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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.