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(**************************************************************************)
(* *)
(* SMTCoq *)
(* Copyright (C) 2011 - 2021 *)
(* *)
(* See file "AUTHORS" for the list of authors *)
(* *)
(* This file is distributed under the terms of the CeCILL-C licence *)
(* *)
(**************************************************************************)
Require Import SMTCoq.
Require Import Bool PArray Int63 List ZArith BVList Logic.
Import ListNotations.
Local Open Scope list_scope.
Local Open Scope int63_scope.
Local Open Scope Z_scope.
Local Open Scope bv_scope.
Infix "-->" := implb (at level 60, right associativity) : bool_scope.
Import BVList.BITVECTOR_LIST.
Require Import FArray.
Section BV.
Import BVList.BITVECTOR_LIST.
Local Open Scope bv_scope.
Goal forall (bv1 bv2 bv3 bv4: bitvector 4),
bv_eq #b|0|0|0|0| bv1 &&
bv_eq #b|1|0|0|0| bv2 &&
bv_eq #b|1|1|0|0| bv3 -->
bv_eq #b|1|1|1|0| bv4 -->
bv_ult bv1 bv2 || bv_ult bv3 bv1 --> bv_ult bv1 bv3 --> bv_ult bv1 bv4 || bv_ult bv4 bv1.
Proof.
smt.
Qed.
Goal forall (a: bitvector 32), bv_eq a a.
Proof.
smt.
Qed.
Goal forall (bv1 bv2: bitvector 4),
(Bool.eqb (bv_eq bv1 bv2) (bv_eq bv2 bv1)).
Proof.
smt.
Qed.
Goal forall (bv1 bv2 bv3 bv4: bitvector 4),
bv_eq #b|0|0|0|0| bv1 &&
bv_eq #b|1|0|0|0| bv2 &&
bv_eq #b|1|1|0|0| bv3 -->
bv_eq #b|1|1|1|0| bv4 -->
bv_ult bv1 bv2 || bv_ult bv3 bv1 && bv_ult bv3 bv4.
Proof.
smt.
Qed.
Goal forall (a b c: bitvector 4),
(bv_eq c (bv_and a b)) -->
(bv_eq (bv_and (bv_and c a) b) c).
Proof.
smt.
Qed.
End BV.
Section Arrays.
Import BVList.BITVECTOR_LIST.
Import FArray.
Local Open Scope farray_scope.
Local Open Scope bv_scope.
Goal forall (a:farray Z Z), equal a a.
Proof.
smt.
Qed.
Goal forall (a b: farray Z Z), Bool.eqb (equal a b) (equal b a).
Proof.
smt.
Qed.
Goal forall (a b: farray (bitvector 8) (bitvector 8)), Bool.eqb (equal a b) (equal b a).
Proof.
smt.
Qed.
Goal forall (a b c d: farray Z Z),
equal b[0 <- 4] c -->
equal d b[0 <- 4][1 <- 4] &&
equal a d[1 <- b[1]] -->
equal a c.
Proof.
smt.
Qed.
Goal forall (bv1 bv2 : bitvector 4)
(a b c d : farray (bitvector 4) Z),
bv_eq #b|0|0|0|0| bv1 -->
bv_eq #b|1|0|0|0| bv2 -->
equal c b[bv1 <- 4] -->
equal d b[bv1 <- 4][bv2 <- 4] -->
equal a d[bv2 <- b[bv2]] -->
equal a c.
Proof.
smt.
Qed.
Goal forall (a b: farray Z Z)
(v w x y: Z)
(g: farray Z Z -> Z)
(f: Z -> Z),
(equal a[x <- v] b) && (equal a[y <- w] b) --> (Z.eqb (f x) (f y)) || (Z.eqb (g a) (g b)).
Proof.
smt.
Qed.
Goal forall (a b: farray Z Z) i,
Z.eqb (select (store (store (store a i 3%Z) 1%Z (select (store b i 4) i)) 2%Z 2%Z) 1%Z) 4.
Proof.
smt.
rewrite read_over_other_write; try easy.
rewrite read_over_same_write; try easy; try apply Z_compdec.
rewrite read_over_same_write; try easy; try apply Z_compdec.
Qed.
End Arrays.
Section UF.
Goal forall
(x y: Z)
(f: Z -> Z),
Z.eqb y x --> Z.eqb (f x) (f y).
Proof.
smt.
Qed.
Goal forall
(x y: Z)
(f: Z -> Z),
Z.eqb (f x) (f y) --> Z.eqb (f y) (f x).
Proof.
smt.
Qed.
Goal forall
(x y: Z)
(f: Z -> Z),
Z.eqb (x + 1) (y + 1) --> Z.eqb (f y) (f x).
Proof.
smt.
Qed.
Goal forall
(x y: Z)
(f: Z -> Z),
Z.eqb x (y + 1) --> Z.eqb (f y) (f (x - 1)).
Proof.
smt.
Qed.
Goal forall (f:Z -> Z -> Z) x y z, (Z.eqb x y) --> Z.eqb (f z x) (f z y).
Proof.
smt.
Qed.
End UF.
Section LIA.
Goal forall (a b: Z), Bool.eqb (Z.eqb a b) (Z.eqb b a).
Proof.
smt.
Qed.
Goal forall (a b: Z), (Z.eqb a a) && (Z.eqb b b).
Proof.
smt.
Qed.
End LIA.
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