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4 (* ||A|| A project by Andrea Asperti *)
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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_lfxs_conf: ∀R,h,o. s_r_confluent1 … (cdeq h o) (lfxs R).
77 #R #h #o #L1 #T1 #T2 #HT12 #L2 *
78 /3 width=5 by frees_tdeq_conf, ex2_intro/
81 lemma tdeq_lfxs_div: ∀R,h,o,T1,T2. T1 ≛[h, o] T2 →
82 ∀L1,L2. L1 ⪤*[R, T2] L2 → L1 ⪤*[R, T1] L2.
83 /3 width=5 by tdeq_lfxs_conf, tdeq_sym/ qed-.
85 lemma tdeq_lfdeq_conf: ∀h,o. s_r_confluent1 … (cdeq h o) (lfdeq h o).
86 /2 width=5 by tdeq_lfxs_conf/ qed-.
88 lemma tdeq_lfdeq_div: ∀h,o,T1,T2. T1 ≛[h, o] T2 →
89 ∀L1,L2. L1 ≛[h, o, T2] L2 → L1 ≛[h, o, T1] L2.
90 /2 width=5 by tdeq_lfxs_div/ qed-.
92 lemma lfdeq_atom: ∀h,o,I. ⋆ ≛[h, o, ⓪{I}] ⋆.
93 /2 width=1 by lfxs_atom/ qed.
95 (* Basic_2A1: uses: lleq_sort *)
96 lemma lfdeq_sort: ∀h,o,I1,I2,L1,L2,s.
97 L1 ≛[h, o, ⋆s] L2 → L1.ⓘ{I1} ≛[h, o, ⋆s] L2.ⓘ{I2}.
98 /2 width=1 by lfxs_sort/ qed.
100 lemma lfdeq_pair: ∀h,o,I,L1,L2,V1,V2. L1 ≛[h, o, V1] L2 → V1 ≛[h, o] V2 →
101 L1.ⓑ{I}V1 ≛[h, o, #0] L2.ⓑ{I}V2.
102 /2 width=1 by lfxs_pair/ qed.
104 lemma lfdeq_unit: ∀h,o,f,I,L1,L2. 𝐈⦃f⦄ → L1 ⪤*[cdeq_ext h o, cfull, f] L2 →
105 L1.ⓤ{I} ≛[h, o, #0] L2.ⓤ{I}.
106 /2 width=3 by lfxs_unit/ qed.
108 lemma lfdeq_lref: ∀h,o,I1,I2,L1,L2,i.
109 L1 ≛[h, o, #i] L2 → L1.ⓘ{I1} ≛[h, o, #↑i] L2.ⓘ{I2}.
110 /2 width=1 by lfxs_lref/ qed.
112 (* Basic_2A1: uses: lleq_gref *)
113 lemma lfdeq_gref: ∀h,o,I1,I2,L1,L2,l.
114 L1 ≛[h, o, §l] L2 → L1.ⓘ{I1} ≛[h, o, §l] L2.ⓘ{I2}.
115 /2 width=1 by lfxs_gref/ qed.
117 lemma lfdeq_bind_repl_dx: ∀h,o,I,I1,L1,L2.∀T:term.
118 L1.ⓘ{I} ≛[h, o, T] L2.ⓘ{I1} →
120 L1.ⓘ{I} ≛[h, o, T] L2.ⓘ{I2}.
121 /2 width=2 by lfxs_bind_repl_dx/ qed-.
123 (* Basic inversion lemmas ***************************************************)
125 lemma lfdeq_inv_atom_sn: ∀h,o,Y2. ∀T:term. ⋆ ≛[h, o, T] Y2 → Y2 = ⋆.
126 /2 width=3 by lfxs_inv_atom_sn/ qed-.
128 lemma lfdeq_inv_atom_dx: ∀h,o,Y1. ∀T:term. Y1 ≛[h, o, T] ⋆ → Y1 = ⋆.
129 /2 width=3 by lfxs_inv_atom_dx/ qed-.
131 lemma lfdeq_inv_zero: ∀h,o,Y1,Y2. Y1 ≛[h, o, #0] Y2 →
133 | ∃∃I,L1,L2,V1,V2. L1 ≛[h, o, V1] L2 & V1 ≛[h, o] V2 &
134 Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2
135 | ∃∃f,I,L1,L2. 𝐈⦃f⦄ & L1 ⪤*[cdeq_ext h o, cfull, f] L2 &
136 Y1 = L1.ⓤ{I} & Y2 = L2.ⓤ{I}.
137 #h #o #Y1 #Y2 #H elim (lfxs_inv_zero … H) -H *
138 /3 width=9 by or3_intro0, or3_intro1, or3_intro2, ex4_5_intro, ex4_4_intro, conj/
141 lemma lfdeq_inv_lref: ∀h,o,Y1,Y2,i. Y1 ≛[h, o, #↑i] Y2 →
143 ∃∃I1,I2,L1,L2. L1 ≛[h, o, #i] L2 &
144 Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
145 /2 width=1 by lfxs_inv_lref/ qed-.
147 (* Basic_2A1: uses: lleq_inv_bind lleq_inv_bind_O *)
148 lemma lfdeq_inv_bind: ∀h,o,p,I,L1,L2,V,T. L1 ≛[h, o, ⓑ{p,I}V.T] L2 →
149 L1 ≛[h, o, V] L2 ∧ L1.ⓑ{I}V ≛[h, o, T] L2.ⓑ{I}V.
150 /2 width=2 by lfxs_inv_bind/ qed-.
152 (* Basic_2A1: uses: lleq_inv_flat *)
153 lemma lfdeq_inv_flat: ∀h,o,I,L1,L2,V,T. L1 ≛[h, o, ⓕ{I}V.T] L2 →
154 L1 ≛[h, o, V] L2 ∧ L1 ≛[h, o, T] L2.
155 /2 width=2 by lfxs_inv_flat/ qed-.
157 (* Advanced inversion lemmas ************************************************)
159 lemma lfdeq_inv_zero_pair_sn: ∀h,o,I,Y2,L1,V1. L1.ⓑ{I}V1 ≛[h, o, #0] Y2 →
160 ∃∃L2,V2. L1 ≛[h, o, V1] L2 & V1 ≛[h, o] V2 & Y2 = L2.ⓑ{I}V2.
161 /2 width=1 by lfxs_inv_zero_pair_sn/ qed-.
163 lemma lfdeq_inv_zero_pair_dx: ∀h,o,I,Y1,L2,V2. Y1 ≛[h, o, #0] L2.ⓑ{I}V2 →
164 ∃∃L1,V1. L1 ≛[h, o, V1] L2 & V1 ≛[h, o] V2 & Y1 = L1.ⓑ{I}V1.
165 /2 width=1 by lfxs_inv_zero_pair_dx/ qed-.
167 lemma lfdeq_inv_lref_bind_sn: ∀h,o,I1,Y2,L1,i. L1.ⓘ{I1} ≛[h, o, #↑i] Y2 →
168 ∃∃I2,L2. L1 ≛[h, o, #i] L2 & Y2 = L2.ⓘ{I2}.
169 /2 width=2 by lfxs_inv_lref_bind_sn/ qed-.
171 lemma lfdeq_inv_lref_bind_dx: ∀h,o,I2,Y1,L2,i. Y1 ≛[h, o, #↑i] L2.ⓘ{I2} →
172 ∃∃I1,L1. L1 ≛[h, o, #i] L2 & Y1 = L1.ⓘ{I1}.
173 /2 width=2 by lfxs_inv_lref_bind_dx/ qed-.
175 (* Basic forward lemmas *****************************************************)
177 lemma lfdeq_fwd_zero_pair: ∀h,o,I,K1,K2,V1,V2.
178 K1.ⓑ{I}V1 ≛[h, o, #0] K2.ⓑ{I}V2 → K1 ≛[h, o, V1] K2.
179 /2 width=3 by lfxs_fwd_zero_pair/ qed-.
181 (* Basic_2A1: uses: lleq_fwd_bind_sn lleq_fwd_flat_sn *)
182 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.
183 /2 width=3 by lfxs_fwd_pair_sn/ qed-.
185 (* Basic_2A1: uses: lleq_fwd_bind_dx lleq_fwd_bind_O_dx *)
186 lemma lfdeq_fwd_bind_dx: ∀h,o,p,I,L1,L2,V,T.
187 L1 ≛[h, o, ⓑ{p,I}V.T] L2 → L1.ⓑ{I}V ≛[h, o, T] L2.ⓑ{I}V.
188 /2 width=2 by lfxs_fwd_bind_dx/ qed-.
190 (* Basic_2A1: uses: lleq_fwd_flat_dx *)
191 lemma lfdeq_fwd_flat_dx: ∀h,o,I,L1,L2,V,T. L1 ≛[h, o, ⓕ{I}V.T] L2 → L1 ≛[h, o, T] L2.
192 /2 width=3 by lfxs_fwd_flat_dx/ qed-.
194 lemma lfdeq_fwd_dx: ∀h,o,I2,L1,K2. ∀T:term. L1 ≛[h, o, T] K2.ⓘ{I2} →
195 ∃∃I1,K1. L1 = K1.ⓘ{I1}.
196 /2 width=5 by lfxs_fwd_dx/ qed-.