- former llpx_sn an lleq reactivated as lfxs and lfeq

[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 dbe74e5fdba94a19fc552eb95792ba23e41faff2..35badac0e3b4517b83c9f1bf3a88da59a0891d5c 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs_lexs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs_lexs.ma
@@ -21,11 +21,13 @@ include "basic_2/relocation/lexs.ma".
 
 (* 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
+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/
@@ -34,10 +36,16 @@ theorem lexs_trans (RN) (RP) (f): lexs_transitive RN RN RN RP →
 ]
 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 →
+                                  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.
-                   lpx_sn_confluent RN1 RN2 RN1 RP1 RN2 RP2 →
-                   lpx_sn_confluent RP1 RP2 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 *
@@ -68,26 +76,28 @@ theorem lexs_canc_dx: ∀RN,RP,f. Transitive … (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.
+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 #_ #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/
+#I #L1 #L2 #V1 #V2 #f1 #_ #HV12 #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)
+| elim (sand_inv_ppx … 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.
+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 #_ #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/
+#I #L1 #L2 #V1 #V2 #f1 #_ #HV12 #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)
+| elim (sor_inv_ppx … Hf) | elim (sor_inv_pnx … Hf)
+] -Hf /3 width=5 by lexs_next, lexs_push/
 qed-.