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 *)
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15 include "basic_2A/notation/relations/rlift_4.ma".
16 include "basic_2A/grammar/term_weight.ma".
17 include "basic_2A/grammar/term_simple.ma".
19 (* BASIC TERM RELOCATION ****************************************************)
22 lift_sort lift_lref_lt lift_lref_ge lift_bind lift_flat
24 inductive lift: relation4 nat nat term term ≝
25 | lift_sort : ∀k,l,m. lift l m (⋆k) (⋆k)
26 | lift_lref_lt: ∀i,l,m. i < l → lift l m (#i) (#i)
27 | lift_lref_ge: ∀i,l,m. l ≤ i → lift l m (#i) (#(i + m))
28 | lift_gref : ∀p,l,m. lift l m (§p) (§p)
29 | lift_bind : ∀a,I,V1,V2,T1,T2,l,m.
30 lift l m V1 V2 → lift (l + 1) m T1 T2 →
31 lift l m (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2)
32 | lift_flat : ∀I,V1,V2,T1,T2,l,m.
33 lift l m V1 V2 → lift l m T1 T2 →
34 lift l m (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
37 interpretation "relocation" 'RLift l m T1 T2 = (lift l m T1 T2).
39 (* Basic inversion lemmas ***************************************************)
41 fact lift_inv_O2_aux: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 → m = 0 → T1 = T2.
42 #l #m #T1 #T2 #H elim H -l -m -T1 -T2 /3 width=1 by eq_f2/
45 lemma lift_inv_O2: ∀l,T1,T2. ⬆[l, 0] T1 ≡ T2 → T1 = T2.
46 /2 width=4 by lift_inv_O2_aux/ qed-.
48 fact lift_inv_sort1_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 → ∀k. T1 = ⋆k → T2 = ⋆k.
49 #l #m #T1 #T2 * -l -m -T1 -T2 //
50 [ #i #l #m #_ #k #H destruct
51 | #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
52 | #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
56 lemma lift_inv_sort1: ∀l,m,T2,k. ⬆[l,m] ⋆k ≡ T2 → T2 = ⋆k.
57 /2 width=5 by lift_inv_sort1_aux/ qed-.
59 fact lift_inv_lref1_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 → ∀i. T1 = #i →
60 (i < l ∧ T2 = #i) ∨ (l ≤ i ∧ T2 = #(i + m)).
61 #l #m #T1 #T2 * -l -m -T1 -T2
62 [ #k #l #m #i #H destruct
63 | #j #l #m #Hj #i #Hi destruct /3 width=1 by or_introl, conj/
64 | #j #l #m #Hj #i #Hi destruct /3 width=1 by or_intror, conj/
65 | #p #l #m #i #H destruct
66 | #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #i #H destruct
67 | #I #V1 #V2 #T1 #T2 #l #m #_ #_ #i #H destruct
71 lemma lift_inv_lref1: ∀l,m,T2,i. ⬆[l,m] #i ≡ T2 →
72 (i < l ∧ T2 = #i) ∨ (l ≤ i ∧ T2 = #(i + m)).
73 /2 width=3 by lift_inv_lref1_aux/ qed-.
75 lemma lift_inv_lref1_lt: ∀l,m,T2,i. ⬆[l,m] #i ≡ T2 → i < l → T2 = #i.
76 #l #m #T2 #i #H elim (lift_inv_lref1 … H) -H * //
77 #Hli #_ #Hil lapply (le_to_lt_to_lt … Hli Hil) -Hli -Hil #Hll
78 elim (lt_refl_false … Hll)
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 lapply (le_to_lt_to_lt … Hli Hil) -Hli -Hil #Hll
84 elim (lt_refl_false … Hll)
87 fact lift_inv_gref1_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 → ∀p. T1 = §p → T2 = §p.
88 #l #m #T1 #T2 * -l -m -T1 -T2 //
89 [ #i #l #m #_ #k #H destruct
90 | #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
91 | #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
95 lemma lift_inv_gref1: ∀l,m,T2,p. ⬆[l,m] §p ≡ T2 → T2 = §p.
96 /2 width=5 by lift_inv_gref1_aux/ qed-.
98 fact lift_inv_bind1_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 →
99 ∀a,I,V1,U1. T1 = ⓑ{a,I} V1.U1 →
100 ∃∃V2,U2. ⬆[l,m] V1 ≡ V2 & ⬆[l+1,m] U1 ≡ U2 &
102 #l #m #T1 #T2 * -l -m -T1 -T2
103 [ #k #l #m #a #I #V1 #U1 #H destruct
104 | #i #l #m #_ #a #I #V1 #U1 #H destruct
105 | #i #l #m #_ #a #I #V1 #U1 #H destruct
106 | #p #l #m #a #I #V1 #U1 #H destruct
107 | #b #J #W1 #W2 #T1 #T2 #l #m #HW #HT #a #I #V1 #U1 #H destruct /2 width=5 by ex3_2_intro/
108 | #J #W1 #W2 #T1 #T2 #l #m #_ #HT #a #I #V1 #U1 #H destruct
112 lemma lift_inv_bind1: ∀l,m,T2,a,I,V1,U1. ⬆[l,m] ⓑ{a,I} V1. U1 ≡ T2 →
113 ∃∃V2,U2. ⬆[l,m] V1 ≡ V2 & ⬆[l+1,m] U1 ≡ U2 &
115 /2 width=3 by lift_inv_bind1_aux/ qed-.
117 fact lift_inv_flat1_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 →
118 ∀I,V1,U1. T1 = ⓕ{I} V1.U1 →
119 ∃∃V2,U2. ⬆[l,m] V1 ≡ V2 & ⬆[l,m] U1 ≡ U2 &
121 #l #m #T1 #T2 * -l -m -T1 -T2
122 [ #k #l #m #I #V1 #U1 #H destruct
123 | #i #l #m #_ #I #V1 #U1 #H destruct
124 | #i #l #m #_ #I #V1 #U1 #H destruct
125 | #p #l #m #I #V1 #U1 #H destruct
126 | #a #J #W1 #W2 #T1 #T2 #l #m #_ #_ #I #V1 #U1 #H destruct
127 | #J #W1 #W2 #T1 #T2 #l #m #HW #HT #I #V1 #U1 #H destruct /2 width=5 by ex3_2_intro/
131 lemma lift_inv_flat1: ∀l,m,T2,I,V1,U1. ⬆[l,m] ⓕ{I} V1. U1 ≡ T2 →
132 ∃∃V2,U2. ⬆[l,m] V1 ≡ V2 & ⬆[l,m] U1 ≡ U2 &
134 /2 width=3 by lift_inv_flat1_aux/ qed-.
136 fact lift_inv_sort2_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 → ∀k. T2 = ⋆k → T1 = ⋆k.
137 #l #m #T1 #T2 * -l -m -T1 -T2 //
138 [ #i #l #m #_ #k #H destruct
139 | #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
140 | #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
144 (* Basic_1: was: lift_gen_sort *)
145 lemma lift_inv_sort2: ∀l,m,T1,k. ⬆[l,m] T1 ≡ ⋆k → T1 = ⋆k.
146 /2 width=5 by lift_inv_sort2_aux/ qed-.
148 fact lift_inv_lref2_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 → ∀i. T2 = #i →
149 (i < l ∧ T1 = #i) ∨ (l + m ≤ i ∧ T1 = #(i - m)).
150 #l #m #T1 #T2 * -l -m -T1 -T2
151 [ #k #l #m #i #H destruct
152 | #j #l #m #Hj #i #Hi destruct /3 width=1 by or_introl, conj/
153 | #j #l #m #Hj #i #Hi destruct <minus_plus_m_m /4 width=1 by monotonic_le_plus_l, or_intror, conj/
154 | #p #l #m #i #H destruct
155 | #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #i #H destruct
156 | #I #V1 #V2 #T1 #T2 #l #m #_ #_ #i #H destruct
160 (* Basic_1: was: lift_gen_lref *)
161 lemma lift_inv_lref2: ∀l,m,T1,i. ⬆[l,m] T1 ≡ #i →
162 (i < l ∧ T1 = #i) ∨ (l + m ≤ i ∧ T1 = #(i - m)).
163 /2 width=3 by lift_inv_lref2_aux/ qed-.
165 (* Basic_1: was: lift_gen_lref_lt *)
166 lemma lift_inv_lref2_lt: ∀l,m,T1,i. ⬆[l,m] T1 ≡ #i → i < l → T1 = #i.
167 #l #m #T1 #i #H elim (lift_inv_lref2 … H) -H * //
168 #Hli #_ #Hil lapply (le_to_lt_to_lt … Hli Hil) -Hli -Hil #Hll
169 elim (lt_inv_plus_l … Hll) -Hll #Hll
170 elim (lt_refl_false … Hll)
173 (* Basic_1: was: lift_gen_lref_false *)
174 lemma lift_inv_lref2_be: ∀l,m,T1,i. ⬆[l,m] T1 ≡ #i →
175 l ≤ i → i < l + m → ⊥.
176 #l #m #T1 #i #H elim (lift_inv_lref2 … H) -H *
177 [ #H1 #_ #H2 #_ | #H2 #_ #_ #H1 ]
178 lapply (le_to_lt_to_lt … H2 H1) -H2 -H1 #H
179 elim (lt_refl_false … H)
182 (* Basic_1: was: lift_gen_lref_ge *)
183 lemma lift_inv_lref2_ge: ∀l,m,T1,i. ⬆[l,m] T1 ≡ #i → l + m ≤ i → T1 = #(i - m).
184 #l #m #T1 #i #H elim (lift_inv_lref2 … H) -H * //
185 #Hil #_ #Hli lapply (le_to_lt_to_lt … Hli Hil) -Hli -Hil #Hll
186 elim (lt_inv_plus_l … Hll) -Hll #Hll
187 elim (lt_refl_false … Hll)
190 fact lift_inv_gref2_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 → ∀p. T2 = §p → T1 = §p.
191 #l #m #T1 #T2 * -l -m -T1 -T2 //
192 [ #i #l #m #_ #k #H destruct
193 | #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
194 | #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
198 lemma lift_inv_gref2: ∀l,m,T1,p. ⬆[l,m] T1 ≡ §p → T1 = §p.
199 /2 width=5 by lift_inv_gref2_aux/ qed-.
201 fact lift_inv_bind2_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 →
202 ∀a,I,V2,U2. T2 = ⓑ{a,I} V2.U2 →
203 ∃∃V1,U1. ⬆[l,m] V1 ≡ V2 & ⬆[l+1,m] U1 ≡ U2 &
205 #l #m #T1 #T2 * -l -m -T1 -T2
206 [ #k #l #m #a #I #V2 #U2 #H destruct
207 | #i #l #m #_ #a #I #V2 #U2 #H destruct
208 | #i #l #m #_ #a #I #V2 #U2 #H destruct
209 | #p #l #m #a #I #V2 #U2 #H destruct
210 | #b #J #W1 #W2 #T1 #T2 #l #m #HW #HT #a #I #V2 #U2 #H destruct /2 width=5 by ex3_2_intro/
211 | #J #W1 #W2 #T1 #T2 #l #m #_ #_ #a #I #V2 #U2 #H destruct
215 (* Basic_1: was: lift_gen_bind *)
216 lemma lift_inv_bind2: ∀l,m,T1,a,I,V2,U2. ⬆[l,m] T1 ≡ ⓑ{a,I} V2. U2 →
217 ∃∃V1,U1. ⬆[l,m] V1 ≡ V2 & ⬆[l+1,m] U1 ≡ U2 &
219 /2 width=3 by lift_inv_bind2_aux/ qed-.
221 fact lift_inv_flat2_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 →
222 ∀I,V2,U2. T2 = ⓕ{I} V2.U2 →
223 ∃∃V1,U1. ⬆[l,m] V1 ≡ V2 & ⬆[l,m] U1 ≡ U2 &
225 #l #m #T1 #T2 * -l -m -T1 -T2
226 [ #k #l #m #I #V2 #U2 #H destruct
227 | #i #l #m #_ #I #V2 #U2 #H destruct
228 | #i #l #m #_ #I #V2 #U2 #H destruct
229 | #p #l #m #I #V2 #U2 #H destruct
230 | #a #J #W1 #W2 #T1 #T2 #l #m #_ #_ #I #V2 #U2 #H destruct
231 | #J #W1 #W2 #T1 #T2 #l #m #HW #HT #I #V2 #U2 #H destruct /2 width=5 by ex3_2_intro/
235 (* Basic_1: was: lift_gen_flat *)
236 lemma lift_inv_flat2: ∀l,m,T1,I,V2,U2. ⬆[l,m] T1 ≡ ⓕ{I} V2. U2 →
237 ∃∃V1,U1. ⬆[l,m] V1 ≡ V2 & ⬆[l,m] U1 ≡ U2 &
239 /2 width=3 by lift_inv_flat2_aux/ qed-.
241 lemma lift_inv_pair_xy_x: ∀l,m,I,V,T. ⬆[l, m] ②{I} V. T ≡ V → ⊥.
242 #l #m #J #V elim V -V
244 [ lapply (lift_inv_sort2 … H) -H #H destruct
245 | elim (lift_inv_lref2 … H) -H * #_ #H destruct
246 | lapply (lift_inv_gref2 … H) -H #H destruct
248 | * [ #a ] #I #W2 #U2 #IHW2 #_ #T #H
249 [ elim (lift_inv_bind2 … H) -H #W1 #U1 #HW12 #_ #H destruct /2 width=2 by/
250 | elim (lift_inv_flat2 … H) -H #W1 #U1 #HW12 #_ #H destruct /2 width=2 by/
255 (* Basic_1: was: thead_x_lift_y_y *)
256 lemma lift_inv_pair_xy_y: ∀I,T,V,l,m. ⬆[l, m] ②{I} V. T ≡ T → ⊥.
259 [ lapply (lift_inv_sort2 … H) -H #H destruct
260 | elim (lift_inv_lref2 … H) -H * #_ #H destruct
261 | lapply (lift_inv_gref2 … H) -H #H destruct
263 | * [ #a ] #I #W2 #U2 #_ #IHU2 #V #l #m #H
264 [ elim (lift_inv_bind2 … H) -H #W1 #U1 #_ #HU12 #H destruct /2 width=4 by/
265 | elim (lift_inv_flat2 … H) -H #W1 #U1 #_ #HU12 #H destruct /2 width=4 by/
270 (* Basic forward lemmas *****************************************************)
272 lemma lift_fwd_pair1: ∀I,T2,V1,U1,l,m. ⬆[l,m] ②{I}V1.U1 ≡ T2 →
273 ∃∃V2,U2. ⬆[l,m] V1 ≡ V2 & T2 = ②{I}V2.U2.
274 * [ #a ] #I #T2 #V1 #U1 #l #m #H
275 [ elim (lift_inv_bind1 … H) -H /2 width=4 by ex2_2_intro/
276 | elim (lift_inv_flat1 … H) -H /2 width=4 by ex2_2_intro/
280 lemma lift_fwd_pair2: ∀I,T1,V2,U2,l,m. ⬆[l,m] T1 ≡ ②{I}V2.U2 →
281 ∃∃V1,U1. ⬆[l,m] V1 ≡ V2 & T1 = ②{I}V1.U1.
282 * [ #a ] #I #T1 #V2 #U2 #l #m #H
283 [ elim (lift_inv_bind2 … H) -H /2 width=4 by ex2_2_intro/
284 | elim (lift_inv_flat2 … H) -H /2 width=4 by ex2_2_intro/
288 lemma lift_fwd_tw: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 → ♯{T1} = ♯{T2}.
289 #l #m #T1 #T2 #H elim H -l -m -T1 -T2 normalize //
292 lemma lift_simple_dx: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 → 𝐒⦃T1⦄ → 𝐒⦃T2⦄.
293 #l #m #T1 #T2 #H elim H -l -m -T1 -T2 //
294 #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #_ #_ #H
295 elim (simple_inv_bind … H)
298 lemma lift_simple_sn: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 → 𝐒⦃T2⦄ → 𝐒⦃T1⦄.
299 #l #m #T1 #T2 #H elim H -l -m -T1 -T2 //
300 #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #_ #_ #H
301 elim (simple_inv_bind … H)
304 (* Basic properties *********************************************************)
306 (* Basic_1: was: lift_lref_gt *)
307 lemma lift_lref_ge_minus: ∀l,m,i. l + m ≤ i → ⬆[l, m] #(i - m) ≡ #i.
308 #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/
311 lemma lift_lref_ge_minus_eq: ∀l,m,i,j. l + m ≤ i → j = i - m → ⬆[l, m] #j ≡ #i.
312 /2 width=1 by lift_lref_ge_minus/ qed-.
314 (* Basic_1: was: lift_r *)
315 lemma lift_refl: ∀T,l. ⬆[l, 0] T ≡ T.
317 [ * #i // #l elim (lt_or_ge i l) /2 width=1 by lift_lref_lt, lift_lref_ge/
318 | * /2 width=1 by lift_bind, lift_flat/
322 lemma lift_total: ∀T1,l,m. ∃T2. ⬆[l,m] T1 ≡ T2.
324 [ * #i /2 width=2 by lift_sort, lift_gref, ex_intro/
325 #l #m elim (lt_or_ge i l) /3 width=2 by lift_lref_lt, lift_lref_ge, ex_intro/
326 | * [ #a ] #I #V1 #T1 #IHV1 #IHT1 #l #m
327 elim (IHV1 l m) -IHV1 #V2 #HV12
328 [ elim (IHT1 (l+1) m) -IHT1 /3 width=2 by lift_bind, ex_intro/
329 | elim (IHT1 l m) -IHT1 /3 width=2 by lift_flat, ex_intro/
334 (* Basic_1: was: lift_free (right to left) *)
335 lemma lift_split: ∀l1,m2,T1,T2. ⬆[l1, m2] T1 ≡ T2 →
336 ∀l2,m1. l1 ≤ l2 → l2 ≤ l1 + m1 → m1 ≤ m2 →
337 ∃∃T. ⬆[l1, m1] T1 ≡ T & ⬆[l2, m2 - m1] T ≡ T2.
338 #l1 #m2 #T1 #T2 #H elim H -l1 -m2 -T1 -T2
339 [ /3 width=3 by lift_sort, ex2_intro/
340 | #i #l1 #m2 #Hil1 #l2 #m1 #Hl12 #_ #_
341 lapply (lt_to_le_to_lt … Hil1 Hl12) -Hl12 #Hil2 /4 width=3 by lift_lref_lt, ex2_intro/
342 | #i #l1 #m2 #Hil1 #l2 #m1 #_ #Hl21 #Hm12
343 lapply (transitive_le … (i+m1) Hl21 ?) /2 width=1 by monotonic_le_plus_l/ -Hl21 #Hl21
344 >(plus_minus_m_m m2 m1 ?) /3 width=3 by lift_lref_ge, ex2_intro/
345 | /3 width=3 by lift_gref, ex2_intro/
346 | #a #I #V1 #V2 #T1 #T2 #l1 #m2 #_ #_ #IHV #IHT #l2 #m1 #Hl12 #Hl21 #Hm12
347 elim (IHV … Hl12 Hl21 Hm12) -IHV #V0 #HV0a #HV0b
348 elim (IHT (l2+1) … ? ? Hm12) /3 width=5 by lift_bind, le_S_S, ex2_intro/
349 | #I #V1 #V2 #T1 #T2 #l1 #m2 #_ #_ #IHV #IHT #l2 #m1 #Hl12 #Hl21 #Hm12
350 elim (IHV … Hl12 Hl21 Hm12) -IHV #V0 #HV0a #HV0b
351 elim (IHT l2 … ? ? Hm12) /3 width=5 by lift_flat, ex2_intro/
355 (* Basic_1: was only: dnf_dec2 dnf_dec *)
356 lemma is_lift_dec: ∀T2,l,m. Decidable (∃T1. ⬆[l,m] T1 ≡ T2).
358 [ * [1,3: /3 width=2 by lift_sort, lift_gref, ex_intro, or_introl/ ] #i #l #m
359 elim (lt_or_ge i l) #Hli
360 [ /4 width=3 by lift_lref_lt, ex_intro, or_introl/
361 | elim (lt_or_ge i (l + m)) #Hilm
362 [ @or_intror * #T1 #H elim (lift_inv_lref2_be … H Hli Hilm)
363 | -Hli /4 width=2 by lift_lref_ge_minus, ex_intro, or_introl/
366 | * [ #a ] #I #V2 #T2 #IHV2 #IHT2 #l #m
367 [ elim (IHV2 l m) -IHV2
368 [ * #V1 #HV12 elim (IHT2 (l+1) m) -IHT2
369 [ * #T1 #HT12 @or_introl /3 width=2 by lift_bind, ex_intro/
370 | -V1 #HT2 @or_intror * #X #H
371 elim (lift_inv_bind2 … H) -H /3 width=2 by ex_intro/
373 | -IHT2 #HV2 @or_intror * #X #H
374 elim (lift_inv_bind2 … H) -H /3 width=2 by ex_intro/
376 | elim (IHV2 l m) -IHV2
377 [ * #V1 #HV12 elim (IHT2 l m) -IHT2
378 [ * #T1 #HT12 /4 width=2 by lift_flat, ex_intro, or_introl/
379 | -V1 #HT2 @or_intror * #X #H
380 elim (lift_inv_flat2 … H) -H /3 width=2 by ex_intro/
382 | -IHT2 #HV2 @or_intror * #X #H
383 elim (lift_inv_flat2 … H) -H /3 width=2 by ex_intro/
389 (* Basic_1: removed theorems 7:
390 lift_head lift_gen_head
391 lift_weight_map lift_weight lift_weight_add lift_weight_add_O