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/ynat/ynat_plus.ma".
16 include "basic_2/notation/relations/rlift_4.ma".
17 include "basic_2/grammar/term_weight.ma".
18 include "basic_2/grammar/term_simple.ma".
20 (* BASIC TERM RELOCATION ****************************************************)
23 lift_sort lift_lref_lt lift_lref_ge lift_bind lift_flat
25 inductive lift: relation4 ynat nat term term ≝
26 | lift_sort : ∀k,l,m. lift l m (⋆k) (⋆k)
27 | lift_lref_lt: ∀i,l,m. yinj i < l → lift l m (#i) (#i)
28 | lift_lref_ge: ∀i,l,m. l ≤ yinj i → lift l m (#i) (#(i + m))
29 | lift_gref : ∀p,l,m. lift l m (§p) (§p)
30 | lift_bind : ∀a,I,V1,V2,T1,T2,l,m.
31 lift l m V1 V2 → lift (⫯l) m T1 T2 →
32 lift l m (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2)
33 | lift_flat : ∀I,V1,V2,T1,T2,l,m.
34 lift l m V1 V2 → lift l m T1 T2 →
35 lift l m (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
38 interpretation "relocation" 'RLift l m T1 T2 = (lift l m T1 T2).
40 (* Basic inversion lemmas ***************************************************)
42 fact lift_inv_O2_aux: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 → m = 0 → T1 = T2.
43 #l #m #T1 #T2 #H elim H -l -m -T1 -T2 /3 width=1 by eq_f2/
46 lemma lift_inv_O2: ∀l,T1,T2. ⬆[l, 0] T1 ≡ T2 → T1 = T2.
47 /2 width=4 by lift_inv_O2_aux/ qed-.
49 fact lift_inv_sort1_aux: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 → ∀k. T1 = ⋆k → T2 = ⋆k.
50 #l #m #T1 #T2 * -l -m -T1 -T2 //
51 [ #i #l #m #_ #k #H destruct
52 | #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
53 | #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
57 lemma lift_inv_sort1: ∀l,m,T2,k. ⬆[l, m] ⋆k ≡ T2 → T2 = ⋆k.
58 /2 width=5 by lift_inv_sort1_aux/ qed-.
60 fact lift_inv_lref1_aux: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 → ∀i. T1 = #i →
61 (i < l ∧ T2 = #i) ∨ (l ≤ i ∧ T2 = #(i + m)).
62 #l #m #T1 #T2 * -l -m -T1 -T2
63 [ #k #l #m #i #H destruct
64 | #j #l #m #Hj #i #Hi destruct /3 width=1 by or_introl,conj/
65 | #j #l #m #Hj #i #Hi destruct /3 width=1 by or_intror,conj/
66 | #p #l #m #i #H destruct
67 | #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #i #H destruct
68 | #I #V1 #V2 #T1 #T2 #l #m #_ #_ #i #H destruct
72 lemma lift_inv_lref1: ∀l,m,T2,i. ⬆[l, m] #i ≡ T2 →
73 (i < l ∧ T2 = #i) ∨ (l ≤ i ∧ T2 = #(i + m)).
74 /2 width=3 by lift_inv_lref1_aux/ qed-.
76 lemma lift_inv_lref1_lt: ∀l,m,T2,i. ⬆[l, m] #i ≡ T2 → i < l → T2 = #i.
77 #l #m #T2 #i #H elim (lift_inv_lref1 … H) -H * //
78 #Hli #_ #Hil elim (ylt_yle_false … Hli) -Hli //
81 lemma lift_inv_lref1_ge: ∀l,m,T2,i. ⬆[l, m] #i ≡ T2 → l ≤ i → T2 = #(i + m).
82 #l #m #T2 #i #H elim (lift_inv_lref1 … H) -H * //
83 #Hil #_ #Hli elim (ylt_yle_false … Hli) -Hli //
86 fact lift_inv_gref1_aux: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 → ∀p. T1 = §p → T2 = §p.
87 #l #m #T1 #T2 * -l -m -T1 -T2 //
88 [ #i #l #m #_ #k #H destruct
89 | #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
90 | #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
94 lemma lift_inv_gref1: ∀l,m,T2,p. ⬆[l, m] §p ≡ T2 → T2 = §p.
95 /2 width=5 by lift_inv_gref1_aux/ qed-.
97 fact lift_inv_bind1_aux: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 →
98 ∀a,I,V1,U1. T1 = ⓑ{a,I}V1.U1 →
99 ∃∃V2,U2. ⬆[l, m] V1 ≡ V2 & ⬆[⫯l, m] U1 ≡ U2 &
101 #l #m #T1 #T2 * -l -m -T1 -T2
102 [ #k #l #m #a #I #V1 #U1 #H destruct
103 | #i #l #m #_ #a #I #V1 #U1 #H destruct
104 | #i #l #m #_ #a #I #V1 #U1 #H destruct
105 | #p #l #m #a #I #V1 #U1 #H destruct
106 | #b #J #W1 #W2 #T1 #T2 #l #m #HW #HT #a #I #V1 #U1 #H destruct /2 width=5 by ex3_2_intro/
107 | #J #W1 #W2 #T1 #T2 #l #m #_ #HT #a #I #V1 #U1 #H destruct
111 lemma lift_inv_bind1: ∀l,m,T2,a,I,V1,U1. ⬆[l, m] ⓑ{a,I}V1.U1 ≡ T2 →
112 ∃∃V2,U2. ⬆[l, m] V1 ≡ V2 & ⬆[⫯l, m] U1 ≡ U2 &
114 /2 width=3 by lift_inv_bind1_aux/ qed-.
116 fact lift_inv_flat1_aux: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 →
117 ∀I,V1,U1. T1 = ⓕ{I}V1.U1 →
118 ∃∃V2,U2. ⬆[l, m] V1 ≡ V2 & ⬆[l, m] U1 ≡ U2 &
120 #l #m #T1 #T2 * -l -m -T1 -T2
121 [ #k #l #m #I #V1 #U1 #H destruct
122 | #i #l #m #_ #I #V1 #U1 #H destruct
123 | #i #l #m #_ #I #V1 #U1 #H destruct
124 | #p #l #m #I #V1 #U1 #H destruct
125 | #a #J #W1 #W2 #T1 #T2 #l #m #_ #_ #I #V1 #U1 #H destruct
126 | #J #W1 #W2 #T1 #T2 #l #m #HW #HT #I #V1 #U1 #H destruct /2 width=5 by ex3_2_intro/
130 lemma lift_inv_flat1: ∀l,m,T2,I,V1,U1. ⬆[l, m] ⓕ{I}V1.U1 ≡ T2 →
131 ∃∃V2,U2. ⬆[l, m] V1 ≡ V2 & ⬆[l, m] U1 ≡ U2 &
133 /2 width=3 by lift_inv_flat1_aux/ qed-.
135 fact lift_inv_sort2_aux: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 → ∀k. T2 = ⋆k → T1 = ⋆k.
136 #l #m #T1 #T2 * -l -m -T1 -T2 //
137 [ #i #l #m #_ #k #H destruct
138 | #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
139 | #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
143 (* Basic_1: was: lift_gen_sort *)
144 lemma lift_inv_sort2: ∀l,m,T1,k. ⬆[l, m] T1 ≡ ⋆k → T1 = ⋆k.
145 /2 width=5 by lift_inv_sort2_aux/ qed-.
147 fact lift_inv_lref2_aux: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 → ∀i. T2 = #i →
148 (i < l ∧ T1 = #i) ∨ (l + m ≤ i ∧ T1 = #(i - m)).
149 #l #m #T1 #T2 * -l -m -T1 -T2
150 [ #k #l #m #i #H destruct
151 | #j #l #m #Hj #i #Hi destruct /3 width=1 by or_introl, conj/
152 | #j #l #m #Hj #i #Hi destruct <minus_plus_m_m /4 width=1 by monotonic_yle_plus_dx, or_intror, conj/
153 | #p #l #m #i #H destruct
154 | #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #i #H destruct
155 | #I #V1 #V2 #T1 #T2 #l #m #_ #_ #i #H destruct
159 (* Basic_1: was: lift_gen_lref *)
160 lemma lift_inv_lref2: ∀l,m,T1,i. ⬆[l, m] T1 ≡ #i →
161 (i < l ∧ T1 = #i) ∨ (l + m ≤ i ∧ T1 = #(i - m)).
162 /2 width=3 by lift_inv_lref2_aux/ qed-.
164 (* Basic_1: was: lift_gen_lref_lt *)
165 lemma lift_inv_lref2_lt: ∀l,m,T1,i. ⬆[l, m] T1 ≡ #i → i < l → T1 = #i.
166 #l #m #T1 #i #H elim (lift_inv_lref2 … H) -H * //
167 #H #_ #Hil lapply (yle_fwd_plus_sn1 … H) -H
168 #Hli elim (ylt_yle_false … Hli) -Hli //
171 (* Basic_1: was: lift_gen_lref_false *)
172 lemma lift_inv_lref2_be: ∀l,m,T1,i. ⬆[l, m] T1 ≡ #i →
173 l ≤ i → i < l + m → ⊥.
174 #l #m #T1 #i #H elim (lift_inv_lref2 … H) -H *
175 [ #H1 #_ #H2 #_ | #H2 #_ #_ #H1 ]
176 elim (ylt_yle_false … H2) -H2 //
179 (* Basic_1: was: lift_gen_lref_ge *)
180 lemma lift_inv_lref2_ge: ∀l,m,T1,i. ⬆[l, m] T1 ≡ #i → l + m ≤ i → T1 = #(i - m).
181 #l #m #T1 #i #H elim (lift_inv_lref2 … H) -H * //
182 #Hil #_ #H lapply (yle_fwd_plus_sn1 … H) -H
183 #Hli elim (ylt_yle_false … Hli) -Hli //
186 fact lift_inv_gref2_aux: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 → ∀p. T2 = §p → T1 = §p.
187 #l #m #T1 #T2 * -l -m -T1 -T2 //
188 [ #i #l #m #_ #k #H destruct
189 | #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
190 | #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
194 lemma lift_inv_gref2: ∀l,m,T1,p. ⬆[l, m] T1 ≡ §p → T1 = §p.
195 /2 width=5 by lift_inv_gref2_aux/ qed-.
197 fact lift_inv_bind2_aux: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 →
198 ∀a,I,V2,U2. T2 = ⓑ{a,I}V2.U2 →
199 ∃∃V1,U1. ⬆[l, m] V1 ≡ V2 & ⬆[⫯l, m] U1 ≡ U2 &
201 #l #m #T1 #T2 * -l -m -T1 -T2
202 [ #k #l #m #a #I #V2 #U2 #H destruct
203 | #i #l #m #_ #a #I #V2 #U2 #H destruct
204 | #i #l #m #_ #a #I #V2 #U2 #H destruct
205 | #p #l #m #a #I #V2 #U2 #H destruct
206 | #b #J #W1 #W2 #T1 #T2 #l #m #HW #HT #a #I #V2 #U2 #H destruct /2 width=5 by ex3_2_intro/
207 | #J #W1 #W2 #T1 #T2 #l #m #_ #_ #a #I #V2 #U2 #H destruct
211 (* Basic_1: was: lift_gen_bind *)
212 lemma lift_inv_bind2: ∀l,m,T1,a,I,V2,U2. ⬆[l, m] T1 ≡ ⓑ{a,I}V2.U2 →
213 ∃∃V1,U1. ⬆[l, m] V1 ≡ V2 & ⬆[⫯l, m] U1 ≡ U2 &
215 /2 width=3 by lift_inv_bind2_aux/ qed-.
217 fact lift_inv_flat2_aux: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 →
218 ∀I,V2,U2. T2 = ⓕ{I}V2.U2 →
219 ∃∃V1,U1. ⬆[l, m] V1 ≡ V2 & ⬆[l, m] U1 ≡ U2 &
221 #l #m #T1 #T2 * -l -m -T1 -T2
222 [ #k #l #m #I #V2 #U2 #H destruct
223 | #i #l #m #_ #I #V2 #U2 #H destruct
224 | #i #l #m #_ #I #V2 #U2 #H destruct
225 | #p #l #m #I #V2 #U2 #H destruct
226 | #a #J #W1 #W2 #T1 #T2 #l #m #_ #_ #I #V2 #U2 #H destruct
227 | #J #W1 #W2 #T1 #T2 #l #m #HW #HT #I #V2 #U2 #H destruct /2 width=5 by ex3_2_intro/
231 (* Basic_1: was: lift_gen_flat *)
232 lemma lift_inv_flat2: ∀l,m,T1,I,V2,U2. ⬆[l, m] T1 ≡ ⓕ{I}V2.U2 →
233 ∃∃V1,U1. ⬆[l, m] V1 ≡ V2 & ⬆[l, m] U1 ≡ U2 &
235 /2 width=3 by lift_inv_flat2_aux/ qed-.
237 lemma lift_inv_Y1: ∀T1,T2,m. ⬆[∞, m] T1 ≡ T2 → T1 = T2.
239 [ #k #X #m #H lapply (lift_inv_sort1 … H) -H //
240 | #i #X #m #H lapply (lift_inv_lref1_lt … H ?) -H //
241 | #p #X #m #H lapply (lift_inv_gref1 … H) -H //
242 | #a #I #V1 #T1 #IHV1 #IHT1 #X #m #H elim (lift_inv_bind1 … H) -H
243 #V2 #T2 #HV12 #HT12 #H destruct /3 width=2 by eq_f2/
244 | #I #V1 #T1 #IHV1 #IHT1 #X #m #H elim (lift_inv_flat1 … H) -H
245 #V2 #T2 #HV12 #HT12 #H destruct /3 width=2 by eq_f2/
249 lemma lift_inv_pair_xy_x: ∀l,m,I,V,T. ⬆[l, m] ②{I}V.T ≡ V → ⊥.
250 #l #m #J #V elim V -V
252 [ lapply (lift_inv_sort2 … H) -H #H destruct
253 | elim (lift_inv_lref2 … H) -H * #_ #H destruct
254 | lapply (lift_inv_gref2 … H) -H #H destruct
256 | * [ #a ] #I #W2 #U2 #IHW2 #_ #T #H
257 [ elim (lift_inv_bind2 … H) -H #W1 #U1 #HW12 #_ #H destruct /2 width=2 by/
258 | elim (lift_inv_flat2 … H) -H #W1 #U1 #HW12 #_ #H destruct /2 width=2 by/
263 (* Basic_1: was: thead_x_lift_y_y *)
264 lemma lift_inv_pair_xy_y: ∀I,T,V,l,m. ⬆[l, m] ②{I}V.T ≡ T → ⊥.
267 [ lapply (lift_inv_sort2 … H) -H #H destruct
268 | elim (lift_inv_lref2 … H) -H * #_ #H destruct
269 | lapply (lift_inv_gref2 … H) -H #H destruct
271 | * [ #a ] #I #W2 #U2 #_ #IHU2 #V #l #m #H
272 [ elim (lift_inv_bind2 … H) -H #W1 #U1 #_ #HU12 #H destruct /2 width=4 by/
273 | elim (lift_inv_flat2 … H) -H #W1 #U1 #_ #HU12 #H destruct /2 width=4 by/
278 (* Basic forward lemmas *****************************************************)
280 lemma lift_fwd_pair1: ∀I,T2,V1,U1,l,m. ⬆[l, m] ②{I}V1.U1 ≡ T2 →
281 ∃∃V2,U2. ⬆[l, m] V1 ≡ V2 & T2 = ②{I}V2.U2.
282 * [ #a ] #I #T2 #V1 #U1 #l #m #H
283 [ elim (lift_inv_bind1 … H) -H /2 width=4 by ex2_2_intro/
284 | elim (lift_inv_flat1 … H) -H /2 width=4 by ex2_2_intro/
288 lemma lift_fwd_pair2: ∀I,T1,V2,U2,l,m. ⬆[l, m] T1 ≡ ②{I}V2.U2 →
289 ∃∃V1,U1. ⬆[l, m] V1 ≡ V2 & T1 = ②{I}V1.U1.
290 * [ #a ] #I #T1 #V2 #U2 #l #m #H
291 [ elim (lift_inv_bind2 … H) -H /2 width=4 by ex2_2_intro/
292 | elim (lift_inv_flat2 … H) -H /2 width=4 by ex2_2_intro/
296 lemma lift_fwd_tw: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 → ♯{T1} = ♯{T2}.
297 #l #m #T1 #T2 #H elim H -l -m -T1 -T2 normalize //
300 lemma lift_simple_dx: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 → 𝐒⦃T1⦄ → 𝐒⦃T2⦄.
301 #l #m #T1 #T2 #H elim H -l -m -T1 -T2 //
302 #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #_ #_ #H
303 elim (simple_inv_bind … H)
306 lemma lift_simple_sn: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 → 𝐒⦃T2⦄ → 𝐒⦃T1⦄.
307 #l #m #T1 #T2 #H elim H -l -m -T1 -T2 //
308 #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #_ #_ #H
309 elim (simple_inv_bind … H)
312 (* Basic properties *********************************************************)
314 (* Basic_1: was: lift_lref_gt *)
315 lemma lift_lref_ge_minus: ∀l,m,i. l + yinj m ≤ yinj i → ⬆[l, m] #(i - m) ≡ #i.
316 #l #m #i #H >(plus_minus_m_m i m) in ⊢ (? ? ? ? %);
317 elim (yle_inv_plus_inj2 … H) -H #Hlim #H
318 lapply (yle_inv_inj … H) -H /2 width=1 by lift_lref_ge/
321 lemma lift_lref_ge_minus_eq: ∀l,m,i,j. l + yinj m ≤ yinj i → j = i - m → ⬆[l, m] #j ≡ #i.
322 /2 width=1 by lift_lref_ge_minus/ qed-.
324 (* Basic_1: was: lift_r *)
325 lemma lift_refl: ∀T,l. ⬆[l, 0] T ≡ T.
327 [ * #i // #l elim (ylt_split i l) /2 width=1 by lift_lref_lt, lift_lref_ge/
328 | * /2 width=1 by lift_bind, lift_flat/
332 (* Basic_2b: first lemma *)
333 lemma lift_Y1: ∀T,m. ⬆[∞, m] T ≡ T.
334 #T elim T -T * /2 width=1 by lift_lref_lt, lift_bind, lift_flat/
337 lemma lift_total: ∀T1,l,m. ∃T2. ⬆[l, m] T1 ≡ T2.
339 [ * #i /2 width=2 by lift_sort, lift_gref, ex_intro/
340 #l #m elim (ylt_split i l) /3 width=2 by lift_lref_lt, lift_lref_ge, ex_intro/
341 | * [ #a ] #I #V1 #T1 #IHV1 #IHT1 #l #m
342 elim (IHV1 l m) -IHV1 #V2 #HV12
343 [ elim (IHT1 (⫯l) m) -IHT1 /3 width=2 by lift_bind, ex_intro/
344 | elim (IHT1 l m) -IHT1 /3 width=2 by lift_flat, ex_intro/
349 (* Basic_1: was: lift_free (right to left) *)
350 lemma lift_split: ∀l1,m2,T1,T2. ⬆[l1, m2] T1 ≡ T2 →
351 ∀l2,m1. l1 ≤ l2 → l2 ≤ l1 + yinj m1 → m1 ≤ m2 →
352 ∃∃T. ⬆[l1, m1] T1 ≡ T & ⬆[l2, m2 - m1] T ≡ T2.
353 #l1 #m2 #T1 #T2 #H elim H -l1 -m2 -T1 -T2
354 [ /3 width=3 by lift_sort, ex2_intro/
355 | #i #l1 #m2 #Hil1 #l2 #m1 #Hl12 #_ #_
356 lapply (ylt_yle_trans … Hl12 Hil1) -Hl12 #Hil2 /4 width=3 by lift_lref_lt, ex2_intro/
357 | #i #l1 #m2 #Hil1 #l2 #m1 #_ #Hl21 #Hm12
358 lapply (yle_trans … Hl21 (i+m1) ?) /2 width=1 by monotonic_yle_plus_dx/ -Hl21 #Hl21
359 >(plus_minus_m_m m2 m1 ?) /3 width=3 by lift_lref_ge, ex2_intro/
360 | /3 width=3 by lift_gref, ex2_intro/
361 | #a #I #V1 #V2 #T1 #T2 #l1 #m2 #_ #_ #IHV #IHT #l2 #m1 #Hl12 #Hl21 #Hm12
362 elim (IHV … Hl12 Hl21 Hm12) -IHV #V0 #HV0a #HV0b
363 elim (IHT (⫯l2) … ? ? Hm12) /3 width=5 by lift_bind, yle_succ, ex2_intro/
364 | #I #V1 #V2 #T1 #T2 #l1 #m2 #_ #_ #IHV #IHT #l2 #m1 #Hl12 #Hl21 #Hm12
365 elim (IHV … Hl12 Hl21 Hm12) -IHV #V0 #HV0a #HV0b
366 elim (IHT l2 … ? ? Hm12) /3 width=5 by lift_flat, ex2_intro/
370 (* Basic_1: was only: dnf_dec2 dnf_dec *)
371 lemma is_lift_dec: ∀T2,l,m. Decidable (∃T1. ⬆[l, m] T1 ≡ T2).
373 [ * [1,3: /3 width=2 by lift_sort, lift_gref, ex_intro, or_introl/ ] #i #l #m
374 elim (ylt_split i l) #Hli
375 [ /4 width=3 by lift_lref_lt, ex_intro, or_introl/
376 | elim (ylt_split i (l + m)) #Hilm
377 [ @or_intror * #T1 #H elim (lift_inv_lref2_be … H Hli Hilm)
378 | -Hli /4 width=2 by lift_lref_ge_minus, ex_intro, or_introl/
381 | * [ #a ] #I #V2 #T2 #IHV2 #IHT2 #l #m
382 [ elim (IHV2 l m) -IHV2
383 [ * #V1 #HV12 elim (IHT2 (⫯l) m) -IHT2
384 [ * #T1 #HT12 @or_introl /3 width=2 by lift_bind, ex_intro/
385 | -V1 #HT2 @or_intror * #X #H
386 elim (lift_inv_bind2 … H) -H /3 width=2 by ex_intro/
388 | -IHT2 #HV2 @or_intror * #X #H
389 elim (lift_inv_bind2 … H) -H /3 width=2 by ex_intro/
391 | elim (IHV2 l m) -IHV2
392 [ * #V1 #HV12 elim (IHT2 l m) -IHT2
393 [ * #T1 #HT12 /4 width=2 by lift_flat, ex_intro, or_introl/
394 | -V1 #HT2 @or_intror * #X #H
395 elim (lift_inv_flat2 … H) -H /3 width=2 by ex_intro/
397 | -IHT2 #HV2 @or_intror * #X #H
398 elim (lift_inv_flat2 … H) -H /3 width=2 by ex_intro/
404 (* Basic_1: removed theorems 7:
405 lift_head lift_gen_head
406 lift_weight_map lift_weight lift_weight_add lift_weight_add_O