Standard Library | The Coq Proof Assistant

The Coq Proof Assistant

# Library Coq.Init.Logic_Type

This module defines type constructors for types in Type (Datatypes.v and Logic.v defined them for types in Set)
Set Implicit Arguments.
Require Import Datatypes.
Require Export Logic.
Negation of a type in Type
Definition notT (A:Type) := A -> False.
Properties of identity
Section identity_is_a_congruence.
Variables A B : Type.
Variable f : A -> B.
Variables x y z : A.
Lemma identity_sym : identity x y -> identity y x.
Lemma identity_trans : identity x y -> identity y z -> identity x z.
Lemma identity_congr : identity x y -> identity (f x) (f y).
Lemma not_identity_sym : notT (identity x y) -> notT (identity y x).
End identity_is_a_congruence.
Definition identity_ind_r :
forall (A:Type) (a:A) (P:A -> Prop), P a -> forall y:A, identity y a -> P y.
Defined.
Definition identity_rec_r :
forall (A:Type) (a:A) (P:A -> Set), P a -> forall y:A, identity y a -> P y.
Defined.
Definition identity_rect_r :
forall (A:Type) (a:A) (P:A -> Type), P a -> forall y:A, identity y a -> P y.
Defined.
Hint Immediate identity_sym not_identity_sym: core.
Notation refl_id := identity_refl (only parsing).
Notation sym_id := identity_sym (only parsing).
Notation trans_id := identity_trans (only parsing).
Notation sym_not_id := not_identity_sym (only parsing).