(* DEGREE-BASED EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ******)
-definition lfdeq: ∀h. sd h → relation3 term lenv lenv ≝
- λh,o. lfxs (cdeq h o).
+definition lfdeq (h) (o): relation3 term lenv lenv ≝
+ lfxs (cdeq h o).
interpretation
"degree-based equivalence on referred entries (local environment)"
(* Basic properties ***********************************************************)
-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.
+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 →
- ∀T2. T1 ≛[h, o] T2 → L ⊢ 𝐅*⦃T2⦄ ≡ f.
+lemma frees_tdeq_conf (h) (o): ∀f,L,T1. L ⊢ 𝐅*⦃T1⦄ ≘ f →
+ ∀T2. T1 ≛[h, o] T2 → L ⊢ 𝐅*⦃T2⦄ ≘ f.
/4 width=7 by frees_tdeq_conf_lfdeq, lexs_refl, ext2_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.
+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 tdeq_lfxs_conf: ∀R,h,o. s_r_confluent1 … (cdeq h o) (lfxs R).
+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-.
-lemma tdeq_lfxs_div: ∀R,h,o,T1,T2. T1 ≛[h, o] T2 →
- ∀L1,L2. L1 ⪤*[R, T2] L2 → L1 ⪤*[R, T1] L2.
+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 tdeq_lfdeq_conf: ∀h,o. s_r_confluent1 … (cdeq h o) (lfdeq h o).
+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.
+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}] ⋆.
+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 ≛[h, o, ⋆s] L2 → L1.ⓘ{I1} ≛[h, o, ⋆s] L2.ⓘ{I2}.
+lemma lfdeq_sort (h) (o): ∀I1,I2,L1,L2,s.
+ L1 ≛[h, o, ⋆s] L2 → L1.ⓘ{I1} ≛[h, o, ⋆s] L2.ⓘ{I2}.
/2 width=1 by lfxs_sort/ qed.
-lemma lfdeq_pair: ∀h,o,I,L1,L2,V1,V2. L1 ≛[h, o, V1] L2 → V1 ≛[h, o] V2 →
- L1.ⓑ{I}V1 ≛[h, o, #0] L2.ⓑ{I}V2.
+lemma lfdeq_pair (h) (o): ∀I,L1,L2,V1,V2. L1 ≛[h, o, V1] L2 → V1 ≛[h, o] V2 →
+ L1.ⓑ{I}V1 ≛[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.ⓤ{I} ≛[h, o, #0] L2.ⓤ{I}.
+lemma lfdeq_unit (h) (o): ∀f,I,L1,L2. 𝐈⦃f⦄ → L1 ⪤*[cdeq_ext h o, cfull, f] L2 →
+ L1.ⓤ{I} ≛[h, o, #0] L2.ⓤ{I}.
/2 width=3 by lfxs_unit/ qed.
*)
-lemma lfdeq_lref: ∀h,o,I1,I2,L1,L2,i.
- L1 ≛[h, o, #i] L2 → L1.ⓘ{I1} ≛[h, o, #⫯i] L2.ⓘ{I2}.
+lemma lfdeq_lref (h) (o): ∀I1,I2,L1,L2,i.
+ L1 ≛[h, o, #i] L2 → L1.ⓘ{I1} ≛[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 ≛[h, o, §l] L2 → L1.ⓘ{I1} ≛[h, o, §l] L2.ⓘ{I2}.
+lemma lfdeq_gref (h) (o): ∀I1,I2,L1,L2,l.
+ L1 ≛[h, o, §l] L2 → L1.ⓘ{I1} ≛[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.ⓘ{I} ≛[h, o, T] L2.ⓘ{I1} →
- ∀I2. I ≛[h, o] I2 →
- L1.ⓘ{I} ≛[h, o, T] L2.ⓘ{I2}.
+lemma lfdeq_bind_repl_dx (h) (o): ∀I,I1,L1,L2.∀T:term.
+ L1.ⓘ{I} ≛[h, o, T] L2.ⓘ{I1} →
+ ∀I2. I ≛[h, o] I2 →
+ L1.ⓘ{I} ≛[h, o, T] L2.ⓘ{I2}.
/2 width=2 by lfxs_bind_repl_dx/ qed-.
(* Basic inversion lemmas ***************************************************)
-lemma lfdeq_inv_atom_sn: ∀h,o,Y2. ∀T:term. ⋆ ≛[h, o, T] Y2 → Y2 = ⋆.
+lemma lfdeq_inv_atom_sn (h) (o): ∀Y2. ∀T:term. ⋆ ≛[h, o, T] Y2 → Y2 = ⋆.
/2 width=3 by lfxs_inv_atom_sn/ qed-.
-lemma lfdeq_inv_atom_dx: ∀h,o,Y1. ∀T:term. Y1 ≛[h, o, T] ⋆ → Y1 = ⋆.
+lemma lfdeq_inv_atom_dx (h) (o): ∀Y1. ∀T:term. Y1 ≛[h, o, T] ⋆ → Y1 = ⋆.
/2 width=3 by lfxs_inv_atom_dx/ qed-.
(*
-lemma lfdeq_inv_zero: ∀h,o,Y1,Y2. Y1 ≛[h, o, #0] Y2 →
- ∨∨ Y1 = ⋆ ∧ Y2 = ⋆
- | ∃∃I,L1,L2,V1,V2. L1 ≛[h, o, V1] L2 & V1 ≛[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}.
+lemma lfdeq_inv_zero (h) (o): ∀Y1,Y2. Y1 ≛[h, o, #0] Y2 →
+ ∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆
+ | ∃∃I,L1,L2,V1,V2. L1 ≛[h, o, V1] L2 & V1 ≛[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}.
#h #o #Y1 #Y2 #H elim (lfxs_inv_zero … H) -H *
/3 width=9 by or3_intro0, or3_intro1, or3_intro2, ex4_5_intro, ex4_4_intro, conj/
qed-.
*)
-lemma lfdeq_inv_lref: ∀h,o,Y1,Y2,i. Y1 ≛[h, o, #⫯i] Y2 →
- (Y1 = ⋆ ∧ Y2 = ⋆) ∨
- ∃∃I1,I2,L1,L2. L1 ≛[h, o, #i] L2 &
- Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
+lemma lfdeq_inv_lref (h) (o): ∀Y1,Y2,i. Y1 ≛[h, o, #↑i] Y2 →
+ ∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆
+ | ∃∃I1,I2,L1,L2. L1 ≛[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: ∀h,o,p,I,L1,L2,V,T. L1 ≛[h, o, ⓑ{p,I}V.T] L2 →
- L1 ≛[h, o, V] L2 ∧ L1.ⓑ{I}V ≛[h, o, T] L2.ⓑ{I}V.
+lemma lfdeq_inv_bind (h) (o): ∀p,I,L1,L2,V,T. L1 ≛[h, o, ⓑ{p,I}V.T] L2 →
+ ∧∧ L1 ≛[h, o, V] L2 & L1.ⓑ{I}V ≛[h, o, T] L2.ⓑ{I}V.
/2 width=2 by lfxs_inv_bind/ qed-.
(* Basic_2A1: uses: lleq_inv_flat *)
-lemma lfdeq_inv_flat: ∀h,o,I,L1,L2,V,T. L1 ≛[h, o, ⓕ{I}V.T] L2 →
- L1 ≛[h, o, V] L2 ∧ L1 ≛[h, o, T] L2.
+lemma lfdeq_inv_flat (h) (o): ∀I,L1,L2,V,T. L1 ≛[h, o, ⓕ{I}V.T] L2 →
+ ∧∧ L1 ≛[h, o, V] L2 & L1 ≛[h, o, T] L2.
/2 width=2 by lfxs_inv_flat/ qed-.
(* Advanced inversion lemmas ************************************************)
-lemma lfdeq_inv_zero_pair_sn: ∀h,o,I,Y2,L1,V1. L1.ⓑ{I}V1 ≛[h, o, #0] Y2 →
- ∃∃L2,V2. L1 ≛[h, o, V1] L2 & V1 ≛[h, o] V2 & Y2 = L2.ⓑ{I}V2.
+lemma lfdeq_inv_zero_pair_sn (h) (o): ∀I,Y2,L1,V1. L1.ⓑ{I}V1 ≛[h, o, #0] Y2 →
+ ∃∃L2,V2. L1 ≛[h, o, V1] L2 & V1 ≛[h, o] V2 & Y2 = L2.ⓑ{I}V2.
/2 width=1 by lfxs_inv_zero_pair_sn/ qed-.
-lemma lfdeq_inv_zero_pair_dx: ∀h,o,I,Y1,L2,V2. Y1 ≛[h, o, #0] L2.ⓑ{I}V2 →
- ∃∃L1,V1. L1 ≛[h, o, V1] L2 & V1 ≛[h, o] V2 & Y1 = L1.ⓑ{I}V1.
+lemma lfdeq_inv_zero_pair_dx (h) (o): ∀I,Y1,L2,V2. Y1 ≛[h, o, #0] L2.ⓑ{I}V2 →
+ ∃∃L1,V1. L1 ≛[h, o, V1] L2 & V1 ≛[h, o] V2 & Y1 = L1.ⓑ{I}V1.
/2 width=1 by lfxs_inv_zero_pair_dx/ qed-.
-lemma lfdeq_inv_lref_bind_sn: ∀h,o,I1,Y2,L1,i. L1.ⓘ{I1} ≛[h, o, #⫯i] Y2 →
- ∃∃I2,L2. L1 ≛[h, o, #i] L2 & Y2 = L2.ⓘ{I2}.
+lemma lfdeq_inv_lref_bind_sn (h) (o): ∀I1,Y2,L1,i. L1.ⓘ{I1} ≛[h, o, #↑i] Y2 →
+ ∃∃I2,L2. L1 ≛[h, o, #i] L2 & Y2 = L2.ⓘ{I2}.
/2 width=2 by lfxs_inv_lref_bind_sn/ qed-.
-lemma lfdeq_inv_lref_bind_dx: ∀h,o,I2,Y1,L2,i. Y1 ≛[h, o, #⫯i] L2.ⓘ{I2} →
- ∃∃I1,L1. L1 ≛[h, o, #i] L2 & Y1 = L1.ⓘ{I1}.
+lemma lfdeq_inv_lref_bind_dx (h) (o): ∀I2,Y1,L2,i. Y1 ≛[h, o, #↑i] L2.ⓘ{I2} →
+ ∃∃I1,L1. L1 ≛[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.
+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: ∀h,o,I,L1,L2,V,T. L1 ≛[h, o, ②{I}V.T] L2 → L1 ≛[h, o, V] L2.
+lemma lfdeq_fwd_pair_sn (h) (o): ∀I,L1,L2,V,T. L1 ≛[h, o, ②{I}V.T] L2 → L1 ≛[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 ≛[h, o, ⓑ{p,I}V.T] L2 → L1.ⓑ{I}V ≛[h, o, T] L2.ⓑ{I}V.
+lemma lfdeq_fwd_bind_dx (h) (o): ∀p,I,L1,L2,V,T.
+ L1 ≛[h, o, ⓑ{p,I}V.T] L2 → L1.ⓑ{I}V ≛[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: ∀h,o,I,L1,L2,V,T. L1 ≛[h, o, ⓕ{I}V.T] L2 → L1 ≛[h, o, T] L2.
+lemma lfdeq_fwd_flat_dx (h) (o): ∀I,L1,L2,V,T. L1 ≛[h, o, ⓕ{I}V.T] L2 → L1 ≛[h, o, T] L2.
/2 width=3 by lfxs_fwd_flat_dx/ qed-.
-lemma lfdeq_fwd_dx: ∀h,o,I2,L1,K2. ∀T:term. L1 ≛[h, o, T] K2.ⓘ{I2} →
- ∃∃I1,K1. L1 = K1.ⓘ{I1}.
+lemma lfdeq_fwd_dx (h) (o): ∀I2,L1,K2. ∀T:term. L1 ≛[h, o, T] K2.ⓘ{I2} →
+ ∃∃I1,K1. L1 = K1.ⓘ{I1}.
/2 width=5 by lfxs_fwd_dx/ qed-.
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