partial commit in the relocation component to move some files ...

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / relocation / lexs_lexs.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs_lexs.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs_lexs.ma
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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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 "ground_2/relocation/rtmap_sand.ma".
+include "ground_2/relocation/rtmap_sor.ma".
+include "basic_2/grammar/lenv_weight.ma".
+include "basic_2/relocation/lexs.ma".
+
+(* GENERAL ENTRYWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
+
+(* Main properties **********************************************************)
+
+(* Basic_2A1: includes: lpx_sn_trans *)
+theorem lexs_trans (RN) (RP) (f): lexs_transitive RN RN RN RP →
+                                  lexs_transitive RP RP RN RP →
+                                  Transitive … (lexs RN RP f).
+#RN #RP #f #HN #HP #L1 #L0 #H elim H -L1 -L0 -f
+[ #f #L2 #H >(lexs_inv_atom1 … H) -L2 //
+| #I #K1 #K #V1 #V #f #HK1 #HV1 #IH #L2 #H elim (lexs_inv_next1 … H) -H
+  #K2 #V2 #HK2 #HV2 #H destruct /4 width=6 by lexs_next/
+| #I #K1 #K #V1 #V #f #HK1 #HV1 #IH #L2 #H elim (lexs_inv_push1 … H) -H
+  #K2 #V2 #HK2 #HV2 #H destruct /4 width=6 by lexs_push/
+]
+qed-.
+
+(* Basic_2A1: includes: lpx_sn_conf *)
+theorem lexs_conf: ∀RN1,RP1,RN2,RP2.
+                   lpx_sn_confluent RN1 RN2 RN1 RP1 RN2 RP2 →
+                   lpx_sn_confluent RP1 RP2 RN1 RP1 RN2 RP2 →
+                   ∀f. confluent2 … (lexs RN1 RP1 f) (lexs RN2 RP2 f).
+#RN1 #RP1 #RN2 #RP2 #HRN #HRP #f #L0 generalize in match f; -f
+@(f_ind … lw … L0) -L0 #x #IH *
+[ #_ #f #X1 #H1 #X2 #H2 -x
+  >(lexs_inv_atom1 … H1) -X1
+  >(lexs_inv_atom1 … H2) -X2 /2 width=3 by lexs_atom, ex2_intro/
+| #L0 #I #V0 #Hx #f elim (pn_split f) *
+  #g #H #X1 #H1 #X2 #H2 destruct
+  [ elim (lexs_inv_push1 … H1) -H1 #L1 #V1 #HL01 #HV01 #H destruct
+    elim (lexs_inv_push1 … H2) -H2 #L2 #V2 #HL02 #HV02 #H destruct
+    elim (IH … HL01 … HL02) -IH // #L #HL1 #HL2
+    elim (HRP … HV01 … HV02 … HL01 … HL02) -L0 -V0 /3 width=5 by lexs_push, ex2_intro/
+  | elim (lexs_inv_next1 … H1) -H1 #L1 #V1 #HL01 #HV01 #H destruct
+    elim (lexs_inv_next1 … H2) -H2 #L2 #V2 #HL02 #HV02 #H destruct
+    elim (IH … HL01 … HL02) -IH // #L #HL1 #HL2
+    elim (HRN … HV01 … HV02 … HL01 … HL02) -L0 -V0 /3 width=5 by lexs_next, ex2_intro/
+  ]
+]
+qed-.
+
+theorem lexs_canc_sx: ∀RN,RP,f. Transitive … (lexs RN RP f) →
+                                symmetric … (lexs RN RP f) →
+                                left_cancellable … (lexs RN RP f).
+/3 width=3 by/ qed-.
+
+theorem lexs_canc_dx: ∀RN,RP,f. Transitive … (lexs RN RP f) →
+                                symmetric … (lexs RN RP f) →
+                                right_cancellable … (lexs RN RP f).
+/3 width=3 by/ qed-.
+
+theorem lexs_meet: ∀RN,RP,L1,L2,f1. L1 ⦻*[RN, RP, f1] L2 →
+                   ∀f2. L1 ⦻*[RN, RP, f2] L2 →
+                   ∀f. f1 ⋒ f2 ≡ f → L1 ⦻*[RN, RP, f] L2.
+#RN #RP #L1 #L2 #f1 #H elim H -L1 -L2 -f1 //
+#I #L1 #L2 #V1 #V2 #f1 #_ #H1V #IH #f2 elim (pn_split f2) *
+#g2 #H #H2 #f #Hf destruct
+[1,3: elim (lexs_inv_push … H2) |2,4: elim (lexs_inv_next … H2) ] -H2
+#H2 #H2V #_
+[ elim (sand_inv_npx … Hf) | elim (sand_inv_ppx … Hf) | elim (sand_inv_nnx … Hf) | elim (sand_inv_pnx … Hf) ] -Hf
+/3 width=5 by lexs_next, lexs_push/
+qed-.
+
+theorem lexs_join: ∀RN,RP,L1,L2,f1. L1 ⦻*[RN, RP, f1] L2 →
+                   ∀f2. L1 ⦻*[RN, RP, f2] L2 →
+                   ∀f. f1 ⋓ f2 ≡ f → L1 ⦻*[RN, RP, f] L2.
+#RN #RP #L1 #L2 #f1 #H elim H -L1 -L2 -f1 //
+#I #L1 #L2 #V1 #V2 #f1 #_ #H1V #IH #f2 elim (pn_split f2) *
+#g2 #H #H2 #f #Hf destruct
+[1,3: elim (lexs_inv_push … H2) |2,4: elim (lexs_inv_next … H2) ] -H2
+#H2 #H2V #_
+[ elim (sor_inv_npx … Hf) | elim (sor_inv_ppx … Hf) | elim (sor_inv_nnx … Hf) | elim (sor_inv_pnx … Hf) ] -Hf
+/3 width=5 by lexs_next, lexs_push/
+qed-.