2 (**************************************************************************)
5 (* ||A|| A project by Andrea Asperti *)
7 (* ||I|| Developers: *)
8 (* ||T|| The HELM team. *)
9 (* ||A|| http://helm.cs.unibo.it *)
11 (* \ / This file is distributed under the terms of the *)
12 (* v GNU General Public License Version 2 *)
14 (**************************************************************************)
16 include "ground_2/relocation/nstream_after.ma".
17 include "basic_2/notation/relations/rliftstar_3.ma".
18 include "basic_2/syntax/term.ma".
20 (* GENERIC RELOCATION FOR TERMS *********************************************)
23 lift_sort lift_lref_lt lift_lref_ge lift_bind lift_flat
26 inductive lifts: rtmap → relation term ≝
27 | lifts_sort: ∀f,s. lifts f (⋆s) (⋆s)
28 | lifts_lref: ∀f,i1,i2. @⦃i1, f⦄ ≡ i2 → lifts f (#i1) (#i2)
29 | lifts_gref: ∀f,l. lifts f (§l) (§l)
30 | lifts_bind: ∀f,p,I,V1,V2,T1,T2.
31 lifts f V1 V2 → lifts (↑f) T1 T2 →
32 lifts f (ⓑ{p,I}V1.T1) (ⓑ{p,I}V2.T2)
33 | lifts_flat: ∀f,I,V1,V2,T1,T2.
34 lifts f V1 V2 → lifts f T1 T2 →
35 lifts f (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
38 interpretation "uniform relocation (term)"
39 'RLiftStar i T1 T2 = (lifts (uni i) T1 T2).
41 interpretation "generic relocation (term)"
42 'RLiftStar f T1 T2 = (lifts f T1 T2).
44 definition liftable2_sn: predicate (relation term) ≝
45 λR. ∀T1,T2. R T1 T2 → ∀f,U1. ⬆*[f] T1 ≡ U1 →
46 ∃∃U2. ⬆*[f] T2 ≡ U2 & R U1 U2.
48 definition deliftable2_sn: predicate (relation term) ≝
49 λR. ∀U1,U2. R U1 U2 → ∀f,T1. ⬆*[f] T1 ≡ U1 →
50 ∃∃T2. ⬆*[f] T2 ≡ U2 & R T1 T2.
52 definition liftable2_bi: predicate (relation term) ≝
53 λR. ∀T1,T2. R T1 T2 → ∀f,U1. ⬆*[f] T1 ≡ U1 →
54 ∀U2. ⬆*[f] T2 ≡ U2 → R U1 U2.
56 definition deliftable2_bi: predicate (relation term) ≝
57 λR. ∀U1,U2. R U1 U2 → ∀f,T1. ⬆*[f] T1 ≡ U1 →
58 ∀T2. ⬆*[f] T2 ≡ U2 → R T1 T2.
60 (* Basic inversion lemmas ***************************************************)
62 fact lifts_inv_sort1_aux: ∀f,X,Y. ⬆*[f] X ≡ Y → ∀s. X = ⋆s → Y = ⋆s.
63 #f #X #Y * -f -X -Y //
64 [ #f #i1 #i2 #_ #x #H destruct
65 | #f #p #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
66 | #f #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
70 (* Basic_1: was: lift1_sort *)
71 (* Basic_2A1: includes: lift_inv_sort1 *)
72 lemma lifts_inv_sort1: ∀f,Y,s. ⬆*[f] ⋆s ≡ Y → Y = ⋆s.
73 /2 width=4 by lifts_inv_sort1_aux/ qed-.
75 fact lifts_inv_lref1_aux: ∀f,X,Y. ⬆*[f] X ≡ Y → ∀i1. X = #i1 →
76 ∃∃i2. @⦃i1, f⦄ ≡ i2 & Y = #i2.
78 [ #f #s #x #H destruct
79 | #f #i1 #i2 #Hi12 #x #H destruct /2 width=3 by ex2_intro/
80 | #f #l #x #H destruct
81 | #f #p #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
82 | #f #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
86 (* Basic_1: was: lift1_lref *)
87 (* Basic_2A1: includes: lift_inv_lref1 lift_inv_lref1_lt lift_inv_lref1_ge *)
88 lemma lifts_inv_lref1: ∀f,Y,i1. ⬆*[f] #i1 ≡ Y →
89 ∃∃i2. @⦃i1, f⦄ ≡ i2 & Y = #i2.
90 /2 width=3 by lifts_inv_lref1_aux/ qed-.
92 fact lifts_inv_gref1_aux: ∀f,X,Y. ⬆*[f] X ≡ Y → ∀l. X = §l → Y = §l.
93 #f #X #Y * -f -X -Y //
94 [ #f #i1 #i2 #_ #x #H destruct
95 | #f #p #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
96 | #f #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
100 (* Basic_2A1: includes: lift_inv_gref1 *)
101 lemma lifts_inv_gref1: ∀f,Y,l. ⬆*[f] §l ≡ Y → Y = §l.
102 /2 width=4 by lifts_inv_gref1_aux/ qed-.
104 fact lifts_inv_bind1_aux: ∀f,X,Y. ⬆*[f] X ≡ Y →
105 ∀p,I,V1,T1. X = ⓑ{p,I}V1.T1 →
106 ∃∃V2,T2. ⬆*[f] V1 ≡ V2 & ⬆*[↑f] T1 ≡ T2 &
109 [ #f #s #q #J #W1 #U1 #H destruct
110 | #f #i1 #i2 #_ #q #J #W1 #U1 #H destruct
111 | #f #l #b #J #W1 #U1 #H destruct
112 | #f #p #I #V1 #V2 #T1 #T2 #HV12 #HT12 #q #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
113 | #f #I #V1 #V2 #T1 #T2 #_ #_ #q #J #W1 #U1 #H destruct
117 (* Basic_1: was: lift1_bind *)
118 (* Basic_2A1: includes: lift_inv_bind1 *)
119 lemma lifts_inv_bind1: ∀f,p,I,V1,T1,Y. ⬆*[f] ⓑ{p,I}V1.T1 ≡ Y →
120 ∃∃V2,T2. ⬆*[f] V1 ≡ V2 & ⬆*[↑f] T1 ≡ T2 &
122 /2 width=3 by lifts_inv_bind1_aux/ qed-.
124 fact lifts_inv_flat1_aux: ∀f:rtmap. ∀X,Y. ⬆*[f] X ≡ Y →
125 ∀I,V1,T1. X = ⓕ{I}V1.T1 →
126 ∃∃V2,T2. ⬆*[f] V1 ≡ V2 & ⬆*[f] T1 ≡ T2 &
129 [ #f #s #J #W1 #U1 #H destruct
130 | #f #i1 #i2 #_ #J #W1 #U1 #H destruct
131 | #f #l #J #W1 #U1 #H destruct
132 | #f #p #I #V1 #V2 #T1 #T2 #_ #_ #J #W1 #U1 #H destruct
133 | #f #I #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
137 (* Basic_1: was: lift1_flat *)
138 (* Basic_2A1: includes: lift_inv_flat1 *)
139 lemma lifts_inv_flat1: ∀f:rtmap. ∀I,V1,T1,Y. ⬆*[f] ⓕ{I}V1.T1 ≡ Y →
140 ∃∃V2,T2. ⬆*[f] V1 ≡ V2 & ⬆*[f] T1 ≡ T2 &
142 /2 width=3 by lifts_inv_flat1_aux/ qed-.
144 fact lifts_inv_sort2_aux: ∀f,X,Y. ⬆*[f] X ≡ Y → ∀s. Y = ⋆s → X = ⋆s.
145 #f #X #Y * -f -X -Y //
146 [ #f #i1 #i2 #_ #x #H destruct
147 | #f #p #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
148 | #f #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
152 (* Basic_1: includes: lift_gen_sort *)
153 (* Basic_2A1: includes: lift_inv_sort2 *)
154 lemma lifts_inv_sort2: ∀f,X,s. ⬆*[f] X ≡ ⋆s → X = ⋆s.
155 /2 width=4 by lifts_inv_sort2_aux/ qed-.
157 fact lifts_inv_lref2_aux: ∀f,X,Y. ⬆*[f] X ≡ Y → ∀i2. Y = #i2 →
158 ∃∃i1. @⦃i1, f⦄ ≡ i2 & X = #i1.
160 [ #f #s #x #H destruct
161 | #f #i1 #i2 #Hi12 #x #H destruct /2 width=3 by ex2_intro/
162 | #f #l #x #H destruct
163 | #f #p #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
164 | #f #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
168 (* Basic_1: includes: lift_gen_lref lift_gen_lref_lt lift_gen_lref_false lift_gen_lref_ge *)
169 (* Basic_2A1: includes: lift_inv_lref2 lift_inv_lref2_lt lift_inv_lref2_be lift_inv_lref2_ge lift_inv_lref2_plus *)
170 lemma lifts_inv_lref2: ∀f,X,i2. ⬆*[f] X ≡ #i2 →
171 ∃∃i1. @⦃i1, f⦄ ≡ i2 & X = #i1.
172 /2 width=3 by lifts_inv_lref2_aux/ qed-.
174 fact lifts_inv_gref2_aux: ∀f,X,Y. ⬆*[f] X ≡ Y → ∀l. Y = §l → X = §l.
175 #f #X #Y * -f -X -Y //
176 [ #f #i1 #i2 #_ #x #H destruct
177 | #f #p #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
178 | #f #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
182 (* Basic_2A1: includes: lift_inv_gref1 *)
183 lemma lifts_inv_gref2: ∀f,X,l. ⬆*[f] X ≡ §l → X = §l.
184 /2 width=4 by lifts_inv_gref2_aux/ qed-.
186 fact lifts_inv_bind2_aux: ∀f,X,Y. ⬆*[f] X ≡ Y →
187 ∀p,I,V2,T2. Y = ⓑ{p,I}V2.T2 →
188 ∃∃V1,T1. ⬆*[f] V1 ≡ V2 & ⬆*[↑f] T1 ≡ T2 &
191 [ #f #s #q #J #W2 #U2 #H destruct
192 | #f #i1 #i2 #_ #q #J #W2 #U2 #H destruct
193 | #f #l #q #J #W2 #U2 #H destruct
194 | #f #p #I #V1 #V2 #T1 #T2 #HV12 #HT12 #q #J #W2 #U2 #H destruct /2 width=5 by ex3_2_intro/
195 | #f #I #V1 #V2 #T1 #T2 #_ #_ #q #J #W2 #U2 #H destruct
199 (* Basic_1: includes: lift_gen_bind *)
200 (* Basic_2A1: includes: lift_inv_bind2 *)
201 lemma lifts_inv_bind2: ∀f,p,I,V2,T2,X. ⬆*[f] X ≡ ⓑ{p,I}V2.T2 →
202 ∃∃V1,T1. ⬆*[f] V1 ≡ V2 & ⬆*[↑f] T1 ≡ T2 &
204 /2 width=3 by lifts_inv_bind2_aux/ qed-.
206 fact lifts_inv_flat2_aux: ∀f:rtmap. ∀X,Y. ⬆*[f] X ≡ Y →
207 ∀I,V2,T2. Y = ⓕ{I}V2.T2 →
208 ∃∃V1,T1. ⬆*[f] V1 ≡ V2 & ⬆*[f] T1 ≡ T2 &
211 [ #f #s #J #W2 #U2 #H destruct
212 | #f #i1 #i2 #_ #J #W2 #U2 #H destruct
213 | #f #l #J #W2 #U2 #H destruct
214 | #f #p #I #V1 #V2 #T1 #T2 #_ #_ #J #W2 #U2 #H destruct
215 | #f #I #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W2 #U2 #H destruct /2 width=5 by ex3_2_intro/
219 (* Basic_1: includes: lift_gen_flat *)
220 (* Basic_2A1: includes: lift_inv_flat2 *)
221 lemma lifts_inv_flat2: ∀f:rtmap. ∀I,V2,T2,X. ⬆*[f] X ≡ ⓕ{I}V2.T2 →
222 ∃∃V1,T1. ⬆*[f] V1 ≡ V2 & ⬆*[f] T1 ≡ T2 &
224 /2 width=3 by lifts_inv_flat2_aux/ qed-.
226 (* Advanced inversion lemmas ************************************************)
228 lemma lifts_inv_atom1: ∀f,I,Y. ⬆*[f] ⓪{I} ≡ Y →
229 ∨∨ ∃∃s. I = Sort s & Y = ⋆s
230 | ∃∃i,j. @⦃i, f⦄ ≡ j & I = LRef i & Y = #j
231 | ∃∃l. I = GRef l & Y = §l.
233 [ lapply (lifts_inv_sort1 … H)
234 | elim (lifts_inv_lref1 … H)
235 | lapply (lifts_inv_gref1 … H)
236 ] -H /3 width=5 by or3_intro0, or3_intro1, or3_intro2, ex3_2_intro, ex2_intro/
239 lemma lifts_inv_atom2: ∀f,I,X. ⬆*[f] X ≡ ⓪{I} →
240 ∨∨ ∃∃s. X = ⋆s & I = Sort s
241 | ∃∃i,j. @⦃i, f⦄ ≡ j & X = #i & I = LRef j
242 | ∃∃l. X = §l & I = GRef l.
244 [ lapply (lifts_inv_sort2 … H)
245 | elim (lifts_inv_lref2 … H)
246 | lapply (lifts_inv_gref2 … H)
247 ] -H /3 width=5 by or3_intro0, or3_intro1, or3_intro2, ex3_2_intro, ex2_intro/
250 (* Basic_2A1: includes: lift_inv_pair_xy_x *)
251 lemma lifts_inv_pair_xy_x: ∀f,I,V,T. ⬆*[f] ②{I}V.T ≡ V → ⊥.
254 [ lapply (lifts_inv_sort2 … H) -H #H destruct
255 | elim (lifts_inv_lref2 … H) -H
257 | lapply (lifts_inv_gref2 … H) -H #H destruct
259 | * [ #p ] #I #V2 #T2 #IHV2 #_ #U #H
260 [ elim (lifts_inv_bind2 … H) -H #V1 #T1 #HV12 #_ #H destruct /2 width=3 by/
261 | elim (lifts_inv_flat2 … H) -H #V1 #T1 #HV12 #_ #H destruct /2 width=3 by/
266 (* Basic_1: includes: thead_x_lift_y_y *)
267 (* Basic_2A1: includes: lift_inv_pair_xy_y *)
268 lemma lifts_inv_pair_xy_y: ∀I,T,V,f. ⬆*[f] ②{I}V.T ≡ T → ⊥.
271 [ lapply (lifts_inv_sort2 … H) -H #H destruct
272 | elim (lifts_inv_lref2 … H) -H
274 | lapply (lifts_inv_gref2 … H) -H #H destruct
276 | * [ #p ] #I #V2 #T2 #_ #IHT2 #W #f #H
277 [ elim (lifts_inv_bind2 … H) -H #V1 #T1 #_ #HT12 #H destruct /2 width=4 by/
278 | elim (lifts_inv_flat2 … H) -H #V1 #T1 #_ #HT12 #H destruct /2 width=4 by/
283 (* Inversion lemmas with uniform relocations ********************************)
285 lemma lifts_inv_lref1_uni: ∀l,Y,i. ⬆*[l] #i ≡ Y → Y = #(l+i).
286 #l #Y #i1 #H elim (lifts_inv_lref1 … H) -H /4 width=4 by at_mono, eq_f/
289 lemma lifts_inv_lref2_uni: ∀l,X,i2. ⬆*[l] X ≡ #i2 →
290 ∃∃i1. X = #i1 & i2 = l + i1.
291 #l #X #i2 #H elim (lifts_inv_lref2 … H) -H
292 /3 width=3 by at_inv_uni, ex2_intro/
295 lemma lifts_inv_lref2_uni_ge: ∀l,X,i. ⬆*[l] X ≡ #(l + i) → X = #i.
296 #l #X #i2 #H elim (lifts_inv_lref2_uni … H) -H
297 #i1 #H1 #H2 destruct /4 width=2 by injective_plus_r, eq_f, sym_eq/
300 lemma lifts_inv_lref2_uni_lt: ∀l,X,i. ⬆*[l] X ≡ #i → i < l → ⊥.
301 #l #X #i2 #H elim (lifts_inv_lref2_uni … H) -H
302 #i1 #_ #H1 #H2 destruct /2 width=4 by lt_le_false/
305 (* Basic forward lemmas *****************************************************)
307 (* Basic_2A1: includes: lift_inv_O2 *)
308 lemma lifts_fwd_isid: ∀f,T1,T2. ⬆*[f] T1 ≡ T2 → 𝐈⦃f⦄ → T1 = T2.
309 #f #T1 #T2 #H elim H -f -T1 -T2
310 /4 width=3 by isid_inv_at_mono, isid_push, eq_f2, eq_f/
313 (* Basic_2A1: includes: lift_fwd_pair1 *)
314 lemma lifts_fwd_pair1: ∀f:rtmap. ∀I,V1,T1,Y. ⬆*[f] ②{I}V1.T1 ≡ Y →
315 ∃∃V2,T2. ⬆*[f] V1 ≡ V2 & Y = ②{I}V2.T2.
316 #f * [ #p ] #I #V1 #T1 #Y #H
317 [ elim (lifts_inv_bind1 … H) -H /2 width=4 by ex2_2_intro/
318 | elim (lifts_inv_flat1 … H) -H /2 width=4 by ex2_2_intro/
322 (* Basic_2A1: includes: lift_fwd_pair2 *)
323 lemma lifts_fwd_pair2: ∀f:rtmap. ∀I,V2,T2,X. ⬆*[f] X ≡ ②{I}V2.T2 →
324 ∃∃V1,T1. ⬆*[f] V1 ≡ V2 & X = ②{I}V1.T1.
325 #f * [ #p ] #I #V2 #T2 #X #H
326 [ elim (lifts_inv_bind2 … H) -H /2 width=4 by ex2_2_intro/
327 | elim (lifts_inv_flat2 … H) -H /2 width=4 by ex2_2_intro/
331 (* Basic properties *********************************************************)
333 lemma lifts_eq_repl_back: ∀T1,T2. eq_repl_back … (λf. ⬆*[f] T1 ≡ T2).
334 #T1 #T2 #f1 #H elim H -T1 -T2 -f1
335 /4 width=5 by lifts_flat, lifts_bind, lifts_lref, at_eq_repl_back, eq_push/
338 lemma lifts_eq_repl_fwd: ∀T1,T2. eq_repl_fwd … (λf. ⬆*[f] T1 ≡ T2).
339 #T1 #T2 @eq_repl_sym /2 width=3 by lifts_eq_repl_back/ (**) (* full auto fails *)
342 (* Basic_1: includes: lift_r *)
343 (* Basic_2A1: includes: lift_refl *)
344 lemma lifts_refl: ∀T,f. 𝐈⦃f⦄ → ⬆*[f] T ≡ T.
346 /4 width=3 by lifts_flat, lifts_bind, lifts_lref, isid_inv_at, isid_push/
349 (* Basic_2A1: includes: lift_total *)
350 lemma lifts_total: ∀T1,f. ∃T2. ⬆*[f] T1 ≡ T2.
352 /3 width=2 by lifts_lref, lifts_sort, lifts_gref, ex_intro/
353 [ #p ] #I #V1 #T1 #IHV1 #IHT1 #f
354 elim (IHV1 f) -IHV1 #V2 #HV12
355 [ elim (IHT1 (↑f)) -IHT1 /3 width=2 by lifts_bind, ex_intro/
356 | elim (IHT1 f) -IHT1 /3 width=2 by lifts_flat, ex_intro/
360 lemma lift_lref_uni: ∀l,i. ⬆*[l] #i ≡ #(l+i).
361 #l elim l -l /2 width=1 by lifts_lref/
364 (* Basic_1: includes: lift_free (right to left) *)
365 (* Basic_2A1: includes: lift_split *)
366 lemma lifts_split_trans: ∀f,T1,T2. ⬆*[f] T1 ≡ T2 →
367 ∀f1,f2. f2 ⊚ f1 ≡ f →
368 ∃∃T. ⬆*[f1] T1 ≡ T & ⬆*[f2] T ≡ T2.
369 #f #T1 #T2 #H elim H -f -T1 -T2
370 [ /3 width=3 by lifts_sort, ex2_intro/
371 | #f #i1 #i2 #Hi #f1 #f2 #Ht elim (after_at_fwd … Hi … Ht) -Hi -Ht
372 /3 width=3 by lifts_lref, ex2_intro/
373 | /3 width=3 by lifts_gref, ex2_intro/
374 | #f #p #I #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f1 #f2 #Ht
375 elim (IHV … Ht) elim (IHT (↑f1) (↑f2)) -IHV -IHT
376 /3 width=5 by lifts_bind, after_O2, ex2_intro/
377 | #f #I #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f1 #f2 #Ht
378 elim (IHV … Ht) elim (IHT … Ht) -IHV -IHT -Ht
379 /3 width=5 by lifts_flat, ex2_intro/
383 (* Note: apparently, this was missing in Basic_2A1 *)
384 lemma lifts_split_div: ∀f1,T1,T2. ⬆*[f1] T1 ≡ T2 →
386 ∃∃T. ⬆*[f2] T2 ≡ T & ⬆*[f] T1 ≡ T.
387 #f1 #T1 #T2 #H elim H -f1 -T1 -T2
388 [ /3 width=3 by lifts_sort, ex2_intro/
389 | #f1 #i1 #i2 #Hi #f2 #f #Ht elim (after_at1_fwd … Hi … Ht) -Hi -Ht
390 /3 width=3 by lifts_lref, ex2_intro/
391 | /3 width=3 by lifts_gref, ex2_intro/
392 | #f1 #p #I #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f2 #f #Ht
393 elim (IHV … Ht) elim (IHT (↑f2) (↑f)) -IHV -IHT
394 /3 width=5 by lifts_bind, after_O2, ex2_intro/
395 | #f1 #I #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f2 #f #Ht
396 elim (IHV … Ht) elim (IHT … Ht) -IHV -IHT -Ht
397 /3 width=5 by lifts_flat, ex2_intro/
401 (* Basic_1: includes: dnf_dec2 dnf_dec *)
402 (* Basic_2A1: includes: is_lift_dec *)
403 lemma is_lifts_dec: ∀T2,f. Decidable (∃T1. ⬆*[f] T1 ≡ T2).
405 [ * [1,3: /3 width=2 by lifts_sort, lifts_gref, ex_intro, or_introl/ ]
406 #i2 #f elim (is_at_dec f i2) //
407 [ * /4 width=3 by lifts_lref, ex_intro, or_introl/
409 #X #HX elim (lifts_inv_lref2 … HX) -HX
410 /3 width=2 by ex_intro/
412 | * [ #p ] #I #V2 #T2 #IHV2 #IHT2 #f
413 [ elim (IHV2 f) -IHV2
414 [ * #V1 #HV12 elim (IHT2 (↑f)) -IHT2
415 [ * #T1 #HT12 @or_introl /3 width=2 by lifts_bind, ex_intro/
416 | -V1 #HT2 @or_intror * #X #H
417 elim (lifts_inv_bind2 … H) -H /3 width=2 by ex_intro/
419 | -IHT2 #HV2 @or_intror * #X #H
420 elim (lifts_inv_bind2 … H) -H /3 width=2 by ex_intro/
422 | elim (IHV2 f) -IHV2
423 [ * #V1 #HV12 elim (IHT2 f) -IHT2
424 [ * #T1 #HT12 /4 width=2 by lifts_flat, ex_intro, or_introl/
425 | -V1 #HT2 @or_intror * #X #H
426 elim (lifts_inv_flat2 … H) -H /3 width=2 by ex_intro/
428 | -IHT2 #HV2 @or_intror * #X #H
429 elim (lifts_inv_flat2 … H) -H /3 width=2 by ex_intro/
435 (* Properties with uniform relocation ***************************************)
437 lemma lifts_uni: ∀n1,n2,T,U. ⬆*[𝐔❴n1❵∘𝐔❴n2❵] T ≡ U → ⬆*[n1+n2] T ≡ U.
438 /3 width=4 by lifts_eq_repl_back, after_inv_total/ qed.
440 (* Basic_2A1: removed theorems 14:
441 lifts_inv_nil lifts_inv_cons
442 lift_inv_Y1 lift_inv_Y2 lift_inv_lref_Y1 lift_inv_lref_Y2 lift_lref_Y lift_Y1
443 lift_lref_lt_eq lift_lref_ge_eq lift_lref_plus lift_lref_pred
444 lift_lref_ge_minus lift_lref_ge_minus_eq
446 (* Basic_1: removed theorems 8:
448 lift_head lift_gen_head
449 lift_weight_map lift_weight lift_weight_add lift_weight_add_O