- ext2_tc added

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / i_static / tc_lfxs_lfeq.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/i_static/tc_lfxs_lfeq.ma b/matita/matita/contribs/lambdadelta/basic_2/i_static/tc_lfxs_lfeq.ma
new file mode 100644 (file)
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+++ b/matita/matita/contribs/lambdadelta/basic_2/i_static/tc_lfxs_lfeq.ma
@@ -0,0 +1,82 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(*
+include "basic_2/static/lfeq.ma".
+*)
+include "basic_2/static/lfxs_lfxs.ma".
+include "basic_2/i_static/tc_lfxs_fqup.ma".
+
+include "basic_2/notation/relations/lazyeqsn_3.ma".
+
+definition ceq_ext: relation3 lenv bind bind ≝
+                    cext2 (λL,T1,T2. T1 = T2).
+
+definition lfeq: relation3 term lenv lenv ≝
+                 lfxs (λL,T1,T2. T1 = T2).
+
+interpretation
+   "syntactic equivalence on referred entries (local environment)"
+   'LazyEqSn T L1 L2 = (lfeq T L1 L2).
+
+axiom lfeq_lfxs_trans: ∀R,L1,L,T. L1 ≡[T] L →
+                       ∀L2. L ⪤*[R, T] L2 → L1 ⪤*[R, T] L2.
+
+axiom cext2_inv_LTC: ∀R,L,I1,I2. cext2 (LTC … R) L I1 I2 → LTC … (cext2 R) L I1 I2.
+(*
+#R #L #I1 #I2 * -I1 -I2
+[ /2 width=1 by ext2_unit, inj/
+| #I #V1 #V2 #HV12 
+*)
+
+  
+(*
+lemma pippo: ∀RN,RP. (∀L. reflexive … (RP L)) →
+             ∀f,L1,L2. L1 ⪤*[LTC … RN, RP, f] L2 →
+             TC … (lexs RN RP f) L1 L2.
+#RN #RP #HRP #f #L1 #L2 #H elim H -f -L1 -L2
+[ /2 width=1 by lexs_atom, inj/ ]
+#f #I1 #I2 #L1 #L2 #HL12 #HI12 #IH 
+[ @step [3: 
+*)
+
+axiom lexs_frees_confluent_LTC_sn: ∀RN,RP. lexs_frees_confluent RN RP →
+                                   lexs_frees_confluent (LTC … RN) RP.
+(*
+#RN #RP #HR #f1 #L1 #T #Hf1 #L2 #H  
+*)
+(* ITERATED EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ***)
+
+lemma pippo: ∀R. (∀L. reflexive … (R L)) →
+             (lexs_frees_confluent (cext2 R) cfull) →
+             ∀L1,L2,T. L1 ⪤**[R, T] L2 →
+             ∃∃L. L1 ⪤*[LTC … R, T] L & L ≡[T] L2.
+#R #H1R #H2R #L1 #L2 #T #H
+@(tc_lfxs_ind_sn … H1R … H) -H -L2
+[ /4 width=5 by lfxs_refl, inj, ex2_intro/
+| #L0 #L2 #_ #HL02 * #L * #f1 #Hf1 #HL1 #HL0
+  lapply (lexs_co ??? cfull … (cext2_inv_LTC R) … HL1) -HL1 // #HL1
+  lapply (lfeq_lfxs_trans … HL0 … HL02) -L0 #HL2
+  elim (lexs_frees_confluent_LTC_sn … H2R … Hf1 … HL1) #f2 #Hf2 #Hf21
+  lapply (lfxs_inv_frees … HL2 … Hf2) -HL2 #HL2
+  elim (lexs_sle_split … ceq_ext … HL2 … Hf21) -HL2
+  [ #L0 #HL0 #HL02
+  |*: /2 width=1 by ext2_refl/
+  ]
+  lapply (sle_lexs_trans … HL0 … Hf21) -Hf21 // #H
+  elim (H2R … Hf2 … H) -H #f0 #Hf0 #Hf02
+  lapply (sle_lexs_trans … HL02 … Hf02) -f2 // #HL02
+  @(ex2_intro … L0)
+  [ @(ex2_intro … Hf1)
+  | @(ex2_intro … HL02) //