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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 "static_2/notation/relations/stareqsn_3.ma".
16 include "static_2/syntax/tdeq_ext.ma".
17 include "static_2/static/rex.ma".
19 (* SORT-IRRELEVANT EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ***)
21 definition rdeq: relation3 term lenv lenv ≝
25 "sort-irrelevant equivalence on referred entries (local environment)"
26 'StarEqSn T L1 L2 = (rdeq T L1 L2).
29 "sort-irrelevant ranged equivalence (local environment)"
30 'StarEqSn f L1 L2 = (sex cdeq_ext cfull f L1 L2).
32 (* Basic properties ***********************************************************)
34 lemma frees_tdeq_conf_rdeq: ∀f,L1,T1. L1 ⊢ 𝐅*⦃T1⦄ ≘ f → ∀T2. T1 ≛ T2 →
35 ∀L2. L1 ≛[f] L2 → L2 ⊢ 𝐅*⦃T2⦄ ≘ f.
36 #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 #H destruct
39 /2 width=3 by frees_sort/
41 >(tdeq_inv_lref1 … H1) -X #Y #H2
42 >(sex_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 (sex_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 (sex_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 (sex_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, sex_inv_tl, ext2_pair, sle_sex_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_sex_trans, sor_inv_sle_dx, sor_inv_sle_sn/
68 lemma frees_tdeq_conf: ∀f,L,T1. L ⊢ 𝐅*⦃T1⦄ ≘ f →
69 ∀T2. T1 ≛ T2 → L ⊢ 𝐅*⦃T2⦄ ≘ f.
70 /4 width=7 by frees_tdeq_conf_rdeq, sex_refl, ext2_refl/ qed-.
72 lemma frees_rdeq_conf: ∀f,L1,T. L1 ⊢ 𝐅*⦃T⦄ ≘ f →
73 ∀L2. L1 ≛[f] L2 → L2 ⊢ 𝐅*⦃T⦄ ≘ f.
74 /2 width=7 by frees_tdeq_conf_rdeq, tdeq_refl/ qed-.
76 lemma tdeq_rex_conf (R): s_r_confluent1 … cdeq (rex R).
77 #R #L1 #T1 #T2 #HT12 #L2 *
78 /3 width=5 by frees_tdeq_conf, ex2_intro/
81 lemma tdeq_rex_div (R): ∀T1,T2. T1 ≛ T2 →
82 ∀L1,L2. L1 ⪤[R,T2] L2 → L1 ⪤[R,T1] L2.
83 /3 width=5 by tdeq_rex_conf, tdeq_sym/ qed-.
85 lemma tdeq_rdeq_conf: s_r_confluent1 … cdeq rdeq.
86 /2 width=5 by tdeq_rex_conf/ qed-.
88 lemma tdeq_rdeq_div: ∀T1,T2. T1 ≛ T2 →
89 ∀L1,L2. L1 ≛[T2] L2 → L1 ≛[T1] L2.
90 /2 width=5 by tdeq_rex_div/ qed-.
92 lemma rdeq_atom: ∀I. ⋆ ≛[⓪{I}] ⋆.
93 /2 width=1 by rex_atom/ qed.
95 lemma rdeq_sort: ∀I1,I2,L1,L2,s.
96 L1 ≛[⋆s] L2 → L1.ⓘ{I1} ≛[⋆s] L2.ⓘ{I2}.
97 /2 width=1 by rex_sort/ qed.
99 lemma rdeq_pair: ∀I,L1,L2,V1,V2.
100 L1 ≛[V1] L2 → V1 ≛ V2 → L1.ⓑ{I}V1 ≛[#0] L2.ⓑ{I}V2.
101 /2 width=1 by rex_pair/ qed.
103 lemma rdeq_unit: ∀f,I,L1,L2. 𝐈⦃f⦄ → L1 ⪤[cdeq_ext,cfull,f] L2 →
104 L1.ⓤ{I} ≛[#0] L2.ⓤ{I}.
105 /2 width=3 by rex_unit/ qed.
107 lemma rdeq_lref: ∀I1,I2,L1,L2,i.
108 L1 ≛[#i] L2 → L1.ⓘ{I1} ≛[#↑i] L2.ⓘ{I2}.
109 /2 width=1 by rex_lref/ qed.
111 lemma rdeq_gref: ∀I1,I2,L1,L2,l.
112 L1 ≛[§l] L2 → L1.ⓘ{I1} ≛[§l] L2.ⓘ{I2}.
113 /2 width=1 by rex_gref/ qed.
115 lemma rdeq_bind_repl_dx: ∀I,I1,L1,L2.∀T:term.
116 L1.ⓘ{I} ≛[T] L2.ⓘ{I1} →
118 L1.ⓘ{I} ≛[T] L2.ⓘ{I2}.
119 /2 width=2 by rex_bind_repl_dx/ qed-.
121 (* Basic inversion lemmas ***************************************************)
123 lemma rdeq_inv_atom_sn: ∀Y2. ∀T:term. ⋆ ≛[T] Y2 → Y2 = ⋆.
124 /2 width=3 by rex_inv_atom_sn/ qed-.
126 lemma rdeq_inv_atom_dx: ∀Y1. ∀T:term. Y1 ≛[T] ⋆ → Y1 = ⋆.
127 /2 width=3 by rex_inv_atom_dx/ qed-.
129 lemma rdeq_inv_zero: ∀Y1,Y2. Y1 ≛[#0] Y2 →
130 ∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆
131 | ∃∃I,L1,L2,V1,V2. L1 ≛[V1] L2 & V1 ≛ V2 &
132 Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2
133 | ∃∃f,I,L1,L2. 𝐈⦃f⦄ & L1 ⪤[cdeq_ext h o,cfull,f] L2 &
134 Y1 = L1.ⓤ{I} & Y2 = L2.ⓤ{I}.
135 #Y1 #Y2 #H elim (rex_inv_zero … H) -H *
136 /3 width=9 by or3_intro0, or3_intro1, or3_intro2, ex4_5_intro, ex4_4_intro, conj/
139 lemma rdeq_inv_lref: ∀Y1,Y2,i. Y1 ≛[#↑i] Y2 →
140 ∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆
141 | ∃∃I1,I2,L1,L2. L1 ≛[#i] L2 &
142 Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
143 /2 width=1 by rex_inv_lref/ qed-.
145 (* Basic_2A1: uses: lleq_inv_bind lleq_inv_bind_O *)
146 lemma rdeq_inv_bind: ∀p,I,L1,L2,V,T. L1 ≛[ⓑ{p,I}V.T] L2 →
147 ∧∧ L1 ≛[V] L2 & L1.ⓑ{I}V ≛[T] L2.ⓑ{I}V.
148 /2 width=2 by rex_inv_bind/ qed-.
150 (* Basic_2A1: uses: lleq_inv_flat *)
151 lemma rdeq_inv_flat: ∀I,L1,L2,V,T. L1 ≛[ⓕ{I}V.T] L2 →
152 ∧∧ L1 ≛[V] L2 & L1 ≛[T] L2.
153 /2 width=2 by rex_inv_flat/ qed-.
155 (* Advanced inversion lemmas ************************************************)
157 lemma rdeq_inv_zero_pair_sn: ∀I,Y2,L1,V1. L1.ⓑ{I}V1 ≛[#0] Y2 →
158 ∃∃L2,V2. L1 ≛[V1] L2 & V1 ≛ V2 & Y2 = L2.ⓑ{I}V2.
159 /2 width=1 by rex_inv_zero_pair_sn/ qed-.
161 lemma rdeq_inv_zero_pair_dx: ∀I,Y1,L2,V2. Y1 ≛[#0] L2.ⓑ{I}V2 →
162 ∃∃L1,V1. L1 ≛[V1] L2 & V1 ≛ V2 & Y1 = L1.ⓑ{I}V1.
163 /2 width=1 by rex_inv_zero_pair_dx/ qed-.
165 lemma rdeq_inv_lref_bind_sn: ∀I1,Y2,L1,i. L1.ⓘ{I1} ≛[#↑i] Y2 →
166 ∃∃I2,L2. L1 ≛[#i] L2 & Y2 = L2.ⓘ{I2}.
167 /2 width=2 by rex_inv_lref_bind_sn/ qed-.
169 lemma rdeq_inv_lref_bind_dx: ∀I2,Y1,L2,i. Y1 ≛[#↑i] L2.ⓘ{I2} →
170 ∃∃I1,L1. L1 ≛[#i] L2 & Y1 = L1.ⓘ{I1}.
171 /2 width=2 by rex_inv_lref_bind_dx/ qed-.
173 (* Basic forward lemmas *****************************************************)
175 lemma rdeq_fwd_zero_pair: ∀I,K1,K2,V1,V2.
176 K1.ⓑ{I}V1 ≛[#0] K2.ⓑ{I}V2 → K1 ≛[V1] K2.
177 /2 width=3 by rex_fwd_zero_pair/ qed-.
179 (* Basic_2A1: uses: lleq_fwd_bind_sn lleq_fwd_flat_sn *)
180 lemma rdeq_fwd_pair_sn: ∀I,L1,L2,V,T. L1 ≛[②{I}V.T] L2 → L1 ≛[V] L2.
181 /2 width=3 by rex_fwd_pair_sn/ qed-.
183 (* Basic_2A1: uses: lleq_fwd_bind_dx lleq_fwd_bind_O_dx *)
184 lemma rdeq_fwd_bind_dx: ∀p,I,L1,L2,V,T.
185 L1 ≛[ⓑ{p,I}V.T] L2 → L1.ⓑ{I}V ≛[T] L2.ⓑ{I}V.
186 /2 width=2 by rex_fwd_bind_dx/ qed-.
188 (* Basic_2A1: uses: lleq_fwd_flat_dx *)
189 lemma rdeq_fwd_flat_dx: ∀I,L1,L2,V,T. L1 ≛[ⓕ{I}V.T] L2 → L1 ≛[T] L2.
190 /2 width=3 by rex_fwd_flat_dx/ qed-.
192 lemma rdeq_fwd_dx: ∀I2,L1,K2. ∀T:term. L1 ≛[T] K2.ⓘ{I2} →
193 ∃∃I1,K1. L1 = K1.ⓘ{I1}.
194 /2 width=5 by rex_fwd_dx/ qed-.