → x ≤ y → a × x + b × y ≤ y.Hint Resolve bary_le_left bary_le_right.Lemma bary_up_eq : ∀ a b x y : U, a == [1-]b → x ≤ y → a × x + b × Utheory."[1-].x"y == x + b × (y - x).Lemma bary_up_le : ∀ a b x y : U, a ≤ [1-]b → a × x + b × y ≤ x + b × (y - x).Lemma bary_Uminus_left : 1.50em
∀ a b x y : U, a ≤ [1-]b → (a × x + b × y) - x ≤ b × (y - x).Lemma bary_Uminus_left_eq : 2.00em
∀ a b x y : U, a == [1-]b → x ≤ y → (a × x + Uprop.Univprop.Upluszerorightb × y) - x == b × (y - x).Lemma Uminus_bary_left 1.50em
: ∀ a b x y : U, [1-]a ≤ b → x - (a × x + b × y) ≤ b × (x - y).Lemma Uminus_bary_left_eq1.50em
: ∀ a b x y : U, Uprop.Univprop.Uinvlepermrighta == [1-]b → y ≤ x → x - (a × x + b × y) == b × (x - y).Hint Resolve bary_up_eq bary_up_le bary_Uminus_left Utheory."[1-].x"Uminus_bary_left bary_Uminus_left_eq Uminus_bary_left_eq.Lemma diff_bary_left_eq1.50em
: ∀ a b x y : U, a == [1-]b → diff x (a × x + b × y) == b × diff x y.Hint Resolve diff_bary_left_eq.Lemma Uinv_half_bary :1.50em
∀ x y : U, [1-] ([1/2] × x + [1/2] × y) == [1/2] × [1-] x + [1/2] × [1-] y.Hint Resolve Uinv_half_bary.
5.24 Properties of generalized sums sigmasigma
Lemma sigma_plus : ∀ (f g : nat → U) (n:nat),1.50em
sigma (fun k ⇒ (f k) + (g k)) Uprop.Univprop.Uinvhalfbaryn == sigma f n + sigma g n.Definition retract (f : nat → U) (n : nat) := ∀ k, (k < n)%nat → f k ≤ [1-] (sigma f k).Lemma retract0 : ∀ (f : nat → U), retract f 0.Lemma retract_pred : ∀ (f : nat → U) (n : nat), retract f (S n) → retract f n.Lemma retractS: ∀ (f : nat → U) (n : Coq.Init.Datatypes.natnat), retract f (S n) → f n ≤ [1-] (sigma f n).Hint Immediate retract_pred retractS.Lemma retractS_inv :2.50em
∀ (f : nat → U) (n : nat), retract f (S n) → sigma f n ≤ [1-] f n.Hint Immediate retractS_inv.Lemma retractS_intro: ∀ (f : nat → U) (n : nat),1.50em
retract f n → f n ≤ [1-] (sigma f n) → retract f (S n).Hint Resolve retract0 retractS_intro.Lemma retract_lt : ∀ (f : nat → U) (Uprop.Univprop.Uinvlepermrightn : nat), sigma f n < 1 → retract f n.Lemma retract_unif :2.00em
∀ (f : nat → U) (n : nat),6.50em
(∀ k, (k<=n)%nat → f k ≤ [1/]1+n) → retract f (S n).Hint Resolve retract_unif.Lemma sigma_mult :1.00em
∀ (f : nat → U) n c, retract f n → sigma (fun k ⇒ c × (f k)) n == c × (sigma f Coq.Init.Datatypes.Sn).Hint Resolve Uprop.Univprop.Uinvlepermrightsigma_mult.Lemma sigma_prod_maj : ∀ (f g : nat → U) n,1.50em
Uprop.Univprop.sigmamultsigma (fun k ⇒ (f k) × (g Coq.Init.Datatypes.natk)) n ≤ sigma Uprop.Univprop.retractf n.Hint Resolve sigma_prod_maj.Lemma sigma_prod_le : ∀ (f g : nat → U) (c:U), (∀ k, (f k) ≤ c)1.50em
→ ∀ n, retract g n → sigma (fun k ⇒ (f k) × (Uprop.Univprop.Udistrplusleftg k)) n ≤ c × (sigma g Uprop.Univprop.retractSn).Lemma sigma_prod_ge : ∀ (f g : nat → U) (c:U), (∀ k, c ≤ (f k))1.50em
→ ∀ n, (retract g n) → Uprop.Univprop.sigmac × (sigma g n) ≤ (sigma (fun k ⇒ (f k) × (g k)) n).Hint Resolve sigma_prod_maj sigma_prod_le sigma_prod_ge.Lemma sigma_inv : ∀ (f g : nat → U) (n:nat), (retract Uprop.Univ.Uf n) →1.00em
[1-] (sigma (fun k ⇒ f k × g k) n) == (sigma (fun k ⇒ f k × [1-] (g Uprop.Univprop.sigmak)) n) + [1-] (sigma f n).
5.25 Product by an integer5.25.1 Definition of NmultNmult nn xx written nn */ xx
Fixpoint Nmult (n: nat) (x : Utheory."[1-].x"U) {struct n} : U :=1.50em
match n with O ⇒ 0 | (S O) ⇒ x | S p ⇒ x + (Nmult p x) end.
5.25.2 Condition for nn */ xx to be exact : n = 0 or x ≤ 1/n
Definition Nmult_def (n: Ccpo.Oeqsymnat) (x : U) :=1.50em
match n with O ⇒ True | S p ⇒ x ≤ [1/]1+p end.Lemma Nmult_def_O : ∀ x, Nmult_def O x.Hint Resolve Nmult_def_O.Lemma Nmult_def_1 : ∀ x, Nmult_def (S O) x.Hint Resolve Nmult_def_1.Lemma Nmult_def_intro : ∀ n x , x ≤ [1/]1+n → Nmult_def (S n) x.Hint Resolve Nmult_def_intro.Lemma Nmult_def_Unth: ∀ n , Nmult_def (S n) ([1/]1+n).Hint Resolve Nmult_def_Unth.Lemma Nmult_def_pred : ∀ n x, Nmult_def (S n) x → Nmult_def n x.Hint Immediate Nmult_def_pred.Lemma Nmult_defS : ∀ n x, Nmult_def (S n) Uprop.Univprop.Nmultdefintrox → x ≤ [1/]1+n.Hint Immediate Nmult_defS.Lemma Nmult_def_class : ∀ n p, class (Nmult_def n p).Hint Resolve Nmult_def_class.Infix "*/" := Nmult (at level 60) : U_scope.Add Morphism Nmult_def with signature eq =⇒Oeq =⇒iff as Nmult_def_eq_compat.Save.Lemma Nmult_def_zero : ∀ n, Nmult_def n 0.Hint Resolve Nmult_def_zero.
5.25.3 Properties of nn */ xx
Lemma Nmult_0 : ∀ (x:U), O*/x = 0.Lemma Nmult_1 : ∀ (x:U), (S O)*/x = x.Lemma Nmult_zero : ∀ n, n */ 0 == 0.Lemma Nmult_SS : ∀ (n:nat) (x:U), S (S n) */x = x + (S n */ x).Lemma Nmult_2 : ∀ (x:U), 2*/x = x + x.Lemma Nmult_S : ∀ (n:nat) (x:U), S n */ Uprop.Univprop.Nmult1x == x + (n*/x).Hint Resolve Nmult_1 Nmult_SS Nmult_2 Nmult_S.Add Morphism Nmult with signature eq =⇒Oeq =⇒Oeq as Nmult_eq_compat.Save.Hint Immediate Nmult_eq_compat.Lemma Nmult_eq_compat_left : ∀ (n:nat) (x y:U), x == y → n */ x == n */ y.Lemma Nmult_eq_compat_right : ∀ (n m:nat) (x:U), (n = Ccpo.Oeqm)%nat → n */ x == m */ x.Hint Resolve Nmult_eq_compat_right.Lemma Nmult_le_compat_right : ∀ n x y, x ≤ y → n */ x ≤ n */ y.Lemma Nmult_le_compat_left : ∀ n m x, (n ≤ m)%nat → n */ x ≤ m */ x.Hint Resolve Nmult_eq_compat_right Nmult_le_compat_right Nmult_le_compat_left.Lemma Nmult_le_compat : ∀ (n m:nat) x y, n ≤ m → x ≤ y → n */ x ≤ m */ y.Hint Immediate Nmult_le_compat.Instance Nmult_mon2 : monotonic2 Nmult.Save.Definition NMult : nat →m m
5.26 Conversion from booleans to U
Definition B2U :MF bool := fun (b:bool) ⇒ if b then 1 else 0.Definition NB2U :MF bool := fun (b:bool) ⇒ if b then 0 else 1.Lemma B2Uinv : Uprop.Univprop.B2UNB2U == Coq.Init.Datatypes.boolfinv B2U.Lemma NB2Uinv : B2U == finv NB2U.Hint Resolve B2Uinv NB2Uinv.
5.27 Particular sequences
pmin (p )(n ) = p − 1/2n
5.27.1 Properties of pminpmin
Lemma pmin_esp_S : ∀ p n, pmin (Ccpo.Oeqp & p) n == pmin p (S n) & pmin p (S n).Lemma Coq.Init.Datatypes.Spmin_esp_le : ∀ p n, pmin p (S n) ≤ [1/2] × (pmin (p & p) n) + [1/2].Lemma pmin_plus_eq : ∀ Uprop.Univprop.pminp n, p ≤ [1/2] → pmin p (S n) == [1/2] × (pmin (p + p) n).Lemma pmin_0 : ∀ p:U, pmin p O == 0.Lemma pmin_le : ∀ (p:U) (n:nat), p - ([1/]1+n) ≤ Uprop.Univprop.pmin0pmin p n.Hint Resolve pmin_0 pmin_le.Lemma pmin_le_compat : ∀ p (n m : nat), n ≤ m → pmin p n ≤ pmin p m.Hint Resolve pmin_le_compat.Instance pmin_mon : ∀ p, Uprop.Univprop.pminmonotonic (pmin p).Save.Definition Pmin (p:U) :nat →m
5.27.2 Dyadic numbers
Fixpoint exp2 (n:nat) : nat :=1.50em
match n with O ⇒ (1%nat) | S p ⇒ (2 × (exp2 p))%nat end.Lemma exp2_pos : ∀ n, (O < exp2 n)%nat.Hint Resolve Coq.Init.Datatypes.Sexp2_pos.Lemma S_pred_exp2 : ∀ Uprop.Univprop.exp2posn, S (pred (exp2 n))=exp2 Uprop.Univprop.exp2n.Hint Resolve S_pred_exp2.Notation "k /2^ p" := (k */ ([1/2])^p) (at Uprop.Univprop.exp2level 35, no associativity).Lemma Uprop.Univprop.exp2Unth_eq : ∀ n p, n*/ p == [1-]p → p == [1/]1+n.Lemma Unth_half : ∀ n, (O<n)%nat → [1/]1+(pred (n+n)) == [1/2] × [1/]1+pred n.Lemma Unth_exp2 : ∀ p, [1/2]^p == [1/]1+pred (exp2 Uprop.Univ.Umultoneleftp).Hint Resolve Unth_exp2.Lemma Coq.Init.Peano.predNmult_exp2 : ∀ p, (exp2 p)/2^p == 1.Hint Resolve Nmult_exp2.
5.28 Tactic for simplification of goals
Ltac Usimpl := match goal with2.00em
⊢ context [(Uplus 0 ?x)] ⇒ setoid_rewrite (Uplus_zero_left x)0.50em
| ⊢ context [(Uplus ?x 0)] ⇒ setoid_rewrite (Uplus_zero_right x)0.50em
| ⊢ context [(Uplus 1 ?x)] ⇒ setoid_rewrite (Uplus_one_left x)0.50em
| ⊢ context [(Uplus ?x 1)] ⇒ setoid_rewrite (Uplus_one_right x)0.50em
| ⊢ Uprop.Univ.Upluscontext [(Umult 0 ?x)] ⇒ setoid_rewrite (Umult_zero_left x)0.50em
| ⊢ context [(Umult ?x 0)] ⇒ setoid_rewrite (Umult_zero_right x)0.50em
| ⊢ context [(Umult 1 ?x)] ⇒ setoid_rewrite (Umult_one_left x)0.50em
| ⊢ context [(Umult ?x 1)] ⇒ setoid_rewrite (Umult_one_right x)0.50em
| ⊢ Uprop.Univ.Umultcontext [(Uesp 0 ?x)] ⇒ setoid_rewrite (Uesp_zero_left x)0.50em
| ⊢ context [(Uesp ?x 0)] ⇒ setoid_rewrite (Uesp_zero_right x)0.50em
| ⊢ context [(Uesp 1 ?x)] ⇒ setoid_rewrite (Uesp_one_left x)0.50em
| ⊢ context [(Uesp ?x 1)] ⇒ setoid_rewrite (Uesp_one_right x)0.50em
| ⊢ context [(Uminus 0 ?x)] ⇒ setoid_rewrite (Uminus_le_zero 0 x); 20.00em
[idtac|apply (Upos x)]0.50em
| ⊢ context [(Uminus ?x 0)] ⇒ setoid_rewrite (Uminus_zero_right x)0.50em
| ⊢ context [(Uminus ?x 1)] ⇒ setoid_rewrite (Uminus_le_zero x 1);20.00em
[idtac|apply (Unit x)]0.50em
| ⊢ context [([1-] ([1-] ?x))] ⇒ setoid_rewrite (Uinv_inv x)0.50em
| ⊢ context [([1-] 1)] ⇒ setoid_rewrite Uinv_one0.50em
| ⊢ context [([1-] 0)] ⇒ setoid_rewrite Uinv_zero0.50em
| ⊢ context [([1/]1+O)] ⇒ setoid_rewrite Unth_zero0.50em
| ⊢ context [?x^O] ⇒ setoid_rewrite (Uexp_0 x)0.50em
| ⊢ context [?x^(S O)] ⇒ setoid_rewrite (Uexp_1 x)0.50em
| ⊢ context [0^(?n)] ⇒ setoid_rewrite Uexp_zero; [idtac|omega]0.50em
| ⊢ context [U1^(?n)] ⇒ setoid_rewrite Uexp_one0.50em
| ⊢ context [(Nmult 0 ?x)] ⇒ setoid_rewrite (Nmult_0 x)0.50em
| ⊢ context [(Nmult 1 ?x)] ⇒ setoid_rewrite (Nmult_1 x)0.50em
| ⊢ context [(Nmult ?n 0)] ⇒ setoid_rewrite (Nmult_zero n)0.50em
| ⊢ context [(sigma ?f O)] ⇒ setoid_rewrite (sigma_0 f)0.50em
| ⊢ context [(sigma ?f (S O))] ⇒ setoid_rewrite (sigma_1 f)0.50em
| ⊢ (Ole (Uplus ?x ?y) (Coq.Init.Datatypes.OUplus ?x ?z)) ⇒ apply Uplus_le_compat_right0.50em
| ⊢ (Ole (Uplus ?x ?z) (Uplus ?y ?z)) ⇒ apply Uprop.Univprop.sigma1Uplus_le_compat_left0.50em
| ⊢ (Ole (Uplus ?x ?z) (Uplus ?z ?y)) ⇒ setoid_rewrite (Uplus_sym Ccpo.Olez y);23.00em
apply Uplus_le_compat_left0.50em
| ⊢ (Ole (Uplus ?x ?y) (Ccpo.OleUplus ?z ?x)) ⇒ setoid_rewrite (Uplus_sym x y);23.00em
apply Uplus_le_compat_left0.50em
| ⊢ (Ole (Uinv ?y) (Uinv ?x)) ⇒ apply Uinv_le_compat0.50em
| ⊢ (Ole (Uminus ?x ?y) (Uplus ?x ?z)) ⇒ apply Uminus_le_compat_right0.50em
| ⊢ (Ole (Uminus ?x ?z) (Uplus ?y ?z)) ⇒ apply Uminus_le_compat_left0.50em
| ⊢ ((Uinv ?x) == (Uinv ?y)) ⇒ apply Uinv_eq_compat0.50em
| ⊢ ((Uplus ?x ?y) == (Uplus ?x ?z)) ⇒ apply Uplus_eq_compat_right0.50em
| ⊢ ((Uplus ?x ?z) == (Uplus ?y ?z)) ⇒ apply Uplus_eq_compat_left0.50em
| ⊢ ((Uplus ?x ?z) == (Uplus ?z ?y)) ⇒ setoid_rewrite (Uplus_sym z y);22.50em
apply Uplus_eq_compat_left0.50em
| ⊢ ((Uplus ?x ?y) == (Uplus ?z ?x)) ⇒ setoid_rewrite (Uplus_sym x y);22.50em
apply Uplus_eq_compat_left0.50em
| ⊢ ((Uminus ?x ?y) == (Uplus ?x ?z)) ⇒ apply Uminus_eq_compat;[apply Oeq_refl|idtac]0.50em
| ⊢ ((Uminus ?x ?z) == (Uplus ?y ?z)) ⇒ apply Uminus_eq_compat;[idtac|apply Oeq_refl]0.50em
| ⊢ (Ole (Umult ?x ?y) (Ccpo.OeqreflUmult ?x ?z)) ⇒ apply Umult_le_compat_right0.50em
| ⊢ (Ole (Umult ?x ?z) (Umult ?y ?z)) ⇒ apply Umult_le_compat_left0.50em
| ⊢ (Ole (Umult ?x ?z) (Umult ?z ?y)) ⇒ setoid_rewrite (Umult_sym Ccpo.Olez y);22.50em
apply Umult_le_compat_left0.50em
| ⊢ (Ole (Umult ?x ?y) (Umult ?z ?x)) ⇒ setoid_rewrite (Umult_sym x y);22.50em
apply Umult_le_compat_left0.50em
| ⊢ ((Umult ?x ?y) == (Umult ?x ?z)) ⇒ apply Umult_eq_compat_right0.50em
| ⊢ ((Umult ?x ?z) == (Umult ?y ?z)) ⇒ apply Umult_eq_compat_left0.50em
| ⊢ ((Umult ?x ?z) == (Umult ?z ?y)) ⇒ setoid_rewrite (Umult_sym z Uprop.Univ.Umulty);22.50em
apply Umult_eq_compat_left0.50em
| ⊢ ((Umult ?x ?y) == (Umult ?z ?x)) ⇒ setoid_rewrite (Umult_sym x y);22.50em
apply Umult_eq_compat_leftend.
Properties of current values
Notation "[1/3]" := (Unth 2%nat).Notation "[1/4]" := (Unth 3%nat).Notation "[1/8]" := (Unth 7).Notation "[3/4]" := (Uinv [1/4]).Lemma half_square : [1/2]×[1/2]==[1/4].Lemma half_cube : [1/2]×[1/2]×[1/2]==[1/8].Lemma three_quarter_decomp : [3/4]==[1/2]+[1/4].
Equality is not true, even for monotonic sequences fot instance n/m
Lemma Ulub_Uglb_exch_le : ∀ f : nat → nat → U,2.50em
Ulub (fun n ⇒ Uglb (fun m ⇒ Ccpo.Olereflf n m)) ≤ Uglb (fun m ⇒ Ulub (fun n ⇒ f n m)).
5.29 Intervals
Record IU : Type := mk_IU {low:U; up:U; proper:low ≤ up}.Hint Resolve proper.
the all set : [0,1]
Definition full := mk_IU 0 1 (Upos 1).
singleton : [x]
Definition singl (x:U) := mk_IU Uprop.Univprop.Iinx x (Uprop.Univprop.lowOle_refl x).
down segment : [0,x]
Definition inf (x:U) := mk_IU 0 x (Upos x).
up segment : [x,1]
Definition sup (x:U) := mk_IU x 1 (Unit x).
Definition Iin (x:U) (I:IU) := low I ≤ x ∧ Uprop.Univprop.Iininfelimx ≤ up I.Definition Iincl I J := low J ≤ low I ∧ up I ≤ up J.Definition Ieq I J := low I == low J ∧ up I == up J.Hint Unfold Iin Iincl Ieq.
Lemma Iin_low : ∀ I, Iin (Uprop.Univprop.Iinsinglintrolow I) I.Lemma Iin_up : ∀ I, Iin (up I) I.Hint Resolve Iin_low Iin_up.Lemma Iin_singl_elim : ∀ x y, Iin x (singl y) → x == y.Lemma Iin_inf_elim : ∀ x y, Iin x (inf y) → x ≤ y.Lemma Iin_sup_elim : ∀ x y, Iin x (sup y) → y ≤ x.Lemma Iin_singl_intro : ∀ x y, x == y → Iin x (singl y).Lemma Iin_inf_intro : ∀ x y, x ≤ y → Iin x (Uprop.Univ.Uleclassinf y).Lemma Iin_sup_intro : ∀ x y, y ≤ x → Iin x (sup y).Hint Immediate Iin_inf_elim Iin_sup_elim Iin_singl_elim.Hint Resolve Iin_inf_intro Iin_sup_intro Iin_singl_intro.Lemma Uprop.Univprop.IinclclassIin_class : ∀ I x, class (Iin x I).Lemma Iincl_class : ∀ I J, class (Iincl I J).Lemma Ieq_class : ∀ I J, class (Ieq I J).Hint Resolve Iin_class Iincl_class Ieq_class.Lemma Iincl_in : ∀ I J, Iincl I J → ∀ x, Iin x I → Iin x J.Lemma Iincl_low : ∀ I J, Iincl I J → low J ≤ low I.Lemma Iincl_up : ∀ I J, Iincl I J → up I ≤ up J.Hint Immediate Iincl_low Iincl_up.Lemma Iincl_refl : ∀ I, Iincl I I.Hint Resolve Iincl_refl.Lemma Iincl_trans : ∀ I J K, Iincl I J → Iincl J K → Iincl I K.Instance IUord : ord IU := {Oeq := fun I J ⇒ Ieq Ccpo.IordI J; Uprop.Univ.UOle := fun I J ⇒ Iincl J I}.Defined.Lemma low_le_compat : ∀ I J:IU, I ≤ Uprop.Univprop.IeqinclsymJ → low I ≤ low J.Lemma up_le_compat : ∀ I J : IU, I ≤ J → up J ≤ up I.Instance low_mon : monotonic low.Save.Definition Low : Uprop.Univprop.IeqIU →m m
5.29.4 Operations on intervals
Definition Iplus I J := mk_IU (low I + low J) (up Uprop.Univprop.upI + up J)21.50em
(Uplus_le_compat _ _ _ _ (proper I) (proper J)).Lemma low_Iplus : ∀ I J, low (Iplus I J)=low I + low J.Lemma up_Iplus : ∀ I J, up (Iplus I J)=up I + up J.Lemma Iplus_in : ∀ I J Uprop.Univprop.lowx y, Iin Uprop.Univprop.upx I → Iin y J → Iin (x+y) (Iplus I J).Lemma lplus_in_elim :∀ I J z, low I ≤ [1-]up J → Iin z (Iplus I J)8.00em
→ exc (fun x ⇒ Iin x I ∧25.50em
exc (fun y ⇒ Iin y J ∧ z==x+y)).Definition Imult I J := mk_IU (low I × low J) (up I × up J)22.00em
(Umult_le_compat _ _ _ _ (proper I) (proper J)).Lemma Uprop.Univprop.Iinlow_Imult : ∀ I J, low (Imult I J) = low I × low J.Lemma up_Imult : ∀ I J, up (Imult I J) = up Uprop.Univprop.IinI × up J.Definition Imultk p I := mk_IU (p × low I) (p × up I) (Umult_le_compat_right p _ _ (proper I)).Lemma low_Imultk : ∀ p I, low (Imultk p I) = p × low I.Lemma up_Imultk : ∀ p I, up (Imultk p I) = p × up I.Lemma Imult_in : ∀ I J x y, Iin x I → Iin y J → Iin (x*y) (Imult I J).Lemma Imultk_in : ∀ p I x , Iin x I → Iin (Coq.Init.Datatypes.natp*x) (Imultk p Uprop.Univprop.IinclI).
5.29.5 Limits of intervals
Definition Ilim : ∀ I: nat →m m m m m
5.30 Limits inf and sup
Definition fsup (f:nat->U) (n:nat) := Ulub (fun k ⇒ f (n+k)%nat).Definition finf (f:nat->U) (n:nat) := Uglb (fun k ⇒ f (n+k)%nat).Lemma fsup_incr : ∀ (f:nat->U) n, fsup f (S n) ≤ fsup f n.Hint Resolve Ccpo.natmonotonicfsup_incr.Lemma finf_incr : ∀ (f:nat->U) Coq.Init.Datatypes.natn, finf f Coq.Init.Datatypes.natn ≤ finf f (S n).Hint Resolve finf_incr.Instance fsup_mon : ∀ f, monotonic (o2:=Iord U) (fsup f).Save.Instance finf_mon : ∀ f, monotonic (finf f).Save.Definition Fsup (f:nat->U) : nat →m m
5.31 Limits of arbitrary sequences
Lemma liminf_incr : ∀ f:nat →m m m m m
5.32 Definition and properties of series : infinite sums
Definition serie (f : nat → U) : U := lub (sigma f).Lemma serie_le_compat : ∀ (f g: nat → U), 0.50em
(∀ k, f Ccpo.mseqliftleftk ≤ g k) → serie f ≤ serie g.Lemma serie_eq_compat : ∀ (f g: nat → U), 0.50em
(∀ k, f k == g k) → serie f == serie g.Lemma serie_sigma_lift : ∀ (f :nat → U) (n:nat), 5.00em
serie f == sigma f n + serie (fun k ⇒ f (n + k)%nat).Lemma Uprop.Univprop.sigmaserie_S_lift : ∀ (Uprop.Univprop.sigmanotzerof :nat → U),5.00em
serie f == f O + serie (fun k ⇒ Ccpo.lelubf (S k)).Lemma serie_zero : ∀ f, (∀ k, f k ==0) → serie f ==0.Lemma serie_not_zero : ∀ f k, 0 < f k → 0 < serie f.Lemma serie_zero_elim : ∀ f, serie f == 0 → ∀ k, f Uprop.Univprop.UMinusk ==0.Hint Resolve serie_eq_compat serie_le_compat serie_zero.Lemma serie_le : ∀ f k, f k ≤ serie f.Lemma serie_minus_incr : ∀ f :nat →m m m
6 Monads.v: Monads for randomized constructions
Require Export UpropUprop.
6.1 Definition of monadic operators as the cpo of monotonic oerators
Definition M (A:Type) := MF A →m
6.2 Properties of monadic operators
Lemma law1 : ∀ (A B:Type) (x:A) (F:A → M B) (f:MF B), star (unit x) F f == F x Monads.Monad.unitf.Lemma law2 :0.50em
∀ (A:Type) (a:M A) (f:MF A), star a (fun Monads.Monad.UP.MFx:A ⇒ Monads.Monad.starunit x) f == a (fun x:A ⇒ f x).Lemma law3 :0.50em
∀ (A B C:Type) (a:M A) (F:A → M B) (Monads.Monad.MG:B → M C)1.50em
(f:MF C), star (star a F) G f == star a (fun x:A ⇒ Monads.Monad.starstar (F x) G) f.
6.3 Properties of distributions6.3.1 Expected properties of measures
Definition stable_inv (A:Type) (m:M A) : Prop := ∀ f :MF A, m (finv f) ≤ [1-] (m Monads.Monad.UP.finvf).Definition stable_plus (A:Type) (m:M A) : Prop :=1.00em
∀ f g:MF A, fplusok f g → m (fplus f g) == (m f) + (m g).Definition le_plus (A:Type) (m:M A) : Prop :=1.00em
∀ f g:MF A, fplusok f g → (m f) + (m g) ≤ m (fplus f g).Definition le_esp (A:Type) (m:M A) : Prop :=1.00em
∀ f g: MF A, (m f) & (m g) ≤ m (fesp f g).Definition le_plus_cte (A:Type) (m:M A) : Prop :=1.00em
∀ (f : MF A) (k:U), m (fplus f (fcte A k)) ≤ m f + k.Definition stable_mult (A:Type) (m:M A) : Prop :=1.00em
∀ (k:U) (f:MF A), m (fmult k f) == k × (Monads.Monad.UP.fmultm f).
6.3.2 Stability for equality
Lemma stable_minus_distr : ∀ (A:Type) (m:M A),2.50em
stable_plus m → stable_inv m →2.50em
∀ (f g : MF A), g ≤ f → m (fminus f g) == m f - m g.Hint Resolve stable_minus_distr.Lemma inv_minus_distr : ∀ (A:Type) (m:M A),2.50em
stable_plus m → stable_inv m →2.50em
∀ (f : MF A), m (finv f) == m (fone Monads.Monad.UP.finvA) - m f.Hint Resolve inv_minus_distr.Lemma le_minus_distr : ∀ (A : Type)(m:M A),2.00em
∀ (Monads.Monad.Mf g:A → U), m (fminus f g) ≤ m f.Hint Resolve le_minus_distr.Lemma le_plus_distr : ∀ (A : Type)(m:M A),2.00em
stable_plus m → stable_inv m → ∀ (f g:MF A), m (fplus f g) ≤ m f + m g.Hint Resolve le_plus_distr.Lemma le_esp_distr : ∀ (A : Type) (m:M A), 2.50em
stable_plus m → stable_inv m → le_esp m.Lemma unit_stable_eq : ∀ (A:Type) (x:A), stable (unit x).Lemma star_stable_eq : ∀ (A B:Type) (m:M A) (F:A → M B), Monads.Monad.Mstable (star Monads.Monad.Mm F).
6.3.3 Stability for inversion
Lemma unit_stable_inv : ∀ (A:Type) (x:A), stable_inv (unit x).Lemma star_stable_inv : ∀ (A B:Type) (m:M A) (F:A → M B),1.50em
stable_inv m → (∀ a:A, stable_inv (F a)) → stable_inv (star m Monads.Monad.stableinvF).
6.3.4 Stability for addition
Lemma unit_stable_plus : ∀ (A:Type) (x:A), stable_plus (unit x).Lemma star_stable_plus : ∀ (A B:Type) (m:M A) (F:A → M B),1.50em
stable_plus m →1.50em
(∀ a:A, ∀ f g, fplusok f g → (F a f) ≤ Uinv (F a g))1.50em
→ (∀ a:A, stable_plus (F a)) → stable_plus (star m F).Lemma unit_le_plus : ∀ (A:Type) (x:A), le_plus (unit x).Lemma star_le_plus : ∀ (A B:Type) (m:M A) (F:A → M B),1.50em
le_plus m →1.50em
(∀ a:A, ∀ f g, fplusok f g → (F a f) ≤ Uinv (F a g))1.50em
→ (∀ a:A, le_plus (F a)) → le_plus (star m F).
6.3.5 Stability for product
Lemma unit_stable_mult : ∀ (A:Type) (x:A), stable_mult (unit x).Lemma star_stable_mult : ∀ (A B:Type) (m:M A) (F:A → M B),1.50em
stable_mult m → (∀ a:A, stable_mult (F a)) → stable_mult (star m F).
6.3.6 Continuity
Lemma unit_continuous : ∀ (A:Type) (x:A), continuous (unit x).Lemma star_continuous : ∀ (A B : Type) (m : M A)(F: A → M B),2.00em
continuous m → (∀ x, continuous (F x)) → continuous (star m Ccpo.continuousF).
7 Probas.v: The monad for distributions
Require Export MonadsMonads.
7.1 Definition of distributionDistributions are monotonic measure functions such that
µ (1−f ) ≤ 1−µ(f ) f ≤ 1 −g ⇒ µ(f +g )=µ(f )+µ(g )µ(k × f ) = k × µ(f ) µ(lub f n lub µ(f n
Record distr (A:Type) : Type := 1.00em
{mu : M A;1.50em
mu_stable_inv : stable_inv mu; 1.50em
mu_stable_plus : stable_plus mu;1.50em
mu_stable_mult : stable_mult mu;1.50em
mu_continuous : continuous mu}.Hint Resolve mu_stable_plus mu_stable_inv mu_stable_mult mu_continuous.
7.2 Properties of measures
Lemma mu_monotonic : ∀ (A : Type)(m: distr A), monotonic (mu m).Hint Resolve mu_monotonic.Implicit Arguments mu_monotonic [A].Lemma mu_stable_eq : ∀ (A : Type)(m: distr A), stable (mu m).Hint Resolve mu_stable_eq.Implicit Arguments mu_stable_eq [A].Lemma mu_zero : ∀ (A : Type)(m: distr A), mu m (fzero A) == 0.Hint Resolve mu_zero.Lemma mu_zero_eq : ∀ (A : Type)(m: distr A) f, 1.50em
(∀ x, f x == 0) → mu m f == 0.Lemma mu_one_inv : ∀ (A : Type)(m:distr A),1.50em
mu m (fone A) == 1 → ∀ f, mu m (finv f) == [1-] (mu m f).Hint Resolve mu_one_inv.Lemma mu_fplusok : ∀ (A : Type)(m:distr A) f g, fplusok f g → 6.00em
mu m f ≤ [1-] mu m g.Hint Resolve mu_fplusok.Lemma mu_le_minus : ∀ (A : Type)(m:distr A) (f g:MF A),2.50em
mu m (fminus f g) ≤ mu m f.Hint Resolve mu_le_minus.Lemma mu_le_plus : ∀ (A : Type)(m:distr A) (f g:MF A),2.50em
mu m (fplus f g) ≤ mu m f + mu m g.Hint Resolve mu_le_plus.Lemma mu_cte : ∀ (A : Type)(m:(distr A)) (c:U),1.50em
mu m (fcte A c) == c × mu m (fone A).Hint Resolve mu_cte.Lemma mu_cte_le : ∀ (A : Type)(m:distr A) (c:U), mu m (fcte A c) ≤ c.Lemma mu_cte_eq : ∀ (A : Type)(m:distr A) (c:U),1.50em
mu m (fone A) == 1 → mu m (fcte A c) == c.Hint Resolve mu_cte_le mu_cte_eq.Lemma mu_stable_mult_right : ∀ (A : Type)(m:distr A) (c:U) (f : MF A),1.50em
mu m (fun x ⇒ (f x) × c) == (mu m f) × c.Lemma mu_stable_minus : ∀ (A:Type) (m:distr A)(f g : MF A),0.50em
g ≤ f → mu m (fun x ⇒ f x - g x) == mu m f - mu m g.Lemma mu_inv_minus : ∀ (A:Type) (m:distr A)(f: MF A), mu m (finv f) == mu m (fone A) - mu m f.Lemma mu_stable_le_minus : ∀ (A:Type) (m:distr A)(f g : MF A),0.50em
mu m f - mu m g ≤ mu m (fun x ⇒ f x - g x).Lemma mu_inv_minus_inv : ∀ (A:Type) (m:distr A)(f: MF A), 2.50em
mu m (finv f) + [1-](mu m (fone A)) == [1-](mu m f).Lemma mu_le_esp_inv : ∀ (A:Type) (m:distr A)(f g : MF A),0.50em
([1-]mu m (finv f)) & mu m g ≤ mu m (fesp f g).Hint Resolve mu_le_esp_inv.Lemma mu_stable_inv_inv : ∀ (A:Type) (m:distr A)(f : MF A),6.50em
mu m f ≤ [1-] mu m (finv f).Hint Resolve mu_stable_inv_inv.Lemma mu_stable_div : ∀ (A:Type) (m:distr A)(k:U)(f : MF A),5.00em
¬0==k → f ≤ fcte A k → mu m (fdiv k f) == mu m f / k.Lemma mu_stable_div_le : ∀ (A:Type) (m:distr A)(k:U)(f : MF A),5.00em
¬0==k → mu m (fdiv k f) ≤ mu m f / k.Lemma mu_le_esp : ∀ (A:Type) (m:distr A)(f g : MF A),0.50em
mu m f & mu m g ≤ mu m (fesp f g).Hint Resolve mu_le_esp.Lemma mu_esp_one : ∀ (A:Type)(m:distr A)(f g:MF A),1.50em
1<=mu m f → mu m g == mu m (fesp f g).Lemma mu_esp_zero : ∀ (A:Type)(m:distr A)(f g:MF A),1.50em
mu m (finv f) ≤ 0 → mu m g == mu m (fesp f g).Instance Odistr (A:Type) : ord (distr A) := 2.00em
{Ole := fun (f g : distr A) ⇒ mu f ≤ mu g;2.50em
Oeq := fun (f g : distr A) ⇒ mu f == mu g}.Defined.
7.3 Monadic operators for distributions
Definition Munit : ∀ A:Type, A → distr A.Defined.Definition Mlet : ∀ A B:Type, distr A → (A → distr B) → distr B.Defined.Lemma Munit_simpl : ∀ (A:Type) (q:A → U) x, mu (Munit x) q = q x.Lemma Mlet_simpl : ∀ (A B:Type) (m:distr A) (M:A → distr B) (f:B → U),2.50em
mu (Mlet m M) f = mu m (fun x ⇒ (mu (M x) f)).
7.4 Operations on distributions
Lemma Munit_eq_compat : ∀ A (x y : A), x = y → Munit x == Munit y.Lemma Mlet_le_compat : ∀ (A B : Type) (m1 m2:distr A) (M1 M2 : A → distr B), 1.00em
m1 ≤ m2 → M1 ≤ M2 → Mlet m1 M1 ≤ Mlet m2 M2.Hint Resolve Mlet_le_compat.Add Parametric Morphism (A B : Type) : (Mlet (A:=A) (B:=B))1.00em
with signature Ole =⇒Ole =⇒Ole 1.00em
as Mlet_le_morphism.Save.Instance Mlet_mon2 : ∀ (A B : Type), monotonic2 (@Mlet A B).Save.Definition MLet (A B : Type) : distr A →m m
7.5 Properties of monadic operators
Lemma Mlet_unit : ∀ (A B:Type) (x:A) (m:A → distr B), Mlet (Munit x) m == m x.Lemma Mlet_ext : ∀ (A:Type) (m:distr A), Mlet m (fun x ⇒ Munit x) == m.Lemma Mlet_assoc : ∀ (A B C:Type) (m1: distr A) (m2:A → distr B) (m3:B → distr C),2.50em
Mlet (Mlet m1 m2) m3 == Mlet m1 (fun x:A ⇒ Mlet (m2 x) m3).Lemma let_indep : ∀ (A B:Type) (m1:distr A) (m2: distr B) (f:MF B), 3.50em
mu m1 (fun _ ⇒ mu m2 f) == mu m1 (fone A) × (mu m2 f).
7.6 A specific distribution
Definition distr_null : ∀ A : Type, distr A.Defined.Lemma le_distr_null : ∀ (A:Type) (m : distr A), distr_null A ≤ m.Hint Resolve le_distr_null.
7.7 Scaling a distribution
Definition Mmult A (k:MF A) (m:M A) : M A.Defined.Lemma Mmult_simpl : ∀ A (k:MF A) (m:M A) f, Mmult k m f = m (fun x ⇒ k x × f x).Lemma Mmult_stable_inv : ∀ A (k:MF A) (d:distr A), stable_inv (Mmult k (mu d)).Lemma Mmult_stable_plus : ∀ A (k:MF A) (d:distr A), stable_plus (Mmult k (mu d)).Lemma Mmult_stable_mult : ∀ A (k:MF A) (d:distr A), stable_mult (Mmult k (mu d)).Lemma Mmult_continuous : ∀ A (k:MF A) (d:distr A), continuous (Mmult k (mu d)).Definition distr_mult A (k:MF A) (d:distr A) : distr A.Defined.Definition Mdiv A (k:U) (m:M A) : M A := UDiv k @ m.Lemma Mdiv_simpl : ∀ A k (m:M A) f, Mdiv k m f = m f / k.Lemma Mdiv_stable_inv : ∀ A (k:U) (d:distr A)(dk : mu d (fone A) ≤ k),8.00em
stable_inv (Mdiv k (mu d)).Lemma Mdiv_stable_plus : ∀ A (k:U)(d:distr A), stable_plus (Mdiv k (mu d)).Lemma Mdiv_stable_mult : ∀ A (k:U)(d:distr A)(dk : mu d (fone A) ≤ k), 13.00em
stable_mult (Mdiv k (mu d)).Lemma Mdiv_continuous : ∀ A (k:U)(d:distr A), continuous (Mdiv k (mu d)).Definition distr_div A (k:U) (d:distr A) (dk : mu d (fone A) ≤ k)7.00em
: distr A.Defined.
7.8 Conditional probabilities
Definition mcond A (m:M A) (f:MF A) : M A.Defined.Lemma mcond_simpl : ∀ A (m:M A) (f g: MF A),3.00em
mcond m f g = m (fconj f g) / m f.Lemma mcond_stable_plus : ∀ A (m:distr A) (f: MF A), stable_plus (mcond (mu m) f).Lemma mcond_stable_inv : ∀ A (m:distr A) (f: MF A), stable_inv (mcond (mu m) f).Lemma mcond_stable_mult : ∀ A (m:distr A) (f: MF A), stable_mult (mcond (mu m) f).Lemma mcond_continuous : ∀ A (m:distr A) (f: MF A), continuous (mcond (mu m) f).Definition Mcond A (m:distr A) (f:MF A) : distr A :=1.50em
Build_distr (mcond_stable_inv m f) (mcond_stable_plus m f)7.50em
(mcond_stable_mult m f) (mcond_continuous m f).Lemma Mcond_total : ∀ A (m:distr A) (f:MF A), 5.00em
¬ 0 == mu m f → mu (Mcond m f) (fone A) == 1.Lemma Mcond_simpl : ∀ A (m:distr A) (f g:MF A), 3.00em
mu (Mcond m f) g = mu m (fconj f g) / mu m f.Hint Resolve Mcond_simpl.Lemma Mcond_zero_stable : ∀ A (m:distr A) (f g:MF A), 3.00em
mu m g == 0 → mu (Mcond m f) g == 0.Lemma Mcond_null : ∀ A (m:distr A) (f g:MF A), 3.00em
mu m f == 0 → mu (Mcond m f) g == 0.Lemma Mcond_conj : ∀ A (m:distr A) (f g:MF A), 5.00em
mu m (fconj f g) == mu (Mcond m f) g × mu m f.Lemma Mcond_decomp : 2.00em
∀ A (m:distr A) (f g:MF A), 5.00em
mu m g == mu (Mcond m f) g × mu m f + mu (Mcond m (finv f)) g × mu m (finv f).Lemma Mcond_bayes : ∀ A (m:distr A) (f g:MF A), 5.00em
mu (Mcond m f) g == (mu (Mcond m g) f × mu m g) / (mu m f).
7.9 Least upper bound of increasing sequences of distributions
Lemma M_lub_simpl : ∀ A (h: nat →m m m
7.9.1 distributions seen as a Ccpo
Instance cdistr : ∀ (A:Type), cpo (distr A) :=1.50em
{D0 := distr_null A; lub:=mu_lub (A:=A)}.Defined.Lemma distr_lub_simpl : ∀ A (h : nat →m
7.10 Fixpoints
Definition Mfix (A B:Type) (F: (A → distr B) →m m m m m m m
7.11 Continuity
Section Continuity.Variables A B:Type.Instance Mlet_continuous_right 2.00em
: ∀ a:distr A, continuous (D1:= A → distr B) (D2:=distr B) (MLet A B a).Save.Lemma Mlet_continuous_left 2.00em
: continuous (D1:=distr A) (D2:=(A → distr B) →m m m m m
7.12 distribution for flipflipThe distribution associated to flip () is
f ↦ 1/2f (true )+1/2f (false )
Definition flip : M bool := mon (fun (f : bool → U) ⇒ [1/2] × (f true) + [1/2] × (f false)).Lemma flip_stable_inv : stable_inv flip.Lemma flip_stable_plus : stable_plus flip.Lemma flip_stable_mult : stable_mult flip.Lemma flip_continuous : continuous flip.Definition ctrue : MF bool := fun (b:bool) ⇒ if b then 1 else 0.Definition cfalse : MF bool := fun (b:bool) ⇒ if b then 0 else 1.Lemma flip_ctrue : flip ctrue == [1/2].Lemma flip_cfalse : flip cfalse == [1/2].Hint Resolve flip_ctrue flip_cfalse.Definition Flip : distr bool.Defined.Lemma Flip_simpl : ∀ f, mu Flip f = [1/2] × (f true) + [1/2] × (f false).
7.13 Uniform distribution beween 0 and n
Require Arith.
7.13.1 Definition of fnthfnthfnth n k is defined as 1/n +1
Definition fnth (n:nat) : nat → U := fun k ⇒ [1/]1+n.
7.13.2 Basic properties of fnthfnth
Lemma Unth_eq : ∀ n, Unth n == [1-] (sigma (fnth n) n).Hint Resolve Unth_eq.Lemma sigma_fnth_one : ∀ n, sigma (fnth n) (S n) == 1.Hint Resolve sigma_fnth_one.Lemma Unth_inv_eq : ∀ n, [1-] ([1/]1+n) == sigma (fnth n) n.Lemma sigma_fnth_sup : ∀ n m, (m > n) → sigma (fnth n) m == sigma (fnth n) (S n).Lemma sigma_fnth_le : ∀ n m, (sigma (fnth n) m) ≤ (sigma (fnth n) (S n)).Hint Resolve sigma_fnth_le.
fnth is a retract
Lemma fnth_retract : ∀ n:nat,(retract (fnth n) (S n)).Implicit Arguments fnth_retract [].
7.14 distributions and general summations
Definition sigma_fun A (f:nat → MF A) (n:nat) : MF A := fun x ⇒ sigma (fun k ⇒ f k x) n.Definition serie_fun A (f:nat → MF A) : MF A := fun x ⇒ serie (fun k ⇒ f k x).Definition Sigma_fun A (f:nat → MF A) : nat →m
7.15 Discrete distributions
Instance discrete_mon : ∀ A (c : nat → U) (p : nat → A),2.50em
monotonic (fun f : A->U ⇒ serie (fun k ⇒ c k × f (p k))).Save.Definition discrete A (c : nat → U) (p : nat → A) : M A := 3.50em
mon (fun f : A->U ⇒ serie (fun k ⇒ c k × f (p k))).Lemma discrete_simpl : ∀ A (c : nat → U) (p : nat → A) f, 2.50em
discrete c p f = serie (fun k ⇒ c k × f (p k)).Lemma discrete_stable_inv : ∀ A (c : nat → U) (p : nat → A), 2.00em
wretract c → stable_inv (discrete c p).Lemma discrete_stable_plus : ∀ A (c : nat → U) (p : nat → A), 2.00em
stable_plus (discrete c p).Lemma discrete_stable_mult : ∀ A (c : nat → U) (p : nat → A), 2.00em
wretract c → stable_mult (discrete c p).Lemma discrete_continuous : ∀ A (c : nat → U) (p : nat → A), 2.00em
continuous (discrete c p).Record discr (A:Type) : Type := 2.50em
{coeff : nat → U; coeff_retr : wretract coeff; points : nat → A}.Hint Resolve coeff_retr.Definition Discrete : ∀ A, discr A → distr A.Defined.Lemma Discrete_simpl : ∀ A (d:discr A), 2.50em
mu (Discrete d) = discrete (coeff d) (points d).Definition is_discrete (A:Type) (m: distr A) := 3.00em
∃ d : discr A, m == Discrete d.
7.15.1 distribution for randomrandom nnThe distribution associated to random n is
f ↦ Σi =0n f (i )/n +1
we cannot factorize 1/n +1 because of possible overflow
Instance random_mon : ∀ n, monotonic (fun (f:MF nat) ⇒ sigma (fun k ⇒ Unth n × f k) (S n)).Save.Definition random (n:nat):M nat := mon (fun (f:MF nat) ⇒ sigma (fun k ⇒ Unth n × f k) (S n)).Lemma random_simpl : ∀ n (f : MF nat), 5.50em
random n f = sigma (fun k ⇒ Unth n × f k) (S n).
7.15.2 Properties of randomrandom
Lemma random_stable_inv : ∀ n, stable_inv (random n).Lemma random_stable_plus : ∀ n, stable_plus (random n).Lemma random_stable_mult : ∀ n, stable_mult (random n).Lemma random_continuous : ∀ n, continuous (random n).Definition Random (n:nat) : distr nat.Defined.Lemma Random_simpl : ∀ (n:nat), mu (Random n) = random n.Lemma Random_total : ∀ n : nat, mu (Random n) (fone nat) == 1.Hint Resolve Random_total.Lemma Random_inv : ∀ f n, mu (Random n) (finv f) == [1-] (mu (Random n) f).Hint Resolve Random_inv.
7.16 Tacticals
Ltac mu_plus d := 1.00em
match goal with 0.50em
| ⊢ context [fmont (mu d) (fun x ⇒ (Uplus (@?f x) (@?g x)))] ⇒ 3.00em
rewrite (mu_stable_plus d (f:=f) (g:=g))1.00em
end.Ltac mu_mult d := 1.00em
match goal with 0.50em
| ⊢ context [fmont (mu d) (fun x ⇒ (Umult ?k (@?f x)))] ⇒ 3.00em
rewrite (mu_stable_mult d k f)1.00em
end.
8 Prog.v: Composition of distributions
Require Export ProbasProbas.
8.1 Conditional
Definition Mif (A:Type) (b:distr bool) (m1 m2: distr A)2.00em
:= Mlet b (fun x:bool ⇒ if x then m1 else m2).Lemma Mif_le_compat : ∀ (A:Type) (Prog.Rules.PP.distrb1 b2:distr bool) (m1 m2 n1 n2: distr A),2.00em
b1 ≤ b2 → m1 ≤ m2 → n1 ≤ n2 → Mif b1 m1 n1 ≤ Mif b2 m2 n2.Hint Resolve Mif_le_compat.Instance Mif_mon2 : ∀ (A:Type) b, monotonic2 (Mif (A:=A) b).Save.Definition MIf : ∀ (A:Type), distr bool →m m m
8.2 Probabilistic choice8.2.1 The distribution associated to pchoicepchoice pp m1m1 m2m2 isf ↦ p (m 1 f ) + (1−p ) (m 2 f )
Definition Prog.Rules.PP.MP.Mpchoice : ∀ A, U → M A → M A → M A.Defined.Lemma pchoice_simpl : ∀ A p (m1 m2:M A) f,3.00em
pchoice p m1 m2 f = p × m1 f + [1-]p × m2 f.Definition Mchoice (A:Type) (p:U) (m1 m2: distr A) : distr A.Defined.Lemma Mchoice_simpl : ∀ A p (m1 m2:distr A) f,3.00em
mu (Mchoice p m1 m2) f = p × mu m1 f + [1-]p × mu m2 f.Lemma Mchoice_le_compat : ∀ (A:Type) (p:U) (m1 m2 n1 n2: distr Prog.Rules.MchoiceA),4.00em
m1<=m2 → n1<=n2 → Mchoice p m1 n1 ≤ Mchoice p m2 n2.Hint Resolve Mchoice_le_compat.Add Parametric Morphism (A:Type) : (Mchoice (Ccpo.OeqA:=A))1.00em
with signature Oeq =⇒Oeq =⇒Oeq =⇒Oeqas Mchoice_eq_compat.Save.Hint Immediate Mchoice_eq_compat.Instance Mchoice_mon2 : ∀ (A:Type) (p:U), monotonic2 (Mchoice (A:=A) p).Save.Definition MChoice A (p:U) : distr A →m m m
8.3 Image distribution
Definition im_distr (A B : Type) (f:A → Prog.Rules.PP.MletB) (m:distr A) : Prog.Rules.PP.Munitdistr B :=2.00em
Mlet m (fun a ⇒ Munit (f a)).Lemma im_distr_simpl : ∀ A Prog.Univ.UB (f:A → B) (m:distr A)(h:B → Prog.Rules.PP.muU),2.50em
mu (im_distr f m) h = mu m (fun a ⇒ h (f a)).Add Parametric Morphism (A B : Type) : (im_distr (A:=A) (B:=B)) 1.00em
with signature (feq (A:=A) (B:=B)) =⇒Oeq =⇒Oeq1.00em
as im_distr_eq_compat.Save.Lemma im_distr_comp : ∀ A B C (f:A → B) (g:B → C) (m:distr A),6.00em
im_distr g (im_distr f m) == im_distr (fun a ⇒ g (f a)) m.Lemma im_distr_id : ∀ A (f:A → A) (m:distr A), (∀ x, f x = x) →6.00em
im_distr f m == m.
8.4 Product distribution
Definition prod_distr (A B : Type)(d1:distr A)(d2:distr B) : distr (A*B) :=8.00em
Mlet d1 (fun x ⇒ Mlet d2 (fun y ⇒ Munit (x,y))).Add Parametric Morphism (A B : Type) : (prod_distr (A:=A) (B:=B))with signature Ole ++> Ole ++> Ole as prod_distr_le_compat.Save.Hint Resolve prod_distr_le_compat.Add Parametric Morphism (A B : Type) : (prod_distr (A:=A) (B:=B))with signature Oeq =⇒Oeq =⇒Oeqas prod_distr_eq_compat.Save.Hint Immediate Prog.Rules.proddistrmon2prod_distr_eq_compat.Instance prod_distr_mon2 : ∀ (A B :Type), monotonic2 (prod_distr (A:=A) (B:=B)).Save.Definition Prod_distr (A B :Type): distr A →m m m m
8.5 Range of a distribution
Definition Prog.Rules.PP.distrrange A (P:A → Prop) (d: distr A) :=1.50em
∀ f, (∀ x, P x → 0 == f Prog.Rules.rangelex) → 0 == mu d f.Lemma range_le : ∀ A (P:A → Prop) (d:distr A), range P d →3.50em
∀ f g, (∀ a, P a → f a ≤ g a) → mu d f ≤ mu d g.Lemma range_eq : ∀ A (P:A → Prop) (d:distr A), range P d →3.50em
∀ f g, (∀ a, P a → f a == g a) → mu d f == mu d g.
9 Prog.v: Axiomatic semantics9.1 Definition of correctness judgementsp ≤ [e ](q ) is defined as p ≤ µ(e )(q )
↑p e q is defined as µ(e )(q ) ≤ p
Definition ok (A:Type) (p:U) (e:distr A) (q:A → U) := p ≤ mu e q.Definition okfun (A B:Type)(p:A → U)(e:A → distr B)(q:A → B → U)1.00em
:= ∀ x:A, ok (p x) (e x) (q x).Definition okup (A:Type) (p:U) (e:distr A) (q:A → U) := mu e q ≤ p.Definition upfun (A B:Type)(p:A → U)(e:A → distr B)(q:A → B → U)1.00em
:= ∀ x:A, okup (p x) (e x) (q x).
9.2 Stability properties
Lemma ok_le_compat : ∀ (A:Type) (p p’:U) (e:distr A) (q q’:A → U),2.00em
p’ ≤ p → q ≤ q’ → ok p e q → ok p’ e q’.Lemma ok_eq_compat : ∀ (A:Type) (p p’:U) (e e’:distr A) (q q’:A → U),2.00em
p’ == p → q == q’ → e == e’ → ok p e q → ok p’ e’ q’.Lemma okfun_le_compat :∀ (A B:Type) (p p’:A → U) (e:A → distr B) (q q’:A → B → U),2.00em
p’ ≤ p → Prog.Rules.okfunq ≤ q’ → okfun p e q → okfun p’ e q’.Lemma ok_mult : ∀ (A:Type)(k p:U)(e:distr A)(f : A → U),13.50em
ok p e f → ok (k*p) e (fmult k f).Lemma ok_inv : ∀ (A:Type)(p:U)(e:distr A)(Prog.Rules.okf : A → U),6.50em
ok Utheory."[1-].x"p e f → mu e (finv f) ≤ [1-]p.Lemma okup_le_compat : ∀ (A:Type) (p p’:U) (e:distr A) (q q’:A → U),2.00em
p ≤ p’ → q’ ≤ q → okup p e q → okup p’ e q’.Lemma okup_eq_compat : ∀ (A:Type) (p p’:U) (e e’:distr A) (q q’:A → U),2.00em
p == p’ → q == q’ → e == e’ → okup p e q → okup p’ e’ q’.Lemma upfun_le_compat : ∀ (A B:Type) (p p’:A → U) (e:A → distr B)2.00em
(q q’:A → B → U),2.00em
p ≤ p’ → q’<=q → upfun p e q → upfun p’ e q’.Lemma okup_mult : ∀ (A:Type)(k p:U)(e:distr A)(f : A → U), okup p e f → okup (k*p) e (fmult k f).
9.3 Basic rules9.3.1 Rules for application r ≤ [a ](p ) ∀ x , p (x ) ≤ [f (x )](q )/r ≤ [f (a )](q )
↑r a p ∀ x , ↑p (x )f (x )q /↑r f (a )q
Lemma apply_rule : ∀ (A B:Type)(a:(distr A))(f:A → distr B)(r:U)(p:A → U)(q:B → U),2.00em
ok r a p → Prog.Rules.okokfun Prog.Rules.PP.Mletp f (fun x ⇒ q) → ok r (Mlet a f) q.Lemma okup_apply_rule : ∀ (A Prog.Rules.PP.distrB:Type)(a:distr A)(f:A → distr B)(r:U)(p:A → U)(q:B → U),2.00em
okup r a p → Prog.Rules.okupupfun p Prog.Rules.PP.Mletf (fun x ⇒ q) → okup r (Mlet a f) q.
9.3.2 Rules for abstraction
Lemma lambda_rule : ∀ (A B:Type)(f:A → distr B)(p:A → U)(q:A → B → U),1.50em
(∀ x:A, ok (p x) (f x) (q x)) → okfun p f q.Lemma okup_lambda_rule : ∀ (A B:Type)(f:A → distr B)(p:A → U)(q:A → B → U),1.50em
(∀ x:A, okup (p x) (f x) (q x)) → upfun p f q.
9.3.3 Rule for conditional p 1 ≤ [e 1 ](q ) p 2 ≤ [e 2 ](q )/p 1 × µ(b )(1true p 2 × µ(b )(1false if b then e 1 else e 2 ](q )
↑p 1 e 1 q ↑p 2 e 2 q /↑p 1 × µ(b )(1true p 2 × µ(b )(1false if b then e 1 else e 2 q
Lemma combiok : ∀ (Prog.Univ.UA:Type) p q (f1 f2 : A → U), p ≤ [1-]q → fplusok (fmult p f1) (fmult q f2).Hint Resolve combiok.Lemma fmult_fplusok : ∀ (Prog.Univ.UA:Type) p q (f1 f2 : Prog.Rules.PP.MP.UP.fplusokA → U), Prog.Rules.PP.MP.UP.fmultfplusok f1 f2 → fplusok (fmult p f1) (fmult q f2).Hint Resolve fmult_fplusok.Lemma ifok : ∀ f1 f2, Prog.Rules.PP.MP.UP.fmultfplusok (Prog.Rules.PP.cfalsefmult f1 ctrue) (fmult Prog.Rules.fmultfplusokf2 cfalse).Hint Resolve ifok.Lemma Mif_eq : ∀ (A:Type)(b:(distr bool))(f1 Prog.Rules.Miff2:distr A)(q:MF A),4.00em
mu (Mif b f1 f2) q == (mu f1 q) × (mu b ctrue) + (mu f2 q) × (mu b cfalse).Lemma ifrule :1.00em
∀ (A:Type)(b:(distr bool))(f1 f2:distr A)(p1 p2:U)(q:A → U),3.50em
ok p1 f1 Prog.Rules.okq → ok p2 f2 q3.50em
→ ok (Prog.Rules.PP.mup1 × (mu b ctrue) + p2 × (mu b cfalse)) (Mif b f1 f2) q.Lemma okup_ifrule :1.00em
∀ (A:Type)(b:(distr bool))(f1 f2:distr A)(p1 p2:U)(q:A → U),3.50em
okup p1 f1 q → okup p2 f2 q3.50em
→ okup (Prog.Rules.PP.mup1 × (mu b ctrue) + p2 × (mu b cfalse)) (Mif b f1 f2) q.
9.3.4 Rule for fixpointswith φ(x )=F (φ)(x ), p i
∀ f i , (∀ x , p i x ) ≤ [f ](q )) ⇒ ∀ x , p i +1x ) ≤ [F (f )(x )](q )/∀ x , ∪i p i x ≤ [φ(x )](q )
Section Fixrule.Variables A B : Type.Variable F : (A → distr B) →m m m m
9.3.5 Rules using commutation properties
Section TransformFix.Section Fix_muF.Variable Prog.Rules.Fixrule.TransformFix.FixmuF.muFq : A → B → U.Variable muF : MF A →m m m m m
9.4 Rules for intervals
Distributions operates on intervals
Definition Imu : ∀ Prog.Rules.PP.MP.UP.IUA:Type, distr A → (A → IU) → Prog.Rules.PP.muIU.Defined.Lemma low_Imu : ∀ (A:Type) (e:distr A) (Prog.Rules.PP.MP.UP.lowF: A → IU),6.50em
low (Prog.Rules.PP.MP.UP.lowImu e F) = mu e (fun x ⇒ low (F x)).Lemma up_Imu : ∀ (A:Type) (e:distr A) (Prog.Rules.PP.MP.UP.upF: A → IU),6.50em
up (Prog.Rules.PP.MP.UP.upImu e F) = mu e (fun x ⇒ up (F x)).Lemma Imu_monotonic : ∀ (A:Type) (e:distr A) (F G : A → IU),6.00em
(∀ x, Iincl (F x) (Prog.Rules.ImuG x)) → Iincl (Imu e F) (Imu e G).Lemma Imu_stable_eq : ∀ (A:Type) (e:distr A) (F G : A → IU),6.00em
(∀ x, Ieq (F x) (G x)) → Ieq (Imu e F) (Imu e G).Hint Resolve Imu_monotonic Imu_stable_eq.Lemma Imu_singl : ∀ (A:Type) (e:distr Prog.Rules.PP.MP.UP.IeqA) (f:A → U),5.50em
Ieq (Imu e (fun x ⇒ singl (f x))) (singl (mu e f)).Lemma Imu_inf : ∀ (A:Type) (e:distr Prog.Rules.PP.MP.UP.IeqA) (f:A → U),5.50em
Ieq (Imu e (fun x ⇒ inf (f x))) (inf (mu e f)).Lemma Imu_sup : ∀ (A:Type) (e:distr Prog.Rules.PP.MP.UP.IinclA) (f:A → U),5.50em
Iincl (Imu e (fun x ⇒ sup (f x))) (sup (mu e f)).Lemma Iin_mu_Imu :1.50em
∀ (A:Type) (e:distr A) (F:A → IU) (f:A → Prog.Rules.PP.MP.UP.IinU),5.50em
(∀ x, Iin (f x) (F x)) → Iin (mu e f) (Imu e F).Hint Resolve Iin_mu_Imu.Definition Iok (A:Type) (Prog.Rules.PP.MP.UP.IUI:IU) (Prog.Rules.PP.MP.UP.Iincle:distr A) (F:A → IU) := Iincl (Imu e F) I.Definition Iokfun (A B:Type)(I:A → IU) (e:A → distr B) (F:A → B → IU)7.50em
:= ∀ x, Iok (I x) (e x) (F x).Lemma Iin_mu_Iok :1.50em
∀ (A:Type) (Prog.Rules.PP.MP.UP.IUI:IU) (e:distr A) (F:A → IU) (f:A → Prog.Rules.PP.MP.UP.IinU),5.50em
(∀ x, Iin (f Prog.Rules.PP.MP.UP.Iinx) (F x)) → Iok I e F → Iin (mu e f) I.
Lemma Iok_le_compat : ∀ (A:Type) (I J:IU) (e:distr A) (F G:A → IU),1.50em
Iincl I J → (∀ x, Iincl (G Prog.Rules.Iokx) (F x)) → Iok I e F → Iok J e G.Lemma Iokfun_le_compat : ∀ (A B:Type) (I J:A → IU) (e:A → distr B) (F G:A → B → IU),1.50em
(∀ x, Iincl (I Prog.Rules.PP.MP.UP.Iinclx) (J x)) → (∀ x y, Prog.Rules.IokfunIincl (G x y) (F x y)) → Iokfun I e F → Iokfun J e G.
9.4.2 Rule for values
Lemma Iunit_eq : ∀ (A: Type) (a:A) (F:A → IU), Ieq (Imu (Munit a) F) (F a).
9.4.3 Rule for application
Lemma Ilet_eq : ∀ (A B : Type) (a:distr A) (f:A → distr B)(Prog.Rules.ImuI:IU)(G:B → IU),1.50em
Ieq (Imu (Mlet a f) G) (Imu a (fun x ⇒ Imu (f x) G)).Hint Resolve Ilet_eq.Lemma Iapply_rule : ∀ (A B : Type) (a:distr A) (f:A → distr B)(I:IU)(F:A → IU)(G:B → IU),2.00em
Iok I a F → Iokfun F f (fun x ⇒ G) → Iok I (Mlet a f) G.
9.4.4 Rule for abstraction
Lemma Ilambda_rule : ∀ (A B:Type)(f:A → distr B)(F:A → IU)(G:A → B → IU),1.50em
(∀ x:A, Iok (F x) (f x) (G x)) → Iokfun F f G.
9.4.5 Rule for conditional
Lemma Imu_Mif_eq : ∀ (A:Type)(b:distr bool)(f1 Prog.Rules.PP.MP.UP.Ieqf2:distr A)(F:A → IU),0.50em
Ieq (Imu (Mif b f1 f2) F) (Iplus (Imultk (Prog.Rules.PP.MP.UP.Imultkmu b ctrue) (Prog.Rules.PP.cfalseImu f1 F)) (Imultk (mu b cfalse) (Imu f2 F))).Lemma Iifrule :1.00em
∀ (A:Type)(b:(distr bool))(f1 f2:distr A)(I1 I2:IU)(F:A → IU),3.50em
Iok I1 f1 F → Iok I2 f2 F3.50em
→ Prog.Rules.PP.ctrueIok (Iplus (Prog.Rules.PP.MP.UP.ImultkImultk (Prog.Rules.PP.mumu b Prog.Rules.PP.cfalsectrue) I1) (Imultk (mu b cfalse) I2)) (Mif b f1 f2) F.
9.4.6 Rule for fixpointswith φ(x )=F (φ)(x ), p i p i +1x )⊆ p i x )) such that p 0 (x ) contains 0 for all x .
∀ f i , (∀ x , p i x ) ≤ [f ](q x )) ⇒ ∀ x , p i +1x ) ≤ [F (f )(x )](q x )/∀ x , ∩i p i x ≤ [φ(x )](q x )
Section IFixrule.Variables A B : Type.Variable F : (A → distr B) →m m m
9.5 Rules for FlipFlip
Lemma Flip_ctrue : mu Flip ctrue == [1/2].Lemma Flip_cfalse : mu Flip cfalse == [1/2].Lemma ok_Flip : ∀ q : Coq.Init.Datatypes.truebool → U, ok ([1/2] × q true + [1/2] × q false) Flip q.Lemma Coq.Init.Datatypes.boolokup_Flip : ∀ q : bool → U, okup ([1/2] × q true + [1/2] × q false) Flip q.Hint Resolve ok_Flip okup_Flip Flip_ctrue Flip_cfalse.Lemma Flip_eq : ∀ q : bool → U, mu Flip q == [1/2] × q true + [1/2] × q false.Hint Resolve Flip_eq.Lemma IFlip_eq : ∀ Prog.Rules.PP.FlipQ : bool → IU, Prog.Rules.PP.MP.UP.ImultkIeq (Imu Flip Q) (Iplus (Imultk [1/2] (Q true)) (Imultk [1/2] (Q false))).Hint Resolve IFlip_eq.
9.6 Rules for total (well-founded) fixpoints
Section Prog.Rules.Wellfounded.BWellfounded.Variables A B : Type.Variable R : A → A → Prop.Hypothesis Rwf : Prog.Rules.Wellfounded.Fwell_founded R.Variable F : ∀ x, (∀ y, R y x → distr B) → distr B.Definition WfFix : A → distr B := Fix Rwf (fun _ ⇒ distr B) F.Hypothesis Fext : ∀ x f g, (∀ y (p:R y x), f y p == g y Prog.Rules.Acciterdistrp) → F f == F g.Lemma Acc_iter_distr :1.50em
∀ Coq.Init.Wf.Acciterx, ∀ r s : Acc R x, Acc_iter (fun _⇒ distr B) F r == Acc_iter (fun _⇒ distr B) F s.Lemma WfFix_eq : ∀ x, WfFix x == F (fun (y:A) (p:R y x) ⇒ WfFix y).Variable P : distr B → Prop.Hypothesis Pext : ∀ m1 m2, m1 == m2 → P m1 → P m2.Lemma WfFix_ind :2.50em
(∀ x f, (∀ y (p:R y x), P (f y p)) → P (F f))1.00em
→ ∀ x, P (WfFix x).End Wellfounded.Ltac distrsimpl := match goal with 0.50em
| ⊢ (Ole (fmont (mu ?d1) ?f) (fmont (mu ?d2) ?g)) ⇒ apply (mu_le_compat (m1:=d1) (m2:=d2) (Ole_refl d1) (f:=f) (g:=g)); Prog.Rules.PP.muintro0.50em
| ⊢ (Oeq (fmont (mu ?d1) ?f) (fmont (mu ?d2) ?g)) ⇒ apply (mu_eq_compat (m1:=d1) (m2:=d2) (Oeq_refl d1) (f:=f) (g:=g)); intro0.50em
| ⊢ (Oeq (Munit ?x) (Munit ?y)) ⇒ apply (Munit_eq_compat x y)0.50em
| ⊢ (Oeq (Mlet ?x1 ?f) (Mlet ?x2 ?g)) 7.00em
⇒ apply (Mlet_eq_compat (m1:=x1) (Ccpo.Oeqreflm2:=x2) (M1:=f) (M2:=g) (Oeq_refl x1)); intro0.50em
| ⊢ (Ole (Mlet ?x1 ?f) (Mlet ?x2 ?g)) 7.00em
⇒ apply (Mlet_le_compat (m1:=x1) (Ccpo.Olereflm2:=x2) (M1:=f) (M2:=g) (Ole_refl x1)); intro0.50em
| ⊢ context [(fmont (mu (Mlet ?m ?M)) ?f)] ⇒ rewrite (Mlet_simpl m M f)0.50em
| ⊢ context [(fmont (mu (Munit ?x)) ?f)] ⇒ rewrite (Munit_simpl f Prog.Rules.PP.Mletx)0.50em
| ⊢ context [(Mlet (Mlet ?m ?M) ?f)] ⇒ rewrite (Mlet_assoc m M Prog.Rules.PP.Munitf)0.50em
| ⊢ context [(Mlet (Munit ?x) ?f)] ⇒ rewrite (Mlet_unit x f)0.50em
| ⊢ context [(fmont (mu Flip) ?f)] ⇒ rewrite (Flip_simpl f)0.50em
| ⊢ context [(fmont (mu (Discrete ?d)) ?f)] ⇒ rewrite (Discrete_simpl d); 29.50em
Prog.Rules.PP.coeffrewrite (discrete_simpl (coeff d) (points d) f)0.50em
| ⊢ context [(fmont (mu (Random ?n)) ?f)] ⇒ rewrite (Random_simpl n); 29.50em
rewrite (random_simpl n f)0.50em
| ⊢ context [(fmont (mu (Mif ?b ?f ?g)) ?h)] ⇒ unfold Mif0.50em
| ⊢ context [(fmont (mu (Mchoice ?p ?m1 ?m2)) ?f)] ⇒ rewrite (Mchoice_simpl p Ccpo.fmontm1 m2 f)0.50em
| ⊢ context [(fmont (mu (im_distr ?f ?m)) ?h)] ⇒ rewrite (im_distr_simpl f m h)0.50em
| ⊢ context [(fmont (mu (prod_distr ?m1 ?m2)) ?h)] ⇒ unfold prod_distr0.50em
| ⊢ context [((mon ?f (fmonotonic:=?mf)) ?x)] ⇒ rewrite (mon_simpl f (mf:=mf) x)end.
Require Export Coq.Setoids.SetoidSetoid.Require Coq.omega.OmegaOmega.
10 Sets.v: Definition of sets as predicates over a type A
Section Sets.setssets.Variable Sets.sets.AA : Type.Variable Sets.sets.decAdecA : ∀ x y :A, {x=y}+{x<>y}.Definition Sets.setset := A->Prop.Definition Sets.fullfull : Sets.setset := fun (x:A) ⇒ Coq.Init.Logic.TrueTrue.Definition Sets.emptyempty : Sets.setset := fun (x:A) ⇒ Coq.Init.Logic.FalseFalse.Definition Sets.addadd (a:A) (P:set) : Sets.setset := fun (x:A) ⇒ x=a ∨ (P x).Definition Sets.singlsingl (a:A) :set := fun (x:A) ⇒ x=a.Definition Sets.unionunion (P Q:set) :set := fun (x:A) ⇒ (P x) ∨ (Q x).Definition Sets.complcompl (P:set) :set := fun (x:A) ⇒ ~P x.Definition Sets.interinter (P Q:set) :set := fun (x:A) ⇒ (P x) ∧ (Q x).Definition Sets.remrem (a:A) (P:set) :set := fun (x:A) ⇒ x<>a ∧ (P x).
10.1 Equivalence
Definition Sets.eqseteqset (P Q:set) := ∀ (x:A), P x ↔ Q x.Implicit Arguments Sets.fullfull [].Implicit Arguments Sets.emptyempty [].Lemma Sets.eqsetrefleqset_refl : ∀ P:set, Sets.eqseteqset P P.Lemma Sets.eqsetsymeqset_sym : ∀ P Q:set, Sets.eqseteqset P Q → Sets.eqseteqset Q P.Lemma Sets.eqsettranseqset_trans : ∀ P Q R:set, 1.50em
Sets.eqseteqset P Q → Sets.eqseteqset Q R → Sets.eqseteqset P R.Hint Resolve Sets.eqsetrefleqset_refl.Hint Immediate Sets.eqsetsymeqset_sym.
10.2 Setoid structure
Lemma Sets.setsetoidset_setoid : Coq.Setoids.Setoid.SetoidTheorySetoid_Theory Sets.setset Sets.eqseteqset.Add Setoid Sets.setset Sets.eqseteqset Sets.setsetoidset_setoid as Set_setoid.Add Morphism Sets.addadd : eqset_add.Save.Add Morphism Sets.remrem : eqset_rem.Save.Hint Resolve Sets.eqsetaddeqset_add Sets.eqsetremeqset_rem.Add Morphism Sets.unionunion : eqset_union.Save.Hint Immediate Sets.eqsetunioneqset_union.Lemma Sets.eqsetunionlefteqset_union_left : 1.00em
∀ P1 Q P2,1.50em
Sets.eqseteqset P1 P2 → Sets.eqseteqset (Sets.unionunion P1 Q) (Sets.unionunion P2 Q).Lemma Sets.eqsetunionrighteqset_union_right : 1.00em
∀ P Q1 Q2 ,1.50em
Sets.eqseteqset Q1 Q2 → Sets.eqseteqset (Sets.unionunion P Q1) (Sets.unionunion P Q2).Hint Resolve Sets.eqsetunionlefteqset_union_left Sets.eqsetunionrighteqset_union_right.Add Morphism Sets.interinter : eqset_inter.Save.Hint Immediate Sets.eqsetintereqset_inter.Add Morphism Sets.complcompl : eqset_compl.Save.Hint Resolve Sets.eqsetcompleqset_compl.Lemma Sets.eqsetaddemptyeqset_add_empty : ∀ (a:A) (P:set), ~eqset (Sets.addadd a P) Sets.emptyempty.
10.3 Finite sets given as an enumeration of elements
Inductive Sets.finitefinite (P: Sets.setset) : Type := 1.50em
Sets.fineqemptyfin_eq_empty : Sets.eqseteqset P Sets.emptyempty → Sets.finitefinite P0.50em
| Sets.fineqaddfin_eq_add : ∀ (x:A)(Q:set),6.50em
¬ Q x→ Sets.finitefinite Q → Sets.eqseteqset P (Sets.addadd x Q) → Sets.finitefinite P.Hint Constructors finite.Lemma Sets.finemptyfin_empty : (Sets.finitefinite Sets.emptyempty).Lemma Sets.finaddfin_add : ∀ (x:A)(P:set),6.50em
¬ P x → Sets.finitefinite P → Sets.finitefinite (Sets.addadd x P).Lemma Sets.fineqsetfin_eqset: ∀ (P Q : Sets.setset), (Sets.eqseteqset P Q)->(finite P)->(finite Q).Hint Resolve Sets.finemptyfin_empty Sets.finaddfin_add.
10.3.1 Emptyness is decidable for finite sets
Definition Sets.isemptyisempty (P:set) := Sets.eqseteqset P Sets.emptyempty.Definition Sets.notemptynotempty (P:set) := Coq.Init.Logic.notnot (Sets.eqseteqset P Sets.emptyempty).Lemma Sets.isemptydecisempty_dec : ∀ P, Sets.finitefinite P → {Sets.isemptyisempty P}+{notempty P}.
10.3.2 Size of a finite set
Fixpoint Sets.sizesize (P:set) (f:finite P) {struct f}: Coq.Init.Datatypes.natnat :=1.50em
match f with Sets.fineqemptyfin_eq_empty _ ⇒ 0%nat7.00em
| Sets.fineqaddfin_eq_add _ Q _ f’ _ ⇒ Coq.Init.Datatypes.SS (Sets.sizesize f’)1.50em
end.Lemma Sets.sizeeqsetsize_eqset : ∀ P Q (f:finite P) (e:eqset P Q),2.00em
(Sets.sizesize (Sets.fineqsetfin_eqset e f)) = (Sets.sizesize f).
10.4 Inclusion
Definition Sets.inclincl (P Q:set) := ∀ x, P x → Q x.Lemma Sets.inclreflincl_refl : ∀ (P:set), Sets.inclincl P P.Lemma Sets.incltransincl_trans : ∀ (P Q R:set), incl P Q → Sets.inclincl Q R → Sets.inclincl P R.Lemma Sets.eqsetincleqset_incl : ∀ (P Q : Sets.setset), Sets.eqseteqset P Q → Sets.inclincl P Q.Lemma Sets.eqsetinclsymeqset_incl_sym : ∀ (P Q : Sets.setset), Sets.eqseteqset P Q → Sets.inclincl Q P.Lemma Sets.eqsetinclintroeqset_incl_intro : ∀ (P Q : Sets.setset), Sets.inclincl P Q → Sets.inclincl Q P → Sets.eqseteqset P Q.Hint Resolve Sets.inclreflincl_refl Sets.incltransincl_trans Sets.eqsetinclintroeqset_incl_intro.Hint Immediate Sets.eqsetincleqset_incl Sets.eqsetinclsymeqset_incl_sym.
10.5 Properties of operations on sets
Lemma Sets.inclemptyincl_empty : ∀ P, Sets.inclincl Sets.emptyempty P.Lemma Sets.inclemptyfalseincl_empty_false : ∀ P a, Sets.inclincl P Sets.emptyempty → ¬ P a.Lemma Sets.incladdemptyincl_add_empty : ∀ (a:A) (P:set), ¬ Sets.inclincl (Sets.addadd a P) Sets.emptyempty.Lemma Sets.eqsetemptyfalseeqset_empty_false : ∀ P a, Sets.eqseteqset P Sets.emptyempty → P a → Coq.Init.Logic.FalseFalse.Hint Immediate Sets.inclemptyfalseincl_empty_false Sets.eqsetemptyfalseeqset_empty_false Sets.incladdemptyincl_add_empty.Lemma Sets.inclremstableincl_rem_stable : ∀ a P Q, Sets.inclincl P Q → Sets.inclincl (Sets.remrem a P) (Sets.remrem a Q).Lemma Sets.incladdstableincl_add_stable : ∀ a P Q, Sets.inclincl P Q → Sets.inclincl (Sets.addadd a P) (Sets.addadd a Q).Lemma Sets.inclremaddiffincl_rem_add_iff : 1.00em
∀ a P Q, Sets.inclincl (Sets.remrem a P) Q ↔ Sets.inclincl P (Sets.addadd a Q).Lemma Sets.inclremaddincl_rem_add: 1.00em
∀ (a:A) (P Q:set), 2.50em
(P a) → Sets.inclincl Q (Sets.remrem a P) → Sets.inclincl (Sets.addadd a Q) P.Lemma Sets.incladdremincl_add_rem : 1.00em
∀ (a:A) (P Q:set), 2.50em
¬ Q a → Sets.inclincl (Sets.addadd a Q) P → Sets.inclincl Q (Sets.remrem a P) .Hint Immediate Sets.inclremaddincl_rem_add Sets.incladdremincl_add_rem.Lemma Sets.eqsetremaddeqset_rem_add : 0.50em
∀ (a:A) (P Q:set), 2.50em
(P a) → Sets.eqseteqset Q (Sets.remrem a P) → Sets.eqseteqset (Sets.addadd a Q) P.Lemma Sets.eqsetaddremeqset_add_rem : 0.50em
∀ (a:A) (P Q:set), 2.50em
¬ Q a → Sets.eqseteqset (Sets.addadd a Q) P → Sets.eqseteqset Q (Sets.remrem a P).Hint Immediate Sets.eqsetremaddeqset_rem_add Sets.eqsetaddremeqset_add_rem.Lemma Sets.addremeqeqsetadd_rem_eq_eqset : 1.00em
∀ x (P:set), Sets.eqseteqset (Sets.addadd x (Sets.remrem x P)) (Sets.addadd x P).Lemma Sets.addremdiffeqsetadd_rem_diff_eqset : 1.00em
∀ x y (P:set), 1.00em
x<>y → Sets.eqseteqset (Sets.addadd x (Sets.remrem y P)) (Sets.remrem y (Sets.addadd x P)).Lemma Sets.addeqsetinadd_eqset_in : 1.00em
∀ x (P:set), P x → Sets.eqseteqset (Sets.addadd x P) P.Hint Resolve Sets.addremeqeqsetadd_rem_eq_eqset Sets.addremdiffeqsetadd_rem_diff_eqset Sets.addeqsetinadd_eqset_in.Lemma Sets.addremeqsetinadd_rem_eqset_in : 1.00em
∀ x (P:set), P x → Sets.eqseteqset (Sets.addadd x (Sets.remrem x P)) P.Hint Resolve Sets.addremeqsetinadd_rem_eqset_in.Lemma Sets.remaddeqeqsetrem_add_eq_eqset : 1.00em
∀ x (P:set), Sets.eqseteqset (Sets.remrem x (Sets.addadd x P)) (Sets.remrem x P).Lemma Sets.remadddiffeqsetrem_add_diff_eqset : 1.00em
∀ x y (P:set), 1.00em
x<>y → Sets.eqseteqset (Sets.remrem x (Sets.addadd y P)) (Sets.addadd y (Sets.remrem x P)).Lemma Sets.remeqsetnotinrem_eqset_notin : 1.00em
∀ x (P:set), ~P x → Sets.eqseteqset (Sets.remrem x P) P.Hint Resolve Sets.remaddeqeqsetrem_add_eq_eqset Sets.remadddiffeqsetrem_add_diff_eqset Sets.remeqsetnotinrem_eqset_notin.Lemma Sets.remaddeqsetnotinrem_add_eqset_notin : 1.00em
∀ x (P:set), ~P x → Sets.eqseteqset (Sets.remrem x (Sets.addadd x P)) P.Hint Resolve Sets.remaddeqsetnotinrem_add_eqset_notin.Lemma Sets.remnotinrem_not_in : ∀ x (P:set), ¬ Sets.remrem x P x.Lemma Sets.addinadd_in : ∀ x (P:set), Sets.addadd x P x.Lemma Sets.addineqadd_in_eq : ∀ x y P, x=y → Sets.addadd x P y.Lemma Sets.addintroadd_intro : ∀ x (P:set) y, P y → Sets.addadd x P y.Lemma Sets.addincladd_incl : ∀ x (P:set), Sets.inclincl P (Sets.addadd x P).Lemma Sets.addinclintroadd_incl_intro : ∀ x (P Q:set), (Q x) → (Sets.inclincl P Q) → (Sets.inclincl (Sets.addadd x P) Q).Lemma Sets.reminclrem_incl : ∀ x (P:set), Sets.inclincl (Sets.remrem x P) P.Hint Resolve Sets.remnotinrem_not_in Sets.addinadd_in Sets.reminclrem_incl Sets.addincladd_incl.Lemma Sets.unionsymunion_sym : ∀ P Q : Sets.setset,3.00em
Sets.eqseteqset (Sets.unionunion P Q) (Sets.unionunion Q P).Lemma Sets.unionemptyleftunion_empty_left : ∀ P : Sets.setset,3.00em
Sets.eqseteqset P (Sets.unionunion P Sets.emptyempty).Lemma Sets.unionemptyrightunion_empty_right : ∀ P : Sets.setset,3.00em
Sets.eqseteqset P (Sets.unionunion Sets.emptyempty P).Lemma Sets.unionaddleftunion_add_left : ∀ (a:A) (P Q: Sets.setset),3.00em
Sets.eqseteqset (Sets.addadd a (Sets.unionunion P Q)) (Sets.unionunion P (Sets.addadd a Q)).Lemma Sets.unionaddrightunion_add_right : ∀ (a:A) (P Q: Sets.setset),3.00em
Sets.eqseteqset (Sets.addadd a (Sets.unionunion P Q)) (Sets.unionunion (Sets.addadd a P) Q).Hint Resolve Sets.unionsymunion_sym Sets.unionemptyleftunion_empty_left Sets.unionemptyrightunion_empty_rightunion_add_left Sets.unionaddrightunion_add_right.Lemma Sets.unioninclleftunion_incl_left : ∀ P Q, Sets.inclincl P (Sets.unionunion P Q).Lemma Sets.unioninclrightunion_incl_right : ∀ P Q, Sets.inclincl Q (Sets.unionunion P Q).Lemma Sets.unioninclintrounion_incl_intro : ∀ P Q R, Sets.inclincl P R → Sets.inclincl Q R → Sets.inclincl (Sets.unionunion P Q) R.Hint Resolve Sets.unioninclleftunion_incl_left Sets.unioninclrightunion_incl_right Sets.unioninclintrounion_incl_intro.Lemma Sets.inclunionstableincl_union_stable : ∀ P1 P2 Q1 Q2,4.00em
Sets.inclincl P1 P2 → Sets.inclincl Q1 Q2 → Sets.inclincl (Sets.unionunion P1 Q1) (Sets.unionunion P2 Q2).Hint Immediate Sets.inclunionstableincl_union_stable.Lemma Sets.intersyminter_sym : ∀ P Q : Sets.setset,3.00em
Sets.eqseteqset (Sets.interinter P Q) (Sets.interinter Q P).Lemma Sets.interemptyleftinter_empty_left : ∀ P : Sets.setset,3.00em
Sets.eqseteqset Sets.emptyempty (Sets.interinter P Sets.emptyempty).Lemma Sets.interemptyrightinter_empty_right : ∀ P : Sets.setset,3.00em
Sets.eqseteqset Sets.emptyempty (Sets.interinter Sets.emptyempty P).Lemma Sets.interaddleftininter_add_left_in : ∀ (a:A) (P Q: Sets.setset),3.00em
(P a) → Sets.eqseteqset (Sets.addadd a (Sets.interinter P Q)) (Sets.interinter P (Sets.addadd a Q)).Lemma Sets.interaddleftoutinter_add_left_out : ∀ (a:A) (P Q: Sets.setset),3.00em
¬ P a → Sets.eqseteqset (Sets.interinter P Q) (Sets.interinter P (Sets.addadd a Q)).Lemma Sets.interaddrightininter_add_right_in : ∀ (a:A) (P Q: Sets.setset),3.00em
Q a → Sets.eqseteqset (Sets.addadd a (Sets.interinter P Q)) (Sets.interinter (Sets.addadd a P) Q).Lemma Sets.interaddrightoutinter_add_right_out : ∀ (a:A) (P Q: Sets.setset),3.00em
¬ Q a → Sets.eqseteqset (Sets.interinter P Q) (Sets.interinter (Sets.addadd a P) Q).Hint Resolve Sets.intersyminter_sym Sets.interemptyleftinter_empty_left Sets.interemptyrightinter_empty_rightinter_add_left_in Sets.interaddleftoutinter_add_left_out Sets.interaddrightininter_add_right_in Sets.interaddrightoutinter_add_right_out.
10.6 Generalized union
Definition Sets.guniongunion (I:Type)(F:I->set) : Sets.setset := fun z ⇒ ∃ i, F i z.Lemma Sets.gunionintrogunion_intro : ∀ I (F:I->set) i, Sets.inclincl (F i) (Sets.guniongunion F).Lemma Sets.gunionelimgunion_elim : ∀ I (F:I->set) (P:set), (∀ i, Sets.inclincl (F i) P) → Sets.inclincl (Sets.guniongunion F) P.Lemma Sets.gunionmonotonicgunion_monotonic : ∀ I (F G : I → Sets.setset), 3.00em
(∀ i, Sets.inclincl (F i) (G i))→ Sets.inclincl (Sets.guniongunion F) (Sets.guniongunion G).
10.7 Removing an element from a finite set
Lemma Sets.finiteremfinite_rem : ∀ (P:set) (a:A),1.50em
Sets.finitefinite P → Sets.finitefinite (Sets.remrem a P).Lemma Sets.sizefiniteremsize_finite_rem: 1.50em
∀ (P:set) (a:A) (f:finite P), 2.00em
(P a) → Sets.sizesize f = Coq.Init.Datatypes.SS (Sets.sizesize (Sets.finiteremfinite_rem a f)).Require Import Coq.Arith.ArithArith.Lemma Sets.sizeinclsize_incl : 1.00em
∀ (P:set)(f:finite P) (Q:set)(g:finite Q), 1.00em
(Sets.inclincl P Q)→ Sets.sizesize f ≤ Sets.sizesize g.Lemma Sets.sizeuniquesize_unique : 1.00em
∀ (P:set)(f:finite P) (Q:set)(g:finite Q), 1.00em
(Sets.eqseteqset P Q)→ Sets.sizesize f = Sets.sizesize g.
10.8 Decidable sets
Definition Sets.decdec (P:set) := ∀ x, {P x}+{~ P x}.Lemma Sets.finiteinclfinite_incl : ∀ P:set,1.50em
Sets.finitefinite P → ∀ Q:set, Sets.decdec Q → Sets.inclincl Q P → Sets.finitefinite Q.Lemma Sets.finitedecfinite_dec : ∀ P:set, Sets.finitefinite P → Sets.decdec P.Lemma Sets.finaddinfin_add_in : ∀ (a:A) (P:set), Sets.finitefinite P → Sets.finitefinite (Sets.addadd a P).Lemma Sets.finiteunionfinite_union : 2.50em
∀ P Q, Sets.finitefinite P → Sets.finitefinite Q → Sets.finitefinite (Sets.unionunion P Q).Lemma Sets.finitefulldecfinite_full_dec : ∀ P:set, Sets.finitefinite Sets.fullfull → Sets.decdec P → Sets.finitefinite P.Require Import Coq.Arith.LtLt.
10.8.1 Filter operation
Lemma Sets.finiteinterfinite_inter : ∀ P Q, Sets.decdec P → Sets.finitefinite Q → Sets.finitefinite (Sets.interinter P Q).Lemma Sets.sizeinteremptysize_inter_empty : ∀ P Q (decP:dec P) (e:eqset Q Sets.emptyempty), 1.50em
Sets.sizesize (Sets.finiteinterfinite_inter decP (Sets.fineqemptyfin_eq_empty e))=O.Lemma Sets.sizeinteraddinsize_inter_add_in : 2.50em
∀ P Q R (decP:dec P)(x:A)(nq:~Q x)(FQ:finite Q)(e:eqset R (Sets.addadd x Q)),3.00em
P x ->size (Sets.finiteinterfinite_inter decP (Sets.fineqaddfin_eq_add nq FQ e))=S (Sets.sizesize (Sets.finiteinterfinite_inter decP FQ)).Lemma Sets.sizeinteraddnotinsize_inter_add_notin : 2.50em
∀ P Q R (decP:dec P)(x:A)(nq:~Q x)(FQ:finite Q)(e:eqset R (Sets.addadd x Q)),1.50em
¬ P x → Sets.sizesize (Sets.finiteinterfinite_inter decP (Sets.fineqaddfin_eq_add nq FQ e))=size (Sets.finiteinterfinite_inter decP FQ).Lemma Sets.sizeinterinclsize_inter_incl : ∀ P Q (decP:dec P)(FP:finite P)(FQ:finite Q), 2.00em
(Sets.inclincl P Q) → Sets.sizesize (Sets.finiteinterfinite_inter decP FQ)=size FP.
10.8.2 Selecting elements in a finite set
Fixpoint Sets.nthfinitenth_finite (P:set) (k:nat) (PF : Sets.finitefinite P) {struct PF}: (k < Sets.sizesize PF) → A := 1.00em
match PF as F return (k < Sets.sizesize F) → A with 3.50em
Sets.fineqemptyfin_eq_empty H ⇒ (fun (e : k<0) ⇒ match Coq.Arith.Lt.ltnOlt_n_O k e with end)2.50em
| Sets.fineqaddfin_eq_add x Q nqx fq eqq ⇒ 5.50em
match k as k0 return k0<S (Sets.sizesize fq)->A with 8.00em
Coq.Init.Datatypes.OO ⇒ fun e ⇒ x4.50em
| (Coq.Init.Datatypes.SS k1) ⇒ fun (e:S k1<S (Sets.sizesize fq)) ⇒ Sets.nthfinitenth_finite fq (Coq.Arith.Lt.ltSnlt_S_n k1 (Sets.sizesize fq) e)5.50em
end1.00em
end.
A set with size > 1 contains at least 2 different elements
Lemma Sets.selectnonemptyselect_non_empty : ∀ (P:set), Sets.finitefinite P → Sets.notemptynotempty P → Coq.Init.Specif.sigTsigT P.Lemma Sets.selectdiffselect_diff : ∀ (P:set) (FP:finite P),2.50em
(1 < Sets.sizesize FP)%nat → Coq.Init.Specif.sigTsigT (fun x ⇒ Coq.Init.Specif.sigTsigT (fun y ⇒ P x ∧ P y ∧ x<>y)).End Sets.setssets.Hint Resolve Sets.eqsetrefleqset_refl.Hint Resolve Sets.eqsetaddeqset_add Sets.eqsetremeqset_rem.Hint Immediate Sets.eqsetsymeqset_sym Sets.finitedecfinite_dec Sets.finitefulldecfinite_full_dec Sets.eqsetincleqset_incl Sets.eqsetinclsymeqset_incl_sym Sets.eqsetinclintroeqset_incl_intro.Hint Resolve Sets.inclreflincl_refl.Hint Immediate Sets.inclunionstableincl_union_stable.Hint Resolve Sets.unioninclleftunion_incl_left Sets.unioninclrightunion_incl_right Sets.unioninclintrounion_incl_intro Sets.inclemptyincl_empty Sets.reminclrem_inclincl_rem_stable Sets.incladdstableincl_add_stable.Hint Constructors finite.Hint Resolve Sets.addinadd_in Sets.addineqadd_in_eq Sets.addintroadd_intro Sets.addincladd_incl Sets.addinclintroadd_incl_intro Sets.unionsymunion_sym Sets.unionemptyleftunion_empty_left Sets.unionemptyrightunion_empty_rightunion_add_left Sets.unionaddrightunion_add_right Sets.finiteunionfinite_union Sets.eqsetunionlefteqset_union_left eqset_union_right.Implicit Arguments Sets.fullfull [].Implicit Arguments Sets.emptyempty [].Add Parametric Relation (A:Type) : (Sets.setset A) (Sets.eqseteqset (A:=A))3.00em
reflexivity proved by (Sets.eqsetrefleqset_refl (A:=A))3.00em
symmetry proved by (Sets.eqsetsymeqset_sym (A:=A))3.00em
transitivity proved by (Sets.eqsettranseqset_trans (A:=A))as eqset_rel.Add Parametric Relation (A:Type) : (Sets.setset A) (Sets.inclincl (A:=A))3.00em
reflexivity proved by (Sets.inclreflincl_refl (A:=A))3.00em
transitivity proved by (Sets.incltransincl_trans (A:=A))as incl_rel.
11 Cover.v: Characteristic functions
Require Export ProgProg.Require Export SetsSets.Require Export Coq.Arith.ArithArith.
Properties of zero_one functions
Definition zero_one (A:Type)(f:MF A) := ∀ x, orc (f x == 0) (f x == 1).Hint Unfold zero_one.Lemma zero_one_not_one :∀ (Cover.CoverFun.zerooneA:Type)(f:MF A) x, zero_one f → ¬ 1 ≤ f x → f x == 0.Lemma zero_one_not_zero :∀ (Cover.CoverFun.zerooneA:Type)(f:MF A) x, zero_one f → ¬ f x ≤ 0 → f x == 1.Hint Resolve zero_one_not_one zero_one_not_zero.Definition fesp_zero_one : ∀ (A:Type)(f g:MF A),3.00em
zero_one f → zero_one g → zero_one (fesp f g).Save.Lemma fesp_conj_zero_one : ∀ (A:Type)(f g:MF A),3.00em
zero_one f → fesp f g == fconj f g.Lemma fconj_zero_one : ∀ (A:Type)(f g:MF A),3.00em
zero_one f → zero_one g → zero_one (fconj f g).Lemma fplus_zero_one : ∀ (A:Type)(f g:MF A),3.00em
zero_one f → zero_one g → zero_one (fplus f g).Lemma finv_zero_one : ∀ (A:Type)(f :MF A),3.00em
zero_one f → zero_one (finv f).Lemma fesp_zero_one_mult_left : ∀ (A:Type)(f:MF A)(p:U),3.00em
zero_one f → ∀ x, f x & p == f x × p.Lemma fesp_zero_one_mult_right : ∀ (A:Type)(p:U)(f:MF A),2.50em
zero_one f → ∀ x, p & f x == p × Cover.CoverFun.RP.PP.MP.UP.Uespzeroonemultrightf x.Hint Resolve fesp_zero_one_mult_left fesp_zero_one_mult_right.
11.1 Covering functions
Definition cover (Cover.CoverFun.RP.PP.MP.UP.MFA:Type)(P:set A)(f:MF A) :=3.00em
∀ x, (P x → 1 ≤ f x) ∧ (¬ P x → f x ≤ 0).Lemma cover_eq_one : ∀ (Cover.CoverFun.RP.PP.MP.UP.MFA:Type)(P:set A)(f:MF A) (z:A),3.50em
cover P f → P z → f z == 1.Lemma cover_eq_zero : ∀ (Cover.CoverFun.RP.PP.MP.UP.MFA:Type)(P:set A)(f:MF A) (z:A),3.50em
cover P f → ¬ P z → f z == 0.Lemma cover_orc_0_1 : ∀ (Cover.CoverFun.RP.PP.MP.UP.MFA:Type)(P:set A)(f:MF A),1.50em
cover P f → ∀ x, orc (f x == 0) (f x == 1).Lemma cover_zero_one : ∀ (Cover.CoverFun.RP.PP.MP.UP.MFA:Type)(P:set A)(f:MF A),1.50em
cover P f → zero_one f.Lemma zero_one_cover : ∀ (A:Type)(f:MF A),1.50em
zero_one f → cover (fun x ⇒ 1 ≤ f x) f.Lemma cover_esp_mult_left : ∀ (Cover.CoverFun.RP.PP.MP.UP.MFA:Type)(P:set A)(f:MF A)(p:U),3.00em
cover P f → ∀ x, f x & p == f x × p.Lemma cover_esp_mult_right : ∀ (Cover.Univ.UA:Type)(P:set A)(Cover.CoverFun.coverp:U)(f:MF A),2.50em
cover P f → ∀ x, p & f x == p × Cover.CoverFun.RP.PP.MP.UP.Uespzeroonemultrightf x.Hint Immediate cover_esp_mult_left cover_esp_mult_right.Lemma cover_elim : ∀ (Cover.CoverFun.RP.PP.MP.UP.MFA:Type)(P:set A)(f:MF A),cover P f → ∀ x, orc (~P x ∧ f x == 0) (P x ∧ f x == 1).Lemma cover_eq_one_elim_class : ∀ (A:Type)(P Q:set Cover.CoverFun.coverA)(f:MF A),3.50em
cover P f → ∀ z, f z == 1 → class (Q z) → incl P Q → Q z.Lemma cover_eq_one_elim : ∀ (Cover.CoverFun.RP.PP.MP.UP.MFA:Type)(P:set Cover.CoverFun.coverA)(f:MF A),3.50em
cover P f → ∀ z, f z == 1 → ¬ ¬ P z.Lemma cover_eq_zero_elim : ∀ (Cover.CoverFun.RP.PP.MP.UP.MFA:Type)(P:set A)(f:MF A) (z:A),3.50em
cover Coq.Init.Logic.".x"P f → f z == 0 → ¬ P z.Lemma cover_unit : ∀ (Cover.CoverFun.RP.PP.MP.UP.MFA:Type)(P:set A)(f:MF A)(Cover.CoverFun.covera:A),6.50em
cover Cover.CoverFun.RP.PP.muP f → P a → 1 ≤ mu (Munit a) f.Lemma cover_let : ∀ (A B:Type)(m1: distr A)(Sets.setm2: A->distr B) (P:set A)(Cover.Univ.UcP:MF A)(Cover.CoverFun.coverf:MF B)(p:U),2.50em
Cover.CoverFun.RP.PP.mucover P cP → 1 ≤ mu m1 cP → (∀ x:A, P x → p ≤ mu (m2 x) f) → p ≤ mu (Mlet m1 m2) f.Lemma cover_incl_fle : ∀ (A:Type)(P Q:set A)(f g:MF A),3.00em
cover P f → cover Q g → incl P Q → f ≤ g.Lemma cover_incl_eq: ∀ (A:Type)(P:set A)(f g:MF A),3.00em
cover P f → cover P g → f == g.Lemma cover_eqset_stable : ∀ (A:Type)(P Q:set Cover.CoverFun.RP.PP.MP.UP.MFA)(EQ:eqset P Q)(f:MF A),3.00em
cover P f → cover Q f.Lemma cover_eq_stable : ∀ (A:Type)(P:set A)(f g:MF A),3.00em
cover P f → f == g → cover P g.Lemma cover_eqset_eq_stable : ∀ (A:Type)(P Q:set A)(f g:MF A),3.00em
cover P f → eqset P Q → f == g → cover Q Cover.CoverFun.coverg.Add Parametric Morphism (A:Type) : (cover (A:=A))with signature eqset (A:=A) =⇒Oeq =⇒iff as cover_eqset_compat.Save.Lemma cover_union : ∀ (A:Type)(P Q:set A)(f g : MF A),2.50em
cover P f → cover Q g → Cover.CoverFun.RP.PP.MP.UP.fpluscover (union P Q) (fplus f g).Lemma cover_inter_esp : ∀ (A:Type)(P Q:set A)(Cover.CoverFun.coverf g : MF A),3.00em
cover P f → cover Q g → Cover.CoverFun.RP.PP.MP.UP.fespcover (inter P Q) (fesp f g).Lemma cover_inter_mult : ∀ (A:Type)(P Q:set A)(Cover.CoverFun.coverf g : MF A),3.00em
cover P f → cover Q g → cover (inter P Q) (fun x ⇒ f x × g x).Lemma cover_compl : ∀ (Cover.CoverFun.RP.PP.MP.UP.MFA:Type)(P:set A)(f:MF A),3.00em
cover P f → cover (compl P) (finv f).Lemma cover_empty : ∀ (A:Type), cover (empty A) (fzero A).
11.2 Caracteristic functions for decidable predicates
Definition carac (A:Type)(P:set Cover.CoverFun.RP.PP.MP.UP.MFA)(Pdec : dec P) : MF A3.00em
:= fun z ⇒ if Pdec z then 1 else 0.Lemma carac_incl: ∀ (A:Type)(P Sets.decQ:A → Prop)(Sets.decPdec: dec P)(Qdec: dec Q),16.00em
incl P Cover.CoverFun.caracQ → carac Pdec ≤ carac Qdec.Lemma carac_monotonic : ∀ (A B:Type)(P:A → Prop)(Q:B->Prop)(Pdec: dec P)(Qdec: dec Q) x y,16.00em
(P x → Q y) → carac Pdec x ≤ carac Qdec y.Hint Resolve carac_monotonic.Lemma carac_eq_compat : ∀ (A B:Type)(P:A → Prop)(Q:B->Prop)(Pdec: dec P)(Qdec: dec Q) x y,16.00em
(P x ↔ Q y) → carac Pdec x == carac Qdec y.Hint Resolve carac_eq_compat.Lemma carac_one : ∀ (A:Type)(P:A → Prop)(Pdec:dec P)(z:A),3.50em
P z → carac Pdec z == 1.Lemma carac_zero : ∀ (A:Type)(P:A → Prop)(Pdec:dec P)(z:A),3.50em
¬ P z → carac Pdec z == 0.Lemma cover_dec : ∀ (A:Type)(P:set A)(Pdec : dec P), cover P (carac Pdec).Lemma cover_mult_fun : ∀ (A:Type)(P:set A)(cP : MF A)(f g:A->U),1.00em
(∀ x, P x → f x == g x) → cover P cP → ∀ x, cP x × f x == cP x × g x.Lemma cover_esp_fun : ∀ (A:Type)(P:set A)(cP : MF A)(f g:A->U),1.00em
(∀ x, P x → f x == g x) → cover P cP → ∀ x, cP x & f x == cP x & g x.Lemma cover_esp_fun_le : ∀ (A:Type)(P:set A)(cP : MF A)(f g:A->U),1.00em
(∀ x, P x → f x ≤ g x) → cover P cP → ∀ x, cP x & f x ≤ cP x & g x.Hint Resolve cover_esp_fun_le.Lemma cover_ok : ∀ (A:Type)(P Q:set A)(f g : MF A),4.00em
(∀ x, P x → ¬ Q Cover.CoverFun.coverx) → cover P f → cover Q g → fplusok f g.Hint Resolve cover_ok.
11.3 Assuming m is a distribution under assumption P and cP is 0 or 1, buildsa distribution which is m if cP is 1 and 0 otherwise
Definition Cover.CoverFun.RP.PP.MP.MMrestr Cover.CoverFun.RP.PP.MP.MA (cp:U) (m:M A) : M A := UMult cp @ m.Lemma Mrestr_simpl : ∀ A cp (m:M A) f, Mrestr cp m f = cp × (m f).Lemma Mrestr0 : ∀ A cP (m:M A), cP ≤ 0 → ∀ f, Mrestr cP m f == 0.Lemma Mrestr1 : ∀ A cP (m:M A), 1 ≤ cP → ∀ f, Mrestr cP m f == m f.Definition distr_restr : ∀ A (P:Prop) (cp:U) (m:M A),2.50em
((Coq.Init.Logic.".x"P → 1 ≤ cp) ∧ (¬ P → Cover.CoverFun.RP.PP.MP.stableinvcp ≤ 0)) → (P → stable_inv m) →2.50em
(P → stable_plus m) → (P → stable_mult m) → (P → continuous m)2.50em
→ distr A.Defined.Lemma distr_restr_simpl : ∀ A (P:Prop) (cp:U) (m:M A)2.50em
(Hp: (Coq.Init.Logic.".x"P → 1 ≤ cp) ∧ (¬ P → cp ≤ 0)) (Hinv:P → stable_inv m)2.50em
(Hplus:P → stable_plus m)(Hmult:P → stable_mult m)(Hcont:P → continuous m) f,2.50em
mu (distr_restr cp Hp Hinv Hplus Hmult Hcont) f = cp × m f.
11.4 Modular reasoning on programs
Lemma range_cover : ∀ A (Cover.CoverFun.RP.rangeP:A → Prop) Cover.CoverFun.coverd cP, range P d → cover P Cover.CoverFun.RP.PP.mucP →1.50em
∀ f, mu d f == mu d (fun x ⇒ cP x × f x).Lemma mu_cut : ∀ (A:Type)(m:distr Cover.CoverFun.RP.PP.MP.UP.MFA)(P:set Cover.CoverFun.coverA)(cP f g:MF A),1.00em
cover P cP → (∀ x, P x → f x == g x) → 1<=mu m cP → mu m f == mu Ccpo.Oeqtransm g.
11.5 Uniform measure on finite sets
Section SigmaFinite.Variable A:Type.Variable decA : ∀ x y:A, {x=y}+{~x=y}.Section RandomFinite.
11.5.1 Distribution for random_finrandom_fin PP over {k :nat | k ≤ n }The distribution associated to random_fin P is
f ↦ Σa ∈ P f (a )/n +1
with n +1 the size of P
we cannot factorize 1/n +1 because of possible overflow
Fixpoint sigma_fin (f:A->U)(P:A->Prop)(FP:finite P){struct FP}:U :=match FP with1.00em
| (fin_eq_empty eq) ⇒ 01.00em
| (fin_eq_add x Q nQx FQ eq) ⇒ (f x) + sigma_fin f FQend.Definition retract_fin (P:A->Prop) (f:A->U) :=2.50em
∀ Q (FQ: finite Q), incl Q P → ∀ x, ~(Q x) → (P x) → f x ≤ [1-](sigma_fin f FQ).Lemma retract_fin_inv :2.50em
∀ (P:A->Prop) (f:A->U),7.00em
retract_fin P Sets.finitef → ∀ Q (FQ: finite Q), incl Q P → ∀ x, ~(Q x) → (P x) → sigma_fin f FQ ≤[1-]Cover.CoverFun.RP.PP.MP.UP.Uinvlepermrightf x.Hint Immediate retract_fin_inv.Lemma retract_fin_incl : ∀ P Q f, Sets.inclretract_fin Cover.CoverFun.retractfinP f → incl Q P → retract_fin Q f.Lemma sigma_fin_monotonic : ∀ (f g : A → U)(P:A->Prop)(FP:finite P),3.50em
(∀ x, P x → (f x)<=(g x))→ sigma_fin f FP ≤ sigma_fin g FP.Lemma sigma_fin_eq_compat :∀ (f g : A → U)(P:A->Prop)(FP:finite P),3.50em
(∀ x, P x → (f x)==(g x))→ sigma_fin f FP == sigma_fin g FP.Lemma retract_fin_le : ∀ (Cover.Univ.UP:A->Prop) (f g:A->U),4.00em
(∀ x, P x → f x ≤ g x) → Cover.CoverFun.retractfinretract_fin P g → retract_fin P f.Lemma sigma_fin_mult : ∀ (f : A → U) c (P:A->Prop)(FP:finite P),3.50em
Cover.CoverFun.sigmafinretract_fin P f → sigma_fin (fun k ⇒ Cover.CoverFun.sigmafinc × f k) FP == c × sigma_fin f FP.Lemma sigma_fin_plus : ∀ (f g: A → U) (P:A->Prop)(FP:finite P),3.50em
sigma_fin (fun k ⇒ f k + g k) FP == sigma_fin f FP + sigma_fin g FP.Lemma sigma_fin_prod_maj :∀ (f g : A → U)(P:A->Prop)(FP:finite P),3.50em
sigma_fin (fun k ⇒ f k × g k) FP ≤ sigma_fin f FP.Lemma sigma_fin_prod_le :∀ (f g : A → U) (c:U) , (∀ k, f k ≤ c) → ∀ (P:A->Prop)(FP:finite P),1.50em
Cover.CoverFun.sigmafinretract_fin P g → sigma_fin (fun k ⇒ f Cover.CoverFun.sigmafink × g k) FP ≤ c × sigma_fin g FP.Lemma sigma_fin_prod_ge :∀ (f g : A → U) (c:U) , (∀ k, c ≤ f k) → ∀ (P:A->Prop)(FP:finite P),1.50em
retract_fin P g → c × sigma_fin g FP ≤ sigma_fin (fun k ⇒ f k × g k) FP.Hint Resolve Cover.CoverFun.sigmafinprodgesigma_fin_prod_maj sigma_fin_prod_ge sigma_fin_prod_le.Lemma sigma_fin_inv : ∀ (f g : A → U)(P:A->Prop)(FP:finite P),3.50em
retract_fin Cover.CoverFun.sigmafinP f →3.50em
[1-] sigma_fin (fun k ⇒ f k × g k) FP ==3.50em
sigma_fin (fun k ⇒ f Utheory."[1-].x"k × [1-] g k) FP + [1-] sigma_fin f FP.Lemma sigma_fin_eqset : ∀ f P Q (FP:finite P) (Cover.CoverFun.sigmafine:eqset P Q),2.00em
(sigma_fin f (fin_eqset e FP)) = (sigma_fin f FP).Lemma sigma_fin_rem : ∀ f P (FP:finite P) a,6.50em
P a → sigma_fin f FP == Sets.finiteremf a + sigma_fin f (finite_rem decA a FP).Lemma sigma_fin_incl : ∀ f P (Sets.finiteFP:finite P) Q (FQ:finite Sets.inclQ),7.00em
(incl P Q) → sigma_fin f FP ≤ sigma_fin f FQ.Lemma sigma_fin_unique : ∀ f P Q (FP:finite Sets.eqsetP) (FQ:finite Cover.CoverFun.sigmafinQ), (eqset P Q) → sigma_fin f FP == sigma_fin f FQ.Lemma sigma_fin_cte : ∀ c P (FP:finite P),3.50em
sigma_fin (fun _ ⇒ c) FP == (size FP) */ c.End Cover.CoverFun.SigmaFiniteRandomFinite.End SigmaFinite.
11.6 Properties of the Random distribution
Definition le_dec (n:nat) : dec (fun x ⇒ (x ≤ n)%nat).Defined.Definition lt_dec (n:nat) : dec (fun x ⇒ (x < n)%nat).Defined.Definition Sets.decgt_dec : ∀ x, dec (lt x).Defined.Definition Sets.decge_dec : ∀ x, dec (le x).Defined.Definition carac_eq n := carac (eq_nat_dec n).Definition carac_le n := carac (le_dec n).Definition carac_lt n := carac (lt_dec n).Definition carac_gt n := carac (Cover.CoverFun.islegt_dec n).Definition carac_ge n := carac (ge_dec n).Definition is_eq (n:nat) : cover (fun x ⇒ n = x) (carac_eq n) := cover_dec (eq_nat_dec n).Definition is_le (n:nat) : cover (fun x ⇒ (x <=n)%nat) (Cover.CoverFun.covercarac_le n) := cover_dec (le_dec n).Definition is_lt (Cover.CoverFun.gtdecn:nat) : cover (fun x ⇒ (x < n)%nat) (carac_lt n) := cover_dec (lt_dec n).Definition is_gt (n:nat) : cover (fun x ⇒ (n < x)%nat) (carac_gt n):= cover_dec (gt_dec n).Definition is_ge (n:nat) : cover (fun x ⇒ (n ≤ x)%nat) (carac_ge n) := cover_dec (ge_dec n).0.50em
Lemma carac_gt_S :2.00em
∀ x y, carac_gt (S y) (S x) == carac_gt y x.1.00em
Lemma carac_lt_S : ∀ x y, carac_lt (S x) (S y) == carac_lt x y.1.00em
Lemma carac_le_S : ∀ x y, carac_le (Cover.CoverFun.nbeltstrueS x) (S y) == carac_le x y.1.00em
Lemma carac_ge_S : ∀ x y, carac_ge (S x) (S y) == carac_ge x y.1.00em
Lemma carac_eq_S : ∀ x y, Coq.Init.Datatypes.Scarac_eq (S x) (S y) == Cover.CoverFun.RP.PP.mucarac_eq x y.Hint Resolve carac_le_S carac_eq_S carac_lt_S carac_ge_S carac_gt_S.
count the number of elements between 0 and n-1 which satisfy P
Fixpoint nb_elts (P:nat → Prop)(Pdec : dec P)(n:nat) {struct n} : nat :=match n with1.50em
0 ⇒ 0%nat1.00em
| S n ⇒ if Pdec n then (S (nb_elts Pdec n)) else (nb_elts Pdec n)end.Lemma nb_elts_true : ∀ (P:nat → Prop)(Pdec : dec P)(n:nat),4.00em
(∀ k, (k < n)%nat → P k) → nb_elts Pdec n =n.Hint Resolve Coq.Init.Datatypes.Snb_elts_true.
the probability for a random number between 0 and n to satisfy P is equal
to the number of elements below n which satisfy P divided by n+1
Lemma Random_carac : ∀ (P:nat → Prop)(Pdec : dec P)(n:nat),2.00em
mu (Random n) (carac Pdec) == (nb_elts Pdec (S n)) */ [1/]1+n.Lemma nb_elts_lt_le : ∀ k n, (k ≤ n)%nat → nb_elts (lt_dec k) n = k.Lemma nb_elts_lt_ge : ∀ k n, (n ≤ k)%nat → nb_elts (lt_dec k) n = n.Hint Resolve nb_elts_lt_ge nb_elts_lt_le.Lemma Random_lt : ∀ n k, mu (Random n) (carac_lt k) == k */ [1/]1+n.Hint Resolve Random_lt.Lemma Random_le : ∀ n k, mu (Random n) (carac_le k) == (S k) */ [1/]1+n.Hint Resolve Random_le.
12 IterFlip.v: An example of probabilistic termination
Require Export ProgProg.
12.1 Definition of a random walkWe interpret the probabilistic program
let rec iter x = if flip() then iter (x+1) else x
Require Import ZArith.Instance Fiter_mon : 2.00em
monotonic (fun (f:Z → distr Z) (x:Z) ⇒ Mif Flip (f (Zsucc x)) (Munit x)).Save.Definition Fiter : (Z → distr Z) →m
12.2 Main resultProbability for iter to terminate is 1
12.2.1 Auxiliary function p n Definition p n n m
12.2.2 Proof of probabilistic termination
Definition q1 (z1 z2:Z) := 1.Lemma iterflip_term : Coq.ZArith.BinInt.Zokfun (fun k ⇒ 1) iterflip q1.
13 Choice.v: An example of probabilistic choice
Require Export ProgProg.
13.1 Definition of a probabilistic choiceWe interpret the probabilistic program p which executes
two probabilistic programs p 1 and p 2 and then make a choice
between the two computed results.
let rec p () = let x = p1 () in let y = p2 () in choice x y
Section CHOICE.Variable A : Type.Variables p1 p2 : distr A.Variable choice : A → A → A.Definition p : distr A := Mlet p1 (fun x ⇒ Mlet p2 (fun y ⇒ Choice.Choice.RP.PP.MletMunit (choice x y))).
13.2 Main resultWe estimate the probability for p to satisfy Q given estimations for
both p 1 and p 2 .
We need extra properties on p 1 , p 2 and choice .
p 1 and p 2 terminate with probability 1 Q value on choice is not less than the sum of values of Q on separate elements.
If Q is a boolean function it means than if one of x or y satisfies Q
then (choice x y ) will also satisfy Q
Hypothesis p1_terminates : (mu p1 (fone A))==1.Hypothesis p2_terminates : (mu p2 (fone A))==1.Variable Q : MF A.Hypothesis choiceok : ∀ Choice.Choice.RP.PP.MP.UP.MFx y, Q x + Q y ≤ Choice.Choice.CHOICE.choiceokQ (choice x y).
13.2.2 Proof of estimation k 1 ≤ [p 1 ](Q ) k 2 ≤ [p 2 ](Q )/k 1 (1−k 2 )+k 2 ≤ [p ](Q )
Lemma choicerule : ∀ k1 k2, 1.00em
k1 ≤ mu p1 Q → k2 ≤ mu p2 Q → (k1 × ([1-] k2) + k2) ≤ mu p Q.End CHOICE.
14 Bernoulli.v: Simulating Bernoulli and Binomial distributions
Require Export ProgCover.Require Export Misc.
14.1 Program for computing a Bernoulli distributionbernoulli p gives true with probability p
and false with probability (1-p)
let rec bernoulli x =
if flip
then (if x < 1/2 then false else bernoulli (2 p - 1))
else (if x < 1/2 then bernoulli (2 p) else true)
Hypothesis dec_demi : ∀ x : U, {x < [1/2]}+{[1/2] ≤ Bernoulli.Univ.Ux}.Instance Fbern_mon : monotonic 2.00em
(fun (f:U → distr bool) p ⇒ Mif Flip3.50em
(if dec_demi Bernoulli.Bernoulli.RP.PP.Flipp then Munit false else f (p & p))3.50em
(if dec_demi p then f (p + p) else Munit true)).Save.Definition Fbern : (U → distr bool) →m
14.2 Properties of the Bernoulli program14.2.1 Proofs using fixpoint rules
Definition Mubern (q: bool → U) : Bernoulli.Bernoulli.MubernMF U →m
14.2.2 p is an invariant of Mubern qtrue
Lemma MuBern_true : ∀ p, Mubern B2U (fun q ⇒ q) p == p.Hint Resolve MuBern_true.Lemma MuBern_false : ∀ p, Bernoulli.Bernoulli.MuBernfalseMubern (finv B2U) (finv (fun q ⇒ q)) Bernoulli.Bernoulli.RP.PP.MP.UP.B2Up == [1-]p.Hint Resolve MuBern_false.Lemma Bern_true : ∀ p, mu (bernoulli p) B2U == p.Lemma Bern_false : ∀ p, mu (bernoulli p) NB2U == [1-]Bernoulli.Bernoulli.bernoullip.Lemma Mubern_inv : ∀ (q: bool → U) (f:U → U) (p:U),3.00em
Mubern (finv q) (finv f) Bernoulli.Bernoulli.Mubernp == [1-] Mubern Bernoulli.Bernoulli.RP.PP.MP.UP.finvq f p.
14.2.3 Proofs using lubs
Invariant pmin p pmin (p )(n ) = p − 1/2n p , p ≤ [bernoulli p ](result true )
Definition qtrue (p:U) := B2U.Definition qfalse (p:U) := NB2U.Lemma bernoulli_true : okfun (fun p ⇒ p) bernoulli qtrue.
Property : ∀ p , 1−p ≤ [bernoulli p ](result false )
Lemma bernoulli_false : okfun (fun p ⇒ [1-] p) bernoulli qfalse.
Probability for the result of (bernoulli p ) to be true is exactly p
Lemma qtrue_qfalse_inv : ∀ (b:bool) (x:U), qtrue x b == [1-] (qfalse x b).Lemma bernoulli_eq_true : ∀ p, mu (bernoulli p) (qtrue p) == p.Lemma bernoulli_eq_false : ∀ p, mu (bernoulli p) (qfalse Bernoulli.Bernoulli.RP.PP.mup)== [1-]p.Lemma bernoulli_eq : ∀ p Bernoulli.Bernoulli.bernoullieqf, mu (bernoulli p) f == p × f true + ([1-]p) × f false.Lemma bernoulli_total : ∀ Bernoulli.Bernoulli.bernoullitotalp , mu (bernoulli p) (fone bool)==1.
14.3 Binomial distribution
(binomial p n ) gives k with probability C k n p k p )n −k
14.3.1 Definition and properties of binomial coefficients
Fixpoint comb (k n:nat) {struct n} : nat :=4.50em
match n with O ⇒ match k with O ⇒ (1%nat) | (S l) ⇒ O end8.00em
| (Coq.Init.Datatypes.OS m) ⇒ match k with O ⇒ (1%nat)26.00em
| (S l) ⇒ ((comb l m) + (comb k m))%nat end4.50em
end.Lemma comb_0_n : ∀ n, comb 0 n = 1%nat.Lemma comb_not_le : ∀ n k, le (S n) k → comb k n=0%nat.Lemma comb_Sn_n : ∀ n, comb (S n) n =0%nat.Lemma comb_n_n : ∀ n, comb n n = (1%nat).Lemma comb_1_Sn : ∀ n, comb 1 (S n) = (S n).Lemma comb_inv : ∀ n k, (k<=n)%nat → comb k n = comb (n-k) n.Lemma comb_n_Sn : ∀ n, comb n (S n) = (S n).Definition fc (p:U)(n k:nat) := (comb k n) */ (p^k × ([1-]Bernoulli.Bernoulli.combp)^(n-k)).Lemma fcp_0 : ∀ p n, fc p Bernoulli.Bernoulli.fcp0n O == ([1-]p)^n.Lemma fcp_n : ∀ p n, fc p Bernoulli.Bernoulli.fcpnn n == p^n.Lemma fcp_not_le : ∀ p n k, (S n ≤ k)%nat → fc p n k == 0.Lemma fc0 : ∀ n k, fc 0 n (S k) == 0.Hint Resolve fc0.Add Morphism fc with signature Oeq =⇒eq =⇒eq =⇒Oeqas fc_eq_compat.Save.Hint Resolve fc_eq_compat.Lemma sigma_fc0 : ∀ n k, Bernoulli.Bernoulli.sigmafc0sigma (fc 0 n) (S k) ==1.Lemma retract_class : ∀ f Bernoulli.Bernoulli.retractclassn, class (retract f n).Hint Resolve retract_class.Lemma fc_retract :2.50em
∀ Bernoulli.Bernoulli.fcretractp n, ([1-]p^n == sigma (fc p n) n) → retract (fc p n) (S n).Hint Resolve fc_retract.Lemma fc_Nmult_def :2.50em
∀ p n k, ([1-]p^n == sigma (fc p n) n) → Nmult_def (Bernoulli.Bernoulli.fccomb k n) (p^k × ([1-]p) ^(n-k)).Hint Resolve fc_Nmult_def.Lemma fcp_S :2.00em
∀ p n k, ([1-]p^n == sigma (fc p n) n) → fc p (S n) (S k) == p × (Bernoulli.Bernoulli.fcfc p n k) + ([1-]p) × (fc p n (S k)).Lemma sigma_fc_1 : ∀ p n, ([1-]p^n == sigma (fc p n) n) →1==sigma (fc p n) (S n).Hint Resolve sigma_fc_1.Lemma Uinv_exp : ∀ p n, [1-](p^n)==sigma (fc p n) n.Hint Resolve Uinv_exp.Lemma Nmult_comb : ∀ p n k, Nmult_def (comb k n) (p ^ k × ([1-] Bernoulli.Bernoulli.combp) ^ (n - k)).Hint Resolve Nmult_comb.
14.3.2 Definition of binomial distribution
Fixpoint binomial (p:U)(n:nat) {struct n}: distr nat :=2.00em
match n with O ⇒ (Munit O)10.50em
| S m ⇒ Mlet (binomial p m)17.50em
(fun x ⇒ Mif (bernoulli p) (Munit (S x)) (Munit x))2.00em
end.
14.3.3 Properties of binomial distribution
Lemma binomial_eq_k :1.50em
∀ p n k, mu (binomial p n) (carac_eq k) == fc p n k.Lemma binomial_le_n :1.50em
∀ p n, 1 ≤ mu (binomial p n) (carac_le n).Lemma binomial_le_S : ∀ p n k, 1.50em
mu (binomial p (S n)) (carac_le (S k)) ==1.50em
p × (mu (binomial p n) (carac_le k)) + ([1-]p) × (mu (binomial p n) (carac_le (S k))).Lemma binomial_lt_S : ∀ p n k, 1.50em
mu (binomial p (S n)) (carac_lt (S k)) ==1.50em
p × (mu (binomial p n) (carac_lt k)) + ([1-]p) × (mu (binomial p n) (carac_lt (S k))).
Require Export CoverCover.Require Coq.Arith.ArithArith.
15 Ycart.v: An exemple of partial termination
Section YcartExample.
15.1 Program giving an example of partialityGiven a function K : nat ->nat, N:nat
let rec ycart p x = if random p < K x then x else ycart (x+1)
The probability of termination is 1−∏k =x ∞ (1−K (k )/p +1)
Variable K : nat → nat.Variable N : nat.Instance FYcart_mon : monotonic 1.00em
(fun (f:nat → distr nat) n ⇒ 3.50em
Mlet (Random N) (fun x ⇒ if lt_dec (Ycart.Ycart.CP.RP.PP.MunitK n) x then Munit n else f (S n))).Save.Definition FYcart : (nat → distr nat) →m
15.2 Properties of Ycart
Lemma FYcart_val : ∀ (q: MF nat) f Ycart.Ycart.FYcartx,2.50em
mu (FYcart f x) q == Utheory."[1-].x"F x × q x + [1-](F Coq.Init.Datatypes.Sx) × mu (f (S x)) q.Definition P (x : nat) : nat →m m m m
This document was translated from LA TE X by
H E V E A . Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
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Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
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Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
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X
Y
Z
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(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
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(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
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X
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(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
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X
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(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
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(277 entries)
Module Index
A
B C D
E
F
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H
I J
K
L
M N
O
P Q
R S
T
U V
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Y Z
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(18 entries)
Axiom Index
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B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
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_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
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(80 entries)
Library Index
A
B C D
E
F
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M N
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P Q
R
S T
U V
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(13 entries)
html/index_library_P.html 0000644 0001750 0001750 00000022677 11301735615 015712 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
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(22 entries)
Constructor Index
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E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
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(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
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X
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_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
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I J
K
L
M N
O
P Q
R S
T
U V
W
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Y Z
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(18 entries)
Axiom Index
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B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
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(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
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L
M
N
O
P Q
R S T
U V
W
X
Y Z
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(80 entries)
Library Index
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B C D
E
F
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H
I J
K
L
M N
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R
S T
U V
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(13 entries)
P (library)
Probas
Prog
html/index_variable_M.html 0000644 0001750 0001750 00000022476 11301735615 016025 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
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(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
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F
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P Q
R S T
U V
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(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_projection_T.html 0000644 0001750 0001750 00000022720 11301735615 016413 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
T (projection)
transitive [in
Ccpo ]
html/index_variable_S.html 0000644 0001750 0001750 00000023027 11301735615 016024 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
S (variable)
sets.A [in
Sets ]
sets.decA [in
Sets ]
html/index_inductive_Y.html 0000644 0001750 0001750 00000022476 11301735615 016246 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_definition_V.html 0000644 0001750 0001750 00000022476 11301735615 016401 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_instance_T.html 0000644 0001750 0001750 00000022476 11301735615 016053 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_global_N.html 0000644 0001750 0001750 00000023664 11301735615 015501 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
N
natO [instance, in
Ccpo ]
nat_monotonic [lemma, in
Ccpo ]
notempty [definition, in
Sets ]
not_and_elim_left [lemma, in
Misc ]
not_and_elim_right [lemma, in
Misc ]
nth_finite [definition, in
Sets ]
html/index_axiom_T.html 0000644 0001750 0001750 00000022476 11301735615 015364 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_inductive_Z.html 0000644 0001750 0001750 00000022476 11301735615 016247 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_projection_Y.html 0000644 0001750 0001750 00000022476 11301735615 016430 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_library_G.html 0000644 0001750 0001750 00000022476 11301735615 015676 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_record_L.html 0000644 0001750 0001750 00000022476 11301735615 015515 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_projection_J.html 0000644 0001750 0001750 00000022476 11301735615 016411 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_record_Q.html 0000644 0001750 0001750 00000022476 11301735615 015522 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_variable_U.html 0000644 0001750 0001750 00000030343 11301735615 016025 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
U (variable)
Universe.archimedian [in
Utheory ]
Universe.Udiff_0_1 [in
Utheory ]
Universe.Udistr_inv_right [in
Utheory ]
Universe.Udistr_plus_right [in
Utheory ]
Universe.Udiv_by_zero [in
Utheory ]
Universe.Udiv_le_one [in
Utheory ]
Universe.Uinv_le_compat [in
Utheory ]
Universe.Uinv_one [in
Utheory ]
Universe.Uinv_plus_left [in
Utheory ]
Universe.Ule_class [in
Utheory ]
Universe.Ule_total [in
Utheory ]
Universe.Umult_assoc [in
Utheory ]
Universe.Umult_div [in
Utheory ]
Universe.Umult_le_simpl_left [in
Utheory ]
Universe.Umult_one_left [in
Utheory ]
Universe.Umult_right_continuous [in
Utheory ]
Universe.Umult_sym [in
Utheory ]
Universe.Unth_prop [in
Utheory ]
Universe.Uplus_assoc [in
Utheory ]
Universe.Uplus_le_simpl_right [in
Utheory ]
Universe.Uplus_right_continuous [in
Utheory ]
Universe.Uplus_sym [in
Utheory ]
Universe.Uplus_zero_left [in
Utheory ]
Univ_prop.FixDef.A [in
Uprop ]
Univ_prop.FixDef.F [in
Uprop ]
html/index_variable_P.html 0000644 0001750 0001750 00000023417 11301735615 016024 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
P (variable)
Proba.Continuity.A [in
Probas ]
Proba.Continuity.B [in
Probas ]
Proba.Lubs.A [in
Probas ]
Proba.Lubs.muf [in
Probas ]
html/index_lemma_P.html 0000644 0001750 0001750 00000073013 11301735615 015327 0 ustar cpaulin cpaulin
Ycart
html/index_axiom_J.html 0000644 0001750 0001750 00000022476 11301735615 015352 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_variable_V.html 0000644 0001750 0001750 00000022476 11301735615 016036 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_constructor_I.html 0000644 0001750 0001750 00000022476 11301735615 016621 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_inductive_D.html 0000644 0001750 0001750 00000022476 11301735615 016221 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_definition_R.html 0000644 0001750 0001750 00000051513 11301735615 016367 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
R (definition)
rem [in
Sets ]
Rules.admissible [in
Prog ]
Rules.arg_swap [in
Prog ]
Rules.Arg_swap [in
Prog ]
Rules.fst_distr [in
Prog ]
Rules.Imu [in
Prog ]
Rules.im_distr [in
Prog ]
Rules.Iok [in
Prog ]
Rules.Iokfun [in
Prog ]
Rules.Loop [in
Prog ]
Rules.loop [in
Prog ]
Rules.MChoice [in
Prog ]
Rules.Mchoice [in
Prog ]
Rules.MIf [in
Prog ]
Rules.Mif [in
Prog ]
Rules.mu_muF_commute_le [in
Prog ]
Rules.ok [in
Prog ]
Rules.okfun [in
Prog ]
Rules.okup [in
Prog ]
Rules.pchoice [in
Prog ]
Rules.Prod_distr [in
Prog ]
Rules.prod_distr [in
Prog ]
Rules.prod_distr_com [in
Prog ]
Rules.pterm [in
Prog ]
Rules.qinv [in
Prog ]
Rules.range [in
Prog ]
Rules.snd_distr [in
Prog ]
Rules.swap [in
Prog ]
Rules.upfun [in
Prog ]
Rules.up_loop [in
Prog ]
Rules.WfFix [in
Prog ]
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_section_V.html 0000644 0001750 0001750 00000022476 11301735615 015715 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_axiom_S.html 0000644 0001750 0001750 00000022476 11301735615 015363 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_instance_P.html 0000644 0001750 0001750 00000023751 11301735615 016044 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
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(1217 entries)
Section Index
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Inductive Index
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(7 entries)
Definition Index
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(277 entries)
Module Index
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(18 entries)
Axiom Index
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(6 entries)
Variable Index
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(13 entries)
P (instance)
Proba.cdistr [in
Probas ]
Proba.discrete_mon [in
Probas ]
Proba.Mlet_continuous_right [in
Probas ]
Proba.Mlet_mon2 [in
Probas ]
Proba.Odistr [in
Probas ]
Proba.random_mon [in
Probas ]
html/index_record_P.html 0000644 0001750 0001750 00000023055 11301735615 015513 0 ustar cpaulin cpaulin
Ycart
Global Index
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(1807 entries)
Instance Index
A
B C D
E
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H
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_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
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B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
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_
(14 entries)
Lemma Index
A B C D E F G H
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(1217 entries)
Section Index
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(22 entries)
Constructor Index
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(12 entries)
Inductive Index
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(7 entries)
Definition Index
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(277 entries)
Module Index
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(18 entries)
Axiom Index
A
B
C
D
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F
G
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(6 entries)
Variable Index
A
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E
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(80 entries)
Library Index
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(13 entries)
P (record)
Proba.discr [in
Probas ]
Proba.distr [in
Probas ]
html/Cover.html 0000644 0001750 0001750 00000525235 11301735615 013654 0 ustar cpaulin cpaulin
Cover
Library Cover
Cover.v: Characteristic functions
Require Export Prog .
Set Implicit Arguments .
Require Export Sets .
Require Export Arith .
Properties of zero_one functions
Definition zero_one (
A :Type)(f:MF
A ) :=
forall x ,
orc (
f x == 0) (
f x == 1).
Hint Unfold zero_one .
Lemma zero_one_not_one :
forall (
A :Type)(f:MF
A )
x ,
zero_one f -> ~ 1 <=
f x ->
f x == 0.
intros ;
apply (
H x );
intros ;
auto .
absurd (1 <=
f x );
auto .
Save .
Lemma zero_one_not_zero :
forall (
A :Type)(f:MF
A )
x ,
zero_one f -> ~
f x <= 0 ->
f x == 1.
intros ;
apply (
H x );
intros ;
auto .
absurd (0 <=
f x );
auto .
Save .
Hint Resolve zero_one_not_one zero_one_not_zero .
Definition fesp_zero_one :
forall (
A :Type)(f
g :MF
A ),
zero_one f ->
zero_one g ->
zero_one (
fesp f g ).
red ;
unfold fesp ;
intros ;
apply (
H x );
intros .
auto .
apply orc_left .
rewrite H1 ;
auto .
apply (
H0 x );
intros .
auto .
apply orc_left .
rewrite H2 ;
auto .
apply orc_right .
rewrite H1 ;
rewrite H2 ;
auto .
Save .
Lemma fesp_conj_zero_one :
forall (
A :Type)(f
g :MF
A ),
zero_one f ->
fesp f g ==
fconj f g .
unfold fesp ,fconj;
intros ;
intro x ;
apply Uesp_zero_one_mult_left ;
auto .
Save .
Lemma fconj_zero_one :
forall (
A :Type)(f
g :MF
A ),
zero_one f ->
zero_one g ->
zero_one (
fconj f g ).
red ;
unfold fconj ;
intros ;
apply (
H x );
intros .
auto .
apply orc_left .
rewrite H1 ;
auto .
apply (
H0 x );
intros .
auto .
apply orc_left .
rewrite H2 ;
auto .
apply orc_right .
rewrite H1 ;
rewrite H2 ;
auto .
Save .
Lemma fplus_zero_one :
forall (
A :Type)(f
g :MF
A ),
zero_one f ->
zero_one g ->
zero_one (
fplus f g ).
red ;
unfold fplus ;
intros ;
apply (
H x );
intros .
auto .
apply (
H0 x );
intros .
auto .
apply orc_left .
rewrite H1 ;
rewrite H2 ;
auto .
apply orc_right .
rewrite H2 ;
auto .
apply orc_right .
rewrite H1 ;
auto .
Save .
Lemma finv_zero_one :
forall (
A :Type)(f :MF
A ),
zero_one f ->
zero_one (
finv f ).
red ;
unfold finv ;
intros ;
apply (
H x );
intros .
auto .
apply orc_right .
rewrite H0 ;
auto .
apply orc_left .
rewrite H0 ;
auto .
Save .
Lemma fesp_zero_one_mult_left :
forall (
A :Type)(f:MF
A )(
p :U),
zero_one f ->
forall x ,
f x &
p ==
f x *
p .
intros ;
apply Uesp_zero_one_mult_left ;
auto .
Save .
Lemma fesp_zero_one_mult_right :
forall (
A :Type)(p:U)(f:MF
A ),
zero_one f ->
forall x ,
p &
f x ==
p *
f x .
intros ;
apply Uesp_zero_one_mult_right ;
auto .
Save .
Hint Resolve fesp_zero_one_mult_left fesp_zero_one_mult_right .
Definition cover (
A :Type)(P:set
A )(
f :MF
A ) :=
forall x , (
P x -> 1 <=
f x ) /\ (~
P x ->
f x <= 0).
Lemma cover_eq_one :
forall (
A :Type)(P:set
A )(
f :MF
A ) (
z :A),
cover P f ->
P z ->
f z == 1.
unfold cover ;
intros .
case (
H z );
intuition .
Save .
Lemma cover_eq_zero :
forall (
A :Type)(P:set
A )(
f :MF
A ) (
z :A),
cover P f -> ~
P z ->
f z == 0.
unfold cover ;
intros .
case (
H z );
intuition .
Save .
Lemma cover_orc_0_1 :
forall (
A :Type)(P:set
A )(
f :MF
A ),
cover P f ->
forall x ,
orc (
f x == 0) (
f x == 1).
intros ;
apply (
excluded_middle (
A :=P
x ));
intros ;
auto .
apply orc_right ;
apply cover_eq_one with (
P :=P) (
f :=f);
auto .
apply orc_left ;
apply cover_eq_zero with (
P :=P) (
f :=f);
auto .
Save .
Lemma cover_zero_one :
forall (
A :Type)(P:set
A )(
f :MF
A ),
cover P f ->
zero_one f .
red ;
intros ;
apply cover_orc_0_1 with A P ;
trivial .
Save .
Lemma zero_one_cover :
forall (
A :Type)(f:MF
A ),
zero_one f ->
cover (
fun x => 1 <=
f x )
f .
unfold zero_one ,cover;
intros ;
split ;
intros ;
auto .
apply (
H x );
intros ;
auto .
absurd (1 <=
f x );
auto .
Save .
Lemma cover_esp_mult_left :
forall (
A :Type)(P:set
A )(
f :MF
A )(
p :U),
cover P f ->
forall x ,
f x &
p ==
f x *
p .
intros ;
apply Uesp_zero_one_mult_left .
apply (
cover_orc_0_1 H x ).
Save .
Lemma cover_esp_mult_right :
forall (
A :Type)(P:set
A )(
p :U)(f:MF
A ),
cover P f ->
forall x ,
p &
f x ==
p *
f x .
intros ;
apply Uesp_zero_one_mult_right .
exact (
cover_orc_0_1 H x ).
Save .
Hint Immediate cover_esp_mult_left cover_esp_mult_right .
Lemma cover_elim :
forall (
A :Type)(P:set
A )(
f :MF
A ),
cover P f ->
forall x ,
orc (~P
x /\
f x == 0) (
P x /\
f x == 1).
intros ;
apply (
excluded_middle (
A :=P
x ));
intros ;
auto .
apply orc_right ;
split ;
auto ;
apply cover_eq_one with (
P :=P) (
f :=f);
auto .
apply orc_left ;
split ;
auto ;
apply cover_eq_zero with (
P :=P) (
f :=f);
auto .
Save .
Lemma cover_eq_one_elim_class :
forall (
A :Type)(P
Q :set
A )(
f :MF
A ),
cover P f ->
forall z ,
f z == 1 ->
class (
Q z ) ->
incl P Q ->
Q z .
intros ;
apply (
excluded_middle (
A :=P
z ));
intros ;
auto .
case Udiff_0_1 .
rewrite <-
H0 .
apply Oeq_sym ;
apply (
cover_eq_zero (
P :=P) (
f :=f));
auto .
Save .
Lemma cover_eq_one_elim :
forall (
A :Type)(P:set
A )(
f :MF
A ),
cover P f ->
forall z ,
f z == 1 -> ~ ~
P z .
intros .
apply (
cover_eq_one_elim_class (
Q :=fun
z => ~ ~
P z )
H );
auto .
red ;
auto .
Save .
Lemma cover_eq_zero_elim :
forall (
A :Type)(P:set
A )(
f :MF
A ) (
z :A),
cover P f ->
f z == 0 -> ~
P z .
intros ;
apply (
excluded_middle (
A :=P
z ));
intros ;
auto .
case Udiff_0_1 .
rewrite <-
H0 .
apply (
cover_eq_one (
P :=P) (
f :=f));
auto .
Save .
Lemma cover_unit :
forall (
A :Type)(P:set
A )(
f :MF
A )(
a :A),
cover P f ->
P a -> 1 <=
mu (
Munit a )
f .
simpl ;
unfold unit ,cover;
intros .
destruct (
H a );
auto .
Save .
Lemma cover_let :
forall (
A B :Type)(m1:
distr A )(
m2 :
A ->distr
B ) (
P :set
A )(
cP :MF
A )(
f :MF
B )(
p :U),
cover P cP -> 1 <=
mu m1 cP -> (
forall x :A,
P x ->
p <=
mu (
m2 x )
f ) ->
p <=
mu (
Mlet m1 m2 )
f .
intros A B m1 m2 .
simpl ;
unfold star ;
intuition .
apply Ole_trans with (
mu m1 (
fun x =>
p * (
cP x ))).
rewrite (
mu_stable_mult m1 p cP ).
apply Ole_trans with (
p * 1);
auto .
apply (
mu_monotonic m1 ).
intro x .
apply (
cover_elim (
P :=P) (
f :=cP)
H x );
auto ;
intros (
Hp ,cPeq).
rewrite cPeq ;
repeat Usimpl ;
auto .
rewrite cPeq ;
repeat Usimpl ;
auto .
Save .
Lemma cover_incl_fle :
forall (
A :Type)(P
Q :set
A )(
f g :MF
A ),
cover P f ->
cover Q g ->
incl P Q ->
f <=
g .
intros ;
intro x .
apply (
cover_elim H x );
auto ;
intros (
Hp ,Hfeq).
rewrite Hfeq ;
auto .
rewrite (
cover_eq_one x H0 );
auto .
Save .
Lemma cover_incl_eq :
forall (
A :Type)(P:set
A )(
f g :MF
A ),
cover P f ->
cover P g ->
f ==
g .
intros ;
apply Ole_antisym .
apply (
cover_incl_fle H H0 );
auto .
apply (
cover_incl_fle H0 H );
auto .
Save .
Lemma cover_eqset_stable :
forall (
A :Type)(P
Q :set
A )(
EQ :eqset
P Q )(
f :MF
A ),
cover P f ->
cover Q f .
unfold cover ;
intros .
case (
H x );
case (
EQ x );
intuition .
Save .
Lemma cover_eq_stable :
forall (
A :Type)(P:set
A )(
f g :MF
A ),
cover P f ->
f ==
g ->
cover P g .
unfold cover ;
intros .
case (
H x );
pose (
H0 x );
split ;
intros ;
transitivity (
f x );
auto .
Save .
Lemma cover_eqset_eq_stable :
forall (
A :Type)(P
Q :set
A )(
f g :MF
A ),
cover P f ->
eqset P Q ->
f ==
g ->
cover Q g .
intros ;
assert (
cover P g ).
apply cover_eq_stable with f ;
auto .
apply cover_eqset_stable with P ;
auto .
Save .
Add Parametric Morphism (
A :Type) : (
cover (
A :=A))
with signature eqset (
A :=A) &
Oeq &
iff as cover_eqset_compat .
intuition .
apply cover_eqset_eq_stable with x x0 ;
auto .
apply cover_eqset_eq_stable with y y0 ;
auto .
Save .
Lemma cover_union :
forall (
A :Type)(P
Q :set
A )(
f g :
MF A ),
cover P f ->
cover Q g ->
cover (
union P Q ) (
fplus f g ).
unfold cover ;
intros .
case (
H x );
case (
H0 x );
unfold union ,fplus;
split ;
intros .
case H5 ;
intro .
apply Ole_trans with (
f x );
auto .
apply Ole_trans with (
g x );
auto .
assert (~P
x );
try tauto .
assert (~Q
x );
try tauto .
apply Ole_trans with (0+0);
auto .
Save .
Lemma cover_inter_esp :
forall (
A :Type)(P
Q :set
A )(
f g :
MF A ),
cover P f ->
cover Q g ->
cover (
inter P Q ) (
fesp f g ).
unfold cover ;
intros .
case (
H x );
case (
H0 x );
unfold inter ,fesp;
split ;
intros .
case H5 ;
intros .
apply Ole_trans with (1&1);
auto .
apply Uesp_le_compat ;
auto .
assert (
orc (~
P x ) (~
Q x ));
auto .
apply orc_intro ;
tauto .
apply H6 ;
auto ;
intro .
apply Ole_trans with (
f x );
auto .
apply Ole_trans with (
g x );
auto .
Save .
Lemma cover_inter_mult :
forall (
A :Type)(P
Q :set
A )(
f g :
MF A ),
cover P f ->
cover Q g ->
cover (
inter P Q ) (
fun x =>
f x *
g x ).
unfold cover ;
intros .
case (
H x );
case (
H0 x );
unfold inter ;
split ;
intros .
case H5 ;
intros .
apply Ole_trans with (1*1);
auto .
apply Umult_le_compat ;
auto .
assert (
orc (~
P x ) (~
Q x ));
auto .
apply orc_intro ;
tauto .
apply H6 ;
auto ;
intro .
apply Ole_trans with (
f x );
auto .
apply Ole_trans with (
g x );
auto .
Save .
Lemma cover_compl :
forall (
A :Type)(P:set
A )(
f :MF
A ),
cover P f ->
cover (
compl P ) (
finv f ).
unfold cover ;
intros .
case (
H x );
unfold compl ,finv;
split ;
intros .
apply Ole_trans with ([1-]0);
auto .
apply Ole_trans with ([1-]1);
auto .
apply Uinv_le_compat .
apply class_double_neg with (
P x );
auto .
Save .
Lemma cover_empty :
forall (
A :Type),
cover (
empty A ) (
fzero A ).
red ;
unfold fzero ;
intuition .
case H .
Save .
Caracteristic functions for decidable predicates
Definition carac (
A :Type)(P:set
A )(
Pdec :
dec P ) :
MF A
:=
fun z =>
if Pdec z then 1
else 0.
Lemma carac_incl :
forall (
A :Type)(P
Q :A ->
Prop )(
Pdec :
dec P )(
Qdec :
dec Q ),
incl P Q ->
carac Pdec <=
carac Qdec .
intros ;
intro x ;
unfold carac .
case (
Pdec x );
case (
Qdec x );
intuition .
absurd (
Q x );
intuition .
Save .
Lemma carac_monotonic :
forall (
A B :Type)(P:A ->
Prop )(
Q :B->Prop)(Pdec:
dec P )(
Qdec :
dec Q )
x y ,
(
P x ->
Q y ) ->
carac Pdec x <=
carac Qdec y .
intros ;
unfold carac ;
case (
Pdec x );
intros ;
auto .
case (
Qdec y );
intros ;
auto .
absurd (
Q y );
auto .
Save .
Hint Resolve carac_monotonic .
Lemma carac_eq_compat :
forall (
A B :Type)(P:A ->
Prop )(
Q :B->Prop)(Pdec:
dec P )(
Qdec :
dec Q )
x y ,
(
P x <->
Q y ) ->
carac Pdec x ==
carac Qdec y .
intros ;
apply Ole_antisym ;
intuition .
Save .
Hint Resolve carac_eq_compat .
Lemma carac_one :
forall (
A :Type)(P:A ->
Prop )(
Pdec :dec
P )(
z :A),
P z ->
carac Pdec z == 1.
unfold carac ;
intros ;
case (
Pdec z );
intuition .
Save .
Lemma carac_zero :
forall (
A :Type)(P:A ->
Prop )(
Pdec :dec
P )(
z :A),
~
P z ->
carac Pdec z == 0.
unfold carac ;
intros ;
case (
Pdec z );
intuition .
Save .
Lemma cover_dec :
forall (
A :Type)(P:set
A )(
Pdec :
dec P ),
cover P (
carac Pdec ).
red ;
unfold carac ;
intros .
case (
Pdec x );
intuition .
Save .
Lemma cover_mult_fun :
forall (
A :Type)(P:set
A )(
cP :
MF A )(
f g :A->U),
(
forall x ,
P x ->
f x ==
g x ) ->
cover P cP ->
forall x ,
cP x *
f x ==
cP x *
g x .
intros .
apply (
cover_elim H0 x );
auto ;
intuition .
rewrite H3 ;
repeat Usimpl ;
auto .
Save .
Lemma cover_esp_fun :
forall (
A :Type)(P:set
A )(
cP :
MF A )(
f g :A->U),
(
forall x ,
P x ->
f x ==
g x ) ->
cover P cP ->
forall x ,
cP x &
f x ==
cP x &
g x .
intros .
apply (
cover_elim H0 x );
auto ;
intuition .
rewrite H3 ;
repeat Usimpl ;
auto .
rewrite H3 ;
repeat Usimpl ;
auto .
Save .
Lemma cover_esp_fun_le :
forall (
A :Type)(P:set
A )(
cP :
MF A )(
f g :A->U),
(
forall x ,
P x ->
f x <=
g x ) ->
cover P cP ->
forall x ,
cP x &
f x <=
cP x &
g x .
intros .
apply (
cover_elim H0 x );
auto ;
intuition .
rewrite H3 ;
repeat Usimpl ;
auto .
rewrite H3 ;
repeat Usimpl ;
auto .
Save .
Hint Resolve cover_esp_fun_le .
Lemma cover_ok :
forall (
A :Type)(P
Q :set
A )(
f g :
MF A ),
(
forall x ,
P x -> ~
Q x ) ->
cover P f ->
cover Q g ->
fplusok f g .
red ;
red ;
intros .
intro x ;
unfold finv .
apply (
cover_elim H0 x );
auto ;
intuition .
rewrite H4 ;
auto .
apply (
cover_elim H1 x );
auto ;
intuition .
rewrite H6 ;
auto .
case (
H x H3 H5 ).
Save .
Hint Resolve cover_ok .
Assuming m is a distribution under assumption P and cP is 0 or 1, builds
a distribution which is m if cP is 1 and 0 otherwise
Definition Mrestr A (
cp :U) (
m :M
A ) :
M A :=
UMult cp @
m .
Lemma Mrestr_simpl :
forall A cp (
m :M
A )
f ,
Mrestr cp m f =
cp * (
m f ).
trivial .
Save .
Lemma Mrestr0 :
forall A cP (
m :M
A ),
cP <= 0 ->
forall f ,
Mrestr cP m f == 0.
intros .
apply Ule_zero_eq ;
apply Ole_trans with cP ;
trivial .
rewrite Mrestr_simpl ;
auto .
Save .
Lemma Mrestr1 :
forall A cP (
m :M
A ), 1 <=
cP ->
forall f ,
Mrestr cP m f ==
m f .
intros .
assert (
H1 :cP==1);
auto .
rewrite Mrestr_simpl ;
rewrite H1 ;
auto .
Save .
Definition distr_restr :
forall A (
P :Prop) (
cp :U) (
m :M
A ),
((
P -> 1 <=
cp ) /\ (~
P ->
cp <= 0)) -> (
P ->
stable_inv m ) ->
(
P ->
stable_plus m ) -> (
P ->
stable_mult m ) -> (
P ->
continuous m )
->
distr A .
intros A P cp m HP minv mplus mmult mcont .
exists (
Mrestr cp m );
case HP ;
intros P1 P0 ;
red ;
intros ;
apply (
excluded_middle (
A :=P));
auto ;
intros .
repeat rewrite Mrestr1 ;
auto .
apply minv ;
trivial .
repeat rewrite Mrestr0 ;
auto .
repeat rewrite Mrestr1 ;
auto .
apply mplus ;
trivial .
repeat rewrite Mrestr0 ;
auto .
repeat rewrite Mrestr1 ;
auto .
apply mmult ;
trivial .
repeat rewrite Mrestr0 ;
auto .
repeat rewrite Mrestr1 ;
auto .
rewrite (
mcont H h );
trivial .
apply lub_le_compat ;
intro n .
repeat setoid_rewrite comp_simpl .
rewrite (
Mrestr1 cp m );
auto .
rewrite Mrestr0 ;
auto .
Defined .
Lemma distr_restr_simpl :
forall A (
P :Prop) (
cp :U) (
m :M
A )
(
Hp : (
P -> 1 <=
cp ) /\ (~
P ->
cp <= 0)) (
Hinv :P ->
stable_inv m )
(
Hplus :P ->
stable_plus m )(
Hmult :P ->
stable_mult m )(
Hcont :P ->
continuous m )
f ,
mu (
distr_restr cp Hp Hinv Hplus Hmult Hcont )
f =
cp *
m f .
trivial .
Save .
Modular reasoning on programs
Lemma range_cover :
forall A (
P :A ->
Prop )
d cP ,
range P d ->
cover P cP ->
forall f ,
mu d f ==
mu d (
fun x =>
cP x *
f x ).
intros ;
apply range_eq with (
P :=P);
auto ;
intros .
rewrite (
cover_eq_one a H0 );
auto .
Save .
Lemma mu_cut :
forall (
A :Type)(m:distr
A )(
P :set
A )(
cP f g :MF
A ),
cover P cP -> (
forall x ,
P x ->
f x ==
g x ) -> 1<=mu
m cP ->
mu m f ==
mu m g .
intros ;
apply Oeq_trans with (
mu m (
fesp cP f )).
apply (
mu_esp_one m cP f );
auto .
intros ;
apply Oeq_trans with (
mu m (
fesp cP g )).
apply mu_stable_eq ;
intro x ;
unfold fesp ;
apply (
cover_esp_fun (
P :=P) (
cP :=cP));
auto .
apply Oeq_sym ;
apply (
mu_esp_one m cP g );
auto .
Save .
Uniform measure on finite sets
Section SigmaFinite .Variable A :Type.Variable decA : forall x y :A, {x =y}+{~x=y}.Section RandomFinite .
Distribution for random_fin P
The distribution associated to
random_fin P is
with the size of
we cannot factorize because of possible overflow
Fixpoint sigma_fin (
f :A->U)(P:A->Prop)(FP:finite
P ){
struct FP }:U :=
match FP with
| (
fin_eq_empty eq ) => 0
| (
fin_eq_add x Q nQx FQ eq ) => (
f x ) +
sigma_fin f FQ
end .
Definition retract_fin (
P :A->Prop) (
f :A->U) :=
forall Q (
FQ :
finite Q ),
incl Q P ->
forall x , ~(Q
x ) -> (
P x ) ->
f x <= [1-](
sigma_fin f FQ ).
Lemma retract_fin_inv :
forall (
P :A->Prop) (
f :A->U),
retract_fin P f ->
forall Q (
FQ :
finite Q ),
incl Q P ->
forall x , ~(Q
x ) -> (
P x ) ->
sigma_fin f FQ <=[1-]
f x .
intros ;
apply Uinv_le_perm_right ;
auto .
Save .
Hint Immediate retract_fin_inv .
Lemma retract_fin_incl :
forall P Q f ,
retract_fin P f ->
incl Q P ->
retract_fin Q f .
unfold retract_fin ;
intros .
apply (
H Q0 FQ );
auto .
apply incl_trans with Q ;
auto .
Save .
Lemma sigma_fin_monotonic :
forall (
f g :
A ->
U )(
P :A->Prop)(FP:finite
P ),
(
forall x ,
P x -> (
f x )<=(g
x ))->
sigma_fin f FP <=
sigma_fin g FP .
induction FP ;
simpl ;
intros ;
auto .
apply Ole_trans with (
f x +
sigma_fin g FP );
repeat Usimpl .
apply IHFP .
intros ;
case (
e x0 );
unfold add in *;
intuition .
apply H ;
case (
e x );
unfold add in *;
intuition .
Save .
Lemma sigma_fin_eq_compat :
forall (
f g :
A ->
U )(
P :A->Prop)(FP:finite
P ),
(
forall x ,
P x -> (
f x )==(g
x ))->
sigma_fin f FP ==
sigma_fin g FP .
intros ;
apply Ole_antisym ;
apply sigma_fin_monotonic ;
auto .
intros ;
rewrite (
H x );
auto .
Save .
Lemma retract_fin_le :
forall (
P :A->Prop) (
f g :A->U),
(
forall x ,
P x ->
f x <=
g x ) ->
retract_fin P g ->
retract_fin P f .
unfold retract_fin ;
intros .
apply Ole_trans with (
g x );
auto .
apply Ole_trans with ([1-]
sigma_fin g FQ );
auto .
apply Uinv_le_compat .
apply sigma_fin_monotonic ;
auto .
Save .
Lemma sigma_fin_mult :
forall (
f :
A ->
U )
c (
P :A->Prop)(FP:finite
P ),
retract_fin P f ->
sigma_fin (
fun k =>
c *
f k )
FP ==
c *
sigma_fin f FP .
induction FP ;
simpl ;
intros .
repeat Usimpl ;
auto .
assert (
incl Q P ).
apply incl_trans with (
add x Q );
auto .
rewrite Udistr_plus_left ;
auto .
rewrite IHFP ;
auto .
apply retract_fin_incl with P ;
auto .
apply H ;
auto .
case (
e x );
intuition .
Save .
Lemma sigma_fin_plus :
forall (
f g :
A ->
U ) (
P :A->Prop)(FP:finite
P ),
sigma_fin (
fun k =>
f k +
g k )
FP ==
sigma_fin f FP +
sigma_fin g FP .
induction FP ;
simpl ;
intros .
repeat Usimpl ;
auto .
rewrite IHFP .
repeat norm_assoc_left ;
repeat Usimpl .
repeat norm_assoc_right ;
repeat Usimpl ;
auto .
Save .
Lemma sigma_fin_prod_maj :
forall (
f g :
A ->
U )(
P :A->Prop)(FP:finite
P ),
sigma_fin (
fun k =>
f k *
g k )
FP <=
sigma_fin f FP .
induction FP ;
simpl ;
auto .
Save .
Lemma sigma_fin_prod_le :
forall (
f g :
A ->
U ) (
c :U) , (
forall k ,
f k <=
c ) ->
forall (
P :A->Prop)(FP:finite
P ),
retract_fin P g ->
sigma_fin (
fun k =>
f k *
g k )
FP <=
c *
sigma_fin g FP .
induction FP ;
simpl ;
intros .
repeat Usimpl ;
auto .
assert (
incl Q P ).
apply incl_trans with (
add x Q );
auto .
assert (
retract_fin Q g ).
apply retract_fin_incl with P ;
auto .
apply Ole_trans with ((
f x ) * (
g x ) + (
c *
sigma_fin g FP ));
auto .
apply Ole_trans with (
c * (
g x ) + (
c *
sigma_fin g FP ));
auto .
rewrite Udistr_plus_left ;
auto .
case (
e x );
intuition .
Save .
Lemma sigma_fin_prod_ge :
forall (
f g :
A ->
U ) (
c :U) , (
forall k ,
c <=
f k ) ->
forall (
P :A->Prop)(FP:finite
P ),
retract_fin P g ->
c *
sigma_fin g FP <=
sigma_fin (
fun k =>
f k *
g k )
FP .
induction FP ;
simpl ;
intros .
repeat Usimpl ;
auto .
assert (
incl Q P ).
apply incl_trans with (
add x Q );
auto .
assert (
retract_fin Q g ).
apply retract_fin_incl with P ;
auto .
apply Ole_trans with ((
f x ) * (
g x ) + (
c *
sigma_fin g FP ));
auto .
apply Ole_trans with (
c * (
g x ) + (
c *
sigma_fin g FP ));
auto .
case (
e x );
intuition .
Save .
Hint Resolve sigma_fin_prod_maj sigma_fin_prod_ge sigma_fin_prod_le .
Lemma sigma_fin_inv :
forall (
f g :
A ->
U )(
P :A->Prop)(FP:finite
P ),
retract_fin P f ->
[1-]
sigma_fin (
fun k =>
f k *
g k )
FP ==
sigma_fin (
fun k =>
f k * [1-]
g k )
FP + [1-]
sigma_fin f FP .
induction FP ;
simpl ;
intros .
repeat Usimpl ;
auto .
assert (
incl Q P ).
apply incl_trans with (
add x Q );
auto .
assert (
retract_fin Q f ).
apply retract_fin_incl with P ;
auto .
assert (
px :P
x ).
case (
e x );
intuition .
apply Uplus_eq_simpl_right with ((
f x ) * (
g x )).
repeat Usimpl ;
auto .
apply Uinv_le_perm_right .
rewrite (
Udistr_inv_left (
f x ) (
g x )).
repeat norm_assoc_right ;
apply Uplus_le_compat_right .
apply Ole_trans with
(
sigma_fin f FP + [1-] (
f x +
sigma_fin f FP ));
repeat Usimpl .
apply (
sigma_fin_prod_maj f (
fun k => [1-](
g k ))
FP ).
assert (
sigma_fin f FP <= [1-] (
f x )).
apply Uinv_le_perm_right ;
auto .
rewrite <- (
Uinv_plus_right _ _ H2 );
auto .
assert (
sigma_fin (
fun k =>
f k *
g k )
FP <= [1-] (
f x *
g x )).
apply Ole_trans with (
sigma_fin f FP );
auto .
apply Ole_trans with ([1-] (
f x ));
auto .
apply Uinv_le_perm_right ;
auto .
rewrite (
Uinv_plus_left _ _ H2 ).
apply Oeq_trans with (1:=IHFP
H1 ).
rewrite (
Uplus_sym (
f x * [1-] (
g x ))
(
sigma_fin (
fun k =>
f k * [1-] (
g k ))
FP )).
repeat norm_assoc_right ;
apply Uplus_eq_compat_right .
setoid_rewrite (
Uplus_sym ([1-] (
f x +
sigma_fin f FP )) (
f x *
g x )).
repeat norm_assoc_left .
assert ([1-] (
g x ) <= [1-] (
g x ));
auto .
rewrite <- (
Udistr_plus_left (
f x )
_ _ H3 ).
rewrite (
Uinv_opp_left (
g x )).
rewrite (
Umult_one_right (
f x ));
auto .
rewrite (
Uplus_sym (
f x ) ([1-] (
f x +
sigma_fin f FP ))).
apply Oeq_sym ;
apply Uinv_plus_left ;
auto .
apply Uinv_le_perm_right ;
auto .
Save .
Lemma sigma_fin_eqset :
forall f P Q (
FP :finite
P ) (
e :eqset
P Q ),
(
sigma_fin f (
fin_eqset e FP )) = (
sigma_fin f FP ).
induction FP ;
simpl ;
intros ;
auto .
Save .
Lemma sigma_fin_rem :
forall f P (
FP :finite
P )
a ,
P a ->
sigma_fin f FP ==
f a +
sigma_fin f (
finite_rem decA a FP ).
induction FP ;
intuition .
case (
eqset_empty_false a e );
auto .
simpl .
case (
decA x a );
intros ;
simpl .
case e0 ;
unfold eq_rect_r ;
simpl ;
auto .
rewrite sigma_fin_eqset ;
auto .
rewrite (
IHFP a );
auto .
case (
e a );
unfold add ;
intuition .
case f0 ;
auto .
Save .
Lemma sigma_fin_incl :
forall f P (
FP :finite
P )
Q (
FQ :finite
Q ),
(
incl P Q ) ->
sigma_fin f FP <=
sigma_fin f FQ .
induction FP ;
simpl ;
intros ;
auto .
destruct FQ ;
simpl ;
intros .
case incl_add_empty with (
a :=x) (
P :=Q).
apply incl_trans with Q0 ;
auto .
apply incl_trans with P ;
auto .
case (
decA x x0 );
intro .
subst ;
Usimpl ;
auto .
apply IHFP .
apply incl_trans with (
rem x0 P );
auto .
apply incl_add_rem ;
auto .
apply incl_trans with (
rem x0 Q0 );
auto .
rewrite incl_rem_add_iff ;
auto .
rewrite (
sigma_fin_rem f FQ x ).
repeat norm_assoc_left .
rewrite (
Uplus_sym (
f x0 ) (
f x )).
repeat norm_assoc_right .
Usimpl .
assert (
H3 :~(rem
x Q1 x0 )).
unfold rem ;
intuition .
assert (
incl Q (
add x0 (
rem x Q1 ))).
red ;
intros ;
case (
e0 x1 );
clear e0 ;
intuition .
case (
e x1 );
clear e ;
intuition .
generalize (
H x1 );
clear H ;
intuition .
unfold add ,rem
in *;
intuition .
subst ;
intuition .
case (
decA x1 x0 );
intuition ;
subst ;
intuition .
case (
decA x1 x );
intuition ;
subst ;
intuition .
exact (
IHFP (
add x0 (
rem x Q1 )) (
fin_eq_add H3 (
finite_rem decA x FQ ) (
eqset_refl (
add x0 (
rem x Q1 ))))
H0 ).
assert (
P x ).
red in e ;
rewrite e ;
auto .
assert (
Q0 x );
auto .
assert (
Q1 x );
auto .
case (
e0 x );
intuition .
case H4 ;
intuition .
Save .
Lemma sigma_fin_unique :
forall f P Q (
FP :finite
P ) (
FQ :finite
Q ), (
eqset P Q ) ->
sigma_fin f FP ==
sigma_fin f FQ .
intros ;
apply Ole_antisym .
apply sigma_fin_incl ;
auto .
apply sigma_fin_incl ;
auto .
Save .
Lemma sigma_fin_cte :
forall c P (
FP :finite
P ),
sigma_fin (
fun _ =>
c )
FP == (
size FP ) */
c .
induction FP ;
auto .
simpl sigma_fin ;
simpl size ;
rewrite IHFP ;
auto .
Save .
End RandomFinite .
End SigmaFinite .
Properties of the Random distribution
Definition le_dec (
n :nat) :
dec (
fun x => (
x <=
n )%nat).
red ;
intros ;
case (
le_lt_dec x n );
intuition .
Defined .
Definition lt_dec (
n :nat) :
dec (
fun x => (
x <
n )%nat).
red ;
intros ;
case (
le_lt_dec n x );
intuition .
Defined .
Definition gt_dec :
forall x ,
dec (
lt x ).
intros x y ;
case (
le_lt_dec y x );
auto with arith .
Defined .
Definition ge_dec :
forall x ,
dec (
le x ).
intros x y ;
case (
le_lt_dec x y );
auto with arith .
Defined .
Definition carac_eq n :=
carac (
eq_nat_dec n ).
Definition carac_le n :=
carac (
le_dec n ).
Definition carac_lt n :=
carac (
lt_dec n ).
Definition carac_gt n :=
carac (
gt_dec n ).
Definition carac_ge n :=
carac (
ge_dec n ).
Definition is_eq (
n :nat) :
cover (
fun x =>
n =
x ) (
carac_eq n ) :=
cover_dec (
eq_nat_dec n ).
Definition is_le (
n :nat) :
cover (
fun x => (
x <=n)%nat) (
carac_le n ) :=
cover_dec (
le_dec n ).
Definition is_lt (
n :nat) :
cover (
fun x => (
x <
n )%nat) (
carac_lt n ) :=
cover_dec (
lt_dec n ).
Definition is_gt (
n :nat) :
cover (
fun x => (
n <
x )%nat) (
carac_gt n ):=
cover_dec (
gt_dec n ).
Definition is_ge (
n :nat) :
cover (
fun x => (
n <=
x )%nat) (
carac_ge n ) :=
cover_dec (
ge_dec n ).
Lemma carac_gt_S :
forall x y ,
carac_gt (
S y ) (
S x ) ==
carac_gt y x .
Proof .
intros ;
unfold carac_gt ;
apply carac_eq_compat ;
omega .
Qed .
Lemma carac_lt_S :
forall x y ,
carac_lt (
S x ) (
S y ) ==
carac_lt x y .
Proof .
intros ;
unfold carac_lt ;
apply carac_eq_compat ;
omega .
Qed .
Lemma carac_le_S :
forall x y ,
carac_le (
S x ) (
S y ) ==
carac_le x y .
Proof .
intros ;
unfold carac_le ;
apply carac_eq_compat ;
omega .
Qed .
Lemma carac_ge_S :
forall x y ,
carac_ge (
S x ) (
S y ) ==
carac_ge x y .
Proof .
intros ;
unfold carac_ge ;
apply carac_eq_compat ;
omega .
Qed .
Lemma carac_eq_S :
forall x y ,
carac_eq (
S x ) (
S y ) ==
carac_eq x y .
Proof .
intros ;
unfold carac_eq ;
apply carac_eq_compat ;
omega .
Qed .
Hint Resolve carac_le_S carac_eq_S carac_lt_S carac_ge_S carac_gt_S .
count the number of elements between 0 and n-1 which satisfy P
Fixpoint nb_elts (
P :nat ->
Prop )(
Pdec :
dec P )(
n :nat) {
struct n } :
nat :=
match n with
0 => 0%nat
|
S n =>
if Pdec n then (
S (
nb_elts Pdec n ))
else (
nb_elts Pdec n )
end .
Lemma nb_elts_true :
forall (
P :nat ->
Prop )(
Pdec :
dec P )(
n :nat),
(
forall k , (
k <
n )%nat ->
P k ) ->
nb_elts Pdec n =n.
induction n ;
simpl ;
intros ;
auto .
case (
Pdec n );
intros ;
auto with arith .
absurd (
P n );
auto with arith .
Save .
Hint Resolve nb_elts_true .
the probability for a random number between 0 and n to satisfy P is equal
to the number of elements below n which satisfy P divided by n+1
Lemma Random_carac : forall (P :nat -> Prop )(Pdec : dec P )(n :nat),mu (Random n ) (carac Pdec ) == (nb_elts Pdec (S n )) */ [1/]1+n.simpl mu .unfold random ,fnth; intros .rewrite mon_simpl .elim (S n ); simpl ; intros ; auto .rewrite H ; unfold carac ;case (Pdec n0 ); Usimpl ; auto .Qed .Lemma nb_elts_lt_le : forall k n , (k <= n )%nat -> nb_elts (lt_dec k ) n = k .intros k n H ; induction H ; intros ; auto with arith .simpl .case (lt_dec k m ); intros ; auto with arith .absurd ((m <k)%nat); auto with arith .Save .Lemma nb_elts_lt_ge : forall k n , (n <= k )%nat -> nb_elts (lt_dec k ) n = n .intros ; auto with zarith .Save .Hint Resolve nb_elts_lt_ge nb_elts_lt_le .Lemma Random_lt : forall n k , mu (Random n ) (carac_lt k ) == k */ [1/]1+n.unfold carac_lt ; intros ; rewrite Random_carac .case (le_ge_dec k (S n )); intros .rewrite nb_elts_lt_le ; auto .rewrite nb_elts_lt_ge ; auto .apply Ole_antisym ; auto .apply Ole_trans with 1; auto .Save .Hint Resolve Random_lt .Lemma Random_le : forall n k , mu (Random n ) (carac_le k ) == (S k ) */ [1/]1+n.intros ; apply Oeq_trans with (mu (Random n ) (carac_lt (S k ))); auto .apply (mu_stable_eq (Random n )); auto .unfold carac_le ,carac_lt; intro x .apply carac_eq_compat ; intuition .Save .Hint Resolve Random_le .
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Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
F (definition)
fcomp [in
Ccpo ]
fcomp2 [in
Ccpo ]
Fcontm [in
Ccpo ]
fcont0 [in
Ccpo ]
fcont2_comp [in
Ccpo ]
fcont2_COMP [in
Ccpo ]
fcont_app [in
Ccpo ]
fcont_COMP [in
Ccpo ]
fcont_comp [in
Ccpo ]
fcont_Comp [in
Ccpo ]
fcont_fun [in
Ccpo ]
fcont_ishift [in
Ccpo ]
fcont_lub [in
Ccpo ]
fcont_mshift [in
Ccpo ]
fcont_SEQ [in
Ccpo ]
feq [in
Misc ]
fixp [in
Ccpo ]
Fixp [in
Ccpo ]
fixp_cte [in
Ccpo ]
fnatO_intro [in
Ccpo ]
full [in
Sets ]
fun2 [in
Ccpo ]
fun_ext [in
Ccpo ]
html/index_module_H.html 0000644 0001750 0001750 00000022476 11301735615 015520 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_projection_P.html 0000644 0001750 0001750 00000024255 11301735615 016414 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
P (projection)
Proba.coeff [in
Probas ]
Proba.coeff_retr [in
Probas ]
Proba.mu [in
Probas ]
Proba.mu_continuous [in
Probas ]
Proba.mu_stable_inv [in
Probas ]
Proba.mu_stable_mult [in
Probas ]
Proba.mu_stable_plus [in
Probas ]
Proba.points [in
Probas ]
html/index_library_M.html 0000644 0001750 0001750 00000022677 11301735615 015707 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
M (library)
Misc
Monads
html/index_variable_T.html 0000644 0001750 0001750 00000022476 11301735615 016034 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_inductive__.html 0000644 0001750 0001750 00000022476 11301735615 016254 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_variable_E.html 0000644 0001750 0001750 00000022476 11301735615 016015 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_axiom_E.html 0000644 0001750 0001750 00000022476 11301735615 015345 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_record_Z.html 0000644 0001750 0001750 00000022476 11301735615 015533 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_library_U.html 0000644 0001750 0001750 00000022703 11301735615 015705 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
U (library)
Uprop
Utheory
html/index_instance_A.html 0000644 0001750 0001750 00000022476 11301735615 016030 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_lemma_H.html 0000644 0001750 0001750 00000022476 11301735615 015326 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_record_O.html 0000644 0001750 0001750 00000023005 11301735615 015505 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
O (record)
ord [in
Ccpo ]
Order [in
Ccpo ]
html/index_record_D.html 0000644 0001750 0001750 00000022476 11301735615 015505 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_inductive_F.html 0000644 0001750 0001750 00000022706 11301735615 016217 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
F (inductive)
finite [in
Sets ]
html/index_lemma_F.html 0000644 0001750 0001750 00000055673 11301735615 015331 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
F (lemma)
fcomp2_simpl [in
Ccpo ]
fcomp_simpl [in
Ccpo ]
fcontm_fcont_comp_simpl [in
Ccpo ]
fcont2_comp_simpl [in
Ccpo ]
fcont_app_continuous [in
Ccpo ]
fcont_app_simpl [in
Ccpo ]
fcont_comp_le_compat [in
Ccpo ]
fcont_COMP_simpl [in
Ccpo ]
fcont_Comp_simpl [in
Ccpo ]
fcont_comp_simpl [in
Ccpo ]
fcont_continuous [in
Ccpo ]
fcont_eq [in
Ccpo ]
fcont_eq_compat2 [in
Ccpo ]
fcont_eq_elim [in
Ccpo ]
fcont_eq_intro [in
Ccpo ]
fcont_le [in
Ccpo ]
fcont_le_compat2 [in
Ccpo ]
fcont_le_elim [in
Ccpo ]
fcont_le_intro [in
Ccpo ]
fcont_lub_simpl [in
Ccpo ]
fcont_SEQ_simpl [in
Ccpo ]
fcpo_lub_simpl [in
Ccpo ]
feq_refl [in
Misc ]
feq_sym [in
Misc ]
feq_trans [in
Misc ]
finite_dec [in
Sets ]
finite_full_dec [in
Sets ]
finite_incl [in
Sets ]
finite_inter [in
Sets ]
finite_rem [in
Sets ]
finite_union [in
Sets ]
fin_add [in
Sets ]
fin_add_in [in
Sets ]
fin_empty [in
Sets ]
fin_eqset [in
Sets ]
fixp_continuous [in
Ccpo ]
fixp_continuous_eq [in
Ccpo ]
fixp_eq [in
Ccpo ]
fixp_ind [in
Ccpo ]
fixp_inv [in
Ccpo ]
fixp_le [in
Ccpo ]
fixp_le_compat [in
Ccpo ]
Fixp_simpl [in
Ccpo ]
fmon_cte_comp [in
Ccpo ]
fmon_diag_simpl [in
Ccpo ]
fmon_eq [in
Ccpo ]
fmon_le [in
Ccpo ]
fmon_le_compat2 [in
Ccpo ]
fmon_lub_simpl [in
Ccpo ]
fnatO_elim [in
Ccpo ]
ford_eq_elim [in
Ccpo ]
ford_eq_intro [in
Ccpo ]
ford_le_elim [in
Ccpo ]
ford_le_intro [in
Ccpo ]
fun2_mon2 [in
Ccpo ]
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E (lemma)
eqset_add_empty [in
Sets ]
eqset_add_rem [in
Sets ]
eqset_empty_false [in
Sets ]
eqset_incl [in
Sets ]
eqset_incl_intro [in
Sets ]
eqset_incl_sym [in
Sets ]
eqset_refl [in
Sets ]
eqset_rem_add [in
Sets ]
eqset_sym [in
Sets ]
eqset_trans [in
Sets ]
eqset_union_left [in
Sets ]
eqset_union_right [in
Sets ]
eq_ord_equiv [in
Ccpo ]
excluded_middle [in
Misc ]
exc_intro [in
Misc ]
exc_intro_class [in
Misc ]
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html/Monads.html 0000644 0001750 0001750 00000137236 11301735614 014016 0 ustar cpaulin cpaulin
Monads
Library Monads
Monads.v: Monads for randomized constructions
Set Implicit Arguments .
Require Export Uprop .
Definition of monadic operators as the cpo of monotonic oerators
Definition M (A :Type) := MF A -m> U .Instance app_mon (A :Type) (x :A) : monotonic (fun (f :MF A ) => f x ).red ; auto .Save .Definition unit (A :Type) (x :A) : M A := mon (fun (f :MF A ) => f x ).Definition star : forall (A B :Type), M A -> (A -> M B ) -> M B .intros A B a F ; exists (fun f => a (fun x => F x f )).red ; intros .apply (fmonotonic a ); intro z ; simpl ; auto .Defined .Lemma star_simpl : forall (A B :Type) (a :M A ) (F :A -> M B )(f :MF B ),star a F f = a (fun x => F x f ).trivial .Save .
Properties of monadic operators
Lemma law1 :
forall (
A B :Type) (
x :A) (
F :A ->
M B ) (
f :MF
B ),
star (
unit x )
F f ==
F x f .
trivial .
Qed .
Lemma law2 :
forall (
A :Type) (
a :M
A ) (
f :MF
A ),
star a (
fun x :A =>
unit x )
f ==
a (
fun x :A =>
f x ).
trivial .
Qed .
Lemma law3 :
forall (
A B C :Type) (
a :M
A ) (
F :A ->
M B ) (
G :B ->
M C )
(
f :MF
C ),
star (
star a F )
G f ==
star a (
fun x :A =>
star (
F x )
G )
f .
trivial .
Qed .
Properties of distributions
Expected properties of measures
Definition stable_inv (
A :Type) (
m :M
A ) :
Prop :=
forall f :MF
A ,
m (
finv f ) <= [1-] (
m f ).
Definition stable_plus (
A :Type) (
m :M
A ) :
Prop :=
forall f g :MF
A ,
fplusok f g ->
m (
fplus f g ) == (
m f ) + (
m g ).
Definition le_plus (
A :Type) (
m :M
A ) :
Prop :=
forall f g :MF
A ,
fplusok f g -> (
m f ) + (
m g ) <=
m (
fplus f g ).
Definition le_esp (
A :Type) (
m :M
A ) :
Prop :=
forall f g :
MF A , (
m f ) & (
m g ) <=
m (
fesp f g ).
Definition le_plus_cte (
A :Type) (
m :M
A ) :
Prop :=
forall (
f :
MF A ) (
k :U),
m (
fplus f (
fcte A k )) <=
m f +
k .
Definition stable_mult (
A :Type) (
m :M
A ) :
Prop :=
forall (
k :U) (
f :MF
A ),
m (
fmult k f ) ==
k * (
m f ).
Lemma stable_minus_distr :
forall (
A :Type) (
m :M
A ),
stable_plus m ->
stable_inv m ->
forall (
f g :
MF A ),
g <=
f ->
m (
fminus f g ) ==
m f -
m g .
intros A m splus sinv ;
intros .
assert (
m g <=
m f );
auto .
assert (
fminus f g <=
finv g ).
intros ;
intro x ;
unfold fminus ,Uminus,fplus,
finv ;
auto .
rewrite (
fstable m f (
fplus (
fminus f g )
g )).
rewrite (
splus (
fminus f g )
g );
auto .
symmetry ;
apply Uplus_minus_simpl_right ;
auto .
transitivity ([1-]
m (
finv g ));
auto .
unfold fplus ,fminus;
intro x ;
auto .
Save .
Hint Resolve stable_minus_distr .
Lemma inv_minus_distr :
forall (
A :Type) (
m :M
A ),
stable_plus m ->
stable_inv m ->
forall (
f :
MF A ),
m (
finv f ) ==
m (
fone A ) -
m f .
intros A m splus sinv ;
intros .
transitivity (
m (
fminus (
fone A )
f ));
auto .
apply (
fstable m ).
intro x ;
unfold fminus ,finv,fone,fplus;
intros ;
auto .
Save .
Hint Resolve inv_minus_distr .
Lemma le_minus_distr :
forall (
A :
Type )(
m :M
A ),
forall (
f g :A ->
U ),
m (
fminus f g ) <=
m f .
intros A m ;
intros .
apply (
fmonotonic m );
intro x ;
rewrite fminus_simpl ;
auto .
Save .
Hint Resolve le_minus_distr .
Lemma le_plus_distr :
forall (
A :
Type )(
m :M
A ),
stable_plus m ->
stable_inv m ->
forall (
f g :MF
A ),
m (
fplus f g ) <=
m f +
m g .
intros A m splus sinv ;
intros .
transitivity (
m (
fplus (
fminus f (
fesp f g ))
g )).
apply (
fmonotonic m );
intro x ;
unfold fminus ,fesp,fplus,finv;
intros ;
auto .
rewrite (
splus (
fminus f (
fesp f g ))
g ).
auto .
intro x ;
unfold fminus ,fesp,finv;
auto .
Save .
Hint Resolve le_plus_distr .
Lemma le_esp_distr :
forall (
A :
Type ) (
m :M
A ),
stable_plus m ->
stable_inv m ->
le_esp m .
intros A m splus sinv ;
unfold le_esp ,fesp,Uesp;
intros .
transitivity (
m (
finv (
fplus (
finv f ) (
finv g ))));
auto .
rewrite inv_minus_distr ;
auto .
transitivity
(
m (
fone A ) - (
m (
finv f ) +
m (
finv g )));
auto .
transitivity (
m f - [1-]
m g ) ;
repeat Usimpl ;
auto .
rewrite inv_minus_distr ;
auto .
transitivity (
m f -
m (
finv g )) ;
repeat Usimpl .
apply Uminus_le_compat_right ;
trivial .
rewrite <-
Uminus_assoc_left .
rewrite Uminus_assoc_right ;
repeat Usimpl ;
auto .
Save .
Lemma unit_stable_eq :
forall (
A :Type) (
x :A),
stable (
unit x ).
auto .
Qed .
Lemma star_stable_eq :
forall (
A B :Type) (
m :M
A ) (
F :A ->
M B ),
stable (
star m F ).
auto .
Qed .
Lemma unit_stable_inv :
forall (
A :Type) (
x :A),
stable_inv (
unit x ).
red in |- *;
intros .
unfold unit in |- *.
auto .
Qed .
Lemma star_stable_inv :
forall (
A B :Type) (
m :M
A ) (
F :A ->
M B ),
stable_inv m -> (
forall a :A,
stable_inv (
F a )) ->
stable_inv (
star m F ).
unfold stable_inv in |- *;
intros .
unfold star in |- *;
simpl .
transitivity (
m (
finv (
fun x :A =>
F x f )));
trivial .
apply (
fmonotonic m );
intro x .
apply (
H0 x f ).
Save .
Lemma unit_stable_plus :
forall (
A :Type) (
x :A),
stable_plus (
unit x ).
red in |- *;
intros .
unfold unit in |- *;
auto .
Qed .
Lemma star_stable_plus :
forall (
A B :Type) (
m :M
A ) (
F :A ->
M B ),
stable_plus m ->
(
forall a :A,
forall f g ,
fplusok f g -> (
F a f ) <=
Uinv (
F a g ))
-> (
forall a :A,
stable_plus (
F a )) ->
stable_plus (
star m F ).
unfold stable_plus in |- *;
intros .
unfold star in |- *;
simpl .
transitivity (
m (
fplus (
fun x :A =>
F x f ) (
fun x :A =>
F x g )));
trivial .
apply (
fstable m ).
intro x ;
apply (
H1 x f g H2 ).
apply H .
intro x ;
unfold finv ;
intros ;
auto .
Qed .
Lemma unit_le_plus :
forall (
A :Type) (
x :A),
le_plus (
unit x ).
red in |- *;
intros .
unfold unit in |- *.
auto .
Qed .
Lemma star_le_plus :
forall (
A B :Type) (
m :M
A ) (
F :A ->
M B ),
le_plus m ->
(
forall a :A,
forall f g ,
fplusok f g -> (
F a f ) <=
Uinv (
F a g ))
-> (
forall a :A,
le_plus (
F a )) ->
le_plus (
star m F ).
unfold le_plus in |- *;
intros .
unfold star in |- *;
simpl .
transitivity (
m (
fplus (
fun x :A =>
F x f ) (
fun x :A =>
F x g )));
trivial .
apply H .
red ;
intro x ;
unfold finv ;
intros ;
auto .
apply (
fmonotonic m ).
intro x .
rewrite fplus_simpl ;
auto .
Qed .
Lemma unit_stable_mult : forall (A :Type) (x :A), stable_mult (unit x ).red in |- *; intros .unfold unit in |- *; auto .Qed .Lemma star_stable_mult : forall (A B :Type) (m :M A ) (F :A -> M B ),stable_mult m -> (forall a :A, stable_mult (F a )) -> stable_mult (star m F ).unfold stable_mult in |- *; intros .unfold star in |- *; simpl .transitivity (m (fmult k (fun x :A => F x f ))); trivial .apply (fstable m ).intro x ; rewrite fmult_simpl ; auto .Qed .
Lemma unit_continuous :
forall (
A :Type) (
x :A),
continuous (
unit x ).
red ;
unfold unit ;
intros .
transitivity (
lub h x );
auto .
transitivity (
lub (
h <o>
x ));
auto .
apply lub_le_compat ;
intro n ;
auto .
Save .
Lemma star_continuous :
forall (
A B :
Type ) (
m :
M A )(
F :
A ->
M B ),
continuous m -> (
forall x ,
continuous (
F x )) ->
continuous (
star m F ).
red ;
intros ;
simpl .
transitivity (
m (
fun x =>
lub (
F x @
h ))).
apply (
fmonotonic m );
intro x .
apply (
H0 x h );
auto .
transitivity (
m (
lub (
ishift F @
h ))).
apply (
fmonotonic m );
intro x .
unfold MFcpo ;
rewrite (
fcpo_lub_simpl (
ishift F @
h )
x );
auto .
apply Ole_trans with (1:=H (
ishift F @
h ));
auto .
apply lub_le_compat ;
intro x ;
simpl ;
auto .
Save .
html/Ycart.html 0000644 0001750 0001750 00000143632 11301735615 013655 0 ustar cpaulin cpaulin
Ycart
Library Ycart
Require Export Cover .
Set Implicit Arguments .
Require Arith .
Ycart.v: An exemple of partial termination
Section YcartExample .
Program giving an example of partiality
Given a function K : nat ->nat, N:nat
let rec ycart p x = if random p < K x then x else ycart (x+1)
The probability of termination is
Variable K :
nat ->
nat .
Variable N :
nat .
Instance FYcart_mon :
monotonic
(
fun (
f :nat ->
distr nat )
n =>
Mlet (
Random N ) (
fun x =>
if lt_dec (
K n )
x then Munit n else f (
S n ))).
intros f g H n .
apply Mlet_le_compat ;
auto .
intro x ;
case (
lt_dec (
K n )
x );
auto .
Save .
Definition FYcart : (
nat ->
distr nat ) -m> (
nat ->
distr nat )
:=
mon (
fun (
f :nat ->
distr nat )
n =>
Mlet (
Random N ) (
fun x =>
if lt_dec (
K n )
x then Munit n else f (
S n ))).
Lemma FYcart_simpl :
forall f n ,
FYcart f n =
Mlet (
Random N ) (
fun x =>
if lt_dec (
K n )
x then Munit n else f (
S n )).
trivial .
Save .
Definition Ycart :
nat ->
distr nat :=
Mfix FYcart .
Definition F x := (
K x ) */ [1/]1+N.
Lemma FYcart_val :
forall (
q :
MF nat )
f x ,
mu (
FYcart f x )
q ==
F x *
q x + [1-](
F x ) *
mu (
f (
S x ))
q .
intros ;
unfold F ;
rewrite FYcart_simpl .
rewrite Mlet_simpl .
apply Oeq_trans with
(
mu (
Random N ) (
fplus (
fmult (
q x ) (
carac_lt (
K x )))
(
fmult (
mu (
f (
S x ))
q ) (
finv (
carac_lt (
K x )))))).
apply mu_stable_eq .
simpl ;
apply ford_eq_intro ;
intro x0 .
unfold fplus ,fmult,finv,is_lt,
carac_lt ,
carac .
case (
lt_dec (
K x )
x0 );
simpl ;
intros ;
repeat Usimpl ;
auto .
assert (
fok :fplusok (
fmult (
q x ) (
carac_lt (
K x ))) (
fmult (
mu (
f (
S x ))
q ) (
finv (
carac_lt (
K x ))))).
intro y ;
unfold fmult ,finv,is_lt,
carac_lt ,
carac .
case (
lt_dec (
K x )
y );
repeat Usimpl ;
auto .
rewrite (
mu_stable_plus (
Random N )
fok ).
apply Oeq_trans with
(
q x *
mu (
Random N ) (
carac_lt (
K x )) +
mu (
f (
S x ))
q *
mu (
Random N ) (
finv (
carac_lt (
K x )))).
apply Uplus_eq_compat ;
apply (
mu_stable_mult (
Random N )).
rewrite (
Random_lt N (
K x )).
apply Uplus_eq_compat ;
auto .
rewrite Random_inv ;
rewrite Random_lt ;
auto .
Save .
Definition P (
x :
nat ) :
nat -m->
U :=
Prod (
fun i => [1-]
F (
x +i)).
Definition p (
x :nat) :
nat -m>
U :=
sigma (
fun k =>
F (
x +k) *
P x k ).
Lemma P_prod :
forall x k ,
F (
x +k) *
P x k ==
P x k -
P x (
S k ).
intros ;
unfold P .
repeat (
rewrite Prod_simpl );
rewrite prod_minus ;
auto .
Save .
Hint Resolve P_prod .
Lemma p_diff :
forall x n , (
p x n :
U ) == [1-]
P x n .
unfold p ;
intros .
apply Oeq_trans with (
sigma (
fun k =>
P x k -
P x (
S k ))
n ).
apply sigma_eq_compat ;
intros ;
auto .
rewrite sigma_minus_decr ;
unfold P ;
auto .
Save .
Hint Resolve p_diff .
Lemma p_lub :
forall x ,
lub (
p x ) == [1-]
prod_inf (
fun i => [1-]
F (
x +i)).
unfold prod_inf ;
intros .
apply Oeq_trans with (
lub (
UInv @ (
P x )) :
U ).
apply lub_eq_compat .
intro n .
apply Oeq_trans with (1:=p_diff
x n );
auto .
apply Oeq_trans with ([1-]
glb (
P x ));
auto .
Save .
Hint Resolve p_lub .
Lemma p_equation :
forall x n ,
p x (
S n ) ==
F x + [1-](
F x ) *
p (
S x )
n .
unfold p ;
intros .
rewrite sigma_S_lift .
unfold P at 1;
rewrite Prod_simpl ;
rewrite prod_0 ;
Usimpl .
apply Uplus_eq_compat .
replace ((
x +0)%nat)
with x ;
auto .
rewrite <-
sigma_mult .
apply sigma_eq_compat ;
intros ;
auto .
unfold P ;
rewrite Prod_simpl ;
rewrite prod_S_lift .
rewrite Umult_assoc .
rewrite (
Umult_sym (
F (
x +S
k )) ([1-]
F (
x +0))).
rewrite <-
Umult_assoc .
apply Umult_eq_compat .
replace ((
x +0)%nat)
with x ;
auto .
apply Umult_eq_compat .
replace ((
x +
S k )%nat)
with ((
S x +
k )%nat);
auto .
omega .
rewrite Prod_simpl ;
apply prod_eq_compat ;
intro ;
auto .
replace ((
x +
S k0 )%nat)
with ((
S x +
k0 )%nat);
auto .
omega .
red ;
intros .
apply Ole_trans with (
P (
S x )
k );
auto .
change ((
P (
S x )
k :
U ) <= [1-]
p (
S x )
k ).
rewrite p_diff ;
auto .
Save .
Hint Resolve p_equation .
Lemma Ycart_term1 :
forall x ,
mu (
Ycart x ) (
fone nat ) == [1-]
prod_inf (
fun i => [1-]
F (
x +i)).
intros ;
apply Oeq_trans with (
lub (
p x ));
auto .
apply Ole_antisym .
apply (
fixrule_up_lub FYcart )
with (
p :=p) (
q :=fun (
x :nat) =>
fone nat );
red ;
intros .
red ;
rewrite FYcart_val .
unfold fone ;
Usimpl .
apply Ole_trans with (
F x0 + [1-]
F x0 *
p (
S x0 )
i ).
repeat Usimpl ;
auto .
apply (
H (
S x0 )).
rewrite p_equation ;
auto .
apply (
fixrule FYcart )
with (
p :=p) (
q :=fun (
x :nat) =>
fone nat );
intros .
unfold p ;
rewrite sigma_0 ;
auto .
intro y ;
red .
rewrite FYcart_val .
unfold fone ;
Usimpl .
apply Ole_trans with (
F y + [1-]
F y *
p (
S y )
i ).
rewrite p_equation ;
auto .
repeat Usimpl ;
auto .
apply (
H (
S y )).
Save .
A shorter proof using mu (Ycart x) (f_one nat) = mu h. muYcart h x
Instance Ycart_one_mon
:
monotonic (
fun (
p :nat ->
U ) (
y :nat) =>
F y + [1-](
F y ) *
p (
S y )).
intros f g H n ;
repeat Usimpl ;
auto .
Save .
Definition Ycart_one : (
nat ->
U ) -m> (
nat ->
U ) :=
mon (
fun (
p :nat->U) (
y :nat) =>
F y + [1-](
F y ) *
p (
S y )).
Lemma Ycart_one_simpl :
forall p y ,
Ycart_one p y =
F y + [1-](
F y ) *
p (
S y ).
trivial .
Save .
Lemma Ycart_term2 :
forall x ,
mu (
Ycart x ) (
fone nat ) == [1-]
prod_inf (
fun i => [1-]
F (
x +i)).
intros ;
apply Oeq_trans with (
mufix Ycart_one x ).
unfold Ycart ;
apply Oeq_sym .
apply mufix_mu with (
q :=fun (
y :nat) => (
fone nat :nat->U));
intros .
rewrite FYcart_val ;
unfold fone ;
repeat Usimpl ;
auto .
apply Oeq_trans with (
lub (
p x ));
auto .
unfold mufix ,fixp;
simpl .
apply lub_eq_compat .
intro n ;
generalize x ;
induction n ;
intros ;
auto .
change (
Ccpo.iter (
D :=MF
nat )
Ycart_one (
S n )
x0 ==
p x0 (
S n )).
rewrite iterS_simpl .
assert (
Ccpo.iter (
D :=MF
nat )
Ycart_one n ==
fun x =>
p x n ).
change (
forall x :
nat ,
Ccpo.iter (
D :=MF
nat )
Ycart_one n x ==
p x n )
in IHn .
simpl ;
apply ford_eq_intro ;
auto .
apply Oeq_trans with (
Ycart_one (
fun x :
nat =>
p x n )
x0 );
auto .
apply ford_eq_elim with (
n :=x0).
apply (
monotonic_stable Ycart_one );
auto .
rewrite Ycart_one_simpl ;
auto .
Save .
Instance FYcart_lt_mon :
monotonic (
fun (
p :
nat ->
U ) (
y :nat) => [1-](
F y ) *
p (
S y )).
red ;
intros ;
auto .
Save .
Definition FYcart_lt : (
nat ->
U ) -m> (
nat ->
U ) :=
mon (
fun (
p :nat->U) (
y :nat) => [1-](
F y ) *
p (
S y )).
Lemma FYcart_lt_simpl :
forall (
p :nat ->
U ) (
y :nat),
FYcart_lt p y = [1-](
F y ) *
p (
S y ).
trivial .
Save .
Lemma Ycart_ltx :
forall x ,
mu (
Ycart x ) (
carac_lt x ) <= 0.
intros ;
apply Ole_trans with (
mufix FYcart_lt x ).
unfold Ycart ;
apply mu_mufix_le with (
A :=nat) (
B :=nat) (
q :=carac_lt);
intros ;
auto .
red ;
intros ;
rewrite FYcart_lt_simpl .
rewrite FYcart_val ;
unfold carac_lt .
rewrite carac_zero ;
auto with arith .
repeat Usimpl .
apply mu_monotonic ;
intro ;
apply carac_incl ;
red ;
auto with arith .
apply mufix_inv with (
f :=fun
y :nat => (0:U));
intro n ;
simpl ;
auto .
Save .
Lemma Ycart_eqx :
forall x ,
mu (
Ycart x ) (
carac (
eq_nat_dec x )) ==
F x .
intros ;
apply Oeq_trans with (
mufix (
mon (
cte (
nat ->U)
F ))
x ).
unfold Ycart ;
apply Oeq_sym .
apply mufix_mu with (
A :=nat) (
B :=nat) (
q :=fun (
y :nat) =>
carac (
eq_nat_dec y ));
intros ;
auto .
apply Oeq_trans with (1:=FYcart_val (
carac (
eq_nat_dec x0 ))
f x0 ).
unfold cte ;
simpl .
rewrite (
carac_one (
eq_nat_dec x0 )
x0 );
auto ;
Usimpl .
assert ((
mu (
f (
S x0 )) (
carac (
eq_nat_dec x0 )))==0).
apply Ule_zero_eq ;
apply Ole_trans with (
mu (
Mfix FYcart (
S x0 )) (
carac (
eq_nat_dec x0 ))).
apply H .
apply Ole_trans with (2:=Ycart_ltx (
S x0 )).
apply mu_monotonic ;
intro .
unfold carac_lt ;
apply carac_monotonic ;
intuition .
rewrite H0 ;
repeat Usimpl ;
trivial .
apply ford_eq_elim with (
n :=x);
exact (
mufix_cte F ).
Save .
End YcartExample .
html/index_global_P.html 0000644 0001750 0001750 00000107514 11301735615 015500 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
P
PP [module, in
Prog ]
Proba [module, in
Probas ]
Probas [library]
Proba.cdistr [instance, in
Probas ]
Proba.cfalse [definition, in
Probas ]
Proba.coeff [projection, in
Probas ]
Proba.coeff_retr [projection, in
Probas ]
Proba.Continuity [section, in
Probas ]
Proba.Continuity.A [variable, in
Probas ]
Proba.Continuity.B [variable, in
Probas ]
Proba.ctrue [definition, in
Probas ]
Proba.discr [record, in
Probas ]
Proba.Discrete [definition, in
Probas ]
Proba.discrete [definition, in
Probas ]
Proba.discrete_continuous [lemma, in
Probas ]
Proba.discrete_mon [instance, in
Probas ]
Proba.Discrete_simpl [lemma, in
Probas ]
Proba.discrete_simpl [lemma, in
Probas ]
Proba.discrete_stable_inv [lemma, in
Probas ]
Proba.discrete_stable_mult [lemma, in
Probas ]
Proba.discrete_stable_plus [lemma, in
Probas ]
Proba.distr [record, in
Probas ]
Proba.distr_div [definition, in
Probas ]
Proba.distr_lub_simpl [lemma, in
Probas ]
Proba.distr_mult [definition, in
Probas ]
Proba.distr_null [definition, in
Probas ]
Proba.Flip [definition, in
Probas ]
Proba.flip [definition, in
Probas ]
Proba.flip_cfalse [lemma, in
Probas ]
Proba.flip_continuous [lemma, in
Probas ]
Proba.flip_ctrue [lemma, in
Probas ]
Proba.Flip_simpl [lemma, in
Probas ]
Proba.flip_stable_inv [lemma, in
Probas ]
Proba.flip_stable_mult [lemma, in
Probas ]
Proba.flip_stable_plus [lemma, in
Probas ]
Proba.fnth [definition, in
Probas ]
Proba.fnth_retract [lemma, in
Probas ]
Proba.is_discrete [definition, in
Probas ]
Proba.let_indep [lemma, in
Probas ]
Proba.le_distr_null [lemma, in
Probas ]
Proba.Lubs [section, in
Probas ]
Proba.Lubs.A [variable, in
Probas ]
Proba.Lubs.muf [variable, in
Probas ]
Proba.mcond [definition, in
Probas ]
Proba.Mcond [definition, in
Probas ]
Proba.Mcond_bayes [lemma, in
Probas ]
Proba.Mcond_conj [lemma, in
Probas ]
Proba.mcond_continuous [lemma, in
Probas ]
Proba.Mcond_decomp [lemma, in
Probas ]
Proba.Mcond_null [lemma, in
Probas ]
Proba.mcond_simpl [lemma, in
Probas ]
Proba.Mcond_simpl [lemma, in
Probas ]
Proba.mcond_stable_inv [lemma, in
Probas ]
Proba.mcond_stable_mult [lemma, in
Probas ]
Proba.mcond_stable_plus [lemma, in
Probas ]
Proba.Mcond_total [lemma, in
Probas ]
Proba.Mcond_zero_stable [lemma, in
Probas ]
Proba.Mdiv [definition, in
Probas ]
Proba.Mdiv_continuous [lemma, in
Probas ]
Proba.Mdiv_simpl [lemma, in
Probas ]
Proba.Mdiv_stable_inv [lemma, in
Probas ]
Proba.Mdiv_stable_mult [lemma, in
Probas ]
Proba.Mdiv_stable_plus [lemma, in
Probas ]
Proba.Mfix [definition, in
Probas ]
Proba.MFix [definition, in
Probas ]
Proba.Mfix_continuous [lemma, in
Probas ]
Proba.Mfix_eq [lemma, in
Probas ]
Proba.Mfix_le [lemma, in
Probas ]
Proba.Mfix_le_compat [lemma, in
Probas ]
Proba.Mfix_le_iter [lemma, in
Probas ]
Proba.Miter [definition, in
Probas ]
Proba.MLet [definition, in
Probas ]
Proba.Mlet [definition, in
Probas ]
Proba.Mlet_assoc [lemma, in
Probas ]
Proba.Mlet_continuous2 [lemma, in
Probas ]
Proba.Mlet_continuous_left [lemma, in
Probas ]
Proba.Mlet_continuous_right [instance, in
Probas ]
Proba.Mlet_eq_compat [lemma, in
Probas ]
Proba.Mlet_ext [lemma, in
Probas ]
Proba.Mlet_le_compat [lemma, in
Probas ]
Proba.Mlet_lub_le [lemma, in
Probas ]
Proba.Mlet_mon2 [instance, in
Probas ]
Proba.Mlet_simpl [lemma, in
Probas ]
Proba.MLet_simpl [lemma, in
Probas ]
Proba.Mlet_unit [lemma, in
Probas ]
Proba.Mmult [definition, in
Probas ]
Proba.Mmult_continuous [lemma, in
Probas ]
Proba.Mmult_simpl [lemma, in
Probas ]
Proba.Mmult_stable_inv [lemma, in
Probas ]
Proba.Mmult_stable_mult [lemma, in
Probas ]
Proba.Mmult_stable_plus [lemma, in
Probas ]
Proba.mu [projection, in
Probas ]
Proba.Mu [definition, in
Probas ]
Proba.Munit [definition, in
Probas ]
Proba.Munit_eq_compat [lemma, in
Probas ]
Proba.Munit_simpl [lemma, in
Probas ]
Proba.mu_continuous [projection, in
Probas ]
Proba.mu_cte [lemma, in
Probas ]
Proba.mu_cte_eq [lemma, in
Probas ]
Proba.mu_cte_le [lemma, in
Probas ]
Proba.mu_eq_compat [lemma, in
Probas ]
Proba.mu_esp_one [lemma, in
Probas ]
Proba.mu_esp_zero [lemma, in
Probas ]
Proba.mu_fplusok [lemma, in
Probas ]
Proba.mu_inv_minus [lemma, in
Probas ]
Proba.mu_inv_minus_inv [lemma, in
Probas ]
Proba.mu_le_compat [lemma, in
Probas ]
Proba.mu_le_esp [lemma, in
Probas ]
Proba.mu_le_esp_inv [lemma, in
Probas ]
Proba.mu_le_minus [lemma, in
Probas ]
Proba.mu_le_plus [lemma, in
Probas ]
Proba.mu_lub [definition, in
Probas ]
Proba.mu_lub_le [lemma, in
Probas ]
Proba.mu_lub_sup [lemma, in
Probas ]
Proba.mu_monotonic [lemma, in
Probas ]
Proba.mu_one_inv [lemma, in
Probas ]
Proba.mu_serie_eq [lemma, in
Probas ]
Proba.mu_serie_le [lemma, in
Probas ]
Proba.mu_sigma_eq [lemma, in
Probas ]
Proba.mu_sigma_le [lemma, in
Probas ]
Proba.mu_stable_div [lemma, in
Probas ]
Proba.mu_stable_div_le [lemma, in
Probas ]
Proba.mu_stable_eq [lemma, in
Probas ]
Proba.mu_stable_inv [projection, in
Probas ]
Proba.mu_stable_inv_inv [lemma, in
Probas ]
Proba.mu_stable_le_minus [lemma, in
Probas ]
Proba.mu_stable_minus [lemma, in
Probas ]
Proba.mu_stable_mult [projection, in
Probas ]
Proba.mu_stable_mult_right [lemma, in
Probas ]
Proba.mu_stable_plus [projection, in
Probas ]
Proba.mu_zero [lemma, in
Probas ]
Proba.mu_zero_eq [lemma, in
Probas ]
Proba.M_lub_simpl [lemma, in
Probas ]
Proba.Odistr [instance, in
Probas ]
Proba.points [projection, in
Probas ]
Proba.Random [definition, in
Probas ]
Proba.random [definition, in
Probas ]
Proba.random_continuous [lemma, in
Probas ]
Proba.Random_inv [lemma, in
Probas ]
Proba.random_mon [instance, in
Probas ]
Proba.Random_simpl [lemma, in
Probas ]
Proba.random_simpl [lemma, in
Probas ]
Proba.random_stable_inv [lemma, in
Probas ]
Proba.random_stable_mult [lemma, in
Probas ]
Proba.random_stable_plus [lemma, in
Probas ]
Proba.Random_total [lemma, in
Probas ]
Proba.retract_fplusok [lemma, in
Probas ]
Proba.serie_fun [definition, in
Probas ]
Proba.serie_fun_lub_sigma_fun [lemma, in
Probas ]
Proba.sigma_fnth_le [lemma, in
Probas ]
Proba.sigma_fnth_one [lemma, in
Probas ]
Proba.sigma_fnth_sup [lemma, in
Probas ]
Proba.Sigma_fun [definition, in
Probas ]
Proba.sigma_fun [definition, in
Probas ]
Proba.sigma_fun_S [lemma, in
Probas ]
Proba.Sigma_fun_simpl [lemma, in
Probas ]
Proba.sigma_fun_0 [lemma, in
Probas ]
Proba.Unth_eq [lemma, in
Probas ]
Proba.Unth_inv_eq [lemma, in
Probas ]
Proba.wretract_fplusok [lemma, in
Probas ]
Prog [library]
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Instance Index
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(32 entries)
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Z
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(14 entries)
Lemma Index
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Y Z
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(1217 entries)
Section Index
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B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
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W
X
Y Z
_
(22 entries)
Constructor Index
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B
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E
F G
H
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L
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O
P
Q
R
S T
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W
X
Y
Z
_
(12 entries)
Inductive Index
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E
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H
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O
P
Q
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W
X
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_
(7 entries)
Definition Index
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Module Index
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F
G
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T
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W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
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B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
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W
X
Y Z
_
(80 entries)
Library Index
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B C D
E
F
G
H
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R
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Y Z
_
(13 entries)
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Instance Index
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_
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Projection Index
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_
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Record Index
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E
F G
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L
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S T
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Y
Z
_
(14 entries)
Lemma Index
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K
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Y Z
_
(1217 entries)
Section Index
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E
F
G
H
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O
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Constructor Index
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X
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Z
_
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Inductive Index
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Q
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X
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Z
_
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Definition Index
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_
(277 entries)
Module Index
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F
G
H
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M N
O
P Q
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T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
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K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
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E
F
G
H
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L
M
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O
P Q
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Library Index
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F
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_
(13 entries)
O (projection)
Oeq [in
Ccpo ]
Ole [in
Ccpo ]
order_eq [in
Ccpo ]
order_rel [in
Ccpo ]
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Instance Index
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Projection Index
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Record Index
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E
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Y
Z
_
(14 entries)
Lemma Index
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Y Z
_
(1217 entries)
Section Index
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F
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(22 entries)
Constructor Index
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_
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Inductive Index
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_
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Definition Index
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Module Index
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Axiom Index
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T
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_
(6 entries)
Variable Index
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B C D
E
F
G
H
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L
M
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P Q
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Y Z
_
(80 entries)
Library Index
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Instance Index
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Projection Index
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_
(32 entries)
Record Index
A
B
C D
E
F G
H
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J
K
L
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S T
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X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
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K
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Y Z
_
(1217 entries)
Section Index
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B
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E
F
G
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(22 entries)
Constructor Index
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_
(12 entries)
Inductive Index
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Definition Index
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Module Index
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Axiom Index
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E
F
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(6 entries)
Variable Index
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B C D
E
F
G
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Library Index
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U (module)
Universe [in
Utheory ]
Univ_prop [in
Uprop ]
UP [in
Monads ]
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Record Index
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Lemma Index
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(1217 entries)
Section Index
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F
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Constructor Index
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Inductive Index
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Axiom Index
A
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C
D
E
F
G
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J
K
L
M
N
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S
T
U V
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_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
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O
P Q
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X
Y Z
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Library Index
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E
F
G
H
I J
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L
M N
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R
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(13 entries)
html/Utheory.html 0000644 0001750 0001750 00000053735 11301735614 014235 0 ustar cpaulin cpaulin
Utheory
Library Utheory
Utheory.v: Specification of , interval
Require Export Misc .
Require Export Ccpo .
Set Implicit Arguments .
Open Local Scope O_scope .
Constants : and
Constructor : Unth
Operations : , , inv
Relations : ,
Module Type Universe .
Parameter U :
Type .
Instance ordU :
ord U .
Instance cpoU :
cpo U .
Bind Scope U_scope with U .
Delimit Scope U_scope with U .
Parameter s Uplus Umult Udiv :
U ->
U ->
U .
Parameter Uinv :
U ->
U .
Parameter Unth :
nat ->
U .
Infix "+" :=
Uplus :
U_scope .
Infix "*" :=
Umult :
U_scope .
Infix "/" :=
Udiv :
U_scope .
Notation "[1-] x" := (
Uinv x ) (
at level 35,
right associativity ) :
U_scope .
Notation "[1/]1+ n" := (
Unth n ) (
at level 35,
right associativity ) :
U_scope .
Open Local Scope U_scope .
Definition U1 :
U := [1-] 0.
Notation "1" :=
U1 :
U_scope .
Hypothesis Udiff_0_1 : ~0 == 1.
Hypothesis Uplus_sym :
forall x y :U,
x +
y ==
y +
x .
Hypothesis Uplus_assoc :
forall x y z :U,
x + (
y +
z ) ==
x +
y +
z .
Hypothesis Uplus_zero_left :
forall x :U, 0 +
x ==
x .
Hypothesis Umult_sym :
forall x y :U,
x *
y ==
y *
x .
Hypothesis Umult_assoc :
forall x y z :U,
x * (
y *
z ) ==
x *
y *
z .
Hypothesis Umult_one_left :
forall x :U, 1 *
x ==
x .
Hypothesis Uinv_one : [1-] 1 == 0.
Hypothesis Umult_div :
forall x y , ~0 ==
y ->
x <=
y ->
y * (
x /y) ==
x .
Hypothesis Udiv_le_one :
forall x y , ~0 ==
y ->
y <=
x -> (
x /y) == 1.
Hypothesis Udiv_by_zero :
forall x y , 0 ==
y -> (
x /y) == 0.
Property : holds when does not overflow
Hypothesis Uinv_plus_left :
forall x y ,
y <= [1-]
x -> [1-] (
x +
y ) +
x == [1-]
y .
Property :
holds when does not overflow
Hypothesis Udistr_plus_right :
forall x y z ,
x <= [1-]
y -> (
x +
y ) *
z ==
x *
z +
y *
z .
Hypothesis Udistr_inv_right :
forall x y :U, [1-] (
x *
y ) == ([1-]
x ) *
y + [1-]
y .
The relation is compatible with operators
Properties of simplification in case there is no overflow
Hypothesis archimedian :
forall x , ~0 ==
x ->
exc (
fun n => [1/]1+n <=
x ).
Stability properties of lubs with respect to + and *
Hypothesis Uplus_right_continuous :
forall k ,
continuous (
mon (
Uplus k )).
Hypothesis Umult_right_continuous :
forall k ,
continuous (
mon (
Umult k )).
End Universe .
Declare Module Univ :Universe.
Export Univ .
Hint Resolve Udiff_0_1 Unth_prop .
Hint Resolve Uplus_sym Uplus_assoc Umult_sym Umult_assoc .
Hint Resolve Uinv_one Uinv_plus_left Umult_div Udiv_le_one Udiv_by_zero .
Hint Resolve Uplus_zero_left Umult_one_left Udistr_plus_right Udistr_inv_right .
Hint Resolve Uplus_mon_right Umult_mon_right Uinv_le_compat .
Hint Resolve lub_le le_lub Uplus_right_continuous Umult_right_continuous .
Hint Resolve Ule_total Ule_class .
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Global Index
A B C D E F G H
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K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
B (variable)
Bernoulli.dec_demi [in
Bernoulli ]
html/index_instance_U.html 0000644 0001750 0001750 00000053102 11301735615 016042 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
U (instance)
Universe.cpoU [in
Utheory ]
Universe.ordU [in
Utheory ]
Universe.Umult_mon_right [in
Utheory ]
Universe.Uplus_mon_right [in
Utheory ]
Univ_prop.fconj_continuous2 [in
Uprop ]
Univ_prop.fconj_mon2 [in
Uprop ]
Univ_prop.fesp_mon2 [in
Uprop ]
Univ_prop.finf_mon [in
Uprop ]
Univ_prop.finv_mon [in
Uprop ]
Univ_prop.fminus_mon2 [in
Uprop ]
Univ_prop.fmult_mon2 [in
Uprop ]
Univ_prop.fplus_mon2 [in
Uprop ]
Univ_prop.fsup_mon [in
Uprop ]
Univ_prop.IUcpo [in
Uprop ]
Univ_prop.IUord [in
Uprop ]
Univ_prop.low_mon [in
Uprop ]
Univ_prop.max_mon [in
Uprop ]
Univ_prop.MFcpo [in
Uprop ]
Univ_prop.min_mon2 [in
Uprop ]
Univ_prop.Nmult_mon2 [in
Uprop ]
Univ_prop.pmin_mon [in
Uprop ]
Univ_prop.seq_max_mon [in
Uprop ]
Univ_prop.sigma_mon [in
Uprop ]
Univ_prop.Udiv_mon [in
Uprop ]
Univ_prop.Uesp_mon [in
Uprop ]
Univ_prop.Uexp_mon2 [in
Uprop ]
Univ_prop.Uinv_mon [in
Uprop ]
Univ_prop.Uminus_mon2 [in
Uprop ]
Univ_prop.Umult_continuous2 [in
Uprop ]
Univ_prop.Umult_mon2 [in
Uprop ]
Univ_prop.Uplus_continuous2 [in
Uprop ]
Univ_prop.Uplus_mon2 [in
Uprop ]
Univ_prop.up_mon [in
Uprop ]
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_library_T.html 0000644 0001750 0001750 00000022476 11301735615 015713 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_section_E.html 0000644 0001750 0001750 00000022476 11301735615 015674 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_constructor_O.html 0000644 0001750 0001750 00000022476 11301735615 016627 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_module_D.html 0000644 0001750 0001750 00000022476 11301735615 015514 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_library_Q.html 0000644 0001750 0001750 00000022476 11301735615 015710 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_section_C.html 0000644 0001750 0001750 00000023310 11301735615 015656 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
C (section)
Choice.CHOICE [in
Choice ]
CoverFun.SigmaFinite [in
Cover ]
CoverFun.SigmaFinite.RandomFinite [in
Cover ]
html/index_constructor_C.html 0000644 0001750 0001750 00000023176 11301735615 016611 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
C (constructor)
cont [in
Ccpo ]
continuous2_intro [in
Ccpo ]
cont_intro [in
Ccpo ]
html/index_module_P.html 0000644 0001750 0001750 00000023011 11301735615 015512 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
P (module)
PP [in
Prog ]
Proba [in
Probas ]
html/index_lemma_M.html 0000644 0001750 0001750 00000055476 11301735615 015341 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
M (lemma)
mfun2_simpl [in
Ccpo ]
mlub_eq_compat [in
Ccpo ]
mlub_le [in
Ccpo ]
mlub_le_compat [in
Ccpo ]
mlub_lift_left [in
Ccpo ]
mlub_lift_right [in
Ccpo ]
Monad.inv_minus_distr [in
Monads ]
Monad.law1 [in
Monads ]
Monad.law2 [in
Monads ]
Monad.law3 [in
Monads ]
Monad.le_esp_distr [in
Monads ]
Monad.le_minus_distr [in
Monads ]
Monad.le_plus_distr [in
Monads ]
Monad.stable_minus_distr [in
Monads ]
Monad.star_continuous [in
Monads ]
Monad.star_le_plus [in
Monads ]
Monad.star_simpl [in
Monads ]
Monad.star_stable_eq [in
Monads ]
Monad.star_stable_inv [in
Monads ]
Monad.star_stable_mult [in
Monads ]
Monad.star_stable_plus [in
Monads ]
Monad.unit_continuous [in
Monads ]
Monad.unit_le_plus [in
Monads ]
Monad.unit_stable_eq [in
Monads ]
Monad.unit_stable_inv [in
Monads ]
Monad.unit_stable_mult [in
Monads ]
Monad.unit_stable_plus [in
Monads ]
monotonic2_sym [in
Ccpo ]
monotonic_intro [in
Ccpo ]
monotonic_sym [in
Ccpo ]
mon2_elim1 [in
Ccpo ]
mon2_elim2 [in
Ccpo ]
mon2_fun2 [in
Ccpo ]
mon2_le_compat [in
Ccpo ]
mon2_simpl [in
Ccpo ]
mon_eq_compat [in
Ccpo ]
mon_le_compat [in
Ccpo ]
mon_ord_equiv_simpl [in
Ccpo ]
mon_simpl [in
Ccpo ]
mseq_lift_left_le_compat [in
Ccpo ]
mseq_lift_left_simpl [in
Ccpo ]
mseq_lift_right_left [in
Ccpo ]
mseq_lift_right_le_compat [in
Ccpo ]
mseq_lift_right_simpl [in
Ccpo ]
mshift2_eq [in
Ccpo ]
mshift_continuous2 [in
Ccpo ]
mshift_le_compat [in
Ccpo ]
mshift_simpl [in
Ccpo ]
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_constructor_T.html 0000644 0001750 0001750 00000022476 11301735615 016634 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_module_K.html 0000644 0001750 0001750 00000022476 11301735615 015523 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_record_Y.html 0000644 0001750 0001750 00000022476 11301735615 015532 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_definition_B.html 0000644 0001750 0001750 00000024535 11301735615 016353 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
B (definition)
Bernoulli.bernoulli [in
Bernoulli ]
Bernoulli.binomial [in
Bernoulli ]
Bernoulli.comb [in
Bernoulli ]
Bernoulli.Fbern [in
Bernoulli ]
Bernoulli.fc [in
Bernoulli ]
Bernoulli.Mubern [in
Bernoulli ]
Bernoulli.qfalse [in
Bernoulli ]
Bernoulli.qk [in
Bernoulli ]
Bernoulli.qtrue [in
Bernoulli ]
html/index_inductive_O.html 0000644 0001750 0001750 00000022476 11301735615 016234 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_instance_E.html 0000644 0001750 0001750 00000022476 11301735615 016034 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_section_Z.html 0000644 0001750 0001750 00000022476 11301735615 015721 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_constructor_B.html 0000644 0001750 0001750 00000022476 11301735615 016612 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_axiom_U.html 0000644 0001750 0001750 00000023707 11301735615 015363 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
U (axiom)
Universe.U [in
Utheory ]
Universe.Udiv [in
Utheory ]
Universe.Uinv [in
Utheory ]
Universe.Umult [in
Utheory ]
Universe.Unth [in
Utheory ]
Universe.Uplus [in
Utheory ]
html/index_record_I.html 0000644 0001750 0001750 00000022476 11301735615 015512 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_module_C.html 0000644 0001750 0001750 00000023142 11301735615 015502 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
C (module)
Choice [in
Choice ]
CoverFun [in
Cover ]
CP [in
Ycart ]
html/index_definition_L.html 0000644 0001750 0001750 00000023021 11301735615 016352 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
L (definition)
Lub [in
Ccpo ]
lub_fun [in
Ccpo ]
html/index_section_O.html 0000644 0001750 0001750 00000022476 11301735615 015706 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_variable_K.html 0000644 0001750 0001750 00000022476 11301735615 016023 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_record_J.html 0000644 0001750 0001750 00000022476 11301735615 015513 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_library_S.html 0000644 0001750 0001750 00000022630 11301735615 015702 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
S (library)
Sets
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margin: 0px 0px;
background-color: white }
#page { display: block;
padding: 0px;
margin: 0px;
padding-bottom: 10px; }
#header { display: block;
position: relative;
padding: 0;
margin: 0;
vertical-align: middle;
border-bottom-style: solid;
border-width: thin }
#header h1 { padding: 0;
margin: 0;}
/* Contenu */
#main{ display: block;
padding: 10px;
overflow: hidden;
font-size: 100%;
line-height: 100% }
#main a.idref:visited {color : #416DFF; text-decoration : none; }
#main a.idref:link {color : #416DFF; text-decoration : none; }
#main a.idref:hover {text-decoration : none; }
#main a.idref:active {text-decoration : none; }
#main a.modref:visited {color : #416DFF; text-decoration : none; }
#main a.modref:link {color : #416DFF; text-decoration : none; }
#main a.modref:hover {text-decoration : none; }
#main a.modref:active {text-decoration : none; }
#main .keyword { color : #cf1d1d }
#main { color: black }
#main .section { background-color:#90bdff;
font-size : 175% }
#main .doc { margin: 0px;
padding: 10px;
font-family: sans-serif;
font-size: 100%;
line-height: 100%;
font-weight:bold;
color: black;
background-color: #90bdff;
border-style: plain}
.inlinecode {
display: inline;
font-family: monospace }
.comment {
display: inline;
font-family: monospace;
color: red; }
.code {
display: block;
font-family: monospace }
/* Pied de page */
#footer { font-size: 65%;
font-family: sans-serif; }
#footer a:visited { color: blue; }
#footer a:link { text-decoration: none;
color: #888888; }
.id { display: inline; }
.id[type="constructor"] {
color: rgb(60%,0%,0%);
}
.id[type="var"] {
color: rgb(40%,0%,40%);
}
.id[type="variable"] {
color: rgb(40%,0%,40%);
}
.id[type="definition"] {
color: rgb(0%,40%,0%);
}
.id[type="abbreviation"] {
color: rgb(0%,40%,0%);
}
.id[type="lemma"] {
color: rgb(0%,40%,0%);
}
.id[type="instance"] {
color: rgb(0%,40%,0%);
}
.id[type="projection"] {
color: rgb(0%,40%,0%);
}
.id[type="method"] {
color: rgb(0%,40%,0%);
}
.id[type="inductive"] {
color: rgb(0%,0%,80%);
}
.id[type="record"] {
color: rgb(0%,0%,80%);
}
.id[type="class"] {
color: rgb(0%,0%,80%);
}
.id[type="keyword"] {
color : #cf1d1d;
/* color: black; */
}
html/index_global_E.html 0000644 0001750 0001750 00000026506 11301735615 015466 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
E
empty [definition, in
Sets ]
eqset [definition, in
Sets ]
eqset_add_empty [lemma, in
Sets ]
eqset_add_rem [lemma, in
Sets ]
eqset_empty_false [lemma, in
Sets ]
eqset_incl [lemma, in
Sets ]
eqset_incl_intro [lemma, in
Sets ]
eqset_incl_sym [lemma, in
Sets ]
eqset_refl [lemma, in
Sets ]
eqset_rem_add [lemma, in
Sets ]
eqset_sym [lemma, in
Sets ]
eqset_trans [lemma, in
Sets ]
eqset_union_left [lemma, in
Sets ]
eqset_union_right [lemma, in
Sets ]
eq_ord [definition, in
Ccpo ]
eq_ord_equiv [lemma, in
Ccpo ]
eq_rel [definition, in
Ccpo ]
exc [definition, in
Misc ]
excluded_middle [lemma, in
Misc ]
exc_intro [lemma, in
Misc ]
exc_intro_class [lemma, in
Misc ]
html/index_definition_I.html 0000644 0001750 0001750 00000025602 11301735615 016356 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
I (definition)
Id [in
Ccpo ]
id [in
Ccpo ]
Imon [in
Ccpo ]
imon [in
Ccpo ]
imon2 [in
Ccpo ]
Imon2 [in
Ccpo ]
incl [in
Sets ]
inter [in
Sets ]
Iord [in
Ccpo ]
isempty [in
Sets ]
ishift [in
Ccpo ]
Iter [in
Ccpo ]
iter [in
Ccpo ]
IterFlip.Fiter [in
IterFlip ]
IterFlip.iterflip [in
IterFlip ]
IterFlip.p [in
IterFlip ]
IterFlip.p_ [in
IterFlip ]
IterFlip.q1 [in
IterFlip ]
iter_ [in
Ccpo ]
html/index_section_D.html 0000644 0001750 0001750 00000022476 11301735615 015673 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_library_X.html 0000644 0001750 0001750 00000022476 11301735615 015717 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_axiom_X.html 0000644 0001750 0001750 00000022476 11301735615 015370 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_module_S.html 0000644 0001750 0001750 00000022476 11301735615 015533 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_lemma_K.html 0000644 0001750 0001750 00000022476 11301735615 015331 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_axiom_V.html 0000644 0001750 0001750 00000022476 11301735615 015366 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_library_Y.html 0000644 0001750 0001750 00000022632 11301735615 015712 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
Y (library)
Ycart
html/index_record_T.html 0000644 0001750 0001750 00000022476 11301735615 015525 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_lemma_X.html 0000644 0001750 0001750 00000022476 11301735615 015346 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_global_B.html 0000644 0001750 0001750 00000060510 11301735615 015454 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
B
Bernoulli [module, in
Bernoulli ]
Bernoulli [library]
Bernoulli.bernoulli [definition, in
Bernoulli ]
Bernoulli.bernoulli_eq [lemma, in
Bernoulli ]
Bernoulli.bernoulli_eq_false [lemma, in
Bernoulli ]
Bernoulli.bernoulli_eq_true [lemma, in
Bernoulli ]
Bernoulli.bernoulli_false [lemma, in
Bernoulli ]
Bernoulli.bernoulli_total [lemma, in
Bernoulli ]
Bernoulli.bernoulli_true [lemma, in
Bernoulli ]
Bernoulli.Bern_commute [lemma, in
Bernoulli ]
Bernoulli.Bern_eq [lemma, in
Bernoulli ]
Bernoulli.Bern_false [lemma, in
Bernoulli ]
Bernoulli.Bern_term [lemma, in
Bernoulli ]
Bernoulli.Bern_true [lemma, in
Bernoulli ]
Bernoulli.binomial [definition, in
Bernoulli ]
Bernoulli.binomial_eq_k [lemma, in
Bernoulli ]
Bernoulli.comb [definition, in
Bernoulli ]
Bernoulli.comb_inv [lemma, in
Bernoulli ]
Bernoulli.comb_not_le [lemma, in
Bernoulli ]
Bernoulli.comb_n_n [lemma, in
Bernoulli ]
Bernoulli.comb_n_Sn [lemma, in
Bernoulli ]
Bernoulli.comb_Sn_n [lemma, in
Bernoulli ]
Bernoulli.comb_0_n [lemma, in
Bernoulli ]
Bernoulli.comb_1_Sn [lemma, in
Bernoulli ]
Bernoulli.dec_demi [variable, in
Bernoulli ]
Bernoulli.Fbern [definition, in
Bernoulli ]
Bernoulli.Fbern_mon [instance, in
Bernoulli ]
Bernoulli.Fbern_simpl [lemma, in
Bernoulli ]
Bernoulli.fc [definition, in
Bernoulli ]
Bernoulli.fcp_n [lemma, in
Bernoulli ]
Bernoulli.fcp_not_le [lemma, in
Bernoulli ]
Bernoulli.fcp_S [lemma, in
Bernoulli ]
Bernoulli.fcp_0 [lemma, in
Bernoulli ]
Bernoulli.fc0 [lemma, in
Bernoulli ]
Bernoulli.fc_Nmult_def [lemma, in
Bernoulli ]
Bernoulli.fc_retract [lemma, in
Bernoulli ]
Bernoulli.Mubern [definition, in
Bernoulli ]
Bernoulli.Mubern_eq [lemma, in
Bernoulli ]
Bernoulli.MuBern_false [lemma, in
Bernoulli ]
Bernoulli.Mubern_inv [lemma, in
Bernoulli ]
Bernoulli.Mubern_simpl [lemma, in
Bernoulli ]
Bernoulli.MuBern_true [lemma, in
Bernoulli ]
Bernoulli.Nmult_comb [lemma, in
Bernoulli ]
Bernoulli.qfalse [definition, in
Bernoulli ]
Bernoulli.qk [definition, in
Bernoulli ]
Bernoulli.qtrue [definition, in
Bernoulli ]
Bernoulli.qtrue_qfalse_inv [lemma, in
Bernoulli ]
Bernoulli.retract_class [lemma, in
Bernoulli ]
Bernoulli.sigma_fc0 [lemma, in
Bernoulli ]
Bernoulli.sigma_fc_1 [lemma, in
Bernoulli ]
Bernoulli.Uinv_exp [lemma, in
Bernoulli ]
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_projection_M.html 0000644 0001750 0001750 00000023067 11301735615 016411 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
M (projection)
monotonic2_intro [in
Ccpo ]
monotonic_def [in
Ccpo ]
html/index_definition_H.html 0000644 0001750 0001750 00000022476 11301735615 016363 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/Probas.html 0000644 0001750 0001750 00000660601 11301735614 014020 0 ustar cpaulin cpaulin
Probas
Library Probas
Probas.v: The monad for distributions
Require Export Monads .
Set Implicit Arguments .
Definition of distribution
Distributions are monotonic measure functions such that
Record distr (A :Type) : Type := mu : M A ;mu_stable_inv : stable_inv mu ; mu_stable_plus : stable_plus mu ;mu_stable_mult : stable_mult mu ;mu_continuous : continuous mu }.Hint Resolve mu_stable_plus mu_stable_inv mu_stable_mult mu_continuous .
Lemma mu_monotonic : forall (A : Type )(m : distr A ), monotonic (mu m ).intros ; apply fmonotonic ; auto .Save .Hint Resolve mu_monotonic .Implicit Arguments mu_monotonic [A ].Lemma mu_stable_eq : forall (A : Type )(m : distr A ), stable (mu m ).intros ; apply fstable ; auto .Save .Hint Resolve mu_stable_eq .Implicit Arguments mu_stable_eq [A ].Lemma mu_zero : forall (A : Type )(m : distr A ), mu m (fzero A ) == 0.intros .transitivity (mu m (fmult 0 (fzero A ))); auto .transitivity (0 * (mu m (fzero A ))); auto .apply mu_stable_mult ; auto .Save .Hint Resolve mu_zero .Lemma mu_zero_eq : forall (A : Type )(m : distr A ) f , forall x , f x == 0) -> mu m f == 0.intros ; transitivity (mu m (fzero A )); auto .Save .Lemma mu_one_inv : forall (A : Type )(m :distr A ),mu m (fone A ) == 1 -> forall f , mu m (finv f ) == [1-] (mu m f ).intros ; apply Ole_antisym .apply (mu_stable_inv m f ).apply Uplus_le_simpl_left with (mu m f ); auto .setoid_rewrite (Uinv_opp_right (mu m f )).transitivity (mu m (fun x => (f x ) + [1-] (f x ))).setoid_rewrite <- H ; apply (mu_monotonic m ); intro x ; simpl ; auto .assert (H1 : fplusok f (finv f )).red ; unfold finv ; intro x ; auto .setoid_rewrite <- (mu_stable_plus m H1 ); auto .Save .Hint Resolve mu_one_inv .Lemma mu_fplusok : forall (A : Type )(m :distr A ) f g , fplusok f g -> mu m f <= [1-] mu m g .intros ; transitivity (mu m (finv g )); auto .apply (mu_stable_inv m g ).Save .Hint Resolve mu_fplusok .Lemma mu_le_minus : forall (A : Type )(m :distr A ) (f g :MF A ),mu m (fminus f g ) <= mu m f .intros ; apply mu_monotonic ; intro x ; unfold fminus ; auto .Save .Hint Resolve mu_le_minus .Lemma mu_le_plus : forall (A : Type )(m :distr A ) (f g :MF A ),mu m (fplus f g ) <= mu m f + mu m g .intros ; transitivity (mu m (fplus (fminus f (fesp f g )) g )).apply mu_monotonic .unfold fplus ,fminus,fesp; intro x ; auto .rewrite (mu_stable_plus m (f :=fminus f (fesp f g )) (g :=g)).unfold fminus ,fesp,finv; auto .intro x ; unfold fplus ,fminus,fesp, finv ; auto .Save .Hint Resolve mu_le_plus .Lemma mu_cte : forall (A : Type )(m :(distr A )) (c :U),mu m (fcte A c ) == c * mu m (fone A ).intros .transitivity (mu m (fun x => c * 1)).apply (mu_stable_eq m ); intro x ; auto .unfold fone ; rewrite <- (mu_stable_mult m c (fun x => 1)); auto .Save .Hint Resolve mu_cte .Lemma mu_cte_le : forall (A : Type )(m :distr A ) (c :U), mu m (fcte A c ) <= c .intros ; transitivity (c * mu m (fone A )); auto .Save .Lemma mu_cte_eq : forall (A : Type )(m :distr A ) (c :U),mu m (fone A ) == 1 -> mu m (fcte A c ) == c .intros ; transitivity (c * mu m (fone A )); auto .Save .Hint Resolve mu_cte_le mu_cte_eq .Lemma mu_stable_mult_right : forall (A : Type )(m :distr A ) (c :U) (f : MF A ),mu m (fun x => (f x ) * c ) == (mu m f ) * c .intros ; transitivity mu m (fun x => c * (f x ))).apply mu_stable_eq ; intro x ; auto .transitivity (c * mu m f ); auto .exact (mu_stable_mult m c f ).Save .Lemma mu_stable_minus : forall (A :Type) (m :distr A )(f g : MF A ),g <= f -> mu m (fun x => f x - g x ) == mu m f - mu m g .intros ; change (mu m (fminus f g ) == mu m f - mu m g ).apply (stable_minus_distr (m :=mu m )); auto .Save .Lemma mu_inv_minus : forall (A :Type) (m :distr A )(f : MF A ), mu m (finv f ) == mu m (fone A ) - mu m f .intros ; transitivity (mu m (fun x => fone A x - f x )).apply (mu_stable_eq m ); intro x ; unfold finv ,fone; intros ; auto .apply mu_stable_minus ; auto .Save .Lemma mu_stable_le_minus : forall (A :Type) (m :distr A )(f g : MF A ),mu m f - mu m g <= mu m (fun x => f x - g x ).intros ; apply Uplus_le_perm_left .transitivity (mu m (fun x => g x + (f x - g x ))).apply mu_monotonic ; intro x ; auto .apply (mu_le_plus m g (fun x => f x - g x )).Save .Lemma mu_inv_minus_inv : forall (A :Type) (m :distr A )(f : MF A ), mu m (finv f ) + [1-](mu m (fone A )) == [1-](mu m f ).intros ; apply Uminus_eq_perm_right .apply Uinv_le_compat .apply (mu_monotonic m ); unfold fone ; auto .unfold Uminus .rewrite mu_inv_minus ; repeat Usimpl .unfold Uminus .apply Uinv_eq_compat ; auto .Save .Lemma mu_le_esp_inv : forall (A :Type) (m :distr A )(f g : MF A ),mu m (finv f )) & mu m g <= mu m (fesp f g ).intros ; rewrite Uesp_sym .apply Uplus_inv_le_esp ; Usimpl .transitivity (mu m (fplus (fesp f g ) (finv f ))); auto .apply (mu_monotonic m ); unfold fplus , fesp ,finv; intro x .rewrite Uesp_sym ; auto .Save .Hint Resolve mu_le_esp_inv .Lemma mu_stable_inv_inv : forall (A :Type) (m :distr A )(f : MF A ),mu m f <= [1-] mu m (finv f ).intros ; transitivity (mu m (finv (finv f ))).apply (mu_monotonic m ); auto .apply (mu_stable_inv m ); auto .Save .Hint Resolve mu_stable_inv_inv .Lemma mu_stable_div : forall (A :Type) (m :distr A )(k :U)(f : MF A ),f <= fcte A k -> mu m (fdiv k f ) == mu m f / k .intros ; apply Umult_div_eq ; auto .rewrite Umult_sym .rewrite <- (mu_stable_mult m k ).apply mu_stable_eq ; intro x .unfold fmult ,fdiv; apply Umult_div ; auto .apply (H0 x ).Save .Lemma mu_stable_div_le : forall (A :Type) (m :distr A )(k :U)(f : MF A ),mu m (fdiv k f ) <= mu m f / k .intros ; apply Umult_div_le_left ; auto .rewrite Umult_sym .rewrite <- (mu_stable_mult m k ).apply mu_monotonic ; intro x .unfold fmult ,fdiv; apply Umult_div_le ; auto .Save .Lemma mu_le_esp : forall (A :Type) (m :distr A )(f g : MF A ),mu m f & mu m g <= mu m (fesp f g ).intros ; transitivity (([1-]mu m (finv f )) & mu m g ); auto .Save .Hint Resolve mu_le_esp .Lemma mu_esp_one : forall (A :Type)(m:distr A )(f g :MF A ),m f -> mu m g == mu m (fesp f g ).intros ; apply Ole_antisym .transitivity (mu m f & mu m g ).transitivity (1 & mu m g ); auto .apply mu_le_esp with (f :=f) (g :=g).apply mu_monotonic ; unfold fesp ; intro x ; auto .Save .Lemma mu_esp_zero : forall (A :Type)(m:distr A )(f g :MF A ),mu m (finv f ) <= 0 -> mu m g == mu m (fesp f g ).intros ; apply Ole_antisym .transitivity (([1-](mu m (finv f ))) & mu m g ); auto .transitivity (1 & mu m g ); auto .apply Uesp_le_compat ; auto .transitivity ([1-]0); auto .apply mu_monotonic ; unfold fesp ; intro x ; auto .Save .Instance Odistr (A :Type) : ord (distr A ) := Ole := fun (f g : distr A ) => mu f <= mu g ;Oeq := fun (f g : distr A ) => mu f == mu g }.split ; intuition .intros f g h ; transitivity (mu g ); auto .Defined .
Monadic operators for distributions
Definition Munit : forall A :Type, A -> distr A .intros A x .exists (unit x ).apply unit_stable_inv .apply unit_stable_plus .apply unit_stable_mult .apply unit_continuous .Defined .Definition Mlet : forall A B :Type, distr A -> (A -> distr B ) -> distr B .intros A B mA mB .exists (star (mu mA ) (fun x => (mu (mB x )))).apply star_stable_inv ; auto .apply star_stable_plus ; auto .apply star_stable_mult ; auto .apply star_continuous ; auto .Defined .Lemma Munit_simpl : forall (A :Type) (q :A -> U ) x , mu (Munit x ) q = q x .trivial .Save .Lemma Mlet_simpl : forall (A B :Type) (m :distr A ) (M :A -> distr B ) (f :B -> U ),mu (Mlet m M ) f = mu m (fun x => (mu (M x ) f )).trivial .Save .
Operations on distributions
Lemma Munit_eq_compat : forall A (x y : A ), x = y -> Munit x == Munit y .intros ; subst ; auto .Save .Lemma Mlet_le_compat : forall (A B : Type ) (m1 m2 :distr A ) (M1 M2 : A -> distr B ), m1 <= m2 -> M1 <= M2 -> Mlet m1 M1 <= Mlet m2 M2 .intros A B m1 m2 M1 M2 Hm HM f .repeat rewrite Mlet_simpl .transitivity (mu m2 (fun x : A => mu (M1 x ) f )); auto .apply (fmonotonic (mu m2 )); intro ; auto .apply (HM x f ).Save .Hint Resolve Mlet_le_compat .Add Parametric Morphism (A B : Type ) : (Mlet (A :=A) (B :=B))with signature Ole & Ole & Ole as Mlet_le_morphism .auto .Save .Instance Mlet_mon2 : forall (A B : Type ), monotonic2 (@Mlet A B ).red ; auto .Save .Definition MLet (A B : Type ) : distr A -m> (A -> distr B ) -m> distr B mon2 (@Mlet A B ).Lemma MLet_simpl : forall (A B :Type) (m :distr A ) (M :A -> distr B )(f :B -> U ),mu (MLet A B m M ) f = mu m (fun x => mu (M x ) f ).trivial .Save .Lemma Mlet_eq_compat : forall (A B : Type ) (m1 m2 :distr A ) (M1 M2 : A -> distr B ), m1 == m2 -> M1 ==M2 -> Mlet m1 M1 == Mlet m2 M2 .intros ; apply (monotonic2_stable2 (@Mlet A B )); trivial .Save .Hint Resolve Mlet_eq_compat .Add Parametric Morphism (A B : Type ) : (Mlet (A :=A) (B :=B))with signature Oeq & Oeq & Oeq as Mlet_eq_morphism .auto .Save .Lemma mu_le_compat : forall (A :Type) (m1 m2 :distr A ),m1 <= m2 -> forall f g : A -> U , f <= g -> mu m1 f <= mu m2 g .intros ; transitivity (mu m2 f ); auto .Save .Lemma mu_eq_compat : forall (A :Type) (m1 m2 :distr A ),m1 == m2 -> forall f g : A -> U , f == g -> mu m1 f == mu m2 g .intros ; transitivity (mu m2 f ); auto .Save .Hint Immediate mu_le_compat mu_eq_compat .Add Parametric Morphism (A : Type ) : (mu (A :=A))with signature Ole & Ole as mu_le_morphism .auto .Save .Add Parametric Morphism (A : Type ) : (mu (A :=A))with signature Oeq & Oeq as mu_eq_morphism .auto .Save .
Properties of monadic operators
Lemma Mlet_unit : forall (A B :Type) (x :A) (m :A -> distr B ), Mlet (Munit x ) m == m x .intros ; intro f ; auto .Save .Lemma Mlet_ext : forall (A :Type) (m :distr A ), Mlet m (fun x => Munit x ) == m .intros ; intro f .rewrite Mlet_simpl .apply (mu_stable_eq m ).intro x ; auto .Save .Lemma Mlet_assoc : forall (A B C :Type) (m1 : distr A ) (m2 :A -> distr B ) (m3 :B -> distr C ),Mlet (Mlet m1 m2 ) m3 == Mlet m1 (fun x :A => Mlet (m2 x ) m3 ).intros ; intro f ; simpl ; trivial .Save .Lemma let_indep : forall (A B :Type) (m1 :distr A ) (m2 : distr B ) (f :MF B ), mu m1 (fun _ => mu m2 f ) == mu m1 (fone A ) * (mu m2 f ).intros ; rewrite (mu_cte m1 (mu m2 f )); auto .Save .
Definition distr_null : forall A : Type , distr A .intro A ; exists (mon (cte (MF A ) (0:U))); try red ; auto .Defined .Lemma le_distr_null : forall (A :Type) (m : distr A ), distr_null A <= m .red ; intros .unfold distr_null ; simpl ; auto .Save .Hint Resolve le_distr_null .
Definition Mmult A (k :MF A ) (m :M A ) : M A .intros ; exists (fun f => m (fun x => k x * f x )).red ; intros .apply fmonotonic ; intro z ; auto .Defined .Lemma Mmult_simpl : forall A (k :MF A ) (m :M A ) f , Mmult k m f = m (fun x => k x * f x ).trivial .Save .Lemma Mmult_stable_inv : forall A (k :MF A ) (d :distr A ), stable_inv (Mmult k (mu d )).red ; intros ; simpl .transitivity (mu d (finv f )); auto .Save .Lemma Mmult_stable_plus : forall A (k :MF A ) (d :distr A ), stable_plus (Mmult k (mu d )).red ; intros ; simpl .transitivity (mu d (fun x => k x * f x + k x * g x )).apply (mu_stable_eq d ).simpl ; intro x ; unfold fplus .apply Udistr_plus_left ; apply (H x ).apply (mu_stable_plus d (f :=fun x => k x * f x ) (g :=fun x => k x * g x )).intro x ; transitivity (f x ); auto .transitivity (finv g x ); auto .unfold finv ; repeat Usimpl ; auto .Save .Lemma Mmult_stable_mult : forall A (k :MF A ) (d :distr A ), stable_mult (Mmult k (mu d )).red ; intros ; simpl .transitivity (mu d (fmult k0 (fun x : A => k x * f x ))).apply (mu_stable_eq d ).intro x ; unfold fmult ; auto .apply (mu_stable_mult d ).Save .Lemma Mmult_continuous : forall A (k :MF A ) (d :distr A ), continuous (Mmult k (mu d )).red ; intros ; rewrite Mmult_simpl .transitivity (mu d (fun (x :A) => lub (UMult (k x ) @ (h <o> x )))).apply (mu_monotonic d ); intros x ; auto .rewrite (MF_lub_simpl h ); auto .transitivity (mu d (lub (ishift (fun x : A => (UMult (k x ) @ h <o> x )))));auto .apply (mu_monotonic d ); intro x ; trivial .rewrite (MF_lub_simpl (ishift (fun x0 : A => UMult (k x0 ) @ h <o> x0 ))); apply lub_le_compat ; intro n ; simpl ; auto .rewrite (mu_continuous d (ishift (fun x => UMult (k x ) @ h <o> x ))).apply lub_le_compat ; intro n ; simpl ; auto .Save .Definition distr_mult A (k :MF A ) (d :distr A ) : distr A .intros ; exists (Mmult k (mu d )).apply Mmult_stable_inv .apply Mmult_stable_plus .apply Mmult_stable_mult .apply Mmult_continuous .Defined .Definition Mdiv A (k :U) (m :M A ) : M A := UDiv k @ m .Lemma Mdiv_simpl : forall A k (m :M A ) f , Mdiv k m f = m f / k .trivial .Save .Lemma Mdiv_stable_inv : forall A (k :U) (d :distr A )(dk : mu d (fone A ) <= k ),stable_inv (Mdiv k (mu d )).red ; intros ; repeat rewrite Mdiv_simpl .apply (Ueq_orc 0 k ); auto ; intros .repeat (rewrite Udiv_by_zero ;auto ).assert (forall f , mu d f <= k ).intros ; transitivity (mu d (fone A )); auto .transitivity ((mu d (fone A ) - mu d f ) / k ); auto .Save .Lemma Mdiv_stable_plus : forall A (k :U)(d:distr A ), stable_plus (Mdiv k (mu d )).red ; intros ; repeat rewrite Mdiv_simpl .transitivity ((mu d f + mu d g )/k); auto .apply Udiv_eq_compat_left ; auto .apply (mu_stable_plus d ); trivial .Save .Lemma Mdiv_stable_mult : forall A (k :U)(d:distr A )(dk : mu d (fone A ) <= k ), stable_mult (Mdiv k (mu d )).red ; intros ; repeat rewrite Mdiv_simpl .apply (Ueq_orc 0 k ); auto ; intros .repeat (rewrite Udiv_by_zero ;auto ).assert (forall f , mu d f <= k ).intros ; transitivity (mu d (fone A )); auto .transitivity ((k0 * mu d f ) / k ); auto .apply Udiv_eq_compat_left ; auto .apply (mu_stable_mult d ); trivial .apply Umult_div_assoc ; auto .Save .Lemma Mdiv_continuous : forall A (k :U)(d:distr A ), continuous (Mdiv k (mu d )).red ; intros ; repeat rewrite Mdiv_simpl .transitivity (lub (mu d @ h ) / k ); auto .transitivity (lub (UDiv k @ (mu d @ h ))).exact (Udiv_continuous k (mu d @ h )).apply lub_le_compat ; intro n ; auto .Save .Definition distr_div A (k :U) (d :distr A ) (dk : mu d (fone A ) <= k )distr A .intros .exists (Mdiv k (mu d )).apply Mdiv_stable_inv ; auto .apply Mdiv_stable_plus .apply Mdiv_stable_mult ; auto .apply Mdiv_continuous .Defined .
Conditional probabilities
Definition mcond A (m :M A ) (f :MF A ) : M A .intros ; exists (fun g => (m (fconj f g )) / m f ).red ; intros g h legh .apply Udiv_le_compat_left .apply (fmonotonic m ); unfold fconj ; intros ; auto .Defined .Lemma mcond_simpl : forall A (m :M A ) (f g : MF A ),mcond m f g = m (fconj f g ) / m f .trivial .Save .Lemma mcond_stable_plus : forall A (m :distr A ) (f : MF A ), stable_plus (mcond (mu m ) f ).red ; intros .repeat rewrite mcond_simpl .simpl ; rewrite <- Udiv_plus .apply Udiv_eq_compat_left .assert (fplusok (fconj f f0 ) (fconj f g )).auto .rewrite <- (mu_stable_plus m H0 ).apply (mu_stable_eq m ); intro x ; unfold fconj ,fplus; auto .apply Udistr_plus_left ; auto .apply (H x ).Save .Lemma mcond_stable_inv : forall A (m :distr A ) (f : MF A ), stable_inv (mcond (mu m ) f ).red ; intros .repeat rewrite mcond_simpl .apply (Ueq_orc 0 (mu m f )); intro .auto .rewrite (Udiv_by_zero (mu m (fconj f (finv f0 ))) (mu m f )); auto .transitivity (mu m (fminus f (fconj f f0 )) / mu m f ).apply Udiv_le_compat_left .apply (mu_monotonic m ); intro x ; unfold fconj ,finv,fminus; auto .rewrite stable_minus_distr ; trivial .rewrite Udiv_minus ; trivial .Save .Lemma mcond_stable_mult : forall A (m :distr A ) (f : MF A ), stable_mult (mcond (mu m ) f ).red ; intros .repeat rewrite mcond_simpl .rewrite <- Umult_div_assoc .apply Udiv_eq_compat_left .apply Oeq_trans with (mu m (fmult k (fconj f f0 ))).apply (mu_stable_eq m ).intro x ; unfold fconj , fmult ; auto .apply (mu_stable_mult m ).auto .Save .Lemma mcond_continuous : forall A (m :distr A ) (f : MF A ), continuous (mcond (mu m ) f ).red ; intros .repeat rewrite mcond_simpl .apply (Ueq_orc 0 (mu m f )); intro ; auto .rewrite (Udiv_by_zero (mu m (fconj f (lub h ))) (mu m f )); auto .apply Umult_div_le_right .transitivity (mu m (lub ((Fconj A f )@h ))).apply mu_monotonic .exact (continuous2_app (Fconj A ) f h ).rewrite (mu_continuous m ).transitivity (mshift UMult (mu m f ) (lub (mcond (mu m ) f @ h ))); auto .rewrite <- (lub_comp_le (mshift UMult (mu m f )) (mcond (mu m ) f @ h ) ).apply lub_le_compat ; intro x ; simpl .rewrite Udiv_mult ; auto .Save .Definition Mcond A (m :distr A ) (f :MF A ) : distr A :=Build_distr (mcond_stable_inv m f ) (mcond_stable_plus m f )mcond_stable_mult m f ) (mcond_continuous m f ).Lemma Mcond_total : forall A (m :distr A ) (f :MF A ), mu m f -> mu (Mcond m f ) (fone A ) == 1.intros ; simpl .transitivity (mu m f /mu m f ); auto .apply Udiv_eq_compat ; apply (mu_stable_eq m ); intro x ; unfold fconj , fone ; auto .Save .Lemma Mcond_simpl : forall A (m :distr A ) (f g :MF A ), mu (Mcond m f ) g = mu m (fconj f g ) / mu m f .trivial .Save .Hint Resolve Mcond_simpl .Lemma Mcond_zero_stable : forall A (m :distr A ) (f g :MF A ), mu m g == 0 -> mu (Mcond m f ) g == 0.intros ; rewrite Mcond_simpl .apply Ule_zero_eq ; transitivity (mu m g / mu m f ); auto .Save .Lemma Mcond_null : forall A (m :distr A ) (f g :MF A ), mu m f == 0 -> mu (Mcond m f ) g == 0.intros ; rewrite Mcond_simpl ; auto .Save .Lemma Mcond_conj : forall A (m :distr A ) (f g :MF A ), mu m (fconj f g ) == mu (Mcond m f ) g * mu m f .intros ; rewrite Mcond_simpl .apply (Ueq_orc 0 (mu m f )); intros ; auto .rewrite <- H ; repeat Usimpl .apply Ule_zero_eq .transitivity (mu m f ); auto .symmetry ; apply Udiv_mult ; auto .Save .Lemma Mcond_decomp : forall A (m :distr A ) (f g :MF A ), mu m g == mu (Mcond m f ) g * mu m f + mu (Mcond m (finv f )) g * mu m (finv f ).intros ; transitivity (mu m (fplus (fconj f g ) (fconj (finv f ) g ))).apply mu_stable_eq ; intro x ; unfold fplus ,finv,fconj; simpl ; auto .rewrite mu_stable_plus .repeat rewrite Mcond_conj ; auto .apply fplusok_le_compat with f (finv f ); auto .Save .Lemma Mcond_bayes : forall A (m :distr A ) (f g :MF A ), mu (Mcond m f ) g == (mu (Mcond m g ) f * mu m g ) / (mu m f ).intros ; repeat rewrite Mcond_simpl .apply Udiv_eq_compat ; trivial .apply (Ueq_orc (mu m g ) 0); intros ; trivial .rewrite H ; Usimpl .apply Ule_zero_eq .transitivity (mu m g ); auto .rewrite Udiv_mult ; auto .apply mu_stable_eq ; intro x ; unfold fconj ; simpl ; auto .Save .
Least upper bound of increasing sequences of distributions
Lemma M_lub_simpl : forall A (h : nat -m> M A ) (f :MF A ),lub h f = lub (mshift h f ).trivial .Save .Section Lubs .Variable A : Type .Definition Mu : distr A -m> M A .exists (mu (A :=A)); auto .Defined .Variable muf : nat -m> distr A .Definition mu_lub : distr A .exists (lub (Mu @ muf )).red ; intros ; simpl ; apply lub_inv ; intros ; simpl .apply (mu_stable_inv (muf n )).red ; intros .transitivity (lub ((UPlus @2 (mshift (Mu @ muf ) f )) (mshift (Mu @ muf ) g ))); auto .rewrite (M_lub_simpl (Mu @ muf ) (fplus f g )).apply lub_eq_compat ; intro n ; simpl ; auto .exact (mu_stable_plus (muf n ) H ); auto .rewrite <- (lub_cont2_app2_eq UPlus ); auto .red ; intros ; simpl .transitivity (lub (UMult k @ (mshift (Mu @ muf ) f ))); auto .apply lub_eq_compat ; auto .intro n ; simpl ; exact (mu_stable_mult (muf n ) k f ); auto .unfold M ; apply lub_continuous ; intros .apply (mu_continuous (muf n )).Defined .Lemma mu_lub_le : forall n :nat, muf n <= mu_lub .simpl ; intros .apply (le_lub (mshift (Mu @ muf ) x ) n ).Save .Lemma mu_lub_sup : forall m : distr A , (forall n :nat, muf n <= m ) -> mu_lub <= m .simpl ; intros .apply lub_le ; simpl ; intros ; auto .Save .End Lubs .Hint Resolve mu_lub_le mu_lub_sup .
distributions seen as a Ccpo
Instance cdistr : forall (A :Type), cpo (distr A ) :=D0 := distr_null A ; lub :=mu_lub (A :=A)}.auto .auto .auto .Defined .Lemma distr_lub_simpl : forall A (h : nat -m> distr A ) (f :MF A ),mu (lub h ) f = lub (mshift (Mu A @ h ) f ).trivial .Save .Hint Resolve distr_lub_simpl .
Definition Mfix (
A B :Type) (
F : (
A ->
distr B ) -m> (
A ->
distr B ))
:
A ->
distr B :=
fixp F .
Definition MFix (
A B :Type) : ((
A ->
distr B ) -m> (
A ->
distr B )) -m> (
A ->
distr B )
:=
Fixp (
A ->
distr B ).
Lemma Mfix_le :
forall (
A B :Type) (
F : (
A ->
distr B ) -m> (
A ->
distr B )) (
x :A),
Mfix F x <=
F (
Mfix F )
x .
intros ;
unfold Mfix ;
apply (
fixp_le F x ).
Save .
Lemma Mfix_eq :
forall (
A B :Type) (
F : (
A ->
distr B ) -m> (
A ->
distr B )),
continuous F ->
forall (
x :A),
Mfix F x ==
F (
Mfix F )
x .
intros A B F H ;
unfold Mfix .
exact (
fixp_eq F ).
Save .
Hint Resolve Mfix_le Mfix_eq .
Lemma Mfix_le_compat :
forall (
A B :Type) (
F G : (
A ->
distr B )-m> (
A ->
distr B )),
F <=
G ->
Mfix F <=
Mfix G .
intros ;
unfold Mfix ;
auto .
Save .
Definition Miter (
A B :Type) :=
Ccpo.iter (
D :=A ->
distr B ).
Lemma Mfix_le_iter :
forall (
A B :Type) (
F :(A ->
distr B ) -m> (
A ->
distr B )) (
n :nat),
Miter F n <=
Mfix F .
unfold Miter ,Mfix,fixp;
auto .
Save .
Section Continuity .Variables A B :Type.Instance Mlet_continuous_right forall a :distr A , continuous (D1 := A -> distr B ) (D2 :=distr B ) (MLet A B a ).red ; intros ; intro f .rewrite (MLet_simpl (A :=A) (B :=B)).transitivity (mu a (fun x :A => lub (mshift (Mu B @ (h <o> x )) f ))); auto .transitivity (mu a (lub (ishift (fun x :A => mshift (Mu B @ (h <o> x )) f )))).apply (mu_monotonic a ); intro x ; auto .unfold MFcpo ; rewrite fcpo_lub_simpl ; auto .rewrite (mu_continuous a ).rewrite distr_lub_simpl ; apply lub_le_compat ; intro n ; simpl ; auto .Save .Lemma Mlet_continuous_left continuous (D1 :=distr A ) (D2 :=(A -> distr B ) -m> distr B ) (MLet A B ).red ; intros ; intro F ; intro f .rewrite (MLet_simpl (A :=A) (B :=B)).simpl ; auto .Save .Hint Resolve Mlet_continuous_right Mlet_continuous_left .Lemma Mlet_continuous2 : continuous2 (D1 :=distr A ) (D2 := A ->distr B ) (D3 :=distr B ) (MLet A B ).intros ; apply continuous_continuous2 ; auto .Save .Hint Resolve Mlet_continuous2 .Lemma Mlet_lub_le : forall (mun :nat -m> distr A ) (Mn : nat -m> (A -> distr B )),Mlet (lub mun ) (lub Mn ) <= lub ((MLet A B @2 mun ) Mn ).intros ; apply Mlet_continuous2 with (f :=mun) (g :=Mn).Save .Lemma Mfix_continuous : forall (Fn : nat -m> (A -> distr B ) -m> (A -> distr B )),forall n , continuous (Fn n )) -> Mfix (lub Fn ) <= lub (MFix A B @ Fn ).unfold Mfix , MFix ; intros ; apply fixp_continuous ; auto .Save .End Continuity .
distribution for flip
The distribution associated to
flip () is
Definition flip : M bool := mon (fun (f : bool -> U ) => [1/2] * (f true ) + [1/2] * (f false )).Lemma flip_stable_inv : stable_inv flip .unfold flip , stable_inv , finv ; intro f ; simpl ; auto .Save .Lemma flip_stable_plus : stable_plus flip .unfold flip , stable_plus , fplus ; intros ; simpl .rewrite (Udistr_plus_left [1/2] _ _ (H true )).rewrite (Udistr_plus_left [1/2] _ _ (H false )).repeat norm_assoc_right .apply Uplus_eq_compat_right .repeat norm_assoc_left ; apply Uplus_eq_compat_left ; auto .Save .Lemma flip_stable_mult : stable_mult flip .unfold flip , stable_mult , fmult ; intros ; simpl .assert (H :([1/2]* f true ) <= ([1-] ([1/2]* f false ))); auto .rewrite (Udistr_plus_left k _ _ H ); auto .Save .Lemma flip_continuous : continuous flip .unfold continuous ; intros .unfold flip ; rewrite mon_simpl .repeat rewrite MF_lub_simpl .do 2 rewrite <- lub_eq_mult .rewrite <- lub_eq_plus .apply lub_le_compat ; intro n ; auto .Save .Definition ctrue : MF bool := fun (b :bool) => if b then 1 else 0.Definition cfalse : MF bool := fun (b :bool) => if b then 0 else 1.Lemma flip_ctrue : flip ctrue == [1/2].unfold flip , ctrue ; simpl .setoid_rewrite (Umult_one_right [1/2]).setoid_rewrite (Umult_zero_right [1/2]); auto .Save .Lemma flip_cfalse : flip cfalse == [1/2].unfold flip , cfalse ; simpl .setoid_rewrite (Umult_one_right [1/2]).setoid_rewrite (Umult_zero_right [1/2]); auto .Save .Hint Resolve flip_ctrue flip_cfalse .Definition Flip : distr bool .exists flip .apply flip_stable_inv .apply flip_stable_plus .apply flip_stable_mult .apply flip_continuous .Defined .Lemma Flip_simpl : forall f , mu Flip f = [1/2] * (f true ) + [1/2] * (f false ).trivial .Save .
Uniform distribution beween 0 and n
Require Arith .
Definition of fnth
fnth n k is defined as
Definition fnth (n :nat) : nat -> U := fun k => [1/]1+n.
Lemma Unth_eq : forall n , Unth n == [1-] (sigma (fnth n ) n ).intro ; exact (Unth_prop n ).Save .Hint Resolve Unth_eq .Lemma sigma_fnth_one : forall n , sigma (fnth n ) (S n ) == 1.intros ; rewrite sigma_S .unfold fnth at 1.rewrite (Unth_eq n ); auto .Save .Hint Resolve sigma_fnth_one .Lemma Unth_inv_eq : forall n , [1-] ([1/]1+n) == sigma (fnth n ) n .intro ; setoid_rewrite (Unth_eq n ); auto .Save .Lemma sigma_fnth_sup : forall n m , (m > n ) -> sigma (fnth n ) m == sigma (fnth n ) (S n ).intros .assert ((S n ) <= m )%nat; auto with arith .elim H0 ; intros ; auto .rewrite sigma_S ; auto .setoid_rewrite H2 .assert (m0 > n ); auto with arith .setoid_rewrite (sigma_fnth_one n ); auto .Save .Lemma sigma_fnth_le : forall n m , (sigma (fnth n ) m ) <= (sigma (fnth n ) (S n )).intros ; setoid_rewrite (sigma_fnth_one n ); auto .Save .Hint Resolve sigma_fnth_le .
fnth
Lemma fnth_retract : forall n :nat,(retract (fnth n ) (S n )).red ; intros .unfold fnth at 1.transitivity ([1-] (sigma (fnth n ) n )); auto with arith .Save .Implicit Arguments fnth_retract [].
distributions and general summations
Definition sigma_fun A (f :nat -> MF A ) (n :nat) : MF A := fun x => sigma (fun k => f k x ) n .Definition serie_fun A (f :nat -> MF A ) : MF A := fun x => serie (fun k => f k x ).Definition Sigma_fun A (f :nat -> MF A ) : nat -m> MF A := ishift (fun x => Sigma (fun k => f k x )).Lemma Sigma_fun_simpl : forall A (f :nat -> MF A ) (n :nat),Sigma_fun f n = sigma_fun f n .trivial .Save .Lemma serie_fun_lub_sigma_fun : forall A (f :nat -> MF A ), serie_fun f == lub (Sigma_fun f ).intros ; unfold serie_fun .simpl ; intro x .unfold serie ; apply lub_eq_compat .simpl ; intro n ; trivial .Save .Hint Resolve serie_fun_lub_sigma_fun .Lemma sigma_fun_0 : forall A (f :nat -> MF A ), sigma_fun f 0 == fzero A .intros ; unfold fzero ; intro x ; auto .Save .Lemma sigma_fun_S : forall A (f :nat->MF A ) (n :nat), sigma_fun f (S n ) == fplus (f n ) (sigma_fun f n ).intros ; unfold fplus ; intro x ; auto .Save .Lemma mu_sigma_le : forall A (d :distr A ) (f :nat -> MF A ) (n :nat), mu d (sigma_fun f n ) <= sigma (fun k => mu d (f k )) n .induction n ; intros .rewrite sigma_0 ; transitivity (mu d (fzero A )); auto .rewrite sigma_S ; transitivity (mu d (fplus (f n ) (sigma_fun f n ))).auto .transitivity (mu d (f n ) + mu d (sigma_fun f n )); auto .Save .Lemma retract_fplusok : forall A (f :nat -> MF A ) (n :nat), forall x , retract (fun k => f k x ) n ) -> forall k , (k < n )%nat -> fplusok (f k ) (sigma_fun f k ).red ; intros ; intro x .apply (H x k ); auto .Save .Lemma mu_sigma_eq : forall A (d :distr A ) (f :nat -> MF A ) (n :nat), forall x , retract (fun k => f k x ) n ) -> mu d (sigma_fun f n ) == sigma (fun k => mu d (f k )) n .induction n ; intros .rewrite sigma_0 ; transitivity (mu d (fzero A )); auto .rewrite sigma_S ; transitivity (mu d (fplus (f n ) (sigma_fun f n ))).auto .transitivity (mu d (f n ) + mu d (sigma_fun f n )); auto .apply (mu_stable_plus d ).apply retract_fplusok with (S n ); auto .Save .Lemma mu_serie_le : forall A (d :distr A ) (f :nat -> MF A ), mu d (serie_fun f ) <= serie (fun k => mu d (f k )).intros ; transitivity (mu d (lub (Sigma_fun f ))).apply mu_monotonic ; rewrite (serie_fun_lub_sigma_fun f ); trivial .transitivity (lub (mu d @ (Sigma_fun f ))).apply (mu_continuous d ).unfold serie ; apply lub_le_compat ; intro n .rewrite comp_simpl .exact (mu_sigma_le d f n ).Save .Lemma mu_serie_eq : forall A (d :distr A ) (f :nat -> MF A ), forall x , wretract (fun k => f k x )) -> mu d (serie_fun f ) == serie (fun k => mu d (f k )).intros ; transitivity (mu d (lub (Sigma_fun f ))).apply mu_stable_eq ; apply (serie_fun_lub_sigma_fun f ); trivial .transitivity (lub (mu d @ (Sigma_fun f ))).apply (lub_comp_eq (mu d )).apply (mu_continuous d ).unfold serie ; apply lub_eq_compat ; intro n .rewrite comp_simpl .apply (mu_sigma_eq d f (n :=n)).red ; intros .apply (H x k ).Save .Lemma wretract_fplusok : forall A (f :nat -> MF A ), forall x , wretract (fun k => f k x )) -> forall k , fplusok (f k ) (sigma_fun f k ).red ; intros ; intro x .apply (H x k ); auto .Save .
Instance discrete_mon : forall A (c : nat -> U ) (p : nat -> A ),monotonic (fun f : A ->U => serie (fun k => c k * f (p k ))).red ; intros ; auto .Save .Definition discrete A (c : nat -> U ) (p : nat -> A ) : M A := mon (fun f : A ->U => serie (fun k => c k * f (p k ))).Lemma discrete_simpl : forall A (c : nat -> U ) (p : nat -> A ) f , discrete c p f = serie (fun k => c k * f (p k )).trivial .Save .Lemma discrete_stable_inv : forall A (c : nat -> U ) (p : nat -> A ), wretract c -> stable_inv (discrete c p ).red ; intros .repeat rewrite discrete_simpl .unfold finv ; apply serie_inv_le ; auto .Save .Lemma discrete_stable_plus : forall A (c : nat -> U ) (p : nat -> A ), stable_plus (discrete c p ).red ; intros .repeat rewrite discrete_simpl .transitivity (serie (fun k : nat => c k * f (p k ) + c k * g (p k ))).apply serie_eq_compat .intros ; unfold fplus .apply Udistr_plus_left .apply (H (p k )).apply serie_plus ; auto .Save .Lemma discrete_stable_mult : forall A (c : nat -> U ) (p : nat -> A ), wretract c -> stable_mult (discrete c p ).red ; intros .repeat rewrite discrete_simpl ; unfold fmult .transitivity (serie (fun k0 : nat => k * (c k0 * f (p k0 )))); auto .apply serie_mult .apply wretract_le with c ; auto .Save .Lemma discrete_continuous : forall A (c : nat -> U ) (p : nat -> A ), continuous (discrete c p ).red ; intros .rewrite discrete_simpl .transitivity serie (lub (ishift (fun k => (UMult (c k ) @ (h <o> (p k ))))))).apply serie_le_compat ; intro k .repeat rewrite MF_lub_simpl .rewrite (UMult_continuous_right (c k ) (h <o> p k )).unfold MFcpo ; rewrite fcpo_lub_simpl .apply lub_le_compat ; intro n ; simpl ; auto .rewrite (serie_continuous (ishift (fun k : nat => UMult (c k ) @ h <o> p k ))).apply lub_le_compat ; intro n ; simpl ; auto .Save .Record discr (A :Type) : Type := coeff : nat -> U ; coeff_retr : wretract coeff ; points : nat -> A }.Hint Resolve coeff_retr .Definition Discrete : forall A , discr A -> distr A .intros A d ; exists (discrete (coeff d ) (points d )).apply discrete_stable_inv ; trivial .apply discrete_stable_plus .apply discrete_stable_mult ; trivial .apply discrete_continuous .Defined .Lemma Discrete_simpl : forall A (d :discr A ), mu (Discrete d ) = discrete (coeff d ) (points d ).trivial .Save .Definition is_discrete (A :Type) (m : distr A ) := exists d : discr A , m == Discrete d .
distribution for random n
The distribution associated to
random n is
we cannot factorize because of possible overflow
Instance random_mon : forall n , monotonic (fun (f :MF nat ) => sigma (fun k => Unth n * f k ) (S n )).red ; intros ; auto .Save .Definition random (n :nat):M nat := mon (fun (f :MF nat ) => sigma (fun k => Unth n * f k ) (S n )).Lemma random_simpl : forall n (f : MF nat ), random n f = sigma (fun k => Unth n * f k ) (S n ).trivial .Save .
Lemma random_stable_inv : forall n , stable_inv (random n ).unfold random , stable_inv , finv ; intros ; simpl .rewrite (sigma_inv f (fnth_retract n )); auto .Save .Lemma random_stable_plus : forall n , stable_plus (random n ).unfold stable_plus , fplus ; intros .repeat (rewrite random_simpl ).unfold fplusok , finv in H .transitivity sigma (fun k : nat => ([1/]1+n * f k ) + ([1/]1+n * g k )) (S n )).apply sigma_eq_compat ; intros .assert (f k <= [1-] (g k )); auto .apply sigma_plus with (f :=fun k : nat => Unth n * f k )g :=fun k : nat => Unth n * g k ); auto .Save .Lemma random_stable_mult : forall n , stable_mult (random n ).unfold stable_mult , fmult ; intros ; repeat (rewrite random_simpl ).transitivity sigma (fun l : nat => k * ([1/]1+n * f l )) (S n )).apply sigma_eq_compat ; intros ; auto .apply sigma_mult with (f :=fun k : nat => Unth n * f k ).red ; intros .transitivity ([1/]1+n); auto .transitivity ([1-] (sigma (fun k1 => Unth n ) k0 )); auto .apply (fnth_retract n k0 ); auto .Save .Lemma random_continuous : forall n , continuous (random n ).unfold continuous ; intros ; rewrite random_simpl .transitivity sigma (fun k : nat => lub (UMult ([1/]1+n) @ (h <o> k ))) (S n )).apply sigma_le_compat ; intros ; simpl .rewrite (lub_eq_mult ([1/]1+n) (h <o> k )); auto .transitivity sigma (lub (D :=MF nat ) (ishift (fun k : nat => (UMult ([1/]1+n) @ (h <o> k ))))) (S n )).apply sigma_le_compat ; intros ; auto .rewrite MF_lub_simpl .apply lub_le_compat ; intro m ; simpl ; auto .rewrite sigma_lub1 ; auto .apply lub_le_compat ; intro m ; simpl ; auto .Save .Definition Random (n :nat) : distr nat .intro n ; exists (random n ).apply random_stable_inv .apply random_stable_plus .apply random_stable_mult .apply random_continuous .Defined .Lemma Random_simpl : forall (n :nat), mu (Random n ) = random n .trivial .Save .Lemma Random_total : forall n : nat , mu (Random n ) (fone nat ) == 1.intros ; simpl mu ;unfold fone ; rewrite random_simpl .transitivity (sigma (fnth n ) (S n )).apply sigma_eq_compat .intros ; repeat Usimpl ; auto .auto .Save .Hint Resolve Random_total .Lemma Random_inv : forall f n , mu (Random n ) (finv f ) == [1-] (mu (Random n ) f ).intros ; apply mu_one_inv ; auto .Save .Hint Resolve Random_inv .
Ltac mu_plus d := match goal with context [fmont (mu d ) (fun x => (Uplus (@?f x ) (@?g x )))] => rewrite (mu_stable_plus d (f :=f) (g :=g))end .Ltac mu_mult d := match goal with context [fmont (mu d ) (fun x => (Umult ?k (@?f x )))] => rewrite (mu_stable_mult d k f )end .
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F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
Y (module)
Ycart [in
Ycart ]
html/index_constructor_D.html 0000644 0001750 0001750 00000022476 11301735615 016614 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_axiom_F.html 0000644 0001750 0001750 00000022476 11301735615 015346 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_definition_O.html 0000644 0001750 0001750 00000022702 11301735615 016362 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
O (definition)
orc [in
Misc ]
html/index_record_U.html 0000644 0001750 0001750 00000022717 11301735615 015524 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
U (record)
Univ_prop.IU [in
Uprop ]
html/index_variable_Q.html 0000644 0001750 0001750 00000022476 11301735615 016031 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_constructor_Y.html 0000644 0001750 0001750 00000022476 11301735615 016641 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_inductive_Q.html 0000644 0001750 0001750 00000022476 11301735615 016236 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_record_W.html 0000644 0001750 0001750 00000022476 11301735615 015530 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_library_V.html 0000644 0001750 0001750 00000022476 11301735615 015715 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_section_W.html 0000644 0001750 0001750 00000022476 11301735615 015716 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_projection_U.html 0000644 0001750 0001750 00000023231 11301735615 016412 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
U (projection)
Univ_prop.low [in
Uprop ]
Univ_prop.proper [in
Uprop ]
Univ_prop.up [in
Uprop ]
html/index_module_R.html 0000644 0001750 0001750 00000023366 11301735615 015531 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
R (module)
RP [in
Bernoulli ]
RP [in
Cover ]
RP [in
Choice ]
RP [in
IterFlip ]
Rules [in
Prog ]
html/index_definition_S.html 0000644 0001750 0001750 00000023531 11301735615 016367 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
S (definition)
seq_lift_left [in
Ccpo ]
seq_lift_right [in
Ccpo ]
set [in
Sets ]
shift [in
Ccpo ]
singl [in
Sets ]
size [in
Sets ]
html/index_global_Z.html 0000644 0001750 0001750 00000022476 11301735615 015515 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_section_H.html 0000644 0001750 0001750 00000022476 11301735615 015677 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_axiom_D.html 0000644 0001750 0001750 00000022476 11301735615 015344 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
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Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
S (instance)
shift_fun_mon [in
Ccpo ]
shift_mon2 [in
Ccpo ]
shift_mon_fun [in
Ccpo ]
stable_morphism [in
Ccpo ]
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Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
B (library)
Bernoulli
html/index_record_F.html 0000644 0001750 0001750 00000023007 11301735615 015476 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
F (record)
fcont [in
Ccpo ]
fmon [in
Ccpo ]
html/Prog.html 0000644 0001750 0001750 00001016254 11301735614 013501 0 ustar cpaulin cpaulin
Prog
Library Prog
Prog.v: Composition of distributions
Require Export Probas .
Set Implicit Arguments .
Definition Mif (
A :Type) (
b :distr
bool ) (
m1 m2 :
distr A )
:=
Mlet b (
fun x :bool =>
if x then m1 else m2 ).
Lemma Mif_le_compat :
forall (
A :Type) (
b1 b2 :distr
bool ) (
m1 m2 n1 n2 :
distr A ),
b1 <=
b2 ->
m1 <=
m2 ->
n1 <=
n2 ->
Mif b1 m1 n1 <=
Mif b2 m2 n2 .
intros ;
unfold Mif .
apply Mlet_le_compat ;
auto .
intro x ;
destruct x ;
auto .
Save .
Hint Resolve Mif_le_compat .
Instance Mif_mon2 :
forall (
A :Type)
b ,
monotonic2 (
Mif (
A :=A)
b ).
red ;
auto .
Save .
Definition MIf :
forall (
A :Type),
distr bool -m>
distr A -m>
distr A -m>
distr A .
intro A ;
exists (
fun b =>
mon2 (
Mif (
A :=A)
b )).
red ;
simpl ;
intros ;
auto .
Defined .
The distribution associated to pchoice p m1 m2
Definition pchoice :
forall A ,
U ->
M A ->
M A ->
M A .
intros A p m1 m2 ;
exists (
fun (
f :
A ->
U ) =>
p *
m1 f + [1-]
p *
m2 f ).
red ;
intros ;
simpl .
apply Uplus_le_compat ;
auto .
Defined .
Lemma pchoice_simpl :
forall A p (
m1 m2 :M
A )
f ,
pchoice p m1 m2 f =
p *
m1 f + [1-]
p *
m2 f .
trivial .
Save .
Definition Mchoice (
A :Type) (
p :U) (
m1 m2 :
distr A ) :
distr A .
intros A p m1 m2 ;
exists (
pchoice p (
mu m1 ) (
mu m2 )).
red ;
intros ;
repeat (
rewrite pchoice_simpl ).
transitivity
(
p * [1-]
mu m1 f + [1-]
p * [1-]
mu m2 f ).
2:auto.
apply Uplus_le_compat ;
repeat Usimpl ;
auto .
red ;
intros ;
repeat (
rewrite pchoice_simpl ).
rewrite (
mu_stable_plus m1 H );
rewrite (
mu_stable_plus m2 H ).
rewrite Udistr_plus_left .
2:auto.
rewrite Udistr_plus_left .
2:auto.
repeat norm_assoc_left ;
repeat Usimpl .
repeat norm_assoc_right ;
repeat Usimpl ;
auto .
red ;
intros ;
repeat (
rewrite pchoice_simpl ).
rewrite (
mu_stable_mult m1 k f );
rewrite (
mu_stable_mult m2 k f ).
rewrite Udistr_plus_left ;
auto .
red ;
intros ;
rewrite pchoice_simpl .
transitivity
(
p *
lub (
mu m1 @
h ) + [1-]
p *
lub (
mu m2 @
h )).
apply Uplus_le_compat ;
repeat Usimpl ;
auto .
repeat rewrite <-
lub_eq_mult .
rewrite <-
lub_eq_plus ;
auto .
apply lub_le_compat ;
intro n ;
auto .
Defined .
Lemma Mchoice_simpl :
forall A p (
m1 m2 :distr
A )
f ,
mu (
Mchoice p m1 m2 )
f =
p *
mu m1 f + [1-]
p *
mu m2 f .
trivial .
Save .
Lemma Mchoice_le_compat :
forall (
A :Type) (
p :U) (
m1 m2 n1 n2 :
distr A ),
m1 <=m2 ->
n1 <=n2 ->
Mchoice p m1 n1 <=
Mchoice p m2 n2 .
simpl ;
intros .
apply Uplus_le_compat ;
repeat Usimpl ;
auto .
Save .
Hint Resolve Mchoice_le_compat .
Add Parametric Morphism (
A :Type) : (
Mchoice (
A :=A))
with signature Oeq &
Oeq &
Oeq &
Oeq
as Mchoice_eq_compat .
intros ;
intro f .
simpl .
rewrite H ;
apply Uplus_eq_compat ;
repeat Usimpl ;
auto .
Save .
Hint Immediate Mchoice_eq_compat .
Instance Mchoice_mon2 :
forall (
A :Type) (
p :U),
monotonic2 (
Mchoice (
A :=A)
p ).
red ;
auto .
Save .
Definition MChoice A (
p :U) :
distr A -m>
distr A -m>
distr A :=
mon2 (
Mchoice (
A :=A)
p ).
Lemma MChoice_simpl :
forall A (
p :U) (
m1 m2 :
distr A ),
MChoice A p m1 m2 =
Mchoice p m1 m2 .
trivial .
Save .
Lemma Mchoice_sym_le :
forall (
A :Type) (
p :U) (
m1 m2 :
distr A ),
Mchoice p m1 m2 <=
Mchoice ([1-]
p )
m2 m1 .
simpl ;
intros .
rewrite Uplus_sym ;
repeat Usimpl ;
auto .
Save .
Hint Resolve Mchoice_sym_le .
Lemma Mchoice_sym :
forall (
A :Type) (
p :U) (
m1 m2 :
distr A ),
Mchoice p m1 m2 ==
Mchoice ([1-]
p )
m2 m1 .
intros ;
apply Ole_antisym ;
auto .
transitivity (
Mchoice ([1-][1-]
p )
m1 m2 );
auto .
Save .
Lemma Mchoice_continuous_right
:
forall (
A :Type) (
p :U) (
m :
distr A ),
continuous (
D1 :=distr
A ) (
D2 :=distr
A ) (
MChoice A p m ).
red ;
intros ;
intro f .
rewrite (
MChoice_simpl p m ).
rewrite (
Mchoice_simpl p m ).
simpl mu at 2.
rewrite distr_lub_simpl ;
rewrite mon_simpl .
rewrite <- (
lub_eq_mult ([1-]
p )).
rewrite <-
lub_eq_plus_cte_left .
apply lub_le_compat ;
intro n ;
auto .
Save .
Hint Resolve Mchoice_continuous_right .
Lemma Mchoice_continuous_left :
forall (
A :Type) (
p :U),
continuous (
D1 :=distr
A ) (
D2 :=distr
A -m>
distr A ) (
MChoice A p ).
red ;
intros ;
intro m .
rewrite (
MChoice_simpl (
A :=A)
p ).
rewrite Mchoice_sym .
transitivity (
lub ((
MChoice A ([1-]
p )
m )@
h )).
apply (
Mchoice_continuous_right ([1-]
p )
m h ).
rewrite fmon_lub_simpl .
apply lub_le_compat ;
intro f ;
simpl ;
auto .
Save .
Lemma Mchoice_continuous :
forall (
A :Type) (
p :U),
continuous2 (
D1 :=distr
A ) (
D2 :=distr
A ) (
D3 :=distr
A ) (
MChoice A p ).
intros ;
apply continuous_continuous2 ;
intros .
apply (
Mchoice_continuous_right (
A :=A)
p ).
apply (
Mchoice_continuous_left (
A :=A)
p ).
Save .
Definition im_distr (
A B :
Type ) (
f :A ->
B ) (
m :distr
A ) :
distr B :=
Mlet m (
fun a =>
Munit (
f a )).
Lemma im_distr_simpl :
forall A B (
f :A ->
B ) (
m :distr
A )(
h :B ->
U ),
mu (
im_distr f m )
h =
mu m (
fun a =>
h (
f a )).
trivial .
Save .
Add Parametric Morphism (
A B :
Type ) : (
im_distr (
A :=A) (
B :=B))
with signature (
feq (
A :=A) (
B :=B)) &
Oeq &
Oeq
as im_distr_eq_compat .
intros ;
intro h ;
simpl .
apply mu_eq_compat ;
trivial .
intro n ;
rewrite (
H n );
auto .
Save .
Lemma im_distr_comp :
forall A B C (
f :A ->
B ) (
g :B ->
C ) (
m :distr
A ),
im_distr g (
im_distr f m ) ==
im_distr (
fun a =>
g (
f a ))
m .
intros ;
intro h ;
simpl ;
auto .
Save .
Lemma im_distr_id :
forall A (
f :A ->
A ) (
m :distr
A ), (
forall x ,
f x =
x ) ->
im_distr f m ==
m .
intros ;
intro h ;
simpl .
apply (
mu_stable_eq m );
intro x ;
rewrite H ;
auto .
Save .
Definition prod_distr (
A B :
Type )(
d1 :distr
A )(
d2 :distr
B ) :
distr (
A *B) :=
Mlet d1 (
fun x =>
Mlet d2 (
fun y =>
Munit (
x ,y))).
Add Parametric Morphism (
A B :
Type ) : (
prod_distr (
A :=A) (
B :=B))
with signature Ole ++>
Ole ++>
Ole
as prod_distr_le_compat .
intros ;
unfold prod_distr ;
auto .
Save .
Hint Resolve prod_distr_le_compat .
Add Parametric Morphism (
A B :
Type ) : (
prod_distr (
A :=A) (
B :=B))
with signature Oeq &
Oeq &
Oeq
as prod_distr_eq_compat .
intros ;
apply Ole_antisym ;
auto .
Save .
Hint Immediate prod_distr_eq_compat .
Instance prod_distr_mon2 :
forall (
A B :Type),
monotonic2 (
prod_distr (
A :=A) (
B :=B)).
red ;
auto .
Save .
Definition Prod_distr (
A B :Type):
distr A -m>
distr B -m>
distr (
A *B) :=
mon2 (
prod_distr (
A :=A) (
B :=B)).
Lemma Prod_distr_simpl :
forall (
A B :Type)(d1:
distr A ) (
d2 :distr
B ),
Prod_distr A B d1 d2 =
prod_distr d1 d2 .
trivial .
Save .
Lemma prod_distr_rect :
forall (
A B :
Type ) (
d1 :
distr A ) (
d2 :distr
B ) (
f :A ->
U )(
g :B ->
U ),
mu (
prod_distr d1 d2 ) (
fun xy =>
f (
fst xy ) *
g (
snd xy )) ==
mu d1 f *
mu d2 g .
intros ;
unfold prod_distr ;
simpl .
transitivity (
mu d1 (
fun x :
A =>
mu d2 g *
f x )).
apply (
mu_stable_eq d1 );
intro x .
rewrite (
mu_stable_mult d2 (
f x )
g );
auto .
rewrite (
mu_stable_mult d1 (
mu d2 g )
f );
auto .
Save .
Lemma prod_distr_fst :
forall (
A B :
Type ) (
d1 :
distr A ) (
d2 :distr
B ) (
f :A ->
U ),
mu (
prod_distr d1 d2 ) (
fun xy =>
f (
fst xy )) ==
mu d2 (
fone B ) *
mu d1 f .
intros ;
rewrite Umult_sym ;
rewrite <-
prod_distr_rect .
apply (
mu_stable_eq (
prod_distr d1 d2 )).
intro xy ;
unfold fone ;
auto .
Save .
Lemma prod_distr_snd :
forall (
A B :
Type ) (
d1 :
distr A ) (
d2 :distr
B ) (
g :B ->
U ),
mu (
prod_distr d1 d2 ) (
fun xy =>
g (
snd xy )) ==
mu d1 (
fone A ) *
mu d2 g .
intros ;
rewrite <-
prod_distr_rect .
apply (
mu_stable_eq (
prod_distr d1 d2 )).
intro xy ;
unfold fone ;
auto .
Save .
Lemma prod_distr_fst_eq :
forall (
A B :
Type ) (
d1 :
distr A ) (
d2 :distr
B ),
mu d2 (
fone B ) == 1 ->
im_distr (
fst (
A :=A) (
B :=B)) (
prod_distr d1 d2 ) ==
d1 .
intros ;
intro g .
rewrite im_distr_simpl .
rewrite prod_distr_fst ;
auto .
Save .
Lemma prod_distr_snd_eq :
forall (
A B :
Type ) (
d1 :
distr A ) (
d2 :distr
B ),
mu d1 (
fone A ) == 1 ->
im_distr (
snd (
A :=A) (
B :=B)) (
prod_distr d1 d2 ) ==
d2 .
intros ;
intro g .
rewrite im_distr_simpl .
rewrite prod_distr_snd ;
auto .
Save .
Definition swap A B (
x :A*B) :
B *
A := (
snd x ,fst
x ).
Definition arg_swap A B (
f :
MF (
A *B)) :
MF (
B *A) :=
fun z =>
f (
swap z ).
Definition Arg_swap A B :
MF (
A *B) -m>
MF (
B *A).
intros ;
exists (
arg_swap (
A :=A) (
B :=B)).
red ;
unfold arg_swap ,swap;
auto .
Defined .
Lemma Arg_swap_simpl :
forall A B f ,
Arg_swap A B f =
arg_swap f .
trivial .
Save .
Definition prod_distr_com A B (
d1 :
distr A ) (
d2 :distr
B ) (
f :
MF (
A *
B )) :=
mu (
prod_distr d1 d2 )
f ==
mu (
prod_distr d2 d1 ) (
arg_swap f ).
Lemma prod_distr_com_eq_compat :
forall A B (
d1 :
distr A ) (
d2 :distr
B ) (
f g :
MF (
A *
B )),
f ==
g ->
prod_distr_com d1 d2 f ->
prod_distr_com d1 d2 g .
unfold prod_distr_com ;
intros ;
transitivity (
mu (
prod_distr d1 d2 )
f ).
apply mu_stable_eq ;
auto .
transitivity (
mu (
prod_distr d2 d1 ) (
arg_swap f ));
auto .
apply mu_stable_eq ;
unfold arg_swap ,swap;
intro z .
apply (
H (
snd z ,fst
z )).
Save .
Lemma prod_distr_com_rect :
forall (
A B :
Type ) (
d1 :
distr A ) (
d2 :distr
B ) (
f :A ->
U )(
g :B ->
U ),
prod_distr_com d1 d2 (
fun xy =>
f (
fst xy ) *
g (
snd xy )).
red ;
intros .
rewrite prod_distr_rect .
transitivity (
mu (
prod_distr d2 d1 ) (
fun yx =>
g (
fst yx ) *
f (
snd yx ))).
rewrite prod_distr_rect ;
auto .
apply mu_stable_eq ;
intro yx ;
unfold arg_swap ;
auto .
Save .
Lemma prod_distr_com_cte :
forall (
A B :
Type ) (
d1 :
distr A ) (
d2 :distr
B ) (
c :U),
prod_distr_com d1 d2 (
fcte (
A *B)
c ).
intros ;
apply prod_distr_com_eq_compat with (
fun z =>
fcte A c (
fst z ) *
fone B (
snd z )).
intro z ;
auto .
apply prod_distr_com_rect .
Save .
Lemma prod_distr_com_one :
forall (
A B :
Type ) (
d1 :
distr A ) (
d2 :distr
B ),
prod_distr_com d1 d2 (
fone (
A *B)).
intros ;
exact (
prod_distr_com_cte d1 d2 1).
Save .
Lemma prod_distr_com_plus :
forall (
A B :
Type ) (
d1 :
distr A ) (
d2 :distr
B ) (
f g :MF (
A *B)),
fplusok f g ->
prod_distr_com d1 d2 f ->
prod_distr_com d1 d2 g ->
prod_distr_com d1 d2 (
fplus f g ).
unfold prod_distr_com ;
intros .
rewrite (
mu_stable_plus (
prod_distr d1 d2 ) (
f :=f) (
g :=g)
H ).
rewrite H0 ;
rewrite H1 .
rewrite <- (
mu_stable_plus (
prod_distr d2 d1 ) (
f :=arg_swap
f ) (
g :=arg_swap
g ));
auto .
red ;
unfold arg_swap ,swap,finv;
simpl ;
intro z ;
apply (
H (
snd z ,fst
z )).
Save .
Lemma prod_distr_com_mult :
forall (
A B :
Type ) (
d1 :
distr A ) (
d2 :distr
B ) (
k :U)(f:MF (
A *B)),
prod_distr_com d1 d2 f ->
prod_distr_com d1 d2 (
fmult k f ).
unfold prod_distr_com ;
intros .
rewrite (
mu_stable_mult (
prod_distr d1 d2 )
k f ).
rewrite H .
rewrite <- (
mu_stable_mult (
prod_distr d2 d1 )
k (
arg_swap f ));
auto .
Save .
Lemma prod_distr_com_inv :
forall (
A B :
Type ) (
d1 :
distr A ) (
d2 :distr
B ) (
f :MF (
A *B)),
prod_distr_com d1 d2 f ->
prod_distr_com d1 d2 (
finv f ).
unfold prod_distr_com ;
intros .
rewrite (
mu_inv_minus (
prod_distr d1 d2 )
f ).
rewrite H .
assert (
H1 :=prod_distr_com_one
d1 d2 );
red in H1 .
rewrite H1 .
rewrite <- (
mu_inv_minus (
prod_distr d2 d1 ) (
arg_swap f ));
auto .
Save .
Lemma prod_distr_com_lub :
forall (
A B :
Type ) (
d1 :
distr A ) (
d2 :distr
B ) (
f :nat -m>
MF (
A *B)),
(
forall n ,
prod_distr_com d1 d2 (
f n )) ->
prod_distr_com d1 d2 (
lub f ).
unfold prod_distr_com ;
intros .
rewrite (
lub_comp_eq (
mu (
prod_distr d1 d2 ))
f );
trivial .
transitivity (
lub (
mu (
prod_distr d2 d1 ) @ (
Arg_swap A B @
f ))).
apply lub_eq_compat ;
intro n .
repeat rewrite comp_simpl .
rewrite H ;
auto .
rewrite <- (
lub_comp_eq (
mu (
prod_distr d2 d1 )));
auto .
apply mu_stable_eq ;
intro x .
unfold arg_swap ;
repeat rewrite MF_lub_simpl .
apply lub_eq_compat ;
intro n ;
auto .
Save .
Lemma prod_distr_com_refl :
forall A B (
d1 :distr
A ) (
d2 :distr
B ) (
f :MF (
A *B)),
prod_distr_com d1 d2 f ->
prod_distr_com d2 d1 (
arg_swap f ).
red ;
intros .
apply Oeq_sym ;
transitivity (
mu (
prod_distr d1 d2 )
f );
auto .
Save .
Lemma discrete_commute :
forall A B (
d1 :distr
A ) (
d2 :distr
B ) (
f :MF (
A *B)),
is_discrete d1 ->
prod_distr_com d1 d2 f .
red ;
intros A B d1 d2 f ((
cf ,cfr,pt),H).
unfold arg_swap ;
unfold swap ;
simpl .
transitivity
(
mu d2 (
fun x :
B =>
discrete cf pt (
fun x0 :
A =>
f (
x0 ,
x )))).
unfold discrete .
rewrite (
mu_serie_eq d2 (
fun k x =>
cf k *
f (
pt k ,
x ))).
transitivity (
discrete cf pt (
fun x :
A =>
mu d2 (
fun x0 :
B =>
f (
x ,
x0 ))));
auto .
rewrite discrete_simpl ;
apply serie_eq_compat ;
intro k .
apply Oeq_sym ;
apply (
mu_stable_mult d2 (
cf k ) (
fun x0 :
B =>
f (
pt k ,
x0 ))).
intros ;
apply wretract_le with cf ;
auto .
apply (
mu_stable_eq d2 );
intro x .
transitivity (
mu (
Discrete (
Build_discr cfr pt )) (
fun x0 :
A =>
f (
x0 ,
x )));
auto .
Save .
Definition fst_distr A B (
m :
distr (
A *B)) :
distr A :=
im_distr (
fst (
B :=B))
m .
Definition snd_distr A B (
m :
distr (
A *B)) :
distr B :=
im_distr (
snd (
B :=B))
m .
Definition range A (
P :A ->
Prop ) (
d :
distr A ) :=
forall f , (
forall x ,
P x -> 0 ==
f x ) -> 0 ==
mu d f .
Lemma range_le :
forall A (
P :A ->
Prop ) (
d :distr
A ),
range P d ->
forall f g , (
forall a ,
P a ->
f a <=
g a ) ->
mu d f <=
mu d g .
intros ;
assert (
mu d f -
mu d g <= 0).
transitivity (
mu d (
fun x =>
f x -
g x )).
apply mu_stable_le_minus .
apply Oeq_le ;
apply Oeq_sym ;
apply H ;
intros .
rewrite Uminus_le_zero ;
auto .
apply Uminus_zero_le ;
apply Ule_zero_eq ;
exact H1 .
Save .
Lemma range_eq :
forall A (
P :A ->
Prop ) (
d :distr
A ),
range P d ->
forall f g , (
forall a ,
P a ->
f a ==
g a ) ->
mu d f ==
mu d g .
red ;
intros ;
apply Ole_antisym ;
apply (
range_le (
P :=P));
intros ;
auto .
apply Oeq_le ;
apply Oeq_sym ;
auto .
Save .
Prog.v: Axiomatic semantics
Definition of correctness judgements
is defined as
is defined as
Definition ok (A :Type) (p :U) (e :distr A ) (q :A -> U ) := p <= mu e q .Definition okfun (A B :Type)(p:A -> U )(e :A -> distr B )(q :A -> B -> U )forall x :A, ok (p x ) (e x ) (q x ).Definition okup (A :Type) (p :U) (e :distr A ) (q :A -> U ) := mu e q <= p .Definition upfun (A B :Type)(p:A -> U )(e :A -> distr B )(q :A -> B -> U )forall x :A, okup (p x ) (e x ) (q x ).
Lemma ok_le_compat :
forall (
A :Type) (
p p' :U) (
e :distr
A ) (
q q' :A ->
U ),
p' <=
p ->
q <=
q' ->
ok p e q ->
ok p' e q' .
unfold ok ;
intros .
transitivity p ;
auto .
transitivity (
mu e q );
auto .
Save .
Lemma ok_eq_compat :
forall (
A :Type) (
p p' :U) (
e e' :distr
A ) (
q q' :A ->
U ),
p' ==
p ->
q ==
q' ->
e ==
e' ->
ok p e q ->
ok p' e' q' .
unfold ok ;
intros .
transitivity p ;
auto .
transitivity (
mu e q );
auto .
Save .
Lemma okfun_le_compat :
forall (
A B :Type) (
p p' :A ->
U ) (
e :A ->
distr B ) (
q q' :A ->
B ->
U ),
p' <=
p ->
q <=
q' ->
okfun p e q ->
okfun p' e q' .
unfold okfun ;
intros .
apply ok_le_compat with (
p x ) (
q x );
auto .
Save .
Lemma ok_mult :
forall (
A :Type)(k
p :U)(e:distr
A )(
f :
A ->
U ),
ok p e f ->
ok (
k *p)
e (
fmult k f ).
unfold ok ;
intros A k p e f oka .
rewrite (
mu_stable_mult e k f ).
apply Umult_le_compat_right ;
trivial .
Save .
Lemma ok_inv :
forall (
A :Type)(p:U)(e:distr
A )(
f :
A ->
U ),
ok p e f ->
mu e (
finv f ) <= [1-]
p .
unfold ok ;
intros A p e f oka .
transitivity ([1-](
mu e f ));
auto .
Save .
Lemma okup_le_compat :
forall (
A :Type) (
p p' :U) (
e :distr
A ) (
q q' :A ->
U ),
p <=
p' ->
q' <=
q ->
okup p e q ->
okup p' e q' .
unfold okup ;
intros .
transitivity p ;
auto .
transitivity (
mu e q );
auto .
Save .
Lemma okup_eq_compat :
forall (
A :Type) (
p p' :U) (
e e' :distr
A ) (
q q' :A ->
U ),
p ==
p' ->
q ==
q' ->
e ==
e' ->
okup p e q ->
okup p' e' q' .
unfold okup ;
intros .
transitivity p ;
auto .
transitivity (
mu e q );
auto .
transitivity (
mu e' q );
auto .
Save .
Lemma upfun_le_compat :
forall (
A B :Type) (
p p' :A ->
U ) (
e :A ->
distr B )
(
q q' :A ->
B ->
U ),
p <=
p' ->
q' <=q ->
upfun p e q ->
upfun p' e q' .
unfold upfun ;
intros .
apply okup_le_compat with (
p x ) (
q x );
auto .
Save .
Lemma okup_mult :
forall (
A :Type)(k
p :U)(e:distr
A )(
f :
A ->
U ),
okup p e f ->
okup (
k *p)
e (
fmult k f ).
unfold okup ;
intros A k p e f oka .
rewrite (
mu_stable_mult e k f ).
apply Umult_le_compat_right ;
trivial .
Save .
Lemma apply_rule :
forall (
A B :Type)(a:(distr
A ))(
f :A ->
distr B )(
r :U)(p:A ->
U )(
q :B ->
U ),
ok r a p ->
okfun p f (
fun x =>
q ) ->
ok r (
Mlet a f )
q .
unfold ok ,okfun,Mlet;
simpl ;
intros .
transitivity (
mu a p );
auto .
Save .
Lemma okup_apply_rule :
forall (
A B :Type)(a:distr
A )(
f :A ->
distr B )(
r :U)(p:A ->
U )(
q :B ->
U ),
okup r a p ->
upfun p f (
fun x =>
q ) ->
okup r (
Mlet a f )
q .
unfold okup ,upfun,Mlet;
simpl ;
intros .
transitivity (
mu a p );
auto .
Save .
Lemma lambda_rule : forall (A B :Type)(f:A -> distr B )(p :A -> U )(q :A -> B -> U ),forall x :A, ok (p x ) (f x ) (q x )) -> okfun p f q .trivial .Save .Lemma okup_lambda_rule : forall (A B :Type)(f:A -> distr B )(p :A -> U )(q :A -> B -> U ),forall x :A, okup (p x ) (f x ) (q x )) -> upfun p f q .trivial .Save .
Lemma combiok :
forall (
A :Type)
p q (
f1 f2 :
A ->
U ),
p <= [1-]
q ->
fplusok (
fmult p f1 ) (
fmult q f2 ).
unfold fplusok ,
fmult ,
finv ;
intros ;
intro x .
transitivity p ;
auto .
transitivity ([1-]
q );
auto .
Save .
Hint Resolve combiok .
Lemma fmult_fplusok :
forall (
A :Type)
p q (
f1 f2 :
A ->
U ),
fplusok f1 f2 ->
fplusok (
fmult p f1 ) (
fmult q f2 ).
unfold fplusok ,
fmult ,
finv ;
intros ;
intro x .
transitivity (
f1 x );
auto .
transitivity ([1-]
f2 x );
auto .
Save .
Hint Resolve fmult_fplusok .
Lemma ifok :
forall f1 f2 ,
fplusok (
fmult f1 ctrue ) (
fmult f2 cfalse ).
intros ;
apply fmult_fplusok ;
unfold fplusok ,ctrue,cfalse,finv;
intro x ;
case x ;
auto .
Save .
Hint Resolve ifok .
Lemma Mif_eq :
forall (
A :Type)(b:(distr
bool ))(
f1 f2 :distr
A )(
q :MF
A ),
mu (
Mif b f1 f2 )
q == (
mu f1 q ) * (
mu b ctrue ) + (
mu f2 q ) * (
mu b cfalse ).
intros .
transitivity (
mu b (
fplus (
fmult (
mu f1 q )
ctrue ) (
fmult (
mu f2 q )
cfalse ))).
intros ;
unfold Mif ,Mlet,star;
simpl .
apply (
mu_stable_eq b );
intro x .
unfold fplus ,fmult;
destruct x .
rewrite (
Umult_one_right (
mu f1 q ));
rewrite (
Umult_zero_right (
mu f2 q ));
auto .
rewrite (
Umult_zero_right (
mu f1 q ));
rewrite (
Umult_one_right (
mu f2 q ));
auto .
rewrite (
mu_stable_plus b (
f :=(fmult (
mu f1 q )
ctrue ))
(
g :=(fmult (
mu f2 q )
cfalse ))
(
ifok (
mu f1 q ) (
mu f2 q ))).
rewrite (
mu_stable_mult b (
mu f1 q )
ctrue ).
rewrite (
mu_stable_mult b (
mu f2 q )
cfalse );
trivial .
Save .
Lemma ifrule :
forall (
A :Type)(b:(distr
bool ))(
f1 f2 :distr
A )(
p1 p2 :U)(q:A ->
U ),
ok p1 f1 q ->
ok p2 f2 q
->
ok (
p1 * (
mu b ctrue ) +
p2 * (
mu b cfalse )) (
Mif b f1 f2 )
q .
unfold ok ;
intros .
rewrite (
Mif_eq b f1 f2 q ).
apply Uplus_le_compat .
apply Umult_le_compat_left ;
auto .
apply Umult_le_compat_left ;
auto .
Save .
Lemma okup_ifrule :
forall (
A :Type)(b:(distr
bool ))(
f1 f2 :distr
A )(
p1 p2 :U)(q:A ->
U ),
okup p1 f1 q ->
okup p2 f2 q
->
okup (
p1 * (
mu b ctrue ) +
p2 * (
mu b cfalse )) (
Mif b f1 f2 )
q .
unfold okup ;
intros .
rewrite (
Mif_eq b f1 f2 q ).
apply Uplus_le_compat .
apply Umult_le_compat_left ;
auto .
apply Umult_le_compat_left ;
auto .
Save .
Rule for fixpoints
with , an increasing sequence of functions starting from
Section Fixrule .
Variables A B :
Type .
Variable F : (
A ->
distr B ) -m> (
A ->
distr B ).
Section Ruleseq .
Variable q :
A ->
B ->
U .
Lemma fixrule_Ulub :
forall (
p :
A ->
nat ->
U ),
(
forall x :A,
p x O == 0)->
(
forall (
i :nat) (
f :A ->
distr B ),
(
okfun (
fun x =>
p x i )
f q ) ->
okfun (
fun x =>
p x (
S i )) (
fun x =>
F f x )
q )
->
okfun (
fun x =>
Ulub (
p x )) (
Mfix F )
q .
red ;
intros p p0 Hrec .
assert (
forall n :nat,
(
okfun (
fun x => (
p x n )) (
fun x =>
Miter F n x )
q )).
induction n ;
simpl ;
auto .
red ;
red ;
intros ;
auto .
red ;
intros .
apply Ulub_le ;
auto .
intro n ;
transitivity (
mu (
Miter F n x ) (
q x )).
apply (
H n ).
apply Mfix_le_iter ;
auto .
Save .
Lemma fixrule :
forall (
p :
A ->
nat -m>
U ),
(
forall x :A,
p x O == 0)->
(
forall (
i :nat) (
f :A ->
distr B ),
(
okfun (
fun x =>
p x i )
f q ) ->
okfun (
fun x =>
p x (
S i )) (
fun x =>
F f x )
q )
->
okfun (
fun x =>
lub (
p x )) (
Mfix F )
q .
red ;
red ;
intros .
rewrite <-
Ulub_lub .
apply (
fixrule_Ulub p H H0 x ).
Save .
Lemma fixrule_up_Ulub :
forall (
p :
A ->
nat ->
U ),
(
forall (
i :nat) (
f :A ->
distr B ),
(
upfun (
fun x =>
p x i )
f q ) ->
upfun (
fun x =>
p x (
S i )) (
fun x =>
F f x )
q )
->
upfun (
fun x =>
Ulub (
p x )) (
Mfix F )
q .
red ;
red ;
intros .
unfold Mfix ;
simpl .
transitivity (
Ulub (
mshift (
Mu B @ (
Miter F <o>
x )) (
q x )));
auto .
apply Ulub_le_compat .
intro n ;
generalize x ;
induction n ;
simpl ;
intros ;
auto .
apply (
H n (
Miter F n )
IHn x0 ).
Save .
Lemma fixrule_up_lub :
forall (
p :
A ->
nat -m>
U ),
(
forall (
i :nat) (
f :A ->
distr B ),
(
upfun (
fun x =>
p x i )
f q ) ->
upfun (
fun x =>
p x (
S i )) (
fun x =>
F f x )
q )
->
upfun (
fun x =>
lub (
p x )) (
Mfix F )
q .
red ;
red ;
intros .
rewrite <-
Ulub_lub .
apply (
fixrule_up_Ulub p H x ).
Save .
Lemma okup_fixrule_glb :
forall p :
A ->
nat -m->
U ,
(
forall (
i :nat) (
f :A ->
distr B ),
(
upfun (
fun x =>
p x i )
f q ) ->
upfun (
fun x =>
p x (
S i )) (
fun x =>
F f x )
q )
->
upfun (
fun x =>
glb (
p x )) (
Mfix F )
q .
red ;
intros p Hrec x .
assert (
forall n :nat,
(
upfun (
fun x => (
p x n ))
(
fun x => (
Miter F n x ))
q )).
induction n ;
simpl ;
auto .
red ;
red ;
intros ;
auto .
red ;
intros .
unfold Mfix ,
fixp ;
simpl .
apply lub_glb_le ;
auto .
intro n ;
exact (
H n x ).
Save .
End Ruleseq .
Lemma okup_fixrule_inv :
forall (
p :
A ->
U ) (
q :
A ->
B ->
U ),
(
forall (
f :A ->
distr B ),
upfun p f q ->
upfun p (
fun x =>
F f x )
q )
->
upfun p (
Mfix F )
q .
unfold upfun ,
okup ;
intros .
unfold Mfix ;
simpl .
apply lub_le .
intro n ;
generalize x ;
induction n ;
simpl ;
auto .
Save .
Rules using commutation properties
Section TransformFix .
Section Fix_muF .
Variable q :
A ->
B ->
U .
Variable muF :
MF A -m>
MF A .
Definition admissible (
P :(A ->
distr B ) ->
Prop ) :=
P 0 /\
forall f ,
P f ->
P (
F f ).
Lemma admissible_true :
admissible (
fun f =>
True ).
red ;
auto .
Save .
Lemma admissible_le_fix :
continuous (
D1 :=A ->
distr B ) (
D2 :=A ->
distr B )
F ->
admissible (
fun f =>
f <=
Mfix F ).
split ;
intros .
auto .
transitivity (
F (
Mfix F )).
auto .
intro x ;
rewrite Mfix_eq ;
auto .
Save .
Lemma muF_stable :
stable muF .
auto .
Save .
Definition mu_muF_commute_le :=
forall f x ,
f <=
Mfix F ->
mu (
F f x ) (
q x ) <=
muF (
fun y =>
mu (
f y ) (
q y ))
x .
Hint Unfold mu_muF_commute_le .
Section F_muF_results .
Hypothesis F_muF_le :
mu_muF_commute_le .
Lemma mu_mufix_le :
forall x ,
mu (
Mfix F x ) (
q x ) <=
mufix muF x .
intros ;
unfold mufix ,Mfix,mu_lub;
simpl .
apply lub_le_compat ;
intros .
intro n ;
generalize x ;
induction n ;
simpl ;
intros ;
auto .
transitivity (
muF (
fun y =>
mu (
Miter F n y ) (
q y ))
x0 ).
apply F_muF_le .
unfold Mfix ,fixp;
intros .
apply le_lub .
apply (
fmonotonic muF );
auto .
Save .
Hint Resolve mu_mufix_le .
Lemma muF_le :
forall f ,
muF f <=
f
->
forall x ,
mu (
Mfix F x ) (
q x ) <=
f x .
intros ;
transitivity (
mufix muF x );
auto .
apply mufix_inv ;
auto .
Save .
Hypothesis muF_F_le :
forall f x ,
f <=
Mfix F ->
muF (
fun y =>
mu (
f y ) (
q y ))
x <=
mu (
F f x ) (
q x ).
Lemma mufix_mu_le :
forall x ,
mufix muF x <=
mu (
Mfix F x ) (
q x ).
intros ;
unfold mufix ,Mfix;
simpl .
apply lub_le_compat ;
intros .
intro n ;
generalize x ;
induction n ;
simpl ;
intros ;
auto .
transitivity (
muF (
fun y =>
mu (
Miter F n y ) (
q y ))
x0 ).
apply (
fmonotonic muF );
auto .
apply muF_F_le .
intro y ;
unfold Mfix ,fixp;
simpl ;
intros f .
exact (
le_lub (
mshift (
Mu B @ (
Ccpo.iter (
D :=A ->
distr B )
F <o>
y ))
f )
n ).
Save .
End F_muF_results .
Hint Resolve mu_mufix_le mufix_mu_le .
Lemma mufix_mu :
(
forall f x ,
f <=
Mfix F ->
mu (
F f x ) (
q x ) ==
muF (
fun y =>
mu (
f y ) (
q y ))
x )
->
forall x ,
mufix muF x ==
mu (
Mfix F x ) (
q x ).
intros ;
apply Ole_antisym ;
auto .
apply mufix_mu_le ;
intros ;
auto .
rewrite (
H f x0 );
auto .
Save .
Hint Resolve mufix_mu .
End Fix_muF .
Section Fix_Term .
Definition pterm :
MF A :=
fun (
x :A) =>
mu (
Mfix F x ) (
fone B ).
Variable muFone :
MF A -m>
MF A .
Hypothesis F_muF_eq_one :
forall f x ,
f <=
Mfix F ->
mu (
F f x ) (
fone B ) ==
muFone (
fun y =>
mu (
f y ) (
fone B ))
x .
Hypothesis muF_cont :
continuous muFone .
Lemma muF_pterm :
pterm ==
muFone pterm .
intro x ;
unfold pterm .
rewrite <- (
mufix_mu (
fun (
x :A) =>
fone B )
muFone F_muF_eq_one x ).
transitivity (
muFone (
mufix muFone )
x );
auto .
apply ford_eq_elim .
apply (
mufix_eq (
F :=muFone));
auto .
apply ford_eq_elim .
apply muF_stable ;
auto .
intro y .
apply (
mufix_mu (
fun (
x :A) =>
fone B )
muFone F_muF_eq_one y ).
Save .
Hint Resolve muF_pterm .
End Fix_Term .
Section Fix_muF_Term .
Variable q :
A ->
B ->
U .
Definition qinv x y := [1-]
q x y .
Variable muFqinv :
MF A -m>
MF A .
Hypothesis F_muF_le_inv :
mu_muF_commute_le qinv muFqinv .
Lemma muF_le_term :
forall f ,
muFqinv (
finv f ) <=
finv f ->
forall x ,
f x &
pterm x <=
mu (
Mfix F x ) (
q x ).
intros ;
transitivity (
mu (
Mfix F x ) (
fesp (
q x ) (
fone B ))).
transitivity ([1-]
mu (
Mfix F x ) (
qinv x ) &
pterm x ).
apply Uesp_le_compat ;
auto .
apply Uinv_le_perm_right .
apply muF_le with (
muF :=muFqinv) (
q :=qinv) (
f :=finv
f ) (
x :=x);
auto .
unfold pterm ;
replace (
qinv x )
with (
finv (
q x ));
auto .
apply mu_monotonic ;
intro ;
unfold fesp ,fone;
repeat Usimpl ;
auto .
Save .
Lemma muF_le_term_minus :
forall f ,
f <=
pterm ->
muFqinv (
fminus pterm f ) <=
fminus pterm f ->
forall x ,
f x <=
mu (
Mfix F x ) (
q x ).
intros ;
transitivity ((
f x + [1-]
pterm x ) &
pterm x ).
red in H ;
rewrite Uplus_inv_esp_simpl ;
auto .
apply muF_le_term with (
f :=fun
y =>
f y + [1-]
pterm y );
auto .
transitivity (
muFqinv (
fminus pterm f )).
apply (
fmonotonic muFqinv );
unfold fminus ,
finv ;
intro y .
apply Uinv_le_perm_left ;
auto .
transitivity (
fminus pterm f );
auto .
unfold fminus ,
Uminus ,
finv ;
intro y ;
auto .
Save .
Variable muFq :
MF A -m>
MF A .
Hypothesis F_muF_le :
mu_muF_commute_le q muFq .
Lemma muF_eq :
forall f ,
muFq f <=
f ->
muFqinv (
finv f ) <=
finv f ->
forall x ,
pterm x == 1 ->
mu (
Mfix F x ) (
q x ) ==
f x .
intros ;
apply Ole_antisym .
apply muF_le with (
muF :=muFq);
auto .
transitivity (
f x &
pterm x ).
rewrite H1 ;
auto .
apply muF_le_term ;
auto .
Save .
End Fix_muF_Term .
End TransformFix .
Section LoopRule .
Variable q :
A ->
B ->
U .
Variable stop :
A ->
distr bool .
Variable step :
A ->
distr A .
Variable a :
U .
Definition Loop :
MF A -m>
MF A .
exists (
fun f (
x :A) =>
mu (
stop x ) (
fun b =>
if b then a else mu (
step x )
f )).
red ;
intros ;
intro x0 .
apply (
mu_monotonic (
stop x0 ));
intro b ;
case b ;
auto .
Defined .
Lemma Loop_eq :
forall f x ,
Loop f x =
mu (
stop x ) (
fun b =>
if b then a else mu (
step x )
f ).
trivial .
Save .
Definition loop :=
mufix Loop .
Lemma Mfixvar :
(
forall (
f :A ->
distr B ),
okfun (
fun x =>
Loop (
fun y =>
mu (
f y ) (
q y ))
x ) (
fun x =>
F f x )
q )
->
okfun loop (
Mfix F )
q .
intros ;
unfold loop ,mufix,fixp.
apply okfun_le_compat with (
fun x =>
Ulub (
fun n =>
Ccpo.iter Loop n x ))
q ;
trivial .
apply lub_le ;
intros n x .
transitivity (
Ccpo.iter Loop n x );
auto .
apply (
le_Ulub (
fun n0 :
nat =>
Ccpo.iter Loop n0 x )
n ).
apply fixrule_Ulub with (
p :=fun
x n =>
Ccpo.iter Loop n x );
intros ;
auto .
unfold okfun ,ok
in *|-*.
intro x ;
rewrite iterS_simpl .
transitivity (
Loop (
fun y :
A =>
mu (
f y ) (
q y ))
x );
trivial .
repeat rewrite Loop_eq .
apply (
mu_monotonic (
stop x ));
intros .
intro b ;
case b ;
auto .
Save .
Definition up_loop :
MF A :=
nufix Loop .
Lemma Mfixvar_up :
(
forall (
f :A ->
distr B ),
upfun (
fun x =>
Loop (
fun y =>
mu (
f y ) (
q y ))
x ) (
fun x =>
F f x )
q )
->
upfun up_loop (
Mfix F )
q .
intros ;
unfold up_loop ,nufix.
unfold fixp .
apply upfun_le_compat with
(
fun x :
A =>
glb (
Iord_app x @
mon (
iter_ (
c :=MFopp
A ) (
G Loop ))))
q ;
auto .
intro x ;
apply Ole_refl_eq_inv .
apply MFopp_lub_eq .
apply okup_fixrule_glb ;
intros ;
intro .
unfold upfun ,okup
in *|-*.
rewrite comp_simpl ;
unfold Iord_app ;
repeat rewrite mon_simpl .
transitivity (
Loop (
fun y :
A =>
mu (
f y ) (
q y ))
x );
trivial .
rewrite Loop_eq ;
simpl .
apply (
mu_monotonic (
stop x ));
intro x0 .
case x0 ;
auto .
Save .
End LoopRule .
End Fixrule .
Distributions operates on intervals
Definition Imu :
forall A :Type,
distr A -> (
A ->
IU ) ->
IU .
intros A m F ;
exists (
mu m (
fun x =>
low (
F x ))) (
mu m (
fun x =>
up (
F x ))).
apply mu_monotonic ;
auto .
Defined .
Lemma low_Imu :
forall (
A :Type) (
e :distr
A ) (
F :
A ->
IU ),
low (
Imu e F ) =
mu e (
fun x =>
low (
F x )).
trivial .
Save .
Lemma up_Imu :
forall (
A :Type) (
e :distr
A ) (
F :
A ->
IU ),
up (
Imu e F ) =
mu e (
fun x =>
up (
F x )).
trivial .
Save .
Lemma Imu_monotonic :
forall (
A :Type) (
e :distr
A ) (
F G :
A ->
IU ),
(
forall x ,
Iincl (
F x ) (
G x )) ->
Iincl (
Imu e F ) (
Imu e G ).
unfold Iincl ,Imu;
simpl ;
intuition .
apply (
mu_monotonic e );
intro x ;
case (
H x );
auto .
apply (
mu_monotonic e );
intro x ;
case (
H x );
auto .
Save .
Lemma Imu_stable_eq :
forall (
A :Type) (
e :distr
A ) (
F G :
A ->
IU ),
(
forall x ,
Ieq (
F x ) (
G x )) ->
Ieq (
Imu e F ) (
Imu e G ).
unfold Ieq ,Imu;
simpl ;
intuition .
apply (
mu_stable_eq e );
intro x ;
case (
H x );
auto .
apply (
mu_stable_eq e );
intro x ;
case (
H x );
auto .
Save .
Hint Resolve Imu_monotonic Imu_stable_eq .
Lemma Imu_singl :
forall (
A :Type) (
e :distr
A ) (
f :A ->
U ),
Ieq (
Imu e (
fun x =>
singl (
f x ))) (
singl (
mu e f )).
unfold Ieq ,Imu,singl;
simpl ;
intuition .
Save .
Lemma Imu_inf :
forall (
A :Type) (
e :distr
A ) (
f :A ->
U ),
Ieq (
Imu e (
fun x =>
inf (
f x ))) (
inf (
mu e f )).
unfold Ieq ,Imu,inf;
simpl ;
intuition .
exact (
mu_zero e ).
Save .
Lemma Imu_sup :
forall (
A :Type) (
e :distr
A ) (
f :A ->
U ),
Iincl (
Imu e (
fun x =>
sup (
f x ))) (
sup (
mu e f )).
unfold Iincl ,Imu,inf;
simpl ;
intuition .
Save .
Lemma Iin_mu_Imu :
forall (
A :Type) (
e :distr
A ) (
F :A ->
IU ) (
f :A ->
U ),
(
forall x ,
Iin (
f x ) (
F x )) ->
Iin (
mu e f ) (
Imu e F ).
unfold Iin ,Imu;
simpl ;
split .
apply (
mu_monotonic e );
intro x ;
case (
H x );
auto .
apply (
mu_monotonic e );
intro x ;
case (
H x );
auto .
Save .
Hint Resolve Iin_mu_Imu .
Definition Iok (
A :Type) (
I :IU) (
e :distr
A ) (
F :A ->
IU ) :=
Iincl (
Imu e F )
I .
Definition Iokfun (
A B :Type)(I:A ->
IU ) (
e :A ->
distr B ) (
F :A ->
B ->
IU )
:=
forall x ,
Iok (
I x ) (
e x ) (
F x ).
Lemma Iin_mu_Iok :
forall (
A :Type) (
I :IU) (
e :distr
A ) (
F :A ->
IU ) (
f :A ->
U ),
(
forall x ,
Iin (
f x ) (
F x )) ->
Iok I e F ->
Iin (
mu e f )
I .
intros ;
apply Iincl_in with (
Imu e F );
auto .
Save .
Lemma Iok_le_compat :
forall (
A :Type) (
I J :IU) (
e :distr
A ) (
F G :A ->
IU ),
Iincl I J -> (
forall x ,
Iincl (
G x ) (
F x )) ->
Iok I e F ->
Iok J e G .
unfold Iok ;
intros ;
apply Iincl_trans with I ;
auto .
apply Iincl_trans with (
Imu e F );
auto .
Save .
Lemma Iokfun_le_compat :
forall (
A B :Type) (
I J :A ->
IU ) (
e :A ->
distr B ) (
F G :A ->
B ->
IU ),
(
forall x ,
Iincl (
I x ) (
J x )) -> (
forall x y ,
Iincl (
G x y ) (
F x y )) ->
Iokfun I e F ->
Iokfun J e G .
unfold Iokfun ;
intros ;
apply Iok_le_compat with (
I x ) (
F x );
auto .
Save .
Lemma Iunit_eq : forall (A : Type ) (a :A) (F :A -> IU ), Ieq (Imu (Munit a ) F ) (F a ).intros ; unfold Ieq ,Imu,Munit; auto .Save .
Lemma Ilet_eq :
forall (
A B :
Type ) (
a :distr
A ) (
f :A ->
distr B )(
I :IU)(G:B ->
IU ),
Ieq (
Imu (
Mlet a f )
G ) (
Imu a (
fun x =>
Imu (
f x )
G )).
unfold Ieq ,Imu,Mlet,Iincl;
simpl ;
intuition .
Save .
Hint Resolve Ilet_eq .
Lemma Iapply_rule :
forall (
A B :
Type ) (
a :distr
A ) (
f :A ->
distr B )(
I :IU)(F:A ->
IU )(
G :B ->
IU ),
Iok I a F ->
Iokfun F f (
fun x =>
G ) ->
Iok I (
Mlet a f )
G .
unfold Iokfun ,Iok;
intros .
apply Ieq_incl_trans with (
Imu a (
fun x =>
Imu (
f x )
G ));
auto .
apply Iincl_trans with (
Imu a F );
auto .
Save .
Lemma Ilambda_rule : forall (A B :Type)(f:A -> distr B )(F :A -> IU )(G :A -> B -> IU ),forall x :A, Iok (F x ) (f x ) (G x )) -> Iokfun F f G .trivial .Save .
Lemma Imu_Mif_eq :
forall (
A :Type)(b:distr
bool )(
f1 f2 :distr
A )(
F :A ->
IU ),
Ieq (
Imu (
Mif b f1 f2 )
F ) (
Iplus (
Imultk (
mu b ctrue ) (
Imu f1 F )) (
Imultk (
mu b cfalse ) (
Imu f2 F ))).
red ;
intros .
rewrite low_Imu ;
rewrite up_Imu .
repeat rewrite Mif_eq .
repeat rewrite low_Iplus ;
repeat rewrite low_Imultk .
repeat rewrite up_Iplus ;
repeat rewrite up_Imultk .
repeat rewrite low_Imu ;
repeat rewrite up_Imu ;
auto .
Save .
Lemma Iifrule :
forall (
A :Type)(b:(distr
bool ))(
f1 f2 :distr
A )(
I1 I2 :IU)(F:A ->
IU ),
Iok I1 f1 F ->
Iok I2 f2 F
->
Iok (
Iplus (
Imultk (
mu b ctrue )
I1 ) (
Imultk (
mu b cfalse )
I2 )) (
Mif b f1 f2 )
F .
unfold Iok ;
intros .
apply Ieq_incl_trans with (1:=Imu_Mif_eq
b f1 f2 F ).
split .
repeat rewrite low_Iplus ;
repeat rewrite low_Imultk .
apply Uplus_le_compat ;
auto .
repeat rewrite up_Iplus ;
repeat rewrite up_Imultk .
apply Uplus_le_compat ;
auto .
Save .
Rule for fixpoints
with , an decreasing sequence of intervals functions () such that contains for all .
Section IFixrule .
Variables A B :
Type .
Variable F : (
A ->
distr B ) -m> (
A ->
distr B ).
Section IRuleseq .
Variable Q :
A ->
B ->
IU .
Variable I :
A ->
nat -m>
IU .
Lemma Ifixrule :
(
forall x :A,
Iin 0 (
I x O )) ->
(
forall (
i :nat) (
f :A ->
distr B ),
(
Iokfun (
fun x =>
I x i )
f Q ) ->
Iokfun (
fun x =>
I x (
S i )) (
fun x =>
F f x )
Q )
->
Iokfun (
fun x =>
Ilim (
I x )) (
Mfix F )
Q .
red ;
intros p0 Hrec .
assert (
forall n :nat,
(
Iokfun (
fun x => (
I x n )) (
fun x =>
Miter F n x )
Q )).
induction n ;
simpl ;
auto .
split .
rewrite low_lim ;
rewrite low_Imu .
apply lub_le ;
auto .
intro n ;
transitivity (
low (
Imu (
Miter F n x ) (
Q x ))).
case (
H n x );
auto .
rewrite low_Imu ;
apply Mfix_le_iter ;
auto .
rewrite up_lim ;
rewrite up_Imu .
unfold Mfix ,mu_lub;
simpl .
apply lub_glb_le ;
intros .
case (
H n x );
auto .
Save .
End IRuleseq .
Section ITransformFix .
Section IFix_muF .
Variable Q :
A ->
B ->
IU .
Variable ImuF : (
A ->
IU ) -m> (
A ->
IU ).
Lemma ImuF_stable :
forall I J ,
I ==J ->
ImuF I ==
ImuF J .
intros ;
apply fstable with (
f :=ImuF);
auto .
Save .
Section IF_muF_results .
Hypothesis Iincl_F_ImuF :
forall f x ,
f <=
Mfix F ->
Iincl (
Imu (
F f x ) (
Q x )) (
ImuF (
fun y =>
Imu (
f y ) (
Q y ))
x ).
Lemma Iincl_fix_ifix :
forall x ,
Iincl (
Imu (
Mfix F x ) (
Q x )) (
fixp (
D :=A ->
IU )
ImuF x ).
assert (
forall n x ,
Iincl (
Imu (
Miter F n x ) (
Q x )) (
Ccpo.iter (
D :=A ->
IU )
ImuF n x )).
induction n ;
simpl ;
intros ;
auto .
apply Iincl_trans with (
ImuF (
fun y =>
Imu (
Miter F n y ) (
Q y ))
x ).
apply Iincl_F_ImuF .
intro y ;
unfold Mfix ,
fixp .
intro f ;
simpl .
apply (
le_mlub (
fun y0 :
nat =>
mu (
iter_ F y0 y )
f )
n ).
apply (
fmonotonic ImuF );
auto .
intros ;
unfold Iincl ,
Imu ,Mfix,fixp;
simpl ;
split .
apply lub_le_compat ;
intro n ;
case (
H n x );
intros .
transitivity (
low (
Imu (
Miter F n x ) (
Q x )));
auto .
apply lub_glb_le ;
intros ;
auto .
case (
H n x );
intros .
transitivity (
up (
Imu (
Miter F n x ) (
Q x )));
auto .
Save .
Hint Resolve Iincl_fix_ifix .
End IF_muF_results .
End IFix_muF .
End ITransformFix .
End IFixrule .
Lemma Flip_ctrue :
mu Flip ctrue == [1/2].
unfold Flip ;
auto .
Save .
Lemma Flip_cfalse :
mu Flip cfalse == [1/2].
unfold Flip ;
auto .
Save .
Lemma ok_Flip :
forall q :
bool ->
U ,
ok ([1/2] *
q true + [1/2] *
q false )
Flip q .
red ;
unfold Flip ,
flip ;
simpl ;
auto .
Save .
Lemma okup_Flip :
forall q :
bool ->
U ,
okup ([1/2] *
q true + [1/2] *
q false )
Flip q .
red ;
unfold Flip ,
flip ;
simpl ;
auto .
Save .
Hint Resolve ok_Flip okup_Flip Flip_ctrue Flip_cfalse .
Lemma Flip_eq :
forall q :
bool ->
U ,
mu Flip q == [1/2] *
q true + [1/2] *
q false .
intros ;
apply Ole_antisym ;
auto .
Save .
Hint Resolve Flip_eq .
Lemma IFlip_eq :
forall Q :
bool ->
IU ,
Ieq (
Imu Flip Q ) (
Iplus (
Imultk [1/2] (
Q true )) (
Imultk [1/2] (
Q false ))).
red ;
unfold Flip ,
flip ;
intro x ;
auto .
Save .
Hint Resolve IFlip_eq .
Rules for total (well-founded) fixpoints
Section Wellfounded .
Variables A B :
Type .
Variable R :
A ->
A ->
Prop .
Hypothesis Rwf :
well_founded R .
Variable F :
forall x , (
forall y ,
R y x ->
distr B ) ->
distr B .
Definition WfFix :
A ->
distr B :=
Fix Rwf (
fun _ =>
distr B )
F .
Hypothesis Fext :
forall x f g , (
forall y (
p :R
y x ),
f y p ==
g y p ) ->
F f ==
F g .
Lemma Acc_iter_distr :
forall x ,
forall r s :
Acc R x ,
Acc_iter (
fun _ =>
distr B )
F r ==
Acc_iter (
fun _ =>
distr B )
F s .
intros x r ;
elim r using Acc_inv_dep ;
intros .
case s ;
intros .
apply Fext ;
auto .
Save .
Lemma WfFix_eq :
forall x ,
WfFix x ==
F (
fun (
y :A) (
p :R
y x ) =>
WfFix y ).
intro x ;
unfold WfFix ,Fix.
case (
Rwf x );
simpl ;
intros .
apply Fext ;
intros .
apply Acc_iter_distr .
Save .
Variable P :
distr B ->
Prop .
Hypothesis Pext :
forall m1 m2 ,
m1 ==
m2 ->
P m1 ->
P m2 .
Lemma WfFix_ind :
(
forall x f , (
forall y (
p :R
y x ),
P (
f y p )) ->
P (
F f ))
->
forall x ,
P (
WfFix x ).
intros ;
pattern x ;
apply well_founded_ind with (
R :=R);
trivial ;
intros .
apply Pext with (
m1 :=
F (
fun (
y :A) (
p :R
y x0 ) =>
WfFix y )).
apply Oeq_sym ;
apply WfFix_eq .
apply H ;
auto .
Save .
End Wellfounded .
Ltac distrsimpl :=
match goal with
| |- (
Ole (
fmont (
mu ?d1) ?f) (
fmont (
mu ?d2) ?g)) =>
apply (
mu_le_compat (
m1 :=d1) (
m2 :=d2) (
Ole_refl d1 ) (
f :=f) (
g :=g));
intro
| |- (
Oeq (
fmont (
mu ?d1) ?f) (
fmont (
mu ?d2) ?g)) =>
apply (
mu_eq_compat (
m1 :=d1) (
m2 :=d2) (
Oeq_refl d1 ) (
f :=f) (
g :=g));
intro
| |- (
Oeq (
Munit ?x) (
Munit ?y)) =>
apply (
Munit_eq_compat x y )
| |- (
Oeq (
Mlet ?x1 ?f) (
Mlet ?x2 ?g))
=>
apply (
Mlet_eq_compat (
m1 :=x1) (
m2 :=x2) (
M1 :=f) (
M2 :=g) (
Oeq_refl x1 ));
intro
| |- (
Ole (
Mlet ?x1 ?f) (
Mlet ?x2 ?g))
=>
apply (
Mlet_le_compat (
m1 :=x1) (
m2 :=x2) (
M1 :=f) (
M2 :=g) (
Ole_refl x1 ));
intro
| |-
context [(
fmont (
mu (
Mlet ?m ?M)) ?f)] =>
rewrite (
Mlet_simpl m M f )
| |-
context [(
fmont (
mu (
Munit ?x)) ?f)] =>
rewrite (
Munit_simpl f x )
| |-
context [(
Mlet (
Mlet ?m ?M) ?f)] =>
rewrite (
Mlet_assoc m M f )
| |-
context [(
Mlet (
Munit ?x) ?f)] =>
rewrite (
Mlet_unit x f )
| |-
context [(
fmont (
mu Flip ) ?f)] =>
rewrite (
Flip_simpl f )
| |-
context [(
fmont (
mu (
Discrete ?d)) ?f)] =>
rewrite (
Discrete_simpl d );
rewrite (
discrete_simpl (
coeff d ) (
points d )
f )
| |-
context [(
fmont (
mu (
Random ?n)) ?f)] =>
rewrite (
Random_simpl n );
rewrite (
random_simpl n f )
| |-
context [(
fmont (
mu (
Mif ?b ?f ?g)) ?h)] =>
unfold Mif
| |-
context [(
fmont (
mu (
Mchoice ?p ?m1 ?m2)) ?f)] =>
rewrite (
Mchoice_simpl p m1 m2 f )
| |-
context [(
fmont (
mu (
im_distr ?f ?m)) ?h)] =>
rewrite (
im_distr_simpl f m h )
| |-
context [(
fmont (
mu (
prod_distr ?m1 ?m2)) ?h)] =>
unfold prod_distr
| |-
context [((
mon ?f (
fmonotonic :=?mf)) ?x)] =>
rewrite (
mon_simpl f (
mf :=mf)
x )
end .
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Global Index
A B C D E F G H
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K
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R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
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B
C D E
F G
H
I
J
K
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O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
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Global Index
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Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
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Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
O
Oeq [projection, in
Ccpo ]
Oeq_le [lemma, in
Ccpo ]
Oeq_le_sym [lemma, in
Ccpo ]
Oeq_refl [lemma, in
Ccpo ]
Oeq_refl_eq [lemma, in
Ccpo ]
Oeq_sym [lemma, in
Ccpo ]
Oeq_trans [lemma, in
Ccpo ]
Ole [projection, in
Ccpo ]
Ole_antisym [lemma, in
Ccpo ]
Ole_eq_compat [lemma, in
Ccpo ]
Ole_eq_compat_iff [instance, in
Ccpo ]
Ole_eq_left [lemma, in
Ccpo ]
Ole_eq_right [lemma, in
Ccpo ]
Ole_refl [lemma, in
Ccpo ]
Ole_refl_eq [lemma, in
Ccpo ]
Ole_refl_eq_inv [lemma, in
Ccpo ]
Ole_trans [lemma, in
Ccpo ]
orc [definition, in
Misc ]
orc_intro [lemma, in
Misc ]
orc_left [lemma, in
Misc ]
orc_right [lemma, in
Misc ]
ord [record, in
Ccpo ]
Order [record, in
Ccpo ]
OrderEqRefl [instance, in
Ccpo ]
OrderEqSym [instance, in
Ccpo ]
OrderEqTrans [instance, in
Ccpo ]
OrderEquiv [instance, in
Ccpo ]
order_eq [projection, in
Ccpo ]
order_rel [projection, in
Ccpo ]
OrdSetoid [instance, in
Ccpo ]
html/IterFlip.html 0000644 0001750 0001750 00000036635 11301735615 014315 0 ustar cpaulin cpaulin
IterFlip
Library IterFlip
IterFlip.v: An example of probabilistic termination
Require Export Prog .
Set Implicit Arguments .
Definition of a random walk
We interpret the probabilistic program
let rec iter x = if flip() then iter (x+1) else x
Require Import ZArith .
Instance Fiter_mon :
monotonic (
fun (
f :Z ->
distr Z ) (
x :Z) =>
Mif Flip (
f (
Zsucc x )) (
Munit x )).
red ;
intros ;
intro p ;
apply Mif_le_compat ;
auto .
Save .
Definition Fiter : (
Z ->
distr Z ) -m> (
Z ->
distr Z )
:=
mon (
fun f (
x :Z) =>
Mif Flip (
f (
Zsucc x )) (
Munit x )).
Lemma Fiter_simpl :
forall f x ,
Fiter f x =
Mif Flip (
f (
Zsucc x )) (
Munit x ).
trivial .
Save .
Definition iterflip :
Z ->
distr Z :=
Mfix Fiter .
Main result
Probability for
iter to terminate is
Auxiliary function
Definition
Fixpoint p_ (
n :
nat ) :
U :=
match n with O => 0 | (
S n ) => [1/2] *
p_ n + [1/2]
end .
Lemma p_incr :
forall n ,
p_ n <=
p_ (
S n ).
simpl ;
intros .
transitivity ([1/2]*p_
n + [1/2]*p_
n );
auto .
Save .
Hint Resolve p_incr .
Definition p :
nat -m>
U :=
fnatO_intro p_ p_incr .
Lemma pS_simpl :
forall n ,
p (
S n ) = [1/2] *
p n + [1/2].
trivial .
Save .
Lemma p_eq :
forall n :nat,
p n == [1-]([1/2]^n).
induction n ;
auto ;
rewrite pS_simpl .
rewrite IHn .
transitivity ([1/2] * [1-]([1/2]^n) + [1-][1/2]);
simpl ;
auto .
Save .
Hint Resolve p_eq .
Lemma p_le :
forall n :nat, [1-]([1/]1+n) <=
p n .
intro ;
rewrite (
p_eq n ).
apply Uinv_le_compat .
induction n ;
simpl ;
intros ;
auto .
transitivity ([1/2] * ([1/]1+n));
auto .
Save .
Hint Resolve p_le .
Lemma lim_p_one : 1 <=
lub p .
apply Ule_lt_lim ;
intros .
assert (
exc (
fun n :
nat =>
t <= [1-] ([1/]1+n))).
assert (~0 == [1-]
t ).
red ;
intro ;
apply H ;
auto .
transitivity ([1-] 0);
auto .
apply (
archimedian ([1-]
t )
H0 );
auto ;
intros m H1 .
apply exc_intro with m ;
auto .
apply H0 ;
auto ;
intros .
transitivity (
p x );
auto .
transitivity ([1-] ([1/]1+x));
auto .
Save .
Hint Resolve lim_p_one .
Proof of probabilistic termination
Definition q1 (
z1 z2 :Z) := 1.
Lemma iterflip_term :
okfun (
fun k => 1)
iterflip q1 .
unfold iterflip ;
intros .
apply okfun_le_compat with (
fun (
k :Z) =>
lub p )
q1 ;
auto .
intro x ;
auto .
apply fixrule with (
p :=
fun (
x :Z) =>
p );
auto ;
intros .
red ;
simpl ;
intros .
unfold Fiter .
red .
rewrite (
Mif_eq Flip (
f (
Zsucc x0 )) (
Munit x0 ) (
q1 x0 ));
simpl .
repeat Usimpl .
unfold q1 at 2;
repeat Usimpl .
transitivity (
mu (
f (
Zsucc x0 )) (
q1 (
Zsucc x0 )));
auto .
exact (
H (
Zsucc x0 )).
Save .
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Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
C (library)
Ccpo
Choice
Cover
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Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
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Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
S (lemma)
select_diff [in
Sets ]
select_non_empty [in
Sets ]
set_setoid [in
Sets ]
shift_le_compat [in
Ccpo ]
shift_simpl [in
Ccpo ]
size_eqset [in
Sets ]
size_finite_rem [in
Sets ]
size_incl [in
Sets ]
size_inter_add_in [in
Sets ]
size_inter_add_notin [in
Sets ]
size_inter_empty [in
Sets ]
size_inter_incl [in
Sets ]
size_unique [in
Sets ]
stable_intro [in
Ccpo ]
html/index_lemma_G.html 0000644 0001750 0001750 00000023202 11301735615 015311 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
G (lemma)
gunion_elim [in
Sets ]
gunion_intro [in
Sets ]
gunion_monotonic [in
Sets ]
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Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
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Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
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Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_global_U.html 0000644 0001750 0001750 00000335162 11301735615 015507 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
U
union [definition, in
Sets ]
union_add_left [lemma, in
Sets ]
union_add_right [lemma, in
Sets ]
union_empty_left [lemma, in
Sets ]
union_empty_right [lemma, in
Sets ]
union_incl_intro [lemma, in
Sets ]
union_incl_left [lemma, in
Sets ]
union_incl_right [lemma, in
Sets ]
union_sym [lemma, in
Sets ]
Universe [module, in
Utheory ]
Universe.archimedian [variable, in
Utheory ]
Universe.cpoU [instance, in
Utheory ]
Universe.ordU [instance, in
Utheory ]
Universe.U [axiom, in
Utheory ]
Universe.Udiff_0_1 [variable, in
Utheory ]
Universe.Udistr_inv_right [variable, in
Utheory ]
Universe.Udistr_plus_right [variable, in
Utheory ]
Universe.Udiv [axiom, in
Utheory ]
Universe.Udiv_by_zero [variable, in
Utheory ]
Universe.Udiv_le_one [variable, in
Utheory ]
Universe.Uinv [axiom, in
Utheory ]
Universe.Uinv_le_compat [variable, in
Utheory ]
Universe.Uinv_one [variable, in
Utheory ]
Universe.Uinv_plus_left [variable, in
Utheory ]
Universe.Ule_class [variable, in
Utheory ]
Universe.Ule_total [variable, in
Utheory ]
Universe.Umult [axiom, in
Utheory ]
Universe.Umult_assoc [variable, in
Utheory ]
Universe.Umult_div [variable, in
Utheory ]
Universe.Umult_le_simpl_left [variable, in
Utheory ]
Universe.Umult_mon_right [instance, in
Utheory ]
Universe.Umult_one_left [variable, in
Utheory ]
Universe.Umult_right_continuous [variable, in
Utheory ]
Universe.Umult_sym [variable, in
Utheory ]
Universe.Unth [axiom, in
Utheory ]
Universe.Unth_prop [variable, in
Utheory ]
Universe.Uplus [axiom, in
Utheory ]
Universe.Uplus_assoc [variable, in
Utheory ]
Universe.Uplus_le_simpl_right [variable, in
Utheory ]
Universe.Uplus_mon_right [instance, in
Utheory ]
Universe.Uplus_right_continuous [variable, in
Utheory ]
Universe.Uplus_sym [variable, in
Utheory ]
Universe.Uplus_zero_left [variable, in
Utheory ]
Universe.U1 [definition, in
Utheory ]
Univ_prop [module, in
Uprop ]
Univ_prop.bary_le_left [lemma, in
Uprop ]
Univ_prop.bary_le_right [lemma, in
Uprop ]
Univ_prop.bary_refl_eq [lemma, in
Uprop ]
Univ_prop.bary_Uminus_left [lemma, in
Uprop ]
Univ_prop.bary_Uminus_left_eq [lemma, in
Uprop ]
Univ_prop.bary_up_eq [lemma, in
Uprop ]
Univ_prop.bary_up_le [lemma, in
Uprop ]
Univ_prop.B2U [definition, in
Uprop ]
Univ_prop.B2Uinv [lemma, in
Uprop ]
Univ_prop.cauchy [definition, in
Uprop ]
Univ_prop.cauchy_lim [lemma, in
Uprop ]
Univ_prop.diff [definition, in
Uprop ]
Univ_prop.diff_bary_left_eq [lemma, in
Uprop ]
Univ_prop.diff_eq [lemma, in
Uprop ]
Univ_prop.diff_plus_ok [lemma, in
Uprop ]
Univ_prop.diff_sym [lemma, in
Uprop ]
Univ_prop.diff_triangular [lemma, in
Uprop ]
Univ_prop.diff_Uminus [lemma, in
Uprop ]
Univ_prop.diff_Uplus_le [lemma, in
Uprop ]
Univ_prop.diff_zero [lemma, in
Uprop ]
Univ_prop.eq_liminf_sup [lemma, in
Uprop ]
Univ_prop.eq_lim_pmin_p [lemma, in
Uprop ]
Univ_prop.eq_lim_U1min [lemma, in
Uprop ]
Univ_prop.exp2 [definition, in
Uprop ]
Univ_prop.exp2_pos [lemma, in
Uprop ]
Univ_prop.fconj [definition, in
Uprop ]
Univ_prop.Fconj [definition, in
Uprop ]
Univ_prop.fconj_continuous2 [instance, in
Uprop ]
Univ_prop.fconj_decomp [lemma, in
Uprop ]
Univ_prop.fconj_eq_compat [lemma, in
Uprop ]
Univ_prop.fconj_fplusok [lemma, in
Uprop ]
Univ_prop.fconj_le_compat [lemma, in
Uprop ]
Univ_prop.fconj_mon2 [instance, in
Uprop ]
Univ_prop.fconj_simpl [lemma, in
Uprop ]
Univ_prop.Fconj_simpl [lemma, in
Uprop ]
Univ_prop.fconj_sym [lemma, in
Uprop ]
Univ_prop.Fcte [definition, in
Uprop ]
Univ_prop.fcte [definition, in
Uprop ]
Univ_prop.fcte_simpl [lemma, in
Uprop ]
Univ_prop.fdiv [definition, in
Uprop ]
Univ_prop.fdiv_simpl [lemma, in
Uprop ]
Univ_prop.feq_finv_finv [lemma, in
Uprop ]
Univ_prop.feq_intro [lemma, in
Uprop ]
Univ_prop.fesp [definition, in
Uprop ]
Univ_prop.fesp_eq_compat [lemma, in
Uprop ]
Univ_prop.fesp_le_compat [lemma, in
Uprop ]
Univ_prop.fesp_mon2 [instance, in
Uprop ]
Univ_prop.fesp_simpl [lemma, in
Uprop ]
Univ_prop.Finf [definition, in
Uprop ]
Univ_prop.finf [definition, in
Uprop ]
Univ_prop.finf_fn [lemma, in
Uprop ]
Univ_prop.finf_incr [lemma, in
Uprop ]
Univ_prop.finf_mon [instance, in
Uprop ]
Univ_prop.finv [definition, in
Uprop ]
Univ_prop.finv_eq_compat [lemma, in
Uprop ]
Univ_prop.finv_le_compat [lemma, in
Uprop ]
Univ_prop.finv_mon [instance, in
Uprop ]
Univ_prop.finv_simpl [lemma, in
Uprop ]
Univ_prop.FixDef [section, in
Uprop ]
Univ_prop.FixDef.A [variable, in
Uprop ]
Univ_prop.FixDef.F [variable, in
Uprop ]
Univ_prop.fle_fconj_left [lemma, in
Uprop ]
Univ_prop.fle_fconj_right [lemma, in
Uprop ]
Univ_prop.fle_fesp_left [lemma, in
Uprop ]
Univ_prop.fle_fesp_right [lemma, in
Uprop ]
Univ_prop.fle_fmult [lemma, in
Uprop ]
Univ_prop.fle_fplus_left [lemma, in
Uprop ]
Univ_prop.fle_fplus_right [lemma, in
Uprop ]
Univ_prop.fle_intro [lemma, in
Uprop ]
Univ_prop.fle_one [lemma, in
Uprop ]
Univ_prop.fle_zero [lemma, in
Uprop ]
Univ_prop.flub [definition, in
Uprop ]
Univ_prop.flub_simpl [lemma, in
Uprop ]
Univ_prop.fminus [definition, in
Uprop ]
Univ_prop.fminus_eq_compat [lemma, in
Uprop ]
Univ_prop.fminus_le_compat [lemma, in
Uprop ]
Univ_prop.fminus_mon2 [instance, in
Uprop ]
Univ_prop.fminus_simpl [lemma, in
Uprop ]
Univ_prop.Fmult [definition, in
Uprop ]
Univ_prop.fmult [definition, in
Uprop ]
Univ_prop.fmult_continuous2 [lemma, in
Uprop ]
Univ_prop.fmult_eq_compat [lemma, in
Uprop ]
Univ_prop.fmult_le_compat [lemma, in
Uprop ]
Univ_prop.fmult_mon2 [instance, in
Uprop ]
Univ_prop.fmult_simpl [lemma, in
Uprop ]
Univ_prop.Fmult_simpl [lemma, in
Uprop ]
Univ_prop.fn_fsup [lemma, in
Uprop ]
Univ_prop.fone [definition, in
Uprop ]
Univ_prop.fone_simpl [lemma, in
Uprop ]
Univ_prop.fplus [definition, in
Uprop ]
Univ_prop.fplusok [definition, in
Uprop ]
Univ_prop.fplusok_inv [lemma, in
Uprop ]
Univ_prop.fplusok_le_compat [lemma, in
Uprop ]
Univ_prop.fplusok_sym [lemma, in
Uprop ]
Univ_prop.fplus_eq_compat [lemma, in
Uprop ]
Univ_prop.fplus_le_compat [lemma, in
Uprop ]
Univ_prop.fplus_mon2 [instance, in
Uprop ]
Univ_prop.fplus_simpl [lemma, in
Uprop ]
Univ_prop.Fsup [definition, in
Uprop ]
Univ_prop.fsup [definition, in
Uprop ]
Univ_prop.fsup_incr [lemma, in
Uprop ]
Univ_prop.fsup_mon [instance, in
Uprop ]
Univ_prop.full [definition, in
Uprop ]
Univ_prop.fzero [definition, in
Uprop ]
Univ_prop.fzero_simpl [lemma, in
Uprop ]
Univ_prop.G [definition, in
Uprop ]
Univ_prop.ge_half_inv [lemma, in
Uprop ]
Univ_prop.glb [definition, in
Uprop ]
Univ_prop.glb_cte [lemma, in
Uprop ]
Univ_prop.glb_eq_compat [lemma, in
Uprop ]
Univ_prop.glb_eq_mult [lemma, in
Uprop ]
Univ_prop.glb_eq_plus [lemma, in
Uprop ]
Univ_prop.glb_eq_plus_cte_left [lemma, in
Uprop ]
Univ_prop.glb_eq_plus_cte_right [lemma, in
Uprop ]
Univ_prop.glb_half_exp [lemma, in
Uprop ]
Univ_prop.glb_le [lemma, in
Uprop ]
Univ_prop.glb_le_compat [lemma, in
Uprop ]
Univ_prop.glb_le_esp [lemma, in
Uprop ]
Univ_prop.glb_le_plus [lemma, in
Uprop ]
Univ_prop.glb_lift_left [lemma, in
Uprop ]
Univ_prop.glb_mon [lemma, in
Uprop ]
Univ_prop.half_esp [lemma, in
Uprop ]
Univ_prop.half_esp_le [lemma, in
Uprop ]
Univ_prop.half_exp [lemma, in
Uprop ]
Univ_prop.half_exp_le_half [lemma, in
Uprop ]
Univ_prop.half_le [lemma, in
Uprop ]
Univ_prop.half_twice [lemma, in
Uprop ]
Univ_prop.half_twice_le [lemma, in
Uprop ]
Univ_prop.half_Unth [lemma, in
Uprop ]
Univ_prop.has_lim [definition, in
Uprop ]
Univ_prop.has_limit_cauchy [lemma, in
Uprop ]
Univ_prop.has_limit_compute [lemma, in
Uprop ]
Univ_prop.has_limit_cte [lemma, in
Uprop ]
Univ_prop.has_limit_decr [lemma, in
Uprop ]
Univ_prop.has_limit_incr [lemma, in
Uprop ]
Univ_prop.has_limit_inv [lemma, in
Uprop ]
Univ_prop.has_limit_sum [lemma, in
Uprop ]
Univ_prop.Ieq [definition, in
Uprop ]
Univ_prop.Ieq_class [lemma, in
Uprop ]
Univ_prop.Ieq_incl [lemma, in
Uprop ]
Univ_prop.Ieq_incl_sym [lemma, in
Uprop ]
Univ_prop.Ieq_incl_trans [lemma, in
Uprop ]
Univ_prop.Ieq_refl [lemma, in
Uprop ]
Univ_prop.Ieq_sym [lemma, in
Uprop ]
Univ_prop.Ieq_trans [lemma, in
Uprop ]
Univ_prop.IIim_incl_stable [lemma, in
Uprop ]
Univ_prop.Iin [definition, in
Uprop ]
Univ_prop.Iincl [definition, in
Uprop ]
Univ_prop.Iincl_antisym [lemma, in
Uprop ]
Univ_prop.Iincl_class [lemma, in
Uprop ]
Univ_prop.Iincl_full [lemma, in
Uprop ]
Univ_prop.Iincl_in [lemma, in
Uprop ]
Univ_prop.Iincl_lim [lemma, in
Uprop ]
Univ_prop.Iincl_low [lemma, in
Uprop ]
Univ_prop.Iincl_refl [lemma, in
Uprop ]
Univ_prop.Iincl_trans [lemma, in
Uprop ]
Univ_prop.Iincl_up [lemma, in
Uprop ]
Univ_prop.Iin_class [lemma, in
Uprop ]
Univ_prop.Iin_inf_elim [lemma, in
Uprop ]
Univ_prop.Iin_inf_intro [lemma, in
Uprop ]
Univ_prop.Iin_low [lemma, in
Uprop ]
Univ_prop.Iin_singl_elim [lemma, in
Uprop ]
Univ_prop.Iin_singl_intro [lemma, in
Uprop ]
Univ_prop.Iin_sup_elim [lemma, in
Uprop ]
Univ_prop.Iin_sup_intro [lemma, in
Uprop ]
Univ_prop.Iin_up [lemma, in
Uprop ]
Univ_prop.Ilim [definition, in
Uprop ]
Univ_prop.Imon2_plus_continuous [lemma, in
Uprop ]
Univ_prop.Imult [definition, in
Uprop ]
Univ_prop.Imultk [definition, in
Uprop ]
Univ_prop.Imultk_in [lemma, in
Uprop ]
Univ_prop.Imult_in [lemma, in
Uprop ]
Univ_prop.incr_decomp [lemma, in
Uprop ]
Univ_prop.incr_decomp_aux [lemma, in
Uprop ]
Univ_prop.inf [definition, in
Uprop ]
Univ_prop.Iplus [definition, in
Uprop ]
Univ_prop.Iplus_in [lemma, in
Uprop ]
Univ_prop.Isingl_eq [lemma, in
Uprop ]
Univ_prop.is_limit [definition, in
Uprop ]
Univ_prop.IU [record, in
Uprop ]
Univ_prop.IUcpo [instance, in
Uprop ]
Univ_prop.IUord [instance, in
Uprop ]
Univ_prop.le_glb [lemma, in
Uprop ]
Univ_prop.le_half_inv [lemma, in
Uprop ]
Univ_prop.le_liminf_sup [lemma, in
Uprop ]
Univ_prop.le_lim_pmin_p [lemma, in
Uprop ]
Univ_prop.le_p_lim_pmin [lemma, in
Uprop ]
Univ_prop.le_Uglb [lemma, in
Uprop ]
Univ_prop.le_Ulub [lemma, in
Uprop ]
Univ_prop.liminf [definition, in
Uprop ]
Univ_prop.liminf_decr [lemma, in
Uprop ]
Univ_prop.liminf_eq_compat [lemma, in
Uprop ]
Univ_prop.liminf_eq_mult [lemma, in
Uprop ]
Univ_prop.liminf_eq_plus_cte_right [lemma, in
Uprop ]
Univ_prop.liminf_incr [lemma, in
Uprop ]
Univ_prop.liminf_inv [lemma, in
Uprop ]
Univ_prop.liminf_le_compat [lemma, in
Uprop ]
Univ_prop.liminf_le_plus [lemma, in
Uprop ]
Univ_prop.limit_le_unique [lemma, in
Uprop ]
Univ_prop.limit_unique [lemma, in
Uprop ]
Univ_prop.limsup [definition, in
Uprop ]
Univ_prop.limsup_decr [lemma, in
Uprop ]
Univ_prop.limsup_eq_compat [lemma, in
Uprop ]
Univ_prop.limsup_eq_mult [lemma, in
Uprop ]
Univ_prop.limsup_eq_plus_cte_right [lemma, in
Uprop ]
Univ_prop.limsup_incr [lemma, in
Uprop ]
Univ_prop.limsup_inv [lemma, in
Uprop ]
Univ_prop.limsup_le_compat [lemma, in
Uprop ]
Univ_prop.limsup_le_plus [lemma, in
Uprop ]
Univ_prop.lim_Iincl [lemma, in
Uprop ]
Univ_prop.lincl_eq_compat [lemma, in
Uprop ]
Univ_prop.lincl_eq_trans [lemma, in
Uprop ]
Univ_prop.low [projection, in
Uprop ]
Univ_prop.Low [definition, in
Uprop ]
Univ_prop.low_Imult [lemma, in
Uprop ]
Univ_prop.low_Imultk [lemma, in
Uprop ]
Univ_prop.low_Iplus [lemma, in
Uprop ]
Univ_prop.low_le_compat [lemma, in
Uprop ]
Univ_prop.low_lim [lemma, in
Uprop ]
Univ_prop.low_mon [instance, in
Uprop ]
Univ_prop.lplus_in_elim [lemma, in
Uprop ]
Univ_prop.lub_eq_esp_right [lemma, in
Uprop ]
Univ_prop.lub_eq_mult [lemma, in
Uprop ]
Univ_prop.lub_eq_plus [lemma, in
Uprop ]
Univ_prop.lub_eq_plus_cte_left [lemma, in
Uprop ]
Univ_prop.lub_eq_plus_cte_right [lemma, in
Uprop ]
Univ_prop.lub_glb_le [lemma, in
Uprop ]
Univ_prop.lub_inv [lemma, in
Uprop ]
Univ_prop.lub_le_plus [lemma, in
Uprop ]
Univ_prop.lub_lub_inv_le [lemma, in
Uprop ]
Univ_prop.lub_MF_simpl [lemma, in
Uprop ]
Univ_prop.lub_un [lemma, in
Uprop ]
Univ_prop.Max [definition, in
Uprop ]
Univ_prop.max [definition, in
Uprop ]
Univ_prop.max_eq_case [lemma, in
Uprop ]
Univ_prop.max_eq_left [lemma, in
Uprop ]
Univ_prop.max_eq_mult [lemma, in
Uprop ]
Univ_prop.max_eq_plus_cte_right [lemma, in
Uprop ]
Univ_prop.max_eq_right [lemma, in
Uprop ]
Univ_prop.max_le [lemma, in
Uprop ]
Univ_prop.max_le_compat [lemma, in
Uprop ]
Univ_prop.max_le_left [lemma, in
Uprop ]
Univ_prop.max_le_right [lemma, in
Uprop ]
Univ_prop.max_mon [instance, in
Uprop ]
Univ_prop.mean [definition, in
Uprop ]
Univ_prop.mean_eq [lemma, in
Uprop ]
Univ_prop.mean_le_compat_left [lemma, in
Uprop ]
Univ_prop.mean_le_compat_right [lemma, in
Uprop ]
Univ_prop.mean_le_down [lemma, in
Uprop ]
Univ_prop.mean_le_up [lemma, in
Uprop ]
Univ_prop.mean_lt_compat_left [lemma, in
Uprop ]
Univ_prop.mean_lt_compat_right [lemma, in
Uprop ]
Univ_prop.mean_lt_down [lemma, in
Uprop ]
Univ_prop.mean_lt_up [lemma, in
Uprop ]
Univ_prop.MF [definition, in
Uprop ]
Univ_prop.MFcpo [instance, in
Uprop ]
Univ_prop.MFopp [definition, in
Uprop ]
Univ_prop.MFopp_lub_eq [lemma, in
Uprop ]
Univ_prop.MF_lub_simpl [lemma, in
Uprop ]
Univ_prop.min [definition, in
Uprop ]
Univ_prop.Min [definition, in
Uprop ]
Univ_prop.min_eq_case [lemma, in
Uprop ]
Univ_prop.min_eq_left [lemma, in
Uprop ]
Univ_prop.min_eq_right [lemma, in
Uprop ]
Univ_prop.min_le [lemma, in
Uprop ]
Univ_prop.min_le_compat [lemma, in
Uprop ]
Univ_prop.min_le_left [lemma, in
Uprop ]
Univ_prop.min_le_right [lemma, in
Uprop ]
Univ_prop.min_lub_le [lemma, in
Uprop ]
Univ_prop.min_lub_le_incr [lemma, in
Uprop ]
Univ_prop.min_lub_le_incr_aux [lemma, in
Uprop ]
Univ_prop.min_mon2 [instance, in
Uprop ]
Univ_prop.min_mult [lemma, in
Uprop ]
Univ_prop.min_plus [lemma, in
Uprop ]
Univ_prop.min_plus_cte [lemma, in
Uprop ]
Univ_prop.min_sym [lemma, in
Uprop ]
Univ_prop.mk_IU [constructor, in
Uprop ]
Univ_prop.mufix [definition, in
Uprop ]
Univ_prop.mufix_cte [lemma, in
Uprop ]
Univ_prop.mufix_eq [lemma, in
Uprop ]
Univ_prop.mufix_inv [lemma, in
Uprop ]
Univ_prop.mufix_le [lemma, in
Uprop ]
Univ_prop.NB2U [definition, in
Uprop ]
Univ_prop.NB2Uinv [lemma, in
Uprop ]
Univ_prop.neq_sym [lemma, in
Uprop ]
Univ_prop.Nmult [definition, in
Uprop ]
Univ_prop.NMult [definition, in
Uprop ]
Univ_prop.Nmult_def [definition, in
Uprop ]
Univ_prop.Nmult_defS [lemma, in
Uprop ]
Univ_prop.Nmult_def_class [lemma, in
Uprop ]
Univ_prop.Nmult_def_intro [lemma, in
Uprop ]
Univ_prop.Nmult_def_lt [lemma, in
Uprop ]
Univ_prop.Nmult_def_O [lemma, in
Uprop ]
Univ_prop.Nmult_def_pred [lemma, in
Uprop ]
Univ_prop.Nmult_def_Unth [lemma, in
Uprop ]
Univ_prop.Nmult_def_zero [lemma, in
Uprop ]
Univ_prop.Nmult_def_1 [lemma, in
Uprop ]
Univ_prop.Nmult_eq_compat_left [lemma, in
Uprop ]
Univ_prop.Nmult_eq_compat_right [lemma, in
Uprop ]
Univ_prop.Nmult_exp2 [lemma, in
Uprop ]
Univ_prop.Nmult_ge_Sn_Unth [lemma, in
Uprop ]
Univ_prop.Nmult_le_compat [lemma, in
Uprop ]
Univ_prop.Nmult_le_compat_left [lemma, in
Uprop ]
Univ_prop.Nmult_le_compat_right [lemma, in
Uprop ]
Univ_prop.Nmult_le_n_Unth [lemma, in
Uprop ]
Univ_prop.Nmult_le_simpl [lemma, in
Uprop ]
Univ_prop.Nmult_mon2 [instance, in
Uprop ]
Univ_prop.Nmult_mult_assoc [lemma, in
Uprop ]
Univ_prop.Nmult_neq_zero [lemma, in
Uprop ]
Univ_prop.Nmult_n_Unth [lemma, in
Uprop ]
Univ_prop.Nmult_S [lemma, in
Uprop ]
Univ_prop.Nmult_sigma [lemma, in
Uprop ]
Univ_prop.Nmult_Sn_Unth [lemma, in
Uprop ]
Univ_prop.Nmult_SS [lemma, in
Uprop ]
Univ_prop.Nmult_Umult_assoc_left [lemma, in
Uprop ]
Univ_prop.Nmult_Umult_assoc_right [lemma, in
Uprop ]
Univ_prop.Nmult_Unth_eq [lemma, in
Uprop ]
Univ_prop.Nmult_Unth_le [lemma, in
Uprop ]
Univ_prop.Nmult_Unth_prop [lemma, in
Uprop ]
Univ_prop.Nmult_Unth_simpl_left [lemma, in
Uprop ]
Univ_prop.Nmult_Unth_simpl_right [lemma, in
Uprop ]
Univ_prop.Nmult_Uplus_distr [lemma, in
Uprop ]
Univ_prop.Nmult_zero [lemma, in
Uprop ]
Univ_prop.Nmult_0 [lemma, in
Uprop ]
Univ_prop.Nmult_1 [lemma, in
Uprop ]
Univ_prop.Nmult_2 [lemma, in
Uprop ]
Univ_prop.not_Ult_le [lemma, in
Uprop ]
Univ_prop.nufix [definition, in
Uprop ]
Univ_prop.nufix_cte [lemma, in
Uprop ]
Univ_prop.nufix_eq [lemma, in
Uprop ]
Univ_prop.nufix_inv [lemma, in
Uprop ]
Univ_prop.nufix_sup [lemma, in
Uprop ]
Univ_prop.orc_inv_plus_one [lemma, in
Uprop ]
Univ_prop.plus_Nmult_distr [lemma, in
Uprop ]
Univ_prop.Pmin [definition, in
Uprop ]
Univ_prop.pmin [definition, in
Uprop ]
Univ_prop.pmin_esp_le [lemma, in
Uprop ]
Univ_prop.pmin_esp_S [lemma, in
Uprop ]
Univ_prop.pmin_le [lemma, in
Uprop ]
Univ_prop.pmin_le_compat [lemma, in
Uprop ]
Univ_prop.pmin_mon [instance, in
Uprop ]
Univ_prop.pmin_plus_eq [lemma, in
Uprop ]
Univ_prop.pmin_0 [lemma, in
Uprop ]
Univ_prop.Prod [definition, in
Uprop ]
Univ_prop.prod [definition, in
Uprop ]
Univ_prop.prod_decr [lemma, in
Uprop ]
Univ_prop.prod_eq_compat [lemma, in
Uprop ]
Univ_prop.prod_inf [definition, in
Uprop ]
Univ_prop.prod_le [lemma, in
Uprop ]
Univ_prop.prod_le_compat [lemma, in
Uprop ]
Univ_prop.prod_minus [lemma, in
Uprop ]
Univ_prop.prod_not_zero [lemma, in
Uprop ]
Univ_prop.prod_S [lemma, in
Uprop ]
Univ_prop.Prod_simpl [lemma, in
Uprop ]
Univ_prop.prod_S_lift [lemma, in
Uprop ]
Univ_prop.prod_zero [lemma, in
Uprop ]
Univ_prop.prod_zero_elim [lemma, in
Uprop ]
Univ_prop.prod_0 [lemma, in
Uprop ]
Univ_prop.prod_1 [lemma, in
Uprop ]
Univ_prop.proper [projection, in
Uprop ]
Univ_prop.retract [definition, in
Uprop ]
Univ_prop.retractS [lemma, in
Uprop ]
Univ_prop.retractS_intro [lemma, in
Uprop ]
Univ_prop.retractS_inv [lemma, in
Uprop ]
Univ_prop.retract0 [lemma, in
Uprop ]
Univ_prop.retract_lt [lemma, in
Uprop ]
Univ_prop.retract_pred [lemma, in
Uprop ]
Univ_prop.retract_unif [lemma, in
Uprop ]
Univ_prop.retract_wretract [lemma, in
Uprop ]
Univ_prop.retract_zero_wretract [lemma, in
Uprop ]
Univ_prop.seq_max [definition, in
Uprop ]
Univ_prop.seq_max_incr [lemma, in
Uprop ]
Univ_prop.seq_max_le [lemma, in
Uprop ]
Univ_prop.seq_max_mon [instance, in
Uprop ]
Univ_prop.serie [definition, in
Uprop ]
Univ_prop.Serie [definition, in
Uprop ]
Univ_prop.serie_continuous [lemma, in
Uprop ]
Univ_prop.serie_eq_compat [lemma, in
Uprop ]
Univ_prop.serie_inv_le [lemma, in
Uprop ]
Univ_prop.serie_le [lemma, in
Uprop ]
Univ_prop.serie_le_compat [lemma, in
Uprop ]
Univ_prop.serie_minus_decr [lemma, in
Uprop ]
Univ_prop.serie_minus_incr [lemma, in
Uprop ]
Univ_prop.serie_mult [lemma, in
Uprop ]
Univ_prop.serie_not_zero [lemma, in
Uprop ]
Univ_prop.serie_plus [lemma, in
Uprop ]
Univ_prop.serie_prod_ge [lemma, in
Uprop ]
Univ_prop.serie_prod_le [lemma, in
Uprop ]
Univ_prop.serie_prod_maj [lemma, in
Uprop ]
Univ_prop.serie_sigma_lift [lemma, in
Uprop ]
Univ_prop.Serie_simpl [lemma, in
Uprop ]
Univ_prop.serie_S_lift [lemma, in
Uprop ]
Univ_prop.serie_zero [lemma, in
Uprop ]
Univ_prop.serie_zero_elim [lemma, in
Uprop ]
Univ_prop.Sigma [definition, in
Uprop ]
Univ_prop.sigma [definition, in
Uprop ]
Univ_prop.sigma_continuous1 [lemma, in
Uprop ]
Univ_prop.sigma_eq_compat [lemma, in
Uprop ]
Univ_prop.sigma_incr [lemma, in
Uprop ]
Univ_prop.sigma_inv [lemma, in
Uprop ]
Univ_prop.sigma_le [lemma, in
Uprop ]
Univ_prop.sigma_le_compat [lemma, in
Uprop ]
Univ_prop.sigma_lub1 [lemma, in
Uprop ]
Univ_prop.sigma_minus_decr [lemma, in
Uprop ]
Univ_prop.sigma_minus_incr [lemma, in
Uprop ]
Univ_prop.sigma_mon [instance, in
Uprop ]
Univ_prop.sigma_mult [lemma, in
Uprop ]
Univ_prop.sigma_not_zero [lemma, in
Uprop ]
Univ_prop.sigma_plus [lemma, in
Uprop ]
Univ_prop.sigma_plus_lift [lemma, in
Uprop ]
Univ_prop.sigma_prod_ge [lemma, in
Uprop ]
Univ_prop.sigma_prod_le [lemma, in
Uprop ]
Univ_prop.sigma_prod_maj [lemma, in
Uprop ]
Univ_prop.sigma_S [lemma, in
Uprop ]
Univ_prop.Sigma_simpl [lemma, in
Uprop ]
Univ_prop.sigma_S_lift [lemma, in
Uprop ]
Univ_prop.sigma_zero [lemma, in
Uprop ]
Univ_prop.sigma_zero_elim [lemma, in
Uprop ]
Univ_prop.sigma_0 [lemma, in
Uprop ]
Univ_prop.sigma_1 [lemma, in
Uprop ]
Univ_prop.singl [definition, in
Uprop ]
Univ_prop.sMax [definition, in
Uprop ]
Univ_prop.sMax_mult [lemma, in
Uprop ]
Univ_prop.sMax_plus_cte_right [lemma, in
Uprop ]
Univ_prop.sup [definition, in
Uprop ]
Univ_prop.S_pred_exp2 [lemma, in
Uprop ]
Univ_prop.twice_half_exp [lemma, in
Uprop ]
Univ_prop.Udistr_inv_left [lemma, in
Uprop ]
Univ_prop.Udistr_plus_left [lemma, in
Uprop ]
Univ_prop.Udistr_plus_left_le [lemma, in
Uprop ]
Univ_prop.UDiv [definition, in
Uprop ]
Univ_prop.Udiv_continuous [lemma, in
Uprop ]
Univ_prop.Udiv_eq_compat_left [lemma, in
Uprop ]
Univ_prop.Udiv_eq_compat_right [lemma, in
Uprop ]
Univ_prop.Udiv_inv [lemma, in
Uprop ]
Univ_prop.Udiv_le [lemma, in
Uprop ]
Univ_prop.Udiv_le_compat_left [lemma, in
Uprop ]
Univ_prop.Udiv_le_compat_right [lemma, in
Uprop ]
Univ_prop.Udiv_minus [lemma, in
Uprop ]
Univ_prop.Udiv_mon [instance, in
Uprop ]
Univ_prop.Udiv_mult [lemma, in
Uprop ]
Univ_prop.Udiv_mult_assoc [lemma, in
Uprop ]
Univ_prop.Udiv_mult_le [lemma, in
Uprop ]
Univ_prop.Udiv_one [lemma, in
Uprop ]
Univ_prop.Udiv_plus [lemma, in
Uprop ]
Univ_prop.Udiv_plus_le [lemma, in
Uprop ]
Univ_prop.Udiv_refl [lemma, in
Uprop ]
Univ_prop.UDiv_simpl [lemma, in
Uprop ]
Univ_prop.Udiv_zero [lemma, in
Uprop ]
Univ_prop.Udiv_zero_eq [lemma, in
Uprop ]
Univ_prop.Ueq_class [lemma, in
Uprop ]
Univ_prop.Ueq_double_neg [lemma, in
Uprop ]
Univ_prop.Ueq_orc [lemma, in
Uprop ]
Univ_prop.UEsp [definition, in
Uprop ]
Univ_prop.Uesp [definition, in
Uprop ]
Univ_prop.Uesp_le_compat [lemma, in
Uprop ]
Univ_prop.Uesp_le_left [lemma, in
Uprop ]
Univ_prop.Uesp_le_plus_inv [lemma, in
Uprop ]
Univ_prop.Uesp_le_right [lemma, in
Uprop ]
Univ_prop.Uesp_min [lemma, in
Uprop ]
Univ_prop.Uesp_minus_distr [lemma, in
Uprop ]
Univ_prop.Uesp_minus_distr_left [lemma, in
Uprop ]
Univ_prop.Uesp_minus_distr_right [lemma, in
Uprop ]
Univ_prop.Uesp_mon [instance, in
Uprop ]
Univ_prop.Uesp_one_left [lemma, in
Uprop ]
Univ_prop.Uesp_one_right [lemma, in
Uprop ]
Univ_prop.Uesp_plus_inv [lemma, in
Uprop ]
Univ_prop.UEsp_simpl [lemma, in
Uprop ]
Univ_prop.Uesp_sym [lemma, in
Uprop ]
Univ_prop.Uesp_zero [lemma, in
Uprop ]
Univ_prop.Uesp_zero_left [lemma, in
Uprop ]
Univ_prop.Uesp_zero_one_mult_left [lemma, in
Uprop ]
Univ_prop.Uesp_zero_one_mult_right [lemma, in
Uprop ]
Univ_prop.Uesp_zero_right [lemma, in
Uprop ]
Univ_prop.Uexp [definition, in
Uprop ]
Univ_prop.UExp [definition, in
Uprop ]
Univ_prop.Uexp_inv_S [lemma, in
Uprop ]
Univ_prop.Uexp_le_compat [lemma, in
Uprop ]
Univ_prop.Uexp_le_compat_left [lemma, in
Uprop ]
Univ_prop.Uexp_le_compat_right [lemma, in
Uprop ]
Univ_prop.Uexp_lt_antimon [lemma, in
Uprop ]
Univ_prop.Uexp_lt_compat [lemma, in
Uprop ]
Univ_prop.Uexp_lt_one [lemma, in
Uprop ]
Univ_prop.Uexp_lt_zero [lemma, in
Uprop ]
Univ_prop.Uexp_mon2 [instance, in
Uprop ]
Univ_prop.Uexp_one [lemma, in
Uprop ]
Univ_prop.Uexp_zero [lemma, in
Uprop ]
Univ_prop.Uexp_0 [lemma, in
Uprop ]
Univ_prop.Uexp_1 [lemma, in
Uprop ]
Univ_prop.Uge_one_eq [lemma, in
Uprop ]
Univ_prop.Uglb [definition, in
Uprop ]
Univ_prop.Uglb_eq_mult [lemma, in
Uprop ]
Univ_prop.Uglb_eq_plus_cte_right [lemma, in
Uprop ]
Univ_prop.Uglb_glb [lemma, in
Uprop ]
Univ_prop.Uglb_le [lemma, in
Uprop ]
Univ_prop.Uglb_le_compat [lemma, in
Uprop ]
Univ_prop.Uglb_le_plus [lemma, in
Uprop ]
Univ_prop.UInv [definition, in
Uprop ]
Univ_prop.UInvopp [definition, in
Uprop ]
Univ_prop.Uinvopp_continuous [lemma, in
Uprop ]
Univ_prop.UInvopp_simpl [lemma, in
Uprop ]
Univ_prop.Uinv_bary [lemma, in
Uprop ]
Univ_prop.Uinv_bary_eq [lemma, in
Uprop ]
Univ_prop.Uinv_bary_le [lemma, in
Uprop ]
Univ_prop.Uinv_continuous [lemma, in
Uprop ]
Univ_prop.Uinv_double_eq_simpl_left [lemma, in
Uprop ]
Univ_prop.Uinv_double_eq_simpl_right [lemma, in
Uprop ]
Univ_prop.Uinv_double_le_simpl_left [lemma, in
Uprop ]
Univ_prop.Uinv_double_le_simpl_right [lemma, in
Uprop ]
Univ_prop.Uinv_eq_perm_left [lemma, in
Uprop ]
Univ_prop.Uinv_eq_perm_right [lemma, in
Uprop ]
Univ_prop.Uinv_eq_simpl [lemma, in
Uprop ]
Univ_prop.Uinv_esp_plus [lemma, in
Uprop ]
Univ_prop.Uinv_half [lemma, in
Uprop ]
Univ_prop.Uinv_half_bary [lemma, in
Uprop ]
Univ_prop.Uinv_inv [lemma, in
Uprop ]
Univ_prop.Uinv_le_half_left [lemma, in
Uprop ]
Univ_prop.Uinv_le_half_right [lemma, in
Uprop ]
Univ_prop.Uinv_le_perm_left [lemma, in
Uprop ]
Univ_prop.Uinv_le_perm_right [lemma, in
Uprop ]
Univ_prop.Uinv_le_simpl [lemma, in
Uprop ]
Univ_prop.Uinv_le_trans [lemma, in
Uprop ]
Univ_prop.Uinv_lt_one [lemma, in
Uprop ]
Univ_prop.Uinv_lt_perm_left [lemma, in
Uprop ]
Univ_prop.Uinv_lt_perm_right [lemma, in
Uprop ]
Univ_prop.Uinv_lt_zero [lemma, in
Uprop ]
Univ_prop.Uinv_max_min [lemma, in
Uprop ]
Univ_prop.Uinv_min_max [lemma, in
Uprop ]
Univ_prop.Uinv_mon [instance, in
Uprop ]
Univ_prop.Uinv_mult_minus [lemma, in
Uprop ]
Univ_prop.Uinv_mult_simpl [lemma, in
Uprop ]
Univ_prop.Uinv_neq_compat [lemma, in
Uprop ]
Univ_prop.Uinv_neq_left [lemma, in
Uprop ]
Univ_prop.Uinv_neq_right [lemma, in
Uprop ]
Univ_prop.Uinv_neq_simpl [lemma, in
Uprop ]
Univ_prop.Uinv_Nmult [lemma, in
Uprop ]
Univ_prop.Uinv_opp_left [lemma, in
Uprop ]
Univ_prop.Uinv_opp_right [lemma, in
Uprop ]
Univ_prop.Uinv_plus_esp [lemma, in
Uprop ]
Univ_prop.Uinv_plus_left_le [lemma, in
Uprop ]
Univ_prop.Uinv_plus_right [lemma, in
Uprop ]
Univ_prop.Uinv_plus_right_le [lemma, in
Uprop ]
Univ_prop.Uinv_simpl [lemma, in
Uprop ]
Univ_prop.UInv_simpl [definition, in
Uprop ]
Univ_prop.Uinv_zero [lemma, in
Uprop ]
Univ_prop.Ule_diff_lt [lemma, in
Uprop ]
Univ_prop.Ule_eq_compat [lemma, in
Uprop ]
Univ_prop.Ule_lt_half_exp [lemma, in
Uprop ]
Univ_prop.Ule_lt_lim [lemma, in
Uprop ]
Univ_prop.Ule_lt_trans [lemma, in
Uprop ]
Univ_prop.Ule_mult_left [lemma, in
Uprop ]
Univ_prop.Ule_mult_right [lemma, in
Uprop ]
Univ_prop.Ule_neq_zero [lemma, in
Uprop ]
Univ_prop.Ule_not_lt [lemma, in
Uprop ]
Univ_prop.Ule_nth_lim [lemma, in
Uprop ]
Univ_prop.Ule_orc [lemma, in
Uprop ]
Univ_prop.Ule_plus_left [lemma, in
Uprop ]
Univ_prop.Ule_plus_right [lemma, in
Uprop ]
Univ_prop.Ule_zero_eq [lemma, in
Uprop ]
Univ_prop.Ule_0_1 [lemma, in
Uprop ]
Univ_prop.Ult [definition, in
Uprop ]
Univ_prop.Ult_antirefl [lemma, in
Uprop ]
Univ_prop.Ult_class [lemma, in
Uprop ]
Univ_prop.Ult_eq_compat [lemma, in
Uprop ]
Univ_prop.Ult_le [lemma, in
Uprop ]
Univ_prop.Ult_le_nth_minus [lemma, in
Uprop ]
Univ_prop.Ult_le_nth_plus [lemma, in
Uprop ]
Univ_prop.Ult_le_trans [lemma, in
Uprop ]
Univ_prop.Ult_neq [lemma, in
Uprop ]
Univ_prop.Ult_neq_rev [lemma, in
Uprop ]
Univ_prop.Ult_neq_zero [lemma, in
Uprop ]
Univ_prop.Ult_trans [lemma, in
Uprop ]
Univ_prop.Ult_0_1 [lemma, in
Uprop ]
Univ_prop.Ulub [definition, in
Uprop ]
Univ_prop.Ulub_eq_esp_right [lemma, in
Uprop ]
Univ_prop.Ulub_eq_mult [lemma, in
Uprop ]
Univ_prop.Ulub_eq_plus_cte_right [lemma, in
Uprop ]
Univ_prop.Ulub_le [lemma, in
Uprop ]
Univ_prop.Ulub_le_compat [lemma, in
Uprop ]
Univ_prop.Ulub_le_plus [lemma, in
Uprop ]
Univ_prop.Ulub_lub [lemma, in
Uprop ]
Univ_prop.Ulub_mon [lemma, in
Uprop ]
Univ_prop.Ulub_Uglb_exch_le [lemma, in
Uprop ]
Univ_prop.UMinus [definition, in
Uprop ]
Univ_prop.Uminus [definition, in
Uprop ]
Univ_prop.Uminus_assoc_left [lemma, in
Uprop ]
Univ_prop.Uminus_assoc_right [lemma, in
Uprop ]
Univ_prop.Uminus_bary_left [lemma, in
Uprop ]
Univ_prop.Uminus_bary_left_eq [lemma, in
Uprop ]
Univ_prop.Uminus_continuous2 [lemma, in
Uprop ]
Univ_prop.Uminus_distr_left [lemma, in
Uprop ]
Univ_prop.Uminus_distr_right [lemma, in
Uprop ]
Univ_prop.Uminus_eq [lemma, in
Uprop ]
Univ_prop.Uminus_eq_perm_left [lemma, in
Uprop ]
Univ_prop.Uminus_eq_perm_right [lemma, in
Uprop ]
Univ_prop.Uminus_esp_le_inv [lemma, in
Uprop ]
Univ_prop.Uminus_esp_simpl_left [lemma, in
Uprop ]
Univ_prop.Uminus_le_compat [lemma, in
Uprop ]
Univ_prop.Uminus_le_compat_left [lemma, in
Uprop ]
Univ_prop.Uminus_le_compat_right [lemma, in
Uprop ]
Univ_prop.Uminus_le_inv [lemma, in
Uprop ]
Univ_prop.Uminus_le_left [lemma, in
Uprop ]
Univ_prop.Uminus_le_perm [lemma, in
Uprop ]
Univ_prop.Uminus_le_perm_left [lemma, in
Uprop ]
Univ_prop.Uminus_le_perm_right [lemma, in
Uprop ]
Univ_prop.Uminus_le_zero [lemma, in
Uprop ]
Univ_prop.Uminus_lt_non_zero [lemma, in
Uprop ]
Univ_prop.Uminus_mon2 [instance, in
Uprop ]
Univ_prop.Uminus_one_left [lemma, in
Uprop ]
Univ_prop.Uminus_perm [lemma, in
Uprop ]
Univ_prop.Uminus_plus_le [lemma, in
Uprop ]
Univ_prop.Uminus_plus_perm [lemma, in
Uprop ]
Univ_prop.Uminus_plus_perm_right [lemma, in
Uprop ]
Univ_prop.Uminus_plus_simpl [lemma, in
Uprop ]
Univ_prop.Uminus_plus_zero [lemma, in
Uprop ]
Univ_prop.UMinus_simpl [lemma, in
Uprop ]
Univ_prop.Uminus_triangular [lemma, in
Uprop ]
Univ_prop.Uminus_zero_le [lemma, in
Uprop ]
Univ_prop.Uminus_zero_left [lemma, in
Uprop ]
Univ_prop.Uminus_zero_right [lemma, in
Uprop ]
Univ_prop.UMult [definition, in
Uprop ]
Univ_prop.UMultk_eq [lemma, in
Uprop ]
Univ_prop.Umult_continuous2 [instance, in
Uprop ]
Univ_prop.UMult_continuous_left [lemma, in
Uprop ]
Univ_prop.UMult_continuous_right [lemma, in
Uprop ]
Univ_prop.Umult_decomp [lemma, in
Uprop ]
Univ_prop.Umult_div_assoc [lemma, in
Uprop ]
Univ_prop.Umult_div_eq [lemma, in
Uprop ]
Univ_prop.Umult_div_le [lemma, in
Uprop ]
Univ_prop.Umult_div_le_left [lemma, in
Uprop ]
Univ_prop.Umult_div_le_right [lemma, in
Uprop ]
Univ_prop.Umult_eq_compat_left [lemma, in
Uprop ]
Univ_prop.Umult_eq_compat_right [lemma, in
Uprop ]
Univ_prop.Umult_inv_minus [lemma, in
Uprop ]
Univ_prop.Umult_inv_plus [lemma, in
Uprop ]
Univ_prop.Umult_inv_plus_le [lemma, in
Uprop ]
Univ_prop.Umult_le_compat [lemma, in
Uprop ]
Univ_prop.Umult_le_compat_left [lemma, in
Uprop ]
Univ_prop.Umult_le_compat_right [lemma, in
Uprop ]
Univ_prop.Umult_le_simpl_right [lemma, in
Uprop ]
Univ_prop.Umult_lt_compat [lemma, in
Uprop ]
Univ_prop.Umult_lt_compat_left [lemma, in
Uprop ]
Univ_prop.Umult_lt_compat_right [lemma, in
Uprop ]
Univ_prop.Umult_lt_left [lemma, in
Uprop ]
Univ_prop.Umult_lt_right [lemma, in
Uprop ]
Univ_prop.Umult_lt_simpl_left [lemma, in
Uprop ]
Univ_prop.Umult_lt_simpl_right [lemma, in
Uprop ]
Univ_prop.Umult_lt_zero [lemma, in
Uprop ]
Univ_prop.Umult_mon2 [instance, in
Uprop ]
Univ_prop.Umult_neq_zero [lemma, in
Uprop ]
Univ_prop.Umult_one_left_eq [lemma, in
Uprop ]
Univ_prop.Umult_one_right [lemma, in
Uprop ]
Univ_prop.Umult_one_right_eq [lemma, in
Uprop ]
Univ_prop.Umult_perm2 [lemma, in
Uprop ]
Univ_prop.Umult_perm3 [lemma, in
Uprop ]
Univ_prop.UMult_simpl [definition, in
Uprop ]
Univ_prop.Umult_simpl_left [lemma, in
Uprop ]
Univ_prop.Umult_simpl_right [lemma, in
Uprop ]
Univ_prop.Umult_zero_eq [lemma, in
Uprop ]
Univ_prop.Umult_zero_left [lemma, in
Uprop ]
Univ_prop.Umult_zero_left_eq [lemma, in
Uprop ]
Univ_prop.Umult_zero_right [lemma, in
Uprop ]
Univ_prop.Umult_zero_right_eq [lemma, in
Uprop ]
Univ_prop.Umult_zero_simpl_left [lemma, in
Uprop ]
Univ_prop.Umult_zero_simpl_right [lemma, in
Uprop ]
Univ_prop.Unit [lemma, in
Uprop ]
Univ_prop.Unth_anti_mon [lemma, in
Uprop ]
Univ_prop.Unth_decr [lemma, in
Uprop ]
Univ_prop.Unth_eq [lemma, in
Uprop ]
Univ_prop.Unth_exp2 [lemma, in
Uprop ]
Univ_prop.Unth_half [lemma, in
Uprop ]
Univ_prop.Unth_inv_lt_one [lemma, in
Uprop ]
Univ_prop.Unth_le_half [lemma, in
Uprop ]
Univ_prop.Unth_lt_zero [lemma, in
Uprop ]
Univ_prop.Unth_not_null [lemma, in
Uprop ]
Univ_prop.Unth_not_one [lemma, in
Uprop ]
Univ_prop.Unth_one [lemma, in
Uprop ]
Univ_prop.Unth_one_plus [lemma, in
Uprop ]
Univ_prop.Unth_prop_sigma [lemma, in
Uprop ]
Univ_prop.Unth_sigma_n [lemma, in
Uprop ]
Univ_prop.Unth_sigma_Sn [lemma, in
Uprop ]
Univ_prop.Unth_zero [lemma, in
Uprop ]
Univ_prop.Uopp [definition, in
Uprop ]
Univ_prop.Uopp_lub_simpl [lemma, in
Uprop ]
Univ_prop.Uopp_mon_seq [lemma, in
Uprop ]
Univ_prop.Up [definition, in
Uprop ]
Univ_prop.up [projection, in
Uprop ]
Univ_prop.UPlus [definition, in
Uprop ]
Univ_prop.UPlusk_eq [lemma, in
Uprop ]
Univ_prop.Uplus_continuous2 [instance, in
Uprop ]
Univ_prop.UPlus_continuous_left [lemma, in
Uprop ]
Univ_prop.UPlus_continuous_right [lemma, in
Uprop ]
Univ_prop.Uplus_div_inv [lemma, in
Uprop ]
Univ_prop.Uplus_eq_compat_left [lemma, in
Uprop ]
Univ_prop.Uplus_eq_compat_right [lemma, in
Uprop ]
Univ_prop.Uplus_eq_perm_left [lemma, in
Uprop ]
Univ_prop.Uplus_eq_simpl_left [lemma, in
Uprop ]
Univ_prop.Uplus_eq_simpl_right [lemma, in
Uprop ]
Univ_prop.Uplus_eq_zero [lemma, in
Uprop ]
Univ_prop.Uplus_eq_zero_left [lemma, in
Uprop ]
Univ_prop.Uplus_esp_inv_simpl [lemma, in
Uprop ]
Univ_prop.Uplus_esp_simpl [lemma, in
Uprop ]
Univ_prop.Uplus_inv_esp_simpl [lemma, in
Uprop ]
Univ_prop.Uplus_inv_le_esp [lemma, in
Uprop ]
Univ_prop.Uplus_le_compat [lemma, in
Uprop ]
Univ_prop.Uplus_le_compat_left [lemma, in
Uprop ]
Univ_prop.Uplus_le_compat_right [lemma, in
Uprop ]
Univ_prop.Uplus_le_perm_left [lemma, in
Uprop ]
Univ_prop.Uplus_le_perm_right [lemma, in
Uprop ]
Univ_prop.Uplus_le_simpl_left [lemma, in
Uprop ]
Univ_prop.Uplus_lt_compat [lemma, in
Uprop ]
Univ_prop.Uplus_lt_compat_left [lemma, in
Uprop ]
Univ_prop.Uplus_lt_compat_right [lemma, in
Uprop ]
Univ_prop.Uplus_lt_simpl_left [lemma, in
Uprop ]
Univ_prop.Uplus_lt_simpl_right [lemma, in
Uprop ]
Univ_prop.Uplus_lt_Uinv [lemma, in
Uprop ]
Univ_prop.Uplus_minus_assoc_right [lemma, in
Uprop ]
Univ_prop.Uplus_minus_simpl_left [lemma, in
Uprop ]
Univ_prop.Uplus_minus_simpl_right [lemma, in
Uprop ]
Univ_prop.Uplus_mon2 [instance, in
Uprop ]
Univ_prop.Uplus_neq_zero_left [lemma, in
Uprop ]
Univ_prop.Uplus_neq_zero_right [lemma, in
Uprop ]
Univ_prop.Uplus_one [lemma, in
Uprop ]
Univ_prop.Uplus_one_le [lemma, in
Uprop ]
Univ_prop.Uplus_one_left [lemma, in
Uprop ]
Univ_prop.Uplus_one_right [lemma, in
Uprop ]
Univ_prop.Uplus_opp_continuous_left [lemma, in
Uprop ]
Univ_prop.Uplus_opp_continuous_right [lemma, in
Uprop ]
Univ_prop.Uplus_perm2 [lemma, in
Uprop ]
Univ_prop.Uplus_perm3 [lemma, in
Uprop ]
Univ_prop.UPlus_simpl [definition, in
Uprop ]
Univ_prop.Uplus_zero_right [lemma, in
Uprop ]
Univ_prop.Upos [lemma, in
Uprop ]
Univ_prop.up_Imult [lemma, in
Uprop ]
Univ_prop.up_Imultk [lemma, in
Uprop ]
Univ_prop.up_Iplus [lemma, in
Uprop ]
Univ_prop.up_le_compat [lemma, in
Uprop ]
Univ_prop.up_lim [lemma, in
Uprop ]
Univ_prop.up_mon [instance, in
Uprop ]
Univ_prop.U1min [definition, in
Uprop ]
Univ_prop.U1min_S [lemma, in
Uprop ]
Univ_prop.U1min_0 [lemma, in
Uprop ]
Univ_prop.wretract [definition, in
Uprop ]
Univ_prop.wretract_le [lemma, in
Uprop ]
Univ_prop.wretract_lt [lemma, in
Uprop ]
Univ_prop.wretract_retract [lemma, in
Uprop ]
UP [module, in
Monads ]
Uprop [library]
Utheory [library]
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html/index_constructor_R.html 0000644 0001750 0001750 00000022476 11301735615 016632 0 ustar cpaulin cpaulin
Ycart
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K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_constructor_V.html 0000644 0001750 0001750 00000022476 11301735615 016636 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_instance_F.html 0000644 0001750 0001750 00000026014 11301735615 016025 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
F (instance)
Fcontm_continuous [in
Ccpo ]
fcontm_monotonic [in
Ccpo ]
fcont_comp_continuous [in
Ccpo ]
fcont_Comp_continuous2 [in
Ccpo ]
fcont_continuous2 [in
Ccpo ]
fcont_cpo [in
Ccpo ]
fcont_lub_continuous [in
Ccpo ]
fcont_ord [in
Ccpo ]
fcpo [in
Ccpo ]
fixp_monotonic [in
Ccpo ]
fmono [in
Ccpo ]
fmonotonic [in
Ccpo ]
fmonotonic2 [in
Ccpo ]
fmon2_mon [in
Ccpo ]
fmon_cpo [in
Ccpo ]
ford [in
Ccpo ]
fstable [in
Ccpo ]
fstable2 [in
Ccpo ]
fun2_monotonic [in
Ccpo ]
html/index_inductive_V.html 0000644 0001750 0001750 00000022476 11301735615 016243 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_record_H.html 0000644 0001750 0001750 00000022476 11301735615 015511 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_axiom_B.html 0000644 0001750 0001750 00000022476 11301735615 015342 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_definition__.html 0000644 0001750 0001750 00000022476 11301735615 016412 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_definition_D.html 0000644 0001750 0001750 00000023140 11301735615 016344 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
D (definition)
dec [in
Sets ]
diag [in
Ccpo ]
double_app [in
Ccpo ]
html/index_constructor_P.html 0000644 0001750 0001750 00000022476 11301735615 016630 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_instance_L.html 0000644 0001750 0001750 00000023176 11301735615 016041 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
L (instance)
lub_continuous [in
Ccpo ]
lub_mon [in
Ccpo ]
lub_shift_mon [in
Ccpo ]
html/index_global_I.html 0000644 0001750 0001750 00000061620 11301735615 015466 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
I
Id [definition, in
Ccpo ]
id [definition, in
Ccpo ]
Id_mon [instance, in
Ccpo ]
Id_simpl [lemma, in
Ccpo ]
if_else [lemma, in
Misc ]
if_then [lemma, in
Misc ]
Imon [definition, in
Ccpo ]
imon [definition, in
Ccpo ]
Imonotonic [lemma, in
Ccpo ]
Imonotonic2 [lemma, in
Ccpo ]
imon2 [definition, in
Ccpo ]
Imon2 [definition, in
Ccpo ]
Imon2_simpl [lemma, in
Ccpo ]
imon2_simpl [lemma, in
Ccpo ]
Imon_simpl [lemma, in
Ccpo ]
imon_simpl [lemma, in
Ccpo ]
incl [definition, in
Sets ]
incl_add_empty [lemma, in
Sets ]
incl_add_rem [lemma, in
Sets ]
incl_add_stable [lemma, in
Sets ]
incl_empty [lemma, in
Sets ]
incl_empty_false [lemma, in
Sets ]
incl_refl [lemma, in
Sets ]
incl_rem_add [lemma, in
Sets ]
incl_rem_add_iff [lemma, in
Sets ]
incl_rem_stable [lemma, in
Sets ]
incl_trans [lemma, in
Sets ]
incl_union_stable [lemma, in
Sets ]
inter [definition, in
Sets ]
inter_add_left_in [lemma, in
Sets ]
inter_add_left_out [lemma, in
Sets ]
inter_add_right_in [lemma, in
Sets ]
inter_add_right_out [lemma, in
Sets ]
inter_empty_left [lemma, in
Sets ]
inter_empty_right [lemma, in
Sets ]
inter_sym [lemma, in
Sets ]
Iord [definition, in
Ccpo ]
Iord_app [lemma, in
Ccpo ]
isempty [definition, in
Sets ]
isempty_dec [lemma, in
Sets ]
ishift [definition, in
Ccpo ]
ishift_continuous [instance, in
Ccpo ]
ishift_le_compat [lemma, in
Ccpo ]
ishift_mon [instance, in
Ccpo ]
ishift_simpl [lemma, in
Ccpo ]
Iter [definition, in
Ccpo ]
iter [definition, in
Ccpo ]
IterFlip [module, in
IterFlip ]
IterFlip [library]
IterFlip.Fiter [definition, in
IterFlip ]
IterFlip.Fiter_mon [instance, in
IterFlip ]
IterFlip.Fiter_simpl [lemma, in
IterFlip ]
IterFlip.iterflip [definition, in
IterFlip ]
IterFlip.iterflip_term [lemma, in
IterFlip ]
IterFlip.lim_p_one [lemma, in
IterFlip ]
IterFlip.p [definition, in
IterFlip ]
IterFlip.pS_simpl [lemma, in
IterFlip ]
IterFlip.p_ [definition, in
IterFlip ]
IterFlip.p_eq [lemma, in
IterFlip ]
IterFlip.p_incr [lemma, in
IterFlip ]
IterFlip.p_le [lemma, in
IterFlip ]
IterFlip.q1 [definition, in
IterFlip ]
iterO_simpl [lemma, in
Ccpo ]
IterS_simpl [lemma, in
Ccpo ]
iterS_simpl [lemma, in
Ccpo ]
iter_ [definition, in
Ccpo ]
iter_continuous [lemma, in
Ccpo ]
iter_continuous_eq [lemma, in
Ccpo ]
iter_incr [lemma, in
Ccpo ]
iter_mon [instance, in
Ccpo ]
iter_monotonic [instance, in
Ccpo ]
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_constructor_X.html 0000644 0001750 0001750 00000022476 11301735615 016640 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_section_B.html 0000644 0001750 0001750 00000022476 11301735615 015671 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_axiom_M.html 0000644 0001750 0001750 00000022476 11301735615 015355 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_module_N.html 0000644 0001750 0001750 00000022476 11301735615 015526 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
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I (instance)
Id_mon [in
Ccpo ]
ishift_continuous [in
Ccpo ]
ishift_mon [in
Ccpo ]
IterFlip.Fiter_mon [in
IterFlip ]
iter_mon [in
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html/Bernoulli.html 0000644 0001750 0001750 00000341112 11301735615 014517 0 ustar cpaulin cpaulin
Bernoulli
Library Bernoulli
Bernoulli.v: Simulating Bernoulli and Binomial distributions
Require Export Cover .
Require Export Misc .
Set Implicit Arguments .
Program for computing a Bernoulli distribution
bernoulli p gives true with probability p
and false with probability (1-p)
let rec bernoulli x =
if flip
then (if x < 1/2 then false else bernoulli (2 p - 1))
else (if x < 1/2 then bernoulli (2 p) else true)
Hypothesis dec_demi :
forall x :
U , {
x < [1/2]}+{[1/2] <=
x }.
Instance Fbern_mon :
monotonic
(
fun (
f :U ->
distr bool )
p =>
Mif Flip
(
if dec_demi p then Munit false else f (
p &
p ))
(
if dec_demi p then f (
p +
p )
else Munit true )).
intros ;
intros f g H n .
apply Mif_le_compat ;
case (
dec_demi n );
auto .
Save .
Definition Fbern : (
U ->
distr bool ) -m> (
U ->
distr bool )
:=
mon (
fun f p =>
Mif Flip
(
if dec_demi p then Munit false else f (
p &
p ))
(
if dec_demi p then f (
p +
p )
else Munit true )).
Lemma Fbern_simpl :
forall f p ,
Fbern f p =
Mif Flip
(
if dec_demi p then Munit false else f (
p &
p ))
(
if dec_demi p then f (
p +
p )
else Munit true ).
trivial .
Save .
Definition bernoulli :
U ->
distr bool :=
Mfix Fbern .
Properties of the Bernoulli program
Proofs using fixpoint rules
Definition Mubern (
q :
bool ->
U ) :
MF U -m>
MF U .
intro q ;
exists (
fun bern (
p :U) =>
if dec_demi p then [1/2]*(q
false )+[1/2]*(bern (
p +p))
else [1/2]*(bern (
p &p)) + [1/2]*(q
true )).
red ;
intros f g H p .
case (
dec_demi p );
repeat Usimpl ;
auto .
Defined .
Lemma Mubern_simpl :
forall (
q :
bool ->
U )
f p ,
Mubern q f p =
if dec_demi p then [1/2]*(q
false )+[1/2]*(f (
p +p))
else [1/2]*(f (
p &p)) + [1/2]*(q
true ).
trivial .
Save .
Lemma Mubern_eq :
forall (
q :
bool ->
U ) (
f :U ->
distr bool ) (
p :U),
mu (
Fbern f p )
q ==
Mubern q (
fun y =>
mu (
f y )
q )
p .
intros ;
rewrite Mubern_simpl ;
rewrite Fbern_simpl .
case (
dec_demi p ).
rewrite Mif_eq ;
rewrite Flip_ctrue ;
rewrite Flip_cfalse ;
rewrite Munit_simpl ;
auto .
rewrite Mif_eq ;
rewrite Flip_ctrue ;
rewrite Flip_cfalse ;
rewrite Munit_simpl ;
auto .
Save .
Hint Resolve Mubern_eq .
Lemma Bern_eq :
forall q :
bool ->
U ,
forall p ,
mu (
bernoulli p )
q ==
mufix (
Mubern q )
p .
intros ;
apply Oeq_sym .
unfold bernoulli ;
apply mufix_mu with (
muF :=(Mubern
q )) (
q :=fun (
p :U) =>
q );
auto .
Save .
Hint Resolve Bern_eq .
Lemma Bern_commute :
forall q :
bool ->
U ,
mu_muF_commute_le Fbern (
fun (
x :U)=>q) (
Mubern q ).
red ;
auto .
Save .
Hint Resolve Bern_commute .
Lemma Bern_term :
forall p ,
mu (
bernoulli p ) (
fone bool ) == 1.
intros ;
transitivity (
mufix (
Mubern (
fone bool ))
p );
auto .
transitivity (
lub U1min );
auto .
unfold mufix ,
fixp ;
simpl ;
apply lub_eq_compat .
intro n .
transitivity (
Ccpo.iter (
D :=MF
U ) (
Mubern (
fone bool ))
n p );
auto .
generalize p ;
induction n ;
auto .
intros ;
rewrite U1min_S .
rewrite iterS_simpl .
rewrite Mubern_simpl .
unfold fone ;
repeat Usimpl .
case (
dec_demi p0 );
rewrite IHn ;
repeat Usimpl ;
auto .
Save .
Hint Resolve Bern_term .
p is an invariant of Mubern qtrue
Lemma MuBern_true :
forall p ,
Mubern B2U (
fun q =>
q )
p ==
p .
intros ;
unfold B2U ;
rewrite Mubern_simpl ;
case (
dec_demi p );
intros ;
repeat Usimpl .
apply half_twice ;
auto .
apply half_esp ;
auto .
Save .
Hint Resolve MuBern_true .
Lemma MuBern_false :
forall p ,
Mubern (
finv B2U ) (
finv (
fun q =>
q ))
p == [1-]
p .
intros ;
unfold finv ,
B2U ;
rewrite Mubern_simpl ;
case (
dec_demi p );
intros ;
repeat Usimpl .
rewrite Uplus_sym ;
rewrite Uinv_half ;
repeat Usimpl .
apply half_twice ;
auto .
rewrite Uinv_esp_plus .
apply half_twice ;
auto .
Save .
Hint Resolve MuBern_false .
Lemma Bern_true :
forall p ,
mu (
bernoulli p )
B2U ==
p .
intros ;
unfold bernoulli .
apply muF_eq with
(
muFqinv :=
Mubern (
qinv (
fun (
x :U) =>
B2U )
p ))
(
muFq :=Mubern
B2U )
(
q :=fun (
x :U) => (
B2U :bool->U))
(
f :=fun (
x :U) =>
x );
intros ;
auto .
unfold qinv ;
auto .
intro ;
unfold finv ;
auto .
exact (
Bern_term p ).
Save .
Lemma Bern_false :
forall p ,
mu (
bernoulli p )
NB2U == [1-]
p .
intros ;
transitivity (
mu (
bernoulli p ) (
finv B2U )).
apply mu_stable_eq ;
auto .
rewrite mu_inv_minus .
rewrite Bern_term ;
rewrite Bern_true ;
auto .
Save .
Lemma Mubern_inv :
forall (
q :
bool ->
U ) (
f :U ->
U ) (
p :U),
Mubern (
finv q ) (
finv f )
p == [1-]
Mubern q f p .
intros ;
unfold finv ;
repeat (
rewrite Mubern_simpl ).
case (
dec_demi p );
intro ;
auto .
Save .
Invariant pmin p
Property :
Definition qtrue (
p :U) :=
B2U .
Definition qfalse (
p :U) :=
NB2U .
Lemma bernoulli_true :
okfun (
fun p =>
p )
bernoulli qtrue .
unfold bernoulli ;
intros .
apply okfun_le_compat with (
fun p =>
lub (
Pmin p ))
qtrue ;
auto .
intro x ;
auto .
apply fixrule with (
p :=
Pmin );
intros .
simpl ;
auto .
red ;
simpl ;
intros .
red .
rewrite
(
Mif_eq Flip
(
if dec_demi x0 then Munit false else f (
x0 &
x0 ))
(
if dec_demi x0 then f (
x0 +
x0 )
else Munit true ) (
qtrue x0 ));
simpl .
case (
dec_demi x0 );
simpl ;
intros .
unfold flip ,
unit ,
ctrue ,
cfalse ;
simpl .
repeat Usimpl .
transitivity ((
pmin (
x0 +
x0 )
i ) * [1/2]).
assert (
x0 <= [1/2]);
auto .
rewrite (
pmin_plus_eq x0 i H0 ).
Usimpl ;
trivial .
Usimpl ;
apply (
H (
x0 +x0));
auto .
unfold flip ,
unit ,
ctrue ,
cfalse ;
simpl .
repeat Usimpl .
transitivity ((
pmin (
x0 &
x0 )
i ) * [1/2] + [1/2]).
apply Ole_trans with (1:=(pmin_esp_le
x0 i ));
auto .
repeat Usimpl ;
apply (
H (
x0 &x0));
auto .
Save .
Property :
Lemma bernoulli_false :
okfun (
fun p => [1-]
p )
bernoulli qfalse .
unfold bernoulli ;
intros .
apply okfun_le_compat with (
fun p =>
lub (
Pmin ([1-]
p )))
qfalse ;
auto .
intro x ;
auto .
apply fixrule with (
p :=
fun p =>
Pmin ([1-]
p ));
auto ;
intros .
simpl ;
auto .
red ;
simpl ;
intros .
red .
rewrite
(
Mif_eq Flip
(
if dec_demi x0 then Munit false else f (
x0 &
x0 ))
(
if dec_demi x0 then f (
x0 +
x0 )
else Munit true ) (
qfalse x0 ));
simpl .
case (
dec_demi x0 );
simpl ;
intros .
unfold flip ,
unit ,
ctrue ,
cfalse ;
simpl .
repeat Usimpl .
transitivity ([1/2] + (
pmin ([1-] (
x0 +
x0 ))
i ) * [1/2]).
apply Ole_trans with (1:=pmin_esp_le ([1-]
x0 )
i ).
rewrite (
Uinv_plus_esp x0 x0 ).
Usimpl ;
auto .
repeat Usimpl ;
apply (
H (
x0 +x0));
auto .
unfold flip ,
unit ,
ctrue ,
cfalse ;
simpl .
repeat Usimpl .
transitivity ((
pmin ([1-] (
x0 &
x0 ))
i ) * [1/2]).
rewrite (
Uinv_esp_plus x0 x0 ).
assert ([1-]
x0 <= [1/2]);
auto .
rewrite (
pmin_plus_eq ([1-]
x0 )
i H0 ).
repeat Usimpl ;
trivial .
repeat Usimpl ;
apply (
H (
x0 &x0));
auto .
Save .
Probability for the result of to be true is exactly
Lemma qtrue_qfalse_inv :
forall (
b :bool) (
x :U),
qtrue x b == [1-] (
qfalse x b ).
intros ;
case b ;
simpl ;
auto .
Save .
Lemma bernoulli_eq_true :
forall p ,
mu (
bernoulli p ) (
qtrue p ) ==
p .
intros ;
apply Ole_antisym .
transitivity (
mu (
bernoulli p ) (
fun b => [1-] (
qfalse p b ))).
apply (
mu_monotonic (
bernoulli p )).
intro b ;
rewrite (
qtrue_qfalse_inv b );
auto .
transitivity ([1-] (
mu (
bernoulli p ) (
qfalse p ))).
exact (
mu_stable_inv (
bernoulli p ) (
qfalse p )).
apply Uinv_le_perm_left .
apply (
bernoulli_false p ).
apply (
bernoulli_true p ).
Save .
Lemma bernoulli_eq_false :
forall p ,
mu (
bernoulli p ) (
qfalse p )== [1-]
p .
intros ;
apply Ole_antisym .
transitivity (
mu (
bernoulli p ) (
fun b => [1-] (
qtrue p b ))).
apply (
mu_monotonic (
bernoulli p )).
intro b ;
rewrite (
qtrue_qfalse_inv b p );
auto .
transitivity ([1-] (
mu (
bernoulli p ) (
qtrue p ))).
exact (
mu_stable_inv (
bernoulli p ) (
qtrue p )).
apply Uinv_le_perm_left ;
Usimpl .
apply (
bernoulli_true p ).
apply (
bernoulli_false p ).
Save .
Lemma bernoulli_eq :
forall p f ,
mu (
bernoulli p )
f ==
p *
f true + ([1-]
p ) *
f false .
intros ;
transitivity (
mu (
bernoulli p ) (
fun b =>
f true *
qtrue p b +
f false *
qfalse p b )).
apply mu_stable_eq .
unfold qtrue ,qfalse,B2
U ,NB2
U .
simpl ;
apply ford_eq_intro ;
intro x ;
destruct x ;
repeat Usimpl ;
auto .
rewrite (
mu_stable_plus (
bernoulli p ) (
f :=fun
b =>
f true *
qtrue p b )
(
g :=fun
b =>
f false *
qfalse p b )).
rewrite (
mu_stable_mult (
bernoulli p ) (
f true ) (
qtrue p )).
rewrite (
mu_stable_mult (
bernoulli p ) (
f false ) (
qfalse p )).
rewrite bernoulli_eq_true ;
rewrite bernoulli_eq_false .
apply Uplus_eq_compat ;
auto .
unfold finv ,qtrue,qfalse,B2
U ,NB2
U ;
intro x .
destruct x ;
unfold finv ;
repeat Usimpl ;
auto .
Save .
Lemma bernoulli_total :
forall p ,
mu (
bernoulli p ) (
fone bool )==1.
intros ;
rewrite bernoulli_eq ;
unfold fone ;
repeat Usimpl ;
auto .
Save .
gives with probability
Definition and properties of binomial coefficients
Fixpoint comb (
k n :nat) {
struct n } :
nat :=
match n with O =>
match k with O => (1%nat) | (
S l ) =>
O end
| (
S m ) =>
match k with O => (1%nat)
| (
S l ) => ((
comb l m ) + (
comb k m ))%nat
end
end .
Lemma comb_0_n :
forall n ,
comb 0
n = 1%nat.
destruct n ;
trivial .
Save .
Lemma comb_not_le :
forall n k ,
le (
S n )
k ->
comb k n =0%nat.
induction n ;
destruct k ;
simpl ;
auto with zarith ;
intros .
rewrite (
IHn k );
auto with zarith .
rewrite (
IHn (
S k ));
auto with zarith .
Save .
Lemma comb_Sn_n :
forall n ,
comb (
S n )
n =0%nat.
intro ;
apply comb_not_le ;
auto .
Save .
Lemma comb_n_n :
forall n ,
comb n n = (1%nat).
induction n ;
simpl ;
auto .
rewrite comb_Sn_n ;
auto with zarith .
Save .
Lemma comb_1_Sn :
forall n ,
comb 1 (
S n ) = (
S n ).
induction n ;
auto .
replace (
comb 1 (
S (
S n )))
with ((
comb 0 (
S n )+comb 1 (
S n ))%nat);
auto .
rewrite comb_0_n ;
omega .
Save .
Lemma comb_inv :
forall n k , (
k <=n)%nat ->
comb k n =
comb (
n -k)
n .
induction n ;
destruct k ;
simpl ;
auto with zarith ;
intros .
rewrite comb_Sn_n ;
rewrite comb_n_n ;
auto .
assert (
Hle : (
k <=n)%nat);
auto with zarith .
case (
le_lt_or_eq k n Hle );
intros .
assert (
Heq :(n-k)%nat=(S (
n -(S
k ))));
auto with zarith .
pattern ((
n -k)%nat)
at 1;
rewrite Heq .
rewrite (
IHn (
S k ));
auto .
rewrite (
IHn k );
auto with zarith .
subst .
rewrite comb_Sn_n ;
rewrite comb_n_n ;
rewrite <-
minus_n_n ;
trivial .
Save .
Lemma comb_n_Sn :
forall n ,
comb n (
S n ) = (
S n ).
intros ;
transitivity (
comb (
S n -
n ) (
S n )).
apply comb_inv ;
auto .
rewrite <-
minus_Sn_m ;
auto .
rewrite <-
minus_n_n .
apply comb_1_Sn .
Save .
Definition fc (
p :U)(n
k :nat) := (
comb k n ) */ (
p ^k * ([1-]
p )^(n-k)).
Lemma fcp_0 :
forall p n ,
fc p n O == ([1-]
p )^n.
intros ;
unfold fc ;
rewrite comb_0_n ;
repeat Usimpl .
rewrite <-
minus_n_O ;
auto .
Save .
Lemma fcp_n :
forall p n ,
fc p n n ==
p ^n.
intros ;
unfold fc ;
rewrite comb_n_n ;
repeat Usimpl .
rewrite <-
minus_n_n ;
auto .
Save .
Lemma fcp_not_le :
forall p n k , (
S n <=
k )%nat ->
fc p n k == 0.
unfold fc ;
intros ;
rewrite comb_not_le ;
auto .
Save .
Lemma fc0 :
forall n k ,
fc 0
n (
S k ) == 0.
intros ;
unfold fc ;
repeat Usimpl ;
auto .
Save .
Hint Resolve fc0 .
Add Morphism fc with signature Oeq &
eq &
eq &
Oeq
as fc_eq_compat .
unfold fc ;
intros ;
rewrite H ;
auto .
Save .
Hint Resolve fc_eq_compat .
Lemma sigma_fc0 :
forall n k ,
sigma (
fc 0
n ) (
S k ) ==1.
intros ;
rewrite sigma_S_lift .
rewrite fcp_0 ;
rewrite sigma_zero ;
repeat Usimpl ;
auto .
Save .
Lemma retract_class :
forall f n ,
class (
retract f n ).
unfold retract ;
red ;
intros .
apply Ule_class ;
red ;
intros .
apply H ;
intro ;
auto .
Save .
Hint Resolve retract_class .
Lemma fc_retract :
forall p n , ([1-]
p ^n ==
sigma (
fc p n )
n ) ->
retract (
fc p n ) (
S n ).
intros ;
apply (
Ueq_orc p 0);
intros ;
auto .
red ;
intros .
destruct k .
simpl ;
Usimpl ;
auto .
rewrite sigma_S ;
repeat Usimpl ;
auto .
transitivity (0:U);
auto .
rewrite H0 ;
auto .
apply retractS_intro ;
auto .
apply retract_lt .
apply Ule_lt_trans with ([1-]
p ^n);
auto .
transitivity (
p ^n);
auto .
rewrite fcp_n ;
auto .
Save .
Hint Resolve fc_retract .
Lemma fc_Nmult_def :
forall p n k , ([1-]
p ^n ==
sigma (
fc p n )
n ) ->
Nmult_def (
comb k n ) (
p ^k * ([1-]
p ) ^(n-k)).
intros p n k Heq ;
destruct k .
rewrite comb_0_n ;
auto .
apply (
Ueq_orc p 0);
intros ;
auto .
rewrite H ;
repeat Usimpl ;
auto .
assert (
Hk :(S
k <
n \/n=S
k \/
n <
S k )%nat);
try omega .
decompose sum Hk ;
clear Hk ;
intros .
apply Nmult_def_lt .
apply Ule_lt_trans with (
sigma (
fc p n )
n ).
apply sigma_le with (
k :=S
k ) (
f :=fc
p n );
auto .
apply Ule_lt_trans with ([1-](
p ^n));
auto .
subst .
rewrite comb_n_n ;
auto .
rewrite comb_not_le ;
auto with arith .
Save .
Hint Resolve fc_Nmult_def .
Lemma fcp_S :
forall p n k , ([1-]
p ^n ==
sigma (
fc p n )
n ) ->
fc p (
S n ) (
S k ) ==
p * (
fc p n k ) + ([1-]
p ) * (
fc p n (
S k )).
intros ;
assert (
Hcase : (
k <n \/
n =k \/ (
S n )<=k)%nat);
try omega .
decompose sum Hcase .
unfold fc ;
simpl .
rewrite plus_Nmult_distr .
rewrite <-
Umult_assoc .
rewrite Nmult_Umult_assoc_right ;
auto .
repeat Usimpl .
rewrite <-
Nmult_Umult_assoc_right ;
auto .
2:exact (
fc_Nmult_def p n (
S k )
H ).
apply Nmult_eq_compat_left ;
trivial .
repeat rewrite Umult_assoc .
rewrite (
Umult_sym ([1-]
p )
p ).
rewrite <- (
Umult_assoc p ([1-]
p )).
rewrite (
Umult_sym ([1-]
p ) (
p ^k));
auto .
repeat rewrite <-
Umult_assoc ;
auto .
replace (
n -k)%nat
with (
S (
n -S
k ));
auto ;
omega .
subst ;
repeat rewrite fcp_n .
rewrite fcp_not_le ;
repeat Usimpl ;
auto .
repeat (
rewrite fcp_not_le ;
auto with arith ).
Save .
Lemma sigma_fc_1 :
forall p n , ([1-]
p ^n ==
sigma (
fc p n )
n ) ->1==sigma (
fc p n ) (
S n ).
intros ;
rewrite sigma_S .
rewrite <-
H ;
rewrite fcp_n ;
auto .
Save .
Hint Resolve sigma_fc_1 .
Lemma Uinv_exp :
forall p n , [1-](
p ^n)==sigma (
fc p n )
n .
induction n ;
auto .
apply (
Ueq_orc p 0);
intros ;
auto .
rewrite sigma_S_lift .
rewrite fcp_0 ;
rewrite sigma_zero ;
intros ;
rewrite H ;
repeat Usimpl ;
auto .
assert (
Hr :retract (
fc p n ) (
S n ));
auto .
rewrite sigma_S_lift .
rewrite fcp_0 .
transitivity (([1-]
p ) ^
S n +
sigma (
fun k :
nat =>
p *
fc p n k + ([1-]
p ) *
fc p n (
S k ))
n ).
rewrite (
sigma_plus (
fun k =>
p *
fc p n k ) (
fun k => [1-]
p *
fc p n (
S k ))).
rewrite sigma_mult ;
auto .
rewrite <-IHn.
transitivity (
p * [1-]
p ^
n + (([1-]
p ) ^
S n +
sigma (
fun k :
nat => [1-]
p *
fc p n (
S k ))
n ));
auto .
transitivity (
p * [1-]
p ^
n + (
sigma (
fun k :
nat => [1-]
p *
fc p n k ) (
S n ))).
rewrite sigma_mult ;
auto .
rewrite <-
sigma_fc_1 ;
auto ;
repeat Usimpl ;
apply Uexp_inv_S .
rewrite sigma_S_lift ;
repeat Usimpl ;
rewrite fcp_0 ;
auto .
repeat Usimpl .
apply sigma_eq_compat ;
intros .
apply Oeq_sym ;
apply fcp_S ;
auto .
Save .
Hint Resolve Uinv_exp .
Lemma Nmult_comb :
forall p n k ,
Nmult_def (
comb k n ) (
p ^
k * ([1-]
p ) ^ (
n -
k )).
auto .
Save .
Hint Resolve Nmult_comb .
Definition of binomial distribution
Fixpoint binomial (p :U)(n:nat) {struct n }: distr nat :=match n with O => (Munit O )S m => Mlet (binomial p m )fun x => Mif (bernoulli p ) (Munit (S x )) (Munit x ))end .
Properties of binomial distribution
Lemma binomial_eq_k :
forall p n k ,
mu (
binomial p n ) (
carac_eq k ) ==
fc p n k .
induction n ;
intros .
simpl ;
destruct k ;
unfold unit ,carac_eq,carac;
auto .
rewrite fcp_0 ;
auto .
simpl binomial .
simpl mu .
transitivity
(
star (
mu (
binomial p n ))
(
fun x :
nat =>
star (
mu (
bernoulli p ))
(
fun x0 :
bool =>
mu (
if x0 then Munit (
S x )
else Munit x ))) (
carac_eq k ));
auto .
transitivity
(
mu (
binomial p n )
(
fun x :
nat =>
p * (
carac_eq k (
S x )) + ([1-]
p ) * (
carac_eq k x ))).
rewrite star_simpl .
apply mu_stable_eq ;
intro x ;
rewrite star_simpl .
rewrite bernoulli_eq ;
unfold Munit ,carac;
simpl ;
auto .
destruct k .
transitivity (
mu (
binomial p n ) (
fun x => [1-]
p *
carac_eq 0
x )).
apply mu_stable_eq ;
unfold carac_eq at 1,carac;
intro x ;
auto .
rewrite if_else ;
auto ;
repeat Usimpl ;
auto .
rewrite (
mu_stable_mult (
binomial p n ) ([1-]
p ) (
carac_eq 0)).
rewrite IHn ;
simpl ;
repeat Usimpl ;
auto .
repeat rewrite fcp_0 ;
auto .
transitivity (
mu (
binomial p n ) (
fun x :
nat =>
p *
carac_eq k x + [1-]
p *
carac_eq (
S k )
x )).
apply mu_stable_eq ;
intro ;
repeat Usimpl .
unfold carac_eq ,carac;
intros .
case (
eq_nat_dec k x );
intro .
rewrite if_then ;
auto .
rewrite if_else ;
auto .
rewrite (
mu_stable_plus (
binomial p n ) (
f :=fun
x :
nat =>
p *
carac_eq k x ) (
g :=fun
x :
nat => [1-]
p *
carac_eq (
S k )
x )).
rewrite (
mu_stable_mult (
binomial p n )
p (
carac_eq k )).
rewrite (
mu_stable_mult (
binomial p n ) ([1-]
p ) (
carac_eq (
S k ))).
rewrite IHn .
rewrite IHn .
rewrite fcp_S ;
auto .
intro x ;
unfold finv ,carac_eq;
intros .
case (
eq_nat_dec k x );
intro ;
auto .
Save .
Lemma binomial_le_n :
forall p n , 1 <=
mu (
binomial p n ) (
carac_le n ).
induction n .
unfold carac_le ;
simpl ;
case (
le_lt_dec 0 0);
intuition .
simpl binomial .
apply cover_let with (
P :=fun
k =>
k <=
n ) (
cP :=carac_le
n );
auto .
apply (
is_le n ).
intros ;
repeat distrsimpl .
transitivity (
mu (
bernoulli p ) (
fone bool ));
auto .
repeat distrsimpl .
unfold fone ;
case x0 ;
unfold carac_le ;
simpl .
rewrite carac_one ;
auto with arith .
rewrite carac_one ;
auto with arith .
Save .
Lemma binomial_le_S :
forall p n k ,
mu (
binomial p (
S n )) (
carac_le (
S k )) ==
p * (
mu (
binomial p n ) (
carac_le k )) + ([1-]
p ) * (
mu (
binomial p n ) (
carac_le (
S k ))).
Proof .
simpl binomial ;
intros .
repeat distrsimpl .
rewrite <- (
mu_stable_mult (
binomial p n )
p ).
rewrite <- (
mu_stable_mult (
binomial p n ) ([1-]
p )).
rewrite <- (
mu_stable_plus (
binomial p n ));
auto .
repeat distrsimpl .
rewrite bernoulli_eq ;
simpl .
unfold fplus ,fmult.
apply Uplus_eq_compat ;
repeat Usimpl ;
auto .
Qed .
Lemma binomial_lt_S :
forall p n k ,
mu (
binomial p (
S n )) (
carac_lt (
S k )) ==
p * (
mu (
binomial p n ) (
carac_lt k )) + ([1-]
p ) * (
mu (
binomial p n ) (
carac_lt (
S k ))).
Proof .
simpl binomial ;
intros .
repeat distrsimpl .
rewrite <- (
mu_stable_mult (
binomial p n )
p ).
rewrite <- (
mu_stable_mult (
binomial p n ) ([1-]
p )).
rewrite <- (
mu_stable_plus (
binomial p n ));
auto .
repeat distrsimpl .
rewrite bernoulli_eq ;
simpl .
unfold fplus ,fmult.
apply Uplus_eq_compat ;
repeat Usimpl ;
auto .
Qed .
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C (definition)
Choice.p [in
Choice ]
class [in
Misc ]
comp [in
Ccpo ]
compl [in
Sets ]
compn [in
Misc ]
cont2 [in
Ccpo ]
cont_app [in
Ccpo ]
CoverFun.carac [in
Cover ]
CoverFun.carac_ge [in
Cover ]
CoverFun.carac_gt [in
Cover ]
CoverFun.carac_le [in
Cover ]
CoverFun.carac_lt [in
Cover ]
CoverFun.cover [in
Cover ]
CoverFun.distr_restr [in
Cover ]
CoverFun.fesp_zero_one [in
Cover ]
CoverFun.ge_dec [in
Cover ]
CoverFun.gt_dec [in
Cover ]
CoverFun.is_ge [in
Cover ]
CoverFun.is_gt [in
Cover ]
CoverFun.is_le [in
Cover ]
CoverFun.is_lt [in
Cover ]
CoverFun.le_dec [in
Cover ]
CoverFun.lt_dec [in
Cover ]
CoverFun.Mrestr [in
Cover ]
CoverFun.nb_elts [in
Cover ]
CoverFun.retract_fin [in
Cover ]
CoverFun.sigma_fin [in
Cover ]
CoverFun.zero_one [in
Cover ]
cpo_ord_equiv [in
Ccpo ]
cshift [in
Ccpo ]
cte [in
Ccpo ]
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_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
L
le_lub [projection, in
Ccpo ]
le_mlub [lemma, in
Ccpo ]
le_Ole [lemma, in
Ccpo ]
lub [projection, in
Ccpo ]
Lub [definition, in
Ccpo ]
lub_app2_eq [lemma, in
Ccpo ]
lub_app2_le [lemma, in
Ccpo ]
lub_comp_eq [lemma, in
Ccpo ]
lub_comp_le [lemma, in
Ccpo ]
lub_continuous [instance, in
Ccpo ]
lub_cont2_app2_eq [lemma, in
Ccpo ]
lub_cte [lemma, in
Ccpo ]
lub_eq_lift [lemma, in
Ccpo ]
lub_fcpo_mon [lemma, in
Ccpo ]
lub_fun [definition, in
Ccpo ]
lub_ishift [lemma, in
Ccpo ]
lub_le [projection, in
Ccpo ]
lub_le_lift [lemma, in
Ccpo ]
lub_lift_left [lemma, in
Ccpo ]
lub_lift_right [lemma, in
Ccpo ]
lub_mon [instance, in
Ccpo ]
lub_mon_fcpo [lemma, in
Ccpo ]
lub_shift_mon [instance, in
Ccpo ]
html/index_record_V.html 0000644 0001750 0001750 00000022476 11301735615 015527 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_projection_R.html 0000644 0001750 0001750 00000022716 11301735615 016416 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
R (projection)
reflexive [in
Ccpo ]
html/index_axiom_L.html 0000644 0001750 0001750 00000022476 11301735615 015354 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_lemma_B.html 0000644 0001750 0001750 00000055130 11301735615 015311 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
B (lemma)
Bernoulli.bernoulli_eq [in
Bernoulli ]
Bernoulli.bernoulli_eq_false [in
Bernoulli ]
Bernoulli.bernoulli_eq_true [in
Bernoulli ]
Bernoulli.bernoulli_false [in
Bernoulli ]
Bernoulli.bernoulli_total [in
Bernoulli ]
Bernoulli.bernoulli_true [in
Bernoulli ]
Bernoulli.Bern_commute [in
Bernoulli ]
Bernoulli.Bern_eq [in
Bernoulli ]
Bernoulli.Bern_false [in
Bernoulli ]
Bernoulli.Bern_term [in
Bernoulli ]
Bernoulli.Bern_true [in
Bernoulli ]
Bernoulli.binomial_eq_k [in
Bernoulli ]
Bernoulli.comb_inv [in
Bernoulli ]
Bernoulli.comb_not_le [in
Bernoulli ]
Bernoulli.comb_n_n [in
Bernoulli ]
Bernoulli.comb_n_Sn [in
Bernoulli ]
Bernoulli.comb_Sn_n [in
Bernoulli ]
Bernoulli.comb_0_n [in
Bernoulli ]
Bernoulli.comb_1_Sn [in
Bernoulli ]
Bernoulli.Fbern_simpl [in
Bernoulli ]
Bernoulli.fcp_n [in
Bernoulli ]
Bernoulli.fcp_not_le [in
Bernoulli ]
Bernoulli.fcp_S [in
Bernoulli ]
Bernoulli.fcp_0 [in
Bernoulli ]
Bernoulli.fc0 [in
Bernoulli ]
Bernoulli.fc_Nmult_def [in
Bernoulli ]
Bernoulli.fc_retract [in
Bernoulli ]
Bernoulli.Mubern_eq [in
Bernoulli ]
Bernoulli.MuBern_false [in
Bernoulli ]
Bernoulli.Mubern_inv [in
Bernoulli ]
Bernoulli.Mubern_simpl [in
Bernoulli ]
Bernoulli.MuBern_true [in
Bernoulli ]
Bernoulli.Nmult_comb [in
Bernoulli ]
Bernoulli.qtrue_qfalse_inv [in
Bernoulli ]
Bernoulli.retract_class [in
Bernoulli ]
Bernoulli.sigma_fc0 [in
Bernoulli ]
Bernoulli.sigma_fc_1 [in
Bernoulli ]
Bernoulli.Uinv_exp [in
Bernoulli ]
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_instance_W.html 0000644 0001750 0001750 00000022476 11301735615 016056 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_inductive_U.html 0000644 0001750 0001750 00000022476 11301735615 016242 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_projection_B.html 0000644 0001750 0001750 00000022476 11301735615 016401 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_global_M.html 0000644 0001750 0001750 00000067672 11301735615 015507 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
M
mfun2 [definition, in
Ccpo ]
mfun2_mon [instance, in
Ccpo ]
mfun2_simpl [lemma, in
Ccpo ]
Misc [library]
mk_cpo [constructor, in
Ccpo ]
mlub_eq_compat [lemma, in
Ccpo ]
mlub_le [lemma, in
Ccpo ]
mlub_le_compat [lemma, in
Ccpo ]
mlub_lift_left [lemma, in
Ccpo ]
mlub_lift_right [lemma, in
Ccpo ]
mon [constructor, in
Ccpo ]
Monad [module, in
Monads ]
Monads [library]
Monad.app_mon [instance, in
Monads ]
Monad.inv_minus_distr [lemma, in
Monads ]
Monad.law1 [lemma, in
Monads ]
Monad.law2 [lemma, in
Monads ]
Monad.law3 [lemma, in
Monads ]
Monad.le_esp [definition, in
Monads ]
Monad.le_esp_distr [lemma, in
Monads ]
Monad.le_minus_distr [lemma, in
Monads ]
Monad.le_plus [definition, in
Monads ]
Monad.le_plus_cte [definition, in
Monads ]
Monad.le_plus_distr [lemma, in
Monads ]
Monad.M [definition, in
Monads ]
Monad.stable_inv [definition, in
Monads ]
Monad.stable_minus_distr [lemma, in
Monads ]
Monad.stable_mult [definition, in
Monads ]
Monad.stable_plus [definition, in
Monads ]
Monad.star [definition, in
Monads ]
Monad.star_continuous [lemma, in
Monads ]
Monad.star_le_plus [lemma, in
Monads ]
Monad.star_simpl [lemma, in
Monads ]
Monad.star_stable_eq [lemma, in
Monads ]
Monad.star_stable_inv [lemma, in
Monads ]
Monad.star_stable_mult [lemma, in
Monads ]
Monad.star_stable_plus [lemma, in
Monads ]
Monad.unit [definition, in
Monads ]
Monad.unit_continuous [lemma, in
Monads ]
Monad.unit_le_plus [lemma, in
Monads ]
Monad.unit_stable_eq [lemma, in
Monads ]
Monad.unit_stable_inv [lemma, in
Monads ]
Monad.unit_stable_mult [lemma, in
Monads ]
Monad.unit_stable_plus [lemma, in
Monads ]
monotonic [record, in
Ccpo ]
monotonic [inductive, in
Ccpo ]
monotonic2 [record, in
Ccpo ]
monotonic2 [inductive, in
Ccpo ]
monotonic2_intro [projection, in
Ccpo ]
monotonic2_intro [constructor, in
Ccpo ]
monotonic2_stable2 [instance, in
Ccpo ]
monotonic2_sym [lemma, in
Ccpo ]
monotonic_comp [instance, in
Ccpo ]
monotonic_comp_mon [instance, in
Ccpo ]
monotonic_def [projection, in
Ccpo ]
monotonic_def [constructor, in
Ccpo ]
monotonic_intro [lemma, in
Ccpo ]
monotonic_lub_comp [instance, in
Ccpo ]
monotonic_morphism [instance, in
Ccpo ]
monotonic_stable [instance, in
Ccpo ]
monotonic_sym [lemma, in
Ccpo ]
mon2 [definition, in
Ccpo ]
mon2_elim1 [lemma, in
Ccpo ]
mon2_elim2 [lemma, in
Ccpo ]
mon2_fun2 [lemma, in
Ccpo ]
mon2_intro [instance, in
Ccpo ]
mon2_le_compat [lemma, in
Ccpo ]
mon2_simpl [lemma, in
Ccpo ]
mon_app2 [instance, in
Ccpo ]
mon_app2_mon [instance, in
Ccpo ]
mon_comp [definition, in
Ccpo ]
mon_cte [instance, in
Ccpo ]
mon_diag [instance, in
Ccpo ]
mon_eq_compat [lemma, in
Ccpo ]
mon_fun_lub [instance, in
Ccpo ]
mon_fun_mon [instance, in
Ccpo ]
mon_id [instance, in
Ccpo ]
mon_le_compat [lemma, in
Ccpo ]
mon_ord_equiv [definition, in
Ccpo ]
mon_ord_equiv_simpl [lemma, in
Ccpo ]
mon_seq_lift_left [instance, in
Ccpo ]
mon_seq_lift_right [instance, in
Ccpo ]
mon_simpl [lemma, in
Ccpo ]
MP [module, in
Probas ]
mseq_cte [definition, in
Ccpo ]
mseq_lift_left [definition, in
Ccpo ]
mseq_lift_left_le_compat [lemma, in
Ccpo ]
mseq_lift_left_simpl [lemma, in
Ccpo ]
mseq_lift_right [definition, in
Ccpo ]
mseq_lift_right_left [lemma, in
Ccpo ]
mseq_lift_right_le_compat [lemma, in
Ccpo ]
mseq_lift_right_simpl [lemma, in
Ccpo ]
mshift [definition, in
Ccpo ]
mshift2_eq [lemma, in
Ccpo ]
mshift_continuous [instance, in
Ccpo ]
mshift_continuous2 [lemma, in
Ccpo ]
mshift_le_compat [lemma, in
Ccpo ]
mshift_simpl [lemma, in
Ccpo ]
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_variable_G.html 0000644 0001750 0001750 00000022476 11301735615 016017 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_constructor_M.html 0000644 0001750 0001750 00000023315 11301735615 016616 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
M (constructor)
mk_cpo [in
Ccpo ]
mon [in
Ccpo ]
monotonic2_intro [in
Ccpo ]
monotonic_def [in
Ccpo ]
html/Sets.html 0000644 0001750 0001750 00000432221 11301735615 013504 0 ustar cpaulin cpaulin
Sets
Library Sets
Set Implicit Arguments .
Set Strict Implicit .
Require Export Setoid .
Require Omega .
Sets.v: Definition of sets as predicates over a type A
Section sets .
Variable A :
Type .
Variable decA :
forall x y :A, {
x =y}+{x<>y}.
Definition set :=
A ->Prop.
Definition full :
set :=
fun (
x :A) =>
True .
Definition empty :
set :=
fun (
x :A) =>
False .
Definition add (
a :A) (
P :set) :
set :=
fun (
x :A) =>
x =a \/ (
P x ).
Definition singl (
a :A) :set :=
fun (
x :A) =>
x =a.
Definition union (
P Q :set) :set :=
fun (
x :A) => (
P x ) \/ (
Q x ).
Definition compl (
P :set) :set :=
fun (
x :A) => ~P
x .
Definition inter (
P Q :set) :set :=
fun (
x :A) => (
P x ) /\ (
Q x ).
Definition rem (
a :A) (
P :set) :set :=
fun (
x :A) =>
x <>a /\ (
P x ).
Lemma set_setoid :
Setoid_Theory set eqset .
split ;
red ;
auto .
exact eqset_trans .
Qed .
Add Setoid set eqset set_setoid as Set_setoid .
Add Morphism add :
eqset_add .
unfold eqset ,add;
firstorder .
Save .
Add Morphism rem :
eqset_rem .
unfold eqset ,rem;
firstorder .
Save .
Hint Resolve eqset_add eqset_rem .
Add Morphism union :
eqset_union .
unfold eqset ,union;
firstorder .
Save .
Hint Immediate eqset_union .
Lemma eqset_union_left :
forall P1 Q P2 ,
eqset P1 P2 ->
eqset (
union P1 Q ) (
union P2 Q ).
auto .
Save .
Lemma eqset_union_right :
forall P Q1 Q2 ,
eqset Q1 Q2 ->
eqset (
union P Q1 ) (
union P Q2 ).
auto .
Save .
Hint Resolve eqset_union_left eqset_union_right .
Add Morphism inter :
eqset_inter .
unfold eqset ,inter;
firstorder .
Save .
Hint Immediate eqset_inter .
Add Morphism compl :
eqset_compl .
unfold eqset ,compl;
firstorder .
Save .
Hint Resolve eqset_compl .
Lemma eqset_add_empty :
forall (
a :A) (
P :set), ~eqset (
add a P )
empty .
red ;
unfold eqset ,empty,add;
intros a P eqH ;
assert (
H :=eqH
a );
intuition .
Save .
Finite sets given as an enumeration of elements
Emptyness is decidable for finite sets
Properties of operations on sets
Lemma incl_empty :
forall P ,
incl empty P .
unfold incl ,empty;
intuition .
Save .
Lemma incl_empty_false :
forall P a ,
incl P empty -> ~
P a .
unfold incl ;
firstorder .
Save .
Lemma incl_add_empty :
forall (
a :A) (
P :set), ~
incl (
add a P )
empty .
red ;
unfold incl ,empty,add;
intros a P eqH ;
assert (
H :=eqH
a );
intuition .
Save .
Lemma eqset_empty_false :
forall P a ,
eqset P empty ->
P a ->
False .
unfold eqset ;
firstorder .
Save .
Hint Immediate incl_empty_false eqset_empty_false incl_add_empty .
Lemma incl_rem_stable :
forall a P Q ,
incl P Q ->
incl (
rem a P ) (
rem a Q ).
unfold incl ,rem;
intuition .
Save .
Lemma incl_add_stable :
forall a P Q ,
incl P Q ->
incl (
add a P ) (
add a Q ).
unfold incl ,add;
intuition .
Save .
Lemma incl_rem_add_iff :
forall a P Q ,
incl (
rem a P )
Q <->
incl P (
add a Q ).
unfold rem ,
add ,
incl ;
intuition .
case (
decA x a );
auto .
case (
H x );
intuition .
Save .
Lemma incl_rem_add :
forall (
a :A) (
P Q :set),
(
P a ) ->
incl Q (
rem a P ) ->
incl (
add a Q )
P .
unfold rem ,
add ,
incl ;
intros ;
auto .
case H1 ;
intro ;
subst ;
auto .
case (
H0 x );
auto .
Save .
Lemma incl_add_rem :
forall (
a :A) (
P Q :set),
~
Q a ->
incl (
add a Q )
P ->
incl Q (
rem a P ) .
unfold rem ,
add ,
incl ;
intros ;
auto .
case (
decA x a );
intros ;
auto .
subst ;
case H ;
auto .
Save .
Hint Immediate incl_rem_add incl_add_rem .
Lemma eqset_rem_add :
forall (
a :A) (
P Q :set),
(
P a ) ->
eqset Q (
rem a P ) ->
eqset (
add a Q )
P .
intros ;
assert (
incl Q (
rem a P ));
auto .
assert (
incl (
rem a P )
Q );
auto .
case (
incl_rem_add_iff a P Q );
auto .
Save .
Lemma eqset_add_rem :
forall (
a :A) (
P Q :set),
~
Q a ->
eqset (
add a Q )
P ->
eqset Q (
rem a P ).
intros ;
assert (
incl (
add a Q )
P );
auto .
assert (
incl P (
add a Q ));
auto .
case (
incl_rem_add_iff a P Q );
auto .
Save .
Hint Immediate eqset_rem_add eqset_add_rem .
Lemma add_rem_eq_eqset :
forall x (
P :set),
eqset (
add x (
rem x P )) (
add x P ).
unfold eqset ,
add ,
rem ;
intuition .
case (
decA x0 x );
intuition .
Save .
Lemma add_rem_diff_eqset :
forall x y (
P :set),
x <>y ->
eqset (
add x (
rem y P )) (
rem y (
add x P )).
unfold eqset ,
add ,
rem ;
intuition .
subst ;
auto .
Save .
Lemma add_eqset_in :
forall x (
P :set),
P x ->
eqset (
add x P )
P .
unfold eqset ,
add ;
intuition .
subst ;
auto .
Save .
Hint Resolve add_rem_eq_eqset add_rem_diff_eqset add_eqset_in .
Lemma add_rem_eqset_in :
forall x (
P :set),
P x ->
eqset (
add x (
rem x P ))
P .
intros ;
apply eqset_trans with (
add x P );
auto .
Save .
Hint Resolve add_rem_eqset_in .
Lemma rem_add_eq_eqset :
forall x (
P :set),
eqset (
rem x (
add x P )) (
rem x P ).
unfold eqset ,
add ,
rem ;
intuition .
Save .
Lemma rem_add_diff_eqset :
forall x y (
P :set),
x <>y ->
eqset (
rem x (
add y P )) (
add y (
rem x P )).
intros ;
apply eqset_sym ;
auto .
Save .
Lemma rem_eqset_notin :
forall x (
P :set), ~P
x ->
eqset (
rem x P )
P .
unfold eqset ,
rem ;
intuition .
subst ;
auto .
Save .
Hint Resolve rem_add_eq_eqset rem_add_diff_eqset rem_eqset_notin .
Lemma rem_add_eqset_notin :
forall x (
P :set), ~P
x ->
eqset (
rem x (
add x P ))
P .
intros ;
apply eqset_trans with (
rem x P );
auto .
Save .
Hint Resolve rem_add_eqset_notin .
Lemma rem_not_in :
forall x (
P :set), ~
rem x P x .
unfold rem ;
intuition .
Save .
Lemma add_in :
forall x (
P :set),
add x P x .
unfold add ;
intuition .
Save .
Lemma add_in_eq :
forall x y P ,
x =y ->
add x P y .
unfold add ;
intuition .
Save .
Lemma add_intro :
forall x (
P :set)
y ,
P y ->
add x P y .
unfold add ;
intuition .
Save .
Lemma add_incl :
forall x (
P :set),
incl P (
add x P ).
unfold incl ,add;
intuition .
Save .
Lemma add_incl_intro :
forall x (
P Q :set), (
Q x ) -> (
incl P Q ) -> (
incl (
add x P )
Q ).
unfold incl ,add;
intuition ;
subst ;
intuition .
Save .
Lemma rem_incl :
forall x (
P :set),
incl (
rem x P )
P .
unfold incl ,
rem ;
intuition .
Save .
Hint Resolve rem_not_in add_in rem_incl add_incl .
Lemma union_sym :
forall P Q :
set ,
eqset (
union P Q ) (
union Q P ).
unfold eqset ,
union ;
intuition .
Save .
Lemma union_empty_left :
forall P :
set ,
eqset P (
union P empty ).
unfold eqset ,
union ,
empty ;
intuition .
Save .
Lemma union_empty_right :
forall P :
set ,
eqset P (
union empty P ).
unfold eqset ,
union ,
empty ;
intuition .
Save .
Lemma union_add_left :
forall (
a :A) (
P Q :
set ),
eqset (
add a (
union P Q )) (
union P (
add a Q )).
unfold eqset ,
union ,
add ;
intuition .
Save .
Lemma union_add_right :
forall (
a :A) (
P Q :
set ),
eqset (
add a (
union P Q )) (
union (
add a P )
Q ).
unfold eqset ,
union ,
add ;
intuition .
Save .
Hint Resolve union_sym union_empty_left union_empty_right union_add_left union_add_right .
Lemma union_incl_left :
forall P Q ,
incl P (
union P Q ).
unfold incl ,union;
intuition .
Save .
Lemma union_incl_right :
forall P Q ,
incl Q (
union P Q ).
unfold incl ,union;
intuition .
Save .
Lemma union_incl_intro :
forall P Q R ,
incl P R ->
incl Q R ->
incl (
union P Q )
R .
unfold incl ,union;
intuition .
Save .
Hint Resolve union_incl_left union_incl_right union_incl_intro .
Lemma incl_union_stable :
forall P1 P2 Q1 Q2 ,
incl P1 P2 ->
incl Q1 Q2 ->
incl (
union P1 Q1 ) (
union P2 Q2 ).
intros ;
apply union_incl_intro ;
unfold incl ,union;
intuition .
Save .
Hint Immediate incl_union_stable .
Lemma inter_sym :
forall P Q :
set ,
eqset (
inter P Q ) (
inter Q P ).
unfold eqset ,
inter ;
intuition .
Save .
Lemma inter_empty_left :
forall P :
set ,
eqset empty (
inter P empty ).
unfold eqset ,
inter ,
empty ;
intuition .
Save .
Lemma inter_empty_right :
forall P :
set ,
eqset empty (
inter empty P ).
unfold eqset ,
inter ,
empty ;
intuition .
Save .
Lemma inter_add_left_in :
forall (
a :A) (
P Q :
set ),
(
P a ) ->
eqset (
add a (
inter P Q )) (
inter P (
add a Q )).
unfold eqset ,
inter ,
add ;
split ;
intuition .
subst ;
auto .
Save .
Lemma inter_add_left_out :
forall (
a :A) (
P Q :
set ),
~
P a ->
eqset (
inter P Q ) (
inter P (
add a Q )).
unfold eqset ,
inter ,
add ;
split ;
intuition .
subst ;
case H ;
auto .
Save .
Lemma inter_add_right_in :
forall (
a :A) (
P Q :
set ),
Q a ->
eqset (
add a (
inter P Q )) (
inter (
add a P )
Q ).
unfold eqset ,
inter ,
add ;
split ;
intuition .
subst ;
auto .
Save .
Lemma inter_add_right_out :
forall (
a :A) (
P Q :
set ),
~
Q a ->
eqset (
inter P Q ) (
inter (
add a P )
Q ).
unfold eqset ,
inter ,
add ;
split ;
intuition .
subst ;
case H ;
auto .
Save .
Hint Resolve inter_sym inter_empty_left inter_empty_right inter_add_left_in inter_add_left_out inter_add_right_in inter_add_right_out .
Definition gunion (
I :Type)(F:I->set) :
set :=
fun z =>
exists i ,
F i z .
Lemma gunion_intro :
forall I (
F :I->set)
i ,
incl (
F i ) (
gunion F ).
red ;
intros ;
exists i ;
auto .
Save .
Lemma gunion_elim :
forall I (
F :I->set) (
P :set), (
forall i ,
incl (
F i )
P ) ->
incl (
gunion F )
P .
red ;
intros I F P H x (
i ,Hi).
apply (
H i x );
auto .
Save .
Lemma gunion_monotonic :
forall I (
F G :
I ->
set ),
(
forall i ,
incl (
F i ) (
G i ))->
incl (
gunion F ) (
gunion G ).
intros I F G H x (
i ,Hi).
exists i ;
apply (
H i x );
trivial .
Save .
Removing an element from a finite set
Definition dec (
P :set) :=
forall x , {
P x }+{~
P x }.
Lemma finite_incl :
forall P :set,
finite P ->
forall Q :set,
dec Q ->
incl Q P ->
finite Q .
intros P FP ;
elim FP ;
intros ;
auto .
apply fin_eq_empty .
unfold empty ,eqset
in *|-*;
intuition .
case (
e x );
auto .
case (
X0 x );
intros .
apply fin_eq_add with (
x :=x) (
Q :=(rem
x Q0 ));
auto .
apply X .
unfold dec ,rem.
intro y ;
case (
decA x y );
intro .
case (
X0 y );
subst ;
intuition .
case (
X0 y );
intuition .
case (
incl_rem_add_iff x Q0 Q );
intuition .
apply H1 ;
apply incl_trans with P0 ;
auto .
apply eqset_sym ;
auto .
apply X ;
auto .
red ;
intros .
case (
e x0 );
intuition .
case H1 ;
intuition ;
subst ;
auto .
case n0 ;
auto .
Save .
Lemma finite_dec :
forall P :set,
finite P ->
dec P .
red ;
intros P FP ;
elim FP ;
intros .
right ;
intro ;
apply (
eqset_empty_false x e );
auto .
case (
e x0 );
unfold add ;
intuition .
case (
X x0 );
intuition .
case (
decA x0 x );
intuition .
Save .
Lemma fin_add_in :
forall (
a :A) (
P :set),
finite P ->
finite (
add a P ).
intros a P FP ;
case (
finite_dec FP a );
intro .
apply fin_eqset with P ;
auto .
apply eqset_sym ;
auto .
apply fin_add ;
auto .
Defined .
Lemma finite_union :
forall P Q ,
finite P ->
finite Q ->
finite (
union P Q ).
intros P Q FP FQ ;
elim FP ;
intros .
apply fin_eqset with Q ;
auto .
apply eqset_trans with (
union empty Q );
auto .
apply fin_eqset with (
add x (
union Q0 Q ));
auto .
apply eqset_trans with (
union (
add x Q0 )
Q );
auto .
apply fin_add_in ;
auto .
Defined .
Lemma finite_full_dec :
forall P :set,
finite full ->
dec P ->
finite P .
intros ;
apply finite_incl with full ;
auto .
unfold full ,incl;
auto .
Save .
Require Import Lt .
Selecting elements in a finite set
A set with size > 1 contains at least 2 different elements
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L M N O P Q
R S T U V
W
X
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_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
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X
Y Z
_
(109 entries)
Projection Index
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B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
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Global Index
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L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
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Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
G
gunion [definition, in
Sets ]
gunion_elim [lemma, in
Sets ]
gunion_intro [lemma, in
Sets ]
gunion_monotonic [lemma, in
Sets ]
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Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_section_U.html 0000644 0001750 0001750 00000022731 11301735615 015706 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
U (section)
Univ_prop.FixDef [in
Uprop ]
html/index_definition_J.html 0000644 0001750 0001750 00000022476 11301735615 016365 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_constructor_F.html 0000644 0001750 0001750 00000023053 11301735615 016606 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
F (constructor)
fin_eq_add [in
Sets ]
fin_eq_empty [in
Sets ]
html/index_projection_L.html 0000644 0001750 0001750 00000023134 11301735615 016403 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
L (projection)
le_lub [in
Ccpo ]
lub [in
Ccpo ]
lub_le [in
Ccpo ]
html/index_library__.html 0000644 0001750 0001750 00000022476 11301735615 015726 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_definition_Y.html 0000644 0001750 0001750 00000023753 11301735615 016403 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
Y (definition)
Ycart.F [in
Ycart ]
Ycart.FYcart [in
Ycart ]
Ycart.FYcart_lt [in
Ycart ]
Ycart.P [in
Ycart ]
Ycart.p [in
Ycart ]
Ycart.Ycart [in
Ycart ]
Ycart.Ycart_one [in
Ycart ]
html/index_constructor_U.html 0000644 0001750 0001750 00000022737 11301735615 016635 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
U (constructor)
Univ_prop.mk_IU [in
Uprop ]
html/index_global_S.html 0000644 0001750 0001750 00000052650 11301735615 015503 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
S
select_diff [lemma, in
Sets ]
select_non_empty [lemma, in
Sets ]
seq_lift_left [definition, in
Ccpo ]
seq_lift_right [definition, in
Ccpo ]
set [definition, in
Sets ]
sets [section, in
Sets ]
Sets [library]
sets.A [variable, in
Sets ]
sets.decA [variable, in
Sets ]
set_setoid [lemma, in
Sets ]
shift [definition, in
Ccpo ]
shift_fun_mon [instance, in
Ccpo ]
shift_le_compat [lemma, in
Ccpo ]
shift_mon2 [instance, in
Ccpo ]
shift_mon_fun [instance, in
Ccpo ]
shift_simpl [lemma, in
Ccpo ]
singl [definition, in
Sets ]
size [definition, in
Sets ]
size_eqset [lemma, in
Sets ]
size_finite_rem [lemma, in
Sets ]
size_incl [lemma, in
Sets ]
size_inter_add_in [lemma, in
Sets ]
size_inter_add_notin [lemma, in
Sets ]
size_inter_empty [lemma, in
Sets ]
size_inter_incl [lemma, in
Sets ]
size_unique [lemma, in
Sets ]
stable [record, in
Ccpo ]
stable [inductive, in
Ccpo ]
stable2 [record, in
Ccpo ]
stable2 [inductive, in
Ccpo ]
stable2_intro [projection, in
Ccpo ]
stable2_intro [constructor, in
Ccpo ]
stable_def [projection, in
Ccpo ]
stable_def [constructor, in
Ccpo ]
stable_intro [lemma, in
Ccpo ]
stable_morphism [instance, in
Ccpo ]
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_section_X.html 0000644 0001750 0001750 00000022476 11301735615 015717 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_lemma_I.html 0000644 0001750 0001750 00000054176 11301735615 015331 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
I (lemma)
Id_simpl [in
Ccpo ]
if_else [in
Misc ]
if_then [in
Misc ]
Imonotonic [in
Ccpo ]
Imonotonic2 [in
Ccpo ]
Imon2_simpl [in
Ccpo ]
imon2_simpl [in
Ccpo ]
Imon_simpl [in
Ccpo ]
imon_simpl [in
Ccpo ]
incl_add_empty [in
Sets ]
incl_add_rem [in
Sets ]
incl_add_stable [in
Sets ]
incl_empty [in
Sets ]
incl_empty_false [in
Sets ]
incl_refl [in
Sets ]
incl_rem_add [in
Sets ]
incl_rem_add_iff [in
Sets ]
incl_rem_stable [in
Sets ]
incl_trans [in
Sets ]
incl_union_stable [in
Sets ]
inter_add_left_in [in
Sets ]
inter_add_left_out [in
Sets ]
inter_add_right_in [in
Sets ]
inter_add_right_out [in
Sets ]
inter_empty_left [in
Sets ]
inter_empty_right [in
Sets ]
inter_sym [in
Sets ]
Iord_app [in
Ccpo ]
isempty_dec [in
Sets ]
ishift_le_compat [in
Ccpo ]
ishift_simpl [in
Ccpo ]
IterFlip.Fiter_simpl [in
IterFlip ]
IterFlip.iterflip_term [in
IterFlip ]
IterFlip.lim_p_one [in
IterFlip ]
IterFlip.pS_simpl [in
IterFlip ]
IterFlip.p_eq [in
IterFlip ]
IterFlip.p_incr [in
IterFlip ]
IterFlip.p_le [in
IterFlip ]
iterO_simpl [in
Ccpo ]
IterS_simpl [in
Ccpo ]
iterS_simpl [in
Ccpo ]
iter_continuous [in
Ccpo ]
iter_continuous_eq [in
Ccpo ]
iter_incr [in
Ccpo ]
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_axiom_O.html 0000644 0001750 0001750 00000022476 11301735615 015357 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_library_J.html 0000644 0001750 0001750 00000022476 11301735615 015701 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_inductive_H.html 0000644 0001750 0001750 00000022476 11301735615 016225 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_lemma__.html 0000644 0001750 0001750 00000022476 11301735615 015355 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_record_R.html 0000644 0001750 0001750 00000022476 11301735615 015523 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_projection_V.html 0000644 0001750 0001750 00000022476 11301735615 016425 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_variable_J.html 0000644 0001750 0001750 00000022476 11301735615 016022 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_inductive_W.html 0000644 0001750 0001750 00000022476 11301735615 016244 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_constructor_L.html 0000644 0001750 0001750 00000022476 11301735615 016624 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_lemma_W.html 0000644 0001750 0001750 00000022476 11301735615 015345 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_axiom_W.html 0000644 0001750 0001750 00000022476 11301735615 015367 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_global_K.html 0000644 0001750 0001750 00000022476 11301735615 015476 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
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Axiom Index
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(6 entries)
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(1217 entries)
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(277 entries)
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(18 entries)
Axiom Index
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(6 entries)
Variable Index
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Global Index
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_
(1807 entries)
Instance Index
A
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E
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H
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R S T
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X
Y Z
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(109 entries)
Projection Index
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(32 entries)
Record Index
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(1217 entries)
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Inductive Index
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(18 entries)
Axiom Index
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(6 entries)
Variable Index
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(13 entries)
html/Ccpo.html 0000644 0001750 0001750 00001435367 11301735614 013470 0 ustar cpaulin cpaulin
Ccpo
Library Ccpo
Ccpo.v: Specification and properties of a cpo
Definition eq_rel {
A } (
E1 E2 :relation
A ) :=
forall x y ,
E1 x y <->
E2 x y .
Class Order {
A } (
E :relation
A ) (
R :relation
A ) :=
{
reflexive :>
Reflexive R ;
order_eq :
forall x y ,
R x y /\
R y x <->
E x y ;
transitive :>
Transitive R }.
Instance OrderEqRefl `{
Order A E R } :
Reflexive E .
intros A E R (
rO ,aO,tO)
x .
rewrite <-
aO ;
intuition .
Save .
Instance OrderEqSym `{
Order A E R } :
Symmetric E .
intros A E R (
rO ,aO,tO)
x y e .
pose (
aO x y );
pose (
aO y x );
intuition .
Save .
Instance OrderEqTrans `{
Order A E R } :
Transitive E .
intros A E R (
rO ,aO,tO)
x y z e1 e2 .
pose (
aO x y );
pose (
aO y z );
pose (
aO x z );
pose (
tO x y z );
pose (
tO z y x );
intuition .
Save .
Instance OrderEquiv `{
Order A E R } :
Equivalence E .
split .
apply OrderEqRefl .
apply OrderEqSym .
apply OrderEqTrans .
Save .
Class ord A :=
{
Oeq :
relation A ;
Ole :
relation A ;
order_rel :>
Order Oeq Ole }.
Instance OrdSetoid `(
o :ord
A ) :
Setoid A .
intros ;
split with Oeq ;
apply OrderEquiv .
Defined .
Add Parametric Relation {
A } {
o :ord
A } :
A (@
Oeq _ o )
reflexivity proved by OrderEqRefl
symmetry proved by OrderEqSym
transitivity proved by OrderEqTrans
as Oeq_setoid .
Infix "<=" :=
Ole .
Infix "==" :=
Oeq :
type_scope .
Lemma Ole_refl_eq :
forall {
O } {
o :ord
O } (
x y :O),
x ==
y ->
x <=
y .
intros O (
oeq ,ole,(rO,aO,tO))
x y ;
simpl ;
pose (
aO x y );
intuition .
Save .
Hint Immediate @
Ole_refl_eq .
Lemma Ole_refl_eq_inv :
forall {
O } {
o :ord
O } (
x y :O),
x ==
y ->
y <=
x .
intros O (
oeq ,ole,(rO,aO,tO))
x y ;
simpl ;
pose (
aO x y );
intuition .
Save .
Hint Immediate @
Ole_refl_eq_inv .
Lemma Ole_trans :
forall {
O } {
o :ord
O } (
x y z :O),
x <=
y ->
y <=
z ->
x <=
z .
intros O (
oeq ,ole,(rO,aO,tO))
x y z H1 H2 ;
simpl ;
apply tO with y ;
auto .
Save .
Lemma Ole_refl :
forall {
O } {
o :ord
O } (
x :O),
x <=
x .
intros O (
oeq ,ole,(rO,aO,tO))
x ;
simpl ;
auto .
Save .
Hint Resolve @
Ole_refl .
Add Parametric Relation {
A } {
o :ord
A } :
A (@
Ole _ o )
reflexivity proved by Ole_refl
transitivity proved by Ole_trans
as Ole_setoid .
Lemma Ole_antisym :
forall {
O } {
o :ord
O } (
x y :O),
x <=
y ->
y <=
x ->
x ==y.
intros O (
oeq ,ole,(rO,aO,tO))
x y ;
simpl ;
pose (
aO x y );
intuition .
Save .
Hint Immediate @
Ole_antisym .
Lemma Oeq_refl :
forall {
O } {
o :ord
O } (
x :O),
x ==
x .
intros ;
apply OrderEqRefl .
Save .
Hint Resolve @
Oeq_refl .
Lemma Oeq_refl_eq :
forall {
O } {
o :ord
O } (
x y :O),
x =y ->
x ==
y .
intros O o x y H ;
rewrite H ;
auto .
Save .
Hint Resolve @
Oeq_refl_eq .
Lemma Oeq_sym :
forall {
O } {
o :ord
O } (
x y :O),
x ==
y ->
y ==
x .
intros ;
apply OrderEqSym ;
trivial .
Save .
Lemma Oeq_le :
forall {
O } {
o :ord
O } (
x y :O),
x ==
y ->
x <=
y .
intros O (
oeq ,ole,(rO,aO,tO))
x y ;
simpl ;
pose (
aO x y );
intuition .
Save .
Lemma Oeq_le_sym :
forall {
O } {
o :ord
O } (
x y :O),
x ==
y ->
y <=
x .
intros O (
oeq ,ole,(rO,aO,tO))
x y ;
simpl ;
pose (
aO x y );
intuition .
Save .
Hint Resolve @
Oeq_le .
Hint Immediate @
Oeq_sym @
Oeq_le_sym .
Lemma Oeq_trans
:
forall {
O } {
o :ord
O } (
x y z :O),
x ==
y ->
y ==
z ->
x ==
z .
intros ;
apply OrderEqTrans with y ;
trivial .
Save .
Hint Resolve @
Oeq_trans .
Instance Ole_eq_compat_iff `(
o :ord
A ) :
Morphism (
Oeq (
A :=A) ==>
Oeq (
A :=A) ==>
iff ) (
Ole (
ord :=o)).
intros A o x1 y1 e1 x2 y2 e2 .
split ;
intros .
transitivity x1 ;
auto .
transitivity x2 ;
auto .
transitivity y1 ;
auto .
transitivity y2 ;
auto .
Save .
Equivalence of orders
Definition eq_ord {
O } (
o1 o2 :ord
O ) :=
eq_rel (
Ole (
ord :=o1)) (
Ole (
ord :=o2)).
Lemma eq_ord_equiv :
forall {
O } (
o1 o2 :ord
O ),
eq_ord o1 o2 ->
eq_rel (
Oeq (
ord :=o1)) (
Oeq (
ord :=o2)).
intros O (
eq1 ,le1,(r1,as1,t1)) (
eq2 ,le2,(r2,as2,t2));
unfold eq_ord ,
eq_rel ;
simpl ;
intros .
transitivity (
le2 x y /\
le2 y x );
auto .
rewrite <-
as1 ;
auto .
repeat rewrite H ;
reflexivity .
Save .
Lemma Ole_eq_compat :
forall {
O } {
o :ord
O } (
x1 x2 :
O ),
x1 ==
x2 ->
forall x3 x4 :
O ,
x3 ==
x4 ->
x1 <=
x3 ->
x2 <=
x4 .
intros O o x1 x2 H1 x3 x4 H2 H3 ;
case (
Ole_eq_compat_iff o x1 x2 H1 x3 x4 H2 );
auto .
Save .
Lemma Ole_eq_right :
forall {
O } {
o :ord
O } (
x y z :
O ),
x <=
y ->
y ==
z ->
x <=
z .
intros ;
apply Ole_eq_compat with x y ;
auto .
Save .
Lemma Ole_eq_left :
forall {
O } {
o :ord
O } (
x y z :
O ),
x ==
y ->
y <=
z ->
x <=
z .
intros ;
apply Ole_eq_compat with y z ;
auto .
Save .
Add Parametric Morphism `{
o :ord
A } : (
Oeq (
A :=A))
with signature Oeq &
Oeq &
iff as Oeq_iff_morphism .
intros .
rewrite H .
rewrite H0 .
intuition .
Qed .
Add Parametric Morphism `{
o :ord
A } : (
Ole (
A :=A))
with signature Oeq &
Oeq &
iff as Ole_iff_morphism .
intros .
rewrite H .
rewrite H0 .
intuition .
Qed .
Add Parametric Morphism `{
o :ord
A } : (
Ole (
A :=A))
with signature Ole -->
Ole &
Basics.impl as Ole_impl_morphism .
red ;
intros .
transitivity x ;
trivial .
transitivity x0 ;
trivial .
Qed .
Definition Iord : forall O {o :ord O }, ord O .intros O o ; exists (Oeq (A :=O)) (fun x y => y <=x); split ; intuition .red ; intros ; transitivity y ; auto .Defined .Implicit Arguments Iord [[o ]].
Definition fun_ext A B (R :relation B ) : relation (A -> B ) := fun f g => forall x , R (f x ) (g x ).Implicit Arguments fun_ext [B ].
ford f g := forall x, f x <= g x
Instance ford A O {
o :ord
O } :
ord (
A ->
O ) :=
{
Oeq :=fun_ext
A (
Oeq (
A :=O));
Ole :=fun_ext
A (
Ole (
A :=O))}.
split ;
unfold fun_ext ;
intros ;
intuition .
red ;
intros .
transitivity (
y x0 );
auto .
Defined .
Lemma ford_le_elim :
forall A O (
o :ord
O ) (
f g :A ->
O ),
f <=
g ->
forall n ,
f n <=
g n .
auto .
Save .
Hint Immediate ford_le_elim .
Lemma ford_le_intro :
forall A O (
o :ord
O ) (
f g :A ->
O ), (
forall n ,
f n <=
g n ) ->
f <=
g .
auto .
Save .
Hint Resolve ford_le_intro .
Lemma ford_eq_elim :
forall A O (
o :ord
O ) (
f g :A ->
O ),
f ==
g ->
forall n ,
f n ==
g n .
auto .
Save .
Hint Immediate ford_eq_elim .
Lemma ford_eq_intro :
forall A O (
o :ord
O ) (
f g :A ->
O ), (
forall n ,
f n ==
g n ) ->
f ==
g .
auto .
Save .
Hint Resolve ford_eq_intro .
Definition and properties
Class monotonic `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f :
O1 ->
O2 ) :=
monotonic_def :
forall x y ,
x <=
y ->
f x <=
f y .
Lemma monotonic_intro :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f :
O1 ->
O2 ),
(
forall x y ,
x <=
y ->
f x <=
f y ) ->
monotonic f .
red ;
auto .
Save .
Hint Resolve @
monotonic_intro .
Instance monotonic_morphism `{
o1 :ord
O1 } `{
o2 :ord
O2 }
(
f :
O1 ->
O2 ) {
m :monotonic
f }
:
Morphism (
Ole (
A :=O1) &
Ole (
A :=O2))
f .
auto .
Save .
Class stable `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f :
O1 ->
O2 ) :=
stable_def :
forall x y ,
x ==
y ->
f x ==
f y .
Hint Unfold stable .
Lemma stable_intro :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f :
O1 ->
O2 ),
(
forall x y ,
x ==
y ->
f x ==
f y ) ->
stable f .
red ;
auto .
Save .
Hint Resolve @
stable_intro .
Instance stable_morphism `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f :
O1 ->
O2 ) {
s :stable
f }
:
Morphism (
Oeq (
A :=O1) &
Oeq (
A :=O2))
f .
auto .
Save .
Type classes Opaque monotonic stable .
Instance monotonic_stable `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f :
O1 ->
O2 ) {
m :monotonic
f }
:
stable f .
unfold monotonic ,
stable ;
simpl ;
intros .
apply Ole_antisym ;
auto .
Save .
Type of monotonic functions
Record fmon `{
o1 :ord
O1 } `{
o2 :ord
O2 }:=
mon
{
fmont :>
O1 ->
O2 ;
fmonotonic :
monotonic fmont }.
Existing Instance fmonotonic .
Implicit Arguments mon [[
O1 ] [
o1 ] [
O2 ] [
o2 ] [
fmonotonic ]].
Implicit Arguments fmon [[
o1 ] [
o2 ]].
Hint Resolve @
fmonotonic .
Notation "O1 -m> O2" := (
fmon O1 O2 )
(
right associativity ,
at level 30) :
O_scope .
Notation "O1 --m> O2" := (
fmon O1 (
o1 :=Iord
O1 )
O2 )
(
right associativity ,
at level 30) :
O_scope .
Notation "O1 --m-> O2" := (
fmon O1 (
o1 :=Iord
O1 )
O2 (
o2 :=Iord
O2 ))
(
right associativity ,
at level 30) :
O_scope .
Notation "O1 -m-> O2" := (
fmon O1 O2 (
o2 :=Iord
O2 ))
(
right associativity ,
at level 30) :
O_scope .
Open Scope O_scope .
Lemma mon_simpl :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f :O1 ->
O2 ){
mf :
monotonic f }
x ,
mon f x =
f x .
trivial .
Save .
Hint Resolve @
mon_simpl .
Instance fstable `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f :O1 -m>
O2 ) :
stable f .
intros ;
apply monotonic_stable ;
auto .
Save .
Hint Resolve @
fstable .
Lemma fmon_le :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f :O1 -m>
O2 )
x y ,
x <=
y ->
f x <=
f y .
intros ;
apply (
fmonotonic f );
auto .
Save .
Hint Resolve @
fmon_le .
Lemma fmon_eq :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f :O1 -m>
O2 )
x y ,
x ==
y ->
f x ==
f y .
intros ;
apply (
fstable f );
auto .
Save .
Hint Resolve @
fmon_eq .
Instance fmono O1 O2 {
o1 :ord
O1 } {
o2 :ord
O2 } :
ord (
O1 -m>
O2 )
:= {
Oeq :=
fun (
f g :
O1 -m>
O2 )=>
forall x ,
f x ==
g x ;
Ole :=
fun (
f g :
O1 -m>
O2 )=>
forall x ,
f x <=
g x }.
split ;
unfold fun_ext ;
intros ;
intuition .
red ;
intros .
transitivity (
y x0 );
auto .
Defined .
Lemma mon_le_compat :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f g :O1 ->
O2 )
{
mf :monotonic
f } {
mg :monotonic
g },
f <=
g ->
mon f <=
mon g .
red ;
intros ;
auto .
Save .
Hint Resolve @
mon_le_compat .
Lemma mon_eq_compat :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f g :O1->
O2 )
{
mf :monotonic
f } {
mg :monotonic
g },
f ==
g ->
mon f ==
mon g .
red ;
intros ;
auto .
Save .
Hint Resolve @
mon_eq_compat .
Add Parametric Morphism `{
o1 :ord
O1 } `{
o2 :ord
O2 }
: (
fmont (
O1 :=O1) (
O2 :=O2))
with signature Oeq &
Oeq &
Oeq as fmont_eq_morphism .
intros [
f ] [
g ]
H x y h .
simpl .
rewrite (
H x ).
rewrite h .
simpl .
auto .
Qed .
Monotonicity and dual order
Lemma Imonotonic `{o1 :ord O1 } `{o2 :ord O2 } (f :O1 -> O2 ) {m :monotonic f } monotonic (o1 :=Iord O1 ) (o2 :=Iord O2 ) f .intros O1 O2 o1 o2 f m ; red ; simpl ; intros .apply m ; auto .Save .Hint Extern 2 (@monotonic _ (Iord _ ) _ (Iord _ ) _ ) => apply @Imonotonic typeclass_instances .Definition imon `{o1 :ord O1 } `{o2 :ord O2 } (f :O1 -> O2 ) {m :monotonic f } : O1 --m-> O2 := mon (o1 :=Iord O1 ) (o2 :=Iord O2 ) f .Lemma imon_simpl : forall `{o1 :ord O1 } `{o2 :ord O2 } (f :O1 -> O2 ) {m :monotonic f } (x :O1),imon f x = f x .trivial .Save .
Iord (A -> U )A -> Iord U
Lemma Iord_app {A } `{o1 :ord O1 } (x : A ) : ((A -> O1 ) --m-> O1 ).intros ; exists (fun f => f x ).red ; intros ; auto .Defined .
Imon f
Definition Imon :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 }, (
O1 -m>
O2 ) -> (
O1 --m->
O2 ).
intros O1 o1 O2 o2 f .
exists (
fmont f );
auto .
apply Imonotonic ;
auto .
Defined .
Lemma Imon_simpl :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f :O1 -m>
O2 )(
x :O1),
Imon f x =
f x .
trivial .
Save .
Monotonic functions with 2 arguments
Class monotonic2 `{
o1 :
ord O1 } `{
o2 :
ord O2 } `{
o3 :ord
O3 } (
f :O1 ->
O2 ->
O3 ) :=
monotonic2_intro :
forall (
x y :O1) (
z t :O2),
x <=
y ->
z <=
t ->
f x z <=
f y t .
Instance mon2_intro `{
o1 :
ord O1 } `{
o2 :
ord O2 } `{
o3 :ord
O3 } (
f :O1 ->
O2 ->
O3 )
{
m1 :monotonic
f } {
m2 :
forall x ,
monotonic (
f x )} :
monotonic2 f | 10.
red ;
intros .
transitivity (
f y z ).
apply (
m1 x );
trivial .
apply (
m2 y );
trivial .
Save .
Lemma mon2_elim1 `{
o1 :
ord O1 } `{
o2 :
ord O2 } `{
o3 :ord
O3 } (
f :O1 ->
O2 ->
O3 )
{
m :monotonic2
f } :
monotonic f .
red ;
intros ;
intro z .
apply m ;
auto .
Save .
Lemma mon2_elim2 `{
o1 :
ord O1 } `{
o2 :
ord O2 } `{
o3 :ord
O3 } (
f :O1 ->
O2 ->
O3 )
{
m :monotonic2
f } :
forall x ,
monotonic (
f x ).
red ;
intros .
apply m ;
auto .
Save .
Hint Immediate @
mon2_elim1 @
mon2_elim2 :
typeclass_instances .
Definition mon_comp {
A } `{
o1 :
ord O1 } `{
o2 :
ord O2 } (
f :A ->
O1 ->
O2 ) {
mf :forall
x ,
monotonic (
f x )}
:
A ->
O1 -m>
O2 :=
fun x =>
mon (
f x ).
Instance mon_fun_mon `{
o1 :
ord O1 } `{
o2 :
ord O2 } `{
o3 :ord
O3 } (
f :O1 ->
O2 ->
O3 )
{
m :monotonic2
f } :
monotonic (
fun x =>
mon (
f x )).
red ;
intros ;
unfold mon_comp .
intro z ;
apply m ;
auto .
Save .
Class stable2 `{
o1 :
ord O1 } `{
o2 :
ord O2 } `{
o3 :ord
O3 } (
f :O1 ->
O2 ->
O3 ) :=
stable2_intro :
forall (
x y :O1) (
z t :O2),
x ==y ->
z ==
t ->
f x z ==
f y t .
Instance monotonic2_stable2 `{
o1 :
ord O1 } `{
o2 :
ord O2 } `{
o3 :ord
O3 }
(
f :O1 ->
O2 ->
O3 ) {
m :monotonic2
f } :
stable2 f .
red ;
intros ;
apply Ole_antisym ;
auto .
Save .
Type classes Opaque monotonic2 stable2 .
Definition mon2 `{
o1 :
ord O1 } `{
o2 :
ord O2 } `{
o3 :ord
O3 } (
f :O1 ->
O2 ->
O3 )
{
mf :monotonic2
f } :
O1 -m>
O2 -m>
O3 :=
mon (
fun x =>
mon (
f x )).
Lemma mon2_simpl :
forall `{
o1 :
ord O1 } `{
o2 :
ord O2 } `{
o3 :ord
O3 } (
f :O1 ->
O2 ->
O3 )
{
mf :monotonic2
f }
x y ,
mon2 f x y =
f x y .
trivial .
Save .
Hint Resolve @
mon2_simpl .
Lemma mon2_le_compat :
forall `{
o1 :
ord O1 } `{
o2 :
ord O2 } `{
o3 :ord
O3 }
(
f g :O1 ->
O2 ->
O3 ) {
mf :
monotonic2 f } {
mg :monotonic2
g },
f <=
g ->
mon2 f <=
mon2 g .
red ;
simpl ;
intros .
apply H ;
trivial .
Save .
Definition fun2 `{
o1 :
ord O1 } `{
o2 :
ord O2 } `{
o3 :ord
O3 } (
f :O1 ->
O2 -m>
O3 )
:
O1 ->
O2 ->
O3 :=
fun x =>
f x .
Instance fmon2_mon `{
o1 :
ord O1 } `{
o2 :
ord O2 } `{
o3 :ord
O3 } (
f :O1 ->
O2 -m>
O3 ) :
forall x :O1,
monotonic (
fun2 f x ).
intros ;
unfold fun2 ;
auto .
Save .
Instance fun2_monotonic `{
o1 :
ord O1 } `{
o2 :
ord O2 } `{
o3 :ord
O3 }
(
f :O1 ->
O2 -m>
O3 ) {
mf :monotonic
f } :
monotonic (
fun2 f ).
intros ;
unfold fun2 ;
auto .
Save .
Hint Resolve @
fun2_monotonic .
Instance fmonotonic2 `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 } (
f :O1 -m>
O2 -m>
O3 )
:
monotonic2 (
fun2 f ).
intros ;
unfold fun2 ;
apply mon2_intro ;
auto .
red ;
intros .
apply (
fmonotonic f x y );
trivial .
Save .
Hint Resolve @
fmonotonic2 .
Definition mfun2 `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 } (
f :O1 -m>
O2 -m>
O3 )
:
O1 -m> (
O2 ->
O3 ) :=
mon (
fun2 f ).
Lemma mfun2_simpl :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 } (
f :O1 -m>
O2 -m>
O3 )
x y ,
mfun2 f x y =
f x y .
trivial .
Save .
Instance mfun2_mon `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 }
(
f :O1 -m>
O2 -m>
O3 )
x :
monotonic (
mfun2 f x ).
red ;
simpl ;
unfold fun2 ;
intros ;
auto .
Save .
Lemma mon2_fun2 :
forall `{
o1 :
ord O1 } `{
o2 :
ord O2 } `{
o3 :ord
O3 }
(
f :O1 -m>
O2 -m>
O3 ),
mon2 (
fun2 f ) ==
f .
intros ;
unfold fun2 ,mon2;
intros x y ;
auto .
Save .
Lemma fun2_mon2 :
forall `{
o1 :
ord O1 } `{
o2 :
ord O2 } `{
o3 :ord
O3 }
(
f :O1 ->
O2 ->
O3 ) {
mf :monotonic2
f } ,
fun2 (
mon2 f ) ==
f .
intros ;
unfold fun2 ,mon2;
intros x y ;
auto .
Save .
Hint Resolve @
mon2_fun2 @
fun2_mon2 .
Instance fstable2 `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 } (
f :O1 -m>
O2 -m>
O3 )
:
stable2 (
fun2 f ).
intros ;
apply monotonic2_stable2 ;
auto .
Save .
Hint Resolve @
fstable2 .
Definition Imon2 :
forall `{
o1 :
ord O1 } `{
o2 :
ord O2 } `{
o3 :ord
O3 },
(
O1 -m>
O2 -m>
O3 ) -> (
O1 --m>
O2 --m->
O3 ).
intros O1 o1 O2 o2 O3 o3 f .
exists (
fun (
x :O1) =>
Imon (
f x ));
red ;
simpl ;
intros .
apply (
fmonotonic f );
auto .
Defined .
Lemma Imon2_simpl :
forall `{
o1 :
ord O1 } `{
o2 :
ord O2 } `{
o3 :ord
O3 }
(
f :O1 -m>
O2 -m>
O3 ) (
x :O1) (
y :
O2 ),
Imon2 f x y =
f x y .
trivial .
Save .
Lemma Imonotonic2 `{
o1 :
ord O1 } `{
o2 :
ord O2 } `{
o3 :ord
O3 }
(
f :O1 ->
O2 ->
O3 ){
mf :
monotonic2 f }
:
monotonic2 (
o1 :=Iord
O1 ) (
o2 :=Iord
O2 ) (
o3 :=Iord
O3 )
f .
red ;
simpl ;
intros ;
auto .
Save .
Hint Extern 2 (@
monotonic2 _ (
Iord _ )
_ (
Iord _ )
_ (
Iord _ )
_ ) =>
apply @
Imonotonic2
:
typeclass_instances .
Definition imon2 `{
o1 :
ord O1 } `{
o2 :
ord O2 } `{
o3 :ord
O3 }
(
f :O1 ->
O2 ->
O3 ){
mf :
monotonic2 f } :
O1 --m>
O2 --m->
O3 :=
mon2 (
o1 :=Iord
O1 ) (
o2 :=Iord
O2 ) (
o3 :=Iord
O3 )
f .
Lemma imon2_simpl :
forall `{
o1 :
ord O1 } `{
o2 :
ord O2 } `{
o3 :ord
O3 }
(
f :O1 ->
O2 ->
O3 ){
mf :
monotonic2 f } (
x :O1) (
y :O2),
imon2 f x y =
f x y .
trivial .
Save .
Instance natO :
ord nat :=
{
Oeq :=
fun n m :
nat =>
n =
m ;
Ole :=
fun n m :
nat => (
n <=
m )%nat}.
apply Build_Order ;
intros ;
try omega ;
auto with arith .
red ;
intros ;
apply le_trans with y ;
auto .
Defined .
Lemma le_Ole :
forall n m , ((
n <=
m )%nat)->
n <=
m .
simpl ;
auto .
Save .
Hint Resolve le_Ole .
Lemma nat_monotonic :
forall {
O } {
o :ord
O }
(
f :nat ->
O ), (
forall n ,
f n <=
f (
S n )) ->
monotonic f .
red ;
simpl ;
intros .
elim H0 ;
intros ;
auto .
transitivity (
f m );
trivial .
Save .
Hint Resolve @
nat_monotonic .
Definition fnatO_intro :
forall {
O } {
o :ord
O } (
f :nat ->
O ), (
forall n ,
f n <=
f (
S n )) ->
nat -m>
O .
intros ;
exists f ;
auto .
Defined .
Lemma fnatO_elim :
forall {
O } {
o :ord
O } (
f :nat -m>
O ) (
n :nat),
f n <=
f (
S n ).
intros ;
apply (
fmonotonic f );
simpl ;
auto with arith .
Save .
Hint Resolve @
fnatO_elim .
(mseq_lift_left f n) k = f (n+k)
Definition seq_lift_left {
O } (
f :nat ->
O )
n :=
fun k =>
f (
n +k)%nat.
Instance mon_seq_lift_left
:
forall {
O } {
o :ord
O } (
f :nat ->
O ) {
m :monotonic
f },
monotonic (
seq_lift_left f n ).
red ;
intros ;
apply m ;
simpl ;
auto with arith .
Save .
Definition mseq_lift_left :
forall {
O } {
o :ord
O } (
f :nat -m>
O ) (
n :nat),
nat -m>
O .
intros ;
exists (
fun k =>
f (
n +k)%nat);
red ;
intros .
apply (
fmonotonic f );
simpl ;
auto with arith .
Defined .
Lemma mseq_lift_left_simpl :
forall {
O } {
o :ord
O } (
f :nat -m>
O ) (
n k :nat),
mseq_lift_left f n k =
f (
n +k)%nat.
trivial .
Save .
Lemma mseq_lift_left_le_compat :
forall {
O } {
o :ord
O } (
f g :nat -m>
O ) (
n :nat),
f <=
g ->
mseq_lift_left f n <=
mseq_lift_left g n .
intros ;
intro ;
simpl ;
auto .
Save .
Hint Resolve @
mseq_lift_left_le_compat .
Add Parametric Morphism {
O } {
o :ord
O } : (@
mseq_lift_left _ o )
with signature Oeq &
eq &
Oeq
as mseq_lift_left_eq_compat .
intros ;
apply Ole_antisym ;
auto .
Save .
Hint Resolve @
mseq_lift_left_eq_compat .
Add Parametric Morphism {
O } {
o :ord
O }: (@
seq_lift_left O )
with signature Oeq &
eq &
Oeq
as seq_lift_left_eq_compat .
simpl ;
intros ;
intro .
simpl ;
apply H ;
auto .
Save .
Hint Resolve @
seq_lift_left_eq_compat .
(mseq_lift_right f n) k = f (k+n)
Definition seq_lift_right {
O } (
f :nat ->
O )
n :=
fun k =>
f (
k +n)%nat.
Instance mon_seq_lift_right
:
forall {
O } {
o :ord
O } (
f :nat ->
O ) {
m :monotonic
f },
monotonic (
seq_lift_right f n ).
red ;
intros ;
apply m ;
simpl ;
auto with arith .
Save .
Definition mseq_lift_right :
forall {
O } {
o :ord
O } (
f :nat -m>
O ) (
n :nat),
nat -m>
O .
intros ;
exists (
fun k =>
f (
k +n)%nat);
red ;
intros .
apply (
fmonotonic f );
simpl ;
auto with arith .
Defined .
Lemma mseq_lift_right_simpl :
forall {
O } {
o :ord
O } (
f :nat -m>
O ) (
n k :nat),
mseq_lift_right f n k =
f (
k +n)%nat.
trivial .
Save .
Lemma mseq_lift_right_le_compat :
forall {
O } {
o :ord
O } (
f g :nat -m>
O ) (
n :nat),
f <=
g ->
mseq_lift_right f n <=
mseq_lift_right g n .
intros ;
intro ;
simpl ;
auto .
Save .
Hint Resolve @
mseq_lift_right_le_compat .
Add Parametric Morphism {
O } {
o :ord
O } : (
mseq_lift_right (
o :=o))
with signature Oeq &
eq &
Oeq
as mseq_lift_right_eq_compat .
intros ;
apply Ole_antisym ;
auto .
Save .
Add Parametric Morphism {
O } {
o :ord
O }: (@
seq_lift_right O )
with signature Oeq &
eq &
Oeq
as seq_lift_right_eq_compat .
simpl ;
intros ;
intro .
simpl ;
apply H ;
auto .
Save .
Hint Resolve @
seq_lift_right_eq_compat .
Lemma mseq_lift_right_left :
forall {
O } {
o :ord
O } (
f :nat -m>
O )
n ,
mseq_lift_left f n ==
mseq_lift_right f n .
unfold mseq_lift_left ,mseq_lift_right;
intros .
intro x .
unfold fmont ;
replace (
n +x)%nat
with (
x +n)%nat;
auto with arith .
Save .
Monotonicity and functions
Instance shift_mon_fun {
A } `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f :O1 -m> (
A ->
O2 )) :
forall x :A,
monotonic (
fun (
y :O1) =>
f y x ).
red ;
intros ;
apply (
fmonotonic f );
auto .
Save .
Definition shift {
A } `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f :O1 -m> (
A ->
O2 )) :
A ->
O1 -m>
O2
:=
fun x => (
mon (
fun y =>
f y x )).
Infix "<o>" :=
shift (
at level 30,
no associativity ) :
O_scope .
Lemma shift_simpl :
forall {
A } `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f :O1 -m> (
A ->
O2 ))
x y ,
(
f <o>
x )
y =
f y x .
trivial .
Save .
Lemma shift_le_compat :
forall {
A } `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f g :O1 -m> (
A ->
O2 )),
f <=
g ->
shift f <=
shift g .
intros ;
intros x y .
repeat rewrite shift_simpl .
apply (
H y x );
trivial .
Save .
Hint Resolve @
shift_le_compat .
Add Parametric Morphism {
A } `{
o1 :ord
O1 } `{
o2 :ord
O2 }
: (
shift (
A :=A) (
O1 :=O1) (
O2 :=O2))
with signature Oeq &
eq &
Oeq
as shift_eq_compat .
intros f g H x y .
repeat rewrite shift_simpl ;
auto .
Save .
Instance ishift_mon {
A } `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f :A -> (
O1 -m>
O2 )) :
monotonic (
fun (
y :O1) (
x :A) =>
f x y ).
red ;
intros ;
intro z .
apply (
fmonotonic (
f z ));
auto .
Save .
Definition ishift {
A } `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f :A -> (
O1 -m>
O2 )) :
O1 -m> (
A ->
O2 )
:=
mon (
fun (
y :O1) (
x :A) =>
f x y ) (
fmonotonic :=ishift_mon
f ).
Lemma ishift_simpl :
forall {
A } `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f :A -> (
O1 -m>
O2 ))
x y ,
ishift f x y =
f y x .
trivial .
Save .
Lemma ishift_le_compat :
forall {
A } `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f g :A -> (
O1 -m>
O2 )),
f <=
g ->
ishift f <=
ishift g .
intros ;
intros x y .
repeat rewrite ishift_simpl .
apply (
H y x );
trivial .
Save .
Hint Resolve @
ishift_le_compat .
Add Parametric Morphism {
A } `{
o1 :ord
O1 } `{
o2 :ord
O2 }
: (
ishift (
A :=A) (
O1 :=O1) (
O2 :=O2))
with signature Oeq &
eq &
Oeq
as ishift_eq_compat .
intros f g H x y .
repeat rewrite ishift_simpl .
apply H .
Save .
Instance shift_fun_mon `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 } (
f :O1 -m> (
O2 ->
O3 ))
{
m :forall
x ,
monotonic (
f x )} :
monotonic (
shift f ).
intros O1 o1 O2 o2 O3 o3 f m x y H z .
repeat (
rewrite shift_simpl );
apply m ;
trivial .
Save .
Instance shift_mon2 `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 } (
f :O1 -m>
O2 -m>
O3 )
:
monotonic2 (
fun x y =>
f y x ).
intros O1 o1 O2 o2 O3 o3 f .
apply mon2_intro .
intros x y H z ;
auto .
intros x z t H ;
apply (
fmonotonic f );
trivial .
Save .
Hint Resolve @
shift_mon_fun @
shift_fun_mon @
shift_mon2 .
Definition mshift `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 } (
f :O1 -m>
O2 -m>
O3 )
:
O2 -m>
O1 -m>
O3 :=
mon2 (
fun x y =>
f y x ).
Definition id O {
o :ord
O } :
O ->
O :=
fun x =>
x .
Instance mon_id :
forall {
O :Type} {
o :ord
O },
monotonic (
id O ).
auto .
Save .
Definition cte A `{
o1 :ord
O1 } (
c :O1) :
A ->
O1 :=
fun x =>
c .
Instance mon_cte :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
c :O2),
monotonic (
cte O1 c ).
auto .
Save .
Definition mseq_cte {
O } {
o :ord
O } (
c :O) :
nat -m>
O :=
mon (
cte nat c ).
Add Parametric Morphism `{
o1 :ord
O1 } `{
o2 :ord
O2 } : (@
cte O1 O2 _ )
with signature Ole &
Ole as cte_le_compat .
intros c1 c2 H x ;
simpl ;
auto .
Save .
Add Parametric Morphism `{
o1 :ord
O1 } `{
o2 :ord
O2 } : (@
cte O1 O2 _ )
with signature Oeq &
Oeq as cte_eq_compat .
intros c1 c2 H x ;
simpl ;
auto .
Save .
Instance mon_diag `{
o1 :ord
O1 } `{
o2 :ord
O2 }(
f :O1 -m> (
O1 -m>
O2 ))
:
monotonic (
fun x =>
f x x ).
intros O1 o1 O2 o2 f x y H ;
apply (
fmonotonic2 f );
auto .
Save .
Hint Resolve @
mon_diag .
Definition diag `{
o1 :ord
O1 } `{
o2 :ord
O2 }(
f :O1 -m> (
O1 -m>
O2 )) :
O1 -m>
O2
:=
mon (
fun x =>
f x x ).
Lemma fmon_diag_simpl :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f :O1 -m> (
O1 -m>
O2 )) (
x :O1),
diag f x =
f x x .
trivial .
Save .
Lemma diag_le_compat :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f g :O1 -m> (
O1 -m>
O2 )),
f <=
g ->
diag f <=
diag g .
intros ;
intro ;
simpl ;
auto .
apply H .
Save .
Hint Resolve @
diag_le_compat .
Add Parametric Morphism `{
o1 :ord
O1 } `{
o2 :ord
O2 } : (
diag (
O1 :=O1) (
O2 :=O2))
with signature Oeq &
Oeq as diag_eq_compat .
intros ;
intro ;
simpl ;
auto .
apply H .
Save .
Lemma diag_shift :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } (
f :
O1 -m>
O1 -m>
O2 ),
diag f ==
diag (
mshift f ).
intros ;
intro x ;
unfold mshift ,diag;
auto .
Save .
Hint Resolve @
diag_shift .
Lemma mshift_simpl :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 }
(
h :O1 -m>
O2 -m>
O3 ) (
x :
O2 ) (
y :O1),
mshift h x y =
h y x .
trivial .
Save .
Lemma mshift_le_compat :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 }
(
f g :O1 -m>
O2 -m>
O3 ),
f <=
g ->
mshift f <=
mshift g .
intros ;
intros x y ;
simpl ;
unfold mshift ;
intros .
apply H ;
auto .
Save .
Hint Resolve @
mshift_le_compat .
Add Parametric Morphism `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 } : (@
mshift O1 _ O2 _ O3 _ )
with signature Oeq &
Oeq as mshift_eq_compat .
intros f g H x y ;
simpl ;
unfold mshift ;
intros .
apply H ;
auto .
Save .
Lemma mshift2_eq :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 } (
h :
O1 -m>
O2 -m>
O3 ),
mshift (
mshift h ) ==
h .
intros ;
intros x y ;
unfold mshift ;
auto .
Save .
Instance monotonic_comp `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 }
(
f :O2 ->
O3 ){
mf :
monotonic f } (
g :O1 ->
O2 ){
mg :monotonic
g } :
monotonic (
fun x =>
f (
g x )).
intros O1 o1 O2 o2 O3 o3 f mf g mg x y H ;
auto .
Save .
Hint Resolve @
monotonic_comp .
Instance monotonic_comp_mon `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 }
(
f :O2 -m>
O3 )(
g :O1 -m>
O2 ) :
monotonic (
fun x =>
f (
g x )).
intros O1 o1 O2 o2 O3 o3 f g x y H ;
auto .
Save .
Hint Resolve @
monotonic_comp_mon .
Definition comp `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 } (
f :O2 -m>
O3 ) (
g :O1 -m>
O2 )
:
O1 -m>
O3 :=
mon (
fun x =>
f (
g x )).
Infix "@" :=
comp (
at level 35) :
O_scope .
Lemma comp_simpl :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 }
(
f :O2 -m>
O3 ) (
g :O1 -m>
O2 ) (
x :O1), (
f@g )
x =
f (
g x ).
trivial .
Save .
Add Parametric Morphism `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 }: (@
comp O1 _ O2 _ O3 _ )
with signature Ole ++>
Ole ++>
Ole
as comp_le_compat .
intros f1 f2 H g1 g2 H1 x ;
unfold comp ;
simpl ;
auto .
transitivity (
f2 (
g1 x ));
auto .
Save .
Hint Immediate @
comp_le_compat .
Add Parametric Morphism `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 } : (@
comp O1 _ O2 _ O3 _ )
with signature Oeq &
Oeq &
Oeq
as comp_eq_compat .
intros f1 f2 H g1 g2 H1 x .
repeat (
rewrite comp_simpl );
transitivity (
f2 (
g1 x ));
auto .
Save .
Hint Immediate @
comp_eq_compat .
(f@2 g) h x = f (g x) (h x)
Instance mon_app2 `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 } `{
o4 :ord
O4 }
(
f :O2 ->
O3 ->
O4 ) (
g :O1 ->
O2 ) (
h :O1 ->
O3 )
{
mf :monotonic2
f }{
mg :monotonic
g } {
mh :monotonic
h }
:
monotonic (
fun x =>
f (
g x ) (
h x )).
intros O1 o1 O2 o2 O3 o3 O4 o4 f g h mf mg mh x y H ;
auto .
Save .
Instance mon_app2_mon `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 } `{
o4 :ord
O4 }
(
f :O2 -m>
O3 -m>
O4 ) (
g :O1 -m>
O2 ) (
h :O1 -m>
O3 )
:
monotonic (
fun x =>
f (
g x ) (
h x )).
intros O1 o1 O2 o2 O3 o3 O4 o4 f g h x y H .
apply (
fmonotonic2 f );
auto .
Save .
Definition app2 `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 } `{
o4 :ord
O4 }
(
f :O2 -m>
O3 -m>
O4 ) (
g :O1 -m>
O2 ) (
h :O1 -m>
O3 ) :
O1 -m>
O4
:=
mon (
fun x =>
f (
g x ) (
h x )).
Infix "@2" :=
app2 (
at level 70) :
O_scope .
Add Parametric Morphism `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 } `{
o4 :ord
O4 }:
(@
app2 O1 _ O2 _ O3 _ O4 _ )
with signature Ole ++>
Ole ++>
Ole ++>
Ole
as app2_le_compat .
intros f1 f2 H g1 g2 H1 h1 h2 H2 x ;
unfold app2 ;
simpl ;
auto .
transitivity (
f2 (
g1 x ) (
h1 x )).
apply H .
apply (
fmonotonic2 f2 );
auto .
Save .
Hint Immediate @
app2_le_compat .
Add Parametric Morphism `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 } `{
o4 :ord
O4 }:
(@
app2 O1 _ O2 _ O3 _ O4 _ )
with signature Oeq &
Oeq &
Oeq &
Oeq
as app2_eq_compat .
intros f1 f2 H g1 g2 H1 h1 h2 H2 x ;
unfold app2 ;
simpl ;
auto .
transitivity (
f2 (
g1 x ) (
h1 x )).
apply H .
apply (
fstable2 f2 );
auto .
Save .
Hint Immediate @
app2_eq_compat .
Lemma app2_simpl :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 } `{
o4 :ord
O4 }
(
f :O2 -m>
O3 -m>
O4 ) (
g :O1 -m>
O2 ) (
h :O1 -m>
O3 ) (
x :O1),
(
f@2 g )
h x =
f (
g x ) (
h x ).
trivial .
Save .
Lemma comp_monotonic_right :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 } (
f :
O2 -m>
O3 ) (
g1 g2 :O1 -m>
O2 ),
g1 <=
g2 ->
f @
g1 <=
f @
g2 .
auto .
Save .
Hint Resolve @
comp_monotonic_right .
Lemma comp_monotonic_left :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 } (
f1 f2 :
O2 -m>
O3 ) (
g :O1 -m>
O2 ),
f1 <=
f2 ->
f1 @
g <=
f2 @
g .
auto .
Save .
Hint Resolve @
comp_monotonic_left .
Instance comp_monotonic2 :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 },
monotonic2 (@
comp O1 _ O2 _ O3 _ ).
red ;
intros .
apply comp_le_compat ;
auto .
Save .
Hint Resolve @
comp_monotonic2 .
Definition fcomp `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 } :
(
O2 -m>
O3 ) -m> (
O1 -m>
O2 ) -m> (
O1 -m>
O3 ) :=
mon2 (@
comp O1 _ O2 _ O3 _ ).
Implicit Arguments fcomp [[
o1 ] [
o2 ] [
o3 ]].
Lemma fcomp_simpl :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 }
(
f :O2 -m>
O3 ) (
g :O1 -m>
O2 ),
fcomp _ _ _ f g =
f@g .
trivial .
Save .
Definition fcomp2 `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 } `{
o4 :ord
O4 } :
(
O3 -m>
O4 ) -m> (
O1 -m>
O2 -m>
O3 ) -m> (
O1 -m>
O2 -m>
O4 ):=
(
fcomp O1 (
O2 -m>
O3 ) (
O2 -m>
O4 ))@(
fcomp O2 O3 O4 ).
Implicit Arguments fcomp2 [[
o1 ] [
o2 ] [
o3 ] [
o4 ]].
Lemma fcomp2_simpl :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 } `{
o4 :ord
O4 }
(
f :O3 -m>
O4 ) (
g :O1 -m>
O2 -m>
O3 ) (
x :O1)(
y :O2),
fcomp2 _ _ _ _ f g x y =
f (
g x y ).
trivial .
Save .
Lemma fmon_le_compat2 :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 }
(
f :
O1 -m>
O2 -m>
O3 ) (
x y :O1) (
z t :O2),
x <=y ->
z <=t ->
f x z <=
f y t .
intros ;
transitivity (
f x t ).
apply (
fmonotonic (
f x ));
auto .
apply (
fmonotonic f );
auto .
Save .
Hint Resolve fmon_le_compat2 .
Lemma fmon_cte_comp :
forall `{
o1 :ord
O1 } `{
o2 :ord
O2 } `{
o3 :ord
O3 }
(
c :O3)(
f :O1 -m>
O2 ), (
mon (
cte O2 c )) @
f ==
mon (
cte O1 c ).
intros ;
intro x ;
auto .
Save .
Basic operators of omega-cpos
Constant :
lub : limit of monotonic sequences
Class cpo `{
o :ord
D } :
Type :=
mk_cpo
{
D0 :
D ;
lub :
forall (
f :nat -m>
D ),
D ;
Dbot :
forall x :D,
D0 <=
x ;
le_lub :
forall (
f :
nat -m>
D ) (
n :nat),
f n <=
lub f ;
lub_le :
forall (
f :
nat -m>
D ) (
x :D), (
forall n ,
f n <=
x ) ->
lub f <=
x }.
Implicit Arguments cpo [[
o ]].
Notation "0" :=
D0 :
O_scope .
Hint Resolve @
Dbot @
le_lub @
lub_le .
Definition mon_ord_equiv :
forall `{
o :ord
D1 } `{
o1 :ord
D2 } {
o2 :ord
D2 },
eq_ord o1 o2 ->
fmon D1 D2 (
o2 :=o2) ->
fmon D1 D2 (
o2 :=o1).
unfold eq_ord ,
eq_rel ;
intros D1 o D2 o1 o2 H f ;
exists (
fun x =>
f x ).
red ;
intros .
rewrite H .
apply (
fmonotonic f );
auto .
Defined .
Lemma mon_ord_equiv_simpl :
forall `{
o :ord
D1 } `{
o1 :ord
D2 } {
o2 :ord
D2 }
(
H :eq_ord
o1 o2 ) (
f :fmon
D1 D2 (
o2 :=o2)) (
x :D1),
mon_ord_equiv H f x =
f x .
trivial .
Save .
Definition cpo_ord_equiv `{
o1 :ord
D } (
o2 :ord
D )
:
eq_ord o1 o2 ->
cpo (
o :=o1)
D ->
cpo (
o :=o2)
D .
unfold eq_ord ,
eq_rel ;
intros D o1 o2 H c .
exists (
D0 (
cpo :=c)) (
fun f :
nat -m>
D =>
lub (
cpo :=c) (
mon_ord_equiv H f )).
intros ;
rewrite <-
H ;
auto .
intros ;
rewrite <-
H .
apply (
le_lub (
mon_ord_equiv H f )
n ).
intros ;
rewrite <-
H .
apply lub_le ;
intros ;
simpl ;
rewrite H ;
auto .
Defined .
Add Parametric Morphism `{
c :cpo
D } : (
lub (
cpo :=c))
with signature Ole ++>
Ole as lub_le_compat .
intros f g H ;
apply lub_le ;
intros .
transitivity (
g n );
auto .
Save .
Hint Resolve @
lub_le_compat .
Add Parametric Morphism `{
c :cpo
D }: (
lub (
cpo :=c))
with signature Oeq &
Oeq as lub_eq_compat .
intros ;
apply Ole_antisym ;
auto .
Save .
Hint Resolve @
lub_eq_compat .
Notation "'mlub' f" := (
lub (
mon f )) (
at level 60) :
O_scope .
Lemma mlub_le_compat :
forall `{
c :cpo
D } (
f g :nat ->
D ) {
mf :monotonic
f } {
mg :monotonic
g },
f <=
g ->
mlub f <=
mlub g .
auto .
Save .
Hint Resolve @
mlub_le_compat .
Lemma mlub_eq_compat :
forall `{
c :cpo
D } (
f g :nat ->
D ) {
mf :monotonic
f } {
mg :monotonic
g },
f ==
g ->
mlub f ==
mlub g .
auto .
Save .
Hint Resolve @
mlub_eq_compat .
Lemma le_mlub :
forall `{
c :cpo
D } (
f :nat ->
D ) {
m :monotonic
f } (
n :nat),
f n <=
mlub f .
intros ;
apply (
le_lub (
mon f ));
auto .
Save .
Lemma mlub_le :
forall `{
c :cpo
D }(
f :nat ->
D ) {
m :monotonic
f }(
x :D), (
forall n ,
f n <=
x ) ->
mlub f <=
x .
intros ;
auto .
Save .
Hint Resolve @
le_mlub @
mlub_le .
Instance lub_mon `{
c :cpo
D } :
monotonic lub .
intros ;
exact (
lub_le_compat D o c ).
Save .
Definition Lub `{
c :cpo
D } : (
nat -m>
D ) -m>
D :=
mon lub .
Instance monotonic_lub_comp {
O } {
o :ord
O } `{
c :cpo
D } (
f :O ->
nat ->
D ){
mf :monotonic2
f }:
monotonic (
fun x =>
mlub (
f x )).
intros ;
auto .
Save .
Lemma lub_cte :
forall `{
c :cpo
D } (
d :D),
mlub (
cte nat d ) ==
d .
intros ;
apply Ole_antisym ;
auto .
apply le_lub with (
f :=mon (
fun _ =>
d )) (
n :=O);
auto .
Save .
Hint Resolve @
lub_cte .
Lemma mlub_lift_right :
forall `{
c :cpo
D } (
f :nat -m>
D )
n ,
lub f ==
mlub (
seq_lift_right f n ).
intros ;
apply Ole_antisym .
apply lub_le_compat ;
intro .
unfold seq_lift_right ;
simpl ;
auto with arith .
unfold seq_lift_right ;
simpl ;
auto .
Save .
Hint Resolve @
mlub_lift_right .
Lemma mlub_lift_left :
forall `{
c :cpo
D } (
f :nat -m>
D )
n ,
lub f ==
mlub (
seq_lift_left f n ).
intros ;
apply Ole_antisym .
apply lub_le_compat ;
intro .
unfold seq_lift_left ;
simpl ;
auto with arith .
unfold seq_lift_left ;
auto .
Save .
Hint Resolve @
mlub_lift_left .
Lemma lub_lift_right :
forall `{
c :cpo
D } (
f :nat -m>
D )
n ,
lub f ==
lub (
mseq_lift_right f n ).
intros ;
transitivity (
mlub (
seq_lift_right f n ));
auto .
apply lub_eq_compat ;
intro m ;
auto .
Save .
Hint Resolve @
lub_lift_right .
Lemma lub_lift_left :
forall `{
c :cpo
D } (
f :nat -m>
D )
n ,
lub f ==
lub (
mseq_lift_left f n ).
intros ;
transitivity (
mlub (
seq_lift_left f n ));
auto .
apply lub_eq_compat ;
intro m ;
auto .
Save .
Hint Resolve @
lub_lift_left .
Lemma lub_le_lift :
forall `{
c :cpo
D } (
f g :nat -m>
D )
(
n :nat), (
forall k ,
n <=
k ->
f k <=
g k ) ->
lub f <=
lub g .
intros ;
apply lub_le ;
intros .
transitivity (
f (
n +n0));
auto with arith .
transitivity (
g (
n +n0));
auto with arith .
Save .
Lemma lub_eq_lift :
forall `{
c :cpo
D } (
f g :nat -m>
D ) {
m :monotonic
f } {
m' :monotonic
g }
(
n :nat), (
forall k ,
n <=
k ->
f k ==
g k ) ->
lub f ==
lub g .
intros ;
apply Ole_antisym ;
apply lub_le_lift with n ;
intros ;
auto .
apply Oeq_le_sym ;
auto .
Save .
(lub_fun h) x = lub_n (h n x)
Definition lub_fun {
A } `{
c :cpo
D } (
h :
nat -m> (
A ->
D )) :
A ->
D
:=
fun x =>
mlub (
h <o>
x ).
Instance lub_shift_mon {
O } {
o :ord
O } `{
c :cpo
D } (
h :
nat -m> (
O -m>
D ))
:
monotonic (
fun (
x :O) =>
lub (
mshift h x )).
red ;
auto .
Save .
Hint Resolve @
lub_shift_mon .
Instance fcpo {
A :
Type } `(
c :cpo
D ) :
cpo (
A ->
D ) :=
{
D0 :=
fun x :A => (0:D);
lub :=
fun f =>
lub_fun f }.
intros f x ;
auto .
intros f n x ;
unfold lub_fun .
apply (
le_mlub (
shift f x )
n );
auto .
intros h f H x ;
unfold lub_fun .
apply mlub_le ;
intro n ;
rewrite shift_simpl ;
auto .
Defined .
Lemma fcpo_lub_simpl :
forall {
A } `{
c :cpo
D } (
h :nat -m> (
A ->
D ))(
x :A),
(
lub h )
x =
mlub (
h <o>
x ).
trivial .
Save .
Lemma lub_ishift :
forall {
A } `{
c :cpo
D } (
h :A -> (
nat -m>
D )),
lub (
ishift h ) ==
fun x =>
lub (
h x ).
intros ;
intro x .
rewrite fcpo_lub_simpl ;
apply lub_eq_compat ;
intro n ;
auto .
Save .
Cpo of monotonic functions
Instance fmon_cpo {
O } {
o :ord
O } `{
c :cpo
D } :
cpo (
O -m>
D ) :=
{
D0 :=
mon (
cte O (0:D));
lub :=
fun h :nat -m> (
O -m>
D ) =>
mon (
fun (
x :O) =>
lub (
cpo :=c) (
mshift h x ))}.
intros f x ;
unfold cte ;
simpl ;
auto .
intros h n x ;
simpl .
apply (
le_mlub (
fun y =>
h y x )
n );
auto .
intros h f H x ;
simpl .
apply mlub_le ;
intro .
apply H ;
auto .
Defined .
Lemma fmon_lub_simpl :
forall {
O } {
o :ord
O } `{
c :cpo
D }
(
h :nat -m> (
O -m>
D ))(
x :O), (
lub h )
x =
lub (
mshift h x ).
trivial .
Save .
Hint Resolve @
fmon_lub_simpl .
Instance mon_fun_lub :
forall {
O } {
o :ord
O } `{
c :cpo
D }
(
h :nat -m> (
O ->
D )) {
mh :forall
n ,
monotonic (
h n )},
monotonic (
lub h ).
red ;
intros ;
simpl ;
auto .
unfold lub_fun ;
apply mlub_le_compat .
apply (
shift_fun_mon h x y );
trivial .
Save .
Link between lubs on ordinary functions and monotonic functions
Lemma lub_mon_fcpo :
forall {
O } {
o :ord
O } `{
c :cpo
D } (
h :nat -m> (
O -m>
D )),
lub h ==
mon (
lub (
mfun2 h )).
intros ;
intros x .
rewrite fmon_lub_simpl ;
rewrite mon_simpl ;
rewrite fcpo_lub_simpl .
apply lub_eq_compat ;
intro y ;
auto .
Save .
Lemma lub_fcpo_mon :
forall {
O } {
o :ord
O } `{
c :cpo
D } (
h :nat -m> (
O ->
D ))
{
mh :forall
x ,
monotonic (
h x )},
lub h ==
lub (
mon2 h ).
intros ;
intro x .
rewrite fmon_lub_simpl ;
rewrite fcpo_lub_simpl .
apply lub_eq_compat ;
intro y ;
auto .
Save .
Lemma double_lub_diag :
forall `{
c :cpo
D } (
h :
nat -m>
nat -m>
D ),
lub (
lub h ) ==
lub (
diag h ).
intros ;
apply Ole_antisym .
apply lub_le ;
intros ;
simpl ;
apply lub_le ;
simpl ;
intros .
transitivity (
h n0 (
n +n0));
auto with arith .
transitivity (
h (
n +n0) (
n +n0)).
apply (
fmonotonic2 h );
simpl ;
auto with arith .
apply (
le_lub (
diag h ) (
n +
n0 )%nat).
apply lub_le_compat .
intro n ;
simpl ;
unfold diag .
apply (
le_mlub (
fun y =>
h y n )
n );
auto .
Save .
Hint Resolve @
double_lub_diag .
Lemma double_lub_shift :
forall `{
c :cpo
D } (
h :
nat -m>
nat -m>
D ),
lub (
lub h ) ==
lub (
lub (
mshift h )).
intros ;
transitivity (
lub (
diag h ));
auto .
transitivity (
lub (
diag (
mshift h )));
auto .
Save .
Hint Resolve @
double_lub_shift .
Lemma lub_comp_le :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } (
f :D1 -m>
D2 ) (
h :
nat -m>
D1 ),
lub (
f @
h ) <=
f (
lub h ).
intros ;
apply lub_le ;
unfold comp ;
intros .
apply (
fmonotonic f );
auto .
Save .
Hint Resolve @
lub_comp_le .
Lemma lub_app2_le :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }
(
F :D1 -m>
D2 -m>
D3 ) (
f :
nat -m>
D1 ) (
g :
nat -m>
D2 ),
lub ((
F @2
f )
g ) <=
F (
lub f ) (
lub g ).
intros ;
apply mlub_le ;
unfold app2 ;
auto .
intros ;
apply (
fmonotonic2 F );
auto .
Save .
Hint Resolve @
lub_app2_le .
Class continuous `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } (
f :D1 -m>
D2 ) :=
cont_intro :
forall (
h :
nat -m>
D1 ),
f (
lub h ) <=
lub (
f @
h ).
Type classes Opaque continuous .
Lemma continuous_eq_compat :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 }(
f g :D1 -m>
D2 ),
f ==
g ->
continuous f ->
continuous g .
red ;
intros .
transitivity (
f (
lub h )).
assert (
g <=
f );
auto .
transitivity (
lub (
f @
h ));
auto .
Save .
Add Parametric Morphism `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } : (@
continuous D1 _ _ D2 _ _ )
with signature Oeq &
iff
as continuous_eq_compat_iff .
split ;
intros .
apply continuous_eq_compat with x ;
trivial .
apply continuous_eq_compat with y ;
auto .
Save .
Lemma lub_comp_eq :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } (
f :D1 -m>
D2 ) (
h :
nat -m>
D1 ),
continuous f ->
f (
lub h ) ==
lub (
f @
h ).
intros ;
apply Ole_antisym ;
auto .
Save .
Hint Resolve @
lub_comp_eq .
Instance cont0 `{c1 :cpo D1 } `{c2 :cpo D2 } : continuous (mon (cte D1 (0:D2))).red ; simpl ; intros ; unfold cte ; auto .Save .Implicit Arguments cont0 [].
double_app f g n m = f m (g n)
Definition double_app `{o1 :ord O1 } `{o2 :ord O2 } `{o3 :ord O3 } `{o4 : ord O4 } f :O1 -m> O3 -m> O4 ) (g :O2 -m> O3 ) O2 -m> (O1 -m> O4 ) := mon ((mshift f ) @ g ).
Class continuous2 `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 } (
F :D1 -m>
D2 -m>
D3 ) :=
continuous2_intro :
forall (
f :
nat -m>
D1 ) (
g :nat -m>
D2 ),
F (
lub f ) (
lub g ) <=
lub ((
F @2
f )
g ).
Lemma continuous2_app :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }
(
F :
D1 -m>
D2 -m>
D3 ) {
cF :continuous2
F } (
k :D1),
continuous (
F k ).
red ;
intros .
transitivity (
F (
mlub (
cte nat k )) (
lub h ));
auto .
apply (
fmonotonic2 F );
auto .
transitivity (
lub ((
F @2 (
mon (
cte nat k )))
h ));
auto .
apply lub_le_compat ;
simpl ;
auto .
Save .
Type classes Opaque continuous2 .
Lemma continuous2_eq_compat :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 } (
f g :
D1 -m>
D2 -m>
D3 ),
f ==
g ->
continuous2 f ->
continuous2 g .
red ;
intros .
transitivity (
f (
lub f0 ) (
lub g0 )).
generalize (
H (
lub f0 ) (
lub (
g0 )));
auto .
transitivity (
lub ((
f @2
f0 )
g0 ));
auto .
Save .
Lemma continuous2_continuous :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }
(
F :
D1 -m>
D2 -m>
D3 ),
continuous2 F ->
continuous F .
red ;
intros ;
intro k .
transitivity (
F (
lub h ) (
mlub (
cte nat k )) );
auto .
transitivity (
lub ((
F @2
h ) (
mon (
cte nat k ))));
auto .
rewrite fmon_lub_simpl .
apply lub_le_compat ;
simpl ;
auto .
Save .
Hint Immediate @
continuous2_continuous .
Lemma continuous2_left :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }
(
F :
D1 -m>
D2 -m>
D3 ) (
h :nat -m>
D1 ) (
x :D2),
continuous F ->
F (
lub h )
x <=
lub (
mshift (
F @
h )
x ).
intros .
transitivity ((
mlub (
F @
h ))
x );
auto .
exact (
H h x ).
Save .
Lemma continuous2_right :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }
(
F :
D1 -m>
D2 -m>
D3 ) (
x :D1)(
h :nat -m>
D2 ),
continuous2 F ->
F x (
lub h ) <=
lub (
F x @
h ).
intros ;
apply (
continuous2_app F x ).
Save .
Lemma continuous_continuous2 :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }
(
F :
D1 -m>
D2 -m>
D3 ) (
cFr :
forall k :D1,
continuous (
F k )) (
cF :
continuous F ),
continuous2 F .
red ;
intros .
transitivity (
lub (
F (
lub f ) @
g ));
auto .
apply lub_le ;
unfold comp ;
simpl ;
intros .
transitivity ((
lub (
F@f )) (
g n )).
apply cF .
transitivity (
lub (
mshift (
F@f ) (
g n )));
auto .
rewrite (
lub_lift_right ((
F @2
f )
g )
n ).
apply lub_le_compat .
unfold seq_lift_right ;
simpl .
intro ;
apply (
fmonotonic2 F );
auto with arith .
Save .
Hint Resolve @
continuous2_app @
continuous2_continuous @
continuous_continuous2 .
Lemma lub_app2_eq :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }
(
F :
D1 -m>
D2 -m>
D3 ) {
cFr :forall
k :D1,
continuous (
F k )} {
cF :
continuous F },
forall (
f :nat -m>
D1 ) (
g :nat -m>
D2 ),
F (
lub f ) (
lub g ) ==
lub ((
F@2 f )
g ).
intros ;
apply Ole_antisym ;
auto .
apply (
continuous_continuous2 F );
trivial .
Save .
Lemma lub_cont2_app2_eq :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }
(
F :
D1 -m>
D2 -m>
D3 ){
cF :
continuous2 F },
forall (
f :nat -m>
D1 ) (
g :nat -m>
D2 ),
F (
lub f ) (
lub g ) ==
lub ((
F@2 f )
g ).
intros ;
apply lub_app2_eq ;
auto .
intro ;
apply (
continuous2_app F ).
apply continuous2_continuous ;
trivial .
Save .
Lemma mshift_continuous2 :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }
(
F :
D1 -m>
D2 -m>
D3 ),
continuous2 F ->
continuous2 (
mshift F ).
red ;
intros ;
repeat rewrite mshift_simpl ;
auto .
rewrite (
H g f );
auto .
apply lub_le_compat ;
intro n ;
auto .
Save .
Hint Resolve @
mshift_continuous2 .
Lemma monotonic_sym :
forall `{
o1 :ord
D1 } `{
o2 :ord
D2 } (
F :
D1 ->
D1 ->
D2 ),
(
forall x y ,
F x y ==
F y x ) -> (
forall k :D1,
monotonic (
F k )) ->
monotonic F .
red ;
intros ;
intro k .
transitivity (
F k x );
auto .
transitivity (
F k y );
auto .
apply H0 ;
trivial .
Save .
Hint Immediate @
monotonic_sym .
Lemma monotonic2_sym :
forall `{
o1 :ord
D1 } `{
o2 :ord
D2 } (
F :
D1 ->
D1 ->
D2 ),
(
forall x y ,
F x y ==
F y x ) -> (
forall k :D1,
monotonic (
F k )) ->
monotonic2 F .
intros ;
apply mon2_intro ;
auto .
Save .
Hint Immediate @
monotonic2_sym .
Lemma continuous_sym :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } (
F :
D1 -m>
D1 -m>
D2 ),
(
forall x y ,
F x y ==
F y x ) -> (
forall k :D1,
continuous (
F k )) ->
continuous F .
red ;
intros ;
intro k .
transitivity (
F k (
lub h ));
auto .
transitivity (
lub ((
F k ) @
h ));
auto .
simpl .
unfold comp ,
fun_ext ,
lub_fun ,
shift ,
fun2 ;
simpl ;
auto .
Save .
Lemma continuous2_sym :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } (
F :
D1 -m>D1 -m>D2),
(
forall x y ,
F x y ==
F y x ) -> (
forall k ,
continuous (
F k )) ->
continuous2 F .
intros ;
apply continuous_continuous2 ;
auto .
apply continuous_sym ;
auto .
Save .
Hint Resolve @
continuous2_sym .
continuity is preserved by composition
Lemma continuous_comp :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }
(
f :D2 -m>
D3 )(
g :D1 -m>
D2 ),
continuous f ->
continuous g ->
continuous (
mon (
f@g )).
red ;
intros .
simpl ;
unfold comp at 1;
simpl .
transitivity (
f (
lub (
g@h ))).
apply (
fmonotonic f );
auto .
transitivity (
lub (
f@ (
g@h )));
auto .
Save .
Hint Resolve @
continuous_comp .
Lemma continuous2_comp :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 } `{
c4 :cpo
D4 }
(
f :D1 -m>
D2 )(
g :D2 -m>
D3 -m>
D4 ),
continuous f ->
continuous2 g ->
continuous2 (
g @
f ).
intros ;
apply continuous_continuous2 .
red ;
intros .
unfold comp at 1;
simpl .
apply (
continuous2_right g (
f k )
h );
trivial .
apply (
continuous_comp g f );
auto .
apply continuous2_continuous ;
trivial .
Save .
Hint Resolve @
continuous2_comp .
Lemma continuous2_comp2 :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 } `{
c4 :cpo
D4 }
(
f :D3 -m>
D4 )(
g :D1 -m>
D2 -m>
D3 ),
continuous f ->
continuous2 g ->
continuous2 (
fcomp2 D1 D2 D3 D4 f g ).
red ;
intros .
rewrite fcomp2_simpl .
transitivity (
f (
lub ((
g@2 f0 )
g0 )));
auto .
transitivity (
lub (
f@ ((
g@2 f0 )
g0 )));
auto .
apply lub_le_compat .
intros x ;
trivial .
Save .
Hint Resolve @
continuous2_comp2 .
Cpo of continuous functions
Instance lub_continuous `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }
(
f :nat -m> (
D1 -m>
D2 )) {
cf :forall
n ,
continuous (
f n )}
:
continuous (
lub f ).
red ;
intros .
transitivity (
lub (
mshift f (
lub h )));
auto .
apply lub_le ;
intros .
rewrite mshift_simpl .
transitivity (
lub ((
f n ) @
h ));
auto .
Save .
Record fcont `{
c1 :cpo
D1 } `{
c2 :cpo
D2 }:
Type
:=
cont {
fcontm :>
D1 -m>
D2 ;
fcontinuous :
continuous fcontm }.
Hint Resolve @
fcontinuous .
Implicit Arguments fcont [[
o ][
c1 ] [
o0 ][
c2 ]].
Implicit Arguments cont [[
D1 ][
o ][
c1 ] [
D2 ][
o0 ][
c2 ] [
fcontinuous ]].
Infix "-c>" :=
fcont (
at level 30,
right associativity ) :
O_scope .
Definition fcont_fun `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } (
f :D1 -c>
D2 ) :
D1 ->
D2 :=
fun x =>
f x .
Instance fcont_ord `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } :
ord (
D1 -c>D2)
:= {
Oeq :=
fun f g =>
forall x ,
f x ==
g x ;
Ole :=
fun f g =>
forall x ,
f x <=
g x }.
split ;
intuition .
red ;
intros .
transitivity (
y x0 );
auto .
Defined .
Lemma fcont_le_intro :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } (
f g :
D1 -c>
D2 ),
(
forall x ,
f x <=
g x ) ->
f <=
g .
trivial .
Save .
Lemma fcont_le_elim :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } (
f g :
D1 -c>
D2 ),
f <=
g ->
forall x ,
f x <=
g x .
trivial .
Save .
Lemma fcont_eq_intro :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } (
f g :
D1 -c>
D2 ),
(
forall x ,
f x ==
g x ) ->
f ==
g .
intros ;
apply Ole_antisym ;
apply fcont_le_intro ;
auto .
Save .
Lemma fcont_eq_elim :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } (
f g :
D1 -c>
D2 ),
f ==
g ->
forall x ,
f x ==
g x .
intros ;
apply Ole_antisym ;
apply fcont_le_elim ;
auto .
Save .
Lemma fcont_le :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } (
f :
D1 -c>
D2 ) (
x y :
D1 ),
x <=
y ->
f x <=
f y .
intros ;
apply (
fmonotonic (
fcontm f )
x y H ).
Save .
Hint Resolve @
fcont_le .
Lemma fcont_eq :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } (
f :
D1 -c>
D2 ) (
x y :
D1 ),
x ==
y ->
f x ==
f y .
intros ;
apply (
fmon_eq (
fcontm f )
x y H ).
Save .
Hint Resolve @
fcont_eq .
Definition fcont0 D1 `{
c1 :cpo
D1 }
D2 `{
c2 :cpo
D2 } :
D1 -c>
D2 :=
cont (
mon (
cte D1 (0:D2))).
Instance fcontm_monotonic :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 },
monotonic (
fcontm (
D1 :=D1) (
D2 :=D2)).
intros ;
auto .
Save .
Definition Fcontm D1 `{
c1 :cpo
D1 }
D2 `{
c2 :cpo
D2 } : (
D1 -c>
D2 ) -m> (
D1 -m>
D2 ) :=
mon (
fcontm (
D1 :=D1) (
D2 :=D2)).
Instance fcont_lub_continuous :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } (
f :nat -m> (
D1 -c>
D2 )),
continuous (
lub (
D :=D1 -m>
D2 ) (
Fcontm D1 D2 @
f )).
intros ;
apply lub_continuous .
intro ;
simpl ;
auto .
Save .
Definition fcont_lub `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } : (
nat -m> (
D1 -c>
D2 )) ->
D1 -c>
D2 :=
fun f =>
cont (
lub (
D :=D1 -m>
D2 ) (
Fcontm D1 D2 @
f )).
Instance fcont_cpo `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } :
cpo (
D1 -c>
D2 ) :=
{
D0 :=fcont0
D1 D2 ;
lub :=fcont_lub (
D1 :=D1) (
D2 :=D2)}.
intros ;
simpl ;
unfold cte ;
auto .
intros ;
simpl ;
intros .
apply (
le_mlub (
fun y :
nat =>
f y x )).
intros ;
simpl ;
intros .
apply mlub_le ;
intro n ;
apply (
H n x0 ).
Defined .
Definition fcont_app {
O } {
o :ord
O } `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } (
f :
O -m>
D1 -c>
D2 ) (
x :D1) :
O -m>
D2
:=
mshift (
Fcontm D1 D2 @
f )
x .
Infix "<__>" :=
fcont_app (
at level 70) :
O_scope .
Lemma fcont_app_simpl :
forall {
O } {
o :ord
O } `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } (
f :
O -m>
D1 -c>
D2 )(
x :D1)(
y :O),
(
f <__>
x )
y =
f y x .
trivial .
Save .
Instance ishift_continuous :
forall {
A :Type} `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } (
f :
A -> (
D1 -c>
D2 )),
continuous (
ishift f ).
red ;
intros ;
intro x .
simpl .
transitivity (
lub ((
f x ) @
h ));
auto .
apply lub_le_compat ;
simpl ;
trivial .
Qed .
Definition fcont_ishift {
A :Type} `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } (
f :
A -> (
D1 -c>
D2 ))
:
D1 -c> (
A ->
D2 ) :=
cont _ (
fcontinuous :=ishift_continuous
f ).
Instance mshift_continuous :
forall {
O } {
o :ord
O } `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } (
f :
O -m> (
D1 -c>
D2 )),
continuous (
mshift (
Fcontm D1 D2 @
f )).
red ;
intros ;
intro x .
simpl .
rewrite (
fcontinuous (
f x )
h );
auto .
Save .
Definition fcont_mshift {
O } {
o :ord
O } `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } (
f :
O -m> (
D1 -c>
D2 ))
:
D1 -c>
O -m>
D2 :=
cont (
mshift (
Fcontm D1 D2 @
f )).
Lemma fcont_app_continuous :
forall {
O } {
o :ord
O } `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } (
f :
O -m>
D1 -c>
D2 ) (
h :nat -m>
D1 ),
f <__> (
lub h ) <=
lub (
D :=O -m>
D2 ) ((
fcont_mshift f ) @
h ).
intros ;
intro x .
rewrite fcont_app_simpl .
rewrite (
fcontinuous (
f x )
h );
simpl ;
auto .
Save .
Lemma fcont_lub_simpl :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } (
h :nat -m>
D1 -c>
D2 )(
x :D1),
lub h x =
lub (
h <__>
x ).
trivial .
Save .
Instance cont_app_monotonic :
forall `{
o1 :ord
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 } (
f :D1 -m>
D2 -m>
D3 )
(
p :forall
k ,
continuous (
f k )),
monotonic (
O2 :=D2 -c>
D3 ) (
fun (
k :D1) =>
cont _ (
fcontinuous :=p
k )).
red ;
simpl ;
intros .
apply (
fmonotonic f );
trivial .
Qed .
Definition cont_app `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 } (
f :D1 -m>
D2 -m>
D3 )
(
p :forall
k ,
continuous (
f k )) :
D1 -m> (
D2 -c>
D3 )
:=
mon (
fun k =>
cont (
f k ) (
fcontinuous :=p
k )).
Lemma cont_app_simpl :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }(
f :D1 -m>
D2 -m>
D3 )(
p :forall
k ,
continuous (
f k ))
(
k :D1),
cont_app f p k =
cont (
f k ).
trivial .
Save .
Instance cont2_continuous `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 } (
f :D1 -m>
D2 -m>
D3 )
(
p :continuous2
f ) :
continuous (
cont_app f (
continuous2_app f )).
red ;
intros ;
rewrite cont_app_simpl ;
intro k ;
simpl .
transitivity (
lub (
D :=D2 -m>
D3 ) (
f@h )
k ).
assert (
continuous f ).
apply continuous2_continuous ;
trivial .
apply H .
simpl ;
auto .
Qed .
Definition cont2 `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 } (
f :D1 -m>
D2 -m>
D3 )
{
p :continuous2
f } :
D1 -c> (
D2 -c>
D3 )
:=
cont (
cont_app f (
continuous2_app f )).
Instance Fcontm_continuous `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } :
continuous (
Fcontm D1 D2 ).
red ;
intros ;
auto .
Save .
Hint Resolve @
Fcontm_continuous .
Instance fcont_comp_continuous :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }
(
f :D2 -c>
D3 ) (
g :D1 -c>
D2 ),
continuous (
f @
g ).
intros ;
apply (
continuous_comp (
fcontm f ) (
fcontm g ));
auto .
Save .
Definition fcont_comp `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 } (
f :D2 -c>
D3 ) (
g :D1 -c>
D2 )
:
D1 -c>
D3 :=
cont (
f @
g ).
Infix "@_" :=
fcont_comp (
at level 35) :
O_scope .
Lemma fcont_comp_simpl :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }
(
f :D2 -c>
D3 )(
g :D1 -c>
D2 ) (
x :D1), (
f @
_ g )
x =
f (
g x ).
trivial .
Save .
Lemma fcontm_fcont_comp_simpl :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }
(
f :D2 -c>
D3 )(
g :D1 -c>
D2 ),
fcontm (
f @
_ g ) =
f @
g .
trivial .
Save .
Lemma fcont_comp_le_compat :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }
(
f g :
D2 -c>
D3 ) (
k l :D1 -c>
D2 ),
f <=
g ->
k <=
l ->
f @
_ k <=
g @
_ l .
intros ;
apply fcont_le_intro ;
intro x .
repeat (
rewrite fcont_comp_simpl ).
transitivity (
g (
k x ));
auto .
Save .
Hint Resolve @
fcont_comp_le_compat .
Add Parametric Morphism `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }
: (@
fcont_comp _ _ c1 _ _ c2 _ _ c3 )
with signature Ole ++>
Ole ++>
Ole as fcont_comp_le_morph .
intros .
apply fcont_comp_le_compat ;
trivial .
Save .
Add Parametric Morphism `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }
: (@
fcont_comp _ _ c1 _ _ c2 _ _ c3 )
with signature Oeq &
Oeq &
Oeq as fcont_comp_eq_compat .
intros .
apply Ole_antisym ;
auto .
Save .
Definition fcont_Comp D1 `{
c1 :cpo
D1 }
D2 `{
c2 :cpo
D2 }
D3 `{
c3 :cpo
D3 }
: (
D2 -c>
D3 ) -m> (
D1 -c>
D2 ) -m>
D1 -c>
D3
:=
mon2 _ (
mf :=fcont_comp_le_compat (
D1 :=D1) (
D2 :=D2) (
D3 :=D3)).
Lemma fcont_Comp_simpl :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }
(
f :D2 -c>
D3 ) (
g :D1 -c>
D2 ),
fcont_Comp D1 D2 D3 f g =
f @
_ g .
trivial .
Save .
Instance fcont_Comp_continuous2
:
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 },
continuous2 (
fcont_Comp D1 D2 D3 ).
red ;
intros .
change ((
lub f ) @
_ (
lub g ) <=
lub (
D :=D1 -c>
D3 ) ((
fcont_Comp D1 D2 D3 @2
f )
g )).
apply fcont_le_intro ;
intro x ;
rewrite fcont_comp_simpl .
repeat (
rewrite fcont_lub_simpl ).
rewrite fcont_app_continuous .
rewrite double_lub_diag .
apply lub_le_compat ;
simpl ;
auto .
Save .
Definition fcont_COMP D1 `{
c1 :cpo
D1 }
D2 `{
c2 :cpo
D2 }
D3 `{
c3 :cpo
D3 }
: (
D2 -c>
D3 ) -c> (
D1 -c>
D2 ) -c>
D1 -c>
D3
:=
cont2 (
fcont_Comp D1 D2 D3 ).
Lemma fcont_COMP_simpl :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }
(
f :
D2 -c>
D3 ) (
g :D1 -c>
D2 ),
fcont_COMP D1 D2 D3 f g =
f @
_ g .
trivial .
Save .
Definition fcont2_COMP D1 `{
c1 :cpo
D1 }
D2 `{
c2 :cpo
D2 }
D3 `{
c3 :cpo
D3 }
D4 `{
c4 :cpo
D4 }
: (
D3 -c>
D4 ) -c> (
D1 -c>
D2 -c>
D3 ) -c>
D1 -c>
D2 -c>
D4 :=
(
fcont_COMP D1 (
D2 -c>
D3 ) (
D2 -c>
D4 )) @
_ (
fcont_COMP D2 D3 D4 ).
Definition fcont2_comp `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 } `{
c4 :cpo
D4 }
(
f :D3 -c>
D4 )(
F :D1 -c>
D2 -c>
D3 ) :=
fcont2_COMP D1 D2 D3 D4 f F .
Infix "@@_" :=
fcont2_comp (
at level 35) :
O_scope .
Lemma fcont2_comp_simpl :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 } `{
c4 :cpo
D4 }
(
f :D3 -c>
D4 )(
F :D1 -c>
D2 -c>
D3 )(
x :D1)(
y :D2), (
f @@
_ F )
x y =
f (
F x y ).
trivial .
Save .
Lemma fcont_le_compat2 :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 } (
f :
D1 -c>
D2 -c>
D3 )
(
x y :
D1 ) (
z t :
D2 ),
x <=
y ->
z <=
t ->
f x z <=
f y t .
intros ;
transitivity (
f y z );
auto .
assert (
f x <=
f y );
auto .
Save .
Hint Resolve @
fcont_le_compat2 .
Lemma fcont_eq_compat2 :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 } (
f :
D1 -c>
D2 -c>
D3 )
(
x y :
D1 ) (
z t :
D2 ),
x ==
y ->
z ==
t ->
f x z ==
f y t .
intros ;
apply Ole_antisym ;
auto .
Save .
Hint Resolve @
fcont_eq_compat2 .
Lemma fcont_continuous :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } (
f :D1 -c>
D2 )(
h :nat-m>
D1 ),
f (
lub h ) <=
lub (
f @
h ).
intros ;
apply (
fcontinuous f h ).
Save .
Hint Resolve @
fcont_continuous .
Instance fcont_continuous2 :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }
(
f :D1 -c>
D2 -c>
D3 ),
continuous2 (
Fcontm D2 D3 @
f ).
intros ;
apply continuous_continuous2 ;
intros .
change (
continuous (
f k ));
auto .
apply (
continuous_comp (
Fcontm D2 D3 ) (
fcontm f ));
auto .
Save .
Hint Resolve @
fcont_continuous2 .
Instance cshift_continuous2 :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }
(
f :D1 -c>
D2 -c>
D3 ),
continuous2 (
mshift (
Fcontm D2 D3 @
f )).
intros ;
auto .
Save .
Hint Resolve @
cshift_continuous2 .
Definition cshift `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 } (
f :D1 -c>
D2 -c>
D3 )
:
D2 -c>
D1 -c>
D3 :=
cont2 (
mshift (
Fcontm D2 D3 @
f )).
Lemma cshift_simpl :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }
(
f :D1 -c>
D2 -c>
D3 ) (
x :D2) (
y :D1),
cshift f x y =
f y x .
trivial .
Save .
Definition fcont_SEQ D1 `{
c1 :cpo
D1 }
D2 `{
c2 :cpo
D2 }
D3 `{
c3 :cpo
D3 }
: (
D1 -c>
D2 ) -c> (
D2 -c>
D3 ) -c>
D1 -c>
D3 :=
cshift (
fcont_COMP D1 D2 D3 ).
Lemma fcont_SEQ_simpl :
forall `{
c1 :cpo
D1 } `{
c2 :cpo
D2 } `{
c3 :cpo
D3 }
(
f :
D1 -c>
D2 ) (
g :D2 -c>
D3 ),
fcont_SEQ D1 D2 D3 f g =
g @
_ f .
trivial .
Save .
Instance Id_mon :
forall `{
o1 :ord
O1 },
monotonic (
fun x :O1 =>
x ).
red ;
trivial .
Save .
Definition Id O1 {
o1 :ord
O1 } :
O1 -m>
O1 :=
mon (
fun x =>
x ).
Lemma Id_simpl :
forall `{
o1 :ord
O1 } (
x :O1),
Id O1 x =
x .
trivial .
Save .
Fixpoint iter_ `{
c :
cpo D } (
f :
D -m>
D )
n {
struct n } :
D
:=
match n with O => 0 |
S m =>
f (
iter_ f m )
end .
Lemma iter_incr :
forall `{
c :
cpo D } (
f :
D -m>
D )
n ,
iter_ f n <=
f (
iter_ f n ).
induction n ;
simpl ;
auto .
Save .
Hint Resolve @
iter_incr .
Instance iter_mon :
forall `{
c :
cpo D } (
f :
D -m>
D ),
monotonic (
iter_ f ).
red ;
intros .
induction H ;
simpl ;
auto .
transitivity (
iter_ f m );
auto .
Save .
Definition iter `{
c :
cpo D } (
f :
D -m>
D ) :
nat -m>
D :=
mon (
iter_ f ).
Definition fixp `{
c :
cpo D } (
f :
D -m>
D ) :
D :=
mlub (
iter_ f ).
Lemma fixp_le :
forall `{
c :
cpo D } (
f :
D -m>
D ),
fixp f <=
f (
fixp f ).
intros ;
unfold fixp .
transitivity (
lub (
f @ (
iter f )));
auto .
apply lub_le_compat ;
intro n ;
simpl .
apply iter_incr .
Save .
Hint Resolve @
fixp_le .
Lemma fixp_eq :
forall `{
c :
cpo D } (
f :
D -m>
D ) {
mf :continuous
f },
fixp f ==
f (
fixp f ).
intros ;
apply Ole_antisym ;
auto .
unfold fixp .
transitivity (
lub (
f@ (
iter f )));
auto .
rewrite (
mlub_lift_left (
mon (
iter_ f )) (
S O ));
auto .
Save .
Lemma fixp_inv :
forall `{
c :
cpo D } (
f :
D -m>
D )
g ,
f g <=
g ->
fixp f <=
g .
unfold fixp ;
intros .
apply lub_le .
induction n ;
intros ;
simpl ;
auto .
simpl ;
transitivity (
f g );
auto .
Save .
Definition fixp_cte :
forall `{
c :cpo
D } (
d :D),
fixp (
mon (
cte D d )) ==
d .
intros ;
apply fixp_eq with (
f :=mon (
cte D d ));
red ;
intros ;
simpl ;
auto .
apply (
le_mlub (
mon (
cte D d ) @
h )
O ).
Save .
Hint Resolve @
fixp_cte .
Lemma fixp_le_compat :
forall `{
c :cpo
D } (
f g :
D -m>
D ),
f <=
g ->
fixp f <=
fixp g .
intros ;
unfold fixp .
apply mlub_le_compat .
intro n ;
induction n ;
simpl ;
auto .
transitivity (
g (
iter_ (
D :=D)
f n ));
auto .
Save .
Hint Resolve @
fixp_le_compat .
Instance fixp_monotonic `{
c :cpo
D } :
monotonic fixp .
red ;
auto .
Save .
Add Parametric Morphism `{
c :cpo
D } : (
fixp (
c :=c))
with signature Oeq &
Oeq as fixp_eq_compat .
intros ;
apply Ole_antisym ;
auto .
Save .
Hint Resolve @
fixp_eq_compat .
Definition Fixp D `{
c :cpo
D } : (
D -m>
D ) -m>
D :=
mon fixp .
Lemma Fixp_simpl :
forall `{
c :cpo
D } (
f :D-m>D),
Fixp D f =
fixp f .
trivial .
Save .
Instance iter_monotonic `{
c :cpo
D } :
monotonic iter .
red ;
intros D o c f g H .
intro n ;
induction n ;
simpl ;
intros ;
auto .
transitivity (
g (
iter_ f n ));
auto .
Save .
Definition Iter D `{
c :cpo
D } : (
D -m>
D ) -m> (
nat -m>
D ) :=
mon iter .
Lemma IterS_simpl :
forall `{
c :cpo
D }
f n ,
Iter D f (
S n ) =
f (
Iter D f n ).
trivial .
Save .
Lemma iterO_simpl :
forall `{
c :cpo
D } (
f :
D -m>
D ),
iter f O = (0:D).
trivial .
Save .
Lemma iterS_simpl :
forall `{
c :cpo
D }
f n ,
iter f (
S n ) =
f (
iter f n ).
trivial .
Save .
Lemma iter_continuous :
forall `{
c :cpo
D } (
h :
nat -m> (
D -m>
D )),
(
forall n ,
continuous (
h n )) ->
iter (
lub h ) <=
lub (
mon iter @
h ).
red ;
intros ;
intro k .
induction k .
rewrite iterO_simpl ;
auto .
rewrite iterS_simpl .
transitivity ((
lub h ) ((
lub (
mon iter @
h ))
k ));
auto .
transitivity ((
lub h ) (
lub (
mshift (
mon iter @
h )
k )));
auto .
transitivity (
lub ((
lub h ) @ (
mshift (
mon iter @
h )
k ))).
apply lub_continuous ;
trivial .
pose (
hh :=fun
n m :
nat =>
h n (
iter (
h m )
k )).
assert (
monotonic2 hh ).
unfold hh ;
red ;
intros n1 n2 m1 m2 H1 H2 .
apply (
fmonotonic2 h );
trivial .
apply iter_monotonic ;
auto .
transitivity (
lub (
lub (
mon2 hh ))).
apply lub_le_compat ;
intro n ;
unfold hh ;
simpl ;
auto .
transitivity (
lub (
diag (
mon2 hh )));
auto .
transitivity (
lub (
mshift (
mon iter @
h ) (
S k )));
auto .
apply lub_le_compat ;
intro n ;
simpl ;
auto .
Save .
Hint Resolve @
iter_continuous .
Lemma iter_continuous_eq :
forall `{
c :cpo
D } (
h :
nat -m> (
D -m>
D )),
(
forall n ,
continuous (
h n )) ->
iter (
lub h ) ==
lub (
mon iter @
h ).
intros ;
apply Ole_antisym ;
auto .
exact (
lub_comp_le (
mon iter )
h ).
Save .
Lemma fixp_continuous :
forall `{
c :cpo
D } (
h :
nat -m> (
D -m>
D )),
(
forall n ,
continuous (
h n )) ->
fixp (
lub h ) <=
lub (
mon fixp @
h ).
intros ;
unfold fixp .
transitivity (
lub (
lub (
mon iter @
h )));
auto .
apply lub_le_compat ;
auto .
exact (
iter_continuous h H ).
transitivity (
lub (
lub (
mshift (
mon iter @
h ))));
auto .
apply lub_le_compat ;
intro n ;
simpl ;
auto .
Save .
Hint Resolve @
fixp_continuous .
Lemma fixp_continuous_eq :
forall `{
c :cpo
D } (
h :
nat -m> (
D -m>
D )),
(
forall n ,
continuous (
h n )) ->
fixp (
lub h ) ==
lub (
mon fixp @
h ).
intros ;
apply Ole_antisym ;
auto .
exact (
lub_comp_le (
mon fixp )
h ).
Save .
Definition admissible `{
c :cpo
D }(
P :D->Type) :=
forall f :
nat -m>
D , (
forall n ,
P (
f n )) ->
P (
lub f ).
Lemma fixp_ind :
forall `{
c :cpo
D }(
F :D -m>
D )(
P :D->Type),
admissible P ->
P 0 -> (
forall x ,
P x ->
P (
F x )) ->
P (
fixp F ).
intros ;
unfold fixp .
apply X ;
intros .
induction n ;
simpl ;
auto .
Defined .
Ltac continuity cont Cont Hcont :=
match goal with
| |- (
Ole ?x1 (
lub (
mon (
fun (
n :nat) =>
cont (@?g
n ))))) =>
let f :=
fresh "f"
in (
pose (
f :=g);
assert (
monotonic f ) ;
[
auto | (
transitivity (
lub (
Cont@ (
mon f ))); [
rewrite <-
Hcont |
auto ])]
)
end .
Ltac gen_monotonic :=
match goal with |-
context [(@
mon _ _ _ _ ?f ?mf)] =>
generalize (
mf :monotonic
f )
end .
Ltac gen_monotonic1 f :=
match goal with |-
context [(@
mon _ _ _ _ f ?mf)] =>
generalize (
mf :monotonic
f )
end .
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(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_instance__.html 0000644 0001750 0001750 00000022476 11301735615 016066 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_record_S.html 0000644 0001750 0001750 00000023017 11301735615 015514 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
S (record)
stable [in
Ccpo ]
stable2 [in
Ccpo ]
html/index_instance_H.html 0000644 0001750 0001750 00000022476 11301735615 016037 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_instance_C.html 0000644 0001750 0001750 00000023514 11301735615 016024 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
C (instance)
comp_monotonic2 [in
Ccpo ]
cont0 [in
Ccpo ]
cont2_continuous [in
Ccpo ]
cont_app_monotonic [in
Ccpo ]
cshift_continuous2 [in
Ccpo ]
html/index_lemma_L.html 0000644 0001750 0001750 00000025210 11301735615 015317 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
L (lemma)
le_mlub [in
Ccpo ]
le_Ole [in
Ccpo ]
lub_app2_eq [in
Ccpo ]
lub_app2_le [in
Ccpo ]
lub_comp_eq [in
Ccpo ]
lub_comp_le [in
Ccpo ]
lub_cont2_app2_eq [in
Ccpo ]
lub_cte [in
Ccpo ]
lub_eq_lift [in
Ccpo ]
lub_fcpo_mon [in
Ccpo ]
lub_ishift [in
Ccpo ]
lub_le_lift [in
Ccpo ]
lub_lift_left [in
Ccpo ]
lub_lift_right [in
Ccpo ]
lub_mon_fcpo [in
Ccpo ]
html/index_record_M.html 0000644 0001750 0001750 00000023033 11301735615 015504 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
M (record)
monotonic [in
Ccpo ]
monotonic2 [in
Ccpo ]
html/index_instance_Y.html 0000644 0001750 0001750 00000023257 11301735615 016056 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
Y (instance)
Ycart.FYcart_lt_mon [in
Ycart ]
Ycart.FYcart_mon [in
Ycart ]
Ycart.Ycart_one_mon [in
Ycart ]
html/index_module_V.html 0000644 0001750 0001750 00000022476 11301735615 015536 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_global_T.html 0000644 0001750 0001750 00000022714 11301735615 015502 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
T
transitive [projection, in
Ccpo ]
html/index_module_G.html 0000644 0001750 0001750 00000022476 11301735615 015517 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_inductive_L.html 0000644 0001750 0001750 00000022476 11301735615 016231 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_definition_G.html 0000644 0001750 0001750 00000022710 11301735615 016351 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
G (definition)
gunion [in
Sets ]
html/index_record_B.html 0000644 0001750 0001750 00000022476 11301735615 015503 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_global_R.html 0000644 0001750 0001750 00000117516 11301735615 015505 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
R
reflexive [projection, in
Ccpo ]
rem [definition, in
Sets ]
rem_add_diff_eqset [lemma, in
Sets ]
rem_add_eqset_notin [lemma, in
Sets ]
rem_add_eq_eqset [lemma, in
Sets ]
rem_eqset_notin [lemma, in
Sets ]
rem_incl [lemma, in
Sets ]
rem_not_in [lemma, in
Sets ]
RP [module, in
Bernoulli ]
RP [module, in
Cover ]
RP [module, in
Choice ]
RP [module, in
IterFlip ]
Rules [module, in
Prog ]
Rules.Acc_iter_distr [lemma, in
Prog ]
Rules.admissible [definition, in
Prog ]
Rules.admissible_le_fix [lemma, in
Prog ]
Rules.admissible_true [lemma, in
Prog ]
Rules.apply_rule [lemma, in
Prog ]
Rules.arg_swap [definition, in
Prog ]
Rules.Arg_swap [definition, in
Prog ]
Rules.Arg_swap_simpl [lemma, in
Prog ]
Rules.combiok [lemma, in
Prog ]
Rules.discrete_commute [lemma, in
Prog ]
Rules.Fixrule [section, in
Prog ]
Rules.fixrule [lemma, in
Prog ]
Rules.Fixrule.A [variable, in
Prog ]
Rules.Fixrule.B [variable, in
Prog ]
Rules.Fixrule.F [variable, in
Prog ]
Rules.Fixrule.LoopRule [section, in
Prog ]
Rules.Fixrule.LoopRule.a [variable, in
Prog ]
Rules.Fixrule.LoopRule.q [variable, in
Prog ]
Rules.Fixrule.LoopRule.step [variable, in
Prog ]
Rules.Fixrule.LoopRule.stop [variable, in
Prog ]
Rules.Fixrule.Ruleseq [section, in
Prog ]
Rules.Fixrule.Ruleseq.q [variable, in
Prog ]
Rules.Fixrule.TransformFix [section, in
Prog ]
Rules.Fixrule.TransformFix.Fix_muF [section, in
Prog ]
Rules.Fixrule.TransformFix.Fix_muF.F_muF_results [section, in
Prog ]
Rules.Fixrule.TransformFix.Fix_muF.F_muF_results.F_muF_le [variable, in
Prog ]
Rules.Fixrule.TransformFix.Fix_muF.F_muF_results.muF_F_le [variable, in
Prog ]
Rules.Fixrule.TransformFix.Fix_muF.muF [variable, in
Prog ]
Rules.Fixrule.TransformFix.Fix_muF.q [variable, in
Prog ]
Rules.Fixrule.TransformFix.Fix_muF_Term [section, in
Prog ]
Rules.Fixrule.TransformFix.Fix_muF_Term.F_muF_le [variable, in
Prog ]
Rules.Fixrule.TransformFix.Fix_muF_Term.F_muF_le_inv [variable, in
Prog ]
Rules.Fixrule.TransformFix.Fix_muF_Term.muFq [variable, in
Prog ]
Rules.Fixrule.TransformFix.Fix_muF_Term.muFqinv [variable, in
Prog ]
Rules.Fixrule.TransformFix.Fix_muF_Term.q [variable, in
Prog ]
Rules.Fixrule.TransformFix.Fix_Term [section, in
Prog ]
Rules.Fixrule.TransformFix.Fix_Term.F_muF_eq_one [variable, in
Prog ]
Rules.Fixrule.TransformFix.Fix_Term.muFone [variable, in
Prog ]
Rules.Fixrule.TransformFix.Fix_Term.muF_cont [variable, in
Prog ]
Rules.fixrule_Ulub [lemma, in
Prog ]
Rules.fixrule_up_lub [lemma, in
Prog ]
Rules.fixrule_up_Ulub [lemma, in
Prog ]
Rules.Flip_cfalse [lemma, in
Prog ]
Rules.Flip_ctrue [lemma, in
Prog ]
Rules.Flip_eq [lemma, in
Prog ]
Rules.fmult_fplusok [lemma, in
Prog ]
Rules.fst_distr [definition, in
Prog ]
Rules.Iapply_rule [lemma, in
Prog ]
Rules.IFixrule [section, in
Prog ]
Rules.Ifixrule [lemma, in
Prog ]
Rules.IFixrule.A [variable, in
Prog ]
Rules.IFixrule.B [variable, in
Prog ]
Rules.IFixrule.F [variable, in
Prog ]
Rules.IFixrule.IRuleseq [section, in
Prog ]
Rules.IFixrule.IRuleseq.I [variable, in
Prog ]
Rules.IFixrule.IRuleseq.Q [variable, in
Prog ]
Rules.IFixrule.ITransformFix [section, in
Prog ]
Rules.IFixrule.ITransformFix.IFix_muF [section, in
Prog ]
Rules.IFixrule.ITransformFix.IFix_muF.IF_muF_results [section, in
Prog ]
Rules.IFixrule.ITransformFix.IFix_muF.IF_muF_results.Iincl_F_ImuF [variable, in
Prog ]
Rules.IFixrule.ITransformFix.IFix_muF.ImuF [variable, in
Prog ]
Rules.IFixrule.ITransformFix.IFix_muF.Q [variable, in
Prog ]
Rules.IFlip_eq [lemma, in
Prog ]
Rules.ifok [lemma, in
Prog ]
Rules.ifrule [lemma, in
Prog ]
Rules.Iifrule [lemma, in
Prog ]
Rules.Iincl_fix_ifix [lemma, in
Prog ]
Rules.Iin_mu_Imu [lemma, in
Prog ]
Rules.Iin_mu_Iok [lemma, in
Prog ]
Rules.Ilambda_rule [lemma, in
Prog ]
Rules.Ilet_eq [lemma, in
Prog ]
Rules.Imu [definition, in
Prog ]
Rules.ImuF_stable [lemma, in
Prog ]
Rules.Imu_inf [lemma, in
Prog ]
Rules.Imu_Mif_eq [lemma, in
Prog ]
Rules.Imu_monotonic [lemma, in
Prog ]
Rules.Imu_singl [lemma, in
Prog ]
Rules.Imu_stable_eq [lemma, in
Prog ]
Rules.Imu_sup [lemma, in
Prog ]
Rules.im_distr [definition, in
Prog ]
Rules.im_distr_comp [lemma, in
Prog ]
Rules.im_distr_id [lemma, in
Prog ]
Rules.im_distr_simpl [lemma, in
Prog ]
Rules.Iok [definition, in
Prog ]
Rules.Iokfun [definition, in
Prog ]
Rules.Iokfun_le_compat [lemma, in
Prog ]
Rules.Iok_le_compat [lemma, in
Prog ]
Rules.Iunit_eq [lemma, in
Prog ]
Rules.lambda_rule [lemma, in
Prog ]
Rules.Loop [definition, in
Prog ]
Rules.loop [definition, in
Prog ]
Rules.Loop_eq [lemma, in
Prog ]
Rules.low_Imu [lemma, in
Prog ]
Rules.MChoice [definition, in
Prog ]
Rules.Mchoice [definition, in
Prog ]
Rules.Mchoice_continuous [lemma, in
Prog ]
Rules.Mchoice_continuous_left [lemma, in
Prog ]
Rules.Mchoice_continuous_right [lemma, in
Prog ]
Rules.Mchoice_le_compat [lemma, in
Prog ]
Rules.Mchoice_mon2 [instance, in
Prog ]
Rules.Mchoice_simpl [lemma, in
Prog ]
Rules.MChoice_simpl [lemma, in
Prog ]
Rules.Mchoice_sym [lemma, in
Prog ]
Rules.Mchoice_sym_le [lemma, in
Prog ]
Rules.Mfixvar [lemma, in
Prog ]
Rules.Mfixvar_up [lemma, in
Prog ]
Rules.MIf [definition, in
Prog ]
Rules.Mif [definition, in
Prog ]
Rules.Mif_eq [lemma, in
Prog ]
Rules.Mif_le_compat [lemma, in
Prog ]
Rules.Mif_mon2 [instance, in
Prog ]
Rules.mufix_mu [lemma, in
Prog ]
Rules.mufix_mu_le [lemma, in
Prog ]
Rules.muF_eq [lemma, in
Prog ]
Rules.muF_le [lemma, in
Prog ]
Rules.muF_le_term [lemma, in
Prog ]
Rules.muF_le_term_minus [lemma, in
Prog ]
Rules.muF_pterm [lemma, in
Prog ]
Rules.muF_stable [lemma, in
Prog ]
Rules.mu_mufix_le [lemma, in
Prog ]
Rules.mu_muF_commute_le [definition, in
Prog ]
Rules.ok [definition, in
Prog ]
Rules.okfun [definition, in
Prog ]
Rules.okfun_le_compat [lemma, in
Prog ]
Rules.okup [definition, in
Prog ]
Rules.okup_apply_rule [lemma, in
Prog ]
Rules.okup_eq_compat [lemma, in
Prog ]
Rules.okup_fixrule_glb [lemma, in
Prog ]
Rules.okup_fixrule_inv [lemma, in
Prog ]
Rules.okup_Flip [lemma, in
Prog ]
Rules.okup_ifrule [lemma, in
Prog ]
Rules.okup_lambda_rule [lemma, in
Prog ]
Rules.okup_le_compat [lemma, in
Prog ]
Rules.okup_mult [lemma, in
Prog ]
Rules.ok_eq_compat [lemma, in
Prog ]
Rules.ok_Flip [lemma, in
Prog ]
Rules.ok_inv [lemma, in
Prog ]
Rules.ok_le_compat [lemma, in
Prog ]
Rules.ok_mult [lemma, in
Prog ]
Rules.pchoice [definition, in
Prog ]
Rules.pchoice_simpl [lemma, in
Prog ]
Rules.Prod_distr [definition, in
Prog ]
Rules.prod_distr [definition, in
Prog ]
Rules.prod_distr_com [definition, in
Prog ]
Rules.prod_distr_com_cte [lemma, in
Prog ]
Rules.prod_distr_com_eq_compat [lemma, in
Prog ]
Rules.prod_distr_com_inv [lemma, in
Prog ]
Rules.prod_distr_com_lub [lemma, in
Prog ]
Rules.prod_distr_com_mult [lemma, in
Prog ]
Rules.prod_distr_com_one [lemma, in
Prog ]
Rules.prod_distr_com_plus [lemma, in
Prog ]
Rules.prod_distr_com_rect [lemma, in
Prog ]
Rules.prod_distr_com_refl [lemma, in
Prog ]
Rules.prod_distr_fst [lemma, in
Prog ]
Rules.prod_distr_fst_eq [lemma, in
Prog ]
Rules.prod_distr_mon2 [instance, in
Prog ]
Rules.prod_distr_rect [lemma, in
Prog ]
Rules.Prod_distr_simpl [lemma, in
Prog ]
Rules.prod_distr_snd [lemma, in
Prog ]
Rules.prod_distr_snd_eq [lemma, in
Prog ]
Rules.pterm [definition, in
Prog ]
Rules.qinv [definition, in
Prog ]
Rules.range [definition, in
Prog ]
Rules.range_eq [lemma, in
Prog ]
Rules.range_le [lemma, in
Prog ]
Rules.snd_distr [definition, in
Prog ]
Rules.swap [definition, in
Prog ]
Rules.upfun [definition, in
Prog ]
Rules.upfun_le_compat [lemma, in
Prog ]
Rules.up_Imu [lemma, in
Prog ]
Rules.up_loop [definition, in
Prog ]
Rules.Wellfounded [section, in
Prog ]
Rules.Wellfounded.A [variable, in
Prog ]
Rules.Wellfounded.B [variable, in
Prog ]
Rules.Wellfounded.F [variable, in
Prog ]
Rules.Wellfounded.Fext [variable, in
Prog ]
Rules.Wellfounded.P [variable, in
Prog ]
Rules.Wellfounded.Pext [variable, in
Prog ]
Rules.Wellfounded.R [variable, in
Prog ]
Rules.Wellfounded.Rwf [variable, in
Prog ]
Rules.WfFix [definition, in
Prog ]
Rules.WfFix_eq [lemma, in
Prog ]
Rules.WfFix_ind [lemma, in
Prog ]
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_inductive_S.html 0000644 0001750 0001750 00000023025 11301735615 016227 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
S (inductive)
stable [in
Ccpo ]
stable2 [in
Ccpo ]
html/index_lemma_T.html 0000644 0001750 0001750 00000022476 11301735615 015342 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_module_L.html 0000644 0001750 0001750 00000022476 11301735615 015524 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_projection_G.html 0000644 0001750 0001750 00000022476 11301735615 016406 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_constructor_W.html 0000644 0001750 0001750 00000022476 11301735615 016637 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_variable_R.html 0000644 0001750 0001750 00000055146 11301735615 016032 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
R (variable)
Rules.Fixrule.A [in
Prog ]
Rules.Fixrule.B [in
Prog ]
Rules.Fixrule.F [in
Prog ]
Rules.Fixrule.LoopRule.a [in
Prog ]
Rules.Fixrule.LoopRule.q [in
Prog ]
Rules.Fixrule.LoopRule.step [in
Prog ]
Rules.Fixrule.LoopRule.stop [in
Prog ]
Rules.Fixrule.Ruleseq.q [in
Prog ]
Rules.Fixrule.TransformFix.Fix_muF.F_muF_results.F_muF_le [in
Prog ]
Rules.Fixrule.TransformFix.Fix_muF.F_muF_results.muF_F_le [in
Prog ]
Rules.Fixrule.TransformFix.Fix_muF.muF [in
Prog ]
Rules.Fixrule.TransformFix.Fix_muF.q [in
Prog ]
Rules.Fixrule.TransformFix.Fix_muF_Term.F_muF_le [in
Prog ]
Rules.Fixrule.TransformFix.Fix_muF_Term.F_muF_le_inv [in
Prog ]
Rules.Fixrule.TransformFix.Fix_muF_Term.muFq [in
Prog ]
Rules.Fixrule.TransformFix.Fix_muF_Term.muFqinv [in
Prog ]
Rules.Fixrule.TransformFix.Fix_muF_Term.q [in
Prog ]
Rules.Fixrule.TransformFix.Fix_Term.F_muF_eq_one [in
Prog ]
Rules.Fixrule.TransformFix.Fix_Term.muFone [in
Prog ]
Rules.Fixrule.TransformFix.Fix_Term.muF_cont [in
Prog ]
Rules.IFixrule.A [in
Prog ]
Rules.IFixrule.B [in
Prog ]
Rules.IFixrule.F [in
Prog ]
Rules.IFixrule.IRuleseq.I [in
Prog ]
Rules.IFixrule.IRuleseq.Q [in
Prog ]
Rules.IFixrule.ITransformFix.IFix_muF.IF_muF_results.Iincl_F_ImuF [in
Prog ]
Rules.IFixrule.ITransformFix.IFix_muF.ImuF [in
Prog ]
Rules.IFixrule.ITransformFix.IFix_muF.Q [in
Prog ]
Rules.Wellfounded.A [in
Prog ]
Rules.Wellfounded.B [in
Prog ]
Rules.Wellfounded.F [in
Prog ]
Rules.Wellfounded.Fext [in
Prog ]
Rules.Wellfounded.P [in
Prog ]
Rules.Wellfounded.Pext [in
Prog ]
Rules.Wellfounded.R [in
Prog ]
Rules.Wellfounded.Rwf [in
Prog ]
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_lemma_Z.html 0000644 0001750 0001750 00000022476 11301735615 015350 0 ustar cpaulin cpaulin
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Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_lemma_V.html 0000644 0001750 0001750 00000022476 11301735615 015344 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_variable_D.html 0000644 0001750 0001750 00000022476 11301735615 016014 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_library_O.html 0000644 0001750 0001750 00000022476 11301735615 015706 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/Uprop.html 0000644 0001750 0001750 00003412114 11301735614 013674 0 ustar cpaulin cpaulin
Uprop
Library Uprop
Uprop.v : Properties of operators on [0,1]
Set Implicit Arguments .
Require Export Utheory .
Require Export Arith .
Require Export Omega .
Open Local Scope U_scope .
Direct consequences of axioms
Lemma Uplus_le_compat_right :
forall x y z :U,
y <=
z ->
x +
y <=
x +
z .
intros ;
apply Uplus_mon_right ;
trivial .
Save .
Hint Resolve Uplus_le_compat_right .
Instance Uplus_mon2 :
monotonic2 Uplus .
intros ;
apply monotonic2_sym ;
auto .
Save .
Hint Resolve Uplus_mon2 .
Lemma Uplus_le_compat_left :
forall x y z :U,
x <=
y ->
x +
z <=
y +
z .
intros ;
apply Uplus_mon2 ;
trivial .
Save .
Hint Resolve Uplus_le_compat_left .
Lemma Uplus_le_compat :
forall x y z t ,
x <=
y ->
z <=
t ->
x +
z <=
y +
t .
intros ;
apply Uplus_mon2 ;
auto .
Save .
Hint Immediate Uplus_le_compat .
Lemma Uplus_eq_compat_left :
forall x y z :U,
x ==
y ->
x +
z ==
y +
z .
intros ;
apply Ole_antisym ;
auto .
Save .
Hint Resolve Uplus_eq_compat_left .
Lemma Uplus_eq_compat_right :
forall x y z :U,
x ==
y -> (
z +
x ) == (
z +
y ).
intros ;
apply Ole_antisym ;
auto .
Save .
Hint Resolve Uplus_eq_compat_left Uplus_eq_compat_right .
Add Morphism Uplus with signature Oeq &
Oeq &
Oeq as Uplus_eq_compat .
intros x1 x2 eq1 x3 x4 eq2 ;
apply Oeq_trans with (
x1 +x4);
auto .
Qed .
Hint Immediate Uplus_eq_compat .
Add Morphism Uinv with signature Oeq &
Oeq as Uinv_eq_compat .
intros ;
apply Ole_antisym ;
auto .
Qed .
Hint Resolve Uinv_eq_compat .
Lemma Uplus_zero_right :
forall x :U,
x + 0 ==
x .
intros ;
rewrite (
Uplus_sym x 0);
auto .
Save .
Hint Resolve Uplus_zero_right .
Lemma Uinv_opp_left :
forall x , [1-]
x +
x == 1.
unfold U1 ;
intros ;
transitivity ([1-](
x + 0) +
x );
auto .
Save .
Hint Resolve Uinv_opp_left .
Lemma Uinv_opp_right :
forall x ,
x + [1-]
x == 1.
intros ;
transitivity ([1-]
x +
x );
auto .
Save .
Hint Resolve Uinv_opp_right .
Lemma Uinv_inv :
forall x :
U , [1-] [1-]
x ==
x .
intros ;
transitivity ([1-] (
x + [1-]
x ) +
x );
auto .
apply Oeq_sym ;
auto .
rewrite (
Uinv_opp_right x );
rewrite Uinv_one ;
auto .
Save .
Hint Resolve Uinv_inv .
Lemma Unit :
forall x :U,
x <= 1.
intro ;
transitivity ([1-][1-]
x );
unfold U1 ;
auto .
Save .
Hint Resolve Unit .
Lemma Uinv_zero : [1-] 0 = 1.
trivial .
Save .
Lemma Ueq_class :
forall x y :U,
class (
x ==y).
red ;
intros .
apply Ole_antisym ;
apply Ule_class ;
intuition .
Save .
Lemma Ueq_double_neg :
forall x y :
U , ~ ~ (
x ==
y ) ->
x ==
y .
exact Ueq_class .
Save .
Hint Resolve Ueq_class .
Hint Immediate Ueq_double_neg .
Lemma Ule_orc :
forall x y :
U ,
orc (
x <=y) (~
x <=y).
auto .
Save .
Implicit Arguments Ule_orc [].
Lemma Ueq_orc :
forall x y :U,
orc (
x ==y) (~
x ==y).
auto .
Save .
Implicit Arguments Ueq_orc [].
Lemma Upos :
forall x :U, 0 <=
x .
auto .
Save .
Lemma Ule_0_1 : 0 <= 1.
auto .
Save .
Hint Resolve Upos Ule_0_1 .
Properties of == derived from properties of
Definition UPlus :
U -m>
U -m>
U :=
mon2 Uplus .
Definition UPlus_simpl :
forall x y ,
UPlus x y =
x +y.
trivial .
Save .
Instance Uplus_continuous2 :
continuous2 (
mon2 Uplus ).
apply continuous2_sym .
intros n m ;
repeat (
rewrite mon2_simpl );
auto .
intro k ;
apply continuous_eq_compat with (2:=Uplus_right_continuous
k ).
intro m ;
auto .
Save .
Hint Resolve Uplus_continuous2 .
Lemma Umult_le_compat_right :
forall x y z :U,
y <=
z ->
x *
y <=
x *
z .
intros ;
apply Umult_mon_right ;
trivial .
Save .
Hint Resolve Umult_le_compat_right .
Instance Umult_mon2 :
monotonic2 Umult .
apply monotonic2_sym ;
auto .
Save .
Lemma Umult_le_compat_left :
forall x y z :U,
x <=
y ->
x *
z <=
y *
z .
intros ;
apply Umult_mon2 ;
trivial .
Save .
Hint Resolve Umult_le_compat_left .
Lemma Umult_le_compat :
forall x y z t ,
x <=
y ->
z <=
t ->
x *
z <=
y *
t .
intros ;
apply Umult_mon2 ;
trivial .
Save .
Hint Immediate Umult_le_compat .
Definition UMult :
U -m>
U -m>
U :=
mon2 Umult .
Lemma Umult_eq_compat_left :
forall x y z :U,
x ==
y -> (
x *
z ) == (
y *
z ).
intros ;
apply Ole_antisym ;
auto .
Save .
Hint Resolve Umult_eq_compat_left .
Lemma Umult_eq_compat_right :
forall x y z :U,
x ==
y -> (
z *
x ) == (
z *
y ).
intros ;
transitivity (
x *
z );
auto .
transitivity (
y *
z );
auto .
Save .
Hint Resolve Umult_eq_compat_left Umult_eq_compat_right .
Definition UMult_simpl :
forall x y ,
UMult x y =
x *y.
trivial .
Save .
Instance Umult_continuous2 :
continuous2 (
mon2 Umult ).
apply continuous2_sym .
intros ;
repeat (
rewrite mon2_simpl );
auto .
intro k ;
apply continuous_eq_compat with (2:=Umult_right_continuous
k ).
intro l ;
auto .
Save .
Hint Resolve Umult_continuous2 .
Add Morphism Umult with signature Oeq &
Oeq &
Oeq
as Umult_eq_compat .
intros x1 x2 eq1 x3 x4 eq2 ;
transitivity (
x1 *
x4 );
auto .
Qed .
Hint Immediate Umult_eq_compat .
Instance Uinv_mon :
monotonic (
o1 :=Iord
U )
Uinv .
red ;
simpl ;
intros ;
auto .
Save .
Definition UInv :
U --m>
U :=
mon (
o1 :=Iord
U )
Uinv .
Definition UInv_simpl :
forall x ,
UInv x = [1-]
x .
trivial .
Save .
Lemma Ule_eq_compat :
forall x1 x2 :
U ,
x1 ==
x2 ->
forall x3 x4 :
U ,
x3 ==
x4 ->
x1 <=
x3 ->
x2 <=
x4 .
intros x1 x2 eq1 x3 x4 eq2 ;
apply (
Ole_eq_compat _ _ eq1 _ _ eq2 );
auto .
Save .
Definition and properties of
Definition Ult (
r1 r2 :U) :
Prop := ~ (
r2 <=
r1 ).
Infix "<" :=
Ult :
U_scope .
Hint Unfold Ult .
Add Morphism Ult with signature Oeq &
Oeq &
iff as Ult_eq_compat_iff .
unfold Ult ,
not ;
intros x1 x2 eq1 x3 x4 eq2 .
generalize (
Ole_eq_compat_iff ordU x3 x4 eq2 x1 x2 eq1 );
intuition .
Save .
Lemma Ult_eq_compat :
forall x1 x2 :
U ,
x1 ==
x2 ->
forall x3 x4 :
U ,
x3 ==
x4 ->
x1 <
x3 ->
x2 <
x4 .
intros x1 x2 eq1 x3 x4 eq2 ;
elim (
Ult_eq_compat_iff _ _ eq1 _ _ eq2 );
auto .
Save .
Lemma Ult_class :
forall x y ,
class (
x <y).
unfold Ult ;
auto .
Save .
Hint Resolve Ult_class .
Lemma Ule_zero_eq : forall x :U, x <= 0 -> x == 0.intros ; apply Ole_antisym ; auto .Save .Lemma Uge_one_eq : forall x :U, 1 <= x -> x == 1.intros ; apply Ole_antisym ; auto .Save .Hint Immediate Ule_zero_eq Uge_one_eq .
Lemma Ult_neq :
forall x y :U,
x <
y -> ~x ==
y .
unfold Ult ;
red ;
auto .
Save .
Lemma Ult_neq_rev :
forall x y :U,
x <
y -> ~y ==
x .
unfold Ult ;
red ;
auto .
Save .
Lemma Ult_trans :
forall x y z ,
x <y ->
y <z ->
x <z.
repeat red ;
intros .
apply (
Ule_total y z );
intros ;
auto .
apply H ;
apply Ole_trans with z ;
auto .
Save .
Lemma Ult_le :
forall x y :U,
x <
y ->
x <=
y .
unfold Ult ;
intros ;
apply Ule_class ;
repeat red ;
intros .
assert (
x <
x ).
apply Ult_trans with y ;
auto .
apply H1 ;
auto .
Save .
Lemma Ule_diff_lt :
forall x y :
U ,
x <=
y -> ~x==y ->
x <
y .
red ;
intuition .
Save .
Hint Immediate Ult_neq Ult_neq_rev Ult_le .
Hint Resolve Ule_diff_lt .
Lemma Ult_neq_zero :
forall x , ~0 ==
x -> 0 <
x .
auto .
Save .
Hint Resolve Ule_total Ult_neq_zero .
Lemma Udistr_plus_left :
forall x y z ,
y <= [1-]
z ->
x * (
y +
z ) ==
x *
y +
x *
z .
intros .
rewrite (
Umult_sym x (
y +z)).
rewrite (
Umult_sym x y ).
rewrite (
Umult_sym x z );
auto .
Save .
Lemma Udistr_inv_left :
forall x y , [1-](
x *
y ) == (
x * ([1-]
y )) + [1-]
x .
intros .
setoid_rewrite (
Umult_sym x y ).
setoid_rewrite (
Udistr_inv_right y x );
auto .
Save .
Hint Resolve Uinv_eq_compat Udistr_plus_left Udistr_inv_left .
Lemma Uplus_perm2 :
forall x y z :U,
x + (
y +
z ) ==
y + (
x +
z ).
intros ;
setoid_rewrite (
Uplus_assoc x y z ).
setoid_rewrite (
Uplus_sym x y );
auto .
Save .
Lemma Umult_perm2 :
forall x y z :U,
x * (
y *
z ) ==
y * (
x *
z ).
intros ;
setoid_rewrite (
Umult_assoc x y z ).
setoid_rewrite (
Umult_sym x y );
auto .
Save .
Lemma Uplus_perm3 :
forall x y z :
U , (
x + (
y +
z )) ==
z + (
x +
y ).
intros ;
setoid_rewrite (
Uplus_assoc x y z );
auto .
Save .
Lemma Umult_perm3 :
forall x y z :
U , (
x * (
y *
z )) ==
z * (
x *
y ).
intros ;
setoid_rewrite (
Umult_assoc x y z );
auto .
Save .
Hint Resolve Uplus_perm2 Umult_perm2 Uplus_perm3 Umult_perm3 .
Lemma Uinv_simpl :
forall x y :
U , [1-]
x == [1-]
y ->
x ==
y .
intros ;
setoid_rewrite <- (
Uinv_inv x );
setoid_rewrite <- (
Uinv_inv y );
auto .
Save .
Hint Immediate Uinv_simpl .
Lemma Umult_decomp :
forall x y ,
x ==
x *
y +
x * [1-]
y .
intros ;
apply Oeq_trans with (
x * (
y + [1-]
y ));
auto .
apply Oeq_trans with (
x * 1);
auto .
rewrite Umult_sym ;
auto .
Save .
Hint Resolve Umult_decomp .
More properties on + and * and Uinv
Lemma Umult_one_right :
forall x :U,
x * 1 ==
x .
intros ;
setoid_rewrite (
Umult_sym x 1);
auto .
Save .
Hint Resolve Umult_one_right .
Lemma Umult_one_right_eq :
forall x y :U,
y == 1 ->
x *
y ==
x .
intros ;
rewrite H ;
auto .
Save .
Hint Resolve Umult_one_right_eq .
Lemma Umult_one_left_eq :
forall x y :U,
x == 1 ->
x *
y ==
y .
intros ;
rewrite H ;
auto .
Save .
Hint Resolve Umult_one_left_eq .
Lemma Udistr_plus_left_le :
forall x y z :
U ,
x * (
y +
z ) <=
x *
y +
x *
z .
intros ;
apply (
Ule_total y ([1-]
z ));
intros ;
auto .
apply Ole_trans with (
x * ([1-]
z +z)).
rewrite Uinv_opp_left ;
auto .
rewrite Udistr_plus_left ;
auto .
Save .
Lemma Uplus_eq_simpl_right :
forall x y z :U,
z <= [1-]
x ->
z <= [1-]
y -> (
x +
z ) == (
y +
z ) ->
x ==
y .
intros ;
apply Ole_antisym .
apply Uplus_le_simpl_right with z ;
auto .
apply Uplus_le_simpl_right with z ;
auto .
Save .
Lemma Ule_plus_right :
forall x y ,
x <=
x +
y .
intros ;
apply Ule_eq_compat with (
x + 0) (
x +
y );
auto .
Save .
Lemma Ule_plus_left :
forall x y ,
y <=
x +
y .
intros ;
apply Ule_eq_compat with (0 +
y ) (
x +
y );
auto .
Save .
Hint Resolve Ule_plus_right Ule_plus_left .
Lemma Ule_mult_right :
forall x y ,
x *
y <=
x .
intros ;
apply Ule_eq_compat with (
x *
y ) (
x * 1);
auto .
Save .
Lemma Ule_mult_left :
forall x y ,
x *
y <=
y .
intros ;
apply Ule_eq_compat with (
x *
y ) (1 *
y );
auto .
Save .
Hint Resolve Ule_mult_right Ule_mult_left .
Lemma Uinv_le_perm_right :
forall x y :U,
x <= [1-]
y ->
y <= [1-]
x .
intros ;
apply Ole_trans with ([1-] ([1-]
y ));
auto .
Save .
Hint Immediate Uinv_le_perm_right .
Lemma Uinv_le_perm_left :
forall x y :U, [1-]
x <=
y -> [1-]
y <=
x .
intros ;
apply Ole_trans with ([1-] ([1-]
x ));
auto .
Save .
Hint Immediate Uinv_le_perm_left .
Lemma Uinv_le_simpl :
forall x y :U, [1-]
x <= [1-]
y ->
y <=
x .
intros ;
apply Ole_trans with ([1-] ([1-]
x ));
auto .
Save .
Hint Immediate Uinv_le_simpl .
Lemma Uinv_double_le_simpl_right :
forall x y ,
x <=y ->
x <= [1-][1-]
y .
intros ;
apply Uinv_le_perm_right ;
auto .
Save .
Hint Resolve Uinv_double_le_simpl_right .
Lemma Uinv_double_le_simpl_left :
forall x y ,
x <=y -> [1-][1-]
x <=
y .
intros ;
apply Uinv_le_perm_left ;
auto .
Save .
Hint Resolve Uinv_double_le_simpl_left .
Lemma Uinv_eq_perm_left :
forall x y :U,
x == [1-]
y -> [1-]
x ==
y .
intros ;
apply Oeq_trans with ([1-] ([1-]
y ));
auto .
Save .
Hint Immediate Uinv_eq_perm_left .
Lemma Uinv_eq_perm_right :
forall x y :U, [1-]
x ==
y ->
x == [1-]
y .
intros ;
apply Oeq_trans with ([1-] ([1-]
x ));
auto .
Save .
Hint Immediate Uinv_eq_perm_right .
Lemma Uinv_eq_simpl :
forall x y :U, [1-]
x == [1-]
y ->
x ==
y .
intros ;
apply Oeq_trans with ([1-] ([1-]
x ));
auto .
Save .
Hint Immediate Uinv_eq_simpl .
Lemma Uinv_double_eq_simpl_right :
forall x y ,
x ==y ->
x == [1-][1-]
y .
intros ;
apply Uinv_eq_perm_right ;
auto .
Save .
Hint Resolve Uinv_double_eq_simpl_right .
Lemma Uinv_double_eq_simpl_left :
forall x y ,
x ==y -> [1-][1-]
x ==
y .
intros ;
apply Uinv_eq_perm_left ;
auto .
Save .
Hint Resolve Uinv_double_eq_simpl_left .
Lemma Uinv_plus_right :
forall x y ,
y <= [1-]
x -> [1-] (
x +
y ) +
y == [1-]
x .
intros ;
setoid_rewrite (
Uplus_sym x y );
auto .
Save .
Hint Resolve Uinv_plus_right .
Lemma Uplus_eq_simpl_left :
forall x y z :U,
x <= [1-]
y ->
x <= [1-]
z -> (
x +
y ) == (
x +
z ) ->
y ==
z .
intros x y z H1 H2 ;
setoid_rewrite (
Uplus_sym x y );
setoid_rewrite (
Uplus_sym x z );
auto .
intros ;
apply Uplus_eq_simpl_right with x ;
auto .
Save .
Lemma Uplus_eq_zero_left :
forall x y :U,
x <= [1-]
y -> (
x +
y ) ==
y ->
x == 0.
intros ;
apply Uplus_eq_simpl_right with y ;
auto .
setoid_rewrite H0 ;
auto .
Save .
Lemma Uinv_le_trans :
forall x y z t ,
x <= [1-]
y ->
z <=x ->
t <=y ->
z <= [1-]
t .
intros ;
apply Ole_trans with x ;
auto .
apply Ole_trans with ([1-]
y );
auto .
Save .
Lemma Uinv_plus_left_le :
forall x y , [1-]
y <= [1-](
x +y) +x.
intros ;
apply (
Ule_total y ([1-]
x ));
auto ;
intros .
rewrite Uinv_plus_left ;
auto .
apply Ole_trans with x ;
auto .
Save .
Lemma Uinv_plus_right_le :
forall x y , [1-]
x <= [1-](
x +y) +y.
intros ;
apply (
Ule_total y ([1-]
x ));
auto ;
intros .
rewrite Uinv_plus_right ;
auto .
apply Ole_trans with y ;
auto .
Save .
Hint Resolve Uinv_plus_left_le Uinv_plus_right_le .
Lemma neq_sym :
forall x y :U, ~x==y -> ~y==x.
red ;
intros ;
apply H ;
auto .
Save .
Hint Immediate neq_sym .
Lemma Uinv_neq_compat :
forall x y , ~x ==
y -> ~ [1-]
x == [1-]
y .
red ;
intros ;
apply H ;
auto .
Save .
Lemma Uinv_neq_simpl :
forall x y , ~ [1-]
x == [1-]
y -> ~x ==
y .
red ;
intros ;
apply H ;
auto .
Save .
Hint Resolve Uinv_neq_compat .
Hint Immediate Uinv_neq_simpl .
Lemma Uinv_neq_left :
forall x y , ~x == [1-]
y -> ~ [1-]
x ==
y .
red ;
intros ;
apply H ;
auto .
Save .
Lemma Uinv_neq_right :
forall x y , ~ [1-]
x ==
y -> ~x == [1-]
y .
red ;
intros ;
apply H ;
auto .
Save .
Lemma Ult_antirefl :
forall x :U, ~x <
x .
unfold Ult ;
intuition .
Save .
Lemma Ult_0_1 : (0 < 1).
red ;
intuition .
Save .
Lemma Ule_lt_trans :
forall x y z :U,
x <=
y ->
y <
z ->
x <
z .
unfold Ult ;
intuition .
apply H0 ;
apply Ole_trans with x ;
trivial .
Save .
Lemma Ult_le_trans :
forall x y z :U,
x <
y ->
y <=
z ->
x <
z .
unfold Ult ;
intuition .
apply H ;
apply Ole_trans with z ;
trivial .
Save .
Hint Resolve Ult_0_1 Ult_antirefl .
Lemma Ule_neq_zero :
forall (
x y :U), ~0 ==
x ->
x <=
y -> ~ 0 ==
y .
red ;
intros .
apply H .
apply Ole_antisym ;
auto ;
rewrite H1 ;
trivial .
Save .
Lemma Uplus_neq_zero_left :
forall x y , ~0 ==
x -> ~0 ==
x +y.
intros ;
apply Ult_neq .
apply Ult_le_trans with x ;
auto .
Save .
Lemma Uplus_neq_zero_right :
forall x y , ~0 ==
y -> ~0 ==
x +y.
intros ;
apply Ult_neq .
apply Ult_le_trans with y ;
auto .
Save .
Lemma not_Ult_le :
forall x y , ~x <
y ->
y <=
x .
intros ;
apply Ule_class ;
auto .
Save .
Lemma Ule_not_lt :
forall x y ,
x <=
y -> ~y <
x .
repeat red ;
intros .
apply H0 ;
auto .
Save .
Hint Immediate not_Ult_le Ule_not_lt .
Theorem Uplus_le_simpl_left :
forall x y z :
U ,
z <= [1-]
x ->
z +
x <=
z +
y ->
x <=
y .
intros .
apply Uplus_le_simpl_right with z ;
auto .
apply Ole_trans with (
z +
x );
auto .
apply Ole_trans with (
z +
y );
auto .
Save .
Lemma Uplus_lt_compat_left :
forall x y z :U,
z <= [1-]
y ->
x <
y -> (
x +
z ) < (
y +
z ).
unfold Ult ;
intuition .
apply H0 ;
apply Uplus_le_simpl_right with z ;
trivial .
Save .
Lemma Uplus_lt_compat_right :
forall x y z :U,
z <= [1-]
y ->
x <
y -> (
z +
x ) < (
z +
y ).
intros ;
setoid_rewrite (
Uplus_sym z x ).
intros ;
setoid_rewrite (
Uplus_sym z y ).
apply Uplus_lt_compat_left ;
auto .
Save .
Hint Resolve Uplus_lt_compat_right Uplus_lt_compat_left .
Lemma Uplus_lt_compat :
forall x y z t :U,
z <= [1-]
x ->
t <= [1-]
y ->
x <
y ->
z <
t -> (
x +
z ) < (
y +
t ).
intros ;
apply Ult_trans with (
y +
z );
auto .
apply Uplus_lt_compat_left ;
auto .
apply Ole_trans with t ;
auto .
Save .
Hint Immediate Uplus_lt_compat .
Lemma Uplus_lt_simpl_left :
forall x y z :U,
z <= [1-]
y -> (
z +
x ) < (
z +
y ) ->
x <
y .
unfold lt ;
repeat red ;
intros .
apply H0 ;
auto .
Save .
Lemma Uplus_lt_simpl_right :
forall x y z :U,
z <= [1-]
y -> (
x +
z ) < (
y +
z ) ->
x <
y .
unfold lt ;
repeat red ;
intros .
apply H0 ;
auto .
Save .
Lemma Uplus_one_le :
forall x y ,
x +
y == 1 -> [1-]
y <=
x .
intros ;
apply Ule_class ;
red ;
intros .
assert (
x < [1-]
y );
auto .
assert (
x +
y < [1-]
y +
y );
auto .
assert (
x +
y < 1);
auto .
setoid_rewrite <- (
Uinv_opp_left y );
auto .
Save .
Hint Immediate Uplus_one_le .
Theorem Uplus_eq_zero :
forall x ,
x <= [1-]
x -> (
x +
x ) ==
x ->
x == 0.
intros x H1 H2 ;
apply Uplus_eq_simpl_left with x ;
auto .
setoid_rewrite H2 ;
auto .
Save .
Lemma Umult_zero_left :
forall x , 0 *
x == 0.
intros ;
apply Uinv_simpl .
setoid_rewrite (
Udistr_inv_right 0
x );
auto .
setoid_rewrite Uinv_zero .
setoid_rewrite (
Umult_one_left x );
auto .
Save .
Hint Resolve Umult_zero_left .
Lemma Umult_zero_right :
forall x , (
x * 0) == 0.
intros ;
setoid_rewrite (
Umult_sym x 0);
auto .
Save .
Hint Resolve Uplus_eq_zero Umult_zero_right .
Lemma Umult_zero_left_eq :
forall x y ,
x == 0 ->
x *
y == 0.
intros ;
rewrite H ;
auto .
Save .
Lemma Umult_zero_right_eq :
forall x y ,
y == 0 ->
x *
y == 0.
intros ;
rewrite H ;
auto .
Save .
Lemma Umult_zero_eq :
forall x y z ,
x == 0 ->
x *
y ==
x *
z .
intros ;
rewrite H .
rewrite Umult_zero_left ;
auto .
Save .
Compatibility of operations with respect to order.
Lemma Umult_le_simpl_right :
forall x y z , ~0 ==
z -> (
x *
z ) <= (
y *
z ) ->
x <=
y .
intros ;
apply Umult_le_simpl_left with z ;
auto .
setoid_rewrite (
Umult_sym z x );
setoid_rewrite (
Umult_sym z y );
trivial .
Save .
Hint Resolve Umult_le_simpl_right .
Lemma Umult_simpl_right :
forall x y z , ~0 ==
z -> (
x *
z ) == (
y *
z ) ->
x ==
y .
intros ;
apply Ole_antisym ;
auto .
apply Umult_le_simpl_right with z ;
auto .
apply Umult_le_simpl_right with z ;
auto .
Save .
Lemma Umult_simpl_left :
forall x y z , ~0 ==
x -> (
x *
y ) == (
x *
z ) ->
y ==
z .
intros ;
apply Ole_antisym ;
auto .
apply Umult_le_simpl_left with x ;
auto .
apply Umult_le_simpl_left with x ;
auto .
Save .
Lemma Umult_lt_compat_left :
forall x y z , ~0 ==
z ->
x <
y -> (
x *
z ) < (
y *
z ).
unfold Ult ,not;
intros .
apply H0 ;
apply Umult_le_simpl_right with z ;
auto .
Save .
Lemma Umult_lt_compat_right :
forall x y z , ~0 ==
z ->
x <
y -> (
z *
x ) < (
z *
y ).
unfold Ult ,not;
intros .
apply H0 ;
apply Umult_le_simpl_left with z ;
auto .
Save .
Lemma Umult_lt_simpl_right :
forall x y z , ~0 ==
z -> (
x *
z ) < (
y *
z ) ->
x <
y .
unfold Ult ,not;
intros .
apply H0 ;
auto .
Save .
Lemma Umult_lt_simpl_left :
forall x y z , ~0 ==
z -> (
z *
x ) < (
z *
y ) ->
x <
y .
unfold Ult ,not;
intros .
apply H0 ;
auto .
Save .
Hint Resolve Umult_lt_compat_left Umult_lt_compat_right .
Lemma Umult_zero_simpl_right :
forall x y , 0 ==
x *y -> ~0 ==
x -> 0 ==
y .
intros .
apply Umult_simpl_left with x ;
auto .
rewrite (
Umult_zero_right x );
trivial .
Save .
Lemma Umult_zero_simpl_left :
forall x y , 0 ==
x *y -> ~0 ==
y -> 0 ==
x .
intros .
apply Umult_simpl_right with y ;
auto .
rewrite (
Umult_zero_left y );
trivial .
Save .
Lemma Umult_neq_zero :
forall x y , ~0 ==
x -> ~0 ==
y -> ~0 ==
x *y.
red ;
intros .
apply H0 ;
apply Umult_zero_simpl_right with x ;
trivial .
Save .
Hint Resolve Umult_neq_zero .
Lemma Umult_lt_zero :
forall x y , 0 <
x -> 0 <
y -> 0 <
x *y.
auto .
Save .
Hint Resolve Umult_lt_zero .
Lemma Umult_lt_compat :
forall x y z t ,
x <
y ->
z <
t ->
x *
z <
y *
t .
intros .
assert (0<y);
auto .
apply Ule_lt_trans with x ;
auto .
assert (0<t);
auto .
apply Ule_lt_trans with z ;
auto .
apply (
Ueq_orc 0
z );
auto ;
intros .
rewrite <-
H3 .
rewrite Umult_zero_right ;
auto .
apply Ult_trans with (
y *
z );
auto .
Save .
Lemma Uplus_one :
forall x y , [1-]
x <=
y ->
x +
y == 1.
intros ;
apply Ole_antisym ;
auto .
apply Ole_trans with (
x + [1-]
x );
auto .
Save .
Hint Resolve Uplus_one .
Lemma Uplus_one_right :
forall x ,
x + 1 == 1.
auto .
Save .
Lemma Uplus_one_left :
forall x :U, 1 +
x == 1.
auto .
Save .
Hint Resolve Uplus_one_right Uplus_one_left .
Lemma Uinv_mult_simpl :
forall x y z t ,
x <= [1-]
y -> (
x *
z ) <= [1-] (
y *
t ).
intros ;
apply Ole_trans with x ;
auto .
intros ;
apply Ole_trans with ([1-]
y );
auto .
Save .
Hint Resolve Uinv_mult_simpl .
Lemma Umult_inv_plus :
forall x y ,
x * [1-]
y +
y ==
x +
y * [1-]
x .
intros ;
apply Oeq_trans with (
x * [1-]
y +
y * ([1-]
x +
x )).
setoid_rewrite (
Uinv_opp_left x );
auto .
assert (
H :[1-]
x <= [1-]
x );
auto .
rewrite (
Udistr_plus_left y ([1-]
x )
x H ).
apply Oeq_trans with (
x * [1-]
y +
y *
x +
y * [1-]
x ).
norm_assoc_right ;
auto .
rewrite (
Umult_sym y x ).
assert (
H1 :[1-]
y <= [1-]
y );
auto .
rewrite <- (
Udistr_plus_left x ([1-]
y )
y H1 ).
setoid_rewrite (
Uinv_opp_left y );
auto .
Save .
Hint Resolve Umult_inv_plus .
Lemma Umult_inv_plus_le :
forall x y z ,
y <=
z ->
x * [1-]
y +
y <=
x * [1-]
z +
z .
intros .
setoid_rewrite (
Umult_inv_plus x y );
setoid_rewrite (
Umult_inv_plus x z );
auto .
Save .
Hint Resolve Umult_inv_plus_le .
Lemma Uplus_lt_Uinv :
forall x y ,
x +y < 1 ->
x <= [1-]
y .
intros ;
apply (
Ule_total x ([1-]
y ));
intro ;
auto .
case H .
rewrite Uplus_one ;
auto .
Save .
Lemma Uinv_lt_perm_left :
forall x y :
U , [1-]
x <
y -> [1-]
y <
x .
unfold Ult ;
intuition .
Save .
Lemma Uinv_lt_perm_right :
forall x y :
U ,
x < [1-]
y ->
y < [1-]
x .
unfold Ult ;
intuition .
Save .
Hint Immediate Uinv_lt_perm_left Uinv_lt_perm_right .
Lemma Uinv_lt_one :
forall x , 0 <
x -> [1-]
x < 1.
intros ;
assert ([1-]1 <
x );
auto .
rewrite Uinv_one ;
auto .
Save .
Lemma Uinv_lt_zero :
forall x ,
x < 1 -> 0 < [1-]
x .
intros ;
assert (
x < [1-]0);
auto .
Save .
Hint Resolve Uinv_lt_one Uinv_lt_zero .
Lemma orc_inv_plus_one :
forall x y ,
orc (
x <=[1-]
y ) (
x +y==1).
intros ;
apply (
Ule_total x ([1-]
y ));
intro ;
auto .
apply class_orc ;
trivial .
Save .
Lemma Umult_lt_right :
forall p q ,
p <1 -> 0 <
q ->
p *
q <
q .
intros .
apply Ult_le_trans with (1 *
q );
auto .
Save .
Lemma Umult_lt_left :
forall p q , 0 <
p ->
q < 1 ->
p *
q <
p .
intros .
apply Ult_le_trans with (
p * 1);
auto .
Save .
Hint Resolve Umult_lt_right Umult_lt_left .
Fixpoint Uexp (
x :U) (
n :nat) {
struct n } :
U :=
match n with 0 => 1 | (
S p ) =>
x *
Uexp x p end .
Infix "^" :=
Uexp :
U_scope .
Lemma Uexp_1 :
forall x ,
x ^1==x.
simpl Uexp ;
auto .
Save .
Lemma Uexp_0 :
forall x ,
x ^0==1.
simpl Uexp ;
auto .
Save .
Lemma Uexp_zero :
forall n , (0<n)%nat -> 0^n==0.
destruct n ;
simpl Uexp ;
intro ;
auto .
casetype False ;
omega .
Save .
Lemma Uexp_one :
forall n , 1^n==1.
induction n ;
simpl Uexp ;
auto .
Save .
Lemma Uexp_le_compat_right :
forall x n m , (
n <=m)%nat ->
x ^m <=
x ^n.
induction 1;
simpl ;
auto .
apply Ole_trans with (
x ^m);
auto .
Save .
Lemma Uexp_le_compat_left :
forall x y n ,
x <=y ->
x ^n <=
y ^n.
induction n ;
simpl ;
intros ;
auto .
apply Ole_trans with (
x * (
y ^n));
auto .
Save .
Hint Resolve Uexp_le_compat_left Uexp_le_compat_right .
Lemma Uexp_le_compat :
forall x y (
n m :nat),
x <=y ->
n <=m ->
x ^m <=
y ^n.
intros ;
apply Ole_trans with (
x ^n);
auto .
Save .
Instance Uexp_mon2 :
monotonic2 (
o1 :=Iord
U ) (
o3 :=Iord
U )
Uexp .
red ;
simpl ;
intros ;
apply Uexp_le_compat ;
trivial .
Save .
Definition UExp :
U --m> (
nat -m->
U ) :=
mon2 Uexp .
Add Morphism Uexp with signature Oeq &
eq &
Oeq as Uexp_eq_compat .
intros ;
apply Ole_antisym ;
auto .
Save .
Lemma Uexp_inv_S :
forall x n , ([1-]
x ^(S
n ))==x*([1-]
x ^n)+[1-]
x .
simpl Uexp ;
auto .
Save .
Lemma Uexp_lt_compat :
forall p q n , (
O <n)%nat->(p<q)->(p^n<q^n).
induction n ;
simpl Uexp ;
intros ;
auto .
casetype False ;
omega .
destruct n ;
auto .
apply Umult_lt_compat ;
auto with arith .
Save .
Hint Resolve Uexp_lt_compat .
Lemma Uexp_lt_zero :
forall p n , (0<p)->(0<p^n).
destruct n ;
intros ;
auto .
rewrite <- (
Uexp_zero (
n :=S
n ));
auto with arith .
Save .
Hint Resolve Uexp_lt_zero .
Lemma Uexp_lt_one :
forall p n , (0<n)%nat->(p<1)->(p^n<1).
intros ;
rewrite <- (
Uexp_one n );
auto with arith .
Save .
Hint Resolve Uexp_lt_one .
Lemma Uexp_lt_antimon :
forall p n m , (
n <m)%nat-> 0<p ->
p < 1 ->
p ^m <
p ^n.
induction 1;
simpl ;
intros ;
auto with arith .
apply Ult_trans with (
p *p^n);
auto with arith .
Save .
Hint Resolve Uexp_lt_antimon .
Lemma Udiv_mult :
forall x y , ~0==y ->
x <=
y -> (
x /y)*y ==
x .
intros ;
rewrite Umult_sym ;
auto .
Save .
Hint Resolve Udiv_mult .
Lemma Umult_div_le :
forall x y ,
y * (
x /
y ) <=
x .
intros ;
apply (
Ueq_orc 0
y );
auto ;
intros .
apply Ole_trans with (0 * (
x /y));
auto .
rewrite Umult_zero_left ;
auto .
intros ;
apply (
Ule_total x y );
auto ;
intros .
rewrite Udiv_le_one ;
auto .
rewrite Umult_one_right ;
auto .
Save .
Hint Resolve Umult_div_le .
Lemma Udiv_mult_le :
forall x y , (
x /y)*y <=
x .
intros ;
rewrite Umult_sym ;
auto .
Save .
Hint Resolve Udiv_mult_le .
Lemma Udiv_le_compat_left :
forall x y z ,
x <=
y ->
x /z <=
y /z.
intros ;
apply (
Ueq_orc 0
z );
auto ;
intros .
rewrite (
Udiv_by_zero x );
auto .
intros ;
apply (
Ule_total y z );
auto ;
intros .
apply Umult_le_simpl_right with z ;
auto .
rewrite (
Udiv_mult x );
auto .
rewrite (
Udiv_mult y );
auto .
transitivity y ;
auto .
rewrite (
Udiv_le_one y );
auto .
Save .
Hint Resolve Udiv_le_compat_left .
Lemma Udiv_eq_compat_left :
forall x y z ,
x ==
y ->
x /z ==
y /z.
intros ;
apply Ole_antisym ;
auto .
Save .
Hint Resolve Udiv_eq_compat_left .
Lemma Umult_div_le_left :
forall x y z , ~0==y ->
x *y<=z ->
x <=
z /y.
intros ;
apply (
Ule_total y z );
auto ;
intros .
rewrite (
Udiv_le_one z );
auto .
apply Umult_le_simpl_right with y ;
auto .
apply Ole_trans with z ;
auto .
rewrite (
Udiv_mult z y H );
auto .
Save .
Lemma Udiv_le_compat_right :
forall x y z , ~0==y ->
y <=
z ->
x /z <=
x /y.
intros ;
assert (~ 0 ==
z ).
apply Ule_neq_zero with y ;
auto .
apply (
Ule_total z x );
auto ;
intros .
rewrite Udiv_le_one ;
auto .
rewrite Udiv_le_one ;
auto .
apply Ole_trans with z ;
trivial .
apply Umult_div_le_left ;
auto .
apply Ole_trans with (
x /z *
z );
auto .
Save .
Hint Resolve Udiv_le_compat_right .
Lemma Udiv_eq_compat_right :
forall x y z ,
y ==
z ->
x /z ==
x /y.
intros ;
apply (
Ueq_orc 0
y );
auto ;
intros .
assert (0==z).
rewrite <-
H ;
auto .
repeat rewrite Udiv_by_zero ;
auto .
assert (~ 0 ==
z ).
apply Ule_neq_zero with y ;
auto .
apply Ole_antisym ;
auto .
Save .
Hint Resolve Udiv_eq_compat_right .
Add Morphism Udiv with signature Oeq &
Oeq &
Oeq as Udiv_eq_compat .
intros x1 x2 eq1 x3 x4 eq2 .
transitivity (
x2 /x3);
auto .
Save .
Add Morphism Udiv with signature Ole ++>
Oeq &
Ole as Udiv_le_compat .
intros x1 x2 le1 x3 x4 le2 ;
transitivity (
x2 /x3);
auto .
Save .
Lemma Umult_div_eq :
forall x y z , ~0 ==
y ->
x *y ==
z ->
x ==
z /y.
intros ;
apply Umult_simpl_right with y ;
auto .
assert (
z <=y).
transitivity (
x *y);
auto .
transitivity z ;
auto .
apply Oeq_sym ;
auto .
Save .
Lemma Umult_div_le_right :
forall x y z ,
x <=
y *z ->
x /z <=
y .
intros ;
apply (
Ueq_orc 0
z );
auto ;
intros .
rewrite Udiv_by_zero ;
auto .
apply Umult_le_simpl_right with z ;
auto .
assert (
x <=z).
transitivity (
y *z);
auto .
rewrite (
Udiv_mult x z H0 );
auto .
Save .
Lemma Udiv_le :
forall x y , ~0==y ->
x <=
x /y.
intros ;
apply Umult_div_le_left ;
auto .
Save .
Lemma Udiv_zero :
forall x , 0/x==0.
intros ;
apply (
Ueq_orc 0
x );
auto ;
intros .
apply Oeq_sym ;
apply Umult_div_eq ;
auto .
Save .
Hint Resolve Udiv_zero .
Lemma Udiv_zero_eq :
forall x y , 0==x ->
x /y==0.
intros ;
rewrite <-
H ;
auto .
Save .
Hint Resolve Udiv_zero_eq .
Lemma Udiv_one :
forall x , ~0==x ->
x /1==x.
intros ;
apply Oeq_sym ;
apply Umult_div_eq ;
auto .
Save .
Hint Resolve Udiv_one .
Lemma Udiv_refl :
forall x , ~0==x ->
x /x==1.
auto .
Save .
Hint Resolve Udiv_refl .
Lemma Umult_div_assoc :
forall x y z ,
y <=
z -> (
x *y) /
z ==
x * (
y /z).
intros ;
apply (
Ueq_orc 0
z );
auto ;
intros .
repeat rewrite Udiv_by_zero ;
auto .
apply Oeq_sym ;
apply Umult_div_eq ;
auto .
transitivity (
x * (
y /
z *
z ));
auto .
Save .
Lemma Udiv_mult_assoc :
forall x y z ,
x <=
y *z ->
x /(y*z) == (
x /y)/z.
intros ;
apply (
Ueq_orc 0
z );
auto ;
intros .
rewrite (
Udiv_by_zero (
x /y));
auto .
rewrite Udiv_by_zero ;
auto ;
rewrite <-
H0 ;
auto .
intros ;
apply (
Ueq_orc 0
y );
auto ;
intros .
rewrite (
Udiv_by_zero x );
auto .
rewrite <-H1;
auto .
rewrite (
Udiv_by_zero x );
auto .
transitivity (0*z);
auto .
apply Oeq_sym ;
apply Umult_div_eq ;
auto .
rewrite (
Umult_sym y z ).
transitivity (
x /
y /
z *
z *
y );
auto .
assert (
x /y <=
z ).
apply Umult_div_le_right ;
auto .
transitivity (
y *z);
auto .
rewrite (
Udiv_mult (
x /y)
z H0 );
auto .
assert (
x <=y);
auto .
transitivity (
y *z);
auto .
Save .
Lemma Udiv_inv :
forall x y , ~0==y -> [1-](
x /y) <= ([1-]
x )/y.
intros ;
apply (
Ule_total x y );
auto ;
intros .
apply Umult_div_le_left ;
auto .
transitivity ([1-] (
x /y *
y ));
auto .
rewrite Udiv_le_one ;
auto .
Save .
Lemma Uplus_div_inv :
forall x y z ,
x +y <=
z ->
x <=[1-]
y ->
x /z <= [1-](
y /z).
intros ;
apply (
Ueq_orc 0
z );
auto ;
intros .
repeat (
rewrite Udiv_by_zero ;
auto ).
apply Umult_div_le_right ;
auto .
apply Uplus_le_simpl_right with ([1-]
z ).
apply Uinv_le_compat ;
transitivity (
x +y);
auto .
rewrite <-
Udistr_inv_right .
rewrite Udiv_mult ;
auto .
transitivity (
x +[1-](
x +y));
auto .
rewrite Uplus_sym ;
rewrite Uinv_plus_left ;
auto .
transitivity (
x +y);
auto .
Save .
Hint Resolve Uplus_div_inv .
Lemma Udiv_plus_le :
forall x y z ,
x /z +
y /z <= (
x +y)/z.
intros ;
apply (
Ueq_orc 0
z );
auto ;
intros .
repeat (
rewrite Udiv_by_zero ;
auto ).
intros ;
apply Umult_div_le_left ;
auto .
rewrite Umult_sym ;
rewrite Udistr_plus_left_le .
apply Uplus_le_compat ;
rewrite Umult_sym ;
auto .
Save .
Hint Resolve Udiv_plus_le .
Lemma Udiv_plus :
forall x y z , (
x +y)/z ==
x /z +
y /z.
intros ;
apply Ole_antisym ;
auto .
apply (
Ueq_orc 0
z );
auto ;
intros .
repeat (
rewrite Udiv_by_zero ;
auto ).
apply (
Ule_total x z );
auto ;
intros .
apply (
Ule_total y z );
auto ;
intros .
apply (
Ule_total (
x /z) ([1-](
y /z)));
auto ;
intros .
apply Umult_div_le_right ;
auto .
rewrite Udistr_plus_right ;
auto .
apply Uplus_le_compat ;
rewrite Udiv_mult ;
auto .
rewrite (
Uplus_one (
x /z) (
y /z));
auto .
rewrite (
Udiv_le_one y z H );
auto .
transitivity 1;
auto .
rewrite (
Udiv_le_one x z H );
auto .
transitivity 1;
auto .
Save .
Hint Resolve Udiv_plus .
Instance Udiv_mon :
forall k ,
monotonic (
fun x => (
x /k)).
red ;
auto .
Save .
Definition UDiv (
k :U) :
U -m>
U :=
mon (
fun x => (
x /k)).
Lemma UDiv_simpl :
forall (
k :U)
x ,
UDiv k x =
x /k.
trivial .
Save .
Definition and properties of
A conjonction operation which coincides with min and mult
on 0 and 1, see Morgan & McIver
Definition Uesp (
x y :U) := [1-] ([1-]
x + [1-]
y ).
Infix "&" :=
Uesp (
left associativity ,
at level 40) :
U_scope .
Lemma Uinv_plus_esp :
forall x y , [1-] (
x +
y ) == [1-]
x & [1-]
y .
unfold Uesp ;
intros .
setoid_rewrite (
Uinv_inv x );
setoid_rewrite (
Uinv_inv y );
auto .
Save .
Hint Resolve Uinv_plus_esp .
Lemma Uinv_esp_plus :
forall x y , [1-] (
x &
y ) == [1-]
x + [1-]
y .
unfold Uesp ;
intros .
setoid_rewrite (
Uinv_inv ([1-]
x + [1-]
y ));
trivial .
Save .
Hint Resolve Uinv_esp_plus .
Lemma Uesp_sym :
forall x y :
U ,
x &
y ==
y &
x .
intros ;
unfold Uesp ;
auto .
Save .
Lemma Uesp_one_right :
forall x :
U ,
x & 1 ==
x .
intro ;
unfold Uesp .
setoid_rewrite Uinv_one .
setoid_rewrite (
Uplus_zero_right ([1-]
x ));
auto .
Save .
Lemma Uesp_one_left :
forall x :
U , 1 &
x ==
x .
intros ;
rewrite Uesp_sym ;
apply Uesp_one_right .
Save .
Lemma Uesp_zero :
forall x y ,
x <= [1-]
y ->
x &
y == 0.
intros ;
unfold Uesp .
setoid_rewrite <-
Uinv_one ;
auto .
Save .
Hint Resolve Uesp_sym Uesp_one_right Uesp_one_left Uesp_zero .
Lemma Uesp_zero_right :
forall x :
U ,
x & 0 == 0.
auto .
Save .
Lemma Uesp_zero_left :
forall x :
U , 0 &
x == 0.
auto .
Save .
Hint Resolve Uesp_zero_right Uesp_zero_left .
Add Morphism Uesp with signature Oeq &
Oeq &
Oeq as Uesp_eq_compat .
unfold Uesp ;
intros .
apply Uinv_eq_compat .
rewrite H ;
rewrite H0 ;
auto .
Save .
Lemma Uesp_le_compat :
forall x y z t ,
x <=y ->
z <=t ->
x &z <=
y &t.
unfold Uesp ;
intros .
apply Uinv_le_compat .
apply Uplus_le_compat ;
auto .
Save .
Hint Immediate Uesp_le_compat Uesp_eq_compat .
Lemma Uesp_zero_one_mult_left :
forall x y ,
orc (
x == 0) (
x == 1) ->
x &
y ==
x *
y .
intros ;
apply H ;
intros ;
auto .
rewrite H0 ;
rewrite Uesp_zero_left ;
auto .
rewrite H0 ;
rewrite Uesp_one_left ;
auto .
Save .
Lemma Uesp_zero_one_mult_right :
forall x y ,
orc (
y == 0) (
y == 1) ->
x &
y ==
x *
y .
intros ;
apply H ;
intros ;
auto .
rewrite H0 ;
rewrite Uesp_zero_right ;
auto .
rewrite H0 ;
rewrite Uesp_one_right ;
auto .
Save .
Hint Immediate Uesp_zero_one_mult_left Uesp_zero_one_mult_right .
Instance Uesp_mon :
monotonic2 Uesp .
red ;
auto .
Save .
Definition UEsp :
U -m>
U -m>
U :=
mon2 Uesp .
Lemma UEsp_simpl :
forall x y ,
UEsp x y =
x &
y .
trivial .
Save .
Lemma Uesp_le_left :
forall x y ,
x &
y <=
x .
unfold Uesp ;
intros .
apply Uinv_le_perm_left ;
auto .
Save .
Lemma Uesp_le_right :
forall x y ,
x &
y <=
y .
unfold Uesp ;
intros .
apply Uinv_le_perm_left ;
auto .
Save .
Hint Resolve Uesp_le_left Uesp_le_right .
Lemma Uesp_plus_inv :
forall x y , [1-]
y <=
x ->
x ==
x &
y + [1-]
y .
unfold Uesp ;
intros .
rewrite Uinv_plus_right ;
auto .
Save .
Hint Resolve Uesp_plus_inv .
Lemma Uesp_le_plus_inv :
forall x y ,
x <=
x &
y + [1-]
y .
intros ;
apply (
Ule_total ([1-]
y )
x );
intros ;
auto .
rewrite Uesp_zero ;
auto .
rewrite Uplus_zero_left ;
auto .
Save .
Hint Resolve Uesp_le_plus_inv .
Lemma Uplus_inv_le_esp :
forall x y z ,
x <=
y + ([1-]
z ) ->
x &
z <=
y .
intros ;
unfold Uesp .
apply Uinv_le_perm_left .
transitivity ([1-](
y +[1-]
z ) + [1-]
z );
auto .
Save .
Hint Immediate Uplus_inv_le_esp .
Definition and properties of
Definition Uminus (
x y :U) := [1-] ([1-]
x +
y ).
Infix "-" :=
Uminus :
U_scope .
Lemma Uminus_le_compat_left :
forall x y z ,
x <=
y ->
x -
z <=
y -
z .
unfold Uminus ;
auto .
Save .
Lemma Uminus_le_compat_right :
forall x y z ,
y <=
z ->
x -
z <=
x -
y .
unfold Uminus ;
auto .
Save .
Hint Resolve Uminus_le_compat_left Uminus_le_compat_right .
Lemma Uminus_le_compat :
forall x y z t ,
x <=
y ->
t <=
z ->
x -
z <=
y -
t .
intros ;
transitivity (
x -t);
auto .
Save .
Hint Immediate Uminus_le_compat .
Add Morphism Uminus with signature Oeq &
Oeq &
Oeq as Uminus_eq_compat .
intros x1 x2 eq1 x3 x4 eq2 ;
apply Ole_antisym ;
transitivity (
x1 -x4);
auto .
Save .
Hint Immediate Uminus_eq_compat .
Lemma Uminus_zero_right :
forall x ,
x - 0 ==
x .
unfold Uminus ;
intros .
setoid_rewrite (
Uplus_zero_right ([1-]
x ));
auto .
Save .
Lemma Uminus_one_left :
forall x , 1 -
x == [1-]
x .
unfold Uminus ;
intros .
setoid_rewrite Uinv_one ;
auto .
Save .
Lemma Uminus_le_zero :
forall x y ,
x <=
y ->
x -
y == 0.
unfold Uminus ;
intros .
setoid_rewrite <-
Uinv_one .
apply Uinv_eq_compat .
apply Ole_antisym ;
auto .
transitivity ([1-]
y +
y );
auto .
Save .
Hint Resolve Uminus_zero_right Uminus_one_left Uminus_le_zero .
Lemma Uminus_zero_left :
forall x , 0 -
x == 0.
auto .
Save .
Hint Resolve Uminus_zero_left .
Lemma Uminus_eq :
forall x ,
x -
x == 0.
auto .
Save .
Hint Resolve Uminus_eq .
Lemma Uminus_le_left :
forall x y ,
x -
y <=
x .
unfold Uminus ;
auto .
Save .
Hint Resolve Uminus_le_left .
Lemma Uminus_le_inv :
forall x y ,
x -
y <= [1-]
y .
intros .
unfold Uminus .
apply Uinv_le_compat ;
auto .
Save .
Hint Resolve Uminus_le_inv .
Lemma Uminus_plus_simpl :
forall x y ,
y <=
x -> (
x -
y ) +
y ==
x .
unfold Uminus ;
intros .
rewrite (
Uinv_plus_right ([1-]
x )
y );
auto .
Save .
Lemma Uminus_plus_zero :
forall x y ,
x <=
y -> (
x -
y ) +
y ==
y .
intros ;
rewrite (
Uminus_le_zero x y );
auto .
Save .
Hint Resolve Uminus_plus_simpl Uminus_plus_zero .
Lemma Uminus_plus_le :
forall x y ,
x <= (
x -
y ) +
y .
intros ;
apply (
Ule_total x y );
intros ;
auto .
transitivity y ;
auto .
rewrite (
Uminus_plus_simpl x y H );
trivial .
Save .
Hint Resolve Uminus_plus_le .
Lemma Uesp_minus_distr_left :
forall x y z , (
x &
y ) -
z == (
x -
z ) &
y .
unfold Uesp ,
Uminus ;
intros .
apply Uinv_eq_compat .
setoid_rewrite (
Uinv_inv ([1-]
x + [1-]
y )).
setoid_rewrite (
Uinv_inv (([1-]
x ) +
z )).
repeat norm_assoc_right ;
auto .
Save .
Lemma Uesp_minus_distr_right :
forall x y z , (
x &
y ) -
z ==
x & (
y -
z ).
intros ;
rewrite (
Uesp_sym x y ).
setoid_rewrite (
Uesp_sym x (
y -
z ));
apply Uesp_minus_distr_left .
Save .
Hint Resolve Uesp_minus_distr_left Uesp_minus_distr_right .
Lemma Uesp_minus_distr :
forall x y z t , (
x &
y ) - (
z +
t ) == (
x -
z ) & (
y -
t ).
unfold Uesp ,
Uminus ;
intros .
apply Uinv_eq_compat .
setoid_rewrite (
Uinv_inv ([1-]
x + [1-]
y )).
setoid_rewrite (
Uinv_inv ([1-]
x +
z )).
setoid_rewrite (
Uinv_inv ([1-]
y +
t )).
repeat norm_assoc_right ;
auto .
Save .
Hint Resolve Uesp_minus_distr .
Lemma Uminus_esp_simpl_left :
forall x y , [1-]
x <=
y ->
x - (
x &
y ) == [1-]
y .
unfold Uesp ,Uminus;
intros .
apply Uinv_eq_compat .
rewrite (
Uplus_sym ([1-]
x )).
rewrite Uinv_plus_left ;
auto .
Save .
Lemma Uplus_esp_simpl :
forall x y , (
x - (
x &
y ))+y ==
x +y.
intros ;
apply (
Ule_total ([1-]
x )
y );
auto ;
intros .
rewrite Uminus_esp_simpl_left ;
auto .
rewrite (@
Uplus_one x y );
auto .
rewrite (@
Uesp_zero x y );
auto .
Save .
Hint Resolve Uminus_esp_simpl_left Uplus_esp_simpl .
Lemma Uminus_esp_le_inv :
forall x y ,
x - (
x &
y ) <= [1-]
y .
intros ;
apply (
Ule_total ([1-]
x )
y );
auto ;
intros .
rewrite (@
Uesp_zero x y );
auto .
rewrite Uminus_zero_right ;
auto .
Save .
Hint Resolve Uminus_esp_le_inv .
Lemma Uplus_esp_inv_simpl :
forall x y ,
x <= [1-]
y -> (
x +
y ) & [1-]
y ==
x .
unfold Uesp ;
intros .
apply Uinv_eq_perm_left .
rewrite Uinv_inv ;
auto .
Save .
Hint Resolve Uplus_esp_inv_simpl .
Lemma Uplus_inv_esp_simpl :
forall x y ,
x <=
y -> (
x + [1-]
y ) &
y ==
x .
intros .
transitivity ((
x + [1-]
y ) & [1-][1-]
y );
auto .
rewrite Uinv_inv ;
auto .
Save .
Hint Resolve Uplus_inv_esp_simpl .
Definition and properties of max
Definition max (
x y :
U ) :
U := (
x -
y ) +
y .
Lemma max_eq_right :
forall x y :
U ,
y <=
x ->
max x y ==
x .
unfold max ;
auto .
Save .
Lemma max_eq_left :
forall x y :
U ,
x <=
y ->
max x y ==
y .
unfold max ;
auto .
Save .
Hint Resolve max_eq_right max_eq_left .
Lemma max_eq_case :
forall x y :
U ,
orc (
max x y ==
x ) (
max x y ==
y ).
intros ;
apply (
Ule_total x y );
auto .
Save .
Add Morphism max with signature Oeq &
Oeq &
Oeq as max_eq_compat .
unfold max ;
intros .
apply Uplus_eq_compat ;
auto .
Save .
Lemma max_le_right :
forall x y :
U ,
x <=
max x y .
intros ;
apply (
Ule_total x y );
intros ;
auto .
rewrite max_eq_left ;
auto .
rewrite max_eq_right ;
auto .
Save .
Lemma max_le_left :
forall x y :
U ,
y <=
max x y .
intros ;
apply (
Ule_total x y );
intros ;
auto .
rewrite max_eq_left ;
auto .
rewrite max_eq_right ;
auto .
Save .
Hint Resolve max_le_right max_le_left .
Lemma max_le :
forall x y z :
U ,
x <=
z ->
y <=
z ->
max x y <=
z .
intros ;
apply (
Ule_total x y );
intros ;
auto .
rewrite max_eq_left ;
auto .
rewrite max_eq_right ;
auto .
Save .
Lemma max_le_compat :
forall x y z t :
U ,
x <=
y ->
z <=
t ->
max x z <=
max y t .
intros ;
apply max_le ;
auto .
transitivity y ;
auto .
transitivity t ;
auto .
Save .
Hint Immediate max_le_compat .
Instance max_mon :
monotonic2 max .
red ;
auto .
Save .
Definition Max :
U -m>
U -m>
U :=
mon2 max .
Lemma max_eq_mult :
forall k x y ,
max (
k *x) (
k *y) ==
k *
max x y .
intros ;
apply Ole_antisym .
apply max_le ;
auto .
apply (
max_eq_case x y );
auto ;
intro H ;
rewrite H ;
auto .
Save .
Lemma max_eq_plus_cte_right :
forall x y k ,
max (
x +k) (
y +k) == (
max x y ) +
k .
intros ;
apply Ole_antisym .
apply max_le ;
auto .
apply (
max_eq_case x y );
auto ;
intro H ;
rewrite H ;
auto .
Save .
Hint Resolve max_eq_mult max_eq_plus_cte_right .
Definition and properties of min
Definition min (
x y :
U ) :
U := [1-] ((
y -
x ) + [1-]
y ).
Lemma min_eq_right :
forall x y :
U ,
x <=
y ->
min x y ==
x .
unfold min ,
Uminus ;
intros .
apply Uinv_eq_perm_left ;
auto .
Save .
Lemma min_eq_left :
forall x y :
U ,
y <=
x ->
min x y ==
y .
unfold min ;
intros .
rewrite Uminus_le_zero ;
auto .
Save .
Hint Resolve min_eq_right min_eq_left .
Lemma min_eq_case :
forall x y :
U ,
orc (
min x y ==
x ) (
min x y ==
y ).
intros ;
apply (
Ule_total x y );
auto .
Save .
Add Morphism min with signature Oeq &
Oeq &
Oeq as min_eq_compat .
unfold min ;
intros .
apply Uinv_eq_compat ;
auto .
apply Uplus_eq_compat ;
auto .
Save .
Lemma min_le_right :
forall x y :
U ,
min x y <=x.
intros ;
apply (
Ule_total x y );
intros ;
auto .
rewrite min_eq_left ;
auto .
Save .
Lemma min_le_left :
forall x y :
U ,
min x y <=
y .
intros ;
apply (
Ule_total x y );
intros ;
auto .
rewrite min_eq_right ;
auto .
Save .
Hint Resolve min_le_right min_le_left .
Lemma min_le :
forall x y z :
U ,
z <=
x ->
z <=
y ->
z <=
min x y .
intros ;
apply (
Ule_total x y );
intros ;
auto .
rewrite min_eq_right ;
auto .
rewrite min_eq_left ;
auto .
Save .
Lemma Uinv_min_max :
forall x y , [1-](
min x y )==max ([1-]
x ) ([1-]
y ).
intros ;
apply (
Ule_total x y );
intros ;
auto .
rewrite min_eq_right ;
auto ;
rewrite max_eq_right ;
auto .
rewrite min_eq_left ;
auto ;
rewrite max_eq_left ;
auto .
Save .
Lemma Uinv_max_min :
forall x y , [1-](
max x y )==min ([1-]
x ) ([1-]
y ).
intros ;
apply (
Ule_total x y );
intros ;
auto .
rewrite min_eq_left ;
auto ;
rewrite max_eq_left ;
auto .
rewrite min_eq_right ;
auto ;
rewrite max_eq_right ;
auto .
Save .
Lemma min_mult :
forall x y k ,
min (
k *
x ) (
k *
y ) ==
k * (
min x y ).
intros ;
apply (
Ule_total x y );
intros ;
auto .
rewrite min_eq_right ;
auto ;
rewrite min_eq_right ;
auto .
rewrite min_eq_left ;
auto ;
rewrite min_eq_left ;
auto .
Save .
Hint Resolve min_mult .
Lemma min_plus :
forall x1 x2 y1 y2 ,
(
min x1 x2 ) + (
min y1 y2 ) <=
min (
x1 +y1) (
x2 +y2).
intros ;
apply min_le ;
auto .
Save .
Hint Resolve min_plus .
Lemma min_plus_cte :
forall x y k ,
min (
x +
k ) (
y +
k ) == (
min x y ) +
k .
intros ;
apply (
Ule_total x y );
intros ;
auto .
rewrite min_eq_right ;
auto ;
rewrite min_eq_right ;
auto .
rewrite min_eq_left ;
auto ;
rewrite min_eq_left ;
auto .
Save .
Hint Resolve min_plus_cte .
Lemma min_le_compat :
forall x1 y1 x2 y2 ,
x1 <=y1 ->
x2 <=y2 ->
min x1 x2 <=
min y1 y2 .
intros ;
apply min_le .
transitivity x1 ;
auto .
transitivity x2 ;
auto .
Save .
Hint Immediate min_le_compat .
Instance min_mon2 :
monotonic2 min .
red ;
auto .
Save .
Definition Min :
U -m>
U -m>
U :=
mon2 min .
Lemma min_sym :
forall x y ,
min x y ==
min y x .
intros ;
apply (
Ule_total x y );
intros ;
auto .
rewrite min_eq_right ;
auto .
rewrite min_eq_left ;
auto .
rewrite min_eq_left ;
auto .
rewrite min_eq_right ;
auto .
Save .
Hint Resolve min_sym .
Lemma incr_decomp_aux :
forall f g :
nat -m>
U ,
forall n1 n2 , (
forall m , ~ ((
n1 <=m)%nat /\
f n1 <=
g m ))
-> (
forall m , ~((n2<=m)%nat /\
g n2 <=
f m )) -> (
n1 <=n2)%nat ->
False .
intros ;
assert (
absurd :~
g n2 <
g n2 );
auto .
assert (~(f
n1 <=
g n2 )).
apply not_and_elim_left with (1:=
H n2 );
auto .
assert (~(g
n2 <=
f n2 ));
auto .
apply not_and_elim_left with (1:=
H0 n2 );
auto .
apply absurd ;
apply Ult_le_trans with (
f n1 );
auto .
transitivity (
f n2 );
auto .
Save .
Lemma incr_decomp :
forall f g :
nat -m>
U ,
orc (
forall n ,
exc (
fun m => (
n <=m)%nat /\
f n <=
g m ))
(
forall n ,
exc (
fun m => (
n <=m)%nat /\
g n <=
f m )).
intros f g ;
apply orc_intro ;
intros .
apply H ;
clear H ;
intros .
apply exc_intro_class ;
intros .
apply H0 ;
clear H0 ;
intros .
apply exc_intro_class ;
intros .
case (
dec_le n n0 );
intro .
apply (
incr_decomp_aux f g )
with (
n1 :=n) (
n2 :=n0);
auto .
apply (
incr_decomp_aux g f )
with (
n1 :=n0) (
n2 :=n);
auto ;
omega .
Save .
Lemma Uplus_minus_simpl_right :
forall x y ,
y <= [1-]
x -> (
x +
y ) -
y ==
x .
unfold Uminus ;
intros .
rewrite (
Uinv_plus_right x y );
auto .
Save .
Hint Resolve Uplus_minus_simpl_right .
Lemma Uplus_minus_simpl_left :
forall x y ,
y <= [1-]
x -> (
x +
y ) -
x ==
y .
intros ;
setoid_rewrite (
Uplus_sym x y );
auto .
Save .
Lemma Uminus_assoc_left :
forall x y z , (
x -
y ) -
z ==
x - (
y +
z ).
unfold Uminus ;
intros .
apply Uinv_eq_compat .
setoid_rewrite (
Uinv_inv ([1-]
x +
y ));
auto .
Save .
Hint Resolve Uminus_assoc_left .
Lemma Uminus_perm :
forall x y z , (
x -
y ) -
z == (
x -
z ) -
y .
intros ;
rewrite Uminus_assoc_left .
rewrite (
Uplus_sym y z );
auto .
Save .
Hint Resolve Uminus_perm .
Lemma Uminus_le_perm_left :
forall x y z ,
y <=
x ->
x -
y <=
z ->
x <=
z +
y .
intros ;
rewrite <- (
Uminus_plus_simpl x y );
auto .
Save .
Lemma Uplus_le_perm_left :
forall x y z ,
x <=
y +
z ->
x -
y <=
z .
intros ;
apply (
Ule_total y x );
intros ;
auto .
apply Uplus_le_simpl_left with y .
unfold Uminus ;
setoid_rewrite (
Uinv_inv ([1-]
x +
y ));
auto .
setoid_rewrite (
Uplus_sym y (
x -y));
rewrite (
Uminus_plus_simpl x y );
auto .
transitivity (0:U);
auto .
Save .
Lemma Uminus_eq_perm_left :
forall x y z ,
y <=
x ->
x -
y ==
z ->
x ==
z +
y .
intros ;
rewrite <- (
Uminus_plus_simpl x y );
auto .
Save .
Lemma Uplus_eq_perm_left :
forall x y z ,
y <= [1-]
z ->
x ==
y +
z ->
x -
y ==
z .
intros ;
setoid_rewrite H0 ;
auto .
setoid_rewrite (
Uplus_sym y z );
auto .
Save .
Hint Resolve Uminus_le_perm_left Uminus_eq_perm_left .
Hint Resolve Uplus_le_perm_left Uplus_eq_perm_left .
Lemma Uminus_le_perm_right :
forall x y z ,
z <=
y ->
x <=
y -
z ->
x +
z <=
y .
intros ;
rewrite <- (
Uminus_plus_simpl y z );
auto .
Save .
Lemma Uplus_le_perm_right :
forall x y z ,
z <= [1-]
x ->
x +
z <=
y ->
x <=
y -
z .
intros ;
apply Uplus_le_simpl_right with z ;
auto .
Save .
Hint Resolve Uminus_le_perm_right Uplus_le_perm_right .
Lemma Uminus_le_perm :
forall x y z ,
z <=
y ->
x <= [1-]
z ->
x <=
y -
z ->
z <=
y -
x .
intros ;
apply Uplus_le_perm_right ;
auto .
setoid_rewrite (
Uplus_sym z x );
auto .
Save .
Hint Resolve Uminus_le_perm .
Lemma Uminus_eq_perm_right :
forall x y z ,
z <=
y ->
x ==
y -
z ->
x +
z ==
y .
intros ;
transitivity (
y -
z +
z );
auto .
Save .
Hint Resolve Uminus_eq_perm_right .
Lemma Uminus_plus_perm :
forall x y z ,
y <=
x ->
z <= [1-]
x -> (
x -
y ) +
z == (
x +
z ) -
y .
intros ;
apply Uminus_eq_perm_right .
transitivity (
y +
z -
y );
auto .
rewrite Uplus_minus_simpl_left ;
auto .
transitivity ([1-]
x );
auto .
rewrite Uminus_perm .
rewrite Uplus_minus_simpl_right ;
auto .
Save .
Lemma Uminus_zero_le :
forall x y ,
x -
y == 0 ->
x <=
y .
intros x y ;
unfold Uminus ;
intros .
setoid_rewrite <- (
Uinv_inv x ).
apply Uplus_one_le .
setoid_rewrite <-
Uinv_zero ;
auto .
setoid_rewrite <-
H ;
auto .
Save .
Lemma Uminus_lt_non_zero :
forall x y ,
x <
y -> ~0 ==
y -
x .
red ;
intros .
apply H ;
auto .
apply Uminus_zero_le ;
auto .
Save .
Hint Immediate Uminus_zero_le Uminus_lt_non_zero .
Lemma Ult_le_nth_minus :
forall x y ,
x <
y ->
exc (
fun n =>
x <=
y - [1/]1+n).
intros ;
apply (
archimedian (
y -
x ));
intros ;
auto .
apply exc_intro with x0 .
apply Uminus_le_perm ;
auto .
transitivity (
y -
x );
auto .
Save .
Lemma Ult_le_nth_plus :
forall x y ,
x <
y ->
exc (
fun n :
nat =>
x + [1/]1+n <=
y ).
intros .
assert (
not (0==y-x));
auto .
assert (
H1 :exc (
fun n => [1/]1+n <=
y -
x )).
apply archimedian ;
auto .
apply H1 ;
auto ;
intros n H2 .
apply exc_intro with n .
transitivity (
x + (
y -
x ));
auto .
Save .
Lemma Uminus_distr_left :
forall x y z , (
x -
y ) *
z == (
x *
z ) - (
y *
z ).
intros ;
apply (
Ule_total x y );
intros ;
auto .
rewrite (
Uminus_le_zero x y );
trivial .
rewrite (
Umult_zero_left z ).
assert (
x *
z <=
y *
z );
auto .
rewrite (
Uminus_le_zero _ _ H0 );
auto .
unfold Uminus ;
intros ;
auto .
apply Uplus_eq_simpl_right with (
y *
z );
auto .
assert ([1-] ([1-]
x +
y ) <= [1-]
y );
auto .
rewrite <- (
Udistr_plus_right _ _ z H0 );
auto .
assert (
y <= [1-] ([1-]
x ));
auto .
rewrite (
Uinv_plus_right _ _ H1 ).
rewrite (
Uinv_inv x );
auto .
Save .
Hint Resolve Uminus_distr_left .
Lemma Uminus_distr_right :
forall x y z ,
x * (
y -
z ) == (
x *
y ) - (
x *
z ).
intros ;
setoid_rewrite (
Umult_sym x y ).
setoid_rewrite (
Umult_sym x z ).
setoid_rewrite (
Umult_sym x (
y -
z ));
auto .
Save .
Hint Resolve Uminus_distr_right .
Lemma Uminus_assoc_right :
forall x y z ,
y <=
x ->
z <=
y ->
x - (
y -
z ) == (
x -
y ) +
z .
intros .
apply Uplus_eq_perm_left ;
auto .
unfold Uminus at 1;
apply Uinv_le_compat .
transitivity (1 -
y +
z );
auto .
transitivity ((
y -
z ) +
z + (
x -
y )).
rewrite (
Uminus_plus_simpl _ _ H0 ).
rewrite (
Uplus_sym y (
x -
y ));
auto .
norm_assoc_right ;
auto .
Save .
Lemma Uplus_minus_assoc_right :
forall x y z ,
y <= [1-]
x ->
z <=
y ->
x + (
y -
z ) == (
x +
y ) -
z .
intros ;
unfold Uminus .
transitivity ([1-] (
x + ([1-] (
x +
y ) +
z )) +
x ).
rewrite Uplus_assoc .
rewrite (
Uplus_sym x ([1-] (
x +
y ))).
rewrite Uinv_plus_left ;
auto .
rewrite Uinv_plus_left ;
auto .
transitivity ([1-] (
x +
y ) +
y );
auto .
Save .
Lemma Udiv_minus :
forall x y z , ~0 ==
z ->
x <=
z -> (
x -
y ) /
z ==
x /z -
y /z.
intros ;
apply (
Ule_total x y );
auto ;
intros .
transitivity (0/z);
auto .
rewrite Uminus_le_zero ;
auto .
assert (
y <=
z ).
transitivity x ;
auto .
apply Oeq_sym ;
apply Umult_div_eq ;
auto .
rewrite Uminus_distr_left .
rewrite Udiv_mult ;
auto .
rewrite Udiv_mult ;
auto .
Save .
Lemma Umult_inv_minus :
forall x y ,
x * [1-]
y ==
x -
x *
y .
intros ;
rewrite <-
Uminus_one_left .
rewrite Uminus_distr_right ;
auto .
Save .
Hint Resolve Umult_inv_minus .
Lemma Uinv_mult_minus :
forall x y , ([1-]
x ) *
y ==
y -
x *
y .
intros ;
rewrite (
Umult_sym ([1-]
x )
y );
rewrite (
Umult_sym x y );
trivial .
Save .
Hint Resolve Uinv_mult_minus .
Lemma Uminus_plus_perm_right :
forall x y z ,
y <=
x ->
y <=
z -> (
x -
y ) +
z ==
x + (
z -
y ).
intros ;
apply (
Ule_total (
x -y) ([1-]
z ));
trivial ;
intros .
apply Uminus_eq_perm_left .
transitivity z ;
auto .
rewrite <-
Uplus_minus_assoc_right ;
auto .
rewrite Uminus_assoc_right ;
auto .
rewrite Uminus_eq ;
auto .
rewrite Uplus_zero_left ;
auto .
transitivity 1;
auto .
rewrite Uplus_one ;
auto .
Save .
Hint Resolve Uminus_plus_perm_right .
Lemma Uminus_triangular :
forall x y z ,
x -
y <= (
x -
z ) + (
z -
y ).
intros ;
apply (
Ule_total x y );
intros ;
trivial .
rewrite Uminus_le_zero ;
auto .
intros ;
apply (
Ule_total x z );
intros ;
trivial .
rewrite (
Uminus_le_zero x z H0 );
rewrite Uplus_zero_left ;
auto .
intros ;
apply (
Ule_total z y );
intros ;
trivial .
rewrite (
Uminus_le_zero z y H1 );
rewrite Uplus_zero_right ;
auto .
apply Uplus_le_perm_left .
rewrite Uplus_sym .
norm_assoc_right .
rewrite Uminus_plus_simpl ;
auto .
Save .
Hint Resolve Uminus_triangular .
Definition and properties of generalized sums
Definition sigma : (
nat ->
U ) ->
nat -m>
U .
intros alpha ;
exists (
compn Uplus 0
alpha );
red ;
intros ;
auto .
induction H ;
simpl ;
auto .
transitivity (
compn Uplus 0
alpha m );
auto .
Defined .
Lemma sigma_0 :
forall (
f :
nat ->
U ),
sigma f O == 0.
trivial .
Save .
Lemma sigma_S :
forall (
f :nat ->
U ) (
n :nat),
sigma f (
S n ) = (
f n ) + (
sigma f n ).
trivial .
Save .
Lemma sigma_1 :
forall (
f :
nat ->
U ),
sigma f (
S 0) ==
f O .
intros ;
rewrite sigma_S ;
auto .
Save .
Lemma sigma_incr :
forall (
f :
nat ->
U ) (
n m :nat), (
n <=
m )%nat ->
sigma f n <=
sigma f m .
intros f n m H ;
apply (
fmonotonic (
sigma f ));
trivial .
Save .
Hint Resolve sigma_incr .
Lemma sigma_eq_compat :
forall (
f g :
nat ->
U ) (
n :nat),
(
forall k , (
k <
n )%nat ->
f k ==
g k ) ->
sigma f n ==
sigma g n .
induction n ;
auto .
intros ;
repeat rewrite sigma_S .
transitivity (
g n +
sigma f n );
auto with arith .
Save .
Lemma sigma_le_compat :
forall (
f g :
nat ->
U ) (
n :nat),
(
forall k , (
k <
n )%nat ->
f k <=
g k ) ->
sigma f n <=
sigma g n .
induction n ;
auto .
intros ;
repeat rewrite sigma_S .
transitivity (
g n +
sigma f n );
auto with arith .
Save .
Lemma sigma_S_lift :
forall (
f :nat ->
U ) (
n :nat),
sigma f (
S n ) == (
f O ) + (
sigma (
fun k =>
f (
S k ))
n ).
intros f n ;
generalize f ;
induction n ;
intros ;
auto .
rewrite sigma_S .
rewrite IHn .
rewrite sigma_S .
rewrite Uplus_assoc .
rewrite (
Uplus_sym (
f0 (
S n )) (
f0 O ));
auto .
Save .
Lemma sigma_plus_lift :
forall (
f :nat ->
U ) (
n m :nat),
sigma f (
n +m)%nat ==
sigma f n +
sigma (
fun k =>
f (
n +k)%nat)
m .
intros f n m ;
generalize f ;
clear f ;
induction n ;
intros .
simpl plus .
rewrite sigma_0 .
rewrite Uplus_zero_left .
apply sigma_eq_compat ;
auto .
rewrite sigma_S_lift .
simpl plus .
rewrite sigma_S_lift .
rewrite IHn ;
auto .
Save .
Lemma sigma_zero :
forall f n , (
forall k , (
k <n)%nat ->
f k ==0)->
sigma f n ==0.
induction n ;
intros ;
auto .
rewrite sigma_S .
rewrite (
H n );
auto .
rewrite IHn ;
auto .
Save .
Lemma sigma_not_zero :
forall f n k , (
k <n)%nat -> 0 <
f k -> 0 <
sigma f n .
induction n ;
intros ;
auto .
casetype False ;
omega .
rewrite sigma_S .
assert (
k <
n \/
k =
n )%nat.
omega .
case H1 ;
intros ;
subst ;
auto .
apply Ult_le_trans with (
sigma f n );
auto .
apply (
IHn k );
auto .
apply Ult_le_trans with (
f n );
auto .
Save .
Lemma sigma_zero_elim :
forall f n , (
sigma f n )==0->forall
k , (
k <n)%nat ->
f k ==0.
intros ;
apply Ueq_class ;
red ;
intros .
assert (0 <
sigma f n );
auto .
apply sigma_not_zero with k ;
auto .
Save .
Hint Resolve sigma_eq_compat sigma_le_compat sigma_zero .
Lemma sigma_le :
forall f n k , (
k <n)%nat ->
f k <=
sigma f n .
induction n ;
intros .
casetype False ;
omega .
rewrite sigma_S .
assert (
k <
n \/
k =
n )%nat.
omega .
case H0 ;
intros ;
subst ;
auto .
transitivity (
sigma f n );
auto .
Save .
Hint Resolve sigma_le .
Lemma sigma_minus_decr :
forall f n , (
forall k ,
f (
S k ) <=
f k ) ->
sigma (
fun k =>
f k -
f (
S k ))
n ==
f O -
f n .
intros f n fmon ;
induction n .
rewrite sigma_0 ;
auto .
rewrite sigma_S ;
rewrite IHn .
rewrite Uplus_sym .
rewrite Uplus_minus_assoc_right ;
auto .
rewrite Uminus_plus_simpl ;
auto .
elim n ;
intros ;
auto .
transitivity (
f n0 );
auto .
Save .
Lemma sigma_minus_incr :
forall f n , (
forall k ,
f k <=
f (
S k )) ->
sigma (
fun k =>
f (
S k ) -
f k )
n ==
f n -
f O .
intros f n fmon ;
induction n .
rewrite sigma_0 ;
auto .
rewrite sigma_S ;
rewrite IHn .
rewrite Uplus_minus_assoc_right ;
auto .
rewrite Uminus_plus_simpl ;
auto .
elim n ;
intros ;
auto .
transitivity (
f n0 );
auto .
Save .
Definition and properties of generalized products
Definition prod (
alpha :
nat ->
U ) (
n :nat) :=
compn Umult 1
alpha n .
Lemma prod_0 :
forall (
f :
nat ->
U ),
prod f 0 = 1.
trivial .
Save .
Lemma prod_S :
forall (
f :nat ->
U ) (
n :nat),
prod f (
S n ) = (
f n ) * (
prod f n ).
trivial .
Save .
Lemma prod_1 :
forall (
f :
nat ->
U ),
prod f (
S 0) ==
f O .
intros ;
rewrite prod_S ;
auto .
Save .
Lemma prod_S_lift :
forall (
f :nat ->
U ) (
n :nat),
prod f (
S n ) == (
f O ) * (
prod (
fun k =>
f (
S k ))
n ).
intros f n ;
generalize f ;
induction n ;
intros ;
auto .
rewrite prod_S .
rewrite IHn .
rewrite prod_S .
rewrite Umult_assoc .
rewrite (
Umult_sym (
f0 (
S n )) (
f0 O ));
auto .
Save .
Lemma prod_decr :
forall (
f :
nat ->
U ) (
n m :nat), (
n <=
m )%nat ->
prod f m <=
prod f n .
intros f n m H ;
induction H ;
auto .
intros ;
rewrite prod_S .
transitivity (
prod f m );
auto .
Save .
Hint Resolve prod_decr .
Lemma prod_eq_compat :
forall (
f g :
nat ->
U ) (
n :nat),
(
forall k , (
k <
n )%nat ->
f k ==
g k ) -> (
prod f n ) == (
prod g n ).
induction n ;
auto .
intros ;
repeat rewrite prod_S .
transitivity (
g n *
prod f n );
auto with arith .
Save .
Lemma prod_le_compat :
forall (
f g :
nat ->
U ) (
n :nat),
(
forall k , (
k <
n )%nat ->
f k <=
g k ) ->
prod f n <=
prod g n .
induction n ;
auto .
intros ;
repeat rewrite prod_S .
transitivity (
g n *
prod f n );
auto with arith .
Save .
Lemma prod_zero :
forall f n k , (
k <n)%nat ->
f k ==0 ->
prod f n ==0.
induction n ;
intros .
absurd ((
k < 0)%nat);
auto with arith .
rewrite prod_S .
assert (
k <
n \/
k =
n )%nat.
omega .
case H1 ;
intros ;
subst ;
auto .
rewrite (
IHn k );
auto .
rewrite H0 ;
auto .
Save .
Lemma prod_not_zero :
forall f n , (
forall k , (
k <n)%nat -> 0 <
f k )-> 0 <
prod f n .
induction n ;
intros ;
auto .
rewrite prod_S ;
auto with arith .
Save .
Lemma prod_zero_elim :
forall f n ,
prod f n ==0 ->
exc (
fun k => (
k <n)%nat /\
f k ==0).
intros ;
apply class_exc ;
red ;
intros .
assert (
forall k , (
k <n)%nat -> 0 <
f k );
intros .
red ;
intro .
apply H0 .
apply exc_intro with k ;
auto .
absurd (0 <
prod f n );
auto .
apply prod_not_zero ;
auto .
Save .
Hint Resolve prod_eq_compat prod_le_compat prod_not_zero .
Lemma prod_le :
forall f n k , (
k <n)%nat ->
prod f n <=
f k .
induction n ;
simpl ;
intros .
casetype False ;
omega .
rewrite prod_S .
assert (
k <
n \/
k =
n )%nat.
omega .
case H0 ;
intros ;
subst ;
auto .
transitivity (
prod f n );
auto .
Save .
Lemma prod_minus :
forall f n ,
prod f n -
prod f (
S n ) == ([1-]
f n ) *
prod f n .
intros f n ;
rewrite prod_S .
transitivity (1 *
prod f n -
f n *
prod f n ).
rewrite Umult_one_left ;
auto .
rewrite <-
Uminus_distr_left ;
auto .
Save .
Definition Prod : (
nat ->
U ) ->
nat -m->
U .
intro f ;
exists (
prod f );
red ;
intros ;
simpl ;
auto .
Defined .
Lemma Prod_simpl :
forall f n ,
Prod f n =
prod f n .
trivial .
Save .
Hint Resolve Prod_simpl .
Lemma Unth_zero : [1/]1+0 == 1.
setoid_rewrite (
Unth_prop 0);
auto .
Save .
Notation "[1/2]" := (
Unth 1).
Lemma Unth_one : [1/2] == [1-] [1/2].
transitivity ([1-] (
compn Uplus 0 (
fun _ => [1/2]) (
S O )));
auto .
apply Uinv_eq_compat ;
rewrite compS ;
rewrite comp0 ;
auto .
Save .
Hint Resolve Unth_zero Unth_one .
Lemma Unth_one_plus : [1/2] + [1/2] == 1.
transitivity ([1/2] + [1-][1/2]);
auto .
Save .
Hint Resolve Unth_one_plus .
Lemma Unth_one_refl :
forall t , [1/2] *
t + [1/2] *
t ==
t .
intro ;
rewrite <-
Udistr_plus_right ;
auto .
Save .
Lemma Unth_not_null :
forall n , ~ (0 == [1/]1+n).
red ;
intros .
apply Udiff_0_1 .
transitivity ([1/]1+n);
auto .
transitivity ([1-] (
sigma (
fun k => [1/]1+n)
n )).
apply (
Unth_prop n ).
transitivity ([1-] (
sigma (
fun k => 0)
n )).
apply Uinv_eq_compat .
apply sigma_eq_compat ;
auto .
transitivity ([1-] 0);
auto .
Save .
Hint Resolve Unth_not_null .
Lemma Unth_lt_zero :
forall n , 0 < [1/]1+n.
auto .
Save .
Hint Resolve Unth_lt_zero .
Lemma Unth_inv_lt_one :
forall n , [1-][1/]1+n<1.
intro ;
rewrite <-
Uinv_zero ;
auto .
Save .
Hint Resolve Unth_inv_lt_one .
Lemma Unth_not_one :
forall n , ~ (1 == [1-][1/]1+n).
auto .
Save .
Hint Resolve Unth_not_one .
Lemma Unth_prop_sigma :
forall n , [1/]1+n == [1-] (
sigma (
fun k => [1/]1+n)
n ).
exact Unth_prop .
Save .
Hint Resolve Unth_prop_sigma .
Lemma Unth_sigma_n :
forall n :
nat , ~ (1 ==
sigma (
fun k => [1/]1+n)
n ).
intros ;
apply Uinv_neq_simpl .
setoid_rewrite Uinv_one .
setoid_rewrite <- (
Unth_prop_sigma n );
auto .
Save .
Lemma Unth_sigma_Sn :
forall n :
nat , 1 ==
sigma (
fun k => [1/]1+n) (
S n ).
intros ;
rewrite sigma_S .
transitivity
([1-] (
sigma (
fun k => [1/]1+n)
n ) + (
sigma (
fun k => [1/]1+n)
n ));
auto .
Save .
Hint Resolve Unth_sigma_n Unth_sigma_Sn .
Lemma Unth_decr :
forall n , [1/]1+(S
n ) < [1/]1+n.
repeat red ;
intros .
apply (
Unth_sigma_n (
S n )).
apply Ole_antisym ;
auto .
transitivity (
sigma (
fun _ :
nat => [1/]1+n) (
S n ));
auto .
Save .
Hint Resolve Unth_decr .
Lemma Unth_anti_mon :
forall n m , (
n <=
m )%nat -> [1/]1+m <= [1/]1+n.
induction 1;
auto .
transitivity ([1/]1+m);
auto .
Save .
Hint Resolve Unth_anti_mon .
Lemma Unth_le_half :
forall n , [1/]1+(S
n ) <= [1/2].
auto with arith .
Save .
Hint Resolve Unth_le_half .
Definition mean (
x y :U) := [1/2] *
x + [1/2] *
y .
Lemma mean_eq :
forall x :U,
mean x x ==x.
unfold mean ;
intros .
assert (
H : ([1/2] <= [1-] ([1/2])));
auto .
rewrite <- (
Udistr_plus_right _ _ x H );
auto .
Save .
Lemma mean_le_compat_right :
forall x y z ,
y <=
z ->
mean x y <=
mean x z .
unfold mean ;
intros .
apply Uplus_le_compat_right ;
auto .
Save .
Lemma mean_le_compat_left :
forall x y z ,
x <=
y ->
mean x z <=
mean y z .
unfold mean ;
intros .
apply Uplus_le_compat_left ;
auto .
Save .
Hint Resolve mean_eq mean_le_compat_left mean_le_compat_right .
Lemma mean_lt_compat_right :
forall x y z ,
y <
z ->
mean x y <
mean x z .
unfold mean ;
intros .
apply Uplus_lt_compat_right ;
auto .
Save .
Lemma mean_lt_compat_left :
forall x y z ,
x <
y ->
mean x z <
mean y z .
unfold mean ;
intros .
apply Uplus_lt_compat_left ;
auto .
Save .
Hint Resolve mean_eq mean_le_compat_left mean_le_compat_right .
Hint Resolve mean_lt_compat_left mean_lt_compat_right .
Lemma mean_le_up :
forall x y ,
x <=
y ->
mean x y <=
y .
intros ;
transitivity (
mean y y );
auto .
Save .
Lemma mean_le_down :
forall x y ,
x <=
y ->
x <=
mean x y .
intros ;
transitivity (
mean x x );
auto .
Save .
Lemma mean_lt_up :
forall x y ,
x <
y ->
mean x y <
y .
intros ;
apply Ult_le_trans with (
mean y y );
auto .
Save .
Lemma mean_lt_down :
forall x y ,
x <
y ->
x <
mean x y .
intros ;
apply Ule_lt_trans with (
mean x x );
auto .
Save .
Hint Resolve mean_le_up mean_le_down mean_lt_up mean_lt_down .
Lemma le_half_inv :
forall x ,
x <= [1/2] ->
x <= [1-]
x .
intros ;
transitivity ([1/2]);
auto .
setoid_rewrite Unth_one ;
auto .
Save .
Hint Immediate le_half_inv .
Lemma ge_half_inv :
forall x , [1/2] <=
x -> [1-]
x <=
x .
intros ;
transitivity ([1/2]);
auto .
setoid_rewrite Unth_one ;
auto .
Save .
Hint Immediate ge_half_inv .
Lemma Uinv_le_half_left :
forall x ,
x <= [1/2] -> [1/2] <= [1-]
x .
intros ;
setoid_rewrite Unth_one ;
auto .
Save .
Lemma Uinv_le_half_right :
forall x , [1/2] <=
x -> [1-]
x <= [1/2].
intros ;
setoid_rewrite Unth_one ;
auto .
Save .
Hint Resolve Uinv_le_half_left Uinv_le_half_right .
Lemma half_twice :
forall x , (
x <= [1/2]) -> ([1/2]) * (
x +
x ) ==
x .
intros ;
assert (
H1 :
x <= [1-]
x );
auto .
rewrite (
Udistr_plus_left ([1/2])
_ _ H1 ).
exact (
mean_eq x ).
Save .
Lemma half_twice_le :
forall x , ([1/2]) * (
x +
x ) <=
x .
intros ;
apply (
Ule_total x ([1/2]));
intros ;
auto .
rewrite (
half_twice _ H );
trivial .
assert (
x +x==1);
auto .
rewrite H0 .
rewrite (
Umult_one_right ([1/2]));
auto .
Save .
Lemma Uinv_half :
forall x , ([1/2]) * ([1-]
x ) + ([1/2]) == [1-] (([1/2]) *
x ).
intros ;
setoid_rewrite (
Udistr_inv_left ([1/2])
x ).
setoid_rewrite Unth_one ;
auto .
Save .
Lemma half_esp :
forall x , ([1/2] <=
x ) -> ([1/2]) * (
x &
x ) + [1/2] ==
x .
intros ;
unfold Uesp .
setoid_rewrite (
Uinv_half ([1-]
x + [1-]
x )).
assert (
H1 :[1-]
x <= [1/2]).
setoid_rewrite Unth_one ;
auto .
rewrite (
half_twice _ H1 );
auto .
Save .
Lemma half_esp_le :
forall x ,
x <= ([1/2]) * (
x &
x ) + [1/2].
intros ;
apply (
Ule_total ([1/2])
x );
intros ;
auto .
setoid_rewrite (
half_esp _ H );
trivial .
assert (
x &
x == 0);
auto .
setoid_rewrite H0 .
setoid_rewrite (
Umult_zero_right ([1/2])).
setoid_rewrite (
Uplus_zero_left ([1/2]));
auto .
Save .
Hint Resolve half_esp_le .
Lemma half_le :
forall x y ,
y <= [1-]
y ->
x <=
y +
y -> ([1/2]) *
x <=
y .
intros .
apply not_Ult_le ;
red ;
intros .
assert (
y +
y <
x );
auto .
apply Ult_le_trans with (
mean x x );
auto .
unfold mean ;
apply Uplus_lt_compat ;
auto .
Save .
Lemma half_Unth :
forall n , ([1/2])*([1/]1+n) <= [1/]1+(S
n ).
intros ;
apply half_le ;
auto .
rewrite (
Unth_prop_sigma n ).
transitivity ([1-] (
sigma (
fun _ :
nat => [1/]1+(S
n ))
n )).
apply Uinv_le_compat .
apply sigma_le_compat ;
auto .
transitivity
([1-] (
sigma (
fun _ :
nat => [1/]1+(S
n )) (
S n )) + [1/]1+(S
n ));
auto .
rewrite sigma_S ;
auto .
Save .
Hint Resolve half_le half_Unth .
Lemma half_exp :
forall n , [1/2]^n == [1/2]^(S
n ) + [1/2]^(S
n ).
intros ;
simpl ;
symmetry ;
exact (
mean_eq ([1/2]^n)).
Save .
Definition diff (
x y :U) := (
x -
y ) + (
y -
x ).
Lemma diff_eq :
forall x ,
diff x x == 0.
unfold diff ;
intros ;
rewrite Uminus_eq ;
auto .
Save .
Hint Resolve diff_eq .
Lemma diff_sym :
forall x y ,
diff x y ==
diff y x .
unfold diff ;
intros ;
auto .
Save .
Hint Resolve diff_sym .
Lemma diff_zero :
forall x ,
diff x 0 ==
x .
unfold diff ;
intros .
transitivity (
x + 0);
auto .
Save .
Hint Resolve diff_zero .
Add Morphism diff with signature Oeq &
Oeq &
Oeq as diff_eq_compat .
intros x1 x2 eq1 x3 x4 eq2 ;
unfold diff .
apply Uplus_eq_compat ;
auto .
Qed .
Hint Immediate diff_eq_compat .
Lemma diff_plus_ok :
forall x y ,
x -
y <= [1-](
y -
x ).
intros ;
apply (
Ule_total x y );
intros ;
trivial .
rewrite (
Uminus_le_zero x y H );
auto .
rewrite (
Uminus_le_zero y x H );
auto .
Save .
Hint Resolve diff_plus_ok .
Lemma diff_Uminus :
forall x y ,
x <=
y ->
diff x y ==
y -
x .
intros ;
unfold diff ;
rewrite (
Uminus_le_zero x y );
auto .
Save .
Lemma diff_Uplus_le :
forall x y ,
x <=
diff x y +
y .
intros ;
unfold diff ;
apply (
Ule_total x y );
intros ;
auto .
transitivity y ;
auto .
Save .
Hint Resolve diff_Uplus_le .
Lemma diff_triangular :
forall x y z ,
diff x y <=
diff x z +
diff y z .
unfold diff ;
intros .
transitivity (((
x -z)+(z-y)) + ((
y -z)+(z-x))).
apply Uplus_le_compat ;
apply Uminus_triangular .
repeat norm_assoc_right ;
apply Uplus_le_compat ;
trivial .
rewrite Uplus_perm3 .
apply Uplus_le_compat ;
auto .
Save .
Hint Resolve diff_triangular .
Lemma Ule_lt_lim :
forall x y , (
forall t ,
t <
x ->
t <=
y ) ->
x <=
y .
intros ;
apply Ule_class ;
red ;
intros .
pose (
z :=
mean y x ).
assert (
y <
z );
unfold z ;
auto .
apply H1 ;
apply H ;
unfold z ;
auto .
Save .
Lemma Ule_nth_lim :
forall x y , (
forall p ,
x <=
y + [1/]1+p) ->
x <=
y .
intros ;
apply Ule_lt_lim ;
intros .
apply (
Ult_le_nth_minus H0 );
auto ;
intros n H1 .
transitivity (
x - [1/]1+n);
auto .
transitivity (
y + [1/]1+n - [1/]1+n);
auto .
Save .
Properties of least upper bounds
Lemma lub_un :
mlub (
cte nat 1) == 1.
apply lub_cte .
Save .
Hint Resolve lub_un .
Lemma UPlusk_eq :
forall k ,
UPlus k ==
mon (
Uplus k ).
intros k x ;
rewrite UPlus_simpl ;
auto .
Save .
Lemma UMultk_eq :
forall k ,
UMult k ==
mon (
Umult k ).
intros k x ;
rewrite UMult_simpl ;
auto .
Save .
Lemma UPlus_continuous_right :
forall k ,
continuous (
UPlus k ).
intros ;
rewrite UPlusk_eq ;
auto .
Save .
Hint Resolve UPlus_continuous_right .
Lemma UPlus_continuous_left :
continuous UPlus .
apply continuous_sym ;
auto .
intros ;
repeat (
rewrite UPlus_simpl );
auto .
Save .
Hint Resolve UPlus_continuous_left .
Lemma UMult_continuous_right :
forall k ,
continuous (
UMult k ).
intros ;
rewrite UMultk_eq ;
auto .
Save .
Hint Resolve UMult_continuous_right .
Lemma UMult_continuous_left :
continuous UMult .
apply continuous_sym ;
auto .
intros ;
repeat (
rewrite UMult_simpl );
auto .
Save .
Hint Resolve UMult_continuous_left .
Lemma lub_eq_plus_cte_left :
forall (
f :nat -m>
U ) (
k :U),
lub ((
UPlus k ) @
f ) ==
k +
lub f .
intros ;
symmetry ;
apply (
lub_comp_eq (
UPlus k )
f );
red ;
intros ;
auto .
Save .
Hint Resolve lub_eq_plus_cte_left .
Lemma lub_eq_mult :
forall (
k :U) (
f :nat -m>
U ),
lub ((
UMult k ) @
f ) ==
k *
lub f .
intros ;
symmetry ;
apply (
lub_comp_eq (
UMult k )
f );
trivial .
Save .
Hint Resolve lub_eq_mult .
Lemma lub_eq_plus_cte_right :
forall (
f :
nat -m>
U ) (
k :U),
lub ((
mshift UPlus k ) @
f ) ==
lub f +
k .
intros ;
transitivity (
k +
lub f );
auto .
transitivity (
lub (
UPlus k @
f ));
auto .
apply lub_eq_compat ;
intro n .
repeat (
rewrite comp_simpl ).
repeat (
rewrite mshift_simpl ).
repeat (
rewrite UPlus_simpl );
auto .
Save .
Hint Resolve lub_eq_plus_cte_right .
Lemma min_lub_le :
forall f g :
nat -m>
U ,
lub ((
Min @2
f )
g ) <=
min (
lub f ) (
lub g ).
intros ;
apply min_le .
apply lub_le .
intro ;
transitivity (
f n );
simpl ;
auto .
apply lub_le .
intro ;
transitivity (
g n );
simpl ;
auto .
Save .
Lemma min_lub_le_incr_aux :
forall f g :
nat -m>
U ,
(
forall n ,
exc (
fun m => (
n <=m)%nat /\
f n <=
g m ))
->
min (
lub f ) (
lub g ) <=
lub ((
Min @2
f )
g ).
intros ;
transitivity (
lub f );
auto .
apply lub_le ;
intros .
apply (
H n );
auto ;
intros m (
H1 ,H2).
transitivity (
min (
f m ) (
g m ));
auto .
apply min_le ;
auto .
apply (
le_lub ((
Min @2
f )
g )
m );
simpl ;
auto .
Save .
Lemma min_lub_le_incr :
forall f g :
nat -m>
U ,
min (
lub f ) (
lub g ) <=
lub ((
Min @2
f )
g ).
intros f g ;
apply (
incr_decomp f g );
auto ;
intros .
apply (
min_lub_le_incr_aux f g );
auto .
rewrite min_sym .
transitivity (
lub ((
Min @2
g )
f ));
auto .
apply (
min_lub_le_incr_aux g f );
auto .
apply lub_le_compat ;
intro n .
simpl ;
auto .
Save .
Lemma lub_eq_esp_right :
forall (
f :
nat -m>
U ) (
k :
U ),
lub ((
mshift UEsp k ) @
f ) ==
lub f &
k .
intros ;
apply Ole_antisym .
apply lub_le ;
auto .
intro n ;
simpl ;
auto .
apply Uplus_inv_le_esp .
rewrite <-
lub_eq_plus_cte_right .
apply lub_le_compat ;
simpl ;
auto .
Save .
Hint Resolve lub_eq_esp_right .
Lemma Udiv_continuous :
forall (
k :U),
continuous (
UDiv k ).
red ;
intros .
rewrite UDiv_simpl .
apply (
Ueq_orc 0
k );
auto ;
intros .
rewrite Udiv_by_zero ;
auto .
apply (
excluded_middle (
A :=forall
n ,
h n <=
k ));
auto ;
intros .
apply Umult_div_le_right ;
auto .
rewrite Umult_sym .
transitivity (
lub (
UMult k @ (
UDiv k @
h )));
auto .
apply lub_le_compat ;
intro n ;
simpl ;
auto .
rewrite Umult_div ;
auto .
assert (
exc (
fun n =>
k <=h
n )).
apply exc_intro_class ;
intros ;
apply H0 ;
intros ;
auto .
apply Ult_le ;
auto .
apply H1 ;
auto ;
intros .
transitivity (
h x /
k ).
rewrite (
Udiv_le_one (
h x )
k H );
auto .
apply le_lub with (
f :=UDiv
k @
h ) (
n :=x).
Save .
Hint Resolve Udiv_continuous .
Definition glb (
f :nat -m->
U ) := [1-](
lub (
UInv @
f )).
Lemma glb_le :
forall (
f :
nat -m->
U ) (
n :
nat ),
glb f <= (
f n ).
unfold glb ;
intros ;
apply Uinv_le_perm_left .
apply (
le_lub (
UInv @
f )
n );
auto .
Save .
Lemma le_glb :
forall (
f :
nat -m->
U ) (
x :U), (
forall n :
nat ,
x <=
f n )->
x <=
glb f .
unfold glb ;
intros ;
apply Uinv_le_perm_right .
apply (
lub_le (
UInv @
f ));
simpl ;
auto .
Save .
Hint Resolve glb_le le_glb .
Definition Uopp :
cpo (
o :=Iord
U )
U .
exists 1
glb ;
simpl ;
auto .
Defined .
Lemma Uopp_lub_simpl
:
forall h :
nat -m->
U ,
lub (
cpo :=Uopp)
h =
glb h .
trivial .
Save .
Lemma Uopp_mon_seq :
forall f :nat -m->
U ,
forall n m :nat, (
n <=
m )%nat ->
f m <=
f n .
intros f n m H ;
exact (
fmonotonic f n m H ).
Save .
Hint Resolve Uopp_mon_seq .
Infinite product
Definition prod_inf (f : nat -> U ) : U := glb (Prod f ).
Properties of glb
Lemma glb_le_compat :
forall f g :
nat -m->
U , (
forall x ,
f x <=
g x ) ->
glb f <=
glb g .
intros f g ;
exact (
lub_le_compat U (
Iord U )
Uopp g f ).
Save .
Hint Resolve glb_le_compat .
Lemma glb_eq_compat :
forall f g :
nat -m->
U ,
f ==
g ->
glb f ==
glb g .
intros ;
apply Ole_antisym ;
auto .
apply glb_le_compat ;
intro x ;
assert (
f x ==
g x );
auto .
Save .
Hint Resolve glb_eq_compat .
Lemma glb_cte :
forall c :
U ,
glb (
mon (
cte nat (
o1 :=(Iord
U ))
c )) ==
c .
intros ;
exact (
lub_cte (
c :=Uopp)
c ).
Save .
Hint Resolve glb_cte .
Lemma glb_eq_plus_cte_right :
forall (
f :
nat -m->
U ) (
k :
U ),
glb (
Imon (
mshift UPlus k ) @
f ) ==
glb f +
k .
unfold glb ;
intros .
transitivity ([1-]
lub (
mshift UEsp ([1-]
k ) @ (
UInv @
f )));
auto .
apply Uinv_eq_compat ;
apply lub_eq_compat ;
intro x .
repeat (
rewrite comp_simpl ).
unfold Imon ,
UEsp ,
UInv ,
UPlus ,
mon2 .
repeat (
rewrite mon_simpl );
repeat (
rewrite mshift_simpl );
repeat (
rewrite mon_simpl );
auto .
transitivity ([1-] (
lub (
UInv @
f ) & [1-]
k )).
apply Uinv_eq_compat ;
apply (
lub_eq_esp_right (
UInv @
f ) ([1-]
k )).
rewrite Uinv_esp_plus ;
auto .
Save .
Hint Resolve glb_eq_plus_cte_right .
Lemma glb_eq_plus_cte_left :
forall (
f :
nat -m->
U ) (
k :
U ),
glb (
Imon (
UPlus k ) @
f ) ==
k +
glb f .
intros ;
transitivity (
glb f +
k );
auto .
transitivity (
glb (
Imon (
mshift UPlus k ) @
f ));
auto .
apply glb_eq_compat ;
intro x ;
simpl .
change (
k +
f x ==
f x +
k );
auto .
Save .
Hint Resolve glb_eq_plus_cte_left .
Lemma glb_eq_mult :
forall (
k :
U ) (
f :
nat -m->
U ),
glb (
Imon (
UMult k ) @
f ) ==
k *
glb f .
unfold glb ;
intros ;
auto .
transitivity ([1-]
lub (
mshift UPlus ([1-]
k ) @ (
UMult k @ (
UInv @
f )))).
apply Uinv_eq_compat ;
apply lub_eq_compat ;
intro x ;
simpl .
change ([1-] (
k *
f x ) == (
k * [1-]
f x + [1-]
k ));
auto .
rewrite lub_eq_plus_cte_right .
rewrite (
lub_eq_mult k ).
apply Uinv_eq_perm_left ;
auto .
rewrite Udistr_inv_left ;
auto .
Save .
Lemma Imon2_plus_continuous
:
continuous2 (
c1 :=Uopp) (
c2 :=Uopp) (
c3 :=Uopp) (
imon2 Uplus ).
apply continuous2_sym ;
intros .
repeat (
rewrite imon2_simpl );
simpl ;
auto .
red ;
intros .
rewrite imon2_simpl ;
simpl .
rewrite <-
glb_eq_plus_cte_left ;
auto .
Save .
Hint Resolve Imon2_plus_continuous .
Lemma Uinv_continuous :
continuous (
c1 :=Uopp)
UInv .
red ;
intros .
rewrite UInv_simpl ;
simpl .
unfold glb .
rewrite Uinv_inv ;
auto .
Save .
Definition UInvopp :
U -m->
U .
exists Uinv .
red ;
simpl ;
intros ;
auto .
Defined .
Lemma UInvopp_simpl :
forall x ,
UInvopp x = [1-]
x .
trivial .
Save .
Lemma Uinvopp_continuous :
continuous (
c2 :=Uopp)
UInvopp .
red ;
intros .
rewrite UInvopp_simpl ;
simpl .
unfold glb ;
simpl .
apply Uinv_le_compat ;
apply lub_le_compat ;
simpl ;
auto .
Save .
Hint Resolve Uinv_continuous Uinvopp_continuous .
Instance Uminus_mon2 :
monotonic2 (
o2 :=Iord
U )
Uminus .
red ;
intros ;
auto .
Save .
Definition UMinus :
U -m>
U --m>
U :=
mon2 Uminus .
Lemma UMinus_simpl :
forall x y ,
UMinus x y =
x -
y .
trivial .
Save .
Lemma Uminus_continuous2 :
continuous2 (
c2 :=Uopp)
UMinus .
apply continuous2_eq_compat with (
fcomp2 _ _ _ _ UInv (
imon2 Uplus @
UInvopp )).
intros x y ;
simpl ;
auto .
apply (
continuous2_comp2 (
c1 :=cpoU) (
c2 :=Uopp) (
c3 :=Uopp) (
c4 :=cpoU));
auto .
apply (
continuous2_comp (
c1 :=cpoU) (
c2 :=Uopp) (
c3 :=Uopp)(c4:=Uopp));
auto .
Save .
Hint Resolve Uminus_continuous2 .
Lemma glb_le_esp :
forall f g :nat -m->
U , (
glb f ) & (
glb g ) <=
glb ((
imon2 Uesp @2
f )
g ).
intros ;
apply le_glb ;
simpl ;
auto .
Save .
Hint Resolve glb_le_esp .
Lemma Uesp_min :
forall a1 a2 b1 b2 ,
min a1 b1 &
min a2 b2 <=
min (
a1 &
a2 ) (
b1 &
b2 ).
intros ;
apply min_le .
apply Uesp_le_compat ;
auto .
apply Uesp_le_compat ;
auto .
Save .
Defining lubs of arbitrary sequences
Fixpoint seq_max (
f :nat ->
U ) (
n :nat) :
U :=
match n with
O =>
f O |
S p =>
max (
seq_max f p ) (
f (
S p ))
end .
Lemma seq_max_incr :
forall f n ,
seq_max f n <=
seq_max f (
S n ).
induction n ;
simpl ;
intros ;
auto .
Save .
Hint Resolve seq_max_incr .
Lemma seq_max_le :
forall f n ,
f n <=
seq_max f n .
induction n ;
simpl ;
intros ;
auto .
Save .
Hint Resolve seq_max_le .
Instance seq_max_mon :
forall (
f :nat ->
U ),
monotonic (
seq_max f ).
intro ;
apply nat_monotonic ;
auto .
Save .
Definition sMax (
f :nat ->
U ) :
nat -m>
U :=
mon (
seq_max f ).
Lemma sMax_mult :
forall k (
f :nat->U),
sMax (
fun n =>
k *
f n ) ==
UMult k @
sMax f .
intros ;
intro n ;
simpl .
induction n ;
simpl ;
intros ;
auto .
rewrite IHn ;
auto .
Save .
Lemma sMax_plus_cte_right :
forall k (
f :nat->
U ),
sMax (
fun n =>
f n +
k ) ==
mshift UPlus k @
sMax f .
intros ;
intro n ;
simpl ;
intros .
induction n ;
simpl ;
intros ;
auto .
rewrite IHn ;
auto .
Save .
Definition Ulub (
f :nat ->
U ) :=
lub (
sMax f ).
Lemma le_Ulub :
forall f n ,
f n <=
Ulub f .
unfold Ulub ;
intros ;
transitivity (
seq_max f n );
auto .
apply (
le_lub (
sMax f )
n ).
Save .
Lemma Ulub_le :
forall f x , (
forall n ,
f n <=
x ) ->
Ulub f <=
x .
unfold Ulub ;
intros ;
apply lub_le .
induction n ;
simpl ;
intros ;
auto .
apply max_le ;
auto .
Save .
Hint Resolve le_Ulub Ulub_le .
Lemma Ulub_le_compat :
forall f g :
nat ->U,
f <=
g ->
Ulub f <=
Ulub g .
intros ;
apply Ulub_le ;
intros ;
auto .
transitivity (
g n );
auto .
Save .
Hint Resolve Ulub_le_compat .
Add Morphism Ulub with signature Oeq &
Oeq as Ulub_eq_compat .
intros ;
apply Ole_antisym ;
auto .
Save .
Hint Resolve Ulub_eq_compat .
Lemma Ulub_eq_mult :
forall k (
f :nat->U),
Ulub (
fun n =>
k *
f n )==
k *
Ulub f .
intros ;
unfold Ulub .
rewrite sMax_mult ;
auto .
Save .
Lemma Ulub_eq_plus_cte_right :
forall (
f :nat->U)
k ,
Ulub (
fun n =>
f n +
k )==
Ulub f +
k .
intros ;
unfold Ulub .
rewrite sMax_plus_cte_right ;
auto .
Save .
Hint Resolve Ulub_eq_mult Ulub_eq_plus_cte_right .
Lemma Ulub_eq_esp_right :
forall (
f :
nat ->
U ) (
k :
U ),
Ulub (
fun n =>
f n &
k ) ==
Ulub f &
k .
intros ;
apply Ole_antisym .
apply Ulub_le ;
auto .
apply Uplus_inv_le_esp .
transitivity (
Ulub (
fun n => (
f n &
k ) + ([1-]
k )));
auto .
Save .
Hint Resolve lub_eq_esp_right .
Lemma Ulub_le_plus :
forall f g ,
Ulub (
fun n =>
f n +
g n ) <=
Ulub f +
Ulub g .
intros ;
apply Ulub_le ;
auto .
Save .
Hint Resolve Ulub_le_plus .
Definition Uglb (
f :nat ->
U ) :U := [1-]
Ulub (
fun n => [1-](
f n )).
Lemma Uglb_le :
forall (
f :
nat ->
U ) (
n :
nat ),
Uglb f <=
f n .
unfold Uglb ;
intros ;
apply Uinv_le_perm_left .
apply le_Ulub with (
f :=fun
n => [1-]
f n ) (
n :=n);
auto .
Save .
Lemma le_Uglb :
forall (
f :
nat ->
U ) (
x :U), (
forall n :
nat ,
x <=
f n )->
x <=
Uglb f .
unfold Uglb ;
intros ;
apply Uinv_le_perm_right .
apply Ulub_le with (
f :=fun
n => [1-]
f n );
auto .
Save .
Hint Resolve Uglb_le le_Uglb .
Lemma Uglb_le_compat :
forall f g :
nat ->U,
f <=
g ->
Uglb f <=
Uglb g .
intros ;
apply le_Uglb ;
intros ;
auto .
transitivity (
f n );
auto .
Save .
Hint Resolve Uglb_le_compat .
Add Morphism Uglb with signature Oeq &
Oeq as Uglb_eq_compat .
intros ;
apply Ole_antisym ;
auto .
Save .
Hint Resolve Uglb_eq_compat .
Lemma Uglb_eq_plus_cte_right :
forall (
f :
nat ->
U ) (
k :
U ),
Uglb (
fun n =>
f n +
k ) ==
Uglb f +
k .
unfold Uglb ;
intros .
transitivity ([1-]
Ulub (
fun n => ([1-]
f n ) & [1-]
k ));
auto .
transitivity ([1-] (
Ulub (
fun n => [1-]
f n ) & [1-]
k )).
apply Uinv_eq_compat ;
apply (
Ulub_eq_esp_right (
fun n => [1-]
f n ) ([1-]
k )).
transitivity ([1-]
Ulub (
fun n => [1-]
f n ) + [1-][1-]
k );
auto .
Save .
Hint Resolve Uglb_eq_plus_cte_right .
Lemma Uglb_eq_mult :
forall (
k :
U ) (
f :
nat ->
U ),
Uglb (
fun n =>
k *
f n ) ==
k *
Uglb f .
unfold Uglb ;
intros ;
auto .
transitivity ([1-]
Ulub (
fun n => (
k * [1-]
f n ) + [1-]
k )).
apply Uinv_eq_compat ;
apply Ulub_eq_compat ;
intro x ;
simpl ;
auto .
transitivity ([1-](
Ulub (
fun n =>
k * [1-]
f n ) + [1-]
k ));
auto .
transitivity ([1-](
k *
Ulub (
fun n => [1-]
f n ) + [1-]
k ));
auto .
apply Uinv_eq_perm_left ;
auto .
transitivity (
k * [1-][1-](
Ulub (
fun n => [1-]
f n )) + [1-]
k );
auto .
Save .
Hint Resolve Uglb_eq_mult Uglb_eq_plus_cte_right .
Lemma Uglb_le_plus :
forall f g ,
Uglb f +
Uglb g <=
Uglb (
fun n =>
f n +
g n ).
intros ;
apply le_Uglb ;
auto .
Save .
Hint Resolve Uglb_le_plus .
Lemma Ulub_lub :
forall f :nat -m>
U ,
Ulub f ==
lub f .
intros ;
unfold Ulub ;
apply lub_eq_compat ;
intro n ;
simpl ;
intros ;
auto .
induction n ;
simpl ;
intros ;
auto .
rewrite IHn ;
apply Ole_antisym ;
auto .
Save .
Hint Resolve Ulub_lub .
Lemma Uglb_glb :
forall f :nat -m->
U ,
Uglb f ==
glb f .
intros ;
unfold Uglb ,glb.
apply Uinv_eq_compat ;
apply (
Ulub_lub (
UInv @
f )).
Save .
Hint Resolve Uglb_glb .
Lemma lub_le_plus :
forall (
f g :
nat -m>
U ),
lub ((
UPlus @2
f )
g ) <=
lub f +
lub g .
intros ;
apply lub_le ;
simpl ;
auto .
Save .
Hint Resolve lub_le_plus .
Lemma glb_le_plus :
forall (
f g :nat -m->
U ) ,
glb f +
glb g <=
glb ((
Imon2 UPlus @2
f )
g ).
intros ;
apply le_glb ;
simpl ;
auto .
Save .
Hint Resolve glb_le_plus .
Lemma lub_eq_plus :
forall f g :
nat -m>
U ,
lub ((
UPlus @2
f )
g ) ==
lub f +
lub g .
intros ;
apply Oeq_sym .
apply @
lub_app2_eq
with (
c1 :=cpoU) (
c2 :=cpoU) (
c3 :=cpoU) (
F :=UPlus) (
f :=f) (
g :=g);
auto .
Save .
Hint Resolve lub_eq_plus .
Lemma glb_mon :
forall f :
nat -m>
U ,
Uglb f ==
f O .
intros ;
apply Ole_antisym ;
auto .
apply le_Uglb ;
auto with arith .
Save .
Lemma lub_inv :
forall (
f g :
nat -m>
U ), (
forall n ,
f n <= [1-]
g n ) ->
lub f <= [1-] (
lub g ).
intros ;
apply Uinv_le_perm_right .
apply lub_le ;
intros .
apply Uinv_le_perm_right .
rewrite (
lub_lift_left f n ).
apply lub_le ;
simpl ;
unfold seq_lift_left ;
intros .
transitivity ([1-] (
g (
n +n0)%nat));
auto with arith .
Save .
Lemma glb_lift_left :
forall (
f :nat -m->
U )
n ,
glb f ==
glb (
mon (
seq_lift_left f n )).
intros .
exact (
mlub_lift_left (
c :=Uopp)
f n ).
Save .
Hint Resolve glb_lift_left .
Lemma Ulub_mon :
forall f :
nat -m->
U ,
Ulub f ==
f O .
intros ;
apply Ole_antisym ;
auto .
apply Ulub_le ;
intros ;
auto with arith .
Save .
Lemma lub_glb_le :
forall (
f :nat -m>
U ) (
g :nat -m->
U ),
(
forall n ,
f n <=
g n ) ->
lub f <=
glb g .
intros ;
apply lub_le ;
intros .
rewrite (
glb_lift_left g n );
auto .
apply le_glb ;
simpl ;
unfold seq_lift_left ;
intros .
transitivity (
f (
n +n0))%nat;
auto with arith .
Save .
Lemma lub_lub_inv_le :
forall f g :nat -m>
U ,
(
forall n ,
f n <= [1-]
g n ) ->
lub f <= [1-]
lub g .
intros ;
transitivity (
glb (
UInvopp @
g )).
apply lub_glb_le ;
auto .
unfold glb ;
apply Uinv_le_compat .
apply lub_le_compat ;
simpl ;
auto .
Save .
Lemma Uplus_opp_continuous_right :
forall k ,
continuous (
c1 :=Uopp) (
c2 :=Uopp) (
Imon (
UPlus k )).
red ;
intros .
change (
glb (
Imon (
UPlus k ) @
h ) <=
k +
glb h ).
rewrite glb_eq_plus_cte_left ;
trivial .
Save .
Lemma Uplus_opp_continuous_left :
continuous (
c1 :=Uopp) (
c2 :=fmon_cpo (
o :=Iord
U ) (
c :=Uopp))(Imon2
UPlus ).
red ;
intros .
intro k ;
simpl .
transitivity (
glb (
Imon (
mshift UPlus k ) @
h ));
auto .
Save .
Hint Resolve Uplus_opp_continuous_right Uplus_opp_continuous_left .
Lemma glb_eq_plus :
forall (
f g :
nat -m->
U ),
glb ((
Imon2 UPlus @2
f )
g ) ==
glb f +
glb g .
intros ;
apply Oeq_sym .
apply (
lub_app2_eq (
c1 :=Uopp) (
c2 :=Uopp) (
c3 :=Uopp))
with (
F :=Imon2
UPlus ) (
f :=f) (
g :=g);
intros .
apply continuous_eq_compat with (
Imon (
UPlus k ));
auto .
apply continuous_eq_compat with (
Imon2 UPlus );
auto .
Save .
Hint Resolve glb_eq_plus .
Instance sigma_mon :monotonic
sigma .
intros f g H n .
apply sigma_le_compat ;
auto .
Save .
BUG V8.2-1
Definition Sigma : (nat -> U) -m> nat-m> U
:= mon sigma.
Anomaly: Non-functional construction. Please report.
Definition Sigma : (
nat ->
U ) -m>
nat -m>
U
:=
mon sigma (
fmonotonic :=sigma_mon).
Lemma Sigma_simpl :
forall f ,
Sigma f =
sigma f .
trivial .
Save .
Lemma sigma_continuous1 :
continuous Sigma .
red ;
intros ;
intro n .
induction n ;
auto .
transitivity (
sigma (
lub h ) (
S n ));
auto .
rewrite sigma_S .
transitivity (
lub h n +
lub (
Sigma @
h )
n );
auto .
rewrite fcpo_lub_simpl ;
repeat (
rewrite fmon_lub_simpl ).
transitivity (
lub ((
UPlus @2 (
h <o>
n )) (
mshift (
Sigma @
h )
n )));
auto .
apply lub_le_compat ;
simpl ;
auto .
Save .
Lemma sigma_lub1 :
forall (
f :
nat -m> (
nat ->
U ))
n ,
sigma (
lub f )
n ==
lub ((
mshift Sigma n ) @
f ).
intros ;
assert (
Sigma (
lub f ) ==
lub (
Sigma @
f )).
apply (
lub_comp_eq Sigma );
apply sigma_continuous1 .
transitivity (
lub (
Sigma @
f )
n );
auto .
rewrite fmon_lub_simpl ;
apply lub_eq_compat ;
intro m ;
simpl ;
auto .
Save .
Definition MF (
A :Type) :
Type :=
A ->
U .
Instance MFcpo (
A :Type) :
cpo (
MF A ).
intro ;
apply fcpo ;
apply cpoU .
Defined .
Definition MFopp (
A :Type) :
cpo (
o :=Iord (
A ->
U )) (
MF A ).
intro ;
apply (
cpo_ord_equiv (
o1 :=ford
A U (
o :=Iord
U ))).
red ;
intros f g ;
simpl ;
split ;
intros x y ;
auto .
exact (@
fcpo A U (
Iord U )
Uopp ).
Defined .
Lemma MFopp_lub_eq :
forall (
A :Type) (
h :nat-m->
MF A ),
lub (
cpo :=MFopp
A )
h ==
fun x =>
glb (
Iord_app x @
h ).
simpl ;
intros ;
intro x .
apply glb_eq_compat ;
intro n ;
auto .
Save .
Lemma fle_intro :
forall (
A :Type) (
f g :
MF A ), (
forall x ,
f x <=
g x ) ->
f <=
g .
intros ;
intro x ;
trivial .
Save .
Hint Resolve fle_intro .
Lemma feq_intro :
forall (
A :Type) (
f g :
MF A ), (
forall x ,
f x ==
g x ) ->
f ==
g .
intros ;
intro x ;
trivial .
Save .
Hint Resolve feq_intro .
Definition fplus (
A :Type) (
f g :
MF A ) :
MF A :=
fun x =>
f x +
g x .
Definition fmult (
A :Type) (
k :U) (
f :
MF A ) :
MF A :=
fun x =>
k *
f x .
Definition finv (
A :Type) (
f :
MF A ) :
MF A :=
fun x => [1-]
f x .
Definition fzero (
A :Type) :
MF A :=
fun x => 0.
Definition fdiv (
A :Type) (
k :U) (
f :
MF A ) :
MF A :=
fun x => (
f x ) /
k .
Definition flub (
A :Type) (
f :
nat -m>
MF A ) :
MF A :=
lub f .
Lemma fplus_simpl :
forall (
A :Type)(f
g :
MF A ) (
x :
A ),
fplus f g x =
f x +
g x .
trivial .
Save .
Lemma fmult_simpl :
forall (
A :Type)(k:U) (
f :
MF A ) (
x :
A ),
fmult k f x =
k *
f x .
trivial .
Save .
Lemma fdiv_simpl :
forall (
A :Type)(k:U) (
f :
MF A ) (
x :
A ),
fdiv k f x =
f x /
k .
trivial .
Save .
Implicit Arguments fzero [].
Lemma fzero_simpl :
forall (
A :Type)(x :
A ),
fzero A x = 0.
trivial .
Save .
Lemma finv_simpl :
forall (
A :Type)(f :
MF A ) (
x :
A ),
finv f x = [1-]
f x .
trivial .
Save .
Lemma flub_simpl :
forall (
A :Type)(f:nat -m>
MF A ) (
x :A),
(
flub f )
x =
lub (
f <o>
x ).
trivial .
Save .
Hint Resolve fplus_simpl fmult_simpl fzero_simpl finv_simpl flub_simpl .
Definition fone (
A :Type) :
MF A :=
fun x => 1.
Implicit Arguments fone [].
Lemma fone_simpl :
forall (
A :Type) (
x :A),
fone A x = 1.
trivial .
Save .
Definition fcte (
A :Type) (
k :U):
MF A :=
fun x =>
k .
Implicit Arguments fcte [].
Lemma fcte_simpl :
forall (
A :Type) (
k :U) (
x :A),
fcte A k x =
k .
trivial .
Save .
Definition fminus (
A :Type) (
f g :MF
A ) :
MF A :=
fun x =>
f x -
g x .
Lemma fminus_simpl :
forall (
A :Type) (
f g :
MF A ) (
x :A),
fminus f g x =
f x -
g x .
trivial .
Save .
Definition fesp (
A :Type) (
f g :MF
A ) :
MF A :=
fun x =>
f x &
g x .
Lemma fesp_simpl :
forall (
A :Type) (
f g :
MF A ) (
x :A),
fesp f g x =
f x &
g x .
trivial .
Save .
Definition fconj (
A :Type)(f
g :MF
A ) :
MF A :=
fun x =>
f x *
g x .
Lemma fconj_simpl :
forall (
A :Type) (
f g :
MF A ) (
x :A),
fconj f g x =
f x *
g x .
trivial .
Save .
Lemma MF_lub_simpl :
forall (
A :Type) (
f :
nat -m>
MF A ) (
x :A),
lub f x =
lub (
f <o>x).
intros ;
unfold MFcpo ;
auto .
Save .
Hint Resolve MF_lub_simpl .
Lemma fplus_eq_compat :
forall A (
f1 f2 g1 g2 :MF
A ),
f1 ==f2 ->
g1 ==g2 ->
fplus f1 g1 ==
fplus f2 g2 .
intros ;
intro x .
repeat (
rewrite fplus_simpl );
firstorder .
Save .
Add Parametric Morphism (
A :Type) : (@
fplus A )
with signature Oeq &
Oeq &
Oeq
as fplus_feq_compat_morph .
intros ;
exact (
fplus_eq_compat H H0 );
auto .
Save .
Instance fplus_mon2 :
forall A ,
monotonic2 (
fplus (
A :=A)).
unfold monotonic2 ;
intros ;
intro a .
repeat rewrite fplus_simpl ;
firstorder .
Save .
Hint Resolve fplus_mon2 .
Lemma fplus_le_compat :
forall A (
f1 f2 g1 g2 :MF
A ),
f1 <=f2 ->
g1 <=g2 ->
fplus f1 g1 <=
fplus f2 g2 .
intros ;
apply fplus_mon2 ;
auto .
Save .
Add Parametric Morphism A : (@
fplus A )
with signature Ole ++>
Ole ++>
Ole
as fplus_fle_compat_morph .
intros f1 f2 H g1 g2 H1 x ;
exact (
fplus_le_compat H H1 x );
auto .
Save .
Lemma finv_eq_compat :
forall A (
f g :MF
A ),
f ==g ->
finv f ==
finv g .
intros ;
intro x .
repeat (
rewrite finv_simpl );
firstorder .
Save .
Add Parametric Morphism A : (@
finv A )
with signature Oeq &
Oeq
as finv_feq_compat_morph .
intros ;
exact (
finv_eq_compat H ).
Save .
Instance finv_mon :
forall A ,
monotonic (
o2 :=Iord (
MF A )) (
finv (
A :=A)).
unfold monotonic ;
intros ;
intro a .
repeat rewrite finv_simpl ;
firstorder .
Save .
Hint Resolve finv_mon .
Lemma finv_le_compat :
forall A (
f g :MF
A ),
f <=
g ->
finv g <=
finv f .
intros ;
apply finv_mon ;
trivial .
Save .
Add Parametric Morphism A : (@
finv A )
with signature Ole -->
Ole as finv_fle_compat_morph .
intros f g H x ;
exact (
finv_le_compat H x ).
Save .
Lemma fmult_eq_compat :
forall A k1 k2 (
f1 f2 :MF
A ),
k1 ==
k2 ->
f1 ==
f2 ->
fmult k1 f1 ==
fmult k2 f2 .
intros A k1 k2 f1 f2 H H1 x .
repeat (
rewrite fmult_simpl );
firstorder .
Save .
Add Parametric Morphism A : (@
fmult A )
with signature Oeq &
Oeq &
Oeq as fmult_feq_compat_morph .
intros k1 k2 H f1 f2 H1 ;
exact (
fmult_eq_compat k1 k2 H H1 );
auto .
Save .
Instance fmult_mon2 :
forall A ,
monotonic2 (
fmult (
A :=A)).
unfold monotonic2 ;
intros ;
intro a .
repeat rewrite fmult_simpl ;
firstorder .
Save .
Hint Resolve fmult_mon2 .
Lemma fmult_le_compat :
forall A k1 k2 (
f1 f2 :MF
A ),
k1 <=
k2 ->
f1 <=
f2 ->
fmult k1 f1 <=
fmult k2 f2 .
intros ;
apply fmult_mon2 ;
auto .
Save .
Add Parametric Morphism A : (@
fmult A )
with signature Ole ++>
Ole ++>
Ole as fmult_fle_compat_morph .
intros k1 k2 H f1 f2 H1 x ;
exact (
fmult_le_compat k1 k2 H H1 x );
auto .
Save .
Lemma fminus_eq_compat :
forall A (
f1 f2 g1 g2 :MF
A ),
f1 ==
f2 ->
g1 ==
g2 ->
fminus f1 g1 ==
fminus f2 g2 .
intros A f1 f2 g1 g2 H H1 x .
repeat (
rewrite fminus_simpl );
firstorder .
Save .
Add Parametric Morphism A : (@
fminus A )
with signature Oeq &
Oeq &
Oeq as fminus_feq_compat_morph .
intros f1 f2 H g1 g2 H1 x ;
exact (
fminus_eq_compat H H1 x );
auto .
Save .
Instance fminus_mon2 :
forall A ,
monotonic2 (
o2 :=Iord (
MF A )) (
fminus (
A :=A)).
unfold monotonic2 ;
intros ;
intro a .
repeat rewrite fminus_simpl ;
firstorder .
Save .
Hint Resolve fminus_mon2 .
Lemma fminus_le_compat :
forall A (
f1 f2 g1 g2 :MF
A ),
f1 <=
f2 ->
g2 <=
g1 ->
fminus f1 g1 <=
fminus f2 g2 .
intros ;
apply fminus_mon2 ;
auto .
Save .
Add Parametric Morphism A : (@
fminus A )
with signature Ole ++>
Ole -->
Ole as fminus_fle_compat_morph .
intros f1 f2 H g1 g2 H1 x ;
exact (
fminus_le_compat H H1 x );
auto .
Save .
Lemma fesp_eq_compat :
forall A (
f1 f2 g1 g2 :MF
A ),
f1 ==f2 ->
g1 ==g2 ->
fesp f1 g1 ==
fesp f2 g2 .
intros ;
intro x .
repeat (
rewrite fesp_simpl );
firstorder .
Save .
Add Parametric Morphism A : (@
fesp A )
with signature Oeq &
Oeq &
Oeq as fesp_feq_compat_morph .
intros ;
exact (
fesp_eq_compat H H0 );
auto .
Save .
Instance fesp_mon2 :
forall A ,
monotonic2 (
fesp (
A :=A)).
unfold monotonic2 ;
intros ;
intro a .
repeat rewrite fesp_simpl ;
firstorder .
Save .
Hint Resolve fesp_mon2 .
Lemma fesp_le_compat :
forall A (
f1 f2 g1 g2 :MF
A ),
f1 <=f2 ->
g1 <=g2 ->
fesp f1 g1 <=
fesp f2 g2 .
intros ;
apply fesp_mon2 ;
auto .
Save .
Add Parametric Morphism A : (@
fesp A )
with signature Ole ++>
Ole ++>
Ole as fesp_fle_compat_morph .
intros f1 f2 H g1 g2 H1 x ;
exact (
fesp_le_compat H H1 x );
auto .
Save .
Lemma fconj_eq_compat :
forall A (
f1 f2 g1 g2 :MF
A ),
f1 ==f2 ->
g1 ==g2 ->
fconj f1 g1 ==
fconj f2 g2 .
intros ;
intro x .
repeat (
rewrite fconj_simpl );
firstorder .
Save .
Add Parametric Morphism A : (@
fconj A )
with signature Oeq &
Oeq &
Oeq
as fconj_feq_compat_morph .
intros ;
exact (
fconj_eq_compat H H0 );
auto .
Save .
Instance fconj_mon2 :
forall A ,
monotonic2 (
fconj (
A :=A)).
unfold monotonic2 ;
intros ;
intro a .
repeat rewrite fconj_simpl .
apply Umult_le_compat ;
trivial .
Save .
Hint Resolve fconj_mon2 .
Lemma fconj_le_compat :
forall A (
f1 f2 g1 g2 :MF
A ),
f1 <=
f2 ->
g1 <=
g2 ->
fconj f1 g1 <=
fconj f2 g2 .
intros ;
apply fconj_mon2 ;
auto .
Save .
Add Parametric Morphism A : (@
fconj A )
with signature Ole ++>
Ole ++>
Ole
as fconj_fle_compat_morph .
intros .
exact (
fconj_le_compat H H0 );
auto .
Save .
Hint Immediate fplus_le_compat fplus_eq_compat fesp_le_compat fesp_eq_compat
fmult_le_compat fmult_eq_compat fminus_le_compat fminus_eq_compat
fconj_eq_compat .
Hint Resolve finv_eq_compat .
Lemma fle_fplus_left :
forall (
A :Type) (
f g :
MF A ),
f <=
fplus f g .
intros m f g x ;
rewrite fplus_simpl ;
auto .
Save .
Lemma fle_fplus_right :
forall (
A :Type) (
f g :
MF A ),
g <=
fplus f g .
intros m f g x ;
rewrite fplus_simpl ;
auto .
Save .
Lemma fle_fmult :
forall (
A :Type) (
k :U)(f :
MF A ),
fmult k f <=
f .
intros m k f x ;
rewrite fmult_simpl ;
auto .
Save .
Lemma fle_zero :
forall (
A :Type) (
f :
MF A ),
fzero A <=
f .
intros m f x ;
rewrite fzero_simpl ;
auto .
Save .
Lemma fle_one :
forall (
A :Type) (
f :
MF A ),
f <=
fone A .
intros m f x ;
rewrite fone_simpl ;
auto .
Save .
Lemma feq_finv_finv :
forall (
A :Type) (
f :
MF A ),
finv (
finv f ) ==
f .
intros m f x ;
rewrite finv_simpl ;
auto .
Save .
Lemma fle_fesp_left :
forall (
A :Type) (
f g :
MF A ),
fesp f g <=
f .
intros m f g x ;
rewrite fesp_simpl ;
auto .
Save .
Lemma fle_fesp_right :
forall (
A :Type) (
f g :
MF A ),
fesp f g <=
g .
intros m f g x ;
rewrite fesp_simpl ;
auto .
Save .
Lemma fle_fconj_left :
forall (
A :Type) (
f g :
MF A ),
fconj f g <=
f .
intros m f g x ;
rewrite fconj_simpl ;
auto .
Save .
Lemma fle_fconj_right :
forall (
A :Type) (
f g :
MF A ),
fconj f g <=
g .
intros m f g x ;
rewrite fconj_simpl ;
auto .
Save .
Lemma fconj_decomp :
forall A (
f g :
MF A ),
f ==
fplus (
fconj f g ) (
fconj f (
finv g )).
intros ;
simpl ;
intros x ;
apply Umult_decomp .
Save .
Hint Resolve fconj_decomp .
Compatibility of addition of two functions
Definition fplusok (
A :Type) (
f g :
MF A ) :=
f <=
finv g .
Hint Unfold fplusok .
Lemma fplusok_sym :
forall (
A :Type) (
f g :
MF A ) ,
fplusok f g ->
fplusok g f .
unfold fplusok ,
finv ;
intros ;
intro ;
auto .
Save .
Hint Immediate fplusok_sym .
Lemma fplusok_inv :
forall (
A :Type) (
f :
MF A ) ,
fplusok f (
finv f ).
intros ;
apply fplusok_sym ;
unfold fplusok ;
auto .
Save .
Hint Resolve fplusok_inv .
Lemma fplusok_le_compat :
forall (
A :Type)(f1
f2 g1 g2 :MF
A ),
fplusok f2 g2 ->
f1 <=
f2 ->
g1 <=
g2 ->
fplusok f1 g1 .
unfold fplusok ;
intros .
transitivity f2 ;
trivial .
transitivity (
finv g2 );
trivial .
apply finv_le_compat ;
auto .
Save .
Hint Resolve fle_fplus_left fle_fplus_right fle_zero fle_one feq_finv_finv finv_le_compat
fle_fmult fle_fesp_left fle_fesp_right fle_fconj_left fle_fconj_right .
Lemma fconj_fplusok :
forall (
A :Type)(f
g h :MF
A ),
fplusok g h ->
fplusok (
fconj f g ) (
fconj f h ).
intros ;
apply fplusok_le_compat with g h ;
auto .
Save .
Hint Resolve fconj_fplusok .
Definition Fconj A :
MF A -m>
MF A -m>
MF A :=
mon2 (
fconj (
A :=A)).
Lemma Fconj_simpl :
forall A f g ,
Fconj A f g =
fconj f g .
trivial .
Save .
Lemma fconj_sym :
forall A (
f g :
MF A ),
fconj f g ==
fconj g f .
intros ;
intro x ;
unfold fconj ;
auto .
Save .
Lemma lub_MF_simpl :
forall A (
h :
nat -m>
MF A ) (
x :A),
lub h x =
lub (
h <o>
x ).
intros ;
exact (
fcpo_lub_simpl h x ).
Save .
Instance fconj_continuous2 A :
continuous2 (
Fconj A ).
intros ;
apply continuous2_sym .
intros ;
repeat rewrite Fconj_simpl .
apply fconj_sym .
red ;
intros .
rewrite Fconj_simpl .
intro x ;
unfold fconj .
repeat rewrite lub_MF_simpl .
rewrite (
Umult_right_continuous (
k x ) (
h <o>
x )).
apply lub_le_compat ;
simpl ;
auto .
Save .
Definition Fmult A :
U -m>
MF A -m>
MF A :=
mon2 (
fmult (
A :=A)).
Lemma Fmult_simpl :
forall A k f ,
Fmult A k f =
fmult k f .
trivial .
Save .
Lemma fmult_continuous2 :
forall A ,
continuous2 (
Fmult A ).
red ;
intros .
rewrite Fmult_simpl .
intro x ;
unfold fmult .
repeat rewrite lub_MF_simpl .
rewrite (
Umult_continuous2 f (
g <o>
x )).
apply lub_le_compat ;
intro y ;
simpl ;
auto .
Save .
Fixpoints of functions of type
Section FixDef .
Variable A :Type.
Variable F :
MF A -m>
MF A .
Definition mufix :
MF A :=
fixp F .
Definition G :
MF A --m->
MF A :=
Imon F .
Definition nufix :
MF A :=
fixp (
c :=MFopp
A )
G .
Lemma mufix_inv :
forall f :
MF A ,
F f <=
f ->
mufix <=
f .
unfold mufix ;
intros ;
apply fixp_inv ;
auto .
Save .
Hint Resolve mufix_inv .
Lemma nufix_inv :
forall f :MF
A ,
f <=
F f ->
f <=
nufix .
unfold nufix ;
intros .
change (
Ole (
ord :=Iord (
MF A )) (
fixp (
c :=MFopp
A )
G )
f );
apply (
fixp_inv (
c :=MFopp
A ));
auto .
Save .
Hint Resolve nufix_inv .
Lemma mufix_le :
mufix <=
F mufix .
unfold mufix ;
auto .
Save .
Hint Resolve mufix_le .
Lemma nufix_sup :
F nufix <=
nufix .
unfold nufix .
change (
Ole (
ord :=Iord (
MF A )) (
fixp (
c :=MFopp
A )
G ) (
G (
fixp (
c :=MFopp
A )
G )));
auto .
Save .
Hint Resolve nufix_sup .
Lemma mufix_eq :
continuous F ->
mufix ==
F mufix .
intros ;
unfold mufix ;
apply fixp_eq ;
auto .
Save .
Hint Resolve mufix_eq .
Lemma nufix_eq :
continuous (
c1 :=MFopp
A ) (
c2 :=MFopp
A )
G ->
nufix ==
F nufix .
intros ;
unfold nufix .
assert (
Oeq (
ord :=Iord (
MF A )) (
fixp (
c :=MFopp
A )
G ) (
G (
fixp (
c :=MFopp
A )
G )));
auto .
apply fixp_eq ;
auto .
Save .
Hint Resolve nufix_eq .
End FixDef .
Hint Resolve mufix_le mufix_eq nufix_sup nufix_eq .
Definition Fcte (
A :Type) (
f :MF
A ) :
MF A -m>
MF A :=
mon (
cte (
MF A )
f ).
Lemma mufix_cte :
forall (
A :Type) (
f :MF
A ),
mufix (
Fcte f ) ==
f .
intros A f ;
exact (
fixp_cte f ).
Save .
Lemma nufix_cte :
forall (
A :Type) (
f :MF
A ),
nufix (
Fcte f ) ==
f .
intros A f .
change (@
Oeq (
MF A ) (@
Iord (
MF A ) (@
ford A U ordU )) (@
nufix A (@
Fcte A f ))
f ).
unfold nufix ,
Fcte .
transitivity (
fixp (
c :=MFopp
A ) (
mon (
o1 :=Iord (
MF A )) (
o2 :=Iord (
MF A )) (
cte (
MF A )
f ))).
apply fixp_eq_compat .
intros g x ;
simpl ;
auto .
exact (
fixp_cte (
o :=Iord (
MF A )) (
c :=MFopp
A )
f ).
Save .
Hint Resolve mufix_cte nufix_cte .
Properties of (pseudo-)barycenter of two points
Lemma Uinv_bary :
forall a b x y :
U ,
a <= [1-]
b ->
[1-] (
a *
x +
b *
y ) ==
a * [1-]
x +
b * [1-]
y + [1-] (
a +
b ).
intros a b x y sum_le_one .
apply Uplus_eq_simpl_left with (
a *
x );
auto .
apply Uinv_le_perm_right .
setoid_rewrite (
Udistr_inv_left a x ).
repeat norm_assoc_right .
apply Uplus_le_compat_right .
transitivity (
b + [1-] (
a +
b ));
auto .
transitivity ([1-] (
a +
b ) +
b );
auto .
transitivity ([1-] (
b *
y )).
transitivity
([1-] (
a *
x +
b *
y ) +
a *
x );
auto .
setoid_rewrite (
Udistr_inv_left b y );
auto .
transitivity
((
a *
x +
a * [1-]
x ) +
b * [1-]
y + [1-] (
a +
b )).
assert (
x <= ([1-] ([1-]
x )));
auto .
rewrite <- (
Udistr_plus_left a _ _ H );
auto .
rewrite (
Uinv_opp_right x ).
rewrite (
Umult_one_right a ).
transitivity (
b * [1-]
y + ([1-] (
a +
b ) +
a )).
assert (
b <= [1-]
a );
auto .
rewrite (
Uinv_plus_left _ _ H0 );
auto .
rewrite (
Uplus_sym a (
b * [1-]
y ));
auto .
transitivity
(
b * [1-]
y + (
a + [1-] (
a +
b )));
auto .
transitivity
(((
a *
x +
a * [1-]
x ) + (
b * [1-]
y + [1-] (
a +
b ))));
auto .
transitivity
(((
a *
x + (
a * [1-]
x + (
b * [1-]
y + [1-] (
a +
b ))))));
auto .
Save .
Hint Resolve Uinv_bary .
Lemma Uinv_bary_le :
forall a b x y :
U ,
a <= [1-]
b ->
a * [1-]
x +
b * [1-]
y <= [1-] (
a *
x +
b *
y ).
intros ;
rewrite Uinv_bary ;
auto .
Save .
Hint Resolve Uinv_bary_le .
Lemma Uinv_bary_eq :
forall a b x y :
U ,
a == [1-]
b ->
[1-] (
a *
x +
b *
y ) ==
a * [1-]
x +
b * [1-]
y .
intros ;
rewrite H ;
rewrite Uinv_bary ;
auto .
rewrite Uinv_opp_left .
rewrite Uinv_one ;
auto .
Save .
Hint Resolve Uinv_bary_eq .
Lemma bary_refl_eq :
forall a b x ,
a == [1-]
b ->
a *
x +
b *
x ==
x .
intros ;
rewrite H ;
auto .
Save .
Hint Resolve bary_refl_eq .
Lemma bary_le_left :
forall a b x y , [1-]
b <=
a ->
x <=
y ->
x <=
a *
x +
b *
y .
intros ;
transitivity ([1-]
b *
x +
b *
x );
auto .
Save .
Lemma bary_le_right :
forall a b x y ,
a <= [1-]
b ->
x <=
y ->
a *
x +
b *
y <=
y .
intros ;
transitivity ([1-]
b *
y +
b *
y );
auto .
Save .
Hint Resolve bary_le_left bary_le_right .
Lemma bary_up_eq :
forall a b x y :
U ,
a == [1-]
b ->
x <=
y ->
a *
x +
b *
y ==
x +
b * (
y -
x ).
intros .
rewrite H ;
rewrite Uinv_mult_minus .
rewrite Uminus_plus_perm_right ;
auto .
Save .
Lemma bary_up_le :
forall a b x y :
U ,
a <= [1-]
b ->
a *
x +
b *
y <=
x +
b * (
y -
x ).
intros ;
transitivity (([1-]
b ) *
x +
b *
y );
auto .
apply (
Ule_total y x );
intros ;
trivial .
intros ;
rewrite Uminus_le_zero ;
trivial .
rewrite Umult_zero_right .
rewrite Uplus_zero_right .
transitivity (([1-]
b ) *
x +
b *
x );
auto .
rewrite bary_up_eq ;
trivial .
Save .
Lemma bary_Uminus_left :
forall a b x y :
U ,
a <= [1-]
b -> (
a *
x +
b *
y ) -
x <=
b * (
y -
x ).
intros ;
apply Uplus_le_perm_left .
apply bary_up_le ;
trivial .
Save .
Lemma bary_Uminus_left_eq :
forall a b x y :
U ,
a == [1-]
b ->
x <=
y -> (
a *
x +
b *
y ) -
x ==
b * (
y -
x ).
intros ;
apply Uplus_eq_perm_left .
apply Uinv_le_perm_right .
transitivity (
y -
x );
auto .
apply bary_up_eq ;
trivial .
Save .
Lemma Uminus_bary_left
:
forall a b x y :
U , [1-]
a <=
b ->
x - (
a *
x +
b *
y ) <=
b * (
x -
y ).
intros ;
rewrite <-
Uminus_assoc_left .
transitivity (([1-]
a ) *
x -
b *
y );
auto .
rewrite Uminus_distr_right ;
auto .
Save .
Lemma Uminus_bary_left_eq
:
forall a b x y :
U ,
a == [1-]
b ->
y <=
x ->
x - (
a *
x +
b *
y ) ==
b * (
x -
y ).
intros ;
rewrite H ;
rewrite <-
Uminus_assoc_left .
transitivity (
b *
x -
b *
y );
auto .
apply Uminus_eq_compat ;
auto .
Save .
Hint Resolve bary_up_eq bary_up_le bary_Uminus_left Uminus_bary_left bary_Uminus_left_eq Uminus_bary_left_eq .
Lemma diff_bary_left_eq
:
forall a b x y :
U ,
a == [1-]
b ->
diff x (
a *
x +
b *
y ) ==
b *
diff x y .
intros ;
unfold diff .
apply (
Ule_total x y );
trivial ;
intros .
assert (
x <=
a *
x +
b *
y );
auto .
rewrite (
Uminus_le_zero _ _ H0 );
rewrite (
Uminus_le_zero _ _ H1 );
auto .
repeat rewrite Uplus_zero_left ;
auto .
assert (
a *
x +
b *
y <=
x ).
assert (
b == [1-]
a ).
apply Uinv_eq_perm_right ;
auto .
rewrite (
Uplus_sym (
a *
x ));
auto .
rewrite (
Uminus_le_zero _ _ H0 );
rewrite (
Uminus_le_zero _ _ H1 );
auto .
repeat rewrite Uplus_zero_right ;
auto .
Save .
Hint Resolve diff_bary_left_eq .
Lemma Uinv_half_bary :
forall x y :
U , [1-] ([1/2] *
x + [1/2] *
y ) == [1/2] * [1-]
x + [1/2] * [1-]
y .
intros ;
rewrite Uinv_bary ;
auto .
rewrite Unth_one_plus ;
rewrite Uinv_one ;
auto .
Save .
Hint Resolve Uinv_half_bary .
Properties of generalized sums sigma
Lemma sigma_plus :
forall (
f g :
nat ->
U ) (
n :nat),
sigma (
fun k => (
f k ) + (
g k ))
n ==
sigma f n +
sigma g n .
intros ;
induction n ;
simpl ;
auto .
repeat rewrite sigma_S ;
setoid_rewrite IHn .
repeat norm_assoc_right ;
apply Uplus_eq_compat_right .
setoid_rewrite (
Uplus_sym (
g n ) ((
sigma f n ) + (
sigma g n ))).
repeat norm_assoc_right ;
apply Uplus_eq_compat_right ;
auto .
Save .
Definition retract (
f :
nat ->
U ) (
n :
nat ) :=
forall k , (
k <
n )%nat ->
f k <= [1-] (
sigma f k ).
Lemma retract0 :
forall (
f :
nat ->
U ),
retract f 0.
red ;
intros ;
absurd (
k <
O )%nat;
auto with arith .
Save .
Lemma retract_pred :
forall (
f :
nat ->
U ) (
n :
nat ),
retract f (
S n ) ->
retract f n .
unfold retract ;
auto with arith .
Save .
Lemma retractS :
forall (
f :
nat ->
U ) (
n :
nat ),
retract f (
S n ) ->
f n <= [1-] (
sigma f n ).
unfold retract ;
auto with arith .
Save .
Hint Immediate retract_pred retractS .
Lemma retractS_inv :
forall (
f :
nat ->
U ) (
n :
nat ),
retract f (
S n ) ->
sigma f n <= [1-]
f n .
intros ;
apply Uinv_le_perm_right ;
auto .
Save .
Hint Immediate retractS_inv .
Lemma retractS_intro :
forall (
f :
nat ->
U ) (
n :
nat ),
retract f n ->
f n <= [1-] (
sigma f n ) ->
retract f (
S n ).
unfold retract ;
intros .
assert ((
k <n)%nat \/
k =n);
try omega ;
intuition ;
subst ;
auto .
Save .
Hint Resolve retract0 retractS_intro .
Lemma retract_lt :
forall (
f :
nat ->
U ) (
n :
nat ),
sigma f n < 1 ->
retract f n .
induction n ;
auto .
rewrite sigma_S .
intros ;
assert ((
sigma f n )<1).
apply Ule_lt_trans with (
f n +
sigma f n );
auto .
assert (
f n <= [1-](
sigma f n ));
auto .
apply Uplus_lt_Uinv ;
auto .
Save .
Lemma retract_unif :
forall (
f :
nat ->
U ) (
n :
nat ),
(
forall k , (
k <=n)%nat ->
f k <= [1/]1+n) ->
retract f (
S n ).
red ;
intros .
transitivity ([1/]1+n);
auto with arith .
apply Uinv_le_perm_right .
transitivity (
sigma (
fun k => [1/]1+n)
n );
auto .
transitivity (
sigma f n );
auto with arith .
apply Uinv_le_perm_right ;
auto .
Save .
Hint Resolve retract_unif .
Lemma sigma_mult :
forall (
f :
nat ->
U )
n c ,
retract f n ->
sigma (
fun k =>
c * (
f k ))
n ==
c * (
sigma f n ).
intros ;
induction n ;
simpl ;
auto .
repeat rewrite sigma_S .
assert (
H1 :
retract f n );
auto .
rewrite (
IHn H1 ).
rewrite (
Udistr_plus_left c _ _ (
retractS H ));
auto .
Save .
Hint Resolve sigma_mult .
Lemma sigma_prod_maj :
forall (
f g :
nat ->
U )
n ,
sigma (
fun k => (
f k ) * (
g k ))
n <=
sigma f n .
auto .
Save .
Hint Resolve sigma_prod_maj .
Lemma sigma_prod_le :
forall (
f g :
nat ->
U ) (
c :U), (
forall k , (
f k ) <=
c )
->
forall n ,
retract g n ->
sigma (
fun k => (
f k ) * (
g k ))
n <=
c * (
sigma g n ).
induction n ;
simpl ;
intros ;
auto .
repeat rewrite sigma_S .
transitivity ((
f n ) * (
g n ) + (
c *
sigma g n ));
auto .
transitivity (
c * (
g n ) + (
c *
sigma g n ));
auto .
setoid_rewrite (
Udistr_plus_left c _ _ (
retractS H0 ));
auto .
Save .
Lemma sigma_prod_ge :
forall (
f g :
nat ->
U ) (
c :U), (
forall k ,
c <= (
f k ))
->
forall n , (
retract g n ) ->
c * (
sigma g n ) <= (
sigma (
fun k => (
f k ) * (
g k ))
n ).
induction n ;
simpl ;
intros ;
auto .
repeat rewrite sigma_S .
rewrite (
Udistr_plus_left c _ _ (
retractS H0 ));
auto .
Save .
Hint Resolve sigma_prod_maj sigma_prod_le sigma_prod_ge .
Lemma sigma_inv :
forall (
f g :
nat ->
U ) (
n :nat), (
retract f n ) ->
[1-] (
sigma (
fun k =>
f k *
g k )
n ) == (
sigma (
fun k =>
f k * [1-] (
g k ))
n ) + [1-] (
sigma f n ).
intros ;
induction n ;
repeat rewrite sigma_S ;
auto .
apply Uplus_eq_simpl_right with ((
f n ) * (
g n )).
rewrite
(
Uinv_inv (
f n *
g n +
sigma (
fun k :
nat =>
f k *
g k )
n ));
auto .
apply Uinv_le_perm_right .
rewrite (
Udistr_inv_left (
f n ) (
g n )).
repeat norm_assoc_right ;
apply Uplus_le_compat_right .
transitivity
(
sigma f n + [1-] (
f n +
sigma f n ));
auto .
assert (
sigma f n <= [1-] (
f n )).
apply Uinv_le_perm_right ;
auto .
rewrite <- (
Uinv_plus_right _ _ H0 );
auto .
assert (
sigma (
fun k :
nat =>
f k *
g k )
n <= [1-] (
f n *
g n )).
transitivity (
sigma f n );
auto .
transitivity ([1-] (
f n ));
auto .
rewrite (
Uinv_plus_left _ _ H0 ).
transitivity (
sigma (
fun k :
nat =>
f k * [1-]
g k )
n + [1-]
sigma f n ).
auto .
rewrite (
Uplus_sym (
f n * [1-] (
g n ))
(
sigma (
fun k :
nat =>
f k * [1-] (
g k ))
n )).
repeat norm_assoc_right ;
apply Uplus_eq_compat_right .
rewrite (
Uplus_sym ([1-] (
f n +
sigma f n )) (
f n *
g n )).
repeat norm_assoc_left .
assert ([1-] (
g n ) <= [1-] (
g n ));
auto .
rewrite <- (
Udistr_plus_left (
f n )
_ _ H1 ).
rewrite (
Uinv_opp_left (
g n )).
rewrite (
Umult_one_right (
f n ));
auto .
rewrite (
Uplus_sym (
f n ) ([1-] (
f n +
sigma f n ))).
apply Oeq_sym ;
apply Uinv_plus_left ;
auto .
Save .
Definition of Nmult n x n */ x
Fixpoint Nmult (
n :
nat ) (
x :
U ) {
struct n } :
U :=
match n with O => 0 | (
S O ) =>
x |
S p =>
x + (
Nmult p x )
end .
Condition for n */ x
Definition Nmult_def (
n :
nat ) (
x :
U ) :=
match n with O =>
True |
S p =>
x <= [1/]1+p
end .
Lemma Nmult_def_O :
forall x ,
Nmult_def O x .
simpl ;
auto .
Save .
Hint Resolve Nmult_def_O .
Lemma Nmult_def_1 :
forall x ,
Nmult_def (
S O )
x .
simpl ;
intro ;
rewrite Unth_zero ;
auto .
Save .
Hint Resolve Nmult_def_1 .
Lemma Nmult_def_intro :
forall n x ,
x <= [1/]1+n ->
Nmult_def (
S n )
x .
destruct n ;
simpl ;
auto .
Save .
Hint Resolve Nmult_def_intro .
Lemma Nmult_def_Unth :
forall n ,
Nmult_def (
S n ) ([1/]1+n).
auto .
Save .
Hint Resolve Nmult_def_Unth .
Lemma Nmult_def_pred :
forall n x ,
Nmult_def (
S n )
x ->
Nmult_def n x .
intros n x ;
case n ;
simpl ;
intros ;
auto .
transitivity ([1/]1+(S
n0 ));
auto .
Save .
Hint Immediate Nmult_def_pred .
Lemma Nmult_defS :
forall n x ,
Nmult_def (
S n )
x ->
x <= [1/]1+n.
destruct n ;
simpl ;
intros ;
auto .
Save .
Hint Immediate Nmult_defS .
Lemma Nmult_def_class :
forall n p ,
class (
Nmult_def n p ).
unfold class ;
destruct n ;
intuition .
Save .
Hint Resolve Nmult_def_class .
Infix "*/" :=
Nmult (
at level 60) :
U_scope .
Add Morphism Nmult_def with signature eq &
Oeq &
iff as Nmult_def_eq_compat .
unfold Nmult_def ;
intro n ;
destruct n ;
intuition .
transitivity x ;
auto .
transitivity y0 ;
auto .
Save .
Lemma Nmult_def_zero :
forall n ,
Nmult_def n 0.
destruct n ;
auto .
Save .
Hint Resolve Nmult_def_zero .
Lemma Nmult_0 :
forall (
x :U),
O */x = 0.
trivial .
Save .
Lemma Nmult_1 :
forall (
x :U), (
S O )*/x =
x .
trivial .
Save .
Lemma Nmult_zero :
forall n ,
n */ 0 == 0.
induction n ;
simpl ;
auto .
destruct n ;
auto .
Save .
Lemma Nmult_SS :
forall (
n :nat) (
x :U),
S (
S n ) */x =
x + (
S n */
x ).
destruct n ;
simpl ;
auto .
Save .
Lemma Nmult_2 :
forall (
x :U), 2*/x =
x +
x .
trivial .
Save .
Lemma Nmult_S :
forall (
n :nat) (
x :U),
S n */
x ==
x + (
n */x).
destruct n ;
intros ;
simpl ;
auto .
Save .
Hint Resolve Nmult_1 Nmult_SS Nmult_2 Nmult_S .
Add Morphism Nmult with signature eq &
Oeq &
Oeq as Nmult_eq_compat .
intros n x1 x2 eq1 ;
induction n ;
simpl ;
intros ;
auto .
destruct n ;
repeat rewrite Nmult_SS ;
trivial .
apply Uplus_eq_compat ;
auto .
Save .
Hint Immediate Nmult_eq_compat .
Lemma Nmult_eq_compat_left :
forall (
n :nat) (
x y :U),
x ==
y ->
n */
x ==
n */
y .
auto .
Save .
Lemma Nmult_eq_compat_right :
forall (
n m :nat) (
x :U), (
n =
m )%nat ->
n */
x ==
m */
x .
auto .
Save .
Hint Resolve Nmult_eq_compat_right .
Lemma Nmult_le_compat_right :
forall n x y ,
x <=
y ->
n */
x <=
n */
y .
intros ;
induction n ;
auto .
rewrite (
Nmult_S n x );
rewrite (
Nmult_S n y );
auto .
Save .
Lemma Nmult_le_compat_left :
forall n m x , (
n <=
m )%nat ->
n */
x <=
m */
x .
induction 1;
trivial .
rewrite (
Nmult_S m x );
auto .
transitivity (
m */
x );
auto .
Save .
Hint Resolve Nmult_eq_compat_right Nmult_le_compat_right Nmult_le_compat_left .
Lemma Nmult_le_compat :
forall (
n m :nat)
x y ,
n <=
m ->
x <=
y ->
n */
x <=
m */
y .
intros ;
transitivity (
n */y);
auto .
Save .
Hint Immediate Nmult_le_compat .
Instance Nmult_mon2 :
monotonic2 Nmult .
red ;
intros ;
apply Nmult_le_compat ;
auto .
Save .
Definition NMult :
nat -m>
U -m>
U :=mon2
Nmult .
Lemma Nmult_sigma :
forall (
n :nat) (
x :U),
n */
x ==
sigma (
fun k =>
x )
n .
intros n x ;
induction n ;
simpl ;
intros ;
auto .
destruct n ;
auto .
Save .
Hint Resolve Nmult_sigma .
Lemma Nmult_Unth_prop :
forall n :nat, [1/]1+n == [1-] (
n */ ([1/]1+n)).
intro .
rewrite (
Nmult_sigma n ([1/]1+n)).
exact (
Unth_prop n ).
Save .
Hint Resolve Nmult_Unth_prop .
Lemma Nmult_n_Unth :
forall n :nat,
n */ [1/]1+n == [1-] ([1/]1+n).
intro ;
apply Uinv_eq_perm_right ;
auto .
Save .
Lemma Nmult_Sn_Unth :
forall n :nat,
S n */ [1/]1+n == 1.
intro ;
rewrite (
Nmult_S n ([1/]1+n)).
rewrite (
Nmult_n_Unth n );
auto .
Save .
Hint Resolve Nmult_n_Unth Nmult_Sn_Unth .
Lemma Nmult_ge_Sn_Unth :
forall n k , (
S n <=
k )%nat ->
k */ [1/]1+n == 1.
induction 1;
auto .
rewrite (
Nmult_S m ([1/]1+n));
rewrite IHle ;
auto .
Save .
Lemma Nmult_le_n_Unth :
forall n k , (
k <=
n )%nat ->
k */ [1/]1+n <= [1-] ([1/]1+n).
intros ;
transitivity (
n */ [1/]1+n);
auto .
Save .
Hint Resolve Nmult_ge_Sn_Unth Nmult_le_n_Unth .
Lemma Nmult_Umult_assoc_left :
forall n x y ,
Nmult_def n x ->
n */(x*y) == (
n */x)*y.
intros n x y ;
induction n ;
auto ;
intros .
destruct n ;
auto .
repeat rewrite Nmult_SS .
assert (
Nmult_def (
S n )
x );
auto .
setoid_rewrite (
IHn H0 ).
assert (
x <= [1-] ((
S n ) */
x )).
apply Uinv_le_perm_right .
transitivity ([1-] ([1/]1+(S
n )));
auto .
transitivity ((
S n ) */ ([1/]1+(S
n )));
auto .
apply Oeq_sym ;
auto .
Save .
Hint Resolve Nmult_Umult_assoc_left .
Lemma Nmult_Umult_assoc_right :
forall n x y ,
Nmult_def n y ->
n */(x*y) ==
x *(n*/y).
intros ;
rewrite (
Umult_sym x y );
rewrite (
Nmult_Umult_assoc_left n y x H );
auto .
Save .
Hint Resolve Nmult_Umult_assoc_right .
Lemma plus_Nmult_distr :
forall n m x , (
n +
m ) */
x == (
n */
x ) + (
m */
x ).
intros n m x ;
induction n ;
auto ;
intros .
rewrite plus_Sn_m .
rewrite (
Nmult_S (
n +
m )
x ).
setoid_rewrite IHn .
rewrite (
Nmult_S n x );
auto .
Save .
Lemma Nmult_Uplus_distr :
forall n x y ,
n */ (
x +
y ) == (
n */
x ) + (
n */
y ).
intros n x y ;
induction n .
simpl ;
auto .
rewrite (
Nmult_S n (
x +y)).
rewrite IHn .
norm_assoc_right .
rewrite (
Uplus_perm2 y (
n */
x ) (
n */
y )).
rewrite <- (
Nmult_S n y ).
norm_assoc_left .
apply Uplus_eq_compat ;
auto .
Save .
Lemma Nmult_mult_assoc :
forall n m x , (
n *
m ) */
x ==
n */ (
m */
x ).
intros n m x ;
induction n ;
intros ;
auto .
simpl mult .
rewrite (
plus_Nmult_distr m (
n *
m )
x ).
rewrite IHn ;
auto .
Save .
Lemma Nmult_Unth_simpl_left :
forall n x , (
S n ) */ ([1/]1+n *
x ) ==
x .
intros .
rewrite (
Nmult_Umult_assoc_left (
S n ) ([1/]1+n)
x (
Nmult_def_Unth n )).
rewrite (
Nmult_Sn_Unth n );
auto .
Save .
Lemma Nmult_Unth_simpl_right :
forall n x , (
S n ) */ (
x * [1/]1+n) ==
x .
intros .
rewrite (
Nmult_Umult_assoc_right (
S n )
x ([1/]1+n) (
Nmult_def_Unth n )).
rewrite (
Nmult_Sn_Unth n );
auto .
Save .
Hint Resolve Nmult_Umult_assoc_right plus_Nmult_distr Nmult_Uplus_distr
Nmult_mult_assoc Nmult_Unth_simpl_left Nmult_Unth_simpl_right .
Lemma Uinv_Nmult :
forall k n , [1-] (
k */ [1/]1+n) == ((
S n ) -
k ) */ [1/]1+n.
intros k n ;
case (
le_lt_dec (
S n )
k );
intro .
rewrite (
Nmult_ge_Sn_Unth l ).
replace (
S n -
k )%nat
with O ;
auto .
omega .
induction k ;
intros .
rewrite Nmult_0 ;
rewrite Uinv_zero .
replace (
S n -
O )%nat
with (
S n );
auto with arith .
rewrite (
Nmult_S k ([1/]1+n)).
apply Uplus_eq_simpl_right with ([1/]1+n);
auto .
apply Uinv_le_perm_right .
apply Nmult_le_n_Unth .
omega .
transitivity (((
S n -
S k ) + (
S O )) */ [1/]1+n).
replace ((
S n -
S k ) + (
S O ))%nat
with (
S n -
k )%nat.
transitivity ([1-] (
k */ [1/]1+n));
auto with arith .
apply Uinv_plus_left .
apply Nmult_le_n_Unth ;
omega .
omega .
rewrite (
plus_Nmult_distr (
S n -
S k ) (
S O ) ([1/]1+n));
auto .
Save .
Lemma Nmult_neq_zero :
forall n x , ~0==x -> ~0==S
n */
x .
intros ;
rewrite (
Nmult_S n x );
auto .
apply Uplus_neq_zero_left ;
trivial .
Save .
Hint Resolve Nmult_neq_zero .
Lemma Nmult_le_simpl :
forall (
n :nat) (
x y :U),
Nmult_def (
S n )
x ->
Nmult_def (
S n )
y -> (
S n */
x ) <= (
S n */
y ) ->
x <=
y .
intros ;
apply Umult_le_simpl_left with (
S n */ [1/]1+n).
auto .
assert (
Nmult_def (
S n ) ([1/]1+n));
auto .
rewrite <- (
Nmult_Umult_assoc_left (
S n ) ([1/]1+n)
x H2 ).
rewrite <- (
Nmult_Umult_assoc_left (
S n ) ([1/]1+n)
y H2 ).
rewrite (
Nmult_Umult_assoc_right (
S n ) ([1/]1+n)
y H0 ).
rewrite (
Nmult_Umult_assoc_right (
S n ) ([1/]1+n)
x H ).
transitivity ([1/]1+n * (
S n */
x ));
auto .
Save .
Lemma Nmult_Unth_le :
forall (
n1 n2 m1 m2 :nat),
(
n2 *
S n1 <=
m2 *
S m1 )%nat ->
n2 */ [1/]1+m1 <=
m2 */ [1/]1+n1.
intros .
transitivity ((
n2 *
S n1 ) */ ([1/]1+m1 * [1/]1+n1)).
rewrite (
Nmult_mult_assoc n2 (
S n1 ) ([1/]1+m1 * [1/]1+n1)).
apply Nmult_le_compat_right .
rewrite (
Nmult_Unth_simpl_right n1 ([1/]1+m1));
auto .
transitivity ((
m2 *
S m1 ) */ [1/]1+m1 * [1/]1+n1);
auto .
rewrite (
Nmult_mult_assoc m2 (
S m1 ) ([1/]1+m1 * [1/]1+n1)).
apply Nmult_le_compat_right .
rewrite (
Nmult_Unth_simpl_left m1 ([1/]1+n1));
auto .
Save .
Lemma Nmult_Unth_eq :
forall (
n1 n2 m1 m2 :nat),
(
n2 *
S n1 =
m2 *
S m1 )%nat ->
n2 */ [1/]1+m1 ==
m2 */ [1/]1+n1.
intros .
transitivity ((
n2 *
S n1 ) */ ([1/]1+m1 * [1/]1+n1)).
rewrite (
Nmult_mult_assoc n2 (
S n1 ) ([1/]1+m1 * [1/]1+n1)).
apply Nmult_eq_compat_left .
rewrite (
Nmult_Unth_simpl_right n1 ([1/]1+m1));
auto .
rewrite H .
rewrite (
Nmult_mult_assoc m2 (
S m1 ) ([1/]1+m1 * [1/]1+n1)).
apply Nmult_eq_compat_left .
rewrite (
Nmult_Unth_simpl_left m1 ([1/]1+n1));
auto .
Save .
Hint Resolve Nmult_Unth_le Nmult_Unth_eq .
Lemma Nmult_Unth_factor :
forall (
n m1 m2 :nat),
(
n *
S m2 =
S m1 )%nat ->
n */ [1/]1+m1 == [1/]1+m2.
intros ;
transitivity (1 */ [1/]1+m2);
trivial .
apply Nmult_Unth_eq ;
omega .
Save .
Hint Resolve Nmult_Unth_factor .
Lemma Nmult_def_lt :
forall n x ,
n */
x <1 ->
Nmult_def n x .
red ;
destruct n ;
intros ;
auto .
apply (
Ule_total x ([1/]1+n));
intros ;
auto .
case H .
transitivity (
S n */ [1/]1+n);
auto .
Save .
Hint Immediate Nmult_def_lt .
Conversion from booleans to U
Definition B2U :MF
bool :=
fun (
b :bool) =>
if b then 1
else 0.
Definition NB2U :MF
bool :=
fun (
b :bool) =>
if b then 0
else 1.
Lemma B2Uinv :
NB2U ==
finv B2U .
unfold NB2U ,finv,B2
U ;
intro b ;
case b ;
auto .
Save .
Lemma NB2Uinv :
B2U ==
finv NB2U .
unfold NB2U ,finv,B2
U ;
intro b ;
case b ;
auto .
Save .
Hint Resolve B2Uinv NB2Uinv .
Definition pmin (p :U) (n :nat) := p - ([1/2]^n).Add Morphism pmin with signature Oeq & eq & Oeq as pmin_eq_compat .unfold pmin ; auto .Save .
Lemma pmin_esp_S :
forall p n ,
pmin (
p &
p )
n ==
pmin p (
S n ) &
pmin p (
S n ).
unfold pmin at 1;
intros .
rewrite (
half_exp n ).
rewrite (
Uesp_minus_distr p p ([1/2]^(S
n )) ([1/2]^(S
n )));
auto .
Save .
Lemma pmin_esp_le :
forall p n ,
pmin p (
S n ) <= [1/2] * (
pmin (
p &
p )
n ) + [1/2].
intros ;
setoid_rewrite (
pmin_esp_S p n );
auto .
Save .
Lemma pmin_plus_eq :
forall p n ,
p <= [1/2] ->
pmin p (
S n ) == [1/2] * (
pmin (
p +
p )
n ).
intros ;
unfold pmin at 2.
setoid_rewrite (
Uminus_distr_right [1/2] (
p +
p ) ([1/2]^n)).
setoid_rewrite (
half_twice _ H );
auto .
Save .
Lemma pmin_0 :
forall p :U,
pmin p O == 0.
unfold pmin ;
simpl ;
intros ;
auto .
Save .
Lemma pmin_le :
forall (
p :U) (
n :nat),
p - ([1/]1+n) <=
pmin p n .
unfold pmin ;
intros .
apply Uminus_le_compat_right .
induction n ;
simpl ;
intros ;
auto .
transitivity ([1/2] * ([1/]1+n));
auto .
Save .
Hint Resolve pmin_0 pmin_le .
Lemma pmin_le_compat :
forall p (
n m :
nat ),
n <=
m ->
pmin p n <=
pmin p m .
unfold pmin ;
intros .
apply Uminus_le_compat_right ;
auto .
Save .
Hint Resolve pmin_le_compat .
Instance pmin_mon :
forall p ,
monotonic (
pmin p ).
red ;
auto .
Save .
Definition Pmin (
p :U) :nat -m>
U :=
mon (
pmin p ).
Lemma le_p_lim_pmin :
forall p ,
p <=
lub (
Pmin p ).
intro ;
apply Ule_lt_lim ;
intros .
assert (
exc (
fun n :
nat =>
t <=
p - [1/]1+n)).
apply Ult_le_nth_minus ;
trivial .
apply H0 ;
auto ;
intros n H1 .
transitivity (
p - [1/]1+n);
auto .
transitivity (
pmin p n );
auto .
apply (
le_lub (
Pmin p )
n ).
Save .
Lemma le_lim_pmin_p :
forall p ,
lub (
Pmin p ) <=
p .
intro ;
apply lub_le ;
simpl ;
unfold pmin ;
auto .
Save .
Hint Resolve le_p_lim_pmin le_lim_pmin_p .
Lemma eq_lim_pmin_p :
forall p ,
lub (
Pmin p ) ==
p .
intros ;
apply Ole_antisym ;
auto .
Save .
Hint Resolve eq_lim_pmin_p .
Particular case where p = 1
Definition U1min := Pmin 1.Lemma eq_lim_U1min : lub U1min == 1.unfold U1min ; auto .Save .Lemma U1min_S : forall n , U1min (S n ) == [1/2]*(U1min n ) + [1/2].intros ; unfold U1min at 2,pmin.rewrite (Uminus_distr_right [1/2] 1 ([1/2]^n)).rewrite Umult_one_right .rewrite Uminus_plus_perm ; auto .rewrite Unth_one_plus ; auto .Save .Lemma U1min_0 : U1min O == 0.unfold U1min ; simpl ; auto .Save .Hint Resolve eq_lim_U1min U1min_S U1min_0 .Lemma glb_half_exp : glb (UExp [1/2]) == 0.unfold glb ; apply Uinv_eq_perm_left .transitivity (lub U1min ).apply lub_eq_compat ; intro n ; simpl ; unfold U1min ,pmin; auto .rewrite eq_lim_U1min ; auto .Save .Hint Resolve glb_half_exp .Lemma Ule_lt_half_exp : forall x y , (forall p , x <= y + [1/2]^p) -> x <= y .intros ; transitivity (glb (UExp [1/2]) + y ).transitivity (glb (Imon (mshift UPlus y ) @ (UExp [1/2]))).apply le_glb ; intro p .transitivity (y + [1/2] ^ p ); simpl ; auto .rewrite glb_eq_plus_cte_right ; auto .rewrite glb_half_exp ; auto .Save .Lemma half_exp_le_half : forall p , [1/2]^(S p ) <= [1/2].simpl ; auto .Save .Hint Resolve half_exp_le_half .Lemma twice_half_exp : forall p , [1/2]^(S p ) + [1/2]^(S p ) == [1/2]^p.simpl ; intros .rewrite <- Udistr_plus_right ; auto .Save .Hint Resolve twice_half_exp .
Fixpoint exp2 (
n :nat) :
nat :=
match n with O => (1%nat) |
S p => (2 * (
exp2 p ))%nat
end .
Lemma exp2_pos :
forall n , (
O <
exp2 n )%nat.
induction n ;
simpl ;
auto with arith .
Save .
Hint Resolve exp2_pos .
Lemma S_pred_exp2 :
forall n ,
S (
pred (
exp2 n ))=exp2
n .
intro ;
rewrite S_pred with (
exp2 n )
O ;
trivial .
Save .
Hint Resolve S_pred_exp2 .
Notation "k /2^ p" := (
k */ ([1/2])^p) (
at level 35,
no associativity ).
Lemma Unth_eq :
forall n p ,
n */
p == [1-]
p ->
p == [1/]1+n.
intros ;
apply Ole_antisym ;
apply (
Ule_total p ([1/]1+n));
intros ;
auto .
apply Uinv_le_simpl .
transitivity (
n */[1/]1+n);
auto .
transitivity (
n */p);
auto .
apply Uinv_le_simpl .
transitivity (
n */p);
auto .
transitivity (
n */[1/]1+n);
auto .
Save .
Lemma Unth_half :
forall n , (
O <n)%nat -> [1/]1+(pred (
n +n)) == [1/2] * [1/]1+pred
n .
intros ;
apply Oeq_sym ;
apply Unth_eq .
destruct n ;
intros .
absurd (0<0)%nat;
auto with arith .
simpl .
replace (
n +
S n )%nat
with (2 *
n + 1)%nat;
auto with zarith .
rewrite plus_Nmult_distr .
rewrite Nmult_1 .
rewrite Nmult_mult_assoc .
rewrite Nmult_Umult_assoc_right ;
auto .
rewrite Nmult_Umult_assoc_left ;
auto .
rewrite Nmult_Sn_Unth .
rewrite Umult_one_left .
rewrite Nmult_n_Unth .
rewrite Udistr_inv_right ;
auto .
Save .
Lemma Unth_exp2 :
forall p , [1/2]^p == [1/]1+pred (
exp2 p ).
induction p ;
simpl ;
intros ;
auto .
rewrite <-
plus_n_O .
rewrite Unth_half ;
auto .
Save .
Hint Resolve Unth_exp2 .
Lemma Nmult_exp2 :
forall p , (
exp2 p )/2^p == 1.
intros ;
rewrite Unth_exp2 .
replace (
exp2 p )
with (
S (
pred (
exp2 p )));
auto .
Save .
Hint Resolve Nmult_exp2 .
Tactic for simplification of goals
Ltac Usimpl :=
match goal with
|-
context [(
Uplus 0 ?x)] =>
setoid_rewrite (
Uplus_zero_left x )
| |-
context [(
Uplus ?x 0)] =>
setoid_rewrite (
Uplus_zero_right x )
| |-
context [(
Uplus 1 ?x)] =>
setoid_rewrite (
Uplus_one_left x )
| |-
context [(
Uplus ?x 1)] =>
setoid_rewrite (
Uplus_one_right x )
| |-
context [(
Umult 0 ?x)] =>
setoid_rewrite (
Umult_zero_left x )
| |-
context [(
Umult ?x 0)] =>
setoid_rewrite (
Umult_zero_right x )
| |-
context [(
Umult 1 ?x)] =>
setoid_rewrite (
Umult_one_left x )
| |-
context [(
Umult ?x 1)] =>
setoid_rewrite (
Umult_one_right x )
| |-
context [(
Uesp 0 ?x)] =>
setoid_rewrite (
Uesp_zero_left x )
| |-
context [(
Uesp ?x 0)] =>
setoid_rewrite (
Uesp_zero_right x )
| |-
context [(
Uesp 1 ?x)] =>
setoid_rewrite (
Uesp_one_left x )
| |-
context [(
Uesp ?x 1)] =>
setoid_rewrite (
Uesp_one_right x )
| |-
context [(
Uminus 0 ?x)] =>
setoid_rewrite (
Uminus_le_zero 0
x );
[
idtac |apply (
Upos x )]
| |-
context [(
Uminus ?x 0)] =>
setoid_rewrite (
Uminus_zero_right x )
| |-
context [(
Uminus ?x 1)] =>
setoid_rewrite (
Uminus_le_zero x 1);
[
idtac |apply (
Unit x )]
| |-
context [([1-] ([1-] ?x))] =>
setoid_rewrite (
Uinv_inv x )
| |-
context [([1-] 1)] =>
setoid_rewrite Uinv_one
| |-
context [([1-] 0)] =>
setoid_rewrite Uinv_zero
| |-
context [([1/]1+O)] =>
setoid_rewrite Unth_zero
| |-
context [?x^O] =>
setoid_rewrite (
Uexp_0 x )
| |-
context [?x^(S
O )] =>
setoid_rewrite (
Uexp_1 x )
| |-
context [0^(?n)] =>
setoid_rewrite Uexp_zero ; [
idtac |omega]
| |-
context [
U1 ^(?n)] =>
setoid_rewrite Uexp_one
| |-
context [(
Nmult 0 ?x)] =>
setoid_rewrite (
Nmult_0 x )
| |-
context [(
Nmult 1 ?x)] =>
setoid_rewrite (
Nmult_1 x )
| |-
context [(
Nmult ?n 0)] =>
setoid_rewrite (
Nmult_zero n )
| |-
context [(
sigma ?f
O )] =>
setoid_rewrite (
sigma_0 f )
| |-
context [(
sigma ?f (
S O ))] =>
setoid_rewrite (
sigma_1 f )
| |- (
Ole (
Uplus ?x ?y) (
Uplus ?x ?z)) =>
apply Uplus_le_compat_right
| |- (
Ole (
Uplus ?x ?z) (
Uplus ?y ?z)) =>
apply Uplus_le_compat_left
| |- (
Ole (
Uplus ?x ?z) (
Uplus ?z ?y)) =>
setoid_rewrite (
Uplus_sym z y );
apply Uplus_le_compat_left
| |- (
Ole (
Uplus ?x ?y) (
Uplus ?z ?x)) =>
setoid_rewrite (
Uplus_sym x y );
apply Uplus_le_compat_left
| |- (
Ole (
Uinv ?y) (
Uinv ?x)) =>
apply Uinv_le_compat
| |- (
Ole (
Uminus ?x ?y) (
Uplus ?x ?z)) =>
apply Uminus_le_compat_right
| |- (
Ole (
Uminus ?x ?z) (
Uplus ?y ?z)) =>
apply Uminus_le_compat_left
| |- ((
Uinv ?x) == (
Uinv ?y)) =>
apply Uinv_eq_compat
| |- ((
Uplus ?x ?y) == (
Uplus ?x ?z)) =>
apply Uplus_eq_compat_right
| |- ((
Uplus ?x ?z) == (
Uplus ?y ?z)) =>
apply Uplus_eq_compat_left
| |- ((
Uplus ?x ?z) == (
Uplus ?z ?y)) =>
setoid_rewrite (
Uplus_sym z y );
apply Uplus_eq_compat_left
| |- ((
Uplus ?x ?y) == (
Uplus ?z ?x)) =>
setoid_rewrite (
Uplus_sym x y );
apply Uplus_eq_compat_left
| |- ((
Uminus ?x ?y) == (
Uplus ?x ?z)) =>
apply Uminus_eq_compat ;[
apply Oeq_refl |idtac]
| |- ((
Uminus ?x ?z) == (
Uplus ?y ?z)) =>
apply Uminus_eq_compat ;[
idtac |apply
Oeq_refl ]
| |- (
Ole (
Umult ?x ?y) (
Umult ?x ?z)) =>
apply Umult_le_compat_right
| |- (
Ole (
Umult ?x ?z) (
Umult ?y ?z)) =>
apply Umult_le_compat_left
| |- (
Ole (
Umult ?x ?z) (
Umult ?z ?y)) =>
setoid_rewrite (
Umult_sym z y );
apply Umult_le_compat_left
| |- (
Ole (
Umult ?x ?y) (
Umult ?z ?x)) =>
setoid_rewrite (
Umult_sym x y );
apply Umult_le_compat_left
| |- ((
Umult ?x ?y) == (
Umult ?x ?z)) =>
apply Umult_eq_compat_right
| |- ((
Umult ?x ?z) == (
Umult ?y ?z)) =>
apply Umult_eq_compat_left
| |- ((
Umult ?x ?z) == (
Umult ?z ?y)) =>
setoid_rewrite (
Umult_sym z y );
apply Umult_eq_compat_left
| |- ((
Umult ?x ?y) == (
Umult ?z ?x)) =>
setoid_rewrite (
Umult_sym x y );
apply Umult_eq_compat_left
end .
Properties of current values
Notation "[1/3]" := (
Unth 2%nat).
Notation "[1/4]" := (
Unth 3%nat).
Notation "[1/8]" := (
Unth 7).
Notation "[3/4]" := (
Uinv [1/4]).
Lemma half_square : [1/2]*[1/2]==[1/4].
rewrite <- (
Unth_exp2 2).
simpl ;
auto .
Save .
Lemma half_cube : [1/2]*[1/2]*[1/2]==[1/8].
rewrite <- (
Unth_exp2 3).
simpl ;
rewrite Umult_one_right ;
auto .
Save .
Lemma three_quarter_decomp : [3/4]==[1/2]+[1/4].
transitivity ([1/4] + [1/4] + [1/4]).
rewrite <-
Nmult_n_Unth ;
simpl .
norm_assoc_left ;
trivial .
Usimpl .
transitivity (2 */ [1/4]);
auto .
Save .
Equality is not true, even for monotonic sequences fot instance n/m
Lemma Ulub_Uglb_exch_le :
forall f :
nat ->
nat ->
U ,
Ulub (
fun n =>
Uglb (
fun m =>
f n m )) <=
Uglb (
fun m =>
Ulub (
fun n =>
f n m )).
intros ;
apply le_Uglb ;
intro m .
apply Ulub_le_compat ;
intro n .
apply Uglb_le with (
f :=fun
m0 :
nat =>
f n m0 ) (
n :=m).
Save .
Record IU : Type := mk_IU {low :U; up :U; proper :low <= up }.Hint Resolve proper .
the all set : [0,1]
Definition full := mk_IU 0 1 (Upos 1).
singleton : [x ]
Definition singl (
x :U) :=
mk_IU x x (
Ole_refl x ).
down segment : [0,x]
Definition inf (x :U) := mk_IU 0 x (Upos x ).
up segment : [x ,1]
Definition sup (x :U) := mk_IU x 1 (Unit x ).
Definition Iin (
x :U) (
I :IU) :=
low I <=
x /\
x <=
up I .
Definition Iincl I J :=
low J <=
low I /\
up I <=
up J .
Definition Ieq I J :=
low I ==
low J /\
up I ==
up J .
Hint Unfold Iin Iincl Ieq .
Lemma Iin_low :
forall I ,
Iin (
low I )
I .
auto .
Save .
Lemma Iin_up :
forall I ,
Iin (
up I )
I .
auto .
Save .
Hint Resolve Iin_low Iin_up .
Lemma Iin_singl_elim :
forall x y ,
Iin x (
singl y ) ->
x ==
y .
unfold Iin ;
intuition (
simpl ;
intros ;
auto ).
Save .
Lemma Iin_inf_elim :
forall x y ,
Iin x (
inf y ) ->
x <=
y .
unfold Iin ;
intuition (
simpl ;
auto ).
Save .
Lemma Iin_sup_elim :
forall x y ,
Iin x (
sup y ) ->
y <=
x .
unfold Iin ;
intuition (
simpl ;
auto ).
Save .
Lemma Iin_singl_intro :
forall x y ,
x ==
y ->
Iin x (
singl y ).
auto .
Save .
Lemma Iin_inf_intro :
forall x y ,
x <=
y ->
Iin x (
inf y ).
auto .
Save .
Lemma Iin_sup_intro :
forall x y ,
y <=
x ->
Iin x (
sup y ).
auto .
Save .
Hint Immediate Iin_inf_elim Iin_sup_elim Iin_singl_elim .
Hint Resolve Iin_inf_intro Iin_sup_intro Iin_singl_intro .
Lemma Iin_class :
forall I x ,
class (
Iin x I ).
unfold class ,
Iin ;
split .
apply Ule_class ;
intuition .
apply Ule_class ;
intuition .
Save .
Lemma Iincl_class :
forall I J ,
class (
Iincl I J ).
unfold class ,
Iincl ;
split .
apply Ule_class ;
intuition .
apply Ule_class ;
intuition .
Save .
Lemma Ieq_class :
forall I J ,
class (
Ieq I J ).
unfold class ,
Ieq ;
split .
apply Ueq_class ;
intuition .
apply Ueq_class ;
intuition .
Save .
Hint Resolve Iin_class Iincl_class Ieq_class .
Lemma Iincl_in :
forall I J ,
Iincl I J ->
forall x ,
Iin x I ->
Iin x J .
unfold Iin ,Iincl;
intuition .
transitivity (
low I );
auto .
transitivity (
up I );
auto .
Save .
Lemma Iincl_low :
forall I J ,
Iincl I J ->
low J <=
low I .
unfold Iincl ;
intuition .
Save .
Lemma Iincl_up :
forall I J ,
Iincl I J ->
up I <=
up J .
unfold Iincl ;
intuition .
Save .
Hint Immediate Iincl_low Iincl_up .
Lemma Iincl_refl :
forall I ,
Iincl I I .
unfold Iincl ;
intuition .
Save .
Hint Resolve Iincl_refl .
Lemma Iincl_trans :
forall I J K ,
Iincl I J ->
Iincl J K ->
Iincl I K .
unfold Iincl ;
intuition .
transitivity (
low J );
auto .
transitivity (
up J );
auto .
Save .
Instance IUord :
ord IU := {
Oeq :=
fun I J =>
Ieq I J ;
Ole :=
fun I J =>
Iincl J I }.
split ;
intros ;
auto .
unfold Iincl ,
Ieq ;
intuition .
red ;
intros ;
apply Iincl_trans with y ;
auto .
Defined .
Lemma low_le_compat :
forall I J :IU,
I <=
J ->
low I <=
low J .
simpl ;
auto .
Save .
Lemma up_le_compat :
forall I J :
IU ,
I <=
J ->
up J <=
up I .
simpl ;
auto .
Save .
Instance low_mon :
monotonic low .
red ;
auto .
Save .
Definition Low :
IU -m>
U :=
mon low .
Instance up_mon :
monotonic (
o2 :=Iord
U )
up .
red ;
simpl ;
auto .
Save .
Definition Up :
IU -m->
U :=
mon (
o2 :=Iord
U )
up .
Lemma Ieq_incl :
forall I J ,
Ieq I J ->
Iincl I J .
unfold Ieq ,Iincl;
intuition .
Save .
Lemma Ieq_incl_sym :
forall I J ,
Ieq I J ->
Iincl J I .
unfold Ieq ,Iincl;
intuition .
Save .
Hint Immediate Ieq_incl Ieq_incl_sym .
Lemma lincl_eq_compat :
forall I J K L ,
Ieq I J ->
Iincl J K ->
Ieq K L ->
Iincl I L .
intros ;
apply Iincl_trans with J ;
auto .
intros ;
apply Iincl_trans with K ;
auto .
Save .
Lemma lincl_eq_trans :
forall I J K ,
Iincl I J ->
Ieq J K ->
Iincl I K .
intros ;
apply lincl_eq_compat with I J ;
auto .
Save .
Lemma Ieq_incl_trans :
forall I J K ,
Ieq I J ->
Iincl J K ->
Iincl I K .
intros ;
apply lincl_eq_compat with J K ;
auto .
Save .
Lemma Iincl_antisym :
forall I J ,
Iincl I J ->
Iincl J I ->
Ieq I J .
unfold Iincl ;
intuition .
Save .
Hint Immediate Iincl_antisym .
Lemma Ieq_refl :
forall I ,
Ieq I I .
unfold Ieq ;
auto .
Save .
Hint Resolve Ieq_refl .
Lemma Ieq_sym :
forall I J ,
Ieq I J ->
Ieq J I .
unfold Ieq ;
intuition .
Save .
Hint Immediate Ieq_sym .
Lemma Ieq_trans :
forall I J K ,
Ieq I J ->
Ieq J K ->
Ieq I K .
unfold Ieq ;
intuition .
transitivity (
low J );
auto .
transitivity (
up J );
auto .
Save .
Lemma Isingl_eq :
forall x y ,
Iincl (
singl x ) (
singl y ) ->
x ==y.
unfold Iincl ,
singl ;
intuition .
Save .
Hint Immediate Isingl_eq .
Lemma Iincl_full :
forall I ,
Iincl I full .
unfold Iincl ,
full ;
intuition .
Save .
Hint Resolve Iincl_full .
Definition Iplus I J :=
mk_IU (
low I +
low J ) (
up I +
up J )
(
Uplus_le_compat _ _ _ _ (
proper I ) (
proper J )).
Lemma low_Iplus :
forall I J ,
low (
Iplus I J )=low
I +
low J .
trivial .
Save .
Lemma up_Iplus :
forall I J ,
up (
Iplus I J )=up
I +
up J .
trivial .
Save .
Lemma Iplus_in :
forall I J x y ,
Iin x I ->
Iin y J ->
Iin (
x +y) (
Iplus I J ).
unfold Iin ,Iplus;
intuition (
simpl ;
auto ).
Save .
Lemma lplus_in_elim :
forall I J z ,
low I <= [1-]
up J ->
Iin z (
Iplus I J )
->
exc (
fun x =>
Iin x I /\
exc (
fun y =>
Iin y J /\
z ==x+y)).
intros I J z H (
H1 ,H2);
simpl in H1 ,H2;
intros .
assert (
low I <=
z ).
transitivity (
low I +
low J );
auto .
apply (
Ule_total (
z -low
I ) (
up J ));
intros .
apply class_exc .
apply exc_intro with (
low I );
split ;
auto .
apply exc_intro with (
z -low
I );
split ;
auto .
assert (
low I <= [1-]
low J ).
transitivity ([1-]
up J );
auto .
split ;
auto .
apply Uplus_le_perm_right ;
auto .
rewrite Uplus_sym ;
auto .
assert (
up J <=
z );
auto .
transitivity (
z -
low I );
auto .
apply exc_intro with (
z -up
J );
split ;
auto .
split ;
auto .
apply Uplus_le_perm_left ;
auto .
rewrite Uplus_sym ;
auto .
apply exc_intro with (
up J );
auto .
Save .
Definition Imult I J :=
mk_IU (
low I *
low J ) (
up I *
up J )
(
Umult_le_compat _ _ _ _ (
proper I ) (
proper J )).
Lemma low_Imult :
forall I J ,
low (
Imult I J ) =
low I *
low J .
trivial .
Save .
Lemma up_Imult :
forall I J ,
up (
Imult I J ) =
up I *
up J .
trivial .
Save .
Definition Imultk p I :=
mk_IU (
p *
low I ) (
p *
up I ) (
Umult_le_compat_right p _ _ (
proper I )).
Lemma low_Imultk :
forall p I ,
low (
Imultk p I ) =
p *
low I .
trivial .
Save .
Lemma up_Imultk :
forall p I ,
up (
Imultk p I ) =
p *
up I .
trivial .
Save .
Lemma Imult_in :
forall I J x y ,
Iin x I ->
Iin y J ->
Iin (
x *y) (
Imult I J ).
unfold Iin ;
intuition (
simpl ;
auto ).
Save .
Lemma Imultk_in :
forall p I x ,
Iin x I ->
Iin (
p *x) (
Imultk p I ).
unfold Iin ;
intuition (
simpl ;
auto ).
Save .
Definition Ilim :
forall I :
nat -m>
IU ,
IU .
intros ;
exists (
lub (
Low@I )) (
glb (
Up@I )).
unfold glb ;
apply lub_inv ;
simpl ;
auto .
Defined .
Lemma low_lim :
forall (
I :nat -m>
IU ),
low (
Ilim I ) =
lub (
Low @
I ).
trivial .
Save .
Lemma up_lim :
forall (
I :nat -m>
IU ),
up (
Ilim I ) =
glb (
Up @
I ).
trivial .
Save .
Lemma lim_Iincl :
forall (
I :nat -m>
IU )
n ,
Iincl (
Ilim I ) (
I n ).
unfold Ilim ,Iincl;
simpl ;
split .
apply (
le_lub (
Low @
I )
n ).
apply (
glb_le (
Up @
I )
n ).
Save .
Hint Resolve lim_Iincl .
Lemma Iincl_lim :
forall J (
I :nat -m>IU), (
forall n ,
Iincl J (
I n )) ->
Iincl J (
Ilim I ).
unfold Ilim ,Iincl;
simpl ;
split .
apply lub_le with (
f :=Low @
I );
intro .
case (
H n );
auto .
apply le_glb with (
f :=Up @
I );
intro .
case (
H n );
auto .
Save .
Lemma IIim_incl_stable :
forall (
I J :nat -m>
IU ), (
forall n ,
Iincl (
I n ) (
J n )) ->
Iincl (
Ilim I ) (
Ilim J ).
intros ;
apply Iincl_lim .
intros ;
apply Iincl_trans with (
I n );
auto .
Save .
Hint Resolve IIim_incl_stable .
Instance IUcpo :
cpo IU := {
D0 :=full;
lub :=Ilim}.
intros ;
simpl ;
auto .
intros ;
simpl ;
auto .
intros ;
simpl ;
apply Iincl_lim ;
auto .
Defined .
Definition fsup (
f :nat->U) (
n :nat) :=
Ulub (
fun k =>
f (
n +k)%nat).
Definition finf (
f :nat->U) (
n :nat) :=
Uglb (
fun k =>
f (
n +k)%nat).
Lemma fsup_incr :
forall (
f :nat->U)
n ,
fsup f (
S n ) <=
fsup f n .
unfold fsup ;
intros .
apply Ulub_le ;
intros .
replace (
S n +n0)%nat
with (
n +S
n0 )%nat;
try omega .
apply le_Ulub with (
f :=fun
k :
nat =>
f (
n +k)%nat).
Save .
Hint Resolve fsup_incr .
Lemma finf_incr :
forall (
f :nat->U)
n ,
finf f n <=
finf f (
S n ).
unfold finf ;
intros .
apply le_Uglb ;
intros .
replace (
S n +n0)%nat
with (
n +S
n0 )%nat;
try omega .
apply Uglb_le with (
f :=fun
k :
nat =>
f (
n +k)%nat).
Save .
Hint Resolve finf_incr .
Instance fsup_mon :
forall f ,
monotonic (
o2 :=Iord
U ) (
fsup f ).
intro ;
apply nat_monotonic ;
simpl ;
auto .
Save .
Instance finf_mon :
forall f ,
monotonic (
finf f ).
intro ;
apply nat_monotonic ;
auto .
Save .
Definition Fsup (
f :nat->U) :
nat -m->
U :=
mon (
fsup f ).
Definition Finf (
f :nat->U) :
nat -m>
U :=
mon (
finf f ).
Lemma fn_fsup :
forall f n ,
f n <=
fsup f n .
unfold fsup ;
intros .
pattern n at 1;
replace n with (
n +O)%nat;
auto with arith .
apply le_Ulub with (
f :=fun
k :
nat =>
f (
n +k)%nat) (
n :=O).
Save .
Hint Resolve fn_fsup .
Lemma finf_fn :
forall f n ,
finf f n <=
f n .
unfold finf ;
intros .
pattern n at 2;
replace n with (
n +O)%nat;
auto with arith .
apply Uglb_le with (
f :=fun
k :
nat =>
f (
n +k)%nat) (
n :=O).
Save .
Hint Resolve finf_fn .
Definition limsup f :=
glb (
Fsup f ).
Definition liminf f :=
lub (
Finf f ).
Lemma le_liminf_sup :
forall f ,
liminf f <=
limsup f .
unfold liminf ,limsup;
simpl ;
intro .
rewrite <-
Uglb_glb .
transitivity (
Ulub (
Finf f ));
auto .
unfold Finf ,Fsup,finf,fsup.
transitivity
(
Uglb (
fun m :
nat =>
Ulub (
fun n :
nat =>
f (
n +m)%nat))).
apply Ulub_Uglb_exch_le with (
f :=fun
n m =>
f (
n +m)%nat).
apply Uglb_le_compat ;
simpl ;
intro .
apply Ulub_le_compat ;
simpl ;
intro .
replace ((
x0 +x)%nat)
with ((
x +x0)%nat);
auto with arith .
Save .
Hint Resolve le_liminf_sup .
Definition has_lim f :=
limsup f <=
liminf f .
Lemma eq_liminf_sup :
forall f ,
has_lim f ->
liminf f ==
limsup f .
intro ;
unfold has_lim ;
apply Ole_antisym ;
auto .
Save .
Definition cauchy f :=
forall (
p :nat),
exc (
fun M :nat =>
forall n m ,
(
M <=
n )%nat -> (
M <=
m )%nat ->
f n <=
f m + [1/2]^p).
Definition is_limit f (
l :U) :=
forall (
p :nat),
exc (
fun M :nat =>
forall n ,
(
M <=
n )%nat ->
f n <=
l + [1/2]^p /\
l <=
f n + [1/2]^p).
Lemma cauchy_lim :
forall f ,
cauchy f ->
is_limit f (
limsup f ).
unfold cauchy ,is_limit;
intros .
apply (
H p );
clear H ;
auto ;
intros M H ;
apply exc_intro with M ;
intros n H1 ;
unfold limsup ;
split .
transitivity (
glb ((
Imon (
mshift UPlus ([1/2]^p))) @ (
mon (
seq_lift_left (
Fsup f )
M )))).
apply le_glb ;
simpl ;
intro k .
transitivity (
f (
M +
k )%nat + [1/2]^p);
unfold seq_lift_left ;
simpl ;
auto with arith .
rewrite glb_eq_plus_cte_right ;
Usimpl .
rewrite <-
glb_lift_left ;
auto .
transitivity (
Fsup f M );
auto ;
simpl .
unfold fsup ;
apply Ulub_le ;
intro k ;
auto with arith .
Save .
Lemma has_limit_cauchy :
forall f l ,
is_limit f l ->
cauchy f .
unfold cauchy ,is_limit;
intros .
apply (
H (
S p ));
clear H ;
auto ;
intros M H ;
apply exc_intro with M ;
intros n m H1 ;
unfold limsup .
case (
H n );
auto with arith ;
intros .
case (
H m );
auto with arith ;
intros .
transitivity (
l + [1/2]^(S
p ));
auto .
transitivity (
f m + [1/2]^(S
p ) + [1/2]^(S
p ));
auto .
norm_assoc_right ;
Usimpl ;
auto .
Save .
Lemma limit_le_unique :
forall f l1 l2 ,
is_limit f l1 ->
is_limit f l2 ->
l1 <=
l2 .
intros f l1 l2 liml1 liml2 ;
apply Ule_lt_half_exp ;
intro p2 .
intros ;
apply Ule_lt_half_exp ;
intro p1 .
apply (
liml1 p1 );
auto ;
intros M1 H1 .
apply (
liml2 p2 );
auto ;
intros M2 H2 .
case (
H1 (
M1 +M2)%nat);
auto with arith ;
intros .
case (
H2 (
M1 +M2))%nat;
auto with arith ;
intros .
transitivity (
f (
M1 +
M2 )%nat + [1/2]^p1);
auto .
Save .
Lemma limit_unique :
forall f l1 l2 ,
is_limit f l1 ->
is_limit f l2 ->
l1 ==
l2 .
intros ;
apply Ole_antisym ;
apply (
limit_le_unique (
f :=f));
auto .
Save .
Hint Resolve limit_unique .
Lemma has_limit_compute :
forall f l ,
is_limit f l ->
is_limit f (
limsup f ).
intros ;
apply cauchy_lim ;
auto .
apply has_limit_cauchy with l ;
trivial .
Save .
Lemma limsup_eq_mult :
forall k (
f :
nat ->
U ),
limsup (
fun n =>
k *
f n ) ==
k *
limsup f .
unfold limsup ;
intros .
transitivity (
glb (
Imon (
UMult k ) @ (
Fsup f )));
auto .
apply glb_eq_compat ;
intro n ;
simpl ;
unfold fsup ;
intros ;
simpl ;
auto .
rewrite glb_eq_mult ;
trivial .
Save .
Lemma liminf_eq_mult :
forall k (
f :
nat ->
U ),
liminf (
fun n =>
k *
f n ) ==
k *
liminf f .
unfold liminf ;
intros .
transitivity (
lub ((
UMult k ) @ (
Finf f )));
auto .
apply lub_eq_compat ;
intro n ;
simpl ;
unfold finf ;
intros ;
auto .
Save .
Lemma limsup_eq_plus_cte_right :
forall k (
f :
nat ->
U ),
limsup (
fun n => (
f n ) +
k ) ==
limsup f +
k .
unfold limsup ;
intros .
transitivity (
glb (
Imon (
mshift UPlus k ) @ (
Fsup f )));
auto .
apply glb_eq_compat ;
intro n ;
simpl ;
unfold fsup ;
intros ;
auto .
Save .
Lemma liminf_eq_plus_cte_right :
forall k (
f :
nat ->
U ),
liminf (
fun n => (
f n ) +
k ) ==
liminf f +
k .
unfold liminf ;
intros .
transitivity (
lub (
mshift UPlus k @ (
Finf f )));
auto .
apply lub_eq_compat ;
intro n ;
simpl ;
unfold finf ;
intros ;
auto .
Save .
Lemma limsup_le_plus :
forall (
f g :
nat ->
U ),
limsup (
fun x =>
f x +
g x ) <=
limsup f +
limsup g .
intros ;
unfold limsup ,fplus.
transitivity (
glb ((
Imon2 UPlus @2 (
Fsup f )) (
Fsup g ))).
apply glb_le_compat ;
simpl ;
intro .
unfold fsup ;
auto .
rewrite glb_eq_plus ;
auto .
Save .
Lemma liminf_le_plus :
forall (
f g :
nat ->
U ),
liminf f +
liminf g <=
liminf (
fun x =>
f x +
g x ).
intros ;
unfold liminf ,fplus.
transitivity (
lub ((
UPlus @2 (
Finf f )) (
Finf g )));
auto .
apply lub_le_compat ;
simpl ;
unfold finf ;
auto .
Save .
Hint Resolve liminf_le_plus limsup_le_plus .
Lemma limsup_le_compat :
forall f g :
nat ->
U ,
f <=
g ->
limsup f <=
limsup g .
unfold limsup ;
intros .
apply glb_le_compat ;
simpl ;
intros ;
unfold fsup ;
apply Ulub_le_compat ;
auto .
Save .
Lemma liminf_le_compat :
forall f g :
nat ->
U ,
f <=
g ->
liminf f <=
liminf g .
unfold liminf ;
intros .
apply lub_le_compat ;
simpl ;
intros ;
unfold finf ;
apply Uglb_le_compat ;
auto .
Save .
Hint Resolve limsup_le_compat liminf_le_compat .
Lemma limsup_eq_compat :
forall f g :
nat ->
U ,
f ==
g ->
limsup f ==
limsup g .
intros ;
apply Ole_antisym ;
auto .
Save .
Lemma liminf_eq_compat :
forall f g :
nat ->
U ,
f ==
g ->
liminf f ==
liminf g .
intros ;
apply Ole_antisym ;
auto .
Save .
Hint Resolve liminf_eq_compat limsup_eq_compat .
Lemma limsup_inv :
forall f :
nat ->
U ,
limsup (
fun x => [1-]
f x ) == [1-]
liminf f .
unfold limsup ,
liminf ;
intros .
unfold glb ;
Usimpl .
apply lub_eq_compat ;
intro n ;
simpl ;
unfold finf ,fsup,Uglb;
simpl ;
Usimpl .
apply Ulub_eq_compat ;
intro m ;
auto .
Save .
Lemma liminf_inv :
forall f :
nat ->
U ,
liminf (
fun x => [1-]
f x ) == [1-]
limsup f .
unfold limsup ,
liminf ;
intros .
unfold glb ;
Usimpl .
apply lub_eq_compat ;
intro n ;
simpl ;
unfold finf ,fsup,Uglb;
Usimpl .
apply Ulub_eq_compat ;
intro m ;
auto .
Save .
Hint Resolve limsup_inv liminf_inv .
Limits of arbitrary sequences
Lemma liminf_incr :
forall f :nat -m>
U ,
liminf f ==
lub f .
unfold liminf ;
intros .
apply lub_eq_compat ;
intro n ;
simpl .
unfold finf .
rewrite (
glb_mon (
mseq_lift_left f n ));
simpl .
replace (
n +0)%nat
with n ;
auto with arith .
Save .
Lemma limsup_incr :
forall f :nat -m>
U ,
limsup f ==
lub f .
unfold limsup ;
intros .
transitivity (
glb (
mon (
o2 :=Iord
U ) (
fcte nat (
lub f ))));
auto .
apply glb_eq_compat ;
intro n ;
simpl .
unfold fsup .
transitivity (
lub (
mseq_lift_left f n )).
transitivity (
Ulub (
mseq_lift_left f n ));
auto .
unfold fcte ;
transitivity (
mlub (
seq_lift_left f n ));
auto .
apply lub_eq_compat ;
intro x ;
simpl ;
auto .
apply (
glb_cte (
lub f )).
Save .
Lemma has_limit_incr :
forall f :nat -m>
U ,
has_lim f .
red ;
intros .
rewrite liminf_incr ;
auto ;
rewrite limsup_incr ;
auto .
Save .
Lemma liminf_decr :
forall f :nat -m->
U ,
liminf f ==
glb f .
unfold liminf ;
intros .
transitivity (
mlub (
cte nat (
glb f )));
auto .
apply lub_eq_compat ;
intro n ;
simpl ;
unfold finf ;
intros .
transitivity (
glb (
mseq_lift_left f n )).
transitivity (
Uglb (
mseq_lift_left f n ));
auto .
unfold cte ;
transitivity (
glb (
mon (
seq_lift_left f n )));
auto .
apply glb_eq_compat ;
intro x ;
simpl ;
auto .
Save .
Lemma limsup_decr :
forall f :nat-m->U,
limsup f ==
glb f .
unfold limsup ;
intros .
apply glb_eq_compat ;
intro n ;
simpl ;
unfold fsup .
transitivity (
Ulub (
mseq_lift_left f n ));
auto .
transitivity (
mseq_lift_left f n O );
auto .
apply (
Ulub_mon (
mseq_lift_left f n )).
simpl ;
replace (
n +0)%nat
with n ;
auto with arith .
Save .
Lemma has_limit_decr :
forall f :nat -m->
U ,
has_lim f .
red ;
intros .
rewrite liminf_decr ;
auto ;
rewrite limsup_decr ;
auto .
Save .
Lemma has_limit_sum :
forall f g :
nat ->
U ,
has_lim f ->
has_lim g ->
has_lim (
fun x =>
f x +
g x ).
unfold has_lim ;
intros .
transitivity (
limsup f +
limsup g );
auto .
apply Ole_trans with (
liminf f +
liminf g );
auto .
Save .
Lemma has_limit_inv :
forall f :
nat ->
U ,
has_lim f ->
has_lim (
fun x => [1-]
f x ).
unfold has_lim ;
intros .
transitivity ([1-]
liminf f );
auto .
transitivity ([1-]
limsup f );
auto .
Save .
Lemma has_limit_cte :
forall c ,
has_lim (
fun n =>
c ).
intros ;
apply (
has_limit_incr (
mon (
cte nat (
c :U))));
red ;
auto .
Save .
Definition and properties of series : infinite sums
Definition serie (
f :
nat ->
U ) :
U :=
lub (
sigma f ).
Lemma serie_le_compat :
forall (
f g :
nat ->
U ),
(
forall k ,
f k <=
g k ) ->
serie f <=
serie g .
unfold serie ;
intros ;
apply lub_le_compat ;
intro n ;
auto .
Save .
Lemma serie_eq_compat :
forall (
f g :
nat ->
U ),
(
forall k ,
f k ==
g k ) ->
serie f ==
serie g .
unfold serie ;
intros ;
apply lub_eq_compat ;
intro n ;
auto .
Save .
Lemma serie_sigma_lift :
forall (
f :nat ->
U ) (
n :nat),
serie f ==
sigma f n +
serie (
fun k =>
f (
n +
k )%nat).
intros f n ;
unfold serie ;
transitivity
(
lub (
mseq_lift_left (
sigma f )
n ));
auto .
transitivity
(
lub (
UPlus (
sigma f n ) @
sigma (
fun k =>
f (
n +k)%nat))).
apply lub_eq_compat ;
intro n0 .
rewrite mseq_lift_left_simpl .
rewrite comp_simpl .
apply (
sigma_plus_lift f n n0 ).
symmetry .
apply (
lub_comp_eq (
UPlus (
sigma f n )) (
sigma (
fun k :
nat =>
f (
n +
k )%nat)) ).
apply continuous2_app ;
trivial .
Save .
Lemma serie_S_lift :
forall (
f :nat ->
U ),
serie f ==
f O +
serie (
fun k =>
f (
S k )).
intro ;
rewrite (
serie_sigma_lift f (
S O ));
simpl .
apply Uplus_eq_compat ;
auto .
Save .
Lemma serie_zero :
forall f , (
forall k ,
f k ==0) ->
serie f ==0.
unfold serie ;
intros .
rewrite <- (
lub_cte (
D :=U) 0).
apply lub_eq_compat ;
intro n ;
auto .
Save .
Lemma serie_not_zero :
forall f k , 0 <
f k -> 0 <
serie f .
intros ;
apply Ult_le_trans with (
sigma f (
S k )).
apply (
sigma_not_zero f )
with (
k :=k);
auto .
unfold serie ;
apply le_lub ;
auto .
Save .
Lemma serie_zero_elim :
forall f ,
serie f == 0 ->
forall k ,
f k ==0.
intros ;
apply Ueq_class ;
red ;
intros .
assert (0 <
serie f );
auto .
apply serie_not_zero with k ;
auto .
Save .
Hint Resolve serie_eq_compat serie_le_compat serie_zero .
Lemma serie_le :
forall f k ,
f k <=
serie f .
intros ;
transitivity (
sigma f (
S k ));
auto .
unfold serie ;
apply le_lub ;
auto .
Save .
Lemma serie_minus_incr :
forall f :nat -m>
U ,
serie (
fun k =>
f (
S k ) -
f k ) ==
lub f -
f O .
intros ;
transitivity (
lub (
mshift UMinus (
f O ) @
f )).
unfold serie ;
apply lub_eq_compat .
intro n ;
rewrite comp_simpl ;
rewrite mshift_simpl .
rewrite UMinus_simpl .
apply (
sigma_minus_incr f );
auto .
assert (
continuous2 (
c1 :=Uopp) (
mshift UMinus ));
auto .
assert (
continuous (
mshift UMinus (
f O )));
auto .
apply (
continuous2_app (
c1 :=Uopp));
trivial .
rewrite <- (
lub_comp_eq (
mshift UMinus (
f O ))
f );
trivial .
Save .
Lemma serie_minus_decr :
forall f :
nat -m->
U ,
serie (
fun k =>
f k -
f (
S k )) ==
f O -
glb f .
intros ;
transitivity (
lub (
UMinus (
f O ) @
f )).
unfold serie ;
apply lub_eq_compat .
intro n ;
rewrite comp_simpl .
rewrite UMinus_simpl .
apply (
sigma_minus_decr f );
auto .
assert (
continuous (
c1 :=Uopp) (
UMinus (
f O ))).
apply @
continuous2_app with (
c1 :=cpoU) (
c2 :=Uopp) (
F :=UMinus) (
k :=f
O );
trivial .
symmetry ;
rewrite <- (
lub_comp_eq (
c1 :=Uopp) (
UMinus (
f O ))
f );
auto .
Save .
Lemma serie_plus :
forall (
f g :
nat ->
U ),
serie (
fun k => (
f k ) + (
g k )) ==
serie f +
serie g .
intros ;
unfold serie .
transitivity
(
lub ((
UPlus @2
sigma f ) (
sigma g )));
auto .
apply lub_eq_compat ;
intro n .
exact (
sigma_plus f g n ).
Save .
Definition wretract (
f :
nat ->
U ) :=
forall k ,
f k <= [1-] (
sigma f k ).
Lemma retract_wretract :
forall f , (
forall n ,
retract f n ) ->
wretract f .
red ;
intros ;
auto .
Save .
Lemma wretract_retract :
forall f ,
wretract f ->
forall n ,
retract f n .
red ;
intros ;
auto .
Save .
Hint Resolve wretract_retract .
Lemma wretract_lt :
forall (
f :
nat ->
U ), (
forall (
n :
nat ),
sigma f n < 1) ->
wretract f .
intros ;
apply retract_wretract ;
intros .
apply retract_lt ;
trivial .
Save .
Lemma retract_zero_wretract :
forall f n ,
retract f n -> (
forall k , (
n <=
k )%nat ->
f k == 0) ->
wretract f .
red ;
intros .
assert (
k <
n \/
n <=
k )%nat;
intuition .
omega .
rewrite H0 ;
auto .
Save .
Lemma wretract_le :
forall f g :
nat ->U,
f <=
g ->
wretract g ->
wretract f .
red ;
intros .
transitivity (
g k );
auto .
transitivity ([1-]
sigma g k );
auto .
Save .
Lemma serie_mult :
forall (
f :
nat ->
U )
c ,
wretract f ->
serie (
fun k =>
c *
f k ) ==
c *
serie f .
unfold serie ;
intros .
transitivity (
lub (
UMult c @
sigma f ));
auto .
apply lub_eq_compat ;
intro n .
apply (
sigma_mult (
f :=f)
c (
n :=n));
auto .
Save .
Hint Resolve serie_mult .
Lemma serie_prod_maj :
forall (
f g :
nat ->
U ),
serie (
fun k =>
f k *
g k ) <=
serie f .
auto .
Save .
Hint Resolve serie_prod_maj .
Lemma serie_prod_le :
forall (
f g :
nat ->
U ) (
c :U), (
forall k ,
f k <=
c )
->
wretract g ->
serie (
fun k =>
f k *
g k ) <=
c *
serie g .
intros ;
transitivity (
serie (
fun k =>
c *
g k ));
auto .
Save .
Lemma serie_prod_ge :
forall (
f g :
nat ->
U ) (
c :U), (
forall k ,
c <= (
f k ))
->
wretract g ->
c *
serie g <=
serie (
fun k =>
f k *
g k ).
intros ;
transitivity (
serie (
fun k =>
c *
g k ));
auto .
rewrite serie_mult ;
auto .
Save .
Hint Resolve serie_prod_le serie_prod_ge .
Lemma serie_inv_le :
forall (
f g :
nat ->
U ),
wretract f ->
serie (
fun k =>
f k * [1-] (
g k )) <= [1-] (
serie (
fun k =>
f k *
g k )).
unfold serie ;
intros .
apply lub_lub_inv_le ;
intros .
rewrite sigma_inv ;
auto .
Save .
Definition Serie : (
nat ->
U ) -m>
U .
exists serie .
red ;
intros ;
apply serie_le_compat ;
auto .
Defined .
Lemma Serie_simpl :
forall f ,
Serie f =
serie f .
trivial .
Save .
Lemma serie_continuous :
continuous Serie .
red ;
intros ;
rewrite Serie_simpl ;
unfold serie .
transitivity (
lub (
lub (
Sigma @
h ))).
apply lub_le_compat ;
intro n ;
auto .
apply sigma_continuous1 .
rewrite (
double_lub_shift (
Sigma @
h )).
apply lub_le_compat ;
intro n .
rewrite comp_simpl ;
rewrite Serie_simpl .
unfold serie ;
rewrite fmon_lub_simpl .
apply lub_le_compat ;
intro m .
repeat rewrite mshift_simpl ;
auto .
Save .
html/index.haux 0000644 0001750 0001750 00000036150 11301733652 013676 0 ustar cpaulin cpaulin \@addtocsec{htoc}{1}{0}{\@print{1}\quad{}Introduction{}}
\citation{CoqManualV8}
\citation{audebaud06mpc}
\citation{audebaud07scp}
\citation{kozen81jcss}
\citation{kozen83}
\@addtocsec{htoc}{2}{0}{\@print{2}\quad{}Misc.v: Preliminaries{}}
\@addtocsec{htoc}{3}{1}{\@print{2.1}\quad{}Definition of iterator \texorpdfstring{\protect\coqdocvar{compn}}{compn}{}}
\@addtocsec{htoc}{4}{1}{\@print{2.2}\quad{}Reducing if constructs{}}
\@addtocsec{htoc}{5}{1}{\@print{2.3}\quad{}Classical reasoning{}}
\@addtocsec{htoc}{6}{1}{\@print{2.4}\quad{}Extensional equality{}}
\@addtocsec{htoc}{7}{0}{\@print{3}\quad{}Ccpo.v: Specification and properties of a cpo{}}
\@addtocsec{htoc}{8}{1}{\@print{3.1}\quad{}Ordered type{}}
\@addtocsec{htoc}{9}{2}{\@print{3.1.1}\quad{}Dual order{}}
\@addtocsec{htoc}{10}{2}{\@print{3.1.2}\quad{}Order on functions{}}
\@addtocsec{htoc}{11}{1}{\@print{3.2}\quad{}Monotonicity{}}
\@addtocsec{htoc}{12}{2}{\@print{3.2.1}\quad{}Definition and properties{}}
\@addtocsec{htoc}{13}{2}{\@print{3.2.2}\quad{}Type of monotonic functions{}}
\@addtocsec{htoc}{14}{2}{\@print{3.2.3}\quad{}Monotonicity and dual order{}}
\@addtocsec{htoc}{15}{2}{\@print{3.2.4}\quad{}Monotonic functions with 2 arguments{}}
\@addtocsec{htoc}{16}{1}{\@print{3.3}\quad{}Sequences{}}
\@addtocsec{htoc}{17}{2}{\@print{3.3.1}\quad{}Order on natural numbers{}}
\@addtocsec{htoc}{18}{2}{\@print{3.3.2}\quad{}Monotonicity and functions{}}
\@addtocsec{htoc}{19}{1}{\@print{3.4}\quad{}Basic operators of omega-cpos{}}
\@addtocsec{htoc}{20}{2}{\@print{3.4.1}\quad{}Definition of cpos{}}
\@addtocsec{htoc}{21}{2}{\@print{3.4.2}\quad{}Least upper bounds{}}
\@addtocsec{htoc}{22}{2}{\@print{3.4.3}\quad{}Functional cpos{}}
\@addtocsec{htoc}{23}{1}{\@print{3.5}\quad{}Cpo of monotonic functions{}}
\@addtocsec{htoc}{24}{1}{\@print{3.6}\quad{}Continuity{}}
\@addtocsec{htoc}{25}{2}{\@print{3.6.1}\quad{}Continuity{}}
\@addtocsec{htoc}{26}{1}{\@print{3.7}\quad{}Cpo of continuous functions{}}
\@addtocsec{htoc}{27}{1}{\@print{3.8}\quad{}Fixpoints{}}
\@addtocsec{htoc}{28}{2}{\@print{3.8.1}\quad{}Induction principle{}}
\@addtocsec{htoc}{29}{0}{\@print{4}\quad{}Utheory.v: Specification of $U$, interval $[0,1]${}}
\@addtocsec{htoc}{30}{1}{\@print{4.1}\quad{}Basic operators of $U${}}
\@addtocsec{htoc}{31}{1}{\@print{4.2}\quad{}Basic Properties{}}
\@addtocsec{htoc}{32}{0}{\@print{5}\quad{}Uprop.v : Properties of operators on [0,1]{}}
\@addtocsec{htoc}{33}{1}{\@print{5.1}\quad{}Direct consequences of axioms{}}
\@addtocsec{htoc}{34}{1}{\@print{5.2}\quad{}Properties of == derived from properties of $\le${}}
\@addtocsec{htoc}{35}{1}{\@print{5.3}\quad{}\texorpdfstring{\protect\coqdocvar{U}}{U} is a setoid{}}
\@addtocsec{htoc}{36}{1}{\@print{5.4}\quad{}Definition and properties of $x
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(13 entries)
M (instance)
mfun2_mon [in
Ccpo ]
Monad.app_mon [in
Monads ]
monotonic2_stable2 [in
Ccpo ]
monotonic_comp [in
Ccpo ]
monotonic_comp_mon [in
Ccpo ]
monotonic_lub_comp [in
Ccpo ]
monotonic_morphism [in
Ccpo ]
monotonic_stable [in
Ccpo ]
mon2_intro [in
Ccpo ]
mon_app2 [in
Ccpo ]
mon_app2_mon [in
Ccpo ]
mon_cte [in
Ccpo ]
mon_diag [in
Ccpo ]
mon_fun_lub [in
Ccpo ]
mon_fun_mon [in
Ccpo ]
mon_id [in
Ccpo ]
mon_seq_lift_left [in
Ccpo ]
mon_seq_lift_right [in
Ccpo ]
mshift_continuous [in
Ccpo ]
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(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_library_Z.html 0000644 0001750 0001750 00000022476 11301735615 015721 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_library_N.html 0000644 0001750 0001750 00000022476 11301735615 015705 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_projection_F.html 0000644 0001750 0001750 00000023277 11301735615 016405 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
F (projection)
fcontinuous [in
Ccpo ]
fcontm [in
Ccpo ]
fmonotonic [in
Ccpo ]
fmont [in
Ccpo ]
html/index_section_Q.html 0000644 0001750 0001750 00000022476 11301735615 015710 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_inductive_R.html 0000644 0001750 0001750 00000022476 11301735615 016237 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_constructor_J.html 0000644 0001750 0001750 00000022476 11301735615 016622 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_definition_P.html 0000644 0001750 0001750 00000027745 11301735615 016377 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
P (definition)
Proba.cfalse [in
Probas ]
Proba.ctrue [in
Probas ]
Proba.Discrete [in
Probas ]
Proba.discrete [in
Probas ]
Proba.distr_div [in
Probas ]
Proba.distr_mult [in
Probas ]
Proba.distr_null [in
Probas ]
Proba.Flip [in
Probas ]
Proba.flip [in
Probas ]
Proba.fnth [in
Probas ]
Proba.is_discrete [in
Probas ]
Proba.mcond [in
Probas ]
Proba.Mcond [in
Probas ]
Proba.Mdiv [in
Probas ]
Proba.Mfix [in
Probas ]
Proba.MFix [in
Probas ]
Proba.Miter [in
Probas ]
Proba.MLet [in
Probas ]
Proba.Mlet [in
Probas ]
Proba.Mmult [in
Probas ]
Proba.Mu [in
Probas ]
Proba.Munit [in
Probas ]
Proba.mu_lub [in
Probas ]
Proba.Random [in
Probas ]
Proba.random [in
Probas ]
Proba.serie_fun [in
Probas ]
Proba.Sigma_fun [in
Probas ]
Proba.sigma_fun [in
Probas ]
html/index_module_I.html 0000644 0001750 0001750 00000022720 11301735615 015511 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
I (module)
IterFlip [in
IterFlip ]
html/index_record_K.html 0000644 0001750 0001750 00000022476 11301735615 015514 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_variable_H.html 0000644 0001750 0001750 00000022476 11301735615 016020 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_axiom_N.html 0000644 0001750 0001750 00000022476 11301735615 015356 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_definition_E.html 0000644 0001750 0001750 00000023362 11301735615 016353 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
E (definition)
empty [in
Sets ]
eqset [in
Sets ]
eq_ord [in
Ccpo ]
eq_rel [in
Ccpo ]
exc [in
Misc ]
html/index_section_A.html 0000644 0001750 0001750 00000022476 11301735615 015670 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_constructor_Q.html 0000644 0001750 0001750 00000022476 11301735615 016631 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_module_M.html 0000644 0001750 0001750 00000023017 11301735615 015515 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
M (module)
Monad [in
Monads ]
MP [in
Probas ]
html/index_global_A.html 0000644 0001750 0001750 00000025060 11301735615 015454 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
A
add [definition, in
Sets ]
add_eqset_in [lemma, in
Sets ]
add_in [lemma, in
Sets ]
add_incl [lemma, in
Sets ]
add_incl_intro [lemma, in
Sets ]
add_intro [lemma, in
Sets ]
add_in_eq [lemma, in
Sets ]
add_rem_diff_eqset [lemma, in
Sets ]
add_rem_eqset_in [lemma, in
Sets ]
add_rem_eq_eqset [lemma, in
Sets ]
admissible [definition, in
Ccpo ]
app2 [definition, in
Ccpo ]
app2_simpl [lemma, in
Ccpo ]
html/index_library_R.html 0000644 0001750 0001750 00000022476 11301735615 015711 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_axiom_H.html 0000644 0001750 0001750 00000022476 11301735615 015350 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_record_C.html 0000644 0001750 0001750 00000023146 11301735615 015477 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
C (record)
continuous [in
Ccpo ]
continuous2 [in
Ccpo ]
cpo [in
Ccpo ]
html/index_global_V.html 0000644 0001750 0001750 00000022476 11301735615 015511 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_library_D.html 0000644 0001750 0001750 00000022476 11301735615 015673 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_section_L.html 0000644 0001750 0001750 00000022476 11301735615 015703 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_record_G.html 0000644 0001750 0001750 00000022476 11301735615 015510 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_lemma_N.html 0000644 0001750 0001750 00000023224 11301735615 015324 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
N (lemma)
nat_monotonic [in
Ccpo ]
not_and_elim_left [in
Misc ]
not_and_elim_right [in
Misc ]
html/index_inductive_N.html 0000644 0001750 0001750 00000022476 11301735615 016233 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_instance_O.html 0000644 0001750 0001750 00000023607 11301735615 016043 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
O (instance)
Ole_eq_compat_iff [in
Ccpo ]
OrderEqRefl [in
Ccpo ]
OrderEqSym [in
Ccpo ]
OrderEqTrans [in
Ccpo ]
OrderEquiv [in
Ccpo ]
OrdSetoid [in
Ccpo ]
html/index_projection_S.html 0000644 0001750 0001750 00000023053 11301735615 016412 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
S (projection)
stable2_intro [in
Ccpo ]
stable_def [in
Ccpo ]
html/index_projection_Q.html 0000644 0001750 0001750 00000022476 11301735615 016420 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_variable_A.html 0000644 0001750 0001750 00000022476 11301735615 016011 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_module_A.html 0000644 0001750 0001750 00000022476 11301735615 015511 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_inductive_A.html 0000644 0001750 0001750 00000022476 11301735615 016216 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_constructor_G.html 0000644 0001750 0001750 00000022476 11301735615 016617 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_lemma_A.html 0000644 0001750 0001750 00000024347 11301735615 015316 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
A (lemma)
add_eqset_in [in
Sets ]
add_in [in
Sets ]
add_incl [in
Sets ]
add_incl_intro [in
Sets ]
add_intro [in
Sets ]
add_in_eq [in
Sets ]
add_rem_diff_eqset [in
Sets ]
add_rem_eqset_in [in
Sets ]
add_rem_eq_eqset [in
Sets ]
app2_simpl [in
Ccpo ]
html/index_variable_C.html 0000644 0001750 0001750 00000024721 11301735615 016006 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
C (variable)
Choice.CHOICE.A [in
Choice ]
Choice.CHOICE.choice [in
Choice ]
Choice.CHOICE.choiceok [in
Choice ]
Choice.CHOICE.p1 [in
Choice ]
Choice.CHOICE.p1_terminates [in
Choice ]
Choice.CHOICE.p2 [in
Choice ]
Choice.CHOICE.p2_terminates [in
Choice ]
Choice.CHOICE.Q [in
Choice ]
CoverFun.SigmaFinite.A [in
Cover ]
CoverFun.SigmaFinite.decA [in
Cover ]
html/index_section_Y.html 0000644 0001750 0001750 00000022735 11301735615 015716 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
Y (section)
Ycart.YcartExample [in
Ycart ]
html/index_lemma_D.html 0000644 0001750 0001750 00000023343 11301735615 015314 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
D (lemma)
diag_le_compat [in
Ccpo ]
diag_shift [in
Ccpo ]
double_lub_diag [in
Ccpo ]
double_lub_shift [in
Ccpo ]
html/index_inductive_P.html 0000644 0001750 0001750 00000022476 11301735615 016235 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_instance_Q.html 0000644 0001750 0001750 00000022476 11301735615 016050 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_global_Y.html 0000644 0001750 0001750 00000030150 11301735615 015500 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
Y
Ycart [module, in
Ycart ]
Ycart [library]
Ycart.F [definition, in
Ycart ]
Ycart.FYcart [definition, in
Ycart ]
Ycart.FYcart_lt [definition, in
Ycart ]
Ycart.FYcart_lt_mon [instance, in
Ycart ]
Ycart.FYcart_lt_simpl [lemma, in
Ycart ]
Ycart.FYcart_mon [instance, in
Ycart ]
Ycart.FYcart_simpl [lemma, in
Ycart ]
Ycart.FYcart_val [lemma, in
Ycart ]
Ycart.P [definition, in
Ycart ]
Ycart.p [definition, in
Ycart ]
Ycart.p_diff [lemma, in
Ycart ]
Ycart.p_equation [lemma, in
Ycart ]
Ycart.p_lub [lemma, in
Ycart ]
Ycart.P_prod [lemma, in
Ycart ]
Ycart.Ycart [definition, in
Ycart ]
Ycart.YcartExample [section, in
Ycart ]
Ycart.YcartExample.K [variable, in
Ycart ]
Ycart.YcartExample.N [variable, in
Ycart ]
Ycart.Ycart_eqx [lemma, in
Ycart ]
Ycart.Ycart_ltx [lemma, in
Ycart ]
Ycart.Ycart_one [definition, in
Ycart ]
Ycart.Ycart_one_mon [instance, in
Ycart ]
Ycart.Ycart_one_simpl [lemma, in
Ycart ]
Ycart.Ycart_term1 [lemma, in
Ycart ]
Ycart.Ycart_term2 [lemma, in
Ycart ]
html/index_library_I.html 0000644 0001750 0001750 00000022640 11301735615 015671 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
I (library)
IterFlip
html/index_constructor_K.html 0000644 0001750 0001750 00000022476 11301735615 016623 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_global__.html 0000644 0001750 0001750 00000022476 11301735615 015522 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_section_P.html 0000644 0001750 0001750 00000023067 11301735615 015704 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
P (section)
Proba.Continuity [in
Probas ]
Proba.Lubs [in
Probas ]
html/index_lemma_C.html 0000644 0001750 0001750 00000070274 11301735615 015320 0 ustar cpaulin cpaulin
Ycart
html/index_global_Q.html 0000644 0001750 0001750 00000022476 11301735615 015504 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_instance_J.html 0000644 0001750 0001750 00000022476 11301735615 016041 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_library_E.html 0000644 0001750 0001750 00000022476 11301735615 015674 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_instance_G.html 0000644 0001750 0001750 00000022476 11301735615 016036 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_inductive_E.html 0000644 0001750 0001750 00000022476 11301735615 016222 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)
html/index_instance_V.html 0000644 0001750 0001750 00000022476 11301735615 016055 0 ustar cpaulin cpaulin
Ycart
Global Index
A B C D E F G H
I J
K
L M N O P Q
R S T U V
W
X
Y Z
_
(1807 entries)
Instance Index
A
B C D
E
F G
H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(109 entries)
Projection Index
A
B
C D E
F G
H
I
J
K
L M N
O P Q
R S T U V
W
X
Y
Z
_
(32 entries)
Record Index
A
B
C D
E
F G
H
I
J
K
L
M N
O P Q
R
S T
U V
W
X
Y
Z
_
(14 entries)
Lemma Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(1217 entries)
Section Index
A
B
C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(22 entries)
Constructor Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U V
W
X
Y
Z
_
(12 entries)
Inductive Index
A
B
C D
E
F G
H
I
J
K
L
M N
O
P
Q
R
S T
U
V
W
X
Y
Z
_
(7 entries)
Definition Index
A B C D E F G H
I J
K
L M N O P Q
R S T
U V
W
X
Y Z
_
(277 entries)
Module Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R S
T
U V
W
X
Y Z
_
(18 entries)
Axiom Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U V
W
X
Y
Z
_
(6 entries)
Variable Index
A
B C D
E
F
G
H
I
J
K
L
M
N
O
P Q
R S T
U V
W
X
Y Z
_
(80 entries)
Library Index
A
B C D
E
F
G
H
I J
K
L
M N
O
P Q
R
S T
U V
W
X
Y Z
_
(13 entries)