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_after.ma".
16 include "static_2/notation/relations/rliftstar_3.ma".
17 include "static_2/syntax/term.ma".
19 (* GENERIC RELOCATION FOR TERMS *********************************************)
22 lift_sort lift_lref_lt lift_lref_ge lift_bind lift_flat
25 inductive lifts: rtmap → relation term ≝
26 | lifts_sort: ∀f,s. lifts f (⋆s) (⋆s)
27 | lifts_lref: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → lifts f (#i1) (#i2)
28 | lifts_gref: ∀f,l. lifts f (§l) (§l)
29 | lifts_bind: ∀f,p,I,V1,V2,T1,T2.
30 lifts f V1 V2 → lifts (⫯f) T1 T2 →
31 lifts f (ⓑ{p,I}V1.T1) (ⓑ{p,I}V2.T2)
32 | lifts_flat: ∀f,I,V1,V2,T1,T2.
33 lifts f V1 V2 → lifts f T1 T2 →
34 lifts f (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
37 interpretation "uniform relocation (term)"
38 'RLiftStar i T1 T2 = (lifts (uni i) T1 T2).
40 interpretation "generic relocation (term)"
41 'RLiftStar f T1 T2 = (lifts f T1 T2).
43 definition liftable2_sn: predicate (relation term) ≝
44 λR. ∀T1,T2. R T1 T2 → ∀f,U1. ⬆*[f] T1 ≘ U1 →
45 ∃∃U2. ⬆*[f] T2 ≘ U2 & R U1 U2.
47 definition deliftable2_sn: predicate (relation term) ≝
48 λR. ∀U1,U2. R U1 U2 → ∀f,T1. ⬆*[f] T1 ≘ U1 →
49 ∃∃T2. ⬆*[f] T2 ≘ U2 & R T1 T2.
51 definition liftable2_bi: predicate (relation term) ≝
52 λR. ∀T1,T2. R T1 T2 → ∀f,U1. ⬆*[f] T1 ≘ U1 →
53 ∀U2. ⬆*[f] T2 ≘ U2 → R U1 U2.
55 definition deliftable2_bi: predicate (relation term) ≝
56 λR. ∀U1,U2. R U1 U2 → ∀f,T1. ⬆*[f] T1 ≘ U1 →
57 ∀T2. ⬆*[f] T2 ≘ U2 → R T1 T2.
59 definition liftable2_dx: predicate (relation term) ≝
60 λR. ∀T1,T2. R T1 T2 → ∀f,U2. ⬆*[f] T2 ≘ U2 →
61 ∃∃U1. ⬆*[f] T1 ≘ U1 & R U1 U2.
63 definition deliftable2_dx: predicate (relation term) ≝
64 λR. ∀U1,U2. R U1 U2 → ∀f,T2. ⬆*[f] T2 ≘ U2 →
65 ∃∃T1. ⬆*[f] T1 ≘ U1 & R T1 T2.
67 (* Basic inversion lemmas ***************************************************)
69 fact lifts_inv_sort1_aux: ∀f,X,Y. ⬆*[f] X ≘ Y → ∀s. X = ⋆s → Y = ⋆s.
70 #f #X #Y * -f -X -Y //
71 [ #f #i1 #i2 #_ #x #H destruct
72 | #f #p #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
73 | #f #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
77 (* Basic_1: was: lift1_sort *)
78 (* Basic_2A1: includes: lift_inv_sort1 *)
79 lemma lifts_inv_sort1: ∀f,Y,s. ⬆*[f] ⋆s ≘ Y → Y = ⋆s.
80 /2 width=4 by lifts_inv_sort1_aux/ qed-.
82 fact lifts_inv_lref1_aux: ∀f,X,Y. ⬆*[f] X ≘ Y → ∀i1. X = #i1 →
83 ∃∃i2. @⦃i1,f⦄ ≘ i2 & Y = #i2.
85 [ #f #s #x #H destruct
86 | #f #i1 #i2 #Hi12 #x #H destruct /2 width=3 by ex2_intro/
87 | #f #l #x #H destruct
88 | #f #p #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
89 | #f #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
93 (* Basic_1: was: lift1_lref *)
94 (* Basic_2A1: includes: lift_inv_lref1 lift_inv_lref1_lt lift_inv_lref1_ge *)
95 lemma lifts_inv_lref1: ∀f,Y,i1. ⬆*[f] #i1 ≘ Y →
96 ∃∃i2. @⦃i1,f⦄ ≘ i2 & Y = #i2.
97 /2 width=3 by lifts_inv_lref1_aux/ qed-.
99 fact lifts_inv_gref1_aux: ∀f,X,Y. ⬆*[f] X ≘ Y → ∀l. X = §l → Y = §l.
100 #f #X #Y * -f -X -Y //
101 [ #f #i1 #i2 #_ #x #H destruct
102 | #f #p #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
103 | #f #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
107 (* Basic_2A1: includes: lift_inv_gref1 *)
108 lemma lifts_inv_gref1: ∀f,Y,l. ⬆*[f] §l ≘ Y → Y = §l.
109 /2 width=4 by lifts_inv_gref1_aux/ qed-.
111 fact lifts_inv_bind1_aux: ∀f,X,Y. ⬆*[f] X ≘ Y →
112 ∀p,I,V1,T1. X = ⓑ{p,I}V1.T1 →
113 ∃∃V2,T2. ⬆*[f] V1 ≘ V2 & ⬆*[⫯f] T1 ≘ T2 &
116 [ #f #s #q #J #W1 #U1 #H destruct
117 | #f #i1 #i2 #_ #q #J #W1 #U1 #H destruct
118 | #f #l #b #J #W1 #U1 #H destruct
119 | #f #p #I #V1 #V2 #T1 #T2 #HV12 #HT12 #q #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
120 | #f #I #V1 #V2 #T1 #T2 #_ #_ #q #J #W1 #U1 #H destruct
124 (* Basic_1: was: lift1_bind *)
125 (* Basic_2A1: includes: lift_inv_bind1 *)
126 lemma lifts_inv_bind1: ∀f,p,I,V1,T1,Y. ⬆*[f] ⓑ{p,I}V1.T1 ≘ Y →
127 ∃∃V2,T2. ⬆*[f] V1 ≘ V2 & ⬆*[⫯f] T1 ≘ T2 &
129 /2 width=3 by lifts_inv_bind1_aux/ qed-.
131 fact lifts_inv_flat1_aux: ∀f:rtmap. ∀X,Y. ⬆*[f] X ≘ Y →
132 ∀I,V1,T1. X = ⓕ{I}V1.T1 →
133 ∃∃V2,T2. ⬆*[f] V1 ≘ V2 & ⬆*[f] T1 ≘ T2 &
136 [ #f #s #J #W1 #U1 #H destruct
137 | #f #i1 #i2 #_ #J #W1 #U1 #H destruct
138 | #f #l #J #W1 #U1 #H destruct
139 | #f #p #I #V1 #V2 #T1 #T2 #_ #_ #J #W1 #U1 #H destruct
140 | #f #I #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
144 (* Basic_1: was: lift1_flat *)
145 (* Basic_2A1: includes: lift_inv_flat1 *)
146 lemma lifts_inv_flat1: ∀f:rtmap. ∀I,V1,T1,Y. ⬆*[f] ⓕ{I}V1.T1 ≘ Y →
147 ∃∃V2,T2. ⬆*[f] V1 ≘ V2 & ⬆*[f] T1 ≘ T2 &
149 /2 width=3 by lifts_inv_flat1_aux/ qed-.
151 fact lifts_inv_sort2_aux: ∀f,X,Y. ⬆*[f] X ≘ Y → ∀s. Y = ⋆s → X = ⋆s.
152 #f #X #Y * -f -X -Y //
153 [ #f #i1 #i2 #_ #x #H destruct
154 | #f #p #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
155 | #f #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
159 (* Basic_1: includes: lift_gen_sort *)
160 (* Basic_2A1: includes: lift_inv_sort2 *)
161 lemma lifts_inv_sort2: ∀f,X,s. ⬆*[f] X ≘ ⋆s → X = ⋆s.
162 /2 width=4 by lifts_inv_sort2_aux/ qed-.
164 fact lifts_inv_lref2_aux: ∀f,X,Y. ⬆*[f] X ≘ Y → ∀i2. Y = #i2 →
165 ∃∃i1. @⦃i1,f⦄ ≘ i2 & X = #i1.
167 [ #f #s #x #H destruct
168 | #f #i1 #i2 #Hi12 #x #H destruct /2 width=3 by ex2_intro/
169 | #f #l #x #H destruct
170 | #f #p #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
171 | #f #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
175 (* Basic_1: includes: lift_gen_lref lift_gen_lref_lt lift_gen_lref_false lift_gen_lref_ge *)
176 (* Basic_2A1: includes: lift_inv_lref2 lift_inv_lref2_lt lift_inv_lref2_be lift_inv_lref2_ge lift_inv_lref2_plus *)
177 lemma lifts_inv_lref2: ∀f,X,i2. ⬆*[f] X ≘ #i2 →
178 ∃∃i1. @⦃i1,f⦄ ≘ i2 & X = #i1.
179 /2 width=3 by lifts_inv_lref2_aux/ qed-.
181 fact lifts_inv_gref2_aux: ∀f,X,Y. ⬆*[f] X ≘ Y → ∀l. Y = §l → X = §l.
182 #f #X #Y * -f -X -Y //
183 [ #f #i1 #i2 #_ #x #H destruct
184 | #f #p #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
185 | #f #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
189 (* Basic_2A1: includes: lift_inv_gref1 *)
190 lemma lifts_inv_gref2: ∀f,X,l. ⬆*[f] X ≘ §l → X = §l.
191 /2 width=4 by lifts_inv_gref2_aux/ qed-.
193 fact lifts_inv_bind2_aux: ∀f,X,Y. ⬆*[f] X ≘ Y →
194 ∀p,I,V2,T2. Y = ⓑ{p,I}V2.T2 →
195 ∃∃V1,T1. ⬆*[f] V1 ≘ V2 & ⬆*[⫯f] T1 ≘ T2 &
198 [ #f #s #q #J #W2 #U2 #H destruct
199 | #f #i1 #i2 #_ #q #J #W2 #U2 #H destruct
200 | #f #l #q #J #W2 #U2 #H destruct
201 | #f #p #I #V1 #V2 #T1 #T2 #HV12 #HT12 #q #J #W2 #U2 #H destruct /2 width=5 by ex3_2_intro/
202 | #f #I #V1 #V2 #T1 #T2 #_ #_ #q #J #W2 #U2 #H destruct
206 (* Basic_1: includes: lift_gen_bind *)
207 (* Basic_2A1: includes: lift_inv_bind2 *)
208 lemma lifts_inv_bind2: ∀f,p,I,V2,T2,X. ⬆*[f] X ≘ ⓑ{p,I}V2.T2 →
209 ∃∃V1,T1. ⬆*[f] V1 ≘ V2 & ⬆*[⫯f] T1 ≘ T2 &
211 /2 width=3 by lifts_inv_bind2_aux/ qed-.
213 fact lifts_inv_flat2_aux: ∀f:rtmap. ∀X,Y. ⬆*[f] X ≘ Y →
214 ∀I,V2,T2. Y = ⓕ{I}V2.T2 →
215 ∃∃V1,T1. ⬆*[f] V1 ≘ V2 & ⬆*[f] T1 ≘ T2 &
218 [ #f #s #J #W2 #U2 #H destruct
219 | #f #i1 #i2 #_ #J #W2 #U2 #H destruct
220 | #f #l #J #W2 #U2 #H destruct
221 | #f #p #I #V1 #V2 #T1 #T2 #_ #_ #J #W2 #U2 #H destruct
222 | #f #I #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W2 #U2 #H destruct /2 width=5 by ex3_2_intro/
226 (* Basic_1: includes: lift_gen_flat *)
227 (* Basic_2A1: includes: lift_inv_flat2 *)
228 lemma lifts_inv_flat2: ∀f:rtmap. ∀I,V2,T2,X. ⬆*[f] X ≘ ⓕ{I}V2.T2 →
229 ∃∃V1,T1. ⬆*[f] V1 ≘ V2 & ⬆*[f] T1 ≘ T2 &
231 /2 width=3 by lifts_inv_flat2_aux/ qed-.
233 (* Advanced inversion lemmas ************************************************)
235 lemma lifts_inv_atom1: ∀f,I,Y. ⬆*[f] ⓪{I} ≘ Y →
236 ∨∨ ∃∃s. I = Sort s & Y = ⋆s
237 | ∃∃i,j. @⦃i,f⦄ ≘ j & I = LRef i & Y = #j
238 | ∃∃l. I = GRef l & Y = §l.
240 [ lapply (lifts_inv_sort1 … H)
241 | elim (lifts_inv_lref1 … H)
242 | lapply (lifts_inv_gref1 … H)
243 ] -H /3 width=5 by or3_intro0, or3_intro1, or3_intro2, ex3_2_intro, ex2_intro/
246 lemma lifts_inv_atom2: ∀f,I,X. ⬆*[f] X ≘ ⓪{I} →
247 ∨∨ ∃∃s. X = ⋆s & I = Sort s
248 | ∃∃i,j. @⦃i,f⦄ ≘ j & X = #i & I = LRef j
249 | ∃∃l. X = §l & I = GRef l.
251 [ lapply (lifts_inv_sort2 … H)
252 | elim (lifts_inv_lref2 … H)
253 | lapply (lifts_inv_gref2 … H)
254 ] -H /3 width=5 by or3_intro0, or3_intro1, or3_intro2, ex3_2_intro, ex2_intro/
257 (* Basic_2A1: includes: lift_inv_pair_xy_x *)
258 lemma lifts_inv_pair_xy_x: ∀f,I,V,T. ⬆*[f] ②{I}V.T ≘ V → ⊥.
261 [ lapply (lifts_inv_sort2 … H) -H #H destruct
262 | elim (lifts_inv_lref2 … H) -H
264 | lapply (lifts_inv_gref2 … H) -H #H destruct
266 | * [ #p ] #I #V2 #T2 #IHV2 #_ #U #H
267 [ elim (lifts_inv_bind2 … H) -H #V1 #T1 #HV12 #_ #H destruct /2 width=3 by/
268 | elim (lifts_inv_flat2 … H) -H #V1 #T1 #HV12 #_ #H destruct /2 width=3 by/
273 (* Basic_1: includes: thead_x_lift_y_y *)
274 (* Basic_2A1: includes: lift_inv_pair_xy_y *)
275 lemma lifts_inv_pair_xy_y: ∀I,T,V,f. ⬆*[f] ②{I}V.T ≘ T → ⊥.
278 [ lapply (lifts_inv_sort2 … H) -H #H destruct
279 | elim (lifts_inv_lref2 … H) -H
281 | lapply (lifts_inv_gref2 … H) -H #H destruct
283 | * [ #p ] #I #V2 #T2 #_ #IHT2 #W #f #H
284 [ elim (lifts_inv_bind2 … H) -H #V1 #T1 #_ #HT12 #H destruct /2 width=4 by/
285 | elim (lifts_inv_flat2 … H) -H #V1 #T1 #_ #HT12 #H destruct /2 width=4 by/
290 (* Inversion lemmas with uniform relocations ********************************)
292 lemma lifts_inv_lref1_uni: ∀l,Y,i. ⬆*[l] #i ≘ Y → Y = #(l+i).
293 #l #Y #i1 #H elim (lifts_inv_lref1 … H) -H /4 width=4 by at_mono, eq_f/
296 lemma lifts_inv_lref2_uni: ∀l,X,i2. ⬆*[l] X ≘ #i2 →
297 ∃∃i1. X = #i1 & i2 = l + i1.
298 #l #X #i2 #H elim (lifts_inv_lref2 … H) -H
299 /3 width=3 by at_inv_uni, ex2_intro/
302 lemma lifts_inv_lref2_uni_ge: ∀l,X,i. ⬆*[l] X ≘ #(l + i) → X = #i.
303 #l #X #i2 #H elim (lifts_inv_lref2_uni … H) -H
304 #i1 #H1 #H2 destruct /4 width=2 by injective_plus_r, eq_f, sym_eq/
307 lemma lifts_inv_lref2_uni_lt: ∀l,X,i. ⬆*[l] X ≘ #i → i < l → ⊥.
308 #l #X #i2 #H elim (lifts_inv_lref2_uni … H) -H
309 #i1 #_ #H1 #H2 destruct /2 width=4 by lt_le_false/
312 (* Basic forward lemmas *****************************************************)
314 (* Basic_2A1: includes: lift_inv_O2 *)
315 lemma lifts_fwd_isid: ∀f,T1,T2. ⬆*[f] T1 ≘ T2 → 𝐈⦃f⦄ → T1 = T2.
316 #f #T1 #T2 #H elim H -f -T1 -T2
317 /4 width=3 by isid_inv_at_mono, isid_push, eq_f2, eq_f/
320 (* Basic_2A1: includes: lift_fwd_pair1 *)
321 lemma lifts_fwd_pair1: ∀f:rtmap. ∀I,V1,T1,Y. ⬆*[f] ②{I}V1.T1 ≘ Y →
322 ∃∃V2,T2. ⬆*[f] V1 ≘ V2 & Y = ②{I}V2.T2.
323 #f * [ #p ] #I #V1 #T1 #Y #H
324 [ elim (lifts_inv_bind1 … H) -H /2 width=4 by ex2_2_intro/
325 | elim (lifts_inv_flat1 … H) -H /2 width=4 by ex2_2_intro/
329 (* Basic_2A1: includes: lift_fwd_pair2 *)
330 lemma lifts_fwd_pair2: ∀f:rtmap. ∀I,V2,T2,X. ⬆*[f] X ≘ ②{I}V2.T2 →
331 ∃∃V1,T1. ⬆*[f] V1 ≘ V2 & X = ②{I}V1.T1.
332 #f * [ #p ] #I #V2 #T2 #X #H
333 [ elim (lifts_inv_bind2 … H) -H /2 width=4 by ex2_2_intro/
334 | elim (lifts_inv_flat2 … H) -H /2 width=4 by ex2_2_intro/
338 (* Basic properties *********************************************************)
340 lemma liftable2_sn_dx (R): symmetric … R → liftable2_sn R → liftable2_dx R.
341 #R #H2R #H1R #T1 #T2 #HT12 #f #U2 #HTU2
342 elim (H1R … T1 … HTU2) -H1R /3 width=3 by ex2_intro/
345 lemma deliftable2_sn_dx (R): symmetric … R → deliftable2_sn R → deliftable2_dx R.
346 #R #H2R #H1R #U1 #U2 #HU12 #f #T2 #HTU2
347 elim (H1R … U1 … HTU2) -H1R /3 width=3 by ex2_intro/
350 lemma lifts_eq_repl_back: ∀T1,T2. eq_repl_back … (λf. ⬆*[f] T1 ≘ T2).
351 #T1 #T2 #f1 #H elim H -T1 -T2 -f1
352 /4 width=5 by lifts_flat, lifts_bind, lifts_lref, at_eq_repl_back, eq_push/
355 lemma lifts_eq_repl_fwd: ∀T1,T2. eq_repl_fwd … (λf. ⬆*[f] T1 ≘ T2).
356 #T1 #T2 @eq_repl_sym /2 width=3 by lifts_eq_repl_back/ (**) (* full auto fails *)
359 (* Basic_1: includes: lift_r *)
360 (* Basic_2A1: includes: lift_refl *)
361 lemma lifts_refl: ∀T,f. 𝐈⦃f⦄ → ⬆*[f] T ≘ T.
363 /4 width=3 by lifts_flat, lifts_bind, lifts_lref, isid_inv_at, isid_push/
366 (* Basic_2A1: includes: lift_total *)
367 lemma lifts_total: ∀T1,f. ∃T2. ⬆*[f] T1 ≘ T2.
369 /3 width=2 by lifts_lref, lifts_sort, lifts_gref, ex_intro/
370 [ #p ] #I #V1 #T1 #IHV1 #IHT1 #f
371 elim (IHV1 f) -IHV1 #V2 #HV12
372 [ elim (IHT1 (⫯f)) -IHT1 /3 width=2 by lifts_bind, ex_intro/
373 | elim (IHT1 f) -IHT1 /3 width=2 by lifts_flat, ex_intro/
377 lemma lifts_push_zero (f): ⬆*[⫯f]#0 ≘ #0.
378 /2 width=1 by lifts_lref/ qed.
380 lemma lifts_push_lref (f) (i1) (i2): ⬆*[f]#i1 ≘ #i2 → ⬆*[⫯f]#(↑i1) ≘ #(↑i2).
382 elim (lifts_inv_lref1 … H) -H #j #Hij #H destruct
383 /3 width=7 by lifts_lref, at_push/
386 lemma lifts_lref_uni: ∀l,i. ⬆*[l] #i ≘ #(l+i).
387 #l elim l -l /2 width=1 by lifts_lref/
390 (* Basic_1: includes: lift_free (right to left) *)
391 (* Basic_2A1: includes: lift_split *)
392 lemma lifts_split_trans: ∀f,T1,T2. ⬆*[f] T1 ≘ T2 →
393 ∀f1,f2. f2 ⊚ f1 ≘ f →
394 ∃∃T. ⬆*[f1] T1 ≘ T & ⬆*[f2] T ≘ T2.
395 #f #T1 #T2 #H elim H -f -T1 -T2
396 [ /3 width=3 by lifts_sort, ex2_intro/
397 | #f #i1 #i2 #Hi #f1 #f2 #Ht elim (after_at_fwd … Hi … Ht) -Hi -Ht
398 /3 width=3 by lifts_lref, ex2_intro/
399 | /3 width=3 by lifts_gref, ex2_intro/
400 | #f #p #I #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f1 #f2 #Ht
401 elim (IHV … Ht) elim (IHT (⫯f1) (⫯f2)) -IHV -IHT
402 /3 width=5 by lifts_bind, after_O2, ex2_intro/
403 | #f #I #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f1 #f2 #Ht
404 elim (IHV … Ht) elim (IHT … Ht) -IHV -IHT -Ht
405 /3 width=5 by lifts_flat, ex2_intro/
409 (* Note: apparently, this was missing in Basic_2A1 *)
410 lemma lifts_split_div: ∀f1,T1,T2. ⬆*[f1] T1 ≘ T2 →
412 ∃∃T. ⬆*[f2] T2 ≘ T & ⬆*[f] T1 ≘ T.
413 #f1 #T1 #T2 #H elim H -f1 -T1 -T2
414 [ /3 width=3 by lifts_sort, ex2_intro/
415 | #f1 #i1 #i2 #Hi #f2 #f #Ht elim (after_at1_fwd … Hi … Ht) -Hi -Ht
416 /3 width=3 by lifts_lref, ex2_intro/
417 | /3 width=3 by lifts_gref, ex2_intro/
418 | #f1 #p #I #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f2 #f #Ht
419 elim (IHV … Ht) elim (IHT (⫯f2) (⫯f)) -IHV -IHT
420 /3 width=5 by lifts_bind, after_O2, ex2_intro/
421 | #f1 #I #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f2 #f #Ht
422 elim (IHV … Ht) elim (IHT … Ht) -IHV -IHT -Ht
423 /3 width=5 by lifts_flat, ex2_intro/
427 (* Basic_1: includes: dnf_dec2 dnf_dec *)
428 (* Basic_2A1: includes: is_lift_dec *)
429 lemma is_lifts_dec: ∀T2,f. Decidable (∃T1. ⬆*[f] T1 ≘ T2).
431 [ * [1,3: /3 width=2 by lifts_sort, lifts_gref, ex_intro, or_introl/ ]
432 #i2 #f elim (is_at_dec f i2) //
433 [ * /4 width=3 by lifts_lref, ex_intro, or_introl/
435 #X #HX elim (lifts_inv_lref2 … HX) -HX
436 /3 width=2 by ex_intro/
438 | * [ #p ] #I #V2 #T2 #IHV2 #IHT2 #f
439 [ elim (IHV2 f) -IHV2
440 [ * #V1 #HV12 elim (IHT2 (⫯f)) -IHT2
441 [ * #T1 #HT12 @or_introl /3 width=2 by lifts_bind, ex_intro/
442 | -V1 #HT2 @or_intror * #X #H
443 elim (lifts_inv_bind2 … H) -H /3 width=2 by ex_intro/
445 | -IHT2 #HV2 @or_intror * #X #H
446 elim (lifts_inv_bind2 … H) -H /3 width=2 by ex_intro/
448 | elim (IHV2 f) -IHV2
449 [ * #V1 #HV12 elim (IHT2 f) -IHT2
450 [ * #T1 #HT12 /4 width=2 by lifts_flat, ex_intro, or_introl/
451 | -V1 #HT2 @or_intror * #X #H
452 elim (lifts_inv_flat2 … H) -H /3 width=2 by ex_intro/
454 | -IHT2 #HV2 @or_intror * #X #H
455 elim (lifts_inv_flat2 … H) -H /3 width=2 by ex_intro/
461 (* Properties with uniform relocation ***************************************)
463 lemma lifts_uni: ∀n1,n2,T,U. ⬆*[𝐔❴n1❵∘𝐔❴n2❵] T ≘ U → ⬆*[n1+n2] T ≘ U.
464 /3 width=4 by lifts_eq_repl_back, after_inv_total/ qed.
466 (* Basic_2A1: removed theorems 14:
467 lifts_inv_nil lifts_inv_cons
468 lift_inv_Y1 lift_inv_Y2 lift_inv_lref_Y1 lift_inv_lref_Y2 lift_lref_Y lift_Y1
469 lift_lref_lt_eq lift_lref_ge_eq lift_lref_plus lift_lref_pred
470 lift_lref_ge_minus lift_lref_ge_minus_eq
472 (* Basic_1: removed theorems 8:
474 lift_head lift_gen_head
475 lift_weight_map lift_weight lift_weight_add lift_weight_add_O