X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Frelocation%2Flexs_lexs.ma;h=753b43b2efdc671f3325d2d7228734e7e685d0c0;hb=5c186c72f508da0849058afeecc6877cd9ed6303;hp=35badac0e3b4517b83c9f1bf3a88da59a0891d5c;hpb=e39d1924cd572acdf0cf8dba08f3b650dfd6abee;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs_lexs.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs_lexs.ma
index 35badac0e..753b43b2e 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs_lexs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs_lexs.ma
@@ -13,55 +13,71 @@
 (**************************************************************************)
 
 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".
+include "basic_2/relocation/drops.ma".
 
 (* GENERIC ENTRYWISE EXTENSION OF CONTEXT-SENSITIVE REALTIONS FOR TERMS *****)
 
 (* Main properties **********************************************************)
 
-theorem lexs_trans_gen (RN1) (RP1) (RN2) (RP2) (RN) (RP) (f):
-                       lexs_transitive RN1 RN2 RN RN1 RP1 →
-                       lexs_transitive RP1 RP2 RP RN1 RP1 →
-                       ∀L1,L0. L1 ⦻*[RN1, RP1, f] L0 →
-                       ∀L2. L0 ⦻*[RN2, RP2, f] L2 →
-                       L1 ⦻*[RN, RP, f] L2.
-#RN1 #RP1 #RN2 #RP2 #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/
+theorem lexs_trans_gen (RN1) (RP1) (RN2) (RP2) (RN) (RP):
+                       ∀L1,f.
+                       (∀g,I,K,n. ⬇*[n] L1 ≘ K.ⓘ{I} → ↑g = ⫱*[n] f → lexs_transitive RN1 RN2 RN RN1 RP1 g K I) →
+                       (∀g,I,K,n. ⬇*[n] L1 ≘ K.ⓘ{I} → ⫯g = ⫱*[n] f → lexs_transitive RP1 RP2 RP RN1 RP1 g K I) →
+                       ∀L0. L1 ⪤*[RN1, RP1, f] L0 →
+                       ∀L2. L0 ⪤*[RN2, RP2, f] L2 →
+                       L1 ⪤*[RN, RP, f] L2.
+#RN1 #RP1 #RN2 #RP2 #RN #RP #L1 elim L1 -L1
+[ #f #_ #_ #L0 #H1 #L2 #H2
+  lapply (lexs_inv_atom1 … H1) -H1 #H destruct
+  lapply (lexs_inv_atom1 … H2) -H2 #H destruct
+  /2 width=1 by lexs_atom/
+| #K1 #I1 #IH #f elim (pn_split f) * #g #H destruct
+  #HN #HP #L0 #H1 #L2 #H2
+  [ elim (lexs_inv_push1 … H1) -H1 #I0 #K0 #HK10 #HI10 #H destruct
+    elim (lexs_inv_push1 … H2) -H2 #I2 #K2 #HK02 #HI02 #H destruct
+    lapply (HP … 0 … HI10 … HK10 … HI02) -HI10 -HI02 /2 width=2 by drops_refl/ #HI12
+    lapply (IH … HK10 … HK02) -IH -K0 /3 width=3 by lexs_push, drops_drop/
+  | elim (lexs_inv_next1 … H1) -H1 #I0 #K0 #HK10 #HI10 #H destruct
+    elim (lexs_inv_next1 … H2) -H2 #I2 #K2 #HK02 #HI02 #H destruct
+    lapply (HN … 0 … HI10 … HK10 … HI02) -HI10 -HI02 /2 width=2 by drops_refl/ #HI12
+    lapply (IH … HK10 … HK02) -IH -K0 /3 width=3 by lexs_next, drops_drop/
+  ]
 ]
 qed-.
 
-(* Basic_2A1: includes: lpx_sn_trans *)
-theorem lexs_trans (RN) (RP) (f): lexs_transitive RN RN RN RN RP →
-                                  lexs_transitive RP RP RP RN RP →
+theorem lexs_trans (RN) (RP) (f): (∀g,I,K. lexs_transitive RN RN RN RN RP g K I) →
+                                  (∀g,I,K. lexs_transitive RP RP RP RN RP g K I) →
                                   Transitive … (lexs RN RP f).
 /2 width=9 by lexs_trans_gen/ qed-.
 
-(* Basic_2A1: includes: lpx_sn_conf *)
-theorem lexs_conf: ∀RN1,RP1,RN2,RP2.
-                   lexs_confluent RN1 RN2 RN1 RP1 RN2 RP2 →
-                   lexs_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/
+theorem lexs_trans_id_cfull: ∀R1,R2,R3,L1,L,f. L1 ⪤*[R1, cfull, f] L → 𝐈⦃f⦄ →
+                             ∀L2. L ⪤*[R2, cfull, f] L2 → L1 ⪤*[R3, cfull, f] L2.
+#R1 #R2 #R3 #L1 #L #f #H elim H -L1 -L -f
+[ #f #Hf #L2 #H >(lexs_inv_atom1 … H) -L2 // ]
+#f #I1 #I #K1 #K #HK1 #_ #IH #Hf #L2 #H
+[ elim (isid_inv_next … Hf) | lapply (isid_inv_push … Hf ??) ] -Hf [5: |*: // ] #Hf
+elim (lexs_inv_push1 … H) -H #I2 #K2 #HK2 #_ #H destruct
+/3 width=1 by lexs_push/
+qed-.
+
+theorem lexs_conf (RN1) (RP1) (RN2) (RP2):
+                  ∀L,f.
+                  (∀g,I,K,n. ⬇*[n] L ≘ K.ⓘ{I} → ↑g = ⫱*[n] f → R_pw_confluent2_lexs RN1 RN2 RN1 RP1 RN2 RP2 g K I) →
+                  (∀g,I,K,n. ⬇*[n] L ≘ K.ⓘ{I} → ⫯g = ⫱*[n] f → R_pw_confluent2_lexs RP1 RP2 RN1 RP1 RN2 RP2 g K I) →
+                  pw_confluent2 … (lexs RN1 RP1 f) (lexs RN2 RP2 f) L.
+#RN1 #RP1 #RN2 #RP2 #L elim L -L
+[ #f #_ #_ #L1 #H1 #L2 #H2 >(lexs_inv_atom1 … H1) >(lexs_inv_atom1 … H2) -H2 -H1
+  /2 width=3 by lexs_atom, ex2_intro/
+| #L #I0 #IH #f elim (pn_split f) * #g #H destruct
+  #HN #HP #Y1 #H1 #Y2 #H2
+  [ elim (lexs_inv_push1 … H1) -H1 #I1 #L1 #HL1 #HI01 #H destruct
+    elim (lexs_inv_push1 … H2) -H2 #I2 #L2 #HL2 #HI02 #H destruct
+    elim (HP … 0 … HI01 … HI02 … HL1 … HL2) -HI01 -HI02 /2 width=2 by drops_refl/ #I #HI1 #HI2
+    elim (IH … HL1 … HL2) -IH -HL1 -HL2 /3 width=5 by drops_drop, lexs_push, ex2_intro/
+  | elim (lexs_inv_next1 … H1) -H1 #I1 #L1 #HL1 #HI01 #H destruct
+    elim (lexs_inv_next1 … H2) -H2 #I2 #L2 #HL2 #HI02 #H destruct
+    elim (HN … 0 … HI01 … HI02 … HL1 … HL2) -HI01 -HI02 /2 width=2 by drops_refl/ #I #HI1 #HI2
+    elim (IH … HL1 … HL2) -IH -HL1 -HL2 /3 width=5 by drops_drop, lexs_next, ex2_intro/
   ]
 ]
 qed-.
@@ -77,11 +93,11 @@ theorem lexs_canc_dx: ∀RN,RP,f. Transitive … (lexs RN RP f) →
 /3 width=3 by/ qed-.
 
 lemma 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 #_ #HV12 #IH #f2 #H #f #Hf
+                 ∀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 -f1 -L1 -L2 //
+#f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H #f #Hf
 elim (pn_split f2) * #g2 #H2 destruct
 try elim (lexs_inv_push … H) try elim (lexs_inv_next … H) -H
 [ elim (sand_inv_npx … Hf) | elim (sand_inv_nnx … Hf)
@@ -90,11 +106,11 @@ try elim (lexs_inv_push … H) try elim (lexs_inv_next … H) -H
 qed-.
 
 lemma 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 #_ #HV12 #IH #f2 #H #f #Hf
+                 ∀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 -f1 -L1 -L2 //
+#f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H #f #Hf
 elim (pn_split f2) * #g2 #H2 destruct
 try elim (lexs_inv_push … H) try elim (lexs_inv_next … H) -H
 [ elim (sor_inv_npx … Hf) | elim (sor_inv_nnx … Hf)