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(* [Require Import SMTCoq.SMTCoq.] loads the SMTCoq library.
If you are using native-coq instead of Coq 8.6, replace it with:
Require Import SMTCoq.
*)
Require Import SMTCoq.SMTCoq.
Require Import Bool.
Local Open Scope int63_scope.
(* Examples that check ZChaff certificates *)
Zchaff_Checker "sat.cnf" "sat.log".
Zchaff_Theorem sat "sat.cnf" "sat.log".
Check sat.
Zchaff_Checker "hole4.cnf" "hole4.log".
(* Example that checks a VeriT certificate, for logic QF_UF *)
Section Verit.
Verit_Checker "euf.smt2" "euf.log".
End Verit.
(* Examples of the zchaff tactic (requires zchaff in your PATH
environment variable):
- with booleans
- with concrete terms *)
Goal forall a b c,
(a || b || c) && ((negb a) || (negb b) || (negb c)) && ((negb a) || b) && ((negb b) || c) && ((negb c) || a) = false.
Proof.
zchaff.
Qed.
Goal forall i j k,
let a := i == j in
let b := j == k in
let c := k == i in
(a || b || c) && ((negb a) || (negb b) || (negb c)) && ((negb a) || b) && ((negb b) || c) && ((negb c) || a) = false.
Proof.
zchaff.
Qed.
(* Examples of the verit tactic (requires verit in your PATH environment
variable):
- with booleans
- in logics QF_UF and QF_LIA *)
Goal forall a b c, ((a || b || c) && ((negb a) || (negb b) || (negb c)) && ((negb a) || b) && ((negb b) || c) && ((negb c) || a)) = false.
Proof.
verit.
Qed.
Goal forall (a b : Z) (P : Z -> bool) (f : Z -> Z),
negb (f a =? b) || negb (P (f a)) || (P b).
Proof.
verit.
Qed.
Goal forall b1 b2 x1 x2,
implb
(ifb b1
(ifb b2 (2*x1+1 =? 2*x2+1) (2*x1+1 =? 2*x2))
(ifb b2 (2*x1 =? 2*x2+1) (2*x1 =? 2*x2)))
((implb b1 b2) && (implb b2 b1) && (x1 =? x2)).
Proof.
verit.
Qed.
(* Examples of using the conversion tactics *)
Local Open Scope positive_scope.
Goal forall (f : positive -> positive) (x y : positive),
implb ((x + 3) =? y)
((f (x + 3)) <=? (f y))
= true.
Proof.
pos_convert.
verit.
Qed.
Goal forall (f : positive -> positive) (x y : positive),
implb ((x + 3) =? y)
((3 <? y) && ((f (x + 3)) <=? (f y)))
= true.
Proof.
pos_convert.
verit.
Qed.
Local Close Scope positive_scope.
Local Open Scope N_scope.
Goal forall (f : N -> N) (x y : N),
implb ((x + 3) =? y)
((f (x + 3)) <=? (f y))
= true.
Proof.
N_convert.
verit.
Qed.
Goal forall (f : N -> N) (x y : N),
implb ((x + 3) =? y)
((2 <? y) && ((f (x + 3)) <=? (f y)))
= true.
Proof.
N_convert.
verit.
Qed.
Local Close Scope N_scope.
Require Import NPeano.
Local Open Scope nat_scope.
Goal forall (f : nat -> nat) (x y : nat),
implb (Nat.eqb (x + 3) y)
((f (x + 3)) <=? (f y))
= true.
Proof.
nat_convert.
verit.
Qed.
Goal forall (f : nat -> nat) (x y : nat),
implb (Nat.eqb (x + 3) y)
((2 <? y) && ((f (x + 3)) <=? (f y)))
= true.
Proof.
nat_convert.
verit.
Qed.
Local Close Scope nat_scope.
Open Scope Z_scope.
(* Some examples of using verit with lemmas. Use <verit_base H1 .. Hn; vauto>
to temporarily add the lemmas H1 .. Hn to the verit environment. *)
Lemma const_fun_is_eq_val_0 :
forall f : Z -> Z,
(forall a b, f a =? f b) ->
forall x, f x =? f 0.
Proof.
intros f Hf.
verit_base Hf; vauto.
Qed.
Section Without_lemmas.
Lemma fSS:
forall (f : Z -> Z) (k : Z) (x : Z),
implb (f (x+1) =? f x + k)
(implb (f (x+2) =? f (x+1) + k)
(f (x+2) =? f x + 2 * k)).
Proof. verit. Qed.
End Without_lemmas.
Section With_lemmas.
Variable f : Z -> Z.
Variable k : Z.
Hypothesis f_k_linear : forall x, f (x + 1) =? f x + k.
Lemma fSS2:
forall x, f (x + 2) =? f x + 2 * k.
Proof. verit_base f_k_linear; vauto. Qed.
End With_lemmas.
(* You can use <Add_lemmas H1 .. Hn> to permanently add the lemmas H1 .. Hn to
the environment. If you did so in a section then, at the end of the section,
you should use <Clear_lemmas> to empty the globally added lemmas because
those lemmas won't be available outside of the section. *)
Section mult3.
Variable mult3 : Z -> Z.
Hypothesis mult3_0 : mult3 0 =? 0.
Hypothesis mult3_Sn : forall n, mult3 (n+1) =? mult3 n + 3.
Add_lemmas mult3_0 mult3_Sn.
Lemma mult3_21 : mult3 7 =? 21.
Proof. verit. Qed.
Clear_lemmas.
End mult3.
Section group.
Variable op : Z -> Z -> Z.
Variable inv : Z -> Z.
Variable e : Z.
Hypothesis associative :
forall a b c : Z, op a (op b c) =? op (op a b) c.
Hypothesis identity :
forall a : Z, (op e a =? a) && (op a e =? a).
Hypothesis inverse :
forall a : Z, (op a (inv a) =? e) && (op (inv a) a =? e).
Add_lemmas associative identity inverse.
Lemma unique_identity e':
(forall z, op e' z =? z) -> e' =? e.
Proof. intros pe'. verit_base pe'; vauto. Qed.
Lemma simplification_right x1 x2 y:
op x1 y =? op x2 y -> x1 =? x2.
Proof. intro H. verit_base H; vauto. Qed.
Lemma simplification_left x1 x2 y:
op y x1 =? op y x2 -> x1 =? x2.
Proof. intro H. verit_base H; vauto. Qed.
Clear_lemmas.
End group.
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