(* *)
(**************************************************************************)
-include "basic_2/notation/relations/lazyeqsn_3.ma".
+include "basic_2/notation/relations/doteqsn_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).
+ 'DotEqSn 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. â\88\80L2,T1,T2. R L2 T1 T2 â\86\92 â\88\80L1. L1 â\89¡[T1] L2 → R L1 T1 T2.
+ λR. â\88\80L2,T1,T2. R L2 T1 T2 â\86\92 â\88\80L1. L1 â\89\90[T1] L2 → R L1 T1 T2.
-(* Basic_properties *********************************************************)
+(* Basic inversion lemmas ***************************************************)
-lemma lfxs_transitive_lfeq: ∀R. lfxs_transitive ceq R R → lfeq_transitive R.
-/2 width=5 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-.
-(* Basic inversion lemmas ***************************************************)
+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_transitive_inv_lfxs: ∀R. lfeq_transitive R → lfxs_transitive ceq R R.
-/2 width=3 by/ qed-.
+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 →
- â\88\80L1,L2,T. L1 â\89¡[T] L2 → L1 ⪤*[R, T] L2.
+ â\88\80L1,L2,T. L1 â\89\90[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_properties *********************************************************)
+
+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
+ >(lfxs_inv_atom_sn … H2) -L2
+ /2 width=1 by frees_atom/
+| #f #I #L1 #V1 #_ #IH #Y #H2
+ elim (lfeq_inv_zero_pair_sn … H2) -H2 #L2 #HL12 #H destruct
+ /3 width=1 by frees_pair/
+| #f #I #L1 #Hf #Y #H2
+ elim (lfxs_inv_zero_unit_sn … H2) -H2 #g #L2 #_ #_ #H destruct
+ /2 width=1 by frees_unit/
+| #f #I #L1 #i #_ #IH #Y #H2
+ elim (lfeq_inv_lref_bind_sn … H2) -H2 #J #L2 #HL12 #H destruct
+ /3 width=1 by frees_lref/
+| /2 width=1 by frees_gref/
+| #f1V #f1T #f1 #p #I #L1 #V1 #T1 #_ #_ #Hf1 #IHV #IHT #L2 #H2
+ elim (lfeq_inv_bind … H2) -H2 /3 width=5 by frees_bind/
+| #f1V #f1T #f1 #I #L1 #V1 #T1 #_ #_ #Hf1 #IHV #IHT #L2 #H2
+ elim (lfeq_inv_flat … H2) -H2 /3 width=5 by frees_flat/
+]
+qed-.
+
(* Basic_2A1: removed theorems 10:
lleq_ind lleq_fwd_lref
lleq_fwd_drop_sn lleq_fwd_drop_dx