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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 *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 include "ground_2/relocation/rtmap_id.ma".
16 include "basic_2/notation/relations/relationstar_4.ma".
17 include "basic_2/syntax/lenv_ext2.ma".
18 include "basic_2/relocation/lexs.ma".
19 include "basic_2/static/frees.ma".
21 (* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
23 definition lfxs (R) (T): relation lenv ≝
24 λL1,L2. ∃∃f. L1 ⊢ 𝐅*⦃T⦄ ≡ f & L1 ⪤*[cext2 R, cfull, f] L2.
26 interpretation "generic extension on referred entries (local environment)"
27 'RelationStar R T L1 L2 = (lfxs R T L1 L2).
29 definition R_frees_confluent: predicate (relation3 …) ≝
31 ∀f1,L,T1. L ⊢ 𝐅*⦃T1⦄ ≡ f1 → ∀T2. RN L T1 T2 →
32 ∃∃f2. L ⊢ 𝐅*⦃T2⦄ ≡ f2 & f2 ⊆ f1.
34 definition lexs_frees_confluent: relation (relation3 …) ≝
36 ∀f1,L1,T. L1 ⊢ 𝐅*⦃T⦄ ≡ f1 →
37 ∀L2. L1 ⪤*[RN, RP, f1] L2 →
38 ∃∃f2. L2 ⊢ 𝐅*⦃T⦄ ≡ f2 & f2 ⊆ f1.
40 definition R_confluent2_lfxs: relation4 (relation3 lenv term term)
41 (relation3 lenv term term) … ≝
43 ∀L0,T0,T1. R1 L0 T0 T1 → ∀T2. R2 L0 T0 T2 →
44 ∀L1. L0 ⪤*[RP1, T0] L1 → ∀L2. L0 ⪤*[RP2, T0] L2 →
45 ∃∃T. R2 L1 T1 T & R1 L2 T2 T.
47 (* Basic inversion lemmas ***************************************************)
49 lemma lfxs_inv_atom_sn: ∀R,Y2,T. ⋆ ⪤*[R, T] Y2 → Y2 = ⋆.
50 #R #Y2 #T * /2 width=4 by lexs_inv_atom1/
53 lemma lfxs_inv_atom_dx: ∀R,Y1,T. Y1 ⪤*[R, T] ⋆ → Y1 = ⋆.
54 #R #I #Y1 * /2 width=4 by lexs_inv_atom2/
57 lemma lfxs_inv_sort: ∀R,Y1,Y2,s. Y1 ⪤*[R, ⋆s] Y2 →
59 | ∃∃I1,I2,L1,L2. L1 ⪤*[R, ⋆s] L2 &
60 Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
61 #R * [ | #Y1 #I1 ] #Y2 #s * #f #H1 #H2
62 [ lapply (lexs_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/
63 | lapply (frees_inv_sort … H1) -H1 #Hf
64 elim (isid_inv_gen … Hf) -Hf #g #Hg #H destruct
65 elim (lexs_inv_push1 … H2) -H2 #I2 #L2 #H12 #_ #H destruct
66 /5 width=7 by frees_sort, ex3_4_intro, ex2_intro, or_intror/
70 lemma lfxs_inv_zero: ∀R,Y1,Y2. Y1 ⪤*[R, #0] Y2 →
72 | ∃∃I,L1,L2,V1,V2. L1 ⪤*[R, V1] L2 & R L1 V1 V2 &
73 Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2
74 | ∃∃f,I,L1,L2. 𝐈⦃f⦄ & L1 ⪤*[cext2 R, cfull, f] L2 &
75 Y1 = L1.ⓤ{I} & Y2 = L2.ⓤ{I}.
76 #R * [ | #Y1 * #I1 [ | #X ] ] #Y2 * #f #H1 #H2
77 [ lapply (lexs_inv_atom1 … H2) -H2 /3 width=1 by or3_intro0, conj/
78 | elim (frees_inv_unit … H1) -H1 #g #HX #H destruct
79 elim (lexs_inv_next1 … H2) -H2 #I2 #L2 #HL12 #H #H2 destruct
80 >(ext2_inv_unit_sn … H) -H /3 width=8 by or3_intro2, ex4_4_intro/
81 | elim (frees_inv_pair … H1) -H1 #g #Hg #H destruct
82 elim (lexs_inv_next1 … H2) -H2 #Z2 #L2 #HL12 #H
83 elim (ext2_inv_pair_sn … H) -H
84 /4 width=9 by or3_intro1, ex4_5_intro, ex2_intro/
88 lemma lfxs_inv_lref: ∀R,Y1,Y2,i. Y1 ⪤*[R, #⫯i] Y2 →
90 | ∃∃I1,I2,L1,L2. L1 ⪤*[R, #i] L2 &
91 Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
92 #R * [ | #Y1 #I1 ] #Y2 #i * #f #H1 #H2
93 [ lapply (lexs_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/
94 | elim (frees_inv_lref … H1) -H1 #g #Hg #H destruct
95 elim (lexs_inv_push1 … H2) -H2
96 /4 width=7 by ex3_4_intro, ex2_intro, or_intror/
100 lemma lfxs_inv_gref: ∀R,Y1,Y2,l. Y1 ⪤*[R, §l] Y2 →
102 | ∃∃I1,I2,L1,L2. L1 ⪤*[R, §l] L2 &
103 Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
104 #R * [ | #Y1 #I1 ] #Y2 #l * #f #H1 #H2
105 [ lapply (lexs_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/
106 | lapply (frees_inv_gref … H1) -H1 #Hf
107 elim (isid_inv_gen … Hf) -Hf #g #Hg #H destruct
108 elim (lexs_inv_push1 … H2) -H2 #I2 #L2 #H12 #_ #H destruct
109 /5 width=7 by frees_gref, ex3_4_intro, ex2_intro, or_intror/
113 (* Basic_2A1: uses: llpx_sn_inv_bind llpx_sn_inv_bind_O *)
114 lemma lfxs_inv_bind: ∀R,p,I,L1,L2,V1,V2,T. L1 ⪤*[R, ⓑ{p,I}V1.T] L2 → R L1 V1 V2 →
115 L1 ⪤*[R, V1] L2 ∧ L1.ⓑ{I}V1 ⪤*[R, T] L2.ⓑ{I}V2.
116 #R #p #I #L1 #L2 #V1 #V2 #T * #f #Hf #HL #HV elim (frees_inv_bind … Hf) -Hf
117 /6 width=6 by sle_lexs_trans, lexs_inv_tl, ext2_pair, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/
120 (* Basic_2A1: uses: llpx_sn_inv_flat *)
121 lemma lfxs_inv_flat: ∀R,I,L1,L2,V,T. L1 ⪤*[R, ⓕ{I}V.T] L2 →
122 L1 ⪤*[R, V] L2 ∧ L1 ⪤*[R, T] L2.
123 #R #I #L1 #L2 #V #T * #f #Hf #HL elim (frees_inv_flat … Hf) -Hf
124 /5 width=6 by sle_lexs_trans, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/
127 (* Advanced inversion lemmas ************************************************)
129 lemma lfxs_inv_sort_bind_sn: ∀R,I1,K1,L2,s. K1.ⓘ{I1} ⪤*[R, ⋆s] L2 →
130 ∃∃I2,K2. K1 ⪤*[R, ⋆s] K2 & L2 = K2.ⓘ{I2}.
131 #R #I1 #K1 #L2 #s #H elim (lfxs_inv_sort … H) -H *
133 | #Z1 #I2 #Y1 #K2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
137 lemma lfxs_inv_sort_bind_dx: ∀R,I2,K2,L1,s. L1 ⪤*[R, ⋆s] K2.ⓘ{I2} →
138 ∃∃I1,K1. K1 ⪤*[R, ⋆s] K2 & L1 = K1.ⓘ{I1}.
139 #R #I2 #K2 #L1 #s #H elim (lfxs_inv_sort … H) -H *
141 | #I1 #Z2 #K1 #Y2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
145 lemma lfxs_inv_zero_pair_sn: ∀R,I,L2,K1,V1. K1.ⓑ{I}V1 ⪤*[R, #0] L2 →
146 ∃∃K2,V2. K1 ⪤*[R, V1] K2 & R K1 V1 V2 &
148 #R #I #L2 #K1 #V1 #H elim (lfxs_inv_zero … H) -H *
150 | #Z #Y1 #K2 #X1 #V2 #HK12 #HV12 #H1 #H2 destruct
151 /2 width=5 by ex3_2_intro/
152 | #f #Z #Y1 #Y2 #_ #_ #H destruct
156 lemma lfxs_inv_zero_pair_dx: ∀R,I,L1,K2,V2. L1 ⪤*[R, #0] K2.ⓑ{I}V2 →
157 ∃∃K1,V1. K1 ⪤*[R, V1] K2 & R K1 V1 V2 &
159 #R #I #L1 #K2 #V2 #H elim (lfxs_inv_zero … H) -H *
161 | #Z #K1 #Y2 #V1 #X2 #HK12 #HV12 #H1 #H2 destruct
162 /2 width=5 by ex3_2_intro/
163 | #f #Z #Y1 #Y2 #_ #_ #_ #H destruct
167 lemma lfxs_inv_zero_unit_sn: ∀R,I,K1,L2. K1.ⓤ{I} ⪤*[R, #0] L2 →
168 ∃∃f,K2. 𝐈⦃f⦄ & K1 ⪤*[cext2 R, cfull, f] K2 &
170 #R #I #K1 #L2 #H elim (lfxs_inv_zero … H) -H *
172 | #Z #Y1 #Y2 #X1 #X2 #_ #_ #H destruct
173 | #f #Z #Y1 #K2 #Hf #HK12 #H1 #H2 destruct /2 width=5 by ex3_2_intro/
177 lemma lfxs_inv_zero_unit_dx: ∀R,I,L1,K2. L1 ⪤*[R, #0] K2.ⓤ{I} →
178 ∃∃f,K1. 𝐈⦃f⦄ & K1 ⪤*[cext2 R, cfull, f] K2 &
180 #R #I #L1 #K2 #H elim (lfxs_inv_zero … H) -H *
182 | #Z #Y1 #Y2 #X1 #X2 #_ #_ #_ #H destruct
183 | #f #Z #K1 #Y2 #Hf #HK12 #H1 #H2 destruct /2 width=5 by ex3_2_intro/
187 lemma lfxs_inv_lref_bind_sn: ∀R,I1,K1,L2,i. K1.ⓘ{I1} ⪤*[R, #⫯i] L2 →
188 ∃∃I2,K2. K1 ⪤*[R, #i] K2 & L2 = K2.ⓘ{I2}.
189 #R #I1 #K1 #L2 #i #H elim (lfxs_inv_lref … H) -H *
191 | #Z1 #I2 #Y1 #K2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/
195 lemma lfxs_inv_lref_bind_dx: ∀R,I2,K2,L1,i. L1 ⪤*[R, #⫯i] K2.ⓘ{I2} →
196 ∃∃I1,K1. K1 ⪤*[R, #i] K2 & L1 = K1.ⓘ{I1}.
197 #R #I2 #K2 #L1 #i #H elim (lfxs_inv_lref … H) -H *
199 | #I1 #Z2 #K1 #Y2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/
203 lemma lfxs_inv_gref_bind_sn: ∀R,I1,K1,L2,l. K1.ⓘ{I1} ⪤*[R, §l] L2 →
204 ∃∃I2,K2. K1 ⪤*[R, §l] K2 & L2 = K2.ⓘ{I2}.
205 #R #I1 #K1 #L2 #l #H elim (lfxs_inv_gref … H) -H *
207 | #Z1 #I2 #Y1 #K2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
211 lemma lfxs_inv_gref_bind_dx: ∀R,I2,K2,L1,l. L1 ⪤*[R, §l] K2.ⓘ{I2} →
212 ∃∃I1,K1. K1 ⪤*[R, §l] K2 & L1 = K1.ⓘ{I1}.
213 #R #I2 #K2 #L1 #l #H elim (lfxs_inv_gref … H) -H *
215 | #I1 #Z2 #K1 #Y2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
219 (* Basic forward lemmas *****************************************************)
221 lemma lfxs_fwd_zero_pair: ∀R,I,K1,K2,V1,V2.
222 K1.ⓑ{I}V1 ⪤*[R, #0] K2.ⓑ{I}V2 → K1 ⪤*[R, V1] K2.
223 #R #I #K1 #K2 #V1 #V2 #H
224 elim (lfxs_inv_zero_pair_sn … H) -H #Y #X #HK12 #_ #H destruct //
227 (* Basic_2A1: uses: llpx_sn_fwd_pair_sn llpx_sn_fwd_bind_sn llpx_sn_fwd_flat_sn *)
228 lemma lfxs_fwd_pair_sn: ∀R,I,L1,L2,V,T. L1 ⪤*[R, ②{I}V.T] L2 → L1 ⪤*[R, V] L2.
229 #R * [ #p ] #I #L1 #L2 #V #T * #f #Hf #HL
230 [ elim (frees_inv_bind … Hf) | elim (frees_inv_flat … Hf) ] -Hf
231 /4 width=6 by sle_lexs_trans, sor_inv_sle_sn, ex2_intro/
234 (* Basic_2A1: uses: llpx_sn_fwd_bind_dx llpx_sn_fwd_bind_O_dx *)
235 lemma lfxs_fwd_bind_dx: ∀R,p,I,L1,L2,V1,V2,T. L1 ⪤*[R, ⓑ{p,I}V1.T] L2 →
236 R L1 V1 V2 → L1.ⓑ{I}V1 ⪤*[R, T] L2.ⓑ{I}V2.
237 #R #p #I #L1 #L2 #V1 #V2 #T #H #HV elim (lfxs_inv_bind … H HV) -H -HV //
240 (* Basic_2A1: uses: llpx_sn_fwd_flat_dx *)
241 lemma lfxs_fwd_flat_dx: ∀R,I,L1,L2,V,T. L1 ⪤*[R, ⓕ{I}V.T] L2 → L1 ⪤*[R, T] L2.
242 #R #I #L1 #L2 #V #T #H elim (lfxs_inv_flat … H) -H //
245 lemma lfxs_fwd_dx: ∀R,I2,L1,K2,T. L1 ⪤*[R, T] K2.ⓘ{I2} →
246 ∃∃I1,K1. L1 = K1.ⓘ{I1}.
247 #R #I2 #L1 #K2 #T * #f elim (pn_split f) * #g #Hg #_ #Hf destruct
248 [ elim (lexs_inv_push2 … Hf) | elim (lexs_inv_next2 … Hf) ] -Hf #I1 #K1 #_ #_ #H destruct
249 /2 width=3 by ex1_2_intro/
252 (* Basic properties *********************************************************)
254 lemma lfxs_atom: ∀R,I. ⋆ ⪤*[R, ⓪{I}] ⋆.
255 #R * /3 width=3 by frees_sort, frees_atom, frees_gref, lexs_atom, ex2_intro/
258 (* Basic_2A1: uses: llpx_sn_sort *)
259 lemma lfxs_sort: ∀R,I1,I2,L1,L2,s.
260 L1 ⪤*[R, ⋆s] L2 → L1.ⓘ{I1} ⪤*[R, ⋆s] L2.ⓘ{I2}.
261 #R #I1 #I2 #L1 #L2 #s * #f #Hf #H12
262 lapply (frees_inv_sort … Hf) -Hf
263 /4 width=3 by frees_sort, lexs_push, isid_push, ex2_intro/
266 lemma lfxs_pair: ∀R,I,L1,L2,V1,V2. L1 ⪤*[R, V1] L2 →
267 R L1 V1 V2 → L1.ⓑ{I}V1 ⪤*[R, #0] L2.ⓑ{I}V2.
268 #R #I1 #I2 #L1 #L2 #V1 *
269 /4 width=3 by ext2_pair, frees_pair, lexs_next, ex2_intro/
272 lemma lfxs_unit: ∀R,f,I,L1,L2. 𝐈⦃f⦄ → L1 ⪤*[cext2 R, cfull, f] L2 →
273 L1.ⓤ{I} ⪤*[R, #0] L2.ⓤ{I}.
274 /4 width=3 by frees_unit, lexs_next, ext2_unit, ex2_intro/ qed.
276 lemma lfxs_lref: ∀R,I1,I2,L1,L2,i.
277 L1 ⪤*[R, #i] L2 → L1.ⓘ{I1} ⪤*[R, #⫯i] L2.ⓘ{I2}.
278 #R #I1 #I2 #L1 #L2 #i * /3 width=3 by lexs_push, frees_lref, ex2_intro/
281 (* Basic_2A1: uses: llpx_sn_gref *)
282 lemma lfxs_gref: ∀R,I1,I2,L1,L2,l.
283 L1 ⪤*[R, §l] L2 → L1.ⓘ{I1} ⪤*[R, §l] L2.ⓘ{I2}.
284 #R #I1 #I2 #L1 #L2 #l * #f #Hf #H12
285 lapply (frees_inv_gref … Hf) -Hf
286 /4 width=3 by frees_gref, lexs_push, isid_push, ex2_intro/
289 lemma lfxs_bind_repl_dx: ∀R,I,I1,L1,L2,T.
290 L1.ⓘ{I} ⪤*[R, T] L2.ⓘ{I1} →
291 ∀I2. cext2 R L1 I I2 →
292 L1.ⓘ{I} ⪤*[R, T] L2.ⓘ{I2}.
293 #R #I #I1 #L1 #L2 #T * #f #Hf #HL12 #I2 #HR
294 /3 width=5 by lexs_pair_repl, ex2_intro/
297 lemma lfxs_sym: ∀R. lexs_frees_confluent (cext2 R) cfull →
298 (∀L1,L2,T1,T2. R L1 T1 T2 → R L2 T2 T1) →
299 ∀T. symmetric … (lfxs R T).
300 #R #H1R #H2R #T #L1 #L2 * #f1 #Hf1 #HL12 elim (H1R … Hf1 … HL12) -Hf1
301 /5 width=5 by sle_lexs_trans, lexs_sym, cext2_sym, ex2_intro/
304 (* Basic_2A1: uses: llpx_sn_co *)
305 lemma lfxs_co: ∀R1,R2. (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) →
306 ∀L1,L2,T. L1 ⪤*[R1, T] L2 → L1 ⪤*[R2, T] L2.
307 #R1 #R2 #HR #L1 #L2 #T * /5 width=7 by lexs_co, cext2_co, ex2_intro/
310 lemma lfxs_isid: ∀R1,R2,L1,L2,T1,T2.
311 (∀f. L1 ⊢ 𝐅*⦃T1⦄ ≡ f → 𝐈⦃f⦄) →
312 (∀f. 𝐈⦃f⦄ → L1 ⊢ 𝐅*⦃T2⦄ ≡ f) →
313 L1 ⪤*[R1, T1] L2 → L1 ⪤*[R2, T2] L2.
314 #R1 #R2 #L1 #L2 #T1 #T2 #H1 #H2 *
315 /4 width=7 by lexs_co_isid, ex2_intro/
318 lemma lfxs_unit_sn: ∀R1,R2,I,K1,L2.
319 K1.ⓤ{I} ⪤*[R1, #0] L2 → K1.ⓤ{I} ⪤*[R2, #0] L2.
320 #R1 #R2 #I #K1 #L2 #H
321 elim (lfxs_inv_zero_unit_sn … H) -H #f #K2 #Hf #HK12 #H destruct
322 /3 width=7 by lfxs_unit, lexs_co_isid/
325 (* Basic_2A1: removed theorems 9:
326 llpx_sn_skip llpx_sn_lref llpx_sn_free
328 llpx_sn_Y llpx_sn_ge_up llpx_sn_ge
329 llpx_sn_fwd_drop_sn llpx_sn_fwd_drop_dx