(* *)
(**************************************************************************)
-include "basic_2/notation/relations/doteqsn_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)"
- 'DotEqSn 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. 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.
+ 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.
(* Basic inversion lemmas ***************************************************)
-lemma lfeq_inv_bind: â\88\80p,I,L1,L2,V,T. L1 â\89\90[ⓑ{p,I}V.T] L2 →
- â\88§â\88§ L1 â\89\90[V] L2 & L1.â\93\91{I}V â\89\90[T] L2.ⓑ{I}V.
+lemma lfeq_inv_bind: â\88\80p,I,L1,L2,V,T. L1 â\89¡[ⓑ{p,I}V.T] L2 →
+ â\88§â\88§ L1 â\89¡[V] L2 & L1.â\93\91{I}V â\89¡[T] L2.ⓑ{I}V.
/2 width=2 by lfxs_inv_bind/ qed-.
-lemma lfeq_inv_flat: â\88\80I,L1,L2,V,T. L1 â\89\90[ⓕ{I}V.T] L2 →
- â\88§â\88§ L1 â\89\90[V] L2 & L1 â\89\90[T] L2.
+lemma lfeq_inv_flat: â\88\80I,L1,L2,V,T. L1 â\89¡[ⓕ{I}V.T] L2 →
+ â\88§â\88§ L1 â\89¡[V] L2 & L1 â\89¡[T] L2.
/2 width=2 by lfxs_inv_flat/ qed-.
(* Advanced inversion lemmas ************************************************)
-lemma lfeq_inv_zero_pair_sn: â\88\80I,L2,K1,V. K1.â\93\91{I}V â\89\90[#0] L2 →
- â\88\83â\88\83K2. K1 â\89\90[V] K2 & L2 = K2.ⓑ{I}V.
+lemma lfeq_inv_zero_pair_sn: â\88\80I,L2,K1,V. K1.â\93\91{I}V â\89¡[#0] L2 →
+ â\88\83â\88\83K2. K1 â\89¡[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: â\88\80I,L1,K2,V. L1 â\89\90[#0] K2.ⓑ{I}V →
- â\88\83â\88\83K1. K1 â\89\90[V] K2 & L1 = K1.ⓑ{I}V.
+lemma lfeq_inv_zero_pair_dx: â\88\80I,L1,K2,V. L1 â\89¡[#0] K2.ⓑ{I}V →
+ â\88\83â\88\83K1. K1 â\89¡[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: â\88\80I1,K1,L2,i. K1.â\93\98{I1} â\89\90[#⫯i] L2 →
- â\88\83â\88\83I2,K2. K1 â\89\90[#i] K2 & L2 = K2.ⓘ{I2}.
+lemma lfeq_inv_lref_bind_sn: â\88\80I1,K1,L2,i. K1.â\93\98{I1} â\89¡[#â\86\91i] L2 →
+ â\88\83â\88\83I2,K2. K1 â\89¡[#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\90[#⫯i] K2.ⓘ{I2} →
- â\88\83â\88\83I1,K1. K1 â\89\90[#i] K2 & L1 = K1.ⓘ{I1}.
+lemma lfeq_inv_lref_bind_dx: â\88\80I2,K2,L1,i. L1 â\89¡[#â\86\91i] K2.ⓘ{I2} →
+ â\88\83â\88\83I1,K1. K1 â\89¡[#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\90[T] L2 → L1 ⪤*[R, T] L2.
+ â\88\80L1,L2,T. L1 â\89¡[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: â\88\80f,L1,T. L1 â\8a¢ ð\9d\90\85*â¦\83Tâ¦\84 â\89¡ f →
- â\88\80L2. L1 â\89\90[T] L2 â\86\92 L2 â\8a¢ ð\9d\90\85*â¦\83Tâ¦\84 â\89¡ f.
+lemma frees_lfeq_conf: â\88\80f,L1,T. L1 â\8a¢ ð\9d\90\85*â¦\83Tâ¦\84 â\89\98 f →
+ â\88\80L2. L1 â\89¡[T] L2 â\86\92 L2 â\8a¢ ð\9d\90\85*â¦\83Tâ¦\84 â\89\98 f.
#f #L1 #T #H elim H -f -L1 -T
[ /2 width=3 by frees_sort/
| #f #i #Hf #L2 #H2