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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(** 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.
Proof.
destruct 1; trivial.
Defined.
Lemma identity_trans : identity x y -> identity y z -> identity x z.
Proof.
destruct 2; trivial.
Defined.
Lemma identity_congr : identity x y -> identity (f x) (f y).
Proof.
destruct 1; trivial.
Defined.
Lemma not_identity_sym : notT (identity x y) -> notT (identity y x).
Proof.
red; intros H H'; apply H; destruct H'; trivial.
Qed.
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.
intros A x P H y H0; case identity_sym with (1 := H0); trivial.
Defined.
Definition identity_rec_r :
forall (A:Type) (a:A) (P:A -> Set), P a -> forall y:A, identity y a -> P y.
intros A x P H y H0; case identity_sym with (1 := H0); trivial.
Defined.
Definition identity_rect_r :
forall (A:Type) (a:A) (P:A -> Type), P a -> forall y:A, identity y a -> P y.
intros A x P H y H0; case identity_sym with (1 := H0); trivial.
Defined.
Hint Immediate identity_sym not_identity_sym: core v62.
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").
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