Classical_Prop.v

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(* File created for Coq V5.10.14b, Oct 1995 *)
(* Classical tactics for proving disjunctions by Julien Narboux, Jul 2005 *)
(* Inferred proof-irrelevance and eq_rect_eq added by Hugo Herbelin, Mar 2006 *)

Classical Propositional Logic

Require Import ClassicalFacts.

Hint Unfold not: core.

Axiom classic : forall P:Prop, P \/ ~ P.

Lemma NNPP : forall p:Prop, ~ ~ p -> p.forall p : Prop, ~ ~ p -> p
Proof.forall p : Prop, ~ ~ p -> p
unfold not; intros; elim (classic p); auto.p:Prop
H:(p -> False) -> False
~ p -> p
intro NP; elim (H NP).
Qed.

Peirce's law states ∀ P Q:Prop, ((P → Q) → P) → P. Thanks to ∀ P, False → P, it is equivalent to the following form

Lemma Peirce : forall P:Prop, ((P -> False) -> P) -> P.forall P : Prop, ((P -> False) -> P) -> P
Proof.forall P : Prop, ((P -> False) -> P) -> P
intros P H; destruct (classic P); auto.
Qed.

Lemma not_imply_elim : forall P Q:Prop, ~ (P -> Q) -> P.forall P Q : Prop, ~ (P -> Q) -> P
Proof.forall P Q : Prop, ~ (P -> Q) -> P
intros; apply NNPP; red.P, Q:Prop
H:~ (P -> Q)
~ P -> False
intro; apply H; intro; absurd P; trivial.
Qed.

Lemma not_imply_elim2 : forall P Q:Prop, ~ (P -> Q) -> ~ Q.forall P Q : Prop, ~ (P -> Q) -> ~ Q
Proof. (* Intuitionistic *)forall P Q : Prop, ~ (P -> Q) -> ~ Q
tauto.
Qed.

Lemma imply_to_or : forall P Q:Prop, (P -> Q) -> ~ P \/ Q.forall P Q : Prop, (P -> Q) -> ~ P \/ Q
Proof.forall P Q : Prop, (P -> Q) -> ~ P \/ Q
intros; elim (classic P); auto.
Qed.

Lemma imply_to_and : forall P Q:Prop, ~ (P -> Q) -> P /\ ~ Q.forall P Q : Prop, ~ (P -> Q) -> P /\ ~ Q
Proof.forall P Q : Prop, ~ (P -> Q) -> P /\ ~ Q
intros; split.P, Q:Prop
H:~ (P -> Q)
P
P, Q:Prop
H:~ (P -> Q)
~ Q
apply not_imply_elim with Q; trivial.P, Q:Prop
H:~ (P -> Q)
~ Q
apply not_imply_elim2 with P; trivial.
Qed.

Lemma or_to_imply : forall P Q:Prop, ~ P \/ Q -> P -> Q.forall P Q : Prop, ~ P \/ Q -> P -> Q
Proof. (* Intuitionistic *)forall P Q : Prop, ~ P \/ Q -> P -> Q
tauto.
Qed.

Lemma not_and_or : forall P Q:Prop, ~ (P /\ Q) -> ~ P \/ ~ Q.forall P Q : Prop, ~ (P /\ Q) -> ~ P \/ ~ Q
Proof.forall P Q : Prop, ~ (P /\ Q) -> ~ P \/ ~ Q
intros; elim (classic P); auto.
Qed.

Lemma or_not_and : forall P Q:Prop, ~ P \/ ~ Q -> ~ (P /\ Q).forall P Q : Prop, ~ P \/ ~ Q -> ~ (P /\ Q)
Proof.forall P Q : Prop, ~ P \/ ~ Q -> ~ (P /\ Q)
simple induction 1; red; simple induction 2; auto.
Qed.

Lemma not_or_and : forall P Q:Prop, ~ (P \/ Q) -> ~ P /\ ~ Q.forall P Q : Prop, ~ (P \/ Q) -> ~ P /\ ~ Q
Proof. (* Intuitionistic *)forall P Q : Prop, ~ (P \/ Q) -> ~ P /\ ~ Q
tauto.
Qed.

Lemma and_not_or : forall P Q:Prop, ~ P /\ ~ Q -> ~ (P \/ Q).forall P Q : Prop, ~ P /\ ~ Q -> ~ (P \/ Q)
Proof. (* Intuitionistic *)forall P Q : Prop, ~ P /\ ~ Q -> ~ (P \/ Q)
tauto.
Qed.

Lemma imply_and_or : forall P Q:Prop, (P -> Q) -> P \/ Q -> Q.forall P Q : Prop, (P -> Q) -> P \/ Q -> Q
Proof. (* Intuitionistic *)forall P Q : Prop, (P -> Q) -> P \/ Q -> Q
tauto.
Qed.

Lemma imply_and_or2 : forall P Q R:Prop, (P -> Q) -> P \/ R -> Q \/ R.forall P Q R : Prop, (P -> Q) -> P \/ R -> Q \/ R
Proof. (* Intuitionistic *)forall P Q R : Prop, (P -> Q) -> P \/ R -> Q \/ R
tauto.
Qed.

Lemma proof_irrelevance : forall (P:Prop) (p1 p2:P), p1 = p2.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
|- ?X \/ _ => (elim (classic X);intro;[left;trivial|right])
end.

Ltac classical_left := match goal with
|- _ \/ ?X => (elim (classic X);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.forall (U : Type) (p : U) (Q : U -> Type) (x : Q p)
  (h : p = p), x = eq_rect p Q x p h
Proof.forall (U : Type) (p : U) (Q : U -> Type) (x : Q p)
  (h : p = p), x = eq_rect p Q x p h
intros; rewrite proof_irrelevance with (p1:=h) (p2:=eq_refl p); reflexivity.
Qed.

End Eq_rect_eq.

Module EqdepTheory := EqdepTheory(Eq_rect_eq).
Export EqdepTheory.