Standard Library | The Coq Proof Assistant

The Coq Proof Assistant

# Library Coq.Logic.Classical_Prop

Classical Propositional Logic
Require Import ClassicalFacts.
#[global]
Hint Unfold not: core.
Axiom classic : forall P:Prop, P \/ ~ P.
Lemma NNPP : forall p:Prop, ~ ~ p -> p.
Register NNPP as core.nnpp.type.
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.