Library abstraction_jfla_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 abstractionj : Set :=
absjmake { absj_b1: Q; absj_b0: Q; absj_r: Q }.
Definition well_formed_abstractionj (a:abstractionj) :=
0 <= absj_r a <= 1.
Definition in_abstractionj (w: ibw) (a:abstractionj) :=
forall i: nat,
forall H_i_ge_1: (i >= 1)%nat,
(w i = true -> ones w (pred i) < absj_r a * i + absj_b1 a)
/\
(w i = false -> ones w (pred i) >= absj_r a * i + absj_b0 a).
Fixpoint ones_early (a:abstractionj) (i:nat) { struct i } :=
match i with
| O => O
| S i' =>
let ones_i' := ones_early a i' in
match Qcompare ones_i' (absj_r a * i + absj_b1 a) with
| Lt => S ones_i'
| Gt | Eq => ones_i'
end
end.
Fixpoint ones_late (a:abstractionj) (i:nat) { struct i } :=
match i with
| O => O
| S i' =>
let ones_i' := ones_late a i' in
match Qcompare (absj_r a * i + absj_b0 a) ones_i' with
| Lt | Eq => ones_i'
| _ => S ones_i'
end
end.
Definition ones_early_alt (a:abstractionj) (i:nat) : nat :=
Zabs_nat (Zmax O (Zmin i (cg (absj_r a * i + absj_b1 a)))).
Definition ones_late_alt (a:abstractionj) (i:nat) : nat :=
Zabs_nat (Zmax O (Zmin i (cg (absj_r a * i + absj_b0 a)))).
Definition w_early (a:abstractionj) : ibw :=
ibw_of_ones (ones_early a).
Definition w_late (a:abstractionj) : ibw :=
ibw_of_ones (ones_late a).
Definition non_empty (a: abstractionj) :=
exists w:ibw, in_abstractionj w a.
Definition non_empty_test (a: abstractionj) :=
absj_b0 a <= absj_b1 a.
Definition absj_included (a1: abstractionj) (a2:abstractionj) :=
forall w: ibw,
forall H_w_in_a1: in_abstractionj w a1,
in_abstractionj w a2.
Definition absj_included_test (a1: abstractionj) (a2:abstractionj) :=
(absj_r a1 == absj_r a2) /\
(absj_b1 a1) <= (absj_b1 a2) /\
(absj_b0 a1) >= (absj_b0 a2).
Definition on_absj_alt (a1: abstractionj) (a2:abstractionj) :=
let absj_b0_1 :=
match Qcompare (absj_b0 a1) 0 with
| Gt => 0
| _ => absj_b0 a1
end
in
let absj_b0_2 :=
match Qcompare (absj_b0 a2) 0 with
| Gt => 0
| _ => absj_b0 a2
end
in
let delta :=
(1 - (1 # Qden (absj_r a1) )) * ((absj_r a2) - (1 # Qden (absj_r a2)))
in
absjmake
(absj_b1 a1 * absj_r a2 + absj_b1 a2 + delta)
(absj_b0_1 * absj_r a2 + absj_b0_2)
(absj_r a1 * absj_r a2).
Definition absj_std a :=
absjmake
(absj_b1 a * (QDen (absj_r a) # Qden (absj_r a)))
(absj_b0 a)
(absj_r a * (QDen (absj_b1 a) # Qden (absj_b1 a))).
Definition on_absj (a1: abstractionj) (a2:abstractionj) :=
let a1_std := absj_std a1 in
let a2_std := absj_std a2 in
on_absj_alt a1_std a2_std.
Definition not_absj (a: abstractionj) :=
let epsilon := 1 - (1 # Qden (absj_r a) ) in
absjmake
(- absj_b0 a - epsilon)
(- absj_b1 a - epsilon)
(1 - absj_r a).
Definition sync_absj (a1 a2: abstractionj) : Prop :=
forall w1: ibw,
forall w2: ibw,
forall H_w1_in_a1: in_abstractionj w1 a1,
forall H_w2_in_a2: in_abstractionj w2 a2,
sync w1 w2.
Definition sync_absj_test (a1 a2: abstractionj) : Prop :=
absj_r a1 == absj_r a2.
Definition prec_absj (a1 a2: abstractionj) : Prop :=
forall w1: ibw,
forall w2: ibw,
forall H_w1_in_a1: in_abstractionj w1 a1,
forall H_w2_in_a2: in_abstractionj w2 a2,
prec w1 w2.
Definition prec_absj_alt (a1 a2: abstractionj) : Prop :=
prec (w_late a1) (w_early a2).
Definition prec_absj_test (a1 a2: abstractionj) : Prop :=
absj_b0 a1 >= absj_b1 a2.
Definition subtyping_absj (a1 a2: abstractionj) : Prop :=
forall w1: ibw,
forall w2: ibw,
forall H_w1_in_a1: in_abstractionj w1 a1,
forall H_w2_in_a2: in_abstractionj w2 a2,
subtyping w1 w2.
Definition subtyping_absj_alt (a1 a2: abstractionj) : Prop :=
prec_absj a1 a2 /\ sync_absj a1 a2.