# Library Coq.Logic.Classical_Prop

``` ```
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. 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. 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. tauto. Qed. Lemma and_not_or : forall P Q:Prop, ~ P /\ ~ Q -> ~ (P \/ Q). Proof. tauto. Qed. Lemma imply_and_or : forall P Q:Prop, (P -> Q) -> P \/ Q -> Q. Proof. tauto. Qed. Lemma imply_and_or2 : forall P Q R:Prop, (P -> Q) -> P \/ R -> Q \/ R. Proof. tauto. Qed. Lemma proof_irrelevance : forall (P:Prop) (p1 p2:P), p1 = p2. Proof proof_irrelevance_cci classic. 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. ```