(**************************************************************************)
include "basic_2/relocation/lexs_length.ma".
-include "basic_2/static/frees_drops.ma".
include "basic_2/static/fsle_fsle.ma".
+include "basic_2/static/lfxs_drops.ma".
include "basic_2/static/lfxs_lfxs.ma".
(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
-definition R_fsle_compatible: predicate (relation3 …) ≝ λRN.
+definition R_fsge_compatible: predicate (relation3 …) ≝ λRN.
∀L,T1,T2. RN L T1 T2 → ⦃L, T2⦄ ⊆ ⦃L, T1⦄.
-definition lfxs_fsle_compatible: predicate (relation3 …) ≝ λRN.
+definition lfxs_fsge_compatible: predicate (relation3 …) ≝ λRN.
∀L1,L2,T. L1 ⪤*[RN, T] L2 → ⦃L2, T⦄ ⊆ ⦃L1, T⦄.
+definition lfxs_fsle_compatible: predicate (relation3 …) ≝ λRN.
+ ∀L1,L2,T. L1 ⪤*[RN, T] L2 → ⦃L1, T⦄ ⊆ ⦃L2, T⦄.
+
(* Basic inversions with free variables inclusion for restricted closures ***)
-lemma frees_lexs_conf: ∀R. lfxs_fsle_compatible R →
- â\88\80L1,T,f1. L1 â\8a¢ ð\9d\90\85*â¦\83Tâ¦\84 â\89¡ f1 →
+lemma frees_lexs_conf: ∀R. lfxs_fsge_compatible R →
+ â\88\80L1,T,f1. L1 â\8a¢ ð\9d\90\85*â¦\83Tâ¦\84 â\89\98 f1 →
∀L2. L1 ⪤*[cext2 R, cfull, f1] L2 →
- â\88\83â\88\83f2. L2 â\8a¢ ð\9d\90\85*â¦\83Tâ¦\84 â\89¡ f2 & f2 ⊆ f1.
+ â\88\83â\88\83f2. L2 â\8a¢ ð\9d\90\85*â¦\83Tâ¦\84 â\89\98 f2 & f2 ⊆ f1.
#R #HR #L1 #T #f1 #Hf1 #L2 #H1L
lapply (HR L1 L2 T ?) /2 width=3 by ex2_intro/ #H2L
@(fsle_frees_trans_eq … H2L … Hf1) /3 width=4 by lexs_fwd_length, sym_eq/
(* Properties with free variables inclusion for restricted closures *********)
(* Note: we just need lveq_inv_refl: ∀L,n1,n2. L ≋ⓧ*[n1, n2] L → ∧∧ 0 = n1 & 0 = n2 *)
-lemma fsle_lfxs_trans: ∀R,L1,T1,T2. ⦃L1, T1⦄ ⊆ ⦃L1, T2⦄ →
+lemma fsge_lfxs_trans: ∀R,L1,T1,T2. ⦃L1, T1⦄ ⊆ ⦃L1, T2⦄ →
∀L2. L1 ⪤*[R, T2] L2 → L1 ⪤*[R, T1] L2.
#R #L1 #T1 #T2 * #n1 #n2 #f1 #f2 #Hf1 #Hf2 #Hn #Hf #L2 #HL12
elim (lveq_inj_length … Hn ?) // #H1 #H2 destruct
/4 width=5 by lfxs_inv_frees, sle_lexs_trans, ex2_intro/
qed-.
-lemma lfxs_sym: ∀R. lfxs_fsle_compatible R →
+lemma lfxs_sym: ∀R. lfxs_fsge_compatible R →
(∀L1,L2,T1,T2. R L1 T1 T2 → R L2 T2 T1) →
∀T. symmetric … (lfxs R T).
#R #H1R #H2R #T #L1 #L2
qed-.
lemma lfxs_pair_sn_split: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) →
- lfxs_fsle_compatible R1 →
+ lfxs_fsge_compatible R1 →
∀L1,L2,V. L1 ⪤*[R1, V] L2 → ∀I,T.
∃∃L. L1 ⪤*[R1, ②{I}V.T] L & L ⪤*[R2, V] L2.
#R1 #R2 #HR1 #HR2 #HR #L1 #L2 #V * #f #Hf #HL12 * [ #p ] #I #T
qed-.
lemma lfxs_flat_dx_split: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) →
- lfxs_fsle_compatible R1 →
+ lfxs_fsge_compatible R1 →
∀L1,L2,T. L1 ⪤*[R1, T] L2 → ∀I,V.
∃∃L. L1 ⪤*[R1, ⓕ{I}V.T] L & L ⪤*[R2, T] L2.
#R1 #R2 #HR1 #HR2 #HR #L1 #L2 #T * #f #Hf #HL12 #I #V
qed-.
lemma lfxs_bind_dx_split: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) →
- lfxs_fsle_compatible R1 →
+ lfxs_fsge_compatible R1 →
∀I,L1,L2,V1,T. L1.ⓑ{I}V1 ⪤*[R1, T] L2 → ∀p.
∃∃L,V. L1 ⪤*[R1, ⓑ{p,I}V1.T] L & L.ⓑ{I}V ⪤*[R2, T] L2 & R1 L1 V1 V.
#R1 #R2 #HR1 #HR2 #HR #I #L1 #L2 #V1 #T * #f #Hf #HL12 #p
qed-.
lemma lfxs_bind_dx_split_void: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) →
- lfxs_fsle_compatible R1 →
+ lfxs_fsge_compatible R1 →
∀L1,L2,T. L1.ⓧ ⪤*[R1, T] L2 → ∀p,I,V.
∃∃L. L1 ⪤*[R1, ⓑ{p,I}V.T] L & L.ⓧ ⪤*[R2, T] L2.
#R1 #R2 #HR1 #HR2 #HR #L1 #L2 #T * #f #Hf #HL12 #p #I #V
(* Main properties with free variables inclusion for restricted closures ****)
theorem lfxs_conf: ∀R1,R2.
- lfxs_fsle_compatible R1 →
- lfxs_fsle_compatible R2 →
+ lfxs_fsge_compatible R1 →
+ lfxs_fsge_compatible R2 →
R_confluent2_lfxs R1 R2 R1 R2 →
∀T. confluent2 … (lfxs R1 T) (lfxs R2 T).
#R1 #R2 #HR1 #HR2 #HR12 #T #L0 #L1 * #f1 #Hf1 #HL01 #L2 * #f #Hf #HL02
]
]
qed-.
+
+theorem lfxs_trans_fsle: ∀R1,R2,R3.
+ lfxs_fsle_compatible R1 → lfxs_transitive_next R1 R2 R3 →
+ ∀L1,L,T. L1 ⪤*[R1, T] L →
+ ∀L2. L ⪤*[R2, T] L2 → L1 ⪤*[R3, T] L2.
+#R1 #R2 #R3 #H1R #H2R #L1 #L #T #H
+lapply (H1R … H) -H1R #H0
+cases H -H #f1 #Hf1 #HL1 #L2 * #f2 #Hf2 #HL2
+lapply (fsle_inv_frees_eq … H0 … Hf1 … Hf2) -H0 -Hf2
+/4 width=14 by lexs_trans_gen, lexs_fwd_length, sle_lexs_trans, ex2_intro/
+qed-.