Library abstraction_hfl_prim_def
Set Implicit Arguments.
Require Import Setoid.
Require Import floor.
Require Import Bool.
Require Import Omega.
Require Import ZArith.
Require Import Znumtheory.
Require Import Z_misc.
Require Import QArith.
Require Import Q_misc.
Require Import comparison.
Require Import misc.
Require Import ibw_def.
Record abstractionh : Set :=
abshmake { absh_b1: Q; absh_b0: Q; absh_r: Q }.
Definition well_formed_abstractionh (a:abstractionh) :=
0 <= absh_r a <= 1.
Definition in_abstractionh (w: ibw) (a:abstractionh) :=
forall i: nat,
forall H_i_ge_1: (i >= 1)%nat,
(w i = true -> ones w i <= absh_r a * i + absh_b1 a)
/\
(w i = false -> ones w i >= absh_r a * i + absh_b0 a).
Fixpoint ones_early (a:abstractionh) (i:nat) { struct i } :=
match i with
| O => O
| S i' =>
let ones_i' := ones_early a i' in
match Qcompare (S ones_i') (absh_r a * i + absh_b1 a) with
| Lt | Eq => S ones_i'
| Gt => ones_i'
end
end.
Fixpoint ones_late (a:abstractionh) (i:nat) { struct i } :=
match i with
| O => O
| S i' =>
let ones_i' := ones_late a i' in
match Qcompare (absh_r a * i + absh_b0 a) ones_i' with
| Lt | Eq => ones_i'
| _ => S ones_i'
end
end.
Definition ones_early_alt (a:abstractionh) (i:nat) : nat :=
Zabs_nat (Zmax O (Zmin i (fl (absh_r a * i + absh_b1 a)))).
Definition ones_late_alt (a:abstractionh) (i:nat) : nat :=
Zabs_nat (Zmax O (Zmin i (cg (absh_r a * i + absh_b0 a)))).
Definition w_early (a:abstractionh) : ibw :=
ibw_of_ones (ones_early a).
Definition w_late (a:abstractionh) : ibw :=
ibw_of_ones (ones_late a).
Definition non_empty (a: abstractionh) :=
exists w:ibw, in_abstractionh w a.
Definition absh_std a :=
abshmake
((Qnum (absh_b1 a) * QDen (absh_r a))%Z # (Qden (absh_b1 a) * Qden (absh_r a)))
(absh_b0 a)
((Qnum (absh_r a) * QDen (absh_b1 a))%Z # (Qden (absh_b1 a) * Qden (absh_r a))).
Definition non_empty_test (a: abstractionh) :=
let a_std := absh_std a in
absh_b1 a_std - absh_b0 a_std >= 1 - (1 # Qden (absh_r a_std)).
Definition non_empty_test_alt (a: abstractionh) :=
Qden (absh_b1 a) = Qden (absh_r a)
/\
absh_b1 a - absh_b0 a >= 1 - (1 # Qden (absh_r a)).
Definition absh_included (a1: abstractionh) (a2:abstractionh) :=
forall w: ibw,
forall H_w_in_a1: in_abstractionh w a1,
in_abstractionh w a2.
Definition absh_included_test (a1: abstractionh) (a2:abstractionh) :=
(absh_r a1 == absh_r a2) /\
(absh_b1 a1) <= (absh_b1 a2) /\
(absh_b0 a1) >= (absh_b0 a2).
Definition on_absh (a1: abstractionh) (a2:abstractionh) :=
let absh_b0_1 :=
match Qcompare (absh_b0 a1) 0 with
| Gt => 0
| _ => absh_b0 a1
end
in
let absh_b0_2 :=
match Qcompare (absh_b0 a2) 0 with
| Gt => 0
| _ => absh_b0 a2
end
in
abshmake
(absh_b1 a1 * absh_r a2 + absh_b1 a2)
(absh_b0_1 * absh_r a2 + absh_b0_2)
(absh_r a1 * absh_r a2).
Definition not_absh (a: abstractionh) :=
abshmake
(- absh_b0 a)
(- absh_b1 a)
(1 - absh_r a).
Definition sync_absh (a1 a2: abstractionh) : Prop :=
forall w1: ibw,
forall w2: ibw,
forall H_w1_in_a1: in_abstractionh w1 a1,
forall H_w2_in_a2: in_abstractionh w2 a2,
sync w1 w2.
Definition sync_absh_test (a1 a2: abstractionh) : Prop :=
absh_r a1 == absh_r a2.
Definition prec_absh (a1 a2: abstractionh) : Prop :=
forall w1: ibw,
forall w2: ibw,
forall H_w1_in_a1: in_abstractionh w1 a1,
forall H_w2_in_a2: in_abstractionh w2 a2,
prec w1 w2.
Definition prec_absh_alt (a1 a2: abstractionh) : Prop :=
prec (w_late a1) (w_early a2).
Definition prec_absh_test (a1 a2: abstractionh) : Prop :=
absh_b1 a2 - absh_b0 a1 < 1.
Definition subtyping_absh (a1 a2: abstractionh) : Prop :=
forall w1: ibw,
forall w2: ibw,
forall H_w1_in_a1: in_abstractionh w1 a1,
forall H_w2_in_a2: in_abstractionh w2 a2,
subtyping w1 w2.
Definition subtyping_absh_alt (a1 a2: abstractionh) : Prop :=
prec_absh a1 a2 /\ sync_absh a1 a2.