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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(*i $Id$ i*)
(** Classical Propositional Logic *)
Require Import ClassicalFacts.
Hint Unfold not: core.
Axiom classic : forall P:Prop, P \/ ~ P.
Lemma NNPP : forall p:Prop, ~ ~ p -> p.
Proof.
unfold not in |- *; intros; elim (classic p); auto.
intro NP; elim (H NP).
Qed.
(** Peirce's law states [forall P Q:Prop, ((P -> Q) -> P) -> P].
Thanks to [forall P, False -> P], it is equivalent to the
following form *)
Lemma Peirce : forall P:Prop, ((P -> False) -> P) -> P.
Proof.
intros P H; destruct (classic P); auto.
Qed.
Lemma not_imply_elim : forall P Q:Prop, ~ (P -> Q) -> P.
Proof.
intros; apply NNPP; red in |- *.
intro; apply H; intro; absurd P; trivial.
Qed.
Lemma not_imply_elim2 : forall P Q:Prop, ~ (P -> Q) -> ~ Q.
Proof. (* Intuitionistic *)
tauto.
Qed.
Lemma imply_to_or : forall P Q:Prop, (P -> Q) -> ~ P \/ Q.
Proof.
intros; elim (classic P); auto.
Qed.
Lemma imply_to_and : forall P Q:Prop, ~ (P -> Q) -> P /\ ~ Q.
Proof.
intros; split.
apply not_imply_elim with Q; trivial.
apply not_imply_elim2 with P; trivial.
Qed.
Lemma or_to_imply : forall P Q:Prop, ~ P \/ Q -> P -> Q.
Proof. (* Intuitionistic *)
tauto.
Qed.
Lemma not_and_or : forall P Q:Prop, ~ (P /\ Q) -> ~ P \/ ~ Q.
Proof.
intros; elim (classic P); auto.
Qed.
Lemma or_not_and : forall P Q:Prop, ~ P \/ ~ Q -> ~ (P /\ Q).
Proof.
simple induction 1; red in |- *; simple induction 2; auto.
Qed.
Lemma not_or_and : forall P Q:Prop, ~ (P \/ Q) -> ~ P /\ ~ Q.
Proof. (* Intuitionistic *)
tauto.
Qed.
Lemma and_not_or : forall P Q:Prop, ~ P /\ ~ Q -> ~ (P \/ Q).
Proof. (* Intuitionistic *)
tauto.
Qed.
Lemma imply_and_or : forall P Q:Prop, (P -> Q) -> P \/ Q -> Q.
Proof. (* Intuitionistic *)
tauto.
Qed.
Lemma imply_and_or2 : forall P Q R:Prop, (P -> Q) -> P \/ R -> Q \/ R.
Proof. (* Intuitionistic *)
tauto.
Qed.
Lemma proof_irrelevance : forall (P:Prop) (p1 p2:P), p1 = p2.
Proof proof_irrelevance_cci classic.
(* classical_left transforms |- A \/ B into ~B |- A *)
(* classical_right transforms |- A \/ B into ~A |- B *)
Ltac classical_right := match goal with
| _:_ |-?X1 \/ _ => (elim (classic X1);intro;[left;trivial|right])
end.
Ltac classical_left := match goal with
| _:_ |- _ \/?X1 => (elim (classic X1);intro;[right;trivial|left])
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.
|
