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Require Import ZArith.
Require Import Classical.
Require Export Reals.
(* real numbers *)
Lemma real_expr: (0 <= 9 * 4)%R.
ergo.
Qed.
Lemma powerRZ_translation: (powerRZ 2 15 < powerRZ 2 17)%R.
ergo.
Qed.
Dp_debug.
Dp_timeout 3.
(* module renamings *)
Module M.
Parameter t : Set.
End M.
Lemma test_module_0 : forall x:M.t, x=x.
ergo.
Qed.
Module N := M.
Lemma test_module_renaming_0 : forall x:N.t, x=x.
ergo.
Qed.
Dp_predefined M.t => "int".
Lemma test_module_renaming_1 : forall x:N.t, x=x.
ergo.
Qed.
(* Coq lists *)
Require Export List.
Lemma test_pol_0 : forall l:list nat, l=l.
ergo.
Qed.
Parameter nlist: list nat -> Prop.
Lemma poly_1 : forall l, nlist l -> True.
intros.
simplify.
Qed.
(* user lists *)
Inductive list (A:Set) : Set :=
| nil : list A
| cons: forall a:A, list A -> list A.
Fixpoint app (A:Set) (l m:list A) {struct l} : list A :=
match l with
| nil => m
| cons a l1 => cons A a (app A l1 m)
end.
Lemma entail: (nil Z) = app Z (nil Z) (nil Z) -> True.
intros; ergo.
Qed.
(* polymorphism *)
Require Import List.
Inductive mylist (A:Set) : Set :=
mynil : mylist A
| mycons : forall a:A, mylist A -> mylist A.
Parameter my_nlist: mylist nat -> Prop.
Goal forall l, my_nlist l -> True.
intros.
simplify.
Qed.
(* First example with the 0 and the equality translated *)
Goal 0 = 0.
simplify.
Qed.
(* Examples in the Propositional Calculus
and theory of equality *)
Parameter A C : Prop.
Goal A -> A.
simplify.
Qed.
Goal A -> (A \/ C).
simplify.
Qed.
Parameter x y z : Z.
Goal x = y -> y = z -> x = z.
ergo.
Qed.
Goal ((((A -> C) -> A) -> A) -> C) -> C.
ergo.
Qed.
(* Arithmetic *)
Open Scope Z_scope.
Goal 1 + 1 = 2.
yices.
Qed.
Goal 2*x + 10 = 18 -> x = 4.
simplify.
Qed.
(* Universal quantifier *)
Goal (forall (x y : Z), x = y) -> 0=1.
try zenon.
ergo.
Qed.
Goal forall (x: nat), (x + 0 = x)%nat.
induction x0; ergo.
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 dp/injection 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.
ergo.
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.
yices.
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 ; yices.
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.
yices.
(*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; ergo.
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.
ergo.
(*
simplify.
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.
yices.
(*zenon.*)
Qed.
(* abstractions *)
Parameter poly_f : forall A:Set, A->A.
Goal forall x:nat, poly_f nat x = poly_f nat x.
ergo.
(*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.
*)
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