# 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.

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.

Lemma not_imply_elim : forall P Q:Prop, ~ (P -> Q) -> P.

Lemma not_imply_elim2 : forall P Q:Prop, ~ (P -> Q) -> ~ Q.

Lemma imply_to_or : forall P Q:Prop, (P -> Q) -> ~ P \/ Q.

Lemma imply_to_and : forall P Q:Prop, ~ (P -> Q) -> P /\ ~ Q.

Lemma or_to_imply : forall P Q:Prop, ~ P \/ Q -> P -> Q.

Lemma not_and_or : forall P Q:Prop, ~ (P /\ Q) -> ~ P \/ ~ Q.

Lemma or_not_and : forall P Q:Prop, ~ P \/ ~ Q -> ~ (P /\ Q).

Lemma not_or_and : forall P Q:Prop, ~ (P \/ Q) -> ~ P /\ ~ Q.

Lemma and_not_or : forall P Q:Prop, ~ P /\ ~ Q -> ~ (P \/ Q).

Lemma imply_and_or : forall P Q:Prop, (P -> Q) -> P \/ Q -> Q.

Lemma imply_and_or2 : forall P Q R:Prop, (P -> Q) -> P \/ R -> Q \/ R.

Lemma proof_irrelevance : forall (P:Prop) (p1 p2:P), p1 = p2.

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.

End Eq_rect_eq.

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