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Diffstat (limited to 'theories/Sets/Relations_3_facts.v')
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diff --git a/theories/Sets/Relations_3_facts.v b/theories/Sets/Relations_3_facts.v new file mode 100755 index 00000000..34322dc7 --- /dev/null +++ b/theories/Sets/Relations_3_facts.v @@ -0,0 +1,171 @@ +(************************************************************************) +(* 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 *) +(************************************************************************) +(****************************************************************************) +(* *) +(* Naive Set Theory in Coq *) +(* *) +(* INRIA INRIA *) +(* Rocquencourt Sophia-Antipolis *) +(* *) +(* Coq V6.1 *) +(* *) +(* Gilles Kahn *) +(* Gerard Huet *) +(* *) +(* *) +(* *) +(* Acknowledgments: This work was started in July 1993 by F. Prost. Thanks *) +(* to the Newton Institute for providing an exceptional work environment *) +(* in Summer 1995. Several developments by E. Ledinot were an inspiration. *) +(****************************************************************************) + +(*i $Id: Relations_3_facts.v,v 1.6.2.1 2004/07/16 19:31:18 herbelin Exp $ i*) + +Require Export Relations_1. +Require Export Relations_1_facts. +Require Export Relations_2. +Require Export Relations_2_facts. +Require Export Relations_3. + +Theorem Rstar_imp_coherent : + forall (U:Type) (R:Relation U) (x y:U), Rstar U R x y -> coherent U R x y. +Proof. +intros U R x y H'; red in |- *. +exists y; auto with sets. +Qed. +Hint Resolve Rstar_imp_coherent. + +Theorem coherent_symmetric : + forall (U:Type) (R:Relation U), Symmetric U (coherent U R). +Proof. +unfold coherent at 1 in |- *. +intros U R; red in |- *. +intros x y H'; elim H'. +intros z H'0; exists z; tauto. +Qed. + +Theorem Strong_confluence : + forall (U:Type) (R:Relation U), Strongly_confluent U R -> Confluent U R. +Proof. +intros U R H'; red in |- *. +intro x; red in |- *; intros a b H'0. +unfold coherent at 1 in |- *. +generalize b; clear b. +elim H'0; clear H'0. +intros x0 b H'1; exists b; auto with sets. +intros x0 y z H'1 H'2 H'3 b H'4. +generalize (Lemma1 U R); intro h; lapply h; + [ intro H'0; generalize (H'0 x0 b); intro h0; lapply h0; + [ intro H'5; generalize (H'5 y); intro h1; lapply h1; + [ intro h2; elim h2; intros z0 h3; elim h3; intros H'6 H'7; + clear h h0 h1 h2 h3 + | clear h h0 h1 ] + | clear h h0 ] + | clear h ]; auto with sets. +generalize (H'3 z0); intro h; lapply h; + [ intro h0; elim h0; intros z1 h1; elim h1; intros H'8 H'9; clear h h0 h1 + | clear h ]; auto with sets. +exists z1; split; auto with sets. +apply Rstar_n with z0; auto with sets. +Qed. + +Theorem Strong_confluence_direct : + forall (U:Type) (R:Relation U), Strongly_confluent U R -> Confluent U R. +Proof. +intros U R H'; red in |- *. +intro x; red in |- *; intros a b H'0. +unfold coherent at 1 in |- *. +generalize b; clear b. +elim H'0; clear H'0. +intros x0 b H'1; exists b; auto with sets. +intros x0 y z H'1 H'2 H'3 b H'4. +cut (ex (fun t:U => Rstar U R y t /\ R b t)). +intro h; elim h; intros t h0; elim h0; intros H'0 H'5; clear h h0. +generalize (H'3 t); intro h; lapply h; + [ intro h0; elim h0; intros z0 h1; elim h1; intros H'6 H'7; clear h h0 h1 + | clear h ]; auto with sets. +exists z0; split; [ assumption | idtac ]. +apply Rstar_n with t; auto with sets. +generalize H'1; generalize y; clear H'1. +elim H'4. +intros x1 y0 H'0; exists y0; auto with sets. +intros x1 y0 z0 H'0 H'1 H'5 y1 H'6. +red in H'. +generalize (H' x1 y0 y1); intro h; lapply h; + [ intro H'7; lapply H'7; + [ intro h0; elim h0; intros z1 h1; elim h1; intros H'8 H'9; + clear h H'7 h0 h1 + | clear h ] + | clear h ]; auto with sets. +generalize (H'5 z1); intro h; lapply h; + [ intro h0; elim h0; intros t h1; elim h1; intros H'7 H'10; clear h h0 h1 + | clear h ]; auto with sets. +exists t; split; auto with sets. +apply Rstar_n with z1; auto with sets. +Qed. + +Theorem Noetherian_contains_Noetherian : + forall (U:Type) (R R':Relation U), + Noetherian U R -> contains U R R' -> Noetherian U R'. +Proof. +unfold Noetherian at 2 in |- *. +intros U R R' H' H'0 x. +elim (H' x); auto with sets. +Qed. + +Theorem Newman : + forall (U:Type) (R:Relation U), + Noetherian U R -> Locally_confluent U R -> Confluent U R. +Proof. +intros U R H' H'0; red in |- *; intro x. +elim (H' x); unfold confluent in |- *. +intros x0 H'1 H'2 y z H'3 H'4. +generalize (Rstar_cases U R x0 y); intro h; lapply h; + [ intro h0; elim h0; + [ clear h h0; intro h1 + | intro h1; elim h1; intros u h2; elim h2; intros H'5 H'6; + clear h h0 h1 h2 ] + | clear h ]; auto with sets. +elim h1; auto with sets. +generalize (Rstar_cases U R x0 z); intro h; lapply h; + [ intro h0; elim h0; + [ clear h h0; intro h1 + | intro h1; elim h1; intros v h2; elim h2; intros H'7 H'8; + clear h h0 h1 h2 ] + | clear h ]; auto with sets. +elim h1; generalize coherent_symmetric; intro t; red in t; auto with sets. +unfold Locally_confluent, locally_confluent, coherent in H'0. +generalize (H'0 x0 u v); intro h; lapply h; + [ intro H'9; lapply H'9; + [ intro h0; elim h0; intros t h1; elim h1; intros H'10 H'11; + clear h H'9 h0 h1 + | clear h ] + | clear h ]; auto with sets. +clear H'0. +unfold coherent at 1 in H'2. +generalize (H'2 u); intro h; lapply h; + [ intro H'0; generalize (H'0 y t); intro h0; lapply h0; + [ intro H'9; lapply H'9; + [ intro h1; elim h1; intros y1 h2; elim h2; intros H'12 H'13; + clear h h0 H'9 h1 h2 + | clear h h0 ] + | clear h h0 ] + | clear h ]; auto with sets. +generalize Rstar_transitive; intro T; red in T. +generalize (H'2 v); intro h; lapply h; + [ intro H'9; generalize (H'9 y1 z); intro h0; lapply h0; + [ intro H'14; lapply H'14; + [ intro h1; elim h1; intros z1 h2; elim h2; intros H'15 H'16; + clear h h0 H'14 h1 h2 + | clear h h0 ] + | clear h h0 ] + | clear h ]; auto with sets. +red in |- *; (exists z1; split); auto with sets. +apply T with y1; auto with sets. +apply T with t; auto with sets. +Qed.
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