(* *)
(**************************************************************************)
-include "basic_2/notation/relations/lazyeqsn_3.ma".
+include "basic_2/notation/relations/ideqsn_3.ma".
include "basic_2/static/lfxs.ma".
(* SYNTACTIC EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES *********)
interpretation
"syntactic equivalence on referred entries (local environment)"
- 'LazyEqSn T L1 L2 = (lfeq T L1 L2).
+ 'IdEqSn T L1 L2 = (lfeq T L1 L2).
+(* Note: "lfeq_transitive R" is equivalent to "lfxs_transitive ceq R R" *)
(* Basic_2A1: uses: lleq_transitive *)
definition lfeq_transitive: predicate (relation3 lenv term term) ≝
λR. ∀L2,T1,T2. R L2 T1 T2 → ∀L1. L1 ≡[T1] L2 → R L1 T1 T2.
(* Basic inversion lemmas ***************************************************)
-lemma lfeq_transitive_inv_lfxs: ∀R. lfeq_transitive R → lfxs_transitive ceq R R.
-/2 width=3 by/ qed-.
-
lemma lfeq_inv_bind: ∀p,I,L1,L2,V,T. L1 ≡[ⓑ{p,I}V.T] L2 →
∧∧ L1 ≡[V] L2 & L1.ⓑ{I}V ≡[T] L2.ⓑ{I}V.
/2 width=2 by lfxs_inv_bind/ qed-.
/2 width=3 by ex2_intro/
qed-.
-lemma lfeq_inv_lref_bind_sn: â\88\80I1,K1,L2,i. K1.â\93\98{I1} â\89¡[#⫯i] L2 →
+lemma lfeq_inv_lref_bind_sn: â\88\80I1,K1,L2,i. K1.â\93\98{I1} â\89¡[#â\86\91i] L2 →
∃∃I2,K2. K1 ≡[#i] K2 & L2 = K2.ⓘ{I2}.
/2 width=2 by lfxs_inv_lref_bind_sn/ qed-.
-lemma lfeq_inv_lref_bind_dx: â\88\80I2,K2,L1,i. L1 â\89¡[#⫯i] K2.ⓘ{I2} →
+lemma lfeq_inv_lref_bind_dx: â\88\80I2,K2,L1,i. L1 â\89¡[#â\86\91i] K2.ⓘ{I2} →
∃∃I1,K1. K1 ≡[#i] K2 & L1 = K1.ⓘ{I1}.
/2 width=2 by lfxs_inv_lref_bind_dx/ qed-.
(* Basic_properties *********************************************************)
-lemma lfxs_transitive_lfeq: ∀R. lfxs_transitive ceq R R → lfeq_transitive R.
-/2 width=5 by/ qed.
-
-lemma frees_lfeq_conf: ∀f,L1,T. L1 ⊢ 𝐅*⦃T⦄ ≡ f →
- ∀L2. L1 ≡[T] L2 → L2 ⊢ 𝐅*⦃T⦄ ≡ f.
+lemma frees_lfeq_conf: ∀f,L1,T. L1 ⊢ 𝐅*⦃T⦄ ≘ f →
+ ∀L2. L1 ≡[T] L2 → L2 ⊢ 𝐅*⦃T⦄ ≘ f.
#f #L1 #T #H elim H -f -L1 -T
[ /2 width=3 by frees_sort/
| #f #i #Hf #L2 #H2