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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
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15 include "basic_2/notation/relations/lazyeq_5.ma".
16 include "basic_2/syntax/tdeq.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 'LazyEq h o T L1 L2 = (lfdeq h o T L1 L2).
29 "degree-based ranged equivalence (local environment)"
30 'LazyEq h o f L1 L2 = (lexs (cdeq h o) cfull f L1 L2).
32 definition lfdeq_transitive: predicate (relation3 lenv term term) ≝
33 λR. ∀L2,T1,T2. R L2 T1 T2 → ∀L1. L1 ≡[h, o, T1] L2 → R L1 T1 T2.
35 (* Basic properties ***********************************************************)
37 lemma frees_tdeq_conf_lexs: ∀h,o,f,L1,T1. L1 ⊢ 𝐅*⦃T1⦄ ≡ f → ∀T2. T1 ≡[h, o] T2 →
38 ∀L2. L1 ≡[h, o, f] L2 → L2 ⊢ 𝐅*⦃T2⦄ ≡ f.
39 #h #o #f #L1 #T1 #H elim H -f -L1 -T1
40 [ #f #I1 #Hf #X #H1 elim (tdeq_fwd_atom1 … H1) -H1
41 #I2 #H1 #Y #H2 lapply (lexs_inv_atom1 … H2) -H2
42 #H2 destruct /2 width=1 by frees_atom/
43 | #f #I #L1 #V1 #s1 #_ #IH #X #H1 elim (tdeq_inv_sort1 … H1) -H1
44 #s2 #d #Hs1 #Hs2 #H1 #Y #H2 elim (lexs_inv_push1 … H2) -H2
45 #L2 #V2 #HL12 #_ #H2 destruct /4 width=3 by frees_sort, tdeq_sort/
46 | #f #I #L1 #V1 #_ #IH #X #H1 >(tdeq_inv_lref1 … H1) -H1
47 #Y #H2 elim (lexs_inv_next1 … H2) -H2
48 #L2 #V2 #HL12 #HV12 #H2 destruct /3 width=1 by frees_zero/
49 | #f #I #L1 #V1 #i #_ #IH #X #H1 >(tdeq_inv_lref1 … H1) -H1
50 #Y #H2 elim (lexs_inv_push1 … H2) -H2
51 #L2 #V2 #HL12 #_ #H2 destruct /3 width=1 by frees_lref/
52 | #f #I #L1 #V1 #l #_ #IH #X #H1 >(tdeq_inv_gref1 … H1) -H1
53 #Y #H2 elim (lexs_inv_push1 … H2) -H2
54 #L2 #V2 #HL12 #_ #H2 destruct /3 width=1 by frees_gref/
55 | #f1V #f1T #f1 #p #I #L1 #V1 #T1 #_ #_ #Hf1 #IHV #IHT #X #H1 elim (tdeq_inv_pair1 … H1) -H1
56 #V2 #T2 #HV12 #HT12 #H1 #L2 #HL12 destruct
57 /6 width=5 by frees_bind, lexs_inv_tl, sle_lexs_trans, sor_inv_sle_dx, sor_inv_sle_sn/
58 | #f1V #f1T #f1 #I #L1 #V1 #T1 #_ #_ #Hf1 #IHV #IHT #X #H1 elim (tdeq_inv_pair1 … H1) -H1
59 #V2 #T2 #HV12 #HT12 #H1 #L2 #HL12 destruct
60 /5 width=5 by frees_flat, sle_lexs_trans, sor_inv_sle_dx, sor_inv_sle_sn/
64 lemma frees_tdeq_conf: ∀h,o,f,L,T1. L ⊢ 𝐅*⦃T1⦄ ≡ f →
65 ∀T2. T1 ≡[h, o] T2 → L ⊢ 𝐅*⦃T2⦄ ≡ f.
66 /3 width=7 by frees_tdeq_conf_lexs, lexs_refl/ qed-.
68 lemma frees_lfdeq_conf_lexs: ∀h,o. lexs_frees_confluent (cdeq h o) cfull.
69 /3 width=7 by frees_tdeq_conf_lexs, ex2_intro/ qed-.
71 lemma tdeq_lfdeq_conf_sn: ∀h,o. s_r_confluent1 … (cdeq h o) (lfdeq h o).
72 #h #o #L1 #T1 #T2 #HT12 #L2 *
73 /3 width=5 by frees_tdeq_conf, ex2_intro/
76 lemma lfdeq_sym: ∀h,o,T. symmetric … (lfdeq h o T).
78 /4 width=7 by frees_tdeq_conf_lexs, lfxs_sym, tdeq_sym, ex2_intro/
81 lemma lfdeq_atom: ∀h,o,I. ⋆ ≡[h, o, ⓪{I}] ⋆.
82 /2 width=1 by lfxs_atom/ qed.
84 lemma lfdeq_sort: ∀h,o,I,L1,L2,V1,V2,s.
85 L1 ≡[h, o, ⋆s] L2 → L1.ⓑ{I}V1 ≡[h, o, ⋆s] L2.ⓑ{I}V2.
86 /2 width=1 by lfxs_sort/ qed.
88 lemma lfdeq_zero: ∀h,o,I,L1,L2,V.
89 L1 ≡[h, o, V] L2 → L1.ⓑ{I}V ≡[h, o, #0] L2.ⓑ{I}V.
90 /2 width=1 by lfxs_zero/ qed.
92 lemma lfdeq_lref: ∀h,o,I,L1,L2,V1,V2,i.
93 L1 ≡[h, o, #i] L2 → L1.ⓑ{I}V1 ≡[h, o, #⫯i] L2.ⓑ{I}V2.
94 /2 width=1 by lfxs_lref/ qed.
96 lemma lfdeq_gref: ∀h,o,I,L1,L2,V1,V2,l.
97 L1 ≡[h, o, §l] L2 → L1.ⓑ{I}V1 ≡[h, o, §l] L2.ⓑ{I}V2.
98 /2 width=1 by lfxs_gref/ qed.
100 lemma lfdeq_pair_repl_dx: ∀h,o,I,L1,L2.∀T:term.∀V,V1.
101 L1.ⓑ{I}V ≡[h, o, T] L2.ⓑ{I}V1 →
103 L1.ⓑ{I}V ≡[h, o, T] L2.ⓑ{I}V2.
104 /2 width=2 by lfxs_pair_repl_dx/ qed-.
106 (* Basic inversion lemmas ***************************************************)
108 lemma lfdeq_inv_atom_sn: ∀h,o,I,Y2. ⋆ ≡[h, o, ⓪{I}] Y2 → Y2 = ⋆.
109 /2 width=3 by lfxs_inv_atom_sn/ qed-.
111 lemma lfdeq_inv_atom_dx: ∀h,o,I,Y1. Y1 ≡[h, o, ⓪{I}] ⋆ → Y1 = ⋆.
112 /2 width=3 by lfxs_inv_atom_dx/ qed-.
114 lemma lfdeq_inv_zero: ∀h,o,Y1,Y2. Y1 ≡[h, o, #0] Y2 →
116 ∃∃I,L1,L2,V1,V2. L1 ≡[h, o, V1] L2 & V1 ≡[h, o] V2 &
117 Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
118 #h #o #Y1 #Y2 #H elim (lfxs_inv_zero … H) -H *
119 /3 width=9 by ex4_5_intro, or_introl, or_intror, conj/
122 lemma lfdeq_inv_lref: ∀h,o,Y1,Y2,i. Y1 ≡[h, o, #⫯i] Y2 →
124 ∃∃I,L1,L2,V1,V2. L1 ≡[h, o, #i] L2 &
125 Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
126 /2 width=1 by lfxs_inv_lref/ qed-.
128 lemma lfdeq_inv_bind: ∀h,o,p,I,L1,L2,V,T. L1 ≡[h, o, ⓑ{p,I}V.T] L2 →
129 L1 ≡[h, o, V] L2 ∧ L1.ⓑ{I}V ≡[h, o, T] L2.ⓑ{I}V.
130 /2 width=2 by lfxs_inv_bind/ qed-.
132 lemma lfdeq_inv_flat: ∀h,o,I,L1,L2,V,T. L1 ≡[h, o, ⓕ{I}V.T] L2 →
133 L1 ≡[h, o, V] L2 ∧ L1 ≡[h, o, T] L2.
134 /2 width=2 by lfxs_inv_flat/ qed-.
136 (* Advanced inversion lemmas ************************************************)
138 lemma lfdeq_inv_zero_pair_sn: ∀h,o,I,Y2,L1,V1. L1.ⓑ{I}V1 ≡[h, o, #0] Y2 →
139 ∃∃L2,V2. L1 ≡[h, o, V1] L2 & V1 ≡[h, o] V2 & Y2 = L2.ⓑ{I}V2.
140 #h #o #I #Y2 #L1 #V1 #H elim (lfxs_inv_zero_pair_sn … H) -H /2 width=5 by ex3_2_intro/
143 lemma lfdeq_inv_zero_pair_dx: ∀h,o,I,Y1,L2,V2. Y1 ≡[h, o, #0] L2.ⓑ{I}V2 →
144 ∃∃L1,V1. L1 ≡[h, o, V1] L2 & V1 ≡[h, o] V2 & Y1 = L1.ⓑ{I}V1.
145 #h #o #I #Y1 #L2 #V2 #H elim (lfxs_inv_zero_pair_dx … H) -H
146 #L1 #V1 #HL12 #HV12 #H destruct /2 width=5 by ex3_2_intro/
149 lemma lfdeq_inv_lref_pair_sn: ∀h,o,I,Y2,L1,V1,i. L1.ⓑ{I}V1 ≡[h, o, #⫯i] Y2 →
150 ∃∃L2,V2. L1 ≡[h, o, #i] L2 & Y2 = L2.ⓑ{I}V2.
151 /2 width=2 by lfxs_inv_lref_pair_sn/ qed-.
153 lemma lfdeq_inv_lref_pair_dx: ∀h,o,I,Y1,L2,V2,i. Y1 ≡[h, o, #⫯i] L2.ⓑ{I}V2 →
154 ∃∃L1,V1. L1 ≡[h, o, #i] L2 & Y1 = L1.ⓑ{I}V1.
155 /2 width=2 by lfxs_inv_lref_pair_dx/ qed-.
157 (* Basic forward lemmas *****************************************************)
159 lemma lfdeq_fwd_bind_sn: ∀h,o,p,I,L1,L2,V,T. L1 ≡[h, o, ⓑ{p,I}V.T] L2 → L1 ≡[h, o, V] L2.
160 /2 width=4 by lfxs_fwd_bind_sn/ qed-.
162 lemma lfdeq_fwd_bind_dx: ∀h,o,p,I,L1,L2,V,T.
163 L1 ≡[h, o, ⓑ{p,I}V.T] L2 → L1.ⓑ{I}V ≡[h, o, T] L2.ⓑ{I}V.
164 /2 width=2 by lfxs_fwd_bind_dx/ qed-.
166 lemma lfdeq_fwd_flat_sn: ∀h,o,I,L1,L2,V,T. L1 ≡[h, o, ⓕ{I}V.T] L2 → L1 ≡[h, o, V] L2.
167 /2 width=3 by lfxs_fwd_flat_sn/ qed-.
169 lemma lfdeq_fwd_flat_dx: ∀h,o,I,L1,L2,V,T. L1 ≡[h, o, ⓕ{I}V.T] L2 → L1 ≡[h, o, T] L2.
170 /2 width=3 by lfxs_fwd_flat_dx/ qed-.
172 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.
173 /2 width=3 by lfxs_fwd_pair_sn/ qed-.
175 lemma lfdeq_fwd_dx: ∀h,o,I,L1,K2,V2. ∀T:term. L1 ≡[h, o, T] K2.ⓑ{I}V2 →
176 ∃∃K1,V1. L1 = K1.ⓑ{I}V1.
177 /2 width=5 by lfxs_fwd_dx/ qed-.
179 (* Basic_2A1: removed theorems 30:
180 lleq_ind lleq_inv_bind lleq_inv_flat lleq_fwd_length lleq_fwd_lref
181 lleq_fwd_drop_sn lleq_fwd_drop_dx
182 lleq_fwd_bind_sn lleq_fwd_bind_dx lleq_fwd_flat_sn lleq_fwd_flat_dx
183 lleq_sort lleq_skip lleq_lref lleq_free lleq_gref lleq_bind lleq_flat
184 lleq_refl lleq_Y lleq_sym lleq_ge_up lleq_ge lleq_bind_O llpx_sn_lrefl
185 lleq_trans lleq_canc_sn lleq_canc_dx lleq_nlleq_trans nlleq_lleq_div