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|
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.
*)
|
