(* *)
(**************************************************************************)
-include "basic_2/notation/relations/lazyeqsn_5.ma".
+include "basic_2/notation/relations/stareqsn_5.ma".
include "basic_2/syntax/tdeq_ext.ma".
include "basic_2/static/lfxs.ma".
interpretation
"degree-based equivalence on referred entries (local environment)"
- 'LazyEqSn h o T L1 L2 = (lfdeq h o T L1 L2).
+ 'StarEqSn h o T L1 L2 = (lfdeq h o T L1 L2).
interpretation
"degree-based ranged equivalence (local environment)"
- 'LazyEqSn h o f L1 L2 = (lexs (cdeq_ext h o) cfull f L1 L2).
-(*
-definition lfdeq_transitive: predicate (relation3 lenv term term) ≝
- λR. ∀L2,T1,T2. R L2 T1 T2 → ∀L1. L1 ≡[h, o, T1] L2 → R L1 T1 T2.
-*)
+ 'StarEqSn h o f L1 L2 = (lexs (cdeq_ext h o) cfull f L1 L2).
+
(* Basic properties ***********************************************************)
-lemma frees_tdeq_conf_lexs: ∀h,o,f,L1,T1. L1 ⊢ 𝐅*⦃T1⦄ ≡ f → ∀T2. T1 ≡[h, o] T2 →
- ∀L2. L1 ≡[h, o, f] L2 → L2 ⊢ 𝐅*⦃T2⦄ ≡ f.
+lemma frees_tdeq_conf_lfdeq: ∀h,o,f,L1,T1. L1 ⊢ 𝐅*⦃T1⦄ ≡ f → ∀T2. T1 ≛[h, o] T2 →
+ ∀L2. L1 ≛[h, o, f] L2 → L2 ⊢ 𝐅*⦃T2⦄ ≡ f.
#h #o #f #L1 #T1 #H elim H -f -L1 -T1
[ #f #L1 #s1 #Hf #X #H1 #L2 #_
elim (tdeq_inv_sort1 … H1) -H1 #s2 #d #_ #_ #H destruct
qed-.
lemma frees_tdeq_conf: ∀h,o,f,L,T1. L ⊢ 𝐅*⦃T1⦄ ≡ f →
- â\88\80T2. T1 â\89¡[h, o] T2 → L ⊢ 𝐅*⦃T2⦄ ≡ f.
-/4 width=7 by frees_tdeq_conf_lexs, lexs_refl, ext2_refl/ qed-.
+ â\88\80T2. T1 â\89\9b[h, o] T2 → L ⊢ 𝐅*⦃T2⦄ ≡ f.
+/4 width=7 by frees_tdeq_conf_lfdeq, lexs_refl, ext2_refl/ qed-.
-lemma frees_lexs_conf: ∀h,o,f,L1,T. L1 ⊢ 𝐅*⦃T⦄ ≡ f →
- ∀L2. L1 ≡[h, o, f] L2 → L2 ⊢ 𝐅*⦃T⦄ ≡ f.
-/2 width=7 by frees_tdeq_conf_lexs, tdeq_refl/ qed-.
+lemma frees_lfdeq_conf: ∀h,o,f,L1,T. L1 ⊢ 𝐅*⦃T⦄ ≡ f →
+ ∀L2. L1 ≛[h, o, f] L2 → L2 ⊢ 𝐅*⦃T⦄ ≡ f.
+/2 width=7 by frees_tdeq_conf_lfdeq, tdeq_refl/ qed-.
-lemma frees_lfdeq_conf_lexs: ∀h,o. lexs_frees_confluent (cdeq_ext h o) cfull.
-/3 width=7 by frees_tdeq_conf_lexs, ex2_intro/ qed-.
-
-lemma tdeq_lfdeq_conf_sn: ∀h,o. s_r_confluent1 … (cdeq h o) (lfdeq h o).
-#h #o #L1 #T1 #T2 #HT12 #L2 *
+lemma tdeq_lfxs_conf: ∀R,h,o. s_r_confluent1 … (cdeq h o) (lfxs R).
+#R #h #o #L1 #T1 #T2 #HT12 #L2 *
/3 width=5 by frees_tdeq_conf, ex2_intro/
qed-.
-(* Basic_2A1: uses: lleq_sym *)
-lemma lfdeq_sym: ∀h,o,T. symmetric … (lfdeq h o T).
-#h #o #T #L1 #L2 *
-/4 width=7 by frees_tdeq_conf_lexs, lfxs_sym, tdeq_sym, ex2_intro/
-qed-.
+lemma tdeq_lfxs_div: ∀R,h,o,T1,T2. T1 ≛[h, o] T2 →
+ ∀L1,L2. L1 ⪤*[R, T2] L2 → L1 ⪤*[R, T1] L2.
+/3 width=5 by tdeq_lfxs_conf, tdeq_sym/ qed-.
-lemma lfdeq_atom: ∀h,o,I. ⋆ ≡[h, o, ⓪{I}] ⋆.
+lemma tdeq_lfdeq_conf: ∀h,o. s_r_confluent1 … (cdeq h o) (lfdeq h o).
+/2 width=5 by tdeq_lfxs_conf/ qed-.
+
+lemma tdeq_lfdeq_div: ∀h,o,T1,T2. T1 ≛[h, o] T2 →
+ ∀L1,L2. L1 ≛[h, o, T2] L2 → L1 ≛[h, o, T1] L2.
+/2 width=5 by tdeq_lfxs_div/ qed-.
+
+lemma lfdeq_atom: ∀h,o,I. ⋆ ≛[h, o, ⓪{I}] ⋆.
/2 width=1 by lfxs_atom/ qed.
(* Basic_2A1: uses: lleq_sort *)
lemma lfdeq_sort: ∀h,o,I1,I2,L1,L2,s.
- L1 â\89¡[h, o, â\8b\86s] L2 â\86\92 L1.â\93\98{I1} â\89¡[h, o, ⋆s] L2.ⓘ{I2}.
+ L1 â\89\9b[h, o, â\8b\86s] L2 â\86\92 L1.â\93\98{I1} â\89\9b[h, o, ⋆s] L2.ⓘ{I2}.
/2 width=1 by lfxs_sort/ qed.
-lemma lfdeq_pair: â\88\80h,o,I,L1,L2,V1,V2. L1 â\89¡[h, o, V1] L2 â\86\92 V1 â\89¡[h, o] V2 →
- L1.â\93\91{I}V1 â\89¡[h, o, #0] L2.ⓑ{I}V2.
+lemma lfdeq_pair: â\88\80h,o,I,L1,L2,V1,V2. L1 â\89\9b[h, o, V1] L2 â\86\92 V1 â\89\9b[h, o] V2 →
+ L1.â\93\91{I}V1 â\89\9b[h, o, #0] L2.ⓑ{I}V2.
/2 width=1 by lfxs_pair/ qed.
(*
lemma lfdeq_unit: ∀h,o,f,I,L1,L2. 𝐈⦃f⦄ → L1 ⪤*[cdeq_ext h o, cfull, f] L2 →
- L1.â\93¤{I} â\89¡[h, o, #0] L2.ⓤ{I}.
+ L1.â\93¤{I} â\89\9b[h, o, #0] L2.ⓤ{I}.
/2 width=3 by lfxs_unit/ qed.
*)
lemma lfdeq_lref: ∀h,o,I1,I2,L1,L2,i.
- L1 â\89¡[h, o, #i] L2 â\86\92 L1.â\93\98{I1} â\89¡[h, o, #⫯i] L2.ⓘ{I2}.
+ L1 â\89\9b[h, o, #i] L2 â\86\92 L1.â\93\98{I1} â\89\9b[h, o, #⫯i] L2.ⓘ{I2}.
/2 width=1 by lfxs_lref/ qed.
(* Basic_2A1: uses: lleq_gref *)
lemma lfdeq_gref: ∀h,o,I1,I2,L1,L2,l.
- L1 â\89¡[h, o, §l] L2 â\86\92 L1.â\93\98{I1} â\89¡[h, o, §l] L2.ⓘ{I2}.
+ L1 â\89\9b[h, o, §l] L2 â\86\92 L1.â\93\98{I1} â\89\9b[h, o, §l] L2.ⓘ{I2}.
/2 width=1 by lfxs_gref/ qed.
lemma lfdeq_bind_repl_dx: ∀h,o,I,I1,L1,L2.∀T:term.
- L1.â\93\98{I} â\89¡[h, o, T] L2.ⓘ{I1} →
- â\88\80I2. I â\89¡[h, o] I2 →
- L1.â\93\98{I} â\89¡[h, o, T] L2.ⓘ{I2}.
+ L1.â\93\98{I} â\89\9b[h, o, T] L2.ⓘ{I1} →
+ â\88\80I2. I â\89\9b[h, o] I2 →
+ L1.â\93\98{I} â\89\9b[h, o, T] L2.ⓘ{I2}.
/2 width=2 by lfxs_bind_repl_dx/ qed-.
(* Basic inversion lemmas ***************************************************)
-lemma lfdeq_inv_atom_sn: â\88\80h,o,Y2. â\88\80T:term. â\8b\86 â\89¡[h, o, T] Y2 → Y2 = ⋆.
+lemma lfdeq_inv_atom_sn: â\88\80h,o,Y2. â\88\80T:term. â\8b\86 â\89\9b[h, o, T] Y2 → Y2 = ⋆.
/2 width=3 by lfxs_inv_atom_sn/ qed-.
-lemma lfdeq_inv_atom_dx: â\88\80h,o,Y1. â\88\80T:term. Y1 â\89¡[h, o, T] ⋆ → Y1 = ⋆.
+lemma lfdeq_inv_atom_dx: â\88\80h,o,Y1. â\88\80T:term. Y1 â\89\9b[h, o, T] ⋆ → Y1 = ⋆.
/2 width=3 by lfxs_inv_atom_dx/ qed-.
(*
-lemma lfdeq_inv_zero: â\88\80h,o,Y1,Y2. Y1 â\89¡[h, o, #0] Y2 →
+lemma lfdeq_inv_zero: â\88\80h,o,Y1,Y2. Y1 â\89\9b[h, o, #0] Y2 →
∨∨ Y1 = ⋆ ∧ Y2 = ⋆
- | â\88\83â\88\83I,L1,L2,V1,V2. L1 â\89¡[h, o, V1] L2 & V1 â\89¡[h, o] V2 &
+ | â\88\83â\88\83I,L1,L2,V1,V2. L1 â\89\9b[h, o, V1] L2 & V1 â\89\9b[h, o] V2 &
Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2
| ∃∃f,I,L1,L2. 𝐈⦃f⦄ & L1 ⪤*[cdeq_ext h o, cfull, f] L2 &
Y1 = L1.ⓤ{I} & Y2 = L2.ⓤ{I}.
/3 width=9 by or3_intro0, or3_intro1, or3_intro2, ex4_5_intro, ex4_4_intro, conj/
qed-.
*)
-lemma lfdeq_inv_lref: â\88\80h,o,Y1,Y2,i. Y1 â\89¡[h, o, #⫯i] Y2 →
+lemma lfdeq_inv_lref: â\88\80h,o,Y1,Y2,i. Y1 â\89\9b[h, o, #⫯i] Y2 →
(Y1 = ⋆ ∧ Y2 = ⋆) ∨
- â\88\83â\88\83I1,I2,L1,L2. L1 â\89¡[h, o, #i] L2 &
+ â\88\83â\88\83I1,I2,L1,L2. L1 â\89\9b[h, o, #i] L2 &
Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
/2 width=1 by lfxs_inv_lref/ qed-.
(* Basic_2A1: uses: lleq_inv_bind lleq_inv_bind_O *)
-lemma lfdeq_inv_bind: â\88\80h,o,p,I,L1,L2,V,T. L1 â\89¡[h, o, ⓑ{p,I}V.T] L2 →
- L1 â\89¡[h, o, V] L2 â\88§ L1.â\93\91{I}V â\89¡[h, o, T] L2.ⓑ{I}V.
+lemma lfdeq_inv_bind: â\88\80h,o,p,I,L1,L2,V,T. L1 â\89\9b[h, o, ⓑ{p,I}V.T] L2 →
+ L1 â\89\9b[h, o, V] L2 â\88§ L1.â\93\91{I}V â\89\9b[h, o, T] L2.ⓑ{I}V.
/2 width=2 by lfxs_inv_bind/ qed-.
(* Basic_2A1: uses: lleq_inv_flat *)
-lemma lfdeq_inv_flat: â\88\80h,o,I,L1,L2,V,T. L1 â\89¡[h, o, ⓕ{I}V.T] L2 →
- L1 â\89¡[h, o, V] L2 â\88§ L1 â\89¡[h, o, T] L2.
+lemma lfdeq_inv_flat: â\88\80h,o,I,L1,L2,V,T. L1 â\89\9b[h, o, ⓕ{I}V.T] L2 →
+ L1 â\89\9b[h, o, V] L2 â\88§ L1 â\89\9b[h, o, T] L2.
/2 width=2 by lfxs_inv_flat/ qed-.
(* Advanced inversion lemmas ************************************************)
-lemma lfdeq_inv_zero_pair_sn: â\88\80h,o,I,Y2,L1,V1. L1.â\93\91{I}V1 â\89¡[h, o, #0] Y2 →
- â\88\83â\88\83L2,V2. L1 â\89¡[h, o, V1] L2 & V1 â\89¡[h, o] V2 & Y2 = L2.ⓑ{I}V2.
+lemma lfdeq_inv_zero_pair_sn: â\88\80h,o,I,Y2,L1,V1. L1.â\93\91{I}V1 â\89\9b[h, o, #0] Y2 →
+ â\88\83â\88\83L2,V2. L1 â\89\9b[h, o, V1] L2 & V1 â\89\9b[h, o] V2 & Y2 = L2.ⓑ{I}V2.
/2 width=1 by lfxs_inv_zero_pair_sn/ qed-.
-lemma lfdeq_inv_zero_pair_dx: â\88\80h,o,I,Y1,L2,V2. Y1 â\89¡[h, o, #0] L2.ⓑ{I}V2 →
- â\88\83â\88\83L1,V1. L1 â\89¡[h, o, V1] L2 & V1 â\89¡[h, o] V2 & Y1 = L1.ⓑ{I}V1.
+lemma lfdeq_inv_zero_pair_dx: â\88\80h,o,I,Y1,L2,V2. Y1 â\89\9b[h, o, #0] L2.ⓑ{I}V2 →
+ â\88\83â\88\83L1,V1. L1 â\89\9b[h, o, V1] L2 & V1 â\89\9b[h, o] V2 & Y1 = L1.ⓑ{I}V1.
/2 width=1 by lfxs_inv_zero_pair_dx/ qed-.
-lemma lfdeq_inv_lref_bind_sn: â\88\80h,o,I1,Y2,L1,i. L1.â\93\98{I1} â\89¡[h, o, #⫯i] Y2 →
- â\88\83â\88\83I2,L2. L1 â\89¡[h, o, #i] L2 & Y2 = L2.ⓘ{I2}.
+lemma lfdeq_inv_lref_bind_sn: â\88\80h,o,I1,Y2,L1,i. L1.â\93\98{I1} â\89\9b[h, o, #⫯i] Y2 →
+ â\88\83â\88\83I2,L2. L1 â\89\9b[h, o, #i] L2 & Y2 = L2.ⓘ{I2}.
/2 width=2 by lfxs_inv_lref_bind_sn/ qed-.
-lemma lfdeq_inv_lref_bind_dx: â\88\80h,o,I2,Y1,L2,i. Y1 â\89¡[h, o, #⫯i] L2.ⓘ{I2} →
- â\88\83â\88\83I1,L1. L1 â\89¡[h, o, #i] L2 & Y1 = L1.ⓘ{I1}.
+lemma lfdeq_inv_lref_bind_dx: â\88\80h,o,I2,Y1,L2,i. Y1 â\89\9b[h, o, #⫯i] L2.ⓘ{I2} →
+ â\88\83â\88\83I1,L1. L1 â\89\9b[h, o, #i] L2 & Y1 = L1.ⓘ{I1}.
/2 width=2 by lfxs_inv_lref_bind_dx/ qed-.
(* Basic forward lemmas *****************************************************)
+lemma lfdeq_fwd_zero_pair: ∀h,o,I,K1,K2,V1,V2.
+ K1.ⓑ{I}V1 ≛[h, o, #0] K2.ⓑ{I}V2 → K1 ≛[h, o, V1] K2.
+/2 width=3 by lfxs_fwd_zero_pair/ qed-.
+
(* Basic_2A1: uses: lleq_fwd_bind_sn lleq_fwd_flat_sn *)
-lemma lfdeq_fwd_pair_sn: â\88\80h,o,I,L1,L2,V,T. L1 â\89¡[h, o, â\91¡{I}V.T] L2 â\86\92 L1 â\89¡[h, o, V] L2.
+lemma lfdeq_fwd_pair_sn: â\88\80h,o,I,L1,L2,V,T. L1 â\89\9b[h, o, â\91¡{I}V.T] L2 â\86\92 L1 â\89\9b[h, o, V] L2.
/2 width=3 by lfxs_fwd_pair_sn/ qed-.
(* Basic_2A1: uses: lleq_fwd_bind_dx lleq_fwd_bind_O_dx *)
lemma lfdeq_fwd_bind_dx: ∀h,o,p,I,L1,L2,V,T.
- L1 â\89¡[h, o, â\93\91{p,I}V.T] L2 â\86\92 L1.â\93\91{I}V â\89¡[h, o, T] L2.ⓑ{I}V.
+ L1 â\89\9b[h, o, â\93\91{p,I}V.T] L2 â\86\92 L1.â\93\91{I}V â\89\9b[h, o, T] L2.ⓑ{I}V.
/2 width=2 by lfxs_fwd_bind_dx/ qed-.
(* Basic_2A1: uses: lleq_fwd_flat_dx *)
-lemma lfdeq_fwd_flat_dx: â\88\80h,o,I,L1,L2,V,T. L1 â\89¡[h, o, â\93\95{I}V.T] L2 â\86\92 L1 â\89¡[h, o, T] L2.
+lemma lfdeq_fwd_flat_dx: â\88\80h,o,I,L1,L2,V,T. L1 â\89\9b[h, o, â\93\95{I}V.T] L2 â\86\92 L1 â\89\9b[h, o, T] L2.
/2 width=3 by lfxs_fwd_flat_dx/ qed-.
-lemma lfdeq_fwd_dx: â\88\80h,o,I2,L1,K2. â\88\80T:term. L1 â\89¡[h, o, T] K2.ⓘ{I2} →
+lemma lfdeq_fwd_dx: â\88\80h,o,I2,L1,K2. â\88\80T:term. L1 â\89\9b[h, o, T] K2.ⓘ{I2} →
∃∃I1,K1. L1 = K1.ⓘ{I1}.
/2 width=5 by lfxs_fwd_dx/ qed-.
-
-(* Basic_2A1: removed theorems 10:
- lleq_ind lleq_fwd_lref
- lleq_fwd_drop_sn lleq_fwd_drop_dx
- lleq_skip lleq_lref lleq_free
- lleq_Y lleq_ge_up lleq_ge
-
-*)
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