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diff --git a/contrib/dp/tests.v b/contrib/dp/tests.v new file mode 100644 index 00000000..52a57a0c --- /dev/null +++ b/contrib/dp/tests.v @@ -0,0 +1,220 @@ + +Require Import ZArith. +Require Import Classical. + +(* First example with the 0 and the equality translated *) + +Goal 0 = 0. +zenon. +Qed. + + +(* Examples in the Propositional Calculus + and theory of equality *) + +Parameter A C : Prop. + +Goal A -> A. +zenon. +Qed. + + +Goal A -> (A \/ C). + +zenon. +Qed. + + +Parameter x y z : Z. + +Goal x = y -> y = z -> x = z. + +zenon. +Qed. + + +Goal ((((A -> C) -> A) -> A) -> C) -> C. + +zenon. +Qed. + + +(* Arithmetic *) +Open Scope Z_scope. + +Goal 1 + 1 = 2. +simplify. +Qed. + + +Goal 2*x + 10 = 18 -> x = 4. + +simplify. +Qed. + + +(* Universal quantifier *) + +Goal (forall (x y : Z), x = y) -> 0=1. +try zenon. +simplify. +Qed. + +Goal forall (x: nat), (x + 0 = x)%nat. + +induction x0. +zenon. +zenon. +Qed. + + +(* No decision procedure can solve this problem + Goal forall (x a b : Z), a * x + b = 0 -> x = - b/a. +*) + + +(* Functions definitions *) + +Definition fst (x y : Z) : Z := x. + +Goal forall (g : Z -> Z) (x y : Z), g (fst x y) = g x. + +simplify. +Qed. + + +(* Eta-expansion example *) + +Definition snd_of_3 (x y z : Z) : Z := y. + +Definition f : Z -> Z -> Z := snd_of_3 0. + +Goal forall (x y z z1 : Z), snd_of_3 x y z = f y z1. + +simplify. +Qed. + + +(* Inductive types definitions - call to incontrib/dp/jection function *) + +Inductive even : Z -> Prop := +| even_0 : even 0 +| even_plus2 : forall z : Z, even z -> even (z + 2). + + +(* Simplify and Zenon can't prove this goal before the timeout + unlike CVC Lite *) + +Goal even 4. +cvcl. +Qed. + + +Definition skip_z (z : Z) (n : nat) := n. + +Definition skip_z1 := skip_z. + +Goal forall (z : Z) (n : nat), skip_z z n = skip_z1 z n. + +zenon. +Qed. + + +(* Axioms definitions and dp_hint *) + +Parameter add : nat -> nat -> nat. +Axiom add_0 : forall (n : nat), add 0%nat n = n. +Axiom add_S : forall (n1 n2 : nat), add (S n1) n2 = S (add n1 n2). + +Dp_hint add_0. +Dp_hint add_S. + +(* Simplify can't prove this goal before the timeout + unlike zenon *) + +Goal forall n : nat, add n 0 = n. + +induction n ; zenon. +Qed. + + +Definition pred (n : nat) : nat := match n with + | 0%nat => 0%nat + | S n' => n' +end. + +Goal forall n : nat, n <> 0%nat -> pred (S n) <> 0%nat. + +zenon. +Qed. + + +Fixpoint plus (n m : nat) {struct n} : nat := + match n with + | 0%nat => m + | S n' => S (plus n' m) +end. + +Goal forall n : nat, plus n 0%nat = n. + +induction n; zenon. +Qed. + + +(* Mutually recursive functions *) + +Fixpoint even_b (n : nat) : bool := match n with + | O => true + | S m => odd_b m +end +with odd_b (n : nat) : bool := match n with + | O => false + | S m => even_b m +end. + +Goal even_b (S (S O)) = true. + +zenon. +Qed. + + +(* sorts issues *) + +Parameter foo : Set. +Parameter ff : nat -> foo -> foo -> nat. +Parameter g : foo -> foo. +Goal (forall x:foo, ff 0 x x = O) -> forall y, ff 0 (g y) (g y) = O. +zenon. +Qed. + + + +(* abstractions *) + +Parameter poly_f : forall A:Set, A->A. + +Goal forall x:nat, poly_f nat x = poly_f nat x. +zenon. +Qed. + + + +(* Anonymous mutually recursive functions : no equations are produced + +Definition mrf := + fix even2 (n : nat) : bool := match n with + | O => true + | S m => odd2 m + end + with odd2 (n : nat) : bool := match n with + | O => false + | S m => even2 m + end for even. + + Thus this goal is unsolvable + +Goal mrf (S (S O)) = true. + +zenon. + +*) |