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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 $Id$ i*)
(****************************************************************************)
(* Bruno Barras, Cristina Cornes *)
(* *)
(* Some of these definitons were taken from : *)
(* Constructing Recursion Operators in Type Theory *)
(* L. Paulson JSC (1986) 2, 325-355 *)
(****************************************************************************)
Require Relation_Definitions.
Require PolyList.
Require PolyListSyntax.
(** Some operators to build relations *)
Section Transitive_Closure.
Variable A: Type.
Variable R: (relation A).
Inductive clos_trans : A->A->Prop :=
t_step: (x,y:A)(R x y)->(clos_trans x y)
| t_trans: (x,y,z:A)(clos_trans x y)->(clos_trans y z)->(clos_trans x z).
End Transitive_Closure.
Section Reflexive_Transitive_Closure.
Variable A: Type.
Variable R: (relation A).
Inductive clos_refl_trans: (relation A) :=
rt_step: (x,y:A)(R x y)->(clos_refl_trans x y)
| rt_refl: (x:A)(clos_refl_trans x x)
| rt_trans: (x,y,z: A)(clos_refl_trans x y)->(clos_refl_trans y z)
->(clos_refl_trans x z).
End Reflexive_Transitive_Closure.
Section Reflexive_Symetric_Transitive_Closure.
Variable A: Type.
Variable R: (relation A).
Inductive clos_refl_sym_trans: (relation A) :=
rst_step: (x,y:A)(R x y)->(clos_refl_sym_trans x y)
| rst_refl: (x:A)(clos_refl_sym_trans x x)
| rst_sym: (x,y:A)(clos_refl_sym_trans x y)->(clos_refl_sym_trans y x)
| rst_trans: (x,y,z:A)(clos_refl_sym_trans x y)->(clos_refl_sym_trans y z)
->(clos_refl_sym_trans x z).
End Reflexive_Symetric_Transitive_Closure.
Section Transposee.
Variable A: Type.
Variable R: (relation A).
Definition transp := [x,y:A](R y x).
End Transposee.
Section Union.
Variable A: Type.
Variable R1,R2: (relation A).
Definition union := [x,y:A](R1 x y)\/(R2 x y).
End Union.
Section Disjoint_Union.
Variable A,B:Set.
Variable leA: A->A->Prop.
Variable leB: B->B->Prop.
Inductive le_AsB : A+B->A+B->Prop :=
le_aa: (x,y:A) (leA x y) -> (le_AsB (inl A B x) (inl A B y))
| le_ab: (x:A)(y:B) (le_AsB (inl A B x) (inr A B y))
| le_bb: (x,y:B) (leB x y) -> (le_AsB (inr A B x) (inr A B y)).
End Disjoint_Union.
Section Lexicographic_Product.
(* Lexicographic order on dependent pairs *)
Variable A:Set.
Variable B:A->Set.
Variable leA: A->A->Prop.
Variable leB: (x:A)(B x)->(B x)->Prop.
Inductive lexprod : (sigS A B) -> (sigS A B) ->Prop :=
left_lex : (x,x':A)(y:(B x)) (y':(B x'))
(leA x x') ->(lexprod (existS A B x y) (existS A B x' y'))
| right_lex : (x:A) (y,y':(B x))
(leB x y y') -> (lexprod (existS A B x y) (existS A B x y')).
End Lexicographic_Product.
Section Symmetric_Product.
Variable A:Set.
Variable B:Set.
Variable leA: A->A->Prop.
Variable leB: B->B->Prop.
Inductive symprod : (A*B) -> (A*B) ->Prop :=
left_sym : (x,x':A)(leA x x')->(y:B)(symprod (x,y) (x',y))
| right_sym : (y,y':B)(leB y y')->(x:A)(symprod (x,y) (x,y')).
End Symmetric_Product.
Section Swap.
Variable A:Set.
Variable R:A->A->Prop.
Inductive swapprod: (A*A)->(A*A)->Prop :=
sp_noswap: (x,x':A*A)(symprod A A R R x x')->(swapprod x x')
| sp_swap: (x,y:A)(p:A*A)(symprod A A R R (x,y) p)->(swapprod (y,x) p).
End Swap.
Section Lexicographic_Exponentiation.
Variable A : Set.
Variable leA : A->A->Prop.
Local Nil := (nil A).
Local List := (list A).
Inductive Ltl : List->List->Prop :=
Lt_nil: (a:A)(x:List)(Ltl Nil (cons a x))
| Lt_hd : (a,b:A) (leA a b)-> (x,y:(list A))(Ltl (cons a x) (cons b y))
| Lt_tl : (a:A)(x,y:List)(Ltl x y) -> (Ltl (cons a x) (cons a y)).
Inductive Desc : List->Prop :=
d_nil : (Desc Nil)
| d_one : (x:A)(Desc (cons x Nil))
| d_conc : (x,y:A)(l:List)(leA x y)
-> (Desc l^(cons y Nil))->(Desc (l^(cons y Nil))^(cons x Nil)).
Definition Pow :Set := (sig List Desc).
Definition lex_exp : Pow -> Pow ->Prop :=
[a,b:Pow](Ltl (proj1_sig List Desc a) (proj1_sig List Desc b)).
End Lexicographic_Exponentiation.
Hints Unfold transp union : sets v62.
Hints Resolve t_step rt_step rt_refl rst_step rst_refl : sets v62.
Hints Immediate rst_sym : sets v62.
|
