1 (**************************************************************************)
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 *)
13 (**************************************************************************)
15 include "ground_2/relocation/nstream_coafter.ma".
16 include "basic_2/relocation/drops_drops.ma".
17 include "basic_2/static/frees_frees.ma".
19 (* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
21 (* Advanced properties ******************************************************)
23 lemma frees_lref_atom: ∀b,L,i. ⬇*[b, 𝐔❴i❵] L ≡ ⋆ →
24 ∀f. 𝐈⦃f⦄ → L ⊢ 𝐅*⦃#i⦄ ≡ f.
25 #b #L elim L -L /2 width=1 by frees_atom/
27 [ #H lapply (drops_fwd_isid … H ?) -H // #H destruct
28 | /5 width=3 by frees_eq_repl_back, frees_lref, drops_inv_drop1, eq_push_inv_isid/
32 lemma frees_lref_pair: ∀f,K,V. K ⊢ 𝐅*⦃V⦄ ≡ f →
33 ∀i,I,L. ⬇*[i] L ≡ K.ⓑ{I}V → L ⊢ 𝐅*⦃#i⦄ ≡ ↑*[i] ⫯f.
34 #f #K #V #Hf #i elim i -i
35 [ #I #L #H lapply (drops_fwd_isid … H ?) -H /2 width=1 by frees_zero/
36 | #i #IH #I #L #H elim (drops_inv_succ … H) -H /3 width=2 by frees_lref/
40 lemma frees_sort_pushs: ∀f,K,s. K ⊢ 𝐅*⦃⋆s⦄ ≡ f →
41 ∀i,L. ⬇*[i] L ≡ K → L ⊢ 𝐅*⦃⋆s⦄ ≡ ↑*[i] f.
42 #f #K #s #Hf #i elim i -i
43 [ #L #H lapply (drops_fwd_isid … H ?) -H //
44 | #i #IH #L #H elim (drops_inv_succ … H) -H /3 width=1 by frees_sort/
48 lemma frees_lref_pushs: ∀f,K,j. K ⊢ 𝐅*⦃#j⦄ ≡ f →
49 ∀i,L. ⬇*[i] L ≡ K → L ⊢ 𝐅*⦃#(i+j)⦄ ≡ ↑*[i] f.
50 #f #K #j #Hf #i elim i -i
51 [ #L #H lapply (drops_fwd_isid … H ?) -H //
52 | #i #IH #L #H elim (drops_inv_succ … H) -H /3 width=1 by frees_lref/
56 lemma frees_gref_pushs: ∀f,K,l. K ⊢ 𝐅*⦃§l⦄ ≡ f →
57 ∀i,L. ⬇*[i] L ≡ K → L ⊢ 𝐅*⦃§l⦄ ≡ ↑*[i] f.
58 #f #K #l #Hf #i elim i -i
59 [ #L #H lapply (drops_fwd_isid … H ?) -H //
60 | #i #IH #L #H elim (drops_inv_succ … H) -H /3 width=1 by frees_gref/
64 (* Advanced inversion lemmas ************************************************)
66 lemma frees_inv_lref_drops: ∀i,f,L. L ⊢ 𝐅*⦃#i⦄ ≡ f →
67 (⬇*[Ⓕ, 𝐔❴i❵] L ≡ ⋆ ∧ 𝐈⦃f⦄) ∨
68 ∃∃g,I,K,V. K ⊢ 𝐅*⦃V⦄ ≡ g &
69 ⬇*[i] L ≡ K.ⓑ{I}V & f = ↑*[i] ⫯g.
71 [ #f #L #H elim (frees_inv_zero … H) -H *
72 /4 width=7 by ex3_4_intro, or_introl, or_intror, conj, drops_refl/
73 | #i #IH #f #L #H elim (frees_inv_lref … H) -H * /3 width=1 by or_introl, conj/
74 #g #I #K #V #Hg #H1 #H2 destruct
75 elim (IH … Hg) -IH -Hg *
76 [ /4 width=3 by or_introl, conj, isid_push, drops_drop/
77 | /4 width=7 by drops_drop, ex3_4_intro, or_intror/
82 (* Properties with generic slicing for local environments *******************)
84 lemma frees_lifts: ∀b,f1,K,T. K ⊢ 𝐅*⦃T⦄ ≡ f1 →
85 ∀f,L. ⬇*[b, f] L ≡ K → ∀U. ⬆*[f] T ≡ U →
86 ∀f2. f ~⊚ f1 ≡ f2 → L ⊢ 𝐅*⦃U⦄ ≡ f2.
87 #b #f1 #K #T #H lapply (frees_fwd_isfin … H) elim H -f1 -K -T
88 [ #f1 #I #Hf1 #_ #f #L #H1 #U #H2 #f2 #H3
89 lapply (coafter_isid_inv_dx … H3 … Hf1) -f1 #Hf2
90 elim (lifts_inv_atom1 … H2) -H2 *
91 /2 width=1 by frees_sort_gen, frees_gref_gen/
92 #i #j #Hij #H #H0 destruct
93 elim (drops_inv_atom2 … H1) -H1 #n #g #H1 #Hf
94 elim (after_at_fwd … Hij … Hf) -f #x #_ #Hj -g -i
95 lapply (at_inv_uni … Hj) -Hj #H destruct
96 /3 width=8 by frees_lref_atom, drops_trans/
97 | #f1 #I #K #V #s #_ #IH #Hf1 #f #L #H1 #U #H2 #f2 #H3
98 lapply (isfin_fwd_push … Hf1 ??) -Hf1 [3: |*: // ] #Hf1
99 lapply (lifts_inv_sort1 … H2) -H2 #H destruct
100 elim (drops_split_trans_pair2 … H1) -H1 [ |*: // ] #Y #W #HLY #HYK #_
101 elim (coafter_fwd_xpx_pushs … H3) [ |*: // ] #g2 #H2 destruct
102 lapply (coafter_tls_succ … H3 ??) -H3 [3: |*: // ] #H3
103 lapply (IH … HYK … H3) -IH -H3 -HYK [1,3: // | skip ]
104 /3 width=5 by drops_isuni_fwd_drop2, frees_sort_pushs/
105 | #f1 #I #K #V #_ #IH #Hf1 #f #L #H1 #U #H2 #f2 #H3
106 lapply (isfin_inv_next … Hf1 ??) -Hf1 [3: |*: // ] #Hf1
107 lapply (lifts_inv_lref1 … H2) -H2 * #j #Hf #H destruct
108 elim (drops_split_trans_pair2 … H1) -H1 [ |*: // ] #Y #W #HLY #HYK #HVW
109 elim (coafter_fwd_xnx_pushs … H3) [ |*: // ] #g2 #H2 destruct
110 lapply (coafter_tls_succ … H3 ??) -H3 [3: |*: // ]
111 <tls_S in ⊢ (???%→?); <tls_pushs <tl_next_rew <tl_next_rew #H3
112 lapply (IH … HYK … HVW … H3) -IH -H3 -HYK -HVW //
113 /2 width=5 by frees_lref_pair/
114 | #f1 #I #K #V #i #_ #IH #Hf1 #f #L #H1 #U #H2 #f2 #H3
115 lapply (isfin_fwd_push … Hf1 ??) -Hf1 [3: |*: // ] #Hf1
116 lapply (lifts_inv_lref1 … H2) -H2 * #x #Hf #H destruct
117 elim (at_inv_nxx … Hf) -Hf [ |*: // ] #j #Hf #H destruct
118 elim (drops_split_trans_pair2 … H1) -H1 [ |*: // ] #Y #W #HLY #HYK #_
119 elim (coafter_fwd_xpx_pushs … H3) [ |*: // ] #g2 #H2 destruct
120 lapply (coafter_tls_succ … H3 ??) -H3 [3: |*: // ] <tls_pushs #H3
121 lapply (drops_isuni_fwd_drop2 … HLY) -HLY // #HLY
122 lapply (IH … HYK … H3) -IH -H3 -HYK [4: |*: /2 width=2 by lifts_lref/ ]
123 >plus_S1 /2 width=3 by frees_lref_pushs/ (**) (* full auto fails *)
124 | #f1 #I #K #V #l #_ #IH #Hf1 #f #L #H1 #U #H2 #f2 #H3
125 lapply (isfin_fwd_push … Hf1 ??) -Hf1 [3: |*: // ] #Hf1
126 lapply (lifts_inv_gref1 … H2) -H2 #H destruct
127 elim (drops_split_trans_pair2 … H1) -H1 [ |*: // ] #Y #W #HLY #HYK #_
128 elim (coafter_fwd_xpx_pushs … H3) [ |*: // ] #g2 #H2 destruct
129 lapply (coafter_tls_succ … H3 ??) -H3 [3: |*: // ] #H3
130 lapply (IH … HYK … H3) -IH -H3 -HYK [1,3: // | skip ]
131 /3 width=5 by drops_isuni_fwd_drop2, frees_gref_pushs/
132 | #f1V #f1T #f1 #p #I #K #V #T #_ #_ #H1f1 #IHV #IHT #H2f1 #f #L #H1 #Y #H2 #f2 #H3
133 elim (sor_inv_isfin3 … H1f1) // #Hf1V #H
134 lapply (isfin_inv_tl … H) -H
135 elim (lifts_inv_bind1 … H2) -H2 #W #U #HVW #HTU #H destruct
136 elim (coafter_sor … H3 … H1f1) /2 width=5 by coafter_isfin2_fwd/ -H3 -H1f1 #f2V #f2T #Hf2V #H
137 elim (coafter_inv_tl1 … H) -H /4 width=5 by frees_bind, drops_skip/
138 | #f1V #f1T #f1 #I #K #V #T #_ #_ #H1f1 #IHV #IHT #H2f1 #f #L #H1 #Y #H2 #f2 #H3
139 elim (sor_inv_isfin3 … H1f1) //
140 elim (lifts_inv_flat1 … H2) -H2 #W #U #HVW #HTU #H destruct
141 elim (coafter_sor … H3 … H1f1)
142 /3 width=5 by coafter_isfin2_fwd, frees_flat/
146 (* Forward lemmas with generic slicing for local environments ***************)
148 lemma frees_fwd_coafter: ∀b,f2,L,U. L ⊢ 𝐅*⦃U⦄ ≡ f2 →
149 ∀f,K. ⬇*[b, f] L ≡ K → ∀T. ⬆*[f] T ≡ U →
150 ∀f1. K ⊢ 𝐅*⦃T⦄ ≡ f1 → f ~⊚ f1 ≡ f2.
151 /4 width=11 by frees_lifts, frees_mono, coafter_eq_repl_back0/ qed-.
153 (* Inversion lemmas with generic slicing for local environments *************)
155 lemma frees_inv_lifts: ∀b,f2,L,U. L ⊢ 𝐅*⦃U⦄ ≡ f2 →
156 ∀f,K. ⬇*[b, f] L ≡ K → ∀T. ⬆*[f] T ≡ U →
157 ∀f1. f ~⊚ f1 ≡ f2 → K ⊢ 𝐅*⦃T⦄ ≡ f1.
158 #b #f2 #L #U #H lapply (frees_fwd_isfin … H) elim H -f2 -L -U
159 [ #f2 #I #Hf2 #_ #f #K #H1 #T #H2 #f1 #H3
160 lapply (coafter_fwd_isid2 … H3 … Hf2) -H3 // -Hf2 #Hf1
161 elim (drops_inv_atom1 … H1) -H1 #H #_ destruct
162 elim (lifts_inv_atom2 … H2) -H2 * /2 width=3 by frees_atom/
163 | #f2 #I #L #W #s #_ #IH #Hf2 #f #Y #H1 #T #H2 #f1 #H3
164 lapply (isfin_fwd_push … Hf2 ??) -Hf2 [3: |*: // ] #Hf2
165 lapply (lifts_inv_sort2 … H2) -H2 #H destruct
166 elim (coafter_inv_xxp … H3) -H3 [1,3: * |*: // ]
167 [ #g #g1 #Hf2 #H #H0 destruct
168 elim (drops_inv_skip1 … H1) -H1 #K #V #HLK #_ #H destruct
169 | #g #Hf2 #H destruct
170 lapply (drops_inv_drop1 … H1) -H1
171 ] /3 width=4 by frees_sort/
172 | #f2 #I #L #W #_ #IH #Hf2 #f #Y #H1 #T #H2 #f1 #H3
173 lapply (isfin_inv_next … Hf2 ??) -Hf2 [3: |*: // ] #Hf2
174 elim (lifts_inv_lref2 … H2) -H2 #i #H2 #H destruct
175 lapply (at_inv_xxp … H2 ?) -H2 // * #g #H #H0 destruct
176 elim (drops_inv_skip1 … H1) -H1 #K #V #HLK #HVW #H destruct
177 elim (coafter_inv_pxn … H3) -H3 [ |*: // ] #g1 #Hf2 #H destruct
178 /3 width=4 by frees_zero/
179 | #f2 #I #L #W #j #_ #IH #Hf2 #f #Y #H1 #T #H2 #f1 #H3
180 lapply (isfin_fwd_push … Hf2 ??) -Hf2 [3: |*: // ] #Hf2
181 elim (lifts_inv_lref2 … H2) -H2 #x #H2 #H destruct
182 elim (coafter_inv_xxp … H3) -H3 [1,3: * |*: // ]
183 [ #g #g1 #Hf2 #H #H0 destruct
184 elim (drops_inv_skip1 … H1) -H1 #K #V #HLK #HVW #H destruct
185 elim (at_inv_xpn … H2) -H2 [ |*: // ] #j #Hg #H destruct
186 | #g #Hf2 #H destruct
187 lapply (drops_inv_drop1 … H1) -H1
188 lapply (at_inv_xnn … H2 ????) -H2 [5: |*: // ]
189 ] /4 width=4 by lifts_lref, frees_lref/
190 | #f2 #I #L #W #l #_ #IH #Hf2 #f #Y #H1 #T #H2 #f1 #H3
191 lapply (isfin_fwd_push … Hf2 ??) -Hf2 [3: |*: // ] #Hf2
192 lapply (lifts_inv_gref2 … H2) -H2 #H destruct
193 elim (coafter_inv_xxp … H3) -H3 [1,3: * |*: // ]
194 [ #g #g1 #Hf2 #H #H0 destruct
195 elim (drops_inv_skip1 … H1) -H1 #K #V #HLK #_ #H destruct
196 | #g #Hf2 #H destruct
197 lapply (drops_inv_drop1 … H1) -H1
198 ] /3 width=4 by frees_gref/
199 | #f2W #f2U #f2 #p #I #L #W #U #_ #_ #H1f2 #IHW #IHU #H2f2 #f #K #H1 #X #H2 #f1 #H3
200 elim (sor_inv_isfin3 … H1f2) // #H1f2W #H
201 lapply (isfin_inv_tl … H) -H
202 elim (lifts_inv_bind2 … H2) -H2 #V #T #HVW #HTU #H destruct
203 elim (coafter_inv_sor … H3 … H1f2) -H3 -H1f2 // #f1W #f1U #H2f2W #H
204 elim (coafter_inv_tl0 … H) -H /4 width=5 by frees_bind, drops_skip/
205 | #f2W #f2U #f2 #I #L #W #U #_ #_ #H1f2 #IHW #IHU #H2f2 #f #K #H1 #X #H2 #f1 #H3
206 elim (sor_inv_isfin3 … H1f2) //
207 elim (lifts_inv_flat2 … H2) -H2 #V #T #HVW #HTU #H destruct
208 elim (coafter_inv_sor … H3 … H1f2) -H3 -H1f2 /3 width=5 by frees_flat/
212 lemma frees_inv_drops: ∀f2,L,U. L ⊢ 𝐅*⦃U⦄ ≡ f2 →
213 ∀f,K. ⬇*[Ⓣ, f] L ≡ K → ∀f1. f ~⊚ f1 ≡ f2 →
214 ∃∃T. K ⊢ 𝐅*⦃T⦄ ≡ f1 & ⬆*[f] T ≡ U.
215 #f2 #L #U #H lapply (frees_fwd_isfin … H) elim H -f2 -L -U
216 [ #f2 #I #Hf2 #_ #f #K #H1 #f1 #H2
217 lapply (coafter_fwd_isid2 … H2 ??) -H2 // -Hf2 #Hf1
218 elim (drops_inv_atom1 … H1) -H1 #H #Hf destruct
219 /4 width=3 by frees_atom, lifts_refl, ex2_intro/
220 | #f2 #I #L #W #s #_ #IH #Hf2 #f #Y #H1 #f1 #H2
221 lapply (isfin_fwd_push … Hf2 ??) -Hf2 [3: |*: // ] #Hf2
222 elim (coafter_inv_xxp … H2) -H2 [1,3: * |*: // ]
223 [ #g #g1 #Hf2 #H #H0 destruct
224 elim (drops_inv_skip1 … H1) -H1 #K #V #HLK #_ #H destruct
225 | #g #Hf2 #H destruct
226 lapply (drops_inv_drop1 … H1) -H1 #HLK
228 elim (IH … HLK … Hf2) -L // -f2 #X #Hg1 #HX
229 lapply (lifts_inv_sort2 … HX) -HX #H destruct
230 /3 width=3 by frees_sort, lifts_sort, ex2_intro/
231 | #f2 #I #L #W #_ #IH #Hf2 #f #Y #H1 #f1 #H2
232 lapply (isfin_inv_next … Hf2 ??) -Hf2 [3: |*: // ] #Hf2
233 elim (coafter_inv_xxn … H2) -H2 [ |*: // ] #g #g1 #Hf2 #H0 #H destruct
234 elim (drops_inv_skip1 … H1) -H1 #K #V #HLK #HVW #H destruct
235 elim (IH … HLK … Hf2) -L // -f2 #X #Hg1 #HX
236 lapply (lifts_inj … HX … HVW) -W #H destruct
237 /3 width=3 by frees_zero, lifts_lref, ex2_intro/
238 | #f2 #I #L #W #j #_ #IH #Hf2 #f #Y #H1 #f1 #H2
239 lapply (isfin_fwd_push … Hf2 ??) -Hf2 [3: |*: // ] #Hf2
240 elim (coafter_inv_xxp … H2) -H2 [1,3: * |*: // ]
241 [ #g #g1 #Hf2 #H #H0 destruct
242 elim (drops_inv_skip1 … H1) -H1 #K #V #HLK #_ #H destruct
243 | #g #Hf2 #H destruct
244 lapply (drops_inv_drop1 … H1) -H1 #HLK
246 elim (IH … HLK … Hf2) -L // -f2 #X #Hg1 #HX
247 elim (lifts_inv_lref2 … HX) -HX #i #Hij #H destruct
248 /4 width=7 by frees_lref, lifts_lref, at_S1, at_next, ex2_intro/
249 | #f2 #I #L #W #l #_ #IH #Hf2 #f #Y #H1 #f1 #H2
250 lapply (isfin_fwd_push … Hf2 ??) -Hf2 [3: |*: // ] #Hf2
251 elim (coafter_inv_xxp … H2) -H2 [1,3: * |*: // ]
252 [ #g #g1 #Hf2 #H #H0 destruct
253 elim (drops_inv_skip1 … H1) -H1 #K #V #HLK #_ #H destruct
254 | #g #Hf2 #H destruct
255 lapply (drops_inv_drop1 … H1) -H1 #HLK
257 elim (IH … HLK … Hf2) -L // -f2 #X #Hg1 #HX
258 lapply (lifts_inv_gref2 … HX) -HX #H destruct
259 /3 width=3 by frees_gref, lifts_gref, ex2_intro/
260 | #f2W #f2U #f2 #p #I #L #W #U #_ #_ #H1f2 #IHW #IHU #H2f2 #f #K #H1 #f1 #H2
261 elim (sor_inv_isfin3 … H1f2) // #H1f2W #H
262 lapply (isfin_inv_tl … H) -H #H1f2U
263 elim (coafter_inv_sor … H2 … H1f2) -H2 -H1f2 // #f1W #f1U #H2f2W #H #Hf1
264 elim (coafter_inv_tl0 … H) -H #g1 #H2f2U #H destruct
265 elim (IHW … H1 … H2f2W) -IHW -H2f2W // -H1f2W #V #Hf1W #HVW
266 elim (IHU … H2f2U) -IHU -H2f2U
267 /3 width=5 by frees_bind, drops_skip, lifts_bind, ex2_intro/
268 | #f2W #f2U #f2 #I #L #W #U #_ #_ #H1f2 #IHW #IHU #H2f2 #f #K #H1 #f1 #H2
269 elim (sor_inv_isfin3 … H1f2) // #H1f2W #H1f2U
270 elim (coafter_inv_sor … H2 … H1f2) -H2 -H1f2 // #f1W #f1U #H2f2W #H2f2U #Hf1
271 elim (IHW … H1 … H2f2W) -IHW -H2f2W // -H1f2W
272 elim (IHU … H1 … H2f2U) -L -H2f2U
273 /3 width=5 by frees_flat, lifts_flat, ex2_intro/