Modularisation des preuves concernant la logique classique, l'indiscernabilité des preuves et l'axiome K

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@8136 85f007b7-540e-0410-9357-904b9bb8a0f7

1 files changed, 29 insertions, 15 deletions

diff --git a/theories/Logic/Classical_Prop.v b/theories/Logic/Classical_Prop.v
index 1df6a354b..d714b3bf7 100755
--- a/theories/Logic/Classical_Prop.v
+++ b/theories/Logic/Classical_Prop.v
@@ -10,7 +10,7 @@
 
 (** Classical Propositional Logic *)
 
-Require Import ProofIrrelevance.
+Require Import ClassicalFacts.
 
 Hint Unfold not: core.
 
@@ -29,8 +29,8 @@ intro; apply H; intro; absurd P; trivial.
 Qed.
 
 Lemma not_imply_elim2 : forall P Q:Prop, ~ (P -> Q) -> ~ Q.
-Proof.
-intros; elim (classic Q); auto.
+Proof. (* Intuitionistic *)
+tauto.
 Qed.
 
 Lemma imply_to_or : forall P Q:Prop, (P -> Q) -> ~ P \/ Q.
@@ -46,9 +46,8 @@ apply not_imply_elim2 with P; trivial.
 Qed.
 
 Lemma or_to_imply : forall P Q:Prop, ~ P \/ Q -> P -> Q.
-Proof.
-simple induction 1; auto.
-intros H1 H2; elim (H1 H2).
+Proof. (* Intuitionistic *)
+tauto.
 Qed.
 
 Lemma not_and_or : forall P Q:Prop, ~ (P /\ Q) -> ~ P \/ ~ Q.
@@ -62,23 +61,23 @@ simple induction 1; red in |- *; simple induction 2; auto.
 Qed.
 
 Lemma not_or_and : forall P Q:Prop, ~ (P \/ Q) -> ~ P /\ ~ Q.
-Proof.
-intros; elim (classic P); auto.
+Proof. (* Intuitionistic *)
+tauto.
 Qed.
 
 Lemma and_not_or : forall P Q:Prop, ~ P /\ ~ Q -> ~ (P \/ Q).
-Proof.
-simple induction 1; red in |- *; simple induction 3; trivial.
+Proof. (* Intuitionistic *)
+tauto.
 Qed.
 
 Lemma imply_and_or : forall P Q:Prop, (P -> Q) -> P \/ Q -> Q.
-Proof.
-simple induction 2; trivial.
+Proof. (* Intuitionistic *)
+tauto.
 Qed.
 
 Lemma imply_and_or2 : forall P Q R:Prop, (P -> Q) -> P \/ R -> Q \/ R.
-Proof.
-simple induction 2; auto.
+Proof. (* Intuitionistic *)
+tauto.
 Qed.
 
 Lemma proof_irrelevance : forall (P:Prop) (p1 p2:P), p1 = p2.
@@ -93,4 +92,19 @@ end.
 
 Ltac classical_left := match goal with 
 | _:_ |- _ \/?X1 => (elim (classic X1);intro;[right;trivial|left])
-end.
\ No newline at end of file
+end.
+
+Require Export EqdepFacts.
+
+Module Eq_rect_eq.
+
+Lemma eq_rect_eq : 
+  forall (U:Type) (p:U) (Q:U -> Type) (x:Q p) (h:p = p), x = eq_rect p Q x p h.
+Proof.
+intros; rewrite proof_irrelevance with (p1:=h) (p2:=refl_equal p); reflexivity.
+Qed.
+
+End Eq_rect_eq.
+
+Module EqdepTheory := EqdepTheory(Eq_rect_eq).
+Export EqdepTheory.