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 (compat "8.3").
Notation sym_id := identity_sym (compat "8.3").
Notation trans_id := identity_trans (compat "8.3").
Notation sym_not_id := not_identity_sym (compat "8.3").