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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
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15 include "basic_2/notation/relations/stareqsn_5.ma".
16 include "basic_2/syntax/tdeq_ext.ma".
17 include "basic_2/static/lfxs.ma".
19 (* DEGREE-BASED EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ******)
21 definition lfdeq: ∀h. sd h → relation3 term lenv lenv ≝
22 λh,o. lfxs (cdeq h o).
25 "degree-based equivalence on referred entries (local environment)"
26 'StarEqSn h o T L1 L2 = (lfdeq h o T L1 L2).
29 "degree-based ranged equivalence (local environment)"
30 'StarEqSn h o f L1 L2 = (lexs (cdeq_ext h o) cfull f L1 L2).
32 (* Basic properties ***********************************************************)
34 lemma frees_tdeq_conf_lfdeq: ∀h,o,f,L1,T1. L1 ⊢ 𝐅*⦃T1⦄ ≡ f → ∀T2. T1 ≛[h, o] T2 →
35 ∀L2. L1 ≛[h, o, f] L2 → L2 ⊢ 𝐅*⦃T2⦄ ≡ f.
36 #h #o #f #L1 #T1 #H elim H -f -L1 -T1
37 [ #f #L1 #s1 #Hf #X #H1 #L2 #_
38 elim (tdeq_inv_sort1 … H1) -H1 #s2 #d #_ #_ #H destruct
39 /2 width=3 by frees_sort/
41 >(tdeq_inv_lref1 … H1) -X #Y #H2
42 >(lexs_inv_atom1 … H2) -Y
43 /2 width=1 by frees_atom/
44 | #f #I #L1 #V1 #_ #IH #X #H1
45 >(tdeq_inv_lref1 … H1) -X #Y #H2
46 elim (lexs_inv_next1 … H2) -H2 #Z #L2 #HL12 #HZ #H destruct
47 elim (ext2_inv_pair_sn … HZ) -HZ #V2 #HV12 #H destruct
48 /3 width=1 by frees_pair/
49 | #f #I #L1 #Hf #X #H1
50 >(tdeq_inv_lref1 … H1) -X #Y #H2
51 elim (lexs_inv_next1 … H2) -H2 #Z #L2 #_ #HZ #H destruct
52 >(ext2_inv_unit_sn … HZ) -Z /2 width=1 by frees_unit/
53 | #f #I #L1 #i #_ #IH #X #H1
54 >(tdeq_inv_lref1 … H1) -X #Y #H2
55 elim (lexs_inv_push1 … H2) -H2 #J #L2 #HL12 #_ #H destruct
56 /3 width=1 by frees_lref/
57 | #f #L1 #l #Hf #X #H1 #L2 #_
58 >(tdeq_inv_gref1 … H1) -X /2 width=1 by frees_gref/
59 | #f1V #f1T #f1 #p #I #L1 #V1 #T1 #_ #_ #Hf1 #IHV #IHT #X #H1
60 elim (tdeq_inv_pair1 … H1) -H1 #V2 #T2 #HV12 #HT12 #H1 #L2 #HL12 destruct
61 /6 width=5 by frees_bind, lexs_inv_tl, ext2_pair, sle_lexs_trans, sor_inv_sle_dx, sor_inv_sle_sn/
62 | #f1V #f1T #f1 #I #L1 #V1 #T1 #_ #_ #Hf1 #IHV #IHT #X #H1
63 elim (tdeq_inv_pair1 … H1) -H1 #V2 #T2 #HV12 #HT12 #H1 #L2 #HL12 destruct
64 /5 width=5 by frees_flat, sle_lexs_trans, sor_inv_sle_dx, sor_inv_sle_sn/
68 lemma frees_tdeq_conf: ∀h,o,f,L,T1. L ⊢ 𝐅*⦃T1⦄ ≡ f →
69 ∀T2. T1 ≛[h, o] T2 → L ⊢ 𝐅*⦃T2⦄ ≡ f.
70 /4 width=7 by frees_tdeq_conf_lfdeq, lexs_refl, ext2_refl/ qed-.
72 lemma frees_lfdeq_conf: ∀h,o,f,L1,T. L1 ⊢ 𝐅*⦃T⦄ ≡ f →
73 ∀L2. L1 ≛[h, o, f] L2 → L2 ⊢ 𝐅*⦃T⦄ ≡ f.
74 /2 width=7 by frees_tdeq_conf_lfdeq, tdeq_refl/ qed-.
76 lemma tdeq_lfdeq_conf_sn: ∀h,o. s_r_confluent1 … (cdeq h o) (lfdeq h o).
77 #h #o #L1 #T1 #T2 #HT12 #L2 *
78 /3 width=5 by frees_tdeq_conf, ex2_intro/
81 lemma lfdeq_atom: ∀h,o,I. ⋆ ≛[h, o, ⓪{I}] ⋆.
82 /2 width=1 by lfxs_atom/ qed.
84 (* Basic_2A1: uses: lleq_sort *)
85 lemma lfdeq_sort: ∀h,o,I1,I2,L1,L2,s.
86 L1 ≛[h, o, ⋆s] L2 → L1.ⓘ{I1} ≛[h, o, ⋆s] L2.ⓘ{I2}.
87 /2 width=1 by lfxs_sort/ qed.
89 lemma lfdeq_pair: ∀h,o,I,L1,L2,V1,V2. L1 ≛[h, o, V1] L2 → V1 ≛[h, o] V2 →
90 L1.ⓑ{I}V1 ≛[h, o, #0] L2.ⓑ{I}V2.
91 /2 width=1 by lfxs_pair/ qed.
93 lemma lfdeq_unit: ∀h,o,f,I,L1,L2. 𝐈⦃f⦄ → L1 ⪤*[cdeq_ext h o, cfull, f] L2 →
94 L1.ⓤ{I} ≛[h, o, #0] L2.ⓤ{I}.
95 /2 width=3 by lfxs_unit/ qed.
97 lemma lfdeq_lref: ∀h,o,I1,I2,L1,L2,i.
98 L1 ≛[h, o, #i] L2 → L1.ⓘ{I1} ≛[h, o, #⫯i] L2.ⓘ{I2}.
99 /2 width=1 by lfxs_lref/ qed.
101 (* Basic_2A1: uses: lleq_gref *)
102 lemma lfdeq_gref: ∀h,o,I1,I2,L1,L2,l.
103 L1 ≛[h, o, §l] L2 → L1.ⓘ{I1} ≛[h, o, §l] L2.ⓘ{I2}.
104 /2 width=1 by lfxs_gref/ qed.
106 lemma lfdeq_bind_repl_dx: ∀h,o,I,I1,L1,L2.∀T:term.
107 L1.ⓘ{I} ≛[h, o, T] L2.ⓘ{I1} →
109 L1.ⓘ{I} ≛[h, o, T] L2.ⓘ{I2}.
110 /2 width=2 by lfxs_bind_repl_dx/ qed-.
112 (* Basic inversion lemmas ***************************************************)
114 lemma lfdeq_inv_atom_sn: ∀h,o,Y2. ∀T:term. ⋆ ≛[h, o, T] Y2 → Y2 = ⋆.
115 /2 width=3 by lfxs_inv_atom_sn/ qed-.
117 lemma lfdeq_inv_atom_dx: ∀h,o,Y1. ∀T:term. Y1 ≛[h, o, T] ⋆ → Y1 = ⋆.
118 /2 width=3 by lfxs_inv_atom_dx/ qed-.
120 lemma lfdeq_inv_zero: ∀h,o,Y1,Y2. Y1 ≛[h, o, #0] Y2 →
122 | ∃∃I,L1,L2,V1,V2. L1 ≛[h, o, V1] L2 & V1 ≛[h, o] V2 &
123 Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2
124 | ∃∃f,I,L1,L2. 𝐈⦃f⦄ & L1 ⪤*[cdeq_ext h o, cfull, f] L2 &
125 Y1 = L1.ⓤ{I} & Y2 = L2.ⓤ{I}.
126 #h #o #Y1 #Y2 #H elim (lfxs_inv_zero … H) -H *
127 /3 width=9 by or3_intro0, or3_intro1, or3_intro2, ex4_5_intro, ex4_4_intro, conj/
130 lemma lfdeq_inv_lref: ∀h,o,Y1,Y2,i. Y1 ≛[h, o, #⫯i] Y2 →
132 ∃∃I1,I2,L1,L2. L1 ≛[h, o, #i] L2 &
133 Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
134 /2 width=1 by lfxs_inv_lref/ qed-.
136 (* Basic_2A1: uses: lleq_inv_bind lleq_inv_bind_O *)
137 lemma lfdeq_inv_bind: ∀h,o,p,I,L1,L2,V,T. L1 ≛[h, o, ⓑ{p,I}V.T] L2 →
138 L1 ≛[h, o, V] L2 ∧ L1.ⓑ{I}V ≛[h, o, T] L2.ⓑ{I}V.
139 /2 width=2 by lfxs_inv_bind/ qed-.
141 (* Basic_2A1: uses: lleq_inv_flat *)
142 lemma lfdeq_inv_flat: ∀h,o,I,L1,L2,V,T. L1 ≛[h, o, ⓕ{I}V.T] L2 →
143 L1 ≛[h, o, V] L2 ∧ L1 ≛[h, o, T] L2.
144 /2 width=2 by lfxs_inv_flat/ qed-.
146 (* Advanced inversion lemmas ************************************************)
148 lemma lfdeq_inv_zero_pair_sn: ∀h,o,I,Y2,L1,V1. L1.ⓑ{I}V1 ≛[h, o, #0] Y2 →
149 ∃∃L2,V2. L1 ≛[h, o, V1] L2 & V1 ≛[h, o] V2 & Y2 = L2.ⓑ{I}V2.
150 /2 width=1 by lfxs_inv_zero_pair_sn/ qed-.
152 lemma lfdeq_inv_zero_pair_dx: ∀h,o,I,Y1,L2,V2. Y1 ≛[h, o, #0] L2.ⓑ{I}V2 →
153 ∃∃L1,V1. L1 ≛[h, o, V1] L2 & V1 ≛[h, o] V2 & Y1 = L1.ⓑ{I}V1.
154 /2 width=1 by lfxs_inv_zero_pair_dx/ qed-.
156 lemma lfdeq_inv_lref_bind_sn: ∀h,o,I1,Y2,L1,i. L1.ⓘ{I1} ≛[h, o, #⫯i] Y2 →
157 ∃∃I2,L2. L1 ≛[h, o, #i] L2 & Y2 = L2.ⓘ{I2}.
158 /2 width=2 by lfxs_inv_lref_bind_sn/ qed-.
160 lemma lfdeq_inv_lref_bind_dx: ∀h,o,I2,Y1,L2,i. Y1 ≛[h, o, #⫯i] L2.ⓘ{I2} →
161 ∃∃I1,L1. L1 ≛[h, o, #i] L2 & Y1 = L1.ⓘ{I1}.
162 /2 width=2 by lfxs_inv_lref_bind_dx/ qed-.
164 (* Basic forward lemmas *****************************************************)
166 lemma lfdeq_fwd_zero_pair: ∀h,o,I,K1,K2,V1,V2.
167 K1.ⓑ{I}V1 ≛[h, o, #0] K2.ⓑ{I}V2 → K1 ≛[h, o, V1] K2.
168 /2 width=3 by lfxs_fwd_zero_pair/ qed-.
170 (* Basic_2A1: uses: lleq_fwd_bind_sn lleq_fwd_flat_sn *)
171 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.
172 /2 width=3 by lfxs_fwd_pair_sn/ qed-.
174 (* Basic_2A1: uses: lleq_fwd_bind_dx lleq_fwd_bind_O_dx *)
175 lemma lfdeq_fwd_bind_dx: ∀h,o,p,I,L1,L2,V,T.
176 L1 ≛[h, o, ⓑ{p,I}V.T] L2 → L1.ⓑ{I}V ≛[h, o, T] L2.ⓑ{I}V.
177 /2 width=2 by lfxs_fwd_bind_dx/ qed-.
179 (* Basic_2A1: uses: lleq_fwd_flat_dx *)
180 lemma lfdeq_fwd_flat_dx: ∀h,o,I,L1,L2,V,T. L1 ≛[h, o, ⓕ{I}V.T] L2 → L1 ≛[h, o, T] L2.
181 /2 width=3 by lfxs_fwd_flat_dx/ qed-.
183 lemma lfdeq_fwd_dx: ∀h,o,I2,L1,K2. ∀T:term. L1 ≛[h, o, T] K2.ⓘ{I2} →
184 ∃∃I1,K1. L1 = K1.ⓘ{I1}.
185 /2 width=5 by lfxs_fwd_dx/ qed-.