From 4d4f08acb5e5f56d38289e5629173bc1b8b5fd57 Mon Sep 17 00:00:00 2001
From: herbelin <herbelin@85f007b7-540e-0410-9357-904b9bb8a0f7>
Date: Wed, 21 Dec 2005 23:50:17 +0000
Subject: Abandon tests syntaxe v7; remplacement des .v par des fichiers en
 syntaxe v8

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@7693 85f007b7-540e-0410-9357-904b9bb8a0f7
---
 test-suite/success/setoid_test.v | 106 +++++++++++++++++++++++++++++++++++++++
 1 file changed, 106 insertions(+)
 create mode 100644 test-suite/success/setoid_test.v

(limited to 'test-suite/success/setoid_test.v')

diff --git a/test-suite/success/setoid_test.v b/test-suite/success/setoid_test.v
new file mode 100644
index 000000000..dd1022f08
--- /dev/null
+++ b/test-suite/success/setoid_test.v
@@ -0,0 +1,106 @@
+Require Import Setoid.
+
+Parameter A : Set.
+
+Axiom eq_dec : forall a b : A, {a = b} + {a <> b}.
+
+Inductive set : Set :=
+  | Empty : set
+  | Add : A -> set -> set.
+
+Fixpoint In (a : A) (s : set) {struct s} : Prop :=
+  match s with
+  | Empty => False
+  | Add b s' => a = b \/ In a s'
+  end.
+
+Definition same (s t : set) : Prop := forall a : A, In a s <-> In a t.
+
+Lemma setoid_set : Setoid_Theory set same.
+
+unfold same in |- *; split.
+red in |- *; auto.
+
+red in |- *.
+intros.
+elim (H a); auto.
+
+intros.
+elim (H a); elim (H0 a).
+split; auto.
+Qed.
+
+Add Setoid set same setoid_set as setsetoid.
+
+Add Morphism In : In_ext.
+unfold same in |- *; intros a s t H; elim (H a); auto.
+Qed.
+
+Lemma add_aux :
+ forall s t : set,
+ same s t -> forall a b : A, In a (Add b s) -> In a (Add b t).
+unfold same in |- *; simple induction 2; intros.
+rewrite H1.
+simpl in |- *; left; reflexivity.
+
+elim (H a).
+intros.
+simpl in |- *; right.
+apply (H2 H1).
+Qed.
+
+Add Morphism Add : Add_ext.
+split; apply add_aux.
+assumption.
+
+rewrite H.
+reflexivity.
+Qed.
+
+Fixpoint remove (a : A) (s : set) {struct s} : set :=
+  match s with
+  | Empty => Empty
+  | Add b t =>
+      match eq_dec a b with
+      | left _ => remove a t
+      | right _ => Add b (remove a t)
+      end
+  end.
+
+Lemma in_rem_not : forall (a : A) (s : set), ~ In a (remove a (Add a Empty)).
+
+intros.
+setoid_replace (remove a (Add a Empty)) with Empty.
+
+auto.
+
+unfold same in |- *.
+split.
+simpl in |- *.
+case (eq_dec a a).
+intros e ff; elim ff.
+
+intros; absurd (a = a); trivial.
+
+simpl in |- *.
+intro H; elim H.
+Qed.
+
+Parameter P : set -> Prop.
+Parameter P_ext : forall s t : set, same s t -> P s -> P t.
+
+Add Morphism P : P_extt.
+intros; split; apply P_ext; (assumption || apply (Seq_sym _ _ setoid_set); assumption).
+Qed.
+
+Lemma test_rewrite :
+ forall (a : A) (s t : set), same s t -> P (Add a s) -> P (Add a t).
+intros.
+rewrite <- H.
+rewrite H.
+setoid_rewrite <- H.
+setoid_rewrite H.
+setoid_rewrite <- H.
+trivial.
+Qed.
+
-- 
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