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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. - |