(* Basic properties ***********************************************************)
-lemma frees_tdeq_conf_lfdeq: â\88\80h,o,f,L1,T1. L1 â\8a¢ ð\9d\90\85*â¦\83T1â¦\84 â\89¡ f → ∀T2. T1 ≛[h, o] T2 →
- â\88\80L2. L1 â\89\9b[h, o, f] L2 â\86\92 L2 â\8a¢ ð\9d\90\85*â¦\83T2â¦\84 â\89¡ f.
+lemma frees_tdeq_conf_lfdeq: â\88\80h,o,f,L1,T1. L1 â\8a¢ ð\9d\90\85*â¦\83T1â¦\84 â\89\98 f → ∀T2. T1 ≛[h, o] T2 →
+ â\88\80L2. L1 â\89\9b[h, o, f] L2 â\86\92 L2 â\8a¢ ð\9d\90\85*â¦\83T2â¦\84 â\89\98 f.
#h #o #f #L1 #T1 #H elim H -f -L1 -T1
[ #f #L1 #s1 #Hf #X #H1 #L2 #_
elim (tdeq_inv_sort1 … H1) -H1 #s2 #d #_ #_ #H destruct
]
qed-.
-lemma frees_tdeq_conf: â\88\80h,o,f,L,T1. L â\8a¢ ð\9d\90\85*â¦\83T1â¦\84 â\89¡ f →
- â\88\80T2. T1 â\89\9b[h, o] T2 â\86\92 L â\8a¢ ð\9d\90\85*â¦\83T2â¦\84 â\89¡ f.
+lemma frees_tdeq_conf: â\88\80h,o,f,L,T1. L â\8a¢ ð\9d\90\85*â¦\83T1â¦\84 â\89\98 f →
+ â\88\80T2. T1 â\89\9b[h, o] T2 â\86\92 L â\8a¢ ð\9d\90\85*â¦\83T2â¦\84 â\89\98 f.
/4 width=7 by frees_tdeq_conf_lfdeq, lexs_refl, ext2_refl/ qed-.
-lemma frees_lfdeq_conf: â\88\80h,o,f,L1,T. L1 â\8a¢ ð\9d\90\85*â¦\83Tâ¦\84 â\89¡ f →
- â\88\80L2. L1 â\89\9b[h, o, f] L2 â\86\92 L2 â\8a¢ ð\9d\90\85*â¦\83Tâ¦\84 â\89¡ f.
+lemma frees_lfdeq_conf: â\88\80h,o,f,L1,T. L1 â\8a¢ ð\9d\90\85*â¦\83Tâ¦\84 â\89\98 f →
+ â\88\80L2. L1 â\89\9b[h, o, f] L2 â\86\92 L2 â\8a¢ ð\9d\90\85*â¦\83Tâ¦\84 â\89\98 f.
/2 width=7 by frees_tdeq_conf_lfdeq, tdeq_refl/ qed-.
-lemma tdeq_lfdeq_conf: ∀h,o. s_r_confluent1 … (cdeq h o) (lfdeq h o).
-#h #o #L1 #T1 #T2 #HT12 #L2 *
+lemma tdeq_lfxs_conf: ∀R,h,o. s_r_confluent1 … (cdeq h o) (lfxs R).
+#R #h #o #L1 #T1 #T2 #HT12 #L2 *
/3 width=5 by frees_tdeq_conf, ex2_intro/
qed-.
+lemma tdeq_lfxs_div: ∀R,h,o,T1,T2. T1 ≛[h, o] T2 →
+ ∀L1,L2. L1 ⪤*[R, T2] L2 → L1 ⪤*[R, T1] L2.
+/3 width=5 by tdeq_lfxs_conf, tdeq_sym/ qed-.
+
+lemma tdeq_lfdeq_conf: ∀h,o. s_r_confluent1 … (cdeq h o) (lfdeq h o).
+/2 width=5 by tdeq_lfxs_conf/ qed-.
+
lemma tdeq_lfdeq_div: ∀h,o,T1,T2. T1 ≛[h, o] T2 →
∀L1,L2. L1 ≛[h, o, T2] L2 → L1 ≛[h, o, T1] L2.
-/3 width=3 by tdeq_lfdeq_conf, tdeq_sym/ qed-.
+/2 width=5 by tdeq_lfxs_div/ qed-.
lemma lfdeq_atom: ∀h,o,I. ⋆ ≛[h, o, ⓪{I}] ⋆.
/2 width=1 by lfxs_atom/ qed.