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Reset Initial.
Require Import ZArith.
Require Import Classical.
(* Premier exemple avec traduction de l'�galit� et du 0 *)
Goal 0 = 0.
simplify.
Qed.
(* Exemples dans le calcul propositionnel *)
Parameter A C : Prop.
Goal A -> A.
simplify.
Qed.
Goal A -> (A \/ C).
zenon.
Qed.
(* Arithm�tique de base *)
Parameter x y z : Z.
Goal x = y -> y = z -> x = z.
zenon.
Qed.
(* Ajo�t de quantificateurs universels *)
Goal (forall (x y : Z), x = y) -> 0 = 1.
simplify.
Qed.
(* D�finition de fonctions *)
Definition fst (x y : Z) : Z := x.
Goal forall (g : Z -> Z) (x y : Z), g (fst x y) = g x.
simplify.
Qed.
(* Exemple d'�ta-expansion *)
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.
(* D�finition de types inductifs - appel de la fonction injection *)
Inductive new : Set :=
| B : new
| N : new -> new.
Inductive even_p : new -> Prop :=
| even_p_B : even_p B
| even_p_N : forall n : new, even_p n -> even_p (N (N n)).
Goal even_p (N (N (N (N B)))).
simplify.
Qed.
Goal even_p (N (N (N (N B)))).
zenon.
Qed.
(* A noter que simplify ne parvient pas � r�soudre ce probl�me
avant le timemout au contraire de zenon *)
Definition skip_z (z : Z) (n : new) := n.
Definition skip_z1 := skip_z.
Goal forall (z : Z) (n : new), skip_z z n = skip_z1 z n.
zenon.
Qed.
(* D�finition d'axiomes et ajo�t de dp_hint *)
Parameter add : new -> new -> new.
Axiom add_B : forall (n : new), add B n = n.
Axiom add_N : forall (n1 n2 : new), add (N n1) n2 = N (add n1 n2).
Dp add_B.
Dp add_N.
(* Encore un exemple que simplify ne r�sout pas avant le timeout *)
Goal forall n : new, add n B = n.
induction n ; zenon.
Qed.
Definition newPred (n : new) : new := match n with
| B => B
| N n' => n'
end.
Goal forall n : new, n <> B -> newPred (N n) <> B.
zenon.
Qed.
Fixpoint newPlus (n m : new) {struct n} : new :=
match n with
| B => m
| N n' => N (newPlus n' m)
end.
Goal forall n : new, newPlus n B = n.
induction n; zenon.
Qed.
Definition h (n : new) (z : Z) : Z := match n with
| B => z + 1
| N n' => z - 1
end.
(* mutually recursive functions *)
Fixpoint even (n:nat) : bool := match n with
| O => true
| S m => odd m
end
with odd (n:nat) : bool := match n with
| O => false
| S m => even m
end.
Goal even (S (S O)) = true.
zenon.
(* anonymous mutually recursive functions : no equations are produced *)
Definition f :=
fix even (n:nat) : bool := match n with
| O => true
| S m => odd m
end
with odd (n:nat) : bool := match n with
| O => false
| S m => even m
end for even.
Goal f (S (S O)) = true.
zenon.
|
