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/arith/nat_le_plus.ma".
16 include "ground/relocation/pr_compose.ma".
17 include "ground/relocation/pr_nat_uni.ma".
18 include "ground/relocation/pr_isi_nat.ma".
19 include "ground/relocation/pr_ist_ist.ma".
20 include "ground/relocation/pr_after_uni.ma".
21 include "ground/relocation/pr_after_nat.ma".
22 include "ground/relocation/pr_after_ist.ma".
23 include "static_2/notation/relations/rliftstar_3.ma".
24 include "static_2/notation/relations/rlift_3.ma".
25 include "static_2/syntax/term.ma".
27 (* GENERIC RELOCATION FOR TERMS *********************************************)
30 lift_sort lift_lref_lt lift_lref_ge lift_bind lift_flat
33 inductive lifts: pr_map → relation term ≝
34 | lifts_sort: ∀f,s. lifts f (⋆s) (⋆s)
35 | lifts_lref: ∀f,i1,i2. @↑❨i1,f❩ ≘ i2 → lifts f (#i1) (#i2)
36 | lifts_gref: ∀f,l. lifts f (§l) (§l)
37 | lifts_bind: ∀f,p,I,V1,V2,T1,T2.
38 lifts f V1 V2 → lifts (⫯f) T1 T2 →
39 lifts f (ⓑ[p,I]V1.T1) (ⓑ[p,I]V2.T2)
40 | lifts_flat: ∀f,I,V1,V2,T1,T2.
41 lifts f V1 V2 → lifts f T1 T2 →
42 lifts f (ⓕ[I]V1.T1) (ⓕ[I]V2.T2)
45 interpretation "generic relocation (term)"
46 'RLiftStar f T1 T2 = (lifts f T1 T2).
48 interpretation "uniform relocation (term)"
49 'RLift i T1 T2 = (lifts (pr_uni i) T1 T2).
51 definition liftable2_sn: 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_sn: 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_bi: predicate (relation term) ≝
60 λR. ∀T1,T2. R T1 T2 → ∀f,U1. ⇧*[f] T1 ≘ U1 →
61 ∀U2. ⇧*[f] T2 ≘ U2 → R U1 U2.
63 definition deliftable2_bi: predicate (relation term) ≝
64 λR. ∀U1,U2. R U1 U2 → ∀f,T1. ⇧*[f] T1 ≘ U1 →
65 ∀T2. ⇧*[f] T2 ≘ U2 → R T1 T2.
67 definition liftable2_dx: predicate (relation term) ≝
68 λR. ∀T1,T2. R T1 T2 → ∀f,U2. ⇧*[f] T2 ≘ U2 →
69 ∃∃U1. ⇧*[f] T1 ≘ U1 & R U1 U2.
71 definition deliftable2_dx: predicate (relation term) ≝
72 λR. ∀U1,U2. R U1 U2 → ∀f,T2. ⇧*[f] T2 ≘ U2 →
73 ∃∃T1. ⇧*[f] T1 ≘ U1 & R T1 T2.
75 (* Basic inversion lemmas ***************************************************)
77 fact lifts_inv_sort1_aux: ∀f,X,Y. ⇧*[f] X ≘ Y → ∀s. X = ⋆s → Y = ⋆s.
78 #f #X #Y * -f -X -Y //
79 [ #f #i1 #i2 #_ #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_sort *)
86 (* Basic_2A1: includes: lift_inv_sort1 *)
87 lemma lifts_inv_sort1: ∀f,Y,s. ⇧*[f] ⋆s ≘ Y → Y = ⋆s.
88 /2 width=4 by lifts_inv_sort1_aux/ qed-.
90 fact lifts_inv_lref1_aux: ∀f,X,Y. ⇧*[f] X ≘ Y → ∀i1. X = #i1 →
91 ∃∃i2. @↑❨i1,f❩ ≘ i2 & Y = #i2.
93 [ #f #s #x #H destruct
94 | #f #i1 #i2 #Hi12 #x #H destruct /2 width=3 by ex2_intro/
95 | #f #l #x #H destruct
96 | #f #p #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
97 | #f #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
101 (* Basic_1: was: lift1_lref *)
102 (* Basic_2A1: includes: lift_inv_lref1 lift_inv_lref1_lt lift_inv_lref1_ge *)
103 lemma lifts_inv_lref1: ∀f,Y,i1. ⇧*[f] #i1 ≘ Y →
104 ∃∃i2. @↑❨i1,f❩ ≘ i2 & Y = #i2.
105 /2 width=3 by lifts_inv_lref1_aux/ qed-.
107 fact lifts_inv_gref1_aux: ∀f,X,Y. ⇧*[f] X ≘ Y → ∀l. X = §l → Y = §l.
108 #f #X #Y * -f -X -Y //
109 [ #f #i1 #i2 #_ #x #H destruct
110 | #f #p #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
111 | #f #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
115 (* Basic_2A1: includes: lift_inv_gref1 *)
116 lemma lifts_inv_gref1: ∀f,Y,l. ⇧*[f] §l ≘ Y → Y = §l.
117 /2 width=4 by lifts_inv_gref1_aux/ qed-.
119 fact lifts_inv_bind1_aux: ∀f,X,Y. ⇧*[f] X ≘ Y →
120 ∀p,I,V1,T1. X = ⓑ[p,I]V1.T1 →
121 ∃∃V2,T2. ⇧*[f] V1 ≘ V2 & ⇧*[⫯f] T1 ≘ T2 &
124 [ #f #s #q #J #W1 #U1 #H destruct
125 | #f #i1 #i2 #_ #q #J #W1 #U1 #H destruct
126 | #f #l #b #J #W1 #U1 #H destruct
127 | #f #p #I #V1 #V2 #T1 #T2 #HV12 #HT12 #q #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
128 | #f #I #V1 #V2 #T1 #T2 #_ #_ #q #J #W1 #U1 #H destruct
132 (* Basic_1: was: lift1_bind *)
133 (* Basic_2A1: includes: lift_inv_bind1 *)
134 lemma lifts_inv_bind1: ∀f,p,I,V1,T1,Y. ⇧*[f] ⓑ[p,I]V1.T1 ≘ Y →
135 ∃∃V2,T2. ⇧*[f] V1 ≘ V2 & ⇧*[⫯f] T1 ≘ T2 &
137 /2 width=3 by lifts_inv_bind1_aux/ qed-.
139 fact lifts_inv_flat1_aux: ∀f,X,Y. ⇧*[f] X ≘ Y →
140 ∀I,V1,T1. X = ⓕ[I]V1.T1 →
141 ∃∃V2,T2. ⇧*[f] V1 ≘ V2 & ⇧*[f] T1 ≘ T2 &
144 [ #f #s #J #W1 #U1 #H destruct
145 | #f #i1 #i2 #_ #J #W1 #U1 #H destruct
146 | #f #l #J #W1 #U1 #H destruct
147 | #f #p #I #V1 #V2 #T1 #T2 #_ #_ #J #W1 #U1 #H destruct
148 | #f #I #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
152 (* Basic_1: was: lift1_flat *)
153 (* Basic_2A1: includes: lift_inv_flat1 *)
154 lemma lifts_inv_flat1: ∀f,I,V1,T1,Y. ⇧*[f] ⓕ[I]V1.T1 ≘ Y →
155 ∃∃V2,T2. ⇧*[f] V1 ≘ V2 & ⇧*[f] T1 ≘ T2 &
157 /2 width=3 by lifts_inv_flat1_aux/ qed-.
159 fact lifts_inv_sort2_aux: ∀f,X,Y. ⇧*[f] X ≘ Y → ∀s. Y = ⋆s → X = ⋆s.
160 #f #X #Y * -f -X -Y //
161 [ #f #i1 #i2 #_ #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_sort *)
168 (* Basic_2A1: includes: lift_inv_sort2 *)
169 lemma lifts_inv_sort2: ∀f,X,s. ⇧*[f] X ≘ ⋆s → X = ⋆s.
170 /2 width=4 by lifts_inv_sort2_aux/ qed-.
172 fact lifts_inv_lref2_aux: ∀f,X,Y. ⇧*[f] X ≘ Y → ∀i2. Y = #i2 →
173 ∃∃i1. @↑❨i1,f❩ ≘ i2 & X = #i1.
175 [ #f #s #x #H destruct
176 | #f #i1 #i2 #Hi12 #x #H destruct /2 width=3 by ex2_intro/
177 | #f #l #x #H destruct
178 | #f #p #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
179 | #f #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
183 (* Basic_1: includes: lift_gen_lref lift_gen_lref_lt lift_gen_lref_false lift_gen_lref_ge *)
184 (* Basic_2A1: includes: lift_inv_lref2 lift_inv_lref2_lt lift_inv_lref2_be lift_inv_lref2_ge lift_inv_lref2_plus *)
185 lemma lifts_inv_lref2: ∀f,X,i2. ⇧*[f] X ≘ #i2 →
186 ∃∃i1. @↑❨i1,f❩ ≘ i2 & X = #i1.
187 /2 width=3 by lifts_inv_lref2_aux/ qed-.
189 fact lifts_inv_gref2_aux: ∀f,X,Y. ⇧*[f] X ≘ Y → ∀l. Y = §l → X = §l.
190 #f #X #Y * -f -X -Y //
191 [ #f #i1 #i2 #_ #x #H destruct
192 | #f #p #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
193 | #f #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
197 (* Basic_2A1: includes: lift_inv_gref1 *)
198 lemma lifts_inv_gref2: ∀f,X,l. ⇧*[f] X ≘ §l → X = §l.
199 /2 width=4 by lifts_inv_gref2_aux/ qed-.
201 fact lifts_inv_bind2_aux: ∀f,X,Y. ⇧*[f] X ≘ Y →
202 ∀p,I,V2,T2. Y = ⓑ[p,I]V2.T2 →
203 ∃∃V1,T1. ⇧*[f] V1 ≘ V2 & ⇧*[⫯f] T1 ≘ T2 &
206 [ #f #s #q #J #W2 #U2 #H destruct
207 | #f #i1 #i2 #_ #q #J #W2 #U2 #H destruct
208 | #f #l #q #J #W2 #U2 #H destruct
209 | #f #p #I #V1 #V2 #T1 #T2 #HV12 #HT12 #q #J #W2 #U2 #H destruct /2 width=5 by ex3_2_intro/
210 | #f #I #V1 #V2 #T1 #T2 #_ #_ #q #J #W2 #U2 #H destruct
214 (* Basic_1: includes: lift_gen_bind *)
215 (* Basic_2A1: includes: lift_inv_bind2 *)
216 lemma lifts_inv_bind2: ∀f,p,I,V2,T2,X. ⇧*[f] X ≘ ⓑ[p,I]V2.T2 →
217 ∃∃V1,T1. ⇧*[f] V1 ≘ V2 & ⇧*[⫯f] T1 ≘ T2 &
219 /2 width=3 by lifts_inv_bind2_aux/ qed-.
221 fact lifts_inv_flat2_aux: ∀f,X,Y. ⇧*[f] X ≘ Y →
222 ∀I,V2,T2. Y = ⓕ[I]V2.T2 →
223 ∃∃V1,T1. ⇧*[f] V1 ≘ V2 & ⇧*[f] T1 ≘ T2 &
226 [ #f #s #J #W2 #U2 #H destruct
227 | #f #i1 #i2 #_ #J #W2 #U2 #H destruct
228 | #f #l #J #W2 #U2 #H destruct
229 | #f #p #I #V1 #V2 #T1 #T2 #_ #_ #J #W2 #U2 #H destruct
230 | #f #I #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W2 #U2 #H destruct /2 width=5 by ex3_2_intro/
234 (* Basic_1: includes: lift_gen_flat *)
235 (* Basic_2A1: includes: lift_inv_flat2 *)
236 lemma lifts_inv_flat2: ∀f,I,V2,T2,X. ⇧*[f] X ≘ ⓕ[I]V2.T2 →
237 ∃∃V1,T1. ⇧*[f] V1 ≘ V2 & ⇧*[f] T1 ≘ T2 &
239 /2 width=3 by lifts_inv_flat2_aux/ qed-.
241 (* Advanced inversion lemmas ************************************************)
243 lemma lifts_inv_atom1: ∀f,I,Y. ⇧*[f] ⓪[I] ≘ Y →
244 ∨∨ ∃∃s. I = Sort s & Y = ⋆s
245 | ∃∃i,j. @↑❨i,f❩ ≘ j & I = LRef i & Y = #j
246 | ∃∃l. I = GRef l & Y = §l.
248 [ lapply (lifts_inv_sort1 … H)
249 | elim (lifts_inv_lref1 … H)
250 | lapply (lifts_inv_gref1 … H)
251 ] -H /3 width=5 by or3_intro0, or3_intro1, or3_intro2, ex3_2_intro, ex2_intro/
254 lemma lifts_inv_atom2: ∀f,I,X. ⇧*[f] X ≘ ⓪[I] →
255 ∨∨ ∃∃s. X = ⋆s & I = Sort s
256 | ∃∃i,j. @↑❨i,f❩ ≘ j & X = #i & I = LRef j
257 | ∃∃l. X = §l & I = GRef l.
259 [ lapply (lifts_inv_sort2 … H)
260 | elim (lifts_inv_lref2 … H)
261 | lapply (lifts_inv_gref2 … H)
262 ] -H /3 width=5 by or3_intro0, or3_intro1, or3_intro2, ex3_2_intro, ex2_intro/
265 (* Basic_2A1: includes: lift_inv_pair_xy_x *)
266 lemma lifts_inv_pair_xy_x: ∀f,I,V,T. ⇧*[f] ②[I]V.T ≘ V → ⊥.
269 [ lapply (lifts_inv_sort2 … H) -H #H destruct
270 | elim (lifts_inv_lref2 … H) -H
272 | lapply (lifts_inv_gref2 … H) -H #H destruct
274 | * [ #p ] #I #V2 #T2 #IHV2 #_ #U #H
275 [ elim (lifts_inv_bind2 … H) -H #V1 #T1 #HV12 #_ #H destruct /2 width=3 by/
276 | elim (lifts_inv_flat2 … H) -H #V1 #T1 #HV12 #_ #H destruct /2 width=3 by/
281 (* Basic_1: includes: thead_x_lift_y_y *)
282 (* Basic_2A1: includes: lift_inv_pair_xy_y *)
283 lemma lifts_inv_pair_xy_y: ∀I,T,V,f. ⇧*[f] ②[I]V.T ≘ T → ⊥.
286 [ lapply (lifts_inv_sort2 … H) -H #H destruct
287 | elim (lifts_inv_lref2 … H) -H
289 | lapply (lifts_inv_gref2 … H) -H #H destruct
291 | * [ #p ] #I #V2 #T2 #_ #IHT2 #W #f #H
292 [ elim (lifts_inv_bind2 … H) -H #V1 #T1 #_ #HT12 #H destruct /2 width=4 by/
293 | elim (lifts_inv_flat2 … H) -H #V1 #T1 #_ #HT12 #H destruct /2 width=4 by/
298 lemma lifts_inv_push_zero_sn (f):
299 ∀X. ⇧*[⫯f]#𝟎 ≘ X → #(𝟎) = X.
301 elim (lifts_inv_lref1 … H) -H #i #Hi #H destruct
302 lapply (pr_pat_inv_unit_push … Hi ???) -Hi //
305 lemma lifts_inv_push_succ_sn (f) (i1):
306 ∀X. ⇧*[⫯f]#(↑i1) ≘ X →
307 ∃∃i2. ⇧*[f]#i1 ≘ #i2 & #(↑i2) = X.
309 elim (lifts_inv_lref1 … H) -H #j #Hij #H destruct
310 elim (pr_nat_inv_succ_push … Hij) -Hij [|*: // ] #i2 #Hi12 #H destruct
311 /3 width=3 by lifts_lref, ex2_intro/
314 (* Inversion lemmas with uniform relocations ********************************)
316 lemma lifts_inv_lref1_uni: ∀l,Y,i. ⇧[l] #i ≘ Y → Y = #(l+i).
317 #l #Y #i1 #H elim (lifts_inv_lref1 … H) -H
319 /4 width=4 by pr_nat_mono, eq_f/
322 lemma lifts_inv_lref2_uni: ∀l,X,i2. ⇧[l] X ≘ #i2 →
323 ∃∃i1. X = #i1 & i1 + l = i2.
324 #l #X #i2 #H elim (lifts_inv_lref2 … H) -H
325 /3 width=3 by pr_nat_inv_uni, ex2_intro/
328 lemma lifts_inv_lref2_uni_ge: ∀l,X,i. ⇧[l] X ≘ #(i+l) → X = #i.
329 #l #X #i2 #H elim (lifts_inv_lref2_uni … H) -H
331 /4 width=2 by eq_inv_nplus_bi_dx, eq_f/
334 lemma lifts_inv_lref2_uni_lt: ∀l,X,i. ⇧[l] X ≘ #i → i < l → ⊥.
335 #l #X #i2 #H elim (lifts_inv_lref2_uni … H) -H
336 #i1 #_ #H1 #H2 destruct
337 /2 width=4 by nlt_ge_false/
340 (* Basic forward lemmas *****************************************************)
342 (* Basic_2A1: includes: lift_inv_O2 *)
343 lemma lifts_fwd_isid: ∀f,T1,T2. ⇧*[f] T1 ≘ T2 → 𝐈❨f❩ → T1 = T2.
344 #f #T1 #T2 #H elim H -f -T1 -T2
345 /4 width=3 by pr_isi_nat_des, pr_isi_push, eq_f2, eq_f/
348 (* Basic_2A1: includes: lift_fwd_pair1 *)
349 lemma lifts_fwd_pair1: ∀f,I,V1,T1,Y. ⇧*[f] ②[I]V1.T1 ≘ Y →
350 ∃∃V2,T2. ⇧*[f] V1 ≘ V2 & Y = ②[I]V2.T2.
351 #f * [ #p ] #I #V1 #T1 #Y #H
352 [ elim (lifts_inv_bind1 … H) -H /2 width=4 by ex2_2_intro/
353 | elim (lifts_inv_flat1 … H) -H /2 width=4 by ex2_2_intro/
357 (* Basic_2A1: includes: lift_fwd_pair2 *)
358 lemma lifts_fwd_pair2: ∀f,I,V2,T2,X. ⇧*[f] X ≘ ②[I]V2.T2 →
359 ∃∃V1,T1. ⇧*[f] V1 ≘ V2 & X = ②[I]V1.T1.
360 #f * [ #p ] #I #V2 #T2 #X #H
361 [ elim (lifts_inv_bind2 … H) -H /2 width=4 by ex2_2_intro/
362 | elim (lifts_inv_flat2 … H) -H /2 width=4 by ex2_2_intro/
366 (* Basic properties *********************************************************)
368 lemma liftable2_sn_dx (R): symmetric … R → liftable2_sn R → liftable2_dx R.
369 #R #H2R #H1R #T1 #T2 #HT12 #f #U2 #HTU2
370 elim (H1R … T1 … HTU2) -H1R /3 width=3 by ex2_intro/
373 lemma deliftable2_sn_dx (R): symmetric … R → deliftable2_sn R → deliftable2_dx R.
374 #R #H2R #H1R #U1 #U2 #HU12 #f #T2 #HTU2
375 elim (H1R … U1 … HTU2) -H1R /3 width=3 by ex2_intro/
378 lemma lifts_eq_repl_back: ∀T1,T2. pr_eq_repl_back … (λf. ⇧*[f] T1 ≘ T2).
379 #T1 #T2 #f1 #H elim H -T1 -T2 -f1
380 /4 width=5 by lifts_flat, lifts_bind, lifts_lref, pr_pat_eq_repl_back, pr_eq_push/
383 lemma lifts_eq_repl_fwd: ∀T1,T2. pr_eq_repl_fwd … (λf. ⇧*[f] T1 ≘ T2).
384 #T1 #T2 @pr_eq_repl_sym /2 width=3 by lifts_eq_repl_back/ (**) (* full auto fails *)
387 (* Basic_1: includes: lift_r *)
388 (* Basic_2A1: includes: lift_refl *)
389 lemma lifts_refl: ∀T,f. 𝐈❨f❩ → ⇧*[f] T ≘ T.
391 /4 width=3 by lifts_flat, lifts_bind, lifts_lref, pr_isi_inv_pat, pr_isi_push/
394 (* Basic_2A1: includes: lift_total *)
395 lemma lifts_total: ∀T1,f. 𝐓❨f❩ → ∃T2. ⇧*[f] T1 ≘ T2.
397 /3 width=2 by lifts_sort, lifts_gref, ex_intro/
398 [ #i #f #Hf elim (Hf (↑i)) -Hf /3 width=2 by ex_intro, lifts_lref/ ]
399 [ #p ] #I #V1 #T1 #IHV1 #IHT1 #f #Hf
400 elim (IHV1 f) -IHV1 // #V2 #HV12
401 [ elim (IHT1 (⫯f)) -IHT1 /3 width=2 by pr_ist_push, ex_intro, lifts_bind/
402 | elim (IHT1 f) -IHT1 /3 width=2 by lifts_flat, ex_intro/
406 lemma lifts_push_zero (f): ⇧*[⫯f]#(𝟎) ≘ #(𝟎).
407 /3 width=2 by pr_nat_refl, lifts_lref/ qed.
409 lemma lifts_push_lref (f) (i1) (i2): ⇧*[f]#i1 ≘ #i2 → ⇧*[⫯f]#(↑i1) ≘ #(↑i2).
411 elim (lifts_inv_lref1 … H) -H #j #Hij #H destruct
412 /3 width=7 by lifts_lref, pr_pat_push/
415 lemma lifts_lref_uni: ∀l,i. ⇧[l] #i ≘ #(l+i).
416 #l elim l -l /2 width=1 by lifts_lref/
419 (* Basic_1: includes: lift_free (right to left) *)
420 (* Basic_2A1: includes: lift_split *)
421 lemma lifts_split_trans: ∀f,T1,T2. ⇧*[f] T1 ≘ T2 →
422 ∀f1,f2. f2 ⊚ f1 ≘ f →
423 ∃∃T. ⇧*[f1] T1 ≘ T & ⇧*[f2] T ≘ T2.
424 #f #T1 #T2 #H elim H -f -T1 -T2
425 [ /3 width=3 by lifts_sort, ex2_intro/
426 | #f #i1 #i2 #Hi #f1 #f2 #Ht elim (pr_after_nat_des … Hi … Ht) -Hi -Ht
427 /3 width=3 by lifts_lref, ex2_intro/
428 | /3 width=3 by lifts_gref, ex2_intro/
429 | #f #p #I #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f1 #f2 #Ht
430 elim (IHV … Ht) elim (IHT (⫯f1) (⫯f2)) -IHV -IHT
431 /3 width=7 by pr_after_refl, ex2_intro, lifts_bind/
432 | #f #I #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f1 #f2 #Ht
433 elim (IHV … Ht) elim (IHT … Ht) -IHV -IHT -Ht
434 /3 width=5 by lifts_flat, ex2_intro/
438 (* Note: apparently, this was missing in Basic_2A1 *)
439 lemma lifts_split_div: ∀f1,T1,T2. ⇧*[f1] T1 ≘ T2 →
440 ∀f2. 𝐓❨f2❩ → ∀f. f2 ⊚ f1 ≘ f →
441 ∃∃T. ⇧*[f2] T2 ≘ T & ⇧*[f] T1 ≘ T.
442 #f1 #T1 #T2 #H elim H -f1 -T1 -T2
443 [ /3 width=3 by lifts_sort, ex2_intro/
444 | #f1 #i1 #i2 #Hi #f2 #Hf2 #f #Ht
445 elim (pr_after_des_ist_nat … Hi … Ht) -Hi -Ht
446 /3 width=3 by lifts_lref, ex2_intro/
447 | /3 width=3 by lifts_gref, ex2_intro/
448 | #f1 #p #I #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f2 #Hf2 #f #Ht
449 elim (IHV … Ht) elim (IHT (⫯f2) … (⫯f)) -IHV -IHT
450 /3 width=7 by pr_ist_push, pr_after_refl, ex2_intro, lifts_bind/
451 | #f1 #I #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f2 #Hf2 #f #Ht
452 elim (IHV … Ht) elim (IHT … Ht) -IHV -IHT -Ht
453 /3 width=5 by lifts_flat, ex2_intro/
457 (* Basic_1: includes: dnf_dec2 dnf_dec *)
458 (* Basic_2A1: includes: is_lift_dec *)
459 lemma is_lifts_dec: ∀T2,f. 𝐓❨f❩ → Decidable (∃T1. ⇧*[f] T1 ≘ T2).
461 [ * [1,3: /3 width=2 by lifts_sort, lifts_gref, ex_intro, or_introl/ ]
462 #i2 #f #Hf elim (is_pr_nat_dec f i2) //
463 [ * /4 width=3 by lifts_lref, ex_intro, or_introl/
465 #X #HX elim (lifts_inv_lref2 … HX) -HX
466 /3 width=2 by ex_intro/
468 | * [ #p ] #I #V2 #T2 #IHV2 #IHT2 #f #Hf
469 [ elim (IHV2 f) -IHV2 //
470 [ * #V1 #HV12 elim (IHT2 (⫯f)) -IHT2 /2 width=1 by pr_ist_push/
471 [ * #T1 #HT12 @or_introl /3 width=2 by lifts_bind, ex_intro/
472 | -V1 #HT2 @or_intror * #X #H
473 elim (lifts_inv_bind2 … H) -H /3 width=2 by ex_intro/
475 | -IHT2 #HV2 @or_intror * #X #H
476 elim (lifts_inv_bind2 … H) -H /3 width=2 by ex_intro/
478 | elim (IHV2 f) -IHV2 //
479 [ * #V1 #HV12 elim (IHT2 f) -IHT2 //
480 [ * #T1 #HT12 /4 width=2 by lifts_flat, ex_intro, or_introl/
481 | -V1 #HT2 @or_intror * #X #H
482 elim (lifts_inv_flat2 … H) -H /3 width=2 by ex_intro/
484 | -IHT2 #HV2 @or_intror * #X #H
485 elim (lifts_inv_flat2 … H) -H /3 width=2 by ex_intro/
491 (* Properties with uniform relocation ***************************************)
493 lemma lifts_uni: ∀n1,n2,T,U. ⇧*[𝐮❨n2❩∘𝐮❨n1❩] T ≘ U → ⇧[n1+n2] T ≘ U.
494 /3 width=4 by lifts_eq_repl_back, pr_after_inv_total/ qed.
496 (* Basic_2A1: removed theorems 14:
497 lifts_inv_nil lifts_inv_cons
498 lift_inv_Y1 lift_inv_Y2 lift_inv_lref_Y1 lift_inv_lref_Y2 lift_lref_Y lift_Y1
499 lift_lref_lt_eq lift_lref_ge_eq lift_lref_plus lift_lref_pred
500 lift_lref_ge_minus lift_lref_ge_minus_eq
502 (* Basic_1: removed theorems 8:
504 lift_head lift_gen_head
505 lift_weight_map lift_weight lift_weight_add lift_weight_add_O