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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/relationstarstar_4.ma".
16 include "basic_2/static/lfxs.ma".
18 (* ITERATED EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ***)
20 definition tc_lfxs (R): term → relation lenv ≝ LTC … (lfxs R).
22 interpretation "iterated extension on referred entries (local environment)"
23 'RelationStarStar R T L1 L2 = (tc_lfxs R T L1 L2).
25 (* Basic properties *********************************************************)
27 lemma tc_lfxs_step_dx: ∀R,L1,L,T. L1 ⦻**[R, T] L →
28 ∀L2. L ⦻*[R, T] L2 → L1 ⦻**[R, T] L2.
29 #R #L1 #L2 #T #HL1 #L2 @step @HL1 (**) (* auto fails *)
32 lemma tc_lfxs_step_sn: ∀R,L1,L,T. L1 ⦻*[R, T] L →
33 ∀L2. L ⦻**[R, T] L2 → L1 ⦻**[R, T] L2.
34 #R #L1 #L2 #T #HL1 #L2 @TC_strap @HL1 (**) (* auto fails *)
37 lemma tc_lfxs_atom: ∀R,I. ⋆ ⦻**[R, ⓪{I}] ⋆.
38 /2 width=1 by inj/ qed.
40 lemma tc_lfxs_sort: ∀R,I,L1,L2,V1,V2,s.
41 L1 ⦻**[R, ⋆s] L2 → L1.ⓑ{I}V1 ⦻**[R, ⋆s] L2.ⓑ{I}V2.
42 #R #I #L1 #L2 #V1 #V2 #s #H elim H -L2
43 /3 width=4 by lfxs_sort, tc_lfxs_step_dx, inj/
46 lemma tc_lfxs_zero: ∀R. (∀L. reflexive … (R L)) →
47 ∀I,L1,L2,V. L1 ⦻**[R, V] L2 →
48 L1.ⓑ{I}V ⦻**[R, #0] L2.ⓑ{I}V.
49 #R #HR #I #L1 #L2 #V #H elim H -L2
50 /3 width=5 by lfxs_zero, tc_lfxs_step_dx, inj/
53 lemma tc_lfxs_lref: ∀R,I,L1,L2,V1,V2,i.
54 L1 ⦻**[R, #i] L2 → L1.ⓑ{I}V1 ⦻**[R, #⫯i] L2.ⓑ{I}V2.
55 #R #I #L1 #L2 #V1 #V2 #i #H elim H -L2
56 /3 width=4 by lfxs_lref, tc_lfxs_step_dx, inj/
59 lemma tc_lfxs_gref: ∀R,I,L1,L2,V1,V2,l.
60 L1 ⦻**[R, §l] L2 → L1.ⓑ{I}V1 ⦻**[R, §l] L2.ⓑ{I}V2.
61 #R #I #L1 #L2 #V1 #V2 #l #H elim H -L2
62 /3 width=4 by lfxs_gref, tc_lfxs_step_dx, inj/
65 lemma tc_lfxs_sym: ∀R. lexs_frees_confluent R cfull →
66 (∀L1,L2,T1,T2. R L1 T1 T2 → R L2 T2 T1) →
67 ∀T. symmetric … (tc_lfxs R T).
68 #R #H1R #H2R #T #L1 #L2 #H elim H -L2
69 /4 width=3 by lfxs_sym, tc_lfxs_step_sn, inj/
72 lemma tc_lfxs_co: ∀R1,R2. (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) →
73 ∀L1,L2,T. L1 ⦻**[R1, T] L2 → L1 ⦻**[R2, T] L2.
74 #R1 #R2 #HR #L1 #L2 #T #H elim H -L2
75 /4 width=5 by lfxs_co, tc_lfxs_step_dx, inj/
78 (* Basic inversion lemmas ***************************************************)
80 (* Basic_2A1: uses: TC_lpx_sn_inv_atom1 *)
81 lemma tc_lfxs_inv_atom_sn: ∀R,I,Y2. ⋆ ⦻**[R, ⓪{I}] Y2 → Y2 = ⋆.
82 #R #I #Y2 #H elim H -Y2 /3 width=3 by inj, lfxs_inv_atom_sn/
85 (* Basic_2A1: uses: TC_lpx_sn_inv_atom2 *)
86 lemma tc_lfxs_inv_atom_dx: ∀R,I,Y1. Y1 ⦻**[R, ⓪{I}] ⋆ → Y1 = ⋆.
87 #R #I #Y1 #H @(TC_ind_dx ??????? H) -Y1
88 /3 width=3 by inj, lfxs_inv_atom_dx/
91 lemma tc_lfxs_inv_sort: ∀R,Y1,Y2,s. Y1 ⦻**[R, ⋆s] Y2 →
93 ∃∃I,L1,L2,V1,V2. L1 ⦻**[R, ⋆s] L2 &
94 Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
95 #R #Y1 #Y2 #s #H elim H -Y2
96 [ #Y2 #H elim (lfxs_inv_sort … H) -H *
97 /4 width=8 by ex3_5_intro, inj, or_introl, or_intror, conj/
98 | #Y #Y2 #_ #H elim (lfxs_inv_sort … H) -H *
99 [ #H #H2 * * /3 width=8 by ex3_5_intro, or_introl, or_intror, conj/
100 | #I #L #L2 #V #V2 #HL2 #H #H2 * *
102 | #I0 #L1 #L0 #V1 #V0 #HL10 #H1 #H0 destruct
103 /4 width=8 by ex3_5_intro, tc_lfxs_step_dx, or_intror/
109 lemma tc_lfxs_inv_gref: ∀R,Y1,Y2,l. Y1 ⦻**[R, §l] Y2 →
111 ∃∃I,L1,L2,V1,V2. L1 ⦻**[R, §l] L2 &
112 Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
113 #R #Y1 #Y2 #l #H elim H -Y2
114 [ #Y2 #H elim (lfxs_inv_gref … H) -H *
115 /4 width=8 by ex3_5_intro, inj, or_introl, or_intror, conj/
116 | #Y #Y2 #_ #H elim (lfxs_inv_gref … H) -H *
117 [ #H #H2 * * /3 width=8 by ex3_5_intro, or_introl, or_intror, conj/
118 | #I #L #L2 #V #V2 #HL2 #H #H2 * *
120 | #I0 #L1 #L0 #V1 #V0 #HL10 #H1 #H0 destruct
121 /4 width=8 by ex3_5_intro, tc_lfxs_step_dx, or_intror/
127 lemma tc_lfxs_inv_bind: ∀R. (∀L. reflexive … (R L)) →
128 ∀p,I,L1,L2,V,T. L1 ⦻**[R, ⓑ{p,I}V.T] L2 →
129 L1 ⦻**[R, V] L2 ∧ L1.ⓑ{I}V ⦻**[R, T] L2.ⓑ{I}V.
130 #R #HR #p #I #L1 #L2 #V #T #H elim H -L2
131 [ #L2 #H elim (lfxs_inv_bind … V ? H) -H /3 width=1 by inj, conj/
132 | #L #L2 #_ #H * elim (lfxs_inv_bind … V ? H) -H /3 width=3 by tc_lfxs_step_dx, conj/
136 lemma tc_lfxs_inv_flat: ∀R,I,L1,L2,V,T. L1 ⦻**[R, ⓕ{I}V.T] L2 →
137 L1 ⦻**[R, V] L2 ∧ L1 ⦻**[R, T] L2.
138 #R #I #L1 #L2 #V #T #H elim H -L2
139 [ #L2 #H elim (lfxs_inv_flat … H) -H /3 width=1 by inj, conj/
140 | #L #L2 #_ #H * elim (lfxs_inv_flat … H) -H /3 width=3 by tc_lfxs_step_dx, conj/
144 (* Advanced inversion lemmas ************************************************)
146 lemma tc_lfxs_inv_sort_pair_sn: ∀R,I,Y2,L1,V1,s. L1.ⓑ{I}V1 ⦻**[R, ⋆s] Y2 →
147 ∃∃L2,V2. L1 ⦻**[R, ⋆s] L2 & Y2 = L2.ⓑ{I}V2.
148 #R #I #Y2 #L1 #V1 #s #H elim (tc_lfxs_inv_sort … H) -H *
150 | #J #Y1 #L2 #X1 #V2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
154 lemma tc_lfxs_inv_sort_pair_dx: ∀R,I,Y1,L2,V2,s. Y1 ⦻**[R, ⋆s] L2.ⓑ{I}V2 →
155 ∃∃L1,V1. L1 ⦻**[R, ⋆s] L2 & Y1 = L1.ⓑ{I}V1.
156 #R #I #Y1 #L2 #V2 #s #H elim (tc_lfxs_inv_sort … H) -H *
158 | #J #L1 #Y2 #V1 #X2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
162 lemma tc_lfxs_inv_gref_pair_sn: ∀R,I,Y2,L1,V1,l. L1.ⓑ{I}V1 ⦻**[R, §l] Y2 →
163 ∃∃L2,V2. L1 ⦻**[R, §l] L2 & Y2 = L2.ⓑ{I}V2.
164 #R #I #Y2 #L1 #V1 #l #H elim (tc_lfxs_inv_gref … H) -H *
166 | #J #Y1 #L2 #X1 #V2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
170 lemma tc_lfxs_inv_gref_pair_dx: ∀R,I,Y1,L2,V2,l. Y1 ⦻**[R, §l] L2.ⓑ{I}V2 →
171 ∃∃L1,V1. L1 ⦻**[R, §l] L2 & Y1 = L1.ⓑ{I}V1.
172 #R #I #Y1 #L2 #V2 #l #H elim (tc_lfxs_inv_gref … H) -H *
174 | #J #L1 #Y2 #V1 #X2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
178 (* Basic forward lemmas *****************************************************)
180 lemma tc_lfxs_fwd_pair_sn: ∀R,I,L1,L2,V,T. L1 ⦻**[R, ②{I}V.T] L2 → L1 ⦻**[R, V] L2.
181 #R #I #L1 #L2 #V #T #H elim H -L2
182 /3 width=5 by lfxs_fwd_pair_sn, tc_lfxs_step_dx, inj/
185 lemma tc_lfxs_fwd_bind_dx: ∀R. (∀L. reflexive … (R L)) →
186 ∀p,I,L1,L2,V,T. L1 ⦻**[R, ⓑ{p,I}V.T] L2 →
187 L1.ⓑ{I}V ⦻**[R, T] L2.ⓑ{I}V.
188 #R #HR #p #I #L1 #L2 #V #T #H elim (tc_lfxs_inv_bind … H) -H //
191 lemma tc_lfxs_fwd_flat_dx: ∀R,I,L1,L2,V,T. L1 ⦻**[R, ⓕ{I}V.T] L2 → L1 ⦻**[R, T] L2.
192 #R #I #L1 #L2 #V #T #H elim (tc_lfxs_inv_flat … H) -H //
195 (* Basic_2A1: removed theorems 2:
196 TC_lpx_sn_inv_pair1 TC_lpx_sn_inv_pair2