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14
15 include "static_2/notation/relations/stareqsn_3.ma".
16 include "static_2/syntax/tdeq_ext.ma".
17 include "static_2/static/rex.ma".
18
19 (* SORT-IRRELEVANT EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ***)
20
21 definition rdeq: relation3 term lenv lenv โ‰
22                  rex cdeq.
23
24 interpretation
25    "sort-irrelevant equivalence on referred entries (local environment)"
26    'StarEqSn T L1 L2 = (rdeq T L1 L2).
27
28 interpretation
29    "sort-irrelevant ranged equivalence (local environment)"
30    'StarEqSn f L1 L2 = (sex cdeq_ext cfull f L1 L2).
31
32 (* Basic properties ***********************************************************)
33
34 lemma frees_tdeq_conf_rdeq: โˆ€f,L1,T1. L1 โŠข ๐…*โฆƒT1โฆ„ โ‰˜ f โ†’ โˆ€T2. T1 โ‰› T2 โ†’
35                             โˆ€L2. L1 โ‰›[f] L2 โ†’ L2 โŠข ๐…*โฆƒT2โฆ„ โ‰˜ f.
36 #f #L1 #T1 #H elim H -f -L1 -T1
37 [ #f #L1 #s1 #Hf #X #H1 #L2 #_
38   elim (tdeq_inv_sort1 โ€ฆ H1) -H1 #s2 #H destruct
39   /2 width=3 by frees_sort/
40 | #f #i #Hf #X #H1
41   >(tdeq_inv_lref1 โ€ฆ H1) -X #Y #H2
42   >(sex_inv_atom1 โ€ฆ H2) -Y
43   /2 width=1 by frees_atom/
44 | #f #I #L1 #V1 #_ #IH #X #H1
45   >(tdeq_inv_lref1 โ€ฆ H1) -X #Y #H2
46   elim (sex_inv_next1 โ€ฆ H2) -H2 #Z #L2 #HL12 #HZ #H destruct
47   elim (ext2_inv_pair_sn โ€ฆ HZ) -HZ #V2 #HV12 #H destruct
48   /3 width=1 by frees_pair/
49 | #f #I #L1 #Hf #X #H1
50   >(tdeq_inv_lref1 โ€ฆ H1) -X #Y #H2
51   elim (sex_inv_next1 โ€ฆ H2) -H2 #Z #L2 #_ #HZ #H destruct
52   >(ext2_inv_unit_sn โ€ฆ HZ) -Z /2 width=1 by frees_unit/
53 | #f #I #L1 #i #_ #IH #X #H1
54   >(tdeq_inv_lref1 โ€ฆ H1) -X #Y #H2
55   elim (sex_inv_push1 โ€ฆ 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 โ€ฆ 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 โ€ฆ 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 โ€ฆ 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/
65 ]
66 qed-.
67
68 lemma frees_tdeq_conf: โˆ€f,L,T1. L โŠข ๐…*โฆƒT1โฆ„ โ‰˜ f โ†’
69                        โˆ€T2. T1 โ‰› T2 โ†’ L โŠข ๐…*โฆƒT2โฆ„ โ‰˜ f.
70 /4 width=7 by frees_tdeq_conf_rdeq, sex_refl, ext2_refl/ qed-.
71
72 lemma frees_rdeq_conf: โˆ€f,L1,T. L1 โŠข ๐…*โฆƒTโฆ„ โ‰˜ f โ†’
73                        โˆ€L2. L1 โ‰›[f] L2 โ†’ L2 โŠข ๐…*โฆƒTโฆ„ โ‰˜ f.
74 /2 width=7 by frees_tdeq_conf_rdeq, tdeq_refl/ qed-.
75
76 lemma tdeq_rex_conf (R): s_r_confluent1 โ€ฆ cdeq (rex R).
77 #R #L1 #T1 #T2 #HT12 #L2 *
78 /3 width=5 by frees_tdeq_conf, ex2_intro/
79 qed-.
80
81 lemma tdeq_rex_div (R): โˆ€T1,T2. T1 โ‰› T2 โ†’
82                         โˆ€L1,L2. L1 โชค[R, T2] L2 โ†’ L1 โชค[R, T1] L2.
83 /3 width=5 by tdeq_rex_conf, tdeq_sym/ qed-.
84
85 lemma tdeq_rdeq_conf: s_r_confluent1 โ€ฆ cdeq rdeq.
86 /2 width=5 by tdeq_rex_conf/ qed-.
87
88 lemma tdeq_rdeq_div: โˆ€T1,T2. T1 โ‰› T2 โ†’
89                      โˆ€L1,L2. L1 โ‰›[T2] L2 โ†’ L1 โ‰›[T1] L2.
90 /2 width=5 by tdeq_rex_div/ qed-.
91
92 lemma rdeq_atom: โˆ€I. โ‹† โ‰›[โ“ช{I}] โ‹†.
93 /2 width=1 by rex_atom/ qed.
94
95 lemma rdeq_sort: โˆ€I1,I2,L1,L2,s.
96                  L1 โ‰›[โ‹†s] L2 โ†’ L1.โ“˜{I1} โ‰›[โ‹†s] L2.โ“˜{I2}.
97 /2 width=1 by rex_sort/ qed.
98
99 lemma rdeq_pair: โˆ€I,L1,L2,V1,V2.
100                  L1 โ‰›[V1] L2 โ†’ V1 โ‰› V2 โ†’ L1.โ“‘{I}V1 โ‰›[#0] L2.โ“‘{I}V2.
101 /2 width=1 by rex_pair/ qed.
102 (*
103 lemma rdeq_unit: โˆ€f,I,L1,L2. ๐ˆโฆƒfโฆ„ โ†’ L1 โชค[cdeq_ext, cfull, f] L2 โ†’
104                  L1.โ“ค{I} โ‰›[#0] L2.โ“ค{I}.
105 /2 width=3 by rex_unit/ qed.
106 *)
107 lemma rdeq_lref: โˆ€I1,I2,L1,L2,i.
108                  L1 โ‰›[#i] L2 โ†’ L1.โ“˜{I1} โ‰›[#โ†‘i] L2.โ“˜{I2}.
109 /2 width=1 by rex_lref/ qed.
110
111 lemma rdeq_gref: โˆ€I1,I2,L1,L2,l.
112                  L1 โ‰›[ยงl] L2 โ†’ L1.โ“˜{I1} โ‰›[ยงl] L2.โ“˜{I2}.
113 /2 width=1 by rex_gref/ qed.
114
115 lemma rdeq_bind_repl_dx: โˆ€I,I1,L1,L2.โˆ€T:term.
116                          L1.โ“˜{I} โ‰›[T] L2.โ“˜{I1} โ†’
117                          โˆ€I2. I โ‰› I2 โ†’
118                          L1.โ“˜{I} โ‰›[T] L2.โ“˜{I2}.
119 /2 width=2 by rex_bind_repl_dx/ qed-.
120
121 (* Basic inversion lemmas ***************************************************)
122
123 lemma rdeq_inv_atom_sn: โˆ€Y2. โˆ€T:term. โ‹† โ‰›[T] Y2 โ†’ Y2 = โ‹†.
124 /2 width=3 by rex_inv_atom_sn/ qed-.
125
126 lemma rdeq_inv_atom_dx: โˆ€Y1. โˆ€T:term. Y1 โ‰›[T] โ‹† โ†’ Y1 = โ‹†.
127 /2 width=3 by rex_inv_atom_dx/ qed-.
128 (*
129 lemma rdeq_inv_zero: โˆ€Y1,Y2. Y1 โ‰›[#0] Y2 โ†’
130                      โˆจโˆจ โˆงโˆง Y1 = โ‹† & Y2 = โ‹†
131                       | โˆƒโˆƒI,L1,L2,V1,V2. L1 โ‰›[V1] L2 & V1 โ‰› V2 &
132                                          Y1 = L1.โ“‘{I}V1 & Y2 = L2.โ“‘{I}V2
133                       | โˆƒโˆƒf,I,L1,L2. ๐ˆโฆƒfโฆ„ & L1 โชค[cdeq_ext h o, cfull, f] L2 &
134                                          Y1 = L1.โ“ค{I} & Y2 = L2.โ“ค{I}.
135 #Y1 #Y2 #H elim (rex_inv_zero โ€ฆ H) -H *
136 /3 width=9 by or3_intro0, or3_intro1, or3_intro2, ex4_5_intro, ex4_4_intro, conj/
137 qed-.
138 *)
139 lemma rdeq_inv_lref: โˆ€Y1,Y2,i. Y1 โ‰›[#โ†‘i] Y2 โ†’
140                      โˆจโˆจ โˆงโˆง Y1 = โ‹† & Y2 = โ‹†
141                       | โˆƒโˆƒI1,I2,L1,L2. L1 โ‰›[#i] L2 &
142                                        Y1 = L1.โ“˜{I1} & Y2 = L2.โ“˜{I2}.
143 /2 width=1 by rex_inv_lref/ qed-.
144
145 (* Basic_2A1: uses: lleq_inv_bind lleq_inv_bind_O *)
146 lemma rdeq_inv_bind: โˆ€p,I,L1,L2,V,T. L1 โ‰›[โ“‘{p,I}V.T] L2 โ†’
147                      โˆงโˆง L1 โ‰›[V] L2 & L1.โ“‘{I}V โ‰›[T] L2.โ“‘{I}V.
148 /2 width=2 by rex_inv_bind/ qed-.
149
150 (* Basic_2A1: uses: lleq_inv_flat *)
151 lemma rdeq_inv_flat: โˆ€I,L1,L2,V,T. L1 โ‰›[โ“•{I}V.T] L2 โ†’
152                      โˆงโˆง L1 โ‰›[V] L2 & L1 โ‰›[T] L2.
153 /2 width=2 by rex_inv_flat/ qed-.
154
155 (* Advanced inversion lemmas ************************************************)
156
157 lemma rdeq_inv_zero_pair_sn: โˆ€I,Y2,L1,V1. L1.โ“‘{I}V1 โ‰›[#0] Y2 โ†’
158                              โˆƒโˆƒL2,V2. L1 โ‰›[V1] L2 & V1 โ‰› V2 & Y2 = L2.โ“‘{I}V2.
159 /2 width=1 by rex_inv_zero_pair_sn/ qed-.
160
161 lemma rdeq_inv_zero_pair_dx: โˆ€I,Y1,L2,V2. Y1 โ‰›[#0] L2.โ“‘{I}V2 โ†’
162                              โˆƒโˆƒL1,V1. L1 โ‰›[V1] L2 & V1 โ‰› V2 & Y1 = L1.โ“‘{I}V1.
163 /2 width=1 by rex_inv_zero_pair_dx/ qed-.
164
165 lemma rdeq_inv_lref_bind_sn: โˆ€I1,Y2,L1,i. L1.โ“˜{I1} โ‰›[#โ†‘i] Y2 โ†’
166                              โˆƒโˆƒI2,L2. L1 โ‰›[#i] L2 & Y2 = L2.โ“˜{I2}.
167 /2 width=2 by rex_inv_lref_bind_sn/ qed-.
168
169 lemma rdeq_inv_lref_bind_dx: โˆ€I2,Y1,L2,i. Y1 โ‰›[#โ†‘i] L2.โ“˜{I2} โ†’
170                              โˆƒโˆƒI1,L1. L1 โ‰›[#i] L2 & Y1 = L1.โ“˜{I1}.
171 /2 width=2 by rex_inv_lref_bind_dx/ qed-.
172
173 (* Basic forward lemmas *****************************************************)
174
175 lemma rdeq_fwd_zero_pair: โˆ€I,K1,K2,V1,V2.
176                           K1.โ“‘{I}V1 โ‰›[#0] K2.โ“‘{I}V2 โ†’ K1 โ‰›[V1] K2.
177 /2 width=3 by rex_fwd_zero_pair/ qed-.
178
179 (* Basic_2A1: uses: lleq_fwd_bind_sn lleq_fwd_flat_sn *)
180 lemma rdeq_fwd_pair_sn: โˆ€I,L1,L2,V,T. L1 โ‰›[โ‘ก{I}V.T] L2 โ†’ L1 โ‰›[V] L2.
181 /2 width=3 by rex_fwd_pair_sn/ qed-.
182
183 (* Basic_2A1: uses: lleq_fwd_bind_dx lleq_fwd_bind_O_dx *)
184 lemma rdeq_fwd_bind_dx: โˆ€p,I,L1,L2,V,T.
185                         L1 โ‰›[โ“‘{p,I}V.T] L2 โ†’ L1.โ“‘{I}V โ‰›[T] L2.โ“‘{I}V.
186 /2 width=2 by rex_fwd_bind_dx/ qed-.
187
188 (* Basic_2A1: uses: lleq_fwd_flat_dx *)
189 lemma rdeq_fwd_flat_dx: โˆ€I,L1,L2,V,T. L1 โ‰›[โ“•{I}V.T] L2 โ†’ L1 โ‰›[T] L2.
190 /2 width=3 by rex_fwd_flat_dx/ qed-.
191
192 lemma rdeq_fwd_dx: โˆ€I2,L1,K2. โˆ€T:term. L1 โ‰›[T] K2.โ“˜{I2} โ†’
193                    โˆƒโˆƒI1,K1. L1 = K1.โ“˜{I1}.
194 /2 width=5 by rex_fwd_dx/ qed-.