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/xoa/ex_3_2.ma".
16 include "basic_2A/notation/relations/rlift_4.ma".
17 include "basic_2A/grammar/term_weight.ma".
18 include "basic_2A/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 nat nat term term ≝
26 | lift_sort : ∀k,l,m. lift l m (⋆k) (⋆k)
27 | lift_lref_lt: ∀i,l,m. i < l → lift l m (#i) (#i)
28 | lift_lref_ge: ∀i,l,m. l ≤ 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 + 1) 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 lapply (le_to_lt_to_lt … Hli Hil) -Hli -Hil #Hll
79 elim (lt_refl_false … Hll)
82 lemma lift_inv_lref1_ge: ∀l,m,T2,i. ⬆[l,m] #i ≡ T2 → l ≤ i → T2 = #(i + m).
83 #l #m #T2 #i #H elim (lift_inv_lref1 … H) -H * //
84 #Hil #_ #Hli lapply (le_to_lt_to_lt … Hli Hil) -Hli -Hil #Hll
85 elim (lt_refl_false … Hll)
88 fact lift_inv_gref1_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 → ∀p. T1 = §p → T2 = §p.
89 #l #m #T1 #T2 * -l -m -T1 -T2 //
90 [ #i #l #m #_ #k #H destruct
91 | #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
92 | #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
96 lemma lift_inv_gref1: ∀l,m,T2,p. ⬆[l,m] §p ≡ T2 → T2 = §p.
97 /2 width=5 by lift_inv_gref1_aux/ qed-.
99 fact lift_inv_bind1_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 →
100 ∀a,I,V1,U1. T1 = ⓑ{a,I} V1.U1 →
101 ∃∃V2,U2. ⬆[l,m] V1 ≡ V2 & ⬆[l+1,m] U1 ≡ U2 &
103 #l #m #T1 #T2 * -l -m -T1 -T2
104 [ #k #l #m #a #I #V1 #U1 #H destruct
105 | #i #l #m #_ #a #I #V1 #U1 #H destruct
106 | #i #l #m #_ #a #I #V1 #U1 #H destruct
107 | #p #l #m #a #I #V1 #U1 #H destruct
108 | #b #J #W1 #W2 #T1 #T2 #l #m #HW #HT #a #I #V1 #U1 #H destruct /2 width=5 by ex3_2_intro/
109 | #J #W1 #W2 #T1 #T2 #l #m #_ #HT #a #I #V1 #U1 #H destruct
113 lemma lift_inv_bind1: ∀l,m,T2,a,I,V1,U1. ⬆[l,m] ⓑ{a,I} V1. U1 ≡ T2 →
114 ∃∃V2,U2. ⬆[l,m] V1 ≡ V2 & ⬆[l+1,m] U1 ≡ U2 &
116 /2 width=3 by lift_inv_bind1_aux/ qed-.
118 fact lift_inv_flat1_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 →
119 ∀I,V1,U1. T1 = ⓕ{I} V1.U1 →
120 ∃∃V2,U2. ⬆[l,m] V1 ≡ V2 & ⬆[l,m] U1 ≡ U2 &
122 #l #m #T1 #T2 * -l -m -T1 -T2
123 [ #k #l #m #I #V1 #U1 #H destruct
124 | #i #l #m #_ #I #V1 #U1 #H destruct
125 | #i #l #m #_ #I #V1 #U1 #H destruct
126 | #p #l #m #I #V1 #U1 #H destruct
127 | #a #J #W1 #W2 #T1 #T2 #l #m #_ #_ #I #V1 #U1 #H destruct
128 | #J #W1 #W2 #T1 #T2 #l #m #HW #HT #I #V1 #U1 #H destruct /2 width=5 by ex3_2_intro/
132 lemma lift_inv_flat1: ∀l,m,T2,I,V1,U1. ⬆[l,m] ⓕ{I} V1. U1 ≡ T2 →
133 ∃∃V2,U2. ⬆[l,m] V1 ≡ V2 & ⬆[l,m] U1 ≡ U2 &
135 /2 width=3 by lift_inv_flat1_aux/ qed-.
137 fact lift_inv_sort2_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 → ∀k. T2 = ⋆k → T1 = ⋆k.
138 #l #m #T1 #T2 * -l -m -T1 -T2 //
139 [ #i #l #m #_ #k #H destruct
140 | #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
141 | #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
145 (* Basic_1: was: lift_gen_sort *)
146 lemma lift_inv_sort2: ∀l,m,T1,k. ⬆[l,m] T1 ≡ ⋆k → T1 = ⋆k.
147 /2 width=5 by lift_inv_sort2_aux/ qed-.
149 fact lift_inv_lref2_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 → ∀i. T2 = #i →
150 (i < l ∧ T1 = #i) ∨ (l + m ≤ i ∧ T1 = #(i - m)).
151 #l #m #T1 #T2 * -l -m -T1 -T2
152 [ #k #l #m #i #H destruct
153 | #j #l #m #Hj #i #Hi destruct /3 width=1 by or_introl, conj/
154 | #j #l #m #Hj #i #Hi destruct <minus_plus_m_m /4 width=1 by monotonic_le_plus_l, or_intror, conj/
155 | #p #l #m #i #H destruct
156 | #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #i #H destruct
157 | #I #V1 #V2 #T1 #T2 #l #m #_ #_ #i #H destruct
161 (* Basic_1: was: lift_gen_lref *)
162 lemma lift_inv_lref2: ∀l,m,T1,i. ⬆[l,m] T1 ≡ #i →
163 (i < l ∧ T1 = #i) ∨ (l + m ≤ i ∧ T1 = #(i - m)).
164 /2 width=3 by lift_inv_lref2_aux/ qed-.
166 (* Basic_1: was: lift_gen_lref_lt *)
167 lemma lift_inv_lref2_lt: ∀l,m,T1,i. ⬆[l,m] T1 ≡ #i → i < l → T1 = #i.
168 #l #m #T1 #i #H elim (lift_inv_lref2 … H) -H * //
169 #Hli #_ #Hil lapply (le_to_lt_to_lt … Hli Hil) -Hli -Hil #Hll
170 elim (lt_inv_plus_l … Hll) -Hll #Hll
171 elim (lt_refl_false … Hll)
174 (* Basic_1: was: lift_gen_lref_false *)
175 lemma lift_inv_lref2_be: ∀l,m,T1,i. ⬆[l,m] T1 ≡ #i →
176 l ≤ i → i < l + m → ⊥.
177 #l #m #T1 #i #H elim (lift_inv_lref2 … H) -H *
178 [ #H1 #_ #H2 #_ | #H2 #_ #_ #H1 ]
179 lapply (le_to_lt_to_lt … H2 H1) -H2 -H1 #H
180 elim (lt_refl_false … H)
183 (* Basic_1: was: lift_gen_lref_ge *)
184 lemma lift_inv_lref2_ge: ∀l,m,T1,i. ⬆[l,m] T1 ≡ #i → l + m ≤ i → T1 = #(i - m).
185 #l #m #T1 #i #H elim (lift_inv_lref2 … H) -H * //
186 #Hil #_ #Hli lapply (le_to_lt_to_lt … Hli Hil) -Hli -Hil #Hll
187 elim (lt_inv_plus_l … Hll) -Hll #Hll
188 elim (lt_refl_false … Hll)
191 fact lift_inv_gref2_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 → ∀p. T2 = §p → T1 = §p.
192 #l #m #T1 #T2 * -l -m -T1 -T2 //
193 [ #i #l #m #_ #k #H destruct
194 | #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
195 | #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
199 lemma lift_inv_gref2: ∀l,m,T1,p. ⬆[l,m] T1 ≡ §p → T1 = §p.
200 /2 width=5 by lift_inv_gref2_aux/ qed-.
202 fact lift_inv_bind2_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 →
203 ∀a,I,V2,U2. T2 = ⓑ{a,I} V2.U2 →
204 ∃∃V1,U1. ⬆[l,m] V1 ≡ V2 & ⬆[l+1,m] U1 ≡ U2 &
206 #l #m #T1 #T2 * -l -m -T1 -T2
207 [ #k #l #m #a #I #V2 #U2 #H destruct
208 | #i #l #m #_ #a #I #V2 #U2 #H destruct
209 | #i #l #m #_ #a #I #V2 #U2 #H destruct
210 | #p #l #m #a #I #V2 #U2 #H destruct
211 | #b #J #W1 #W2 #T1 #T2 #l #m #HW #HT #a #I #V2 #U2 #H destruct /2 width=5 by ex3_2_intro/
212 | #J #W1 #W2 #T1 #T2 #l #m #_ #_ #a #I #V2 #U2 #H destruct
216 (* Basic_1: was: lift_gen_bind *)
217 lemma lift_inv_bind2: ∀l,m,T1,a,I,V2,U2. ⬆[l,m] T1 ≡ ⓑ{a,I} V2. U2 →
218 ∃∃V1,U1. ⬆[l,m] V1 ≡ V2 & ⬆[l+1,m] U1 ≡ U2 &
220 /2 width=3 by lift_inv_bind2_aux/ qed-.
222 fact lift_inv_flat2_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 →
223 ∀I,V2,U2. T2 = ⓕ{I} V2.U2 →
224 ∃∃V1,U1. ⬆[l,m] V1 ≡ V2 & ⬆[l,m] U1 ≡ U2 &
226 #l #m #T1 #T2 * -l -m -T1 -T2
227 [ #k #l #m #I #V2 #U2 #H destruct
228 | #i #l #m #_ #I #V2 #U2 #H destruct
229 | #i #l #m #_ #I #V2 #U2 #H destruct
230 | #p #l #m #I #V2 #U2 #H destruct
231 | #a #J #W1 #W2 #T1 #T2 #l #m #_ #_ #I #V2 #U2 #H destruct
232 | #J #W1 #W2 #T1 #T2 #l #m #HW #HT #I #V2 #U2 #H destruct /2 width=5 by ex3_2_intro/
236 (* Basic_1: was: lift_gen_flat *)
237 lemma lift_inv_flat2: ∀l,m,T1,I,V2,U2. ⬆[l,m] T1 ≡ ⓕ{I} V2. U2 →
238 ∃∃V1,U1. ⬆[l,m] V1 ≡ V2 & ⬆[l,m] U1 ≡ U2 &
240 /2 width=3 by lift_inv_flat2_aux/ qed-.
242 lemma lift_inv_pair_xy_x: ∀l,m,I,V,T. ⬆[l, m] ②{I} V. T ≡ V → ⊥.
243 #l #m #J #V elim V -V
245 [ lapply (lift_inv_sort2 … H) -H #H destruct
246 | elim (lift_inv_lref2 … H) -H * #_ #H destruct
247 | lapply (lift_inv_gref2 … H) -H #H destruct
249 | * [ #a ] #I #W2 #U2 #IHW2 #_ #T #H
250 [ elim (lift_inv_bind2 … H) -H #W1 #U1 #HW12 #_ #H destruct /2 width=2 by/
251 | elim (lift_inv_flat2 … H) -H #W1 #U1 #HW12 #_ #H destruct /2 width=2 by/
256 (* Basic_1: was: thead_x_lift_y_y *)
257 lemma lift_inv_pair_xy_y: ∀I,T,V,l,m. ⬆[l, m] ②{I} V. T ≡ T → ⊥.
260 [ lapply (lift_inv_sort2 … H) -H #H destruct
261 | elim (lift_inv_lref2 … H) -H * #_ #H destruct
262 | lapply (lift_inv_gref2 … H) -H #H destruct
264 | * [ #a ] #I #W2 #U2 #_ #IHU2 #V #l #m #H
265 [ elim (lift_inv_bind2 … H) -H #W1 #U1 #_ #HU12 #H destruct /2 width=4 by/
266 | elim (lift_inv_flat2 … H) -H #W1 #U1 #_ #HU12 #H destruct /2 width=4 by/
271 (* Basic forward lemmas *****************************************************)
273 lemma lift_fwd_pair1: ∀I,T2,V1,U1,l,m. ⬆[l,m] ②{I}V1.U1 ≡ T2 →
274 ∃∃V2,U2. ⬆[l,m] V1 ≡ V2 & T2 = ②{I}V2.U2.
275 * [ #a ] #I #T2 #V1 #U1 #l #m #H
276 [ elim (lift_inv_bind1 … H) -H /2 width=4 by ex2_2_intro/
277 | elim (lift_inv_flat1 … H) -H /2 width=4 by ex2_2_intro/
281 lemma lift_fwd_pair2: ∀I,T1,V2,U2,l,m. ⬆[l,m] T1 ≡ ②{I}V2.U2 →
282 ∃∃V1,U1. ⬆[l,m] V1 ≡ V2 & T1 = ②{I}V1.U1.
283 * [ #a ] #I #T1 #V2 #U2 #l #m #H
284 [ elim (lift_inv_bind2 … H) -H /2 width=4 by ex2_2_intro/
285 | elim (lift_inv_flat2 … H) -H /2 width=4 by ex2_2_intro/
289 lemma lift_fwd_tw: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 → ♯{T1} = ♯{T2}.
290 #l #m #T1 #T2 #H elim H -l -m -T1 -T2 normalize //
293 lemma lift_simple_dx: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 → 𝐒⦃T1⦄ → 𝐒⦃T2⦄.
294 #l #m #T1 #T2 #H elim H -l -m -T1 -T2 //
295 #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #_ #_ #H
296 elim (simple_inv_bind … H)
299 lemma lift_simple_sn: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 → 𝐒⦃T2⦄ → 𝐒⦃T1⦄.
300 #l #m #T1 #T2 #H elim H -l -m -T1 -T2 //
301 #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #_ #_ #H
302 elim (simple_inv_bind … H)
305 (* Basic properties *********************************************************)
307 (* Basic_1: was: lift_lref_gt *)
308 lemma lift_lref_ge_minus: ∀l,m,i. l + m ≤ i → ⬆[l, m] #(i - m) ≡ #i.
309 #l #m #i #H >(plus_minus_m_m i m) in ⊢ (? ? ? ? %); /3 width=2 by lift_lref_ge, le_plus_to_minus_r, le_plus_b/
312 lemma lift_lref_ge_minus_eq: ∀l,m,i,j. l + m ≤ i → j = i - m → ⬆[l, m] #j ≡ #i.
313 /2 width=1 by lift_lref_ge_minus/ qed-.
315 (* Basic_1: was: lift_r *)
316 lemma lift_refl: ∀T,l. ⬆[l, 0] T ≡ T.
318 [ * #i // #l elim (lt_or_ge i l) /2 width=1 by lift_lref_lt, lift_lref_ge/
319 | * /2 width=1 by lift_bind, lift_flat/
323 lemma lift_total: ∀T1,l,m. ∃T2. ⬆[l,m] T1 ≡ T2.
325 [ * #i /2 width=2 by lift_sort, lift_gref, ex_intro/
326 #l #m elim (lt_or_ge i l) /3 width=2 by lift_lref_lt, lift_lref_ge, ex_intro/
327 | * [ #a ] #I #V1 #T1 #IHV1 #IHT1 #l #m
328 elim (IHV1 l m) -IHV1 #V2 #HV12
329 [ elim (IHT1 (l+1) m) -IHT1 /3 width=2 by lift_bind, ex_intro/
330 | elim (IHT1 l m) -IHT1 /3 width=2 by lift_flat, ex_intro/
335 (* Basic_1: was: lift_free (right to left) *)
336 lemma lift_split: ∀l1,m2,T1,T2. ⬆[l1, m2] T1 ≡ T2 →
337 ∀l2,m1. l1 ≤ l2 → l2 ≤ l1 + m1 → m1 ≤ m2 →
338 ∃∃T. ⬆[l1, m1] T1 ≡ T & ⬆[l2, m2 - m1] T ≡ T2.
339 #l1 #m2 #T1 #T2 #H elim H -l1 -m2 -T1 -T2
340 [ /3 width=3 by lift_sort, ex2_intro/
341 | #i #l1 #m2 #Hil1 #l2 #m1 #Hl12 #_ #_
342 lapply (lt_to_le_to_lt … Hil1 Hl12) -Hl12 #Hil2 /4 width=3 by lift_lref_lt, ex2_intro/
343 | #i #l1 #m2 #Hil1 #l2 #m1 #_ #Hl21 #Hm12
344 lapply (transitive_le … (i+m1) Hl21 ?) /2 width=1 by monotonic_le_plus_l/ -Hl21 #Hl21
345 >(plus_minus_m_m m2 m1 ?) /3 width=3 by lift_lref_ge, ex2_intro/
346 | /3 width=3 by lift_gref, ex2_intro/
347 | #a #I #V1 #V2 #T1 #T2 #l1 #m2 #_ #_ #IHV #IHT #l2 #m1 #Hl12 #Hl21 #Hm12
348 elim (IHV … Hl12 Hl21 Hm12) -IHV #V0 #HV0a #HV0b
349 elim (IHT (l2+1) … ? ? Hm12) /3 width=5 by lift_bind, monotonic_le_plus_l, ex2_intro/
350 | #I #V1 #V2 #T1 #T2 #l1 #m2 #_ #_ #IHV #IHT #l2 #m1 #Hl12 #Hl21 #Hm12
351 elim (IHV … Hl12 Hl21 Hm12) -IHV #V0 #HV0a #HV0b
352 elim (IHT l2 … ? ? Hm12) /3 width=5 by lift_flat, ex2_intro/
356 (* Basic_1: was only: dnf_dec2 dnf_dec *)
357 lemma is_lift_dec: ∀T2,l,m. Decidable (∃T1. ⬆[l,m] T1 ≡ T2).
359 [ * [1,3: /3 width=2 by lift_sort, lift_gref, ex_intro, or_introl/ ] #i #l #m
360 elim (lt_or_ge i l) #Hli
361 [ /4 width=3 by lift_lref_lt, ex_intro, or_introl/
362 | elim (lt_or_ge i (l + m)) #Hilm
363 [ @or_intror * #T1 #H elim (lift_inv_lref2_be … H Hli Hilm)
364 | -Hli /4 width=2 by lift_lref_ge_minus, ex_intro, or_introl/
367 | * [ #a ] #I #V2 #T2 #IHV2 #IHT2 #l #m
368 [ elim (IHV2 l m) -IHV2
369 [ * #V1 #HV12 elim (IHT2 (l+1) m) -IHT2
370 [ * #T1 #HT12 @or_introl /3 width=2 by lift_bind, ex_intro/
371 | -V1 #HT2 @or_intror * #X #H
372 elim (lift_inv_bind2 … H) -H /3 width=2 by ex_intro/
374 | -IHT2 #HV2 @or_intror * #X #H
375 elim (lift_inv_bind2 … H) -H /3 width=2 by ex_intro/
377 | elim (IHV2 l m) -IHV2
378 [ * #V1 #HV12 elim (IHT2 l m) -IHT2
379 [ * #T1 #HT12 /4 width=2 by lift_flat, ex_intro, or_introl/
380 | -V1 #HT2 @or_intror * #X #H
381 elim (lift_inv_flat2 … H) -H /3 width=2 by ex_intro/
383 | -IHT2 #HV2 @or_intror * #X #H
384 elim (lift_inv_flat2 … H) -H /3 width=2 by ex_intro/
390 (* Basic_1: removed theorems 7:
391 lift_head lift_gen_head
392 lift_weight_map lift_weight lift_weight_add lift_weight_add_O