(* Basic properties ***********************************************************)
-lemma frees_tdeq_conf_lexs: ∀h,o,f,L1,T1. L1 ⊢ 𝐅*⦃T1⦄ ≡ f → ∀T2. T1 ≛[h, o] T2 →
- ∀L2. L1 ≛[h, o, f] L2 → L2 ⊢ 𝐅*⦃T2⦄ ≡ f.
+lemma frees_tdeq_conf_lfdeq: ∀h,o,f,L1,T1. L1 ⊢ 𝐅*⦃T1⦄ ≡ f → ∀T2. T1 ≛[h, o] T2 →
+ ∀L2. L1 ≛[h, o, f] L2 → L2 ⊢ 𝐅*⦃T2⦄ ≡ 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
lemma frees_tdeq_conf: ∀h,o,f,L,T1. L ⊢ 𝐅*⦃T1⦄ ≡ f →
∀T2. T1 ≛[h, o] T2 → L ⊢ 𝐅*⦃T2⦄ ≡ f.
-/4 width=7 by frees_tdeq_conf_lexs, lexs_refl, ext2_refl/ qed-.
+/4 width=7 by frees_tdeq_conf_lfdeq, lexs_refl, ext2_refl/ qed-.
-lemma frees_lexs_conf: ∀h,o,f,L1,T. L1 ⊢ 𝐅*⦃T⦄ ≡ f →
- ∀L2. L1 ≛[h, o, f] L2 → L2 ⊢ 𝐅*⦃T⦄ ≡ f.
-/2 width=7 by frees_tdeq_conf_lexs, tdeq_refl/ qed-.
-
-lemma frees_lfdeq_conf_lexs: ∀h,o. lexs_frees_confluent (cdeq_ext h o) cfull.
-/3 width=7 by frees_tdeq_conf_lexs, ex2_intro/ qed-.
+lemma frees_lfdeq_conf: ∀h,o,f,L1,T. L1 ⊢ 𝐅*⦃T⦄ ≡ f →
+ ∀L2. L1 ≛[h, o, f] L2 → L2 ⊢ 𝐅*⦃T⦄ ≡ f.
+/2 width=7 by frees_tdeq_conf_lfdeq, tdeq_refl/ qed-.
lemma tdeq_lfdeq_conf_sn: ∀h,o. s_r_confluent1 … (cdeq h o) (lfdeq h o).
#h #o #L1 #T1 #T2 #HT12 #L2 *
/3 width=5 by frees_tdeq_conf, ex2_intro/
qed-.
-(* Basic_2A1: uses: lleq_sym *)
-lemma lfdeq_sym: ∀h,o,T. symmetric … (lfdeq h o T).
-#h #o #T #L1 #L2 *
-/4 width=7 by frees_tdeq_conf_lexs, lfxs_sym, tdeq_sym, ex2_intro/
-qed-.
-
lemma lfdeq_atom: ∀h,o,I. ⋆ ≛[h, o, ⓪{I}] ⋆.
/2 width=1 by lfxs_atom/ qed.