"syntactic equivalence on referred entries (local environment)"
'LazyEqSn T L1 L2 = (lfeq T L1 L2).
-(***************************************************)
+(* 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.
-axiom lfeq_lfxs_trans: ∀R,L1,L,T. L1 ≡[T] L →
- ∀L2. L ⪤*[R, T] L2 → L1 ⪤*[R, T] L2.
+(* Basic_properties *********************************************************)
+
+lemma lfxs_transitive_lfeq: ∀R. lfxs_transitive ceq R R → lfeq_transitive R.
+/2 width=5 by/ qed.
+
+(* 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-.
+
+lemma lfeq_inv_flat: ∀I,L1,L2,V,T. L1 ≡[ⓕ{I}V.T] L2 →
+ ∧∧ L1 ≡[V] L2 & L1 ≡[T] L2.
+/2 width=2 by lfxs_inv_flat/ qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma lfeq_inv_zero_pair_sn: ∀I,L2,K1,V. K1.ⓑ{I}V ≡[#0] L2 →
+ ∃∃K2. K1 ≡[V] K2 & L2 = K2.ⓑ{I}V.
+#I #L2 #K1 #V #H
+elim (lfxs_inv_zero_pair_sn … H) -H #K2 #X #HK12 #HX #H destruct
+/2 width=3 by ex2_intro/
+qed-.
+
+lemma lfeq_inv_zero_pair_dx: ∀I,L1,K2,V. L1 ≡[#0] K2.ⓑ{I}V →
+ ∃∃K1. K1 ≡[V] K2 & L1 = K1.ⓑ{I}V.
+#I #L1 #K2 #V #H
+elim (lfxs_inv_zero_pair_dx … H) -H #K1 #X #HK12 #HX #H destruct
+/2 width=3 by ex2_intro/
+qed-.
+
+lemma lfeq_inv_lref_bind_sn: ∀I1,K1,L2,i. K1.ⓘ{I1} ≡[#⫯i] 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: ∀I2,K2,L1,i. L1 ≡[#⫯i] K2.ⓘ{I2} →
+ ∃∃I1,K1. K1 ≡[#i] K2 & L1 = K1.ⓘ{I1}.
+/2 width=2 by lfxs_inv_lref_bind_dx/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+(* Basic_2A1: was: llpx_sn_lrefl *)
+(* Note: this should have been lleq_fwd_llpx_sn *)
+lemma lfeq_fwd_lfxs: ∀R. c_reflexive … R →
+ ∀L1,L2,T. L1 ≡[T] L2 → L1 ⪤*[R, T] L2.
+#R #HR #L1 #L2 #T * #f #Hf #HL12
+/4 width=7 by lexs_co, cext2_co, ex2_intro/
+qed-.
(* Basic_2A1: removed theorems 10:
lleq_ind lleq_fwd_lref