(* *)
(**************************************************************************)
-include "basic_2/notation/relations/lazyeq_3.ma".
+include "basic_2/notation/relations/ideqsn_3.ma".
include "basic_2/static/lfxs.ma".
-(* EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES *******************)
+(* SYNTACTIC EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES *********)
-definition lfeq: relation3 term lenv lenv ≝ lfxs ceq.
+(* Basic_2A1: was: lleq *)
+definition lfeq: relation3 term lenv lenv ≝
+ lfxs ceq.
interpretation
- "equivalence on referred entries (local environment)"
- 'LazyEq T L1 L2 = (lfeq T L1 L2).
+ "syntactic equivalence on referred entries (local environment)"
+ '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 properties ***********************************************************)
-
-lemma lfeq_atom: ∀I. ⋆ ≡[⓪{I}] ⋆.
-/2 width=1 by lfxs_atom/ qed.
-
-lemma lfeq_sort: ∀I,L1,L2,V1,V2,s.
- L1 ≡[⋆s] L2 → L1.ⓑ{I}V1 ≡[⋆s] L2.ⓑ{I}V2.
-/2 width=1 by lfxs_sort/ qed.
-
-lemma lfeq_zero: ∀I,L1,L2,V.
- L1 ≡[V] L2 → L1.ⓑ{I}V ≡[#0] L2.ⓑ{I}V.
-/2 width=1 by lfxs_zero/ qed.
-
-lemma lfeq_lref: ∀I,L1,L2,V1,V2,i.
- L1 ≡[#i] L2 → L1.ⓑ{I}V1 ≡[#⫯i] L2.ⓑ{I}V2.
-/2 width=1 by lfxs_lref/ qed.
-
-lemma lfeq_gref: ∀I,L1,L2,V1,V2,l.
- L1 ≡[§l] L2 → L1.ⓑ{I}V1 ≡[§l] L2.ⓑ{I}V2.
-/2 width=1 by lfxs_gref/ qed.
-
(* Basic inversion lemmas ***************************************************)
-lemma lfeq_inv_atom_sn: ∀I,Y2. ⋆ ≡[⓪{I}] Y2 → Y2 = ⋆.
-/2 width=3 by lfxs_inv_atom_sn/ qed-.
-
-lemma lfeq_inv_atom_dx: ∀I,Y1. Y1 ≡[⓪{I}] ⋆ → Y1 = ⋆.
-/2 width=3 by lfxs_inv_atom_dx/ qed-.
-
-lemma lfeq_inv_zero: ∀Y1,Y2. Y1 ≡[#0] Y2 →
- (Y1 = ⋆ ∧ Y2 = ⋆) ∨
- ∃∃I,L1,L2,V. L1 ≡[V] L2 &
- Y1 = L1.ⓑ{I}V & Y2 = L2.ⓑ{I}V.
-#Y1 #Y2 #H elim (lfxs_inv_zero … H) -H *
-/3 width=7 by ex3_4_intro, or_introl, or_intror, conj/
-qed-.
-
-lemma lfeq_inv_lref: ∀Y1,Y2,i. Y1 ≡[#⫯i] Y2 →
- (Y1 = ⋆ ∧ Y2 = ⋆) ∨
- ∃∃I,L1,L2,V1,V2. L1 ≡[#i] L2 &
- Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
-#Y1 #Y2 #i #H elim (lfxs_inv_lref … H) -H *
-/3 width=8 by ex3_5_intro, or_introl, or_intror, conj/
-qed-.
-
-lemma lfeq_inv_bind: ∀I,L1,L2,V,T,p. L1 ≡[ⓑ{p,I}V.T] L2 →
- L1 ≡[V] L2 ∧ L1.ⓑ{I}V ≡[T] L2.ⓑ{I}V.
-#I #L1 #L2 #V #T #p #H elim (lfxs_inv_bind … H) -H /2 width=3 by conj/
-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.
-#I #L1 #L2 #V #T #H elim (lfxs_inv_flat … H) -H /2 width=3 by conj/
-qed-.
+ ∧∧ L1 ≡[V] L2 & L1 ≡[T] L2.
+/2 width=2 by lfxs_inv_flat/ qed-.
(* Advanced inversion lemmas ************************************************)
-lemma lfeq_inv_zero_pair_sn: ∀I,Y2,L1,V. L1.ⓑ{I}V ≡[#0] Y2 →
- ∃∃L2. L1 ≡[V] L2 & Y2 = L2.ⓑ{I}V.
-#I #Y2 #L1 #V #H elim (lfxs_inv_zero_pair_sn … H) -H /2 width=3 by ex2_intro/
+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,Y1,L2,V. Y1 ≡[#0] L2.ⓑ{I}V →
- ∃∃L1. L1 ≡[V] L2 & Y1 = L1.ⓑ{I}V.
-#I #Y1 #L2 #V #H elim (lfxs_inv_zero_pair_dx … H) -H
-#L1 #X #HL12 #HX #H destruct /2 width=3 by ex2_intro/
+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_pair_sn: ∀I,Y2,L1,V1,i. L1.ⓑ{I}V1 ≡[#⫯i] Y2 →
- ∃∃L2,V2. L1 ≡[#i] L2 & Y2 = L2.ⓑ{I}V2.
-/2 width=2 by lfxs_inv_lref_pair_sn/ 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_pair_dx: ∀I,Y1,L2,V2,i. Y1 ≡[#⫯i] L2.ⓑ{I}V2 →
- ∃∃L1,V1. L1 ≡[#i] L2 & Y1 = L1.ⓑ{I}V1.
-/2 width=2 by lfxs_inv_lref_pair_dx/ 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 *****************************************************)
-lemma lfeq_fwd_bind_sn: ∀I,L1,L2,V,T,p. L1 ≡[ⓑ{p,I}V.T] L2 → L1 ≡[V] L2.
-/2 width=4 by lfxs_fwd_bind_sn/ qed-.
-
-lemma lfeq_fwd_bind_dx: ∀I,L1,L2,V,T,p.
- L1 ≡[ⓑ{p,I}V.T] L2 → L1.ⓑ{I}V ≡[T] L2.ⓑ{I}V.
-/2 width=2 by lfxs_fwd_bind_dx/ qed-.
-
-lemma lfeq_fwd_flat_sn: ∀I,L1,L2,V,T. L1 ≡[ⓕ{I}V.T] L2 → L1 ≡[V] L2.
-/2 width=3 by lfxs_fwd_flat_sn/ qed-.
-
-lemma lfeq_fwd_flat_dx: ∀I,L1,L2,V,T. L1 ≡[ⓕ{I}V.T] L2 → L1 ≡[T] L2.
-/2 width=3 by lfxs_fwd_flat_dx/ qed-.
-
-lemma lfeq_fwd_pair_sn: ∀I,L1,L2,V,T. L1 ≡[②{I}V.T] L2 → L1 ≡[V] L2.
-/2 width=3 by lfxs_fwd_pair_sn/ qed-.
-
-(* Advanceded forward lemmas with generic extension on referred entries *****)
+(* 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-.
-lemma lfex_fwd_lfxs_refl: ∀R. (∀L. reflexive … (R L)) →
- ∀L1,L2,T. L1 ≡[T] L2 → L1 ⦻*[R, T] L2.
-/2 width=3 by lfxs_co/ 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 30:
- lleq_ind lleq_inv_bind lleq_inv_flat lleq_fwd_length lleq_fwd_lref
+(* Basic_2A1: removed theorems 10:
+ lleq_ind lleq_fwd_lref
lleq_fwd_drop_sn lleq_fwd_drop_dx
- lleq_fwd_bind_sn lleq_fwd_bind_dx lleq_fwd_flat_sn lleq_fwd_flat_dx
- lleq_sort lleq_skip lleq_lref lleq_free lleq_gref lleq_bind lleq_flat
- lleq_refl lleq_Y lleq_sym lleq_ge_up lleq_ge lleq_bind_O llpx_sn_lrefl
- lleq_trans lleq_canc_sn lleq_canc_dx lleq_nlleq_trans nlleq_lleq_div
+ lleq_skip lleq_lref lleq_free
+ lleq_Y lleq_ge_up lleq_ge
+
*)
projects / helm.git / blobdiff