renaming

[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
index b9cd6c4ab58c91f25e3df3a70a8dc8d5208f61b9..753b43b2efdc671f3325d2d7228734e7e685d0c0 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs_lexs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs_lexs.ma
@@ -19,32 +19,51 @@ include "basic_2/relocation/drops.ma".
 
 (* 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 →
+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 #f #HN #HP #L1 #L0 #H elim H -f -L1 -L0
-[ #f #L2 #H >(lexs_inv_atom1 … H) -L2 //
-| #f #I1 #I #K1 #K #HK1 #HI1 #IH #L2 #H elim (lexs_inv_next1 … H) -H
-  #I2 #K2 #HK2 #HI2 #H destruct /4 width=6 by lexs_next/
-| #f #I1 #I #K1 #K #HK1 #HI1 #IH #L2 #H elim (lexs_inv_push1 … H) -H
-  #I2 #K2 #HK2 #HI2 #H destruct /4 width=6 by lexs_push/
+#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_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.
-                  (â\88\80g,I,K,n. â¬\87*[n] L â\89¡ K.â\93\98{I} â\86\92 ⫯g = ⫱*[n] f → R_pw_confluent2_lexs RN1 RN2 RN1 RP1 RN2 RP2 g K I) →
-                  (â\88\80g,I,K,n. â¬\87*[n] L â\89¡ K.â\93\98{I} â\86\92 â\86\91g = ⫱*[n] f → R_pw_confluent2_lexs RP1 RP2 RN1 RP1 RN2 RP2 g K I) →
+                  (â\88\80g,I,K,n. â¬\87*[n] L â\89\98 K.â\93\98{I} â\86\92 â\86\91g = ⫱*[n] f → R_pw_confluent2_lexs RN1 RN2 RN1 RP1 RN2 RP2 g K I) →
+                  (â\88\80g,I,K,n. â¬\87*[n] L â\89\98 K.â\93\98{I} â\86\92 ⫯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
@@ -76,7 +95,7 @@ theorem lexs_canc_dx: ∀RN,RP,f. Transitive … (lexs RN RP f) →
 lemma lexs_meet: ∀RN,RP,L1,L2.
                  ∀f1. L1 ⪤*[RN, RP, f1] L2 →
                  ∀f2. L1 ⪤*[RN, RP, f2] L2 →
-                 â\88\80f. f1 â\8b\92 f2 â\89¡ f → L1 ⪤*[RN, RP, f] L2.
+                 â\88\80f. f1 â\8b\92 f2 â\89\98 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
@@ -89,7 +108,7 @@ qed-.
 lemma lexs_join: ∀RN,RP,L1,L2.
                  ∀f1. L1 ⪤*[RN, RP, f1] L2 →
                  ∀f2. L1 ⪤*[RN, RP, f2] L2 →
-                 â\88\80f. f1 â\8b\93 f2 â\89¡ f → L1 ⪤*[RN, RP, f] L2.
+                 â\88\80f. f1 â\8b\93 f2 â\89\98 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