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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(*i $Id: Logic_Type.v,v 1.19.2.1 2004/07/16 19:31:03 herbelin Exp $ i*)
Set Implicit Arguments.
(** This module defines quantification on the world [Type]
([Logic.v] was defining it on the world [Set]) *)
Require Import Datatypes.
Require Export Logic.
Definition notT (A:Type) := A -> False.
Section identity_is_a_congruence.
Variables A B : Type.
Variable f : A -> B.
Variables x y z : A.
Lemma sym_id : identity x y -> identity y x.
Proof.
destruct 1; trivial.
Qed.
Lemma trans_id : identity x y -> identity y z -> identity x z.
Proof.
destruct 2; trivial.
Qed.
Lemma congr_id : identity x y -> identity (f x) (f y).
Proof.
destruct 1; trivial.
Qed.
Lemma sym_not_id : notT (identity x y) -> notT (identity y x).
Proof.
red in |- *; 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 sym_id 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 sym_id 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 sym_id with (1 := H0); trivial.
Defined.
Inductive prodT (A B:Type) : Type :=
pairT : A -> B -> prodT A B.
Section prodT_proj.
Variables A B : Type.
Definition fstT (H:prodT A B) := match H with
| pairT x _ => x
end.
Definition sndT (H:prodT A B) := match H with
| pairT _ y => y
end.
End prodT_proj.
Definition prodT_uncurry (A B C:Type) (f:prodT A B -> C)
(x:A) (y:B) : C := f (pairT x y).
Definition prodT_curry (A B C:Type) (f:A -> B -> C)
(p:prodT A B) : C := match p with
| pairT x y => f x y
end.
Hint Immediate sym_id sym_not_id: core v62.
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