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Diffstat (limited to 'theories7/Relations/Relation_Operators.v')
| -rwxr-xr-x | theories7/Relations/Relation_Operators.v | 157 |
1 files changed, 0 insertions, 157 deletions
diff --git a/theories7/Relations/Relation_Operators.v b/theories7/Relations/Relation_Operators.v deleted file mode 100755 index 14c2ae30..00000000 --- a/theories7/Relations/Relation_Operators.v +++ /dev/null @@ -1,157 +0,0 @@ -(************************************************************************) -(* 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: Relation_Operators.v,v 1.1.2.1 2004/07/16 19:31:38 herbelin Exp $ 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: Set. - 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: Set. - 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: Set. - 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: Set. - Variable R: (relation A). - - Definition transp := [x,y:A](R y x). -End Transposee. - - -Section Union. - Variable A: Set. - 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. |