diff options
Diffstat (limited to 'theories7/Relations')
| -rwxr-xr-x | theories7/Relations/Newman.v | 115 | ||||
| -rwxr-xr-x | theories7/Relations/Operators_Properties.v | 98 | ||||
| -rwxr-xr-x | theories7/Relations/Relation_Definitions.v | 83 | ||||
| -rwxr-xr-x | theories7/Relations/Relation_Operators.v | 157 | ||||
| -rwxr-xr-x | theories7/Relations/Relations.v | 28 | ||||
| -rwxr-xr-x | theories7/Relations/Rstar.v | 78 |
6 files changed, 0 insertions, 559 deletions
diff --git a/theories7/Relations/Newman.v b/theories7/Relations/Newman.v deleted file mode 100755 index c53db971..00000000 --- a/theories7/Relations/Newman.v +++ /dev/null @@ -1,115 +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: Newman.v,v 1.1.2.1 2004/07/16 19:31:37 herbelin Exp $ i*) - -Require Rstar. - -Section Newman. - -Variable A: Type. -Variable R: A->A->Prop. - -Local Rstar := (Rstar A R). -Local Rstar_reflexive := (Rstar_reflexive A R). -Local Rstar_transitive := (Rstar_transitive A R). -Local Rstar_Rstar' := (Rstar_Rstar' A R). - -Definition coherence := [x:A][y:A] (exT2 ? (Rstar x) (Rstar y)). - -Theorem coherence_intro : (x:A)(y:A)(z:A)(Rstar x z)->(Rstar y z)->(coherence x y). -Proof [x:A][y:A][z:A][h1:(Rstar x z)][h2:(Rstar y z)] - (exT_intro2 A (Rstar x) (Rstar y) z h1 h2). - -(** A very simple case of coherence : *) - -Lemma Rstar_coherence : (x:A)(y:A)(Rstar x y)->(coherence x y). - Proof [x:A][y:A][h:(Rstar x y)](coherence_intro x y y h (Rstar_reflexive y)). - -(** coherence is symmetric *) -Lemma coherence_sym: (x:A)(y:A)(coherence x y)->(coherence y x). - Proof [x:A][y:A][h:(coherence x y)] - (exT2_ind A (Rstar x) (Rstar y) (coherence y x) - [w:A][h1:(Rstar x w)][h2:(Rstar y w)] - (coherence_intro y x w h2 h1) h). - -Definition confluence := - [x:A](y:A)(z:A)(Rstar x y)->(Rstar x z)->(coherence y z). - -Definition local_confluence := - [x:A](y:A)(z:A)(R x y)->(R x z)->(coherence y z). - -Definition noetherian := - (x:A)(P:A->Prop)((y:A)((z:A)(R y z)->(P z))->(P y))->(P x). - -Section Newman_section. - -(** The general hypotheses of the theorem *) - -Hypothesis Hyp1:noetherian. -Hypothesis Hyp2:(x:A)(local_confluence x). - -(** The induction hypothesis *) - -Section Induct. - Variable x:A. - Hypothesis hyp_ind:(u:A)(R x u)->(confluence u). - -(** Confluence in [x] *) - - Variables y,z:A. - Hypothesis h1:(Rstar x y). - Hypothesis h2:(Rstar x z). - -(** particular case [x->u] and [u->*y] *) -Section Newman_. - Variable u:A. - Hypothesis t1:(R x u). - Hypothesis t2:(Rstar u y). - -(** In the usual diagram, we assume also [x->v] and [v->*z] *) - -Theorem Diagram : (v:A)(u1:(R x v))(u2:(Rstar v z))(coherence y z). - -Proof (* We draw the diagram ! *) - [v:A][u1:(R x v)][u2:(Rstar v z)] - (exT2_ind A (Rstar u) (Rstar v) (* local confluence in x for u,v *) - (coherence y z) (* gives w, u->*w and v->*w *) - ([w:A][s1:(Rstar u w)][s2:(Rstar v w)] - (exT2_ind A (Rstar y) (Rstar w) (* confluence in u => coherence(y,w) *) - (coherence y z) (* gives a, y->*a and z->*a *) - ([a:A][v1:(Rstar y a)][v2:(Rstar w a)] - (exT2_ind A (Rstar a) (Rstar z) (* confluence in v => coherence(a,z) *) - (coherence y z) (* gives b, a->*b and z->*b *) - ([b:A][w1:(Rstar a b)][w2:(Rstar z b)] - (coherence_intro y z b (Rstar_transitive y a b v1 w1) w2)) - (hyp_ind v u1 a z (Rstar_transitive v w a s2 v2) u2))) - (hyp_ind u t1 y w t2 s1))) - (Hyp2 x u v t1 u1)). - -Theorem caseRxy : (coherence y z). -Proof (Rstar_Rstar' x z h2 - ([v:A][w:A](coherence y w)) - (coherence_sym x y (Rstar_coherence x y h1)) (*i case x=z i*) - Diagram). (*i case x->v->*z i*) -End Newman_. - -Theorem Ind_proof : (coherence y z). -Proof (Rstar_Rstar' x y h1 ([u:A][v:A](coherence v z)) - (Rstar_coherence x z h2) (*i case x=y i*) - caseRxy). (*i case x->u->*z i*) -End Induct. - -Theorem Newman : (x:A)(confluence x). -Proof [x:A](Hyp1 x confluence Ind_proof). - -End Newman_section. - - -End Newman. - diff --git a/theories7/Relations/Operators_Properties.v b/theories7/Relations/Operators_Properties.v deleted file mode 100755 index 4f1818bc..00000000 --- a/theories7/Relations/Operators_Properties.v +++ /dev/null @@ -1,98 +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: Operators_Properties.v,v 1.1.2.1 2004/07/16 19:31:37 herbelin Exp $ i*) - -(****************************************************************************) -(* Bruno Barras *) -(****************************************************************************) - -Require Relation_Definitions. -Require Relation_Operators. - - -Section Properties. - - Variable A: Set. - Variable R: (relation A). - - Local incl : (relation A)->(relation A)->Prop := - [R1,R2: (relation A)] (x,y:A) (R1 x y) -> (R2 x y). - -Section Clos_Refl_Trans. - - Lemma clos_rt_is_preorder: (preorder A (clos_refl_trans A R)). -Apply Build_preorder. -Exact (rt_refl A R). - -Exact (rt_trans A R). -Qed. - - - -Lemma clos_rt_idempotent: - (incl (clos_refl_trans A (clos_refl_trans A R)) - (clos_refl_trans A R)). -Red. -NewInduction 1; Auto with sets. -Intros. -Apply rt_trans with y; Auto with sets. -Qed. - - Lemma clos_refl_trans_ind_left: (A:Set)(R:A->A->Prop)(M:A)(P:A->Prop) - (P M) - ->((P0,N:A) - (clos_refl_trans A R M P0)->(P P0)->(R P0 N)->(P N)) - ->(a:A)(clos_refl_trans A R M a)->(P a). -Intros. -Generalize H H0 . -Clear H H0. -Elim H1; Intros; Auto with sets. -Apply H2 with x; Auto with sets. - -Apply H3. -Apply H0; Auto with sets. - -Intros. -Apply H5 with P0; Auto with sets. -Apply rt_trans with y; Auto with sets. -Qed. - - -End Clos_Refl_Trans. - - -Section Clos_Refl_Sym_Trans. - - Lemma clos_rt_clos_rst: (inclusion A (clos_refl_trans A R) - (clos_refl_sym_trans A R)). -Red. -NewInduction 1; Auto with sets. -Apply rst_trans with y; Auto with sets. -Qed. - - Lemma clos_rst_is_equiv: (equivalence A (clos_refl_sym_trans A R)). -Apply Build_equivalence. -Exact (rst_refl A R). - -Exact (rst_trans A R). - -Exact (rst_sym A R). -Qed. - - Lemma clos_rst_idempotent: - (incl (clos_refl_sym_trans A (clos_refl_sym_trans A R)) - (clos_refl_sym_trans A R)). -Red. -NewInduction 1; Auto with sets. -Apply rst_trans with y; Auto with sets. -Qed. - -End Clos_Refl_Sym_Trans. - -End Properties. diff --git a/theories7/Relations/Relation_Definitions.v b/theories7/Relations/Relation_Definitions.v deleted file mode 100755 index 1e38e753..00000000 --- a/theories7/Relations/Relation_Definitions.v +++ /dev/null @@ -1,83 +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_Definitions.v,v 1.1.2.1 2004/07/16 19:31:38 herbelin Exp $ i*) - -Section Relation_Definition. - - Variable A: Type. - - Definition relation := A -> A -> Prop. - - Variable R: relation. - - -Section General_Properties_of_Relations. - - Definition reflexive : Prop := (x: A) (R x x). - Definition transitive : Prop := (x,y,z: A) (R x y) -> (R y z) -> (R x z). - Definition symmetric : Prop := (x,y: A) (R x y) -> (R y x). - Definition antisymmetric : Prop := (x,y: A) (R x y) -> (R y x) -> x=y. - - (* for compatibility with Equivalence in ../PROGRAMS/ALG/ *) - Definition equiv := reflexive /\ transitive /\ symmetric. - -End General_Properties_of_Relations. - - - -Section Sets_of_Relations. - - Record preorder : Prop := { - preord_refl : reflexive; - preord_trans : transitive }. - - Record order : Prop := { - ord_refl : reflexive; - ord_trans : transitive; - ord_antisym : antisymmetric }. - - Record equivalence : Prop := { - equiv_refl : reflexive; - equiv_trans : transitive; - equiv_sym : symmetric }. - - Record PER : Prop := { - per_sym : symmetric; - per_trans : transitive }. - -End Sets_of_Relations. - - - -Section Relations_of_Relations. - - Definition inclusion : relation -> relation -> Prop := - [R1,R2: relation] (x,y:A) (R1 x y) -> (R2 x y). - - Definition same_relation : relation -> relation -> Prop := - [R1,R2: relation] (inclusion R1 R2) /\ (inclusion R2 R1). - - Definition commut : relation -> relation -> Prop := - [R1,R2:relation] (x,y:A) (R1 y x) -> (z:A) (R2 z y) - -> (EX y':A |(R2 y' x) & (R1 z y')). - -End Relations_of_Relations. - - -End Relation_Definition. - -Hints Unfold reflexive transitive antisymmetric symmetric : sets v62. - -Hints Resolve Build_preorder Build_order Build_equivalence - Build_PER preord_refl preord_trans - ord_refl ord_trans ord_antisym - equiv_refl equiv_trans equiv_sym - per_sym per_trans : sets v62. - -Hints Unfold inclusion same_relation commut : sets v62. 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. diff --git a/theories7/Relations/Relations.v b/theories7/Relations/Relations.v deleted file mode 100755 index 694d0eec..00000000 --- a/theories7/Relations/Relations.v +++ /dev/null @@ -1,28 +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: Relations.v,v 1.1.2.1 2004/07/16 19:31:38 herbelin Exp $ i*) - -Require Export Relation_Definitions. -Require Export Relation_Operators. -Require Export Operators_Properties. - -Lemma inverse_image_of_equivalence : (A,B:Set)(f:A->B) - (r:(relation B))(equivalence B r)->(equivalence A [x,y:A](r (f x) (f y))). -Intros; Split; Elim H; Red; Auto. -Intros _ equiv_trans _ x y z H0 H1; Apply equiv_trans with (f y); Assumption. -Qed. - -Lemma inverse_image_of_eq : (A,B:Set)(f:A->B) - (equivalence A [x,y:A](f x)=(f y)). -Split; Red; -[ (* reflexivity *) Reflexivity -| (* transitivity *) Intros; Transitivity (f y); Assumption -| (* symmetry *) Intros; Symmetry; Assumption -]. -Qed. diff --git a/theories7/Relations/Rstar.v b/theories7/Relations/Rstar.v deleted file mode 100755 index 3747b45e..00000000 --- a/theories7/Relations/Rstar.v +++ /dev/null @@ -1,78 +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: Rstar.v,v 1.1.2.1 2004/07/16 19:31:38 herbelin Exp $ i*) - -(** Properties of a binary relation [R] on type [A] *) - -Section Rstar. - -Variable A : Type. -Variable R : A->A->Prop. - -(** Definition of the reflexive-transitive closure [R*] of [R] *) -(** Smallest reflexive [P] containing [R o P] *) - -Definition Rstar := [x,y:A](P:A->A->Prop) - ((u:A)(P u u))->((u:A)(v:A)(w:A)(R u v)->(P v w)->(P u w)) -> (P x y). - -Theorem Rstar_reflexive: (x:A)(Rstar x x). - Proof [x:A][P:A->A->Prop] - [h1:(u:A)(P u u)][h2:(u:A)(v:A)(w:A)(R u v)->(P v w)->(P u w)] - (h1 x). - -Theorem Rstar_R: (x:A)(y:A)(z:A)(R x y)->(Rstar y z)->(Rstar x z). - Proof [x:A][y:A][z:A][t1:(R x y)][t2:(Rstar y z)] - [P:A->A->Prop] - [h1:(u:A)(P u u)][h2:(u:A)(v:A)(w:A)(R u v)->(P v w)->(P u w)] - (h2 x y z t1 (t2 P h1 h2)). - -(** We conclude with transitivity of [Rstar] : *) - -Theorem Rstar_transitive: (x:A)(y:A)(z:A)(Rstar x y)->(Rstar y z)->(Rstar x z). - Proof [x:A][y:A][z:A][h:(Rstar x y)] - (h ([u:A][v:A](Rstar v z)->(Rstar u z)) - ([u:A][t:(Rstar u z)]t) - ([u:A][v:A][w:A][t1:(R u v)][t2:(Rstar w z)->(Rstar v z)] - [t3:(Rstar w z)](Rstar_R u v z t1 (t2 t3)))). - -(** Another characterization of [R*] *) -(** Smallest reflexive [P] containing [R o R*] *) - -Definition Rstar' := [x:A][y:A](P:A->A->Prop) - ((P x x))->((u:A)(R x u)->(Rstar u y)->(P x y)) -> (P x y). - -Theorem Rstar'_reflexive: (x:A)(Rstar' x x). - Proof [x:A][P:A->A->Prop][h:(P x x)][h':(u:A)(R x u)->(Rstar u x)->(P x x)]h. - -Theorem Rstar'_R: (x:A)(y:A)(z:A)(R x z)->(Rstar z y)->(Rstar' x y). - Proof [x:A][y:A][z:A][t1:(R x z)][t2:(Rstar z y)] - [P:A->A->Prop][h1:(P x x)] - [h2:(u:A)(R x u)->(Rstar u y)->(P x y)](h2 z t1 t2). - -(** Equivalence of the two definitions: *) - -Theorem Rstar'_Rstar: (x:A)(y:A)(Rstar' x y)->(Rstar x y). - Proof [x:A][y:A][h:(Rstar' x y)] - (h Rstar (Rstar_reflexive x) ([u:A](Rstar_R x u y))). - -Theorem Rstar_Rstar': (x:A)(y:A)(Rstar x y)->(Rstar' x y). - Proof [x:A][y:A][h:(Rstar x y)](h Rstar' ([u:A](Rstar'_reflexive u)) - ([u:A][v:A][w:A][h1:(R u v)][h2:(Rstar' v w)] - (Rstar'_R u w v h1 (Rstar'_Rstar v w h2)))). - - -(** Property of Commutativity of two relations *) - -Definition commut := [A:Set][R1,R2:A->A->Prop] - (x,y:A)(R1 y x)->(z:A)(R2 z y) - ->(EX y':A |(R2 y' x) & (R1 z y')). - - -End Rstar. - |