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 (* Basic inversion lemmas ***************************************************)
61 fact lifts_inv_sort1_aux: ∀f,X,Y. ⬆*[f] X ≘ Y → ∀s. X = ⋆s → Y = ⋆s.
62 #f #X #Y * -f -X -Y //
63 [ #f #i1 #i2 #_ #x #H destruct
64 | #f #p #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
65 | #f #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
69 (* Basic_1: was: lift1_sort *)
70 (* Basic_2A1: includes: lift_inv_sort1 *)
71 lemma lifts_inv_sort1: ∀f,Y,s. ⬆*[f] ⋆s ≘ Y → Y = ⋆s.
72 /2 width=4 by lifts_inv_sort1_aux/ qed-.
74 fact lifts_inv_lref1_aux: ∀f,X,Y. ⬆*[f] X ≘ Y → ∀i1. X = #i1 →
75 ∃∃i2. @⦃i1, f⦄ ≘ i2 & Y = #i2.
77 [ #f #s #x #H destruct
78 | #f #i1 #i2 #Hi12 #x #H destruct /2 width=3 by ex2_intro/
79 | #f #l #x #H destruct
80 | #f #p #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
81 | #f #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
85 (* Basic_1: was: lift1_lref *)
86 (* Basic_2A1: includes: lift_inv_lref1 lift_inv_lref1_lt lift_inv_lref1_ge *)
87 lemma lifts_inv_lref1: ∀f,Y,i1. ⬆*[f] #i1 ≘ Y →
88 ∃∃i2. @⦃i1, f⦄ ≘ i2 & Y = #i2.
89 /2 width=3 by lifts_inv_lref1_aux/ qed-.
91 fact lifts_inv_gref1_aux: ∀f,X,Y. ⬆*[f] X ≘ Y → ∀l. X = §l → Y = §l.
92 #f #X #Y * -f -X -Y //
93 [ #f #i1 #i2 #_ #x #H destruct
94 | #f #p #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
95 | #f #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
99 (* Basic_2A1: includes: lift_inv_gref1 *)
100 lemma lifts_inv_gref1: ∀f,Y,l. ⬆*[f] §l ≘ Y → Y = §l.
101 /2 width=4 by lifts_inv_gref1_aux/ qed-.
103 fact lifts_inv_bind1_aux: ∀f,X,Y. ⬆*[f] X ≘ Y →
104 ∀p,I,V1,T1. X = ⓑ{p,I}V1.T1 →
105 ∃∃V2,T2. ⬆*[f] V1 ≘ V2 & ⬆*[⫯f] T1 ≘ T2 &
108 [ #f #s #q #J #W1 #U1 #H destruct
109 | #f #i1 #i2 #_ #q #J #W1 #U1 #H destruct
110 | #f #l #b #J #W1 #U1 #H destruct
111 | #f #p #I #V1 #V2 #T1 #T2 #HV12 #HT12 #q #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
112 | #f #I #V1 #V2 #T1 #T2 #_ #_ #q #J #W1 #U1 #H destruct
116 (* Basic_1: was: lift1_bind *)
117 (* Basic_2A1: includes: lift_inv_bind1 *)
118 lemma lifts_inv_bind1: ∀f,p,I,V1,T1,Y. ⬆*[f] ⓑ{p,I}V1.T1 ≘ Y →
119 ∃∃V2,T2. ⬆*[f] V1 ≘ V2 & ⬆*[⫯f] T1 ≘ T2 &
121 /2 width=3 by lifts_inv_bind1_aux/ qed-.
123 fact lifts_inv_flat1_aux: ∀f:rtmap. ∀X,Y. ⬆*[f] X ≘ Y →
124 ∀I,V1,T1. X = ⓕ{I}V1.T1 →
125 ∃∃V2,T2. ⬆*[f] V1 ≘ V2 & ⬆*[f] T1 ≘ T2 &
128 [ #f #s #J #W1 #U1 #H destruct
129 | #f #i1 #i2 #_ #J #W1 #U1 #H destruct
130 | #f #l #J #W1 #U1 #H destruct
131 | #f #p #I #V1 #V2 #T1 #T2 #_ #_ #J #W1 #U1 #H destruct
132 | #f #I #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
136 (* Basic_1: was: lift1_flat *)
137 (* Basic_2A1: includes: lift_inv_flat1 *)
138 lemma lifts_inv_flat1: ∀f:rtmap. ∀I,V1,T1,Y. ⬆*[f] ⓕ{I}V1.T1 ≘ Y →
139 ∃∃V2,T2. ⬆*[f] V1 ≘ V2 & ⬆*[f] T1 ≘ T2 &
141 /2 width=3 by lifts_inv_flat1_aux/ qed-.
143 fact lifts_inv_sort2_aux: ∀f,X,Y. ⬆*[f] X ≘ Y → ∀s. Y = ⋆s → X = ⋆s.
144 #f #X #Y * -f -X -Y //
145 [ #f #i1 #i2 #_ #x #H destruct
146 | #f #p #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
147 | #f #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
151 (* Basic_1: includes: lift_gen_sort *)
152 (* Basic_2A1: includes: lift_inv_sort2 *)
153 lemma lifts_inv_sort2: ∀f,X,s. ⬆*[f] X ≘ ⋆s → X = ⋆s.
154 /2 width=4 by lifts_inv_sort2_aux/ qed-.
156 fact lifts_inv_lref2_aux: ∀f,X,Y. ⬆*[f] X ≘ Y → ∀i2. Y = #i2 →
157 ∃∃i1. @⦃i1, f⦄ ≘ i2 & X = #i1.
159 [ #f #s #x #H destruct
160 | #f #i1 #i2 #Hi12 #x #H destruct /2 width=3 by ex2_intro/
161 | #f #l #x #H destruct
162 | #f #p #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
163 | #f #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
167 (* Basic_1: includes: lift_gen_lref lift_gen_lref_lt lift_gen_lref_false lift_gen_lref_ge *)
168 (* Basic_2A1: includes: lift_inv_lref2 lift_inv_lref2_lt lift_inv_lref2_be lift_inv_lref2_ge lift_inv_lref2_plus *)
169 lemma lifts_inv_lref2: ∀f,X,i2. ⬆*[f] X ≘ #i2 →
170 ∃∃i1. @⦃i1, f⦄ ≘ i2 & X = #i1.
171 /2 width=3 by lifts_inv_lref2_aux/ qed-.
173 fact lifts_inv_gref2_aux: ∀f,X,Y. ⬆*[f] X ≘ Y → ∀l. Y = §l → X = §l.
174 #f #X #Y * -f -X -Y //
175 [ #f #i1 #i2 #_ #x #H destruct
176 | #f #p #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
177 | #f #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
181 (* Basic_2A1: includes: lift_inv_gref1 *)
182 lemma lifts_inv_gref2: ∀f,X,l. ⬆*[f] X ≘ §l → X = §l.
183 /2 width=4 by lifts_inv_gref2_aux/ qed-.
185 fact lifts_inv_bind2_aux: ∀f,X,Y. ⬆*[f] X ≘ Y →
186 ∀p,I,V2,T2. Y = ⓑ{p,I}V2.T2 →
187 ∃∃V1,T1. ⬆*[f] V1 ≘ V2 & ⬆*[⫯f] T1 ≘ T2 &
190 [ #f #s #q #J #W2 #U2 #H destruct
191 | #f #i1 #i2 #_ #q #J #W2 #U2 #H destruct
192 | #f #l #q #J #W2 #U2 #H destruct
193 | #f #p #I #V1 #V2 #T1 #T2 #HV12 #HT12 #q #J #W2 #U2 #H destruct /2 width=5 by ex3_2_intro/
194 | #f #I #V1 #V2 #T1 #T2 #_ #_ #q #J #W2 #U2 #H destruct
198 (* Basic_1: includes: lift_gen_bind *)
199 (* Basic_2A1: includes: lift_inv_bind2 *)
200 lemma lifts_inv_bind2: ∀f,p,I,V2,T2,X. ⬆*[f] X ≘ ⓑ{p,I}V2.T2 →
201 ∃∃V1,T1. ⬆*[f] V1 ≘ V2 & ⬆*[⫯f] T1 ≘ T2 &
203 /2 width=3 by lifts_inv_bind2_aux/ qed-.
205 fact lifts_inv_flat2_aux: ∀f:rtmap. ∀X,Y. ⬆*[f] X ≘ Y →
206 ∀I,V2,T2. Y = ⓕ{I}V2.T2 →
207 ∃∃V1,T1. ⬆*[f] V1 ≘ V2 & ⬆*[f] T1 ≘ T2 &
210 [ #f #s #J #W2 #U2 #H destruct
211 | #f #i1 #i2 #_ #J #W2 #U2 #H destruct
212 | #f #l #J #W2 #U2 #H destruct
213 | #f #p #I #V1 #V2 #T1 #T2 #_ #_ #J #W2 #U2 #H destruct
214 | #f #I #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W2 #U2 #H destruct /2 width=5 by ex3_2_intro/
218 (* Basic_1: includes: lift_gen_flat *)
219 (* Basic_2A1: includes: lift_inv_flat2 *)
220 lemma lifts_inv_flat2: ∀f:rtmap. ∀I,V2,T2,X. ⬆*[f] X ≘ ⓕ{I}V2.T2 →
221 ∃∃V1,T1. ⬆*[f] V1 ≘ V2 & ⬆*[f] T1 ≘ T2 &
223 /2 width=3 by lifts_inv_flat2_aux/ qed-.
225 (* Advanced inversion lemmas ************************************************)
227 lemma lifts_inv_atom1: ∀f,I,Y. ⬆*[f] ⓪{I} ≘ Y →
228 ∨∨ ∃∃s. I = Sort s & Y = ⋆s
229 | ∃∃i,j. @⦃i, f⦄ ≘ j & I = LRef i & Y = #j
230 | ∃∃l. I = GRef l & Y = §l.
232 [ lapply (lifts_inv_sort1 … H)
233 | elim (lifts_inv_lref1 … H)
234 | lapply (lifts_inv_gref1 … H)
235 ] -H /3 width=5 by or3_intro0, or3_intro1, or3_intro2, ex3_2_intro, ex2_intro/
238 lemma lifts_inv_atom2: ∀f,I,X. ⬆*[f] X ≘ ⓪{I} →
239 ∨∨ ∃∃s. X = ⋆s & I = Sort s
240 | ∃∃i,j. @⦃i, f⦄ ≘ j & X = #i & I = LRef j
241 | ∃∃l. X = §l & I = GRef l.
243 [ lapply (lifts_inv_sort2 … H)
244 | elim (lifts_inv_lref2 … H)
245 | lapply (lifts_inv_gref2 … H)
246 ] -H /3 width=5 by or3_intro0, or3_intro1, or3_intro2, ex3_2_intro, ex2_intro/
249 (* Basic_2A1: includes: lift_inv_pair_xy_x *)
250 lemma lifts_inv_pair_xy_x: ∀f,I,V,T. ⬆*[f] ②{I}V.T ≘ V → ⊥.
253 [ lapply (lifts_inv_sort2 … H) -H #H destruct
254 | elim (lifts_inv_lref2 … H) -H
256 | lapply (lifts_inv_gref2 … H) -H #H destruct
258 | * [ #p ] #I #V2 #T2 #IHV2 #_ #U #H
259 [ elim (lifts_inv_bind2 … H) -H #V1 #T1 #HV12 #_ #H destruct /2 width=3 by/
260 | elim (lifts_inv_flat2 … H) -H #V1 #T1 #HV12 #_ #H destruct /2 width=3 by/
265 (* Basic_1: includes: thead_x_lift_y_y *)
266 (* Basic_2A1: includes: lift_inv_pair_xy_y *)
267 lemma lifts_inv_pair_xy_y: ∀I,T,V,f. ⬆*[f] ②{I}V.T ≘ T → ⊥.
270 [ lapply (lifts_inv_sort2 … H) -H #H destruct
271 | elim (lifts_inv_lref2 … H) -H
273 | lapply (lifts_inv_gref2 … H) -H #H destruct
275 | * [ #p ] #I #V2 #T2 #_ #IHT2 #W #f #H
276 [ elim (lifts_inv_bind2 … H) -H #V1 #T1 #_ #HT12 #H destruct /2 width=4 by/
277 | elim (lifts_inv_flat2 … H) -H #V1 #T1 #_ #HT12 #H destruct /2 width=4 by/
282 (* Inversion lemmas with uniform relocations ********************************)
284 lemma lifts_inv_lref1_uni: ∀l,Y,i. ⬆*[l] #i ≘ Y → Y = #(l+i).
285 #l #Y #i1 #H elim (lifts_inv_lref1 … H) -H /4 width=4 by at_mono, eq_f/
288 lemma lifts_inv_lref2_uni: ∀l,X,i2. ⬆*[l] X ≘ #i2 →
289 ∃∃i1. X = #i1 & i2 = l + i1.
290 #l #X #i2 #H elim (lifts_inv_lref2 … H) -H
291 /3 width=3 by at_inv_uni, ex2_intro/
294 lemma lifts_inv_lref2_uni_ge: ∀l,X,i. ⬆*[l] X ≘ #(l + i) → X = #i.
295 #l #X #i2 #H elim (lifts_inv_lref2_uni … H) -H
296 #i1 #H1 #H2 destruct /4 width=2 by injective_plus_r, eq_f, sym_eq/
299 lemma lifts_inv_lref2_uni_lt: ∀l,X,i. ⬆*[l] X ≘ #i → i < l → ⊥.
300 #l #X #i2 #H elim (lifts_inv_lref2_uni … H) -H
301 #i1 #_ #H1 #H2 destruct /2 width=4 by lt_le_false/
304 (* Basic forward lemmas *****************************************************)
306 (* Basic_2A1: includes: lift_inv_O2 *)
307 lemma lifts_fwd_isid: ∀f,T1,T2. ⬆*[f] T1 ≘ T2 → 𝐈⦃f⦄ → T1 = T2.
308 #f #T1 #T2 #H elim H -f -T1 -T2
309 /4 width=3 by isid_inv_at_mono, isid_push, eq_f2, eq_f/
312 (* Basic_2A1: includes: lift_fwd_pair1 *)
313 lemma lifts_fwd_pair1: ∀f:rtmap. ∀I,V1,T1,Y. ⬆*[f] ②{I}V1.T1 ≘ Y →
314 ∃∃V2,T2. ⬆*[f] V1 ≘ V2 & Y = ②{I}V2.T2.
315 #f * [ #p ] #I #V1 #T1 #Y #H
316 [ elim (lifts_inv_bind1 … H) -H /2 width=4 by ex2_2_intro/
317 | elim (lifts_inv_flat1 … H) -H /2 width=4 by ex2_2_intro/
321 (* Basic_2A1: includes: lift_fwd_pair2 *)
322 lemma lifts_fwd_pair2: ∀f:rtmap. ∀I,V2,T2,X. ⬆*[f] X ≘ ②{I}V2.T2 →
323 ∃∃V1,T1. ⬆*[f] V1 ≘ V2 & X = ②{I}V1.T1.
324 #f * [ #p ] #I #V2 #T2 #X #H
325 [ elim (lifts_inv_bind2 … H) -H /2 width=4 by ex2_2_intro/
326 | elim (lifts_inv_flat2 … H) -H /2 width=4 by ex2_2_intro/
330 (* Basic properties *********************************************************)
332 lemma lifts_eq_repl_back: ∀T1,T2. eq_repl_back … (λf. ⬆*[f] T1 ≘ T2).
333 #T1 #T2 #f1 #H elim H -T1 -T2 -f1
334 /4 width=5 by lifts_flat, lifts_bind, lifts_lref, at_eq_repl_back, eq_push/
337 lemma lifts_eq_repl_fwd: ∀T1,T2. eq_repl_fwd … (λf. ⬆*[f] T1 ≘ T2).
338 #T1 #T2 @eq_repl_sym /2 width=3 by lifts_eq_repl_back/ (**) (* full auto fails *)
341 (* Basic_1: includes: lift_r *)
342 (* Basic_2A1: includes: lift_refl *)
343 lemma lifts_refl: ∀T,f. 𝐈⦃f⦄ → ⬆*[f] T ≘ T.
345 /4 width=3 by lifts_flat, lifts_bind, lifts_lref, isid_inv_at, isid_push/
348 (* Basic_2A1: includes: lift_total *)
349 lemma lifts_total: ∀T1,f. ∃T2. ⬆*[f] T1 ≘ T2.
351 /3 width=2 by lifts_lref, lifts_sort, lifts_gref, ex_intro/
352 [ #p ] #I #V1 #T1 #IHV1 #IHT1 #f
353 elim (IHV1 f) -IHV1 #V2 #HV12
354 [ elim (IHT1 (⫯f)) -IHT1 /3 width=2 by lifts_bind, ex_intro/
355 | elim (IHT1 f) -IHT1 /3 width=2 by lifts_flat, ex_intro/
359 lemma lifts_lref_uni: ∀l,i. ⬆*[l] #i ≘ #(l+i).
360 #l elim l -l /2 width=1 by lifts_lref/
363 (* Basic_1: includes: lift_free (right to left) *)
364 (* Basic_2A1: includes: lift_split *)
365 lemma lifts_split_trans: ∀f,T1,T2. ⬆*[f] T1 ≘ T2 →
366 ∀f1,f2. f2 ⊚ f1 ≘ f →
367 ∃∃T. ⬆*[f1] T1 ≘ T & ⬆*[f2] T ≘ T2.
368 #f #T1 #T2 #H elim H -f -T1 -T2
369 [ /3 width=3 by lifts_sort, ex2_intro/
370 | #f #i1 #i2 #Hi #f1 #f2 #Ht elim (after_at_fwd … Hi … Ht) -Hi -Ht
371 /3 width=3 by lifts_lref, ex2_intro/
372 | /3 width=3 by lifts_gref, ex2_intro/
373 | #f #p #I #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f1 #f2 #Ht
374 elim (IHV … Ht) elim (IHT (⫯f1) (⫯f2)) -IHV -IHT
375 /3 width=5 by lifts_bind, after_O2, ex2_intro/
376 | #f #I #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f1 #f2 #Ht
377 elim (IHV … Ht) elim (IHT … Ht) -IHV -IHT -Ht
378 /3 width=5 by lifts_flat, ex2_intro/
382 (* Note: apparently, this was missing in Basic_2A1 *)
383 lemma lifts_split_div: ∀f1,T1,T2. ⬆*[f1] T1 ≘ T2 →
385 ∃∃T. ⬆*[f2] T2 ≘ T & ⬆*[f] T1 ≘ T.
386 #f1 #T1 #T2 #H elim H -f1 -T1 -T2
387 [ /3 width=3 by lifts_sort, ex2_intro/
388 | #f1 #i1 #i2 #Hi #f2 #f #Ht elim (after_at1_fwd … Hi … Ht) -Hi -Ht
389 /3 width=3 by lifts_lref, ex2_intro/
390 | /3 width=3 by lifts_gref, ex2_intro/
391 | #f1 #p #I #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f2 #f #Ht
392 elim (IHV … Ht) elim (IHT (⫯f2) (⫯f)) -IHV -IHT
393 /3 width=5 by lifts_bind, after_O2, ex2_intro/
394 | #f1 #I #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f2 #f #Ht
395 elim (IHV … Ht) elim (IHT … Ht) -IHV -IHT -Ht
396 /3 width=5 by lifts_flat, ex2_intro/
400 (* Basic_1: includes: dnf_dec2 dnf_dec *)
401 (* Basic_2A1: includes: is_lift_dec *)
402 lemma is_lifts_dec: ∀T2,f. Decidable (∃T1. ⬆*[f] T1 ≘ T2).
404 [ * [1,3: /3 width=2 by lifts_sort, lifts_gref, ex_intro, or_introl/ ]
405 #i2 #f elim (is_at_dec f i2) //
406 [ * /4 width=3 by lifts_lref, ex_intro, or_introl/
408 #X #HX elim (lifts_inv_lref2 … HX) -HX
409 /3 width=2 by ex_intro/
411 | * [ #p ] #I #V2 #T2 #IHV2 #IHT2 #f
412 [ elim (IHV2 f) -IHV2
413 [ * #V1 #HV12 elim (IHT2 (⫯f)) -IHT2
414 [ * #T1 #HT12 @or_introl /3 width=2 by lifts_bind, ex_intro/
415 | -V1 #HT2 @or_intror * #X #H
416 elim (lifts_inv_bind2 … H) -H /3 width=2 by ex_intro/
418 | -IHT2 #HV2 @or_intror * #X #H
419 elim (lifts_inv_bind2 … H) -H /3 width=2 by ex_intro/
421 | elim (IHV2 f) -IHV2
422 [ * #V1 #HV12 elim (IHT2 f) -IHT2
423 [ * #T1 #HT12 /4 width=2 by lifts_flat, ex_intro, or_introl/
424 | -V1 #HT2 @or_intror * #X #H
425 elim (lifts_inv_flat2 … H) -H /3 width=2 by ex_intro/
427 | -IHT2 #HV2 @or_intror * #X #H
428 elim (lifts_inv_flat2 … H) -H /3 width=2 by ex_intro/
434 (* Properties with uniform relocation ***************************************)
436 lemma lifts_uni: ∀n1,n2,T,U. ⬆*[𝐔❴n1❵∘𝐔❴n2❵] T ≘ U → ⬆*[n1+n2] T ≘ U.
437 /3 width=4 by lifts_eq_repl_back, after_inv_total/ qed.
439 (* Basic_2A1: removed theorems 14:
440 lifts_inv_nil lifts_inv_cons
441 lift_inv_Y1 lift_inv_Y2 lift_inv_lref_Y1 lift_inv_lref_Y2 lift_lref_Y lift_Y1
442 lift_lref_lt_eq lift_lref_ge_eq lift_lref_plus lift_lref_pred
443 lift_lref_ge_minus lift_lref_ge_minus_eq
445 (* Basic_1: removed theorems 8:
447 lift_head lift_gen_head
448 lift_weight_map lift_weight lift_weight_add lift_weight_add_O