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 "basic_2/notation/relations/rlift_4.ma".
16 include "basic_2/grammar/term_weight.ma".
17 include "basic_2/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,d,e. lift d e (⋆k) (⋆k)
26 | lift_lref_lt: ∀i,d,e. i < d → lift d e (#i) (#i)
27 | lift_lref_ge: ∀i,d,e. d ≤ i → lift d e (#i) (#(i + e))
28 | lift_gref : ∀p,d,e. lift d e (§p) (§p)
29 | lift_bind : ∀a,I,V1,V2,T1,T2,d,e.
30 lift d e V1 V2 → lift (d + 1) e T1 T2 →
31 lift d e (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2)
32 | lift_flat : ∀I,V1,V2,T1,T2,d,e.
33 lift d e V1 V2 → lift d e T1 T2 →
34 lift d e (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
37 interpretation "relocation" 'RLift d e T1 T2 = (lift d e T1 T2).
39 (* Basic inversion lemmas ***************************************************)
41 fact lift_inv_O2_aux: ∀d,e,T1,T2. ⇧[d, e] T1 ≡ T2 → e = 0 → T1 = T2.
42 #d #e #T1 #T2 #H elim H -d -e -T1 -T2 // /3 width=1/
45 lemma lift_inv_O2: ∀d,T1,T2. ⇧[d, 0] T1 ≡ T2 → T1 = T2.
46 /2 width=4 by lift_inv_O2_aux/ qed-.
48 fact lift_inv_sort1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀k. T1 = ⋆k → T2 = ⋆k.
49 #d #e #T1 #T2 * -d -e -T1 -T2 //
50 [ #i #d #e #_ #k #H destruct
51 | #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
52 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
56 lemma lift_inv_sort1: ∀d,e,T2,k. ⇧[d,e] ⋆k ≡ T2 → T2 = ⋆k.
57 /2 width=5 by lift_inv_sort1_aux/ qed-.
59 fact lift_inv_lref1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀i. T1 = #i →
60 (i < d ∧ T2 = #i) ∨ (d ≤ i ∧ T2 = #(i + e)).
61 #d #e #T1 #T2 * -d -e -T1 -T2
62 [ #k #d #e #i #H destruct
63 | #j #d #e #Hj #i #Hi destruct /3 width=1/
64 | #j #d #e #Hj #i #Hi destruct /3 width=1/
65 | #p #d #e #i #H destruct
66 | #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
67 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
71 lemma lift_inv_lref1: ∀d,e,T2,i. ⇧[d,e] #i ≡ T2 →
72 (i < d ∧ T2 = #i) ∨ (d ≤ i ∧ T2 = #(i + e)).
73 /2 width=3 by lift_inv_lref1_aux/ qed-.
75 lemma lift_inv_lref1_lt: ∀d,e,T2,i. ⇧[d,e] #i ≡ T2 → i < d → T2 = #i.
76 #d #e #T2 #i #H elim (lift_inv_lref1 … H) -H * //
77 #Hdi #_ #Hid lapply (le_to_lt_to_lt … Hdi Hid) -Hdi -Hid #Hdd
78 elim (lt_refl_false … Hdd)
81 lemma lift_inv_lref1_ge: ∀d,e,T2,i. ⇧[d,e] #i ≡ T2 → d ≤ i → T2 = #(i + e).
82 #d #e #T2 #i #H elim (lift_inv_lref1 … H) -H * //
83 #Hid #_ #Hdi lapply (le_to_lt_to_lt … Hdi Hid) -Hdi -Hid #Hdd
84 elim (lt_refl_false … Hdd)
87 fact lift_inv_gref1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀p. T1 = §p → T2 = §p.
88 #d #e #T1 #T2 * -d -e -T1 -T2 //
89 [ #i #d #e #_ #k #H destruct
90 | #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
91 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
95 lemma lift_inv_gref1: ∀d,e,T2,p. ⇧[d,e] §p ≡ T2 → T2 = §p.
96 /2 width=5 by lift_inv_gref1_aux/ qed-.
98 fact lift_inv_bind1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 →
99 ∀a,I,V1,U1. T1 = ⓑ{a,I} V1.U1 →
100 ∃∃V2,U2. ⇧[d,e] V1 ≡ V2 & ⇧[d+1,e] U1 ≡ U2 &
102 #d #e #T1 #T2 * -d -e -T1 -T2
103 [ #k #d #e #a #I #V1 #U1 #H destruct
104 | #i #d #e #_ #a #I #V1 #U1 #H destruct
105 | #i #d #e #_ #a #I #V1 #U1 #H destruct
106 | #p #d #e #a #I #V1 #U1 #H destruct
107 | #b #J #W1 #W2 #T1 #T2 #d #e #HW #HT #a #I #V1 #U1 #H destruct /2 width=5/
108 | #J #W1 #W2 #T1 #T2 #d #e #_ #HT #a #I #V1 #U1 #H destruct
112 lemma lift_inv_bind1: ∀d,e,T2,a,I,V1,U1. ⇧[d,e] ⓑ{a,I} V1. U1 ≡ T2 →
113 ∃∃V2,U2. ⇧[d,e] V1 ≡ V2 & ⇧[d+1,e] U1 ≡ U2 &
115 /2 width=3 by lift_inv_bind1_aux/ qed-.
117 fact lift_inv_flat1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 →
118 ∀I,V1,U1. T1 = ⓕ{I} V1.U1 →
119 ∃∃V2,U2. ⇧[d,e] V1 ≡ V2 & ⇧[d,e] U1 ≡ U2 &
121 #d #e #T1 #T2 * -d -e -T1 -T2
122 [ #k #d #e #I #V1 #U1 #H destruct
123 | #i #d #e #_ #I #V1 #U1 #H destruct
124 | #i #d #e #_ #I #V1 #U1 #H destruct
125 | #p #d #e #I #V1 #U1 #H destruct
126 | #a #J #W1 #W2 #T1 #T2 #d #e #_ #_ #I #V1 #U1 #H destruct
127 | #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V1 #U1 #H destruct /2 width=5/
131 lemma lift_inv_flat1: ∀d,e,T2,I,V1,U1. ⇧[d,e] ⓕ{I} V1. U1 ≡ T2 →
132 ∃∃V2,U2. ⇧[d,e] V1 ≡ V2 & ⇧[d,e] U1 ≡ U2 &
134 /2 width=3 by lift_inv_flat1_aux/ qed-.
136 fact lift_inv_sort2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀k. T2 = ⋆k → T1 = ⋆k.
137 #d #e #T1 #T2 * -d -e -T1 -T2 //
138 [ #i #d #e #_ #k #H destruct
139 | #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
140 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
144 (* Basic_1: was: lift_gen_sort *)
145 lemma lift_inv_sort2: ∀d,e,T1,k. ⇧[d,e] T1 ≡ ⋆k → T1 = ⋆k.
146 /2 width=5 by lift_inv_sort2_aux/ qed-.
148 fact lift_inv_lref2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀i. T2 = #i →
149 (i < d ∧ T1 = #i) ∨ (d + e ≤ i ∧ T1 = #(i - e)).
150 #d #e #T1 #T2 * -d -e -T1 -T2
151 [ #k #d #e #i #H destruct
152 | #j #d #e #Hj #i #Hi destruct /3 width=1/
153 | #j #d #e #Hj #i #Hi destruct <minus_plus_m_m /4 width=1/
154 | #p #d #e #i #H destruct
155 | #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
156 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
160 (* Basic_1: was: lift_gen_lref *)
161 lemma lift_inv_lref2: ∀d,e,T1,i. ⇧[d,e] T1 ≡ #i →
162 (i < d ∧ T1 = #i) ∨ (d + e ≤ i ∧ T1 = #(i - e)).
163 /2 width=3 by lift_inv_lref2_aux/ qed-.
165 (* Basic_1: was: lift_gen_lref_lt *)
166 lemma lift_inv_lref2_lt: ∀d,e,T1,i. ⇧[d,e] T1 ≡ #i → i < d → T1 = #i.
167 #d #e #T1 #i #H elim (lift_inv_lref2 … H) -H * //
168 #Hdi #_ #Hid lapply (le_to_lt_to_lt … Hdi Hid) -Hdi -Hid #Hdd
169 elim (lt_inv_plus_l … Hdd) -Hdd #Hdd
170 elim (lt_refl_false … Hdd)
173 (* Basic_1: was: lift_gen_lref_false *)
174 lemma lift_inv_lref2_be: ∀d,e,T1,i. ⇧[d,e] T1 ≡ #i →
175 d ≤ i → i < d + e → ⊥.
176 #d #e #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: ∀d,e,T1,i. ⇧[d,e] T1 ≡ #i → d + e ≤ i → T1 = #(i - e).
184 #d #e #T1 #i #H elim (lift_inv_lref2 … H) -H * //
185 #Hid #_ #Hdi lapply (le_to_lt_to_lt … Hdi Hid) -Hdi -Hid #Hdd
186 elim (lt_inv_plus_l … Hdd) -Hdd #Hdd
187 elim (lt_refl_false … Hdd)
190 fact lift_inv_gref2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀p. T2 = §p → T1 = §p.
191 #d #e #T1 #T2 * -d -e -T1 -T2 //
192 [ #i #d #e #_ #k #H destruct
193 | #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
194 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
198 lemma lift_inv_gref2: ∀d,e,T1,p. ⇧[d,e] T1 ≡ §p → T1 = §p.
199 /2 width=5 by lift_inv_gref2_aux/ qed-.
201 fact lift_inv_bind2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 →
202 ∀a,I,V2,U2. T2 = ⓑ{a,I} V2.U2 →
203 ∃∃V1,U1. ⇧[d,e] V1 ≡ V2 & ⇧[d+1,e] U1 ≡ U2 &
205 #d #e #T1 #T2 * -d -e -T1 -T2
206 [ #k #d #e #a #I #V2 #U2 #H destruct
207 | #i #d #e #_ #a #I #V2 #U2 #H destruct
208 | #i #d #e #_ #a #I #V2 #U2 #H destruct
209 | #p #d #e #a #I #V2 #U2 #H destruct
210 | #b #J #W1 #W2 #T1 #T2 #d #e #HW #HT #a #I #V2 #U2 #H destruct /2 width=5/
211 | #J #W1 #W2 #T1 #T2 #d #e #_ #_ #a #I #V2 #U2 #H destruct
215 (* Basic_1: was: lift_gen_bind *)
216 lemma lift_inv_bind2: ∀d,e,T1,a,I,V2,U2. ⇧[d,e] T1 ≡ ⓑ{a,I} V2. U2 →
217 ∃∃V1,U1. ⇧[d,e] V1 ≡ V2 & ⇧[d+1,e] U1 ≡ U2 &
219 /2 width=3 by lift_inv_bind2_aux/ qed-.
221 fact lift_inv_flat2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 →
222 ∀I,V2,U2. T2 = ⓕ{I} V2.U2 →
223 ∃∃V1,U1. ⇧[d,e] V1 ≡ V2 & ⇧[d,e] U1 ≡ U2 &
225 #d #e #T1 #T2 * -d -e -T1 -T2
226 [ #k #d #e #I #V2 #U2 #H destruct
227 | #i #d #e #_ #I #V2 #U2 #H destruct
228 | #i #d #e #_ #I #V2 #U2 #H destruct
229 | #p #d #e #I #V2 #U2 #H destruct
230 | #a #J #W1 #W2 #T1 #T2 #d #e #_ #_ #I #V2 #U2 #H destruct
231 | #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V2 #U2 #H destruct /2 width=5/
235 (* Basic_1: was: lift_gen_flat *)
236 lemma lift_inv_flat2: ∀d,e,T1,I,V2,U2. ⇧[d,e] T1 ≡ ⓕ{I} V2. U2 →
237 ∃∃V1,U1. ⇧[d,e] V1 ≡ V2 & ⇧[d,e] U1 ≡ U2 &
239 /2 width=3 by lift_inv_flat2_aux/ qed-.
241 lemma lift_inv_pair_xy_x: ∀d,e,I,V,T. ⇧[d, e] ②{I} V. T ≡ V → ⊥.
242 #d #e #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/
250 | elim (lift_inv_flat2 … H) -H #W1 #U1 #HW12 #_ #H destruct /2 width=2/
255 (* Basic_1: was: thead_x_lift_y_y *)
256 lemma lift_inv_pair_xy_y: ∀I,T,V,d,e. ⇧[d, e] 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 #d #e #H
264 [ elim (lift_inv_bind2 … H) -H #W1 #U1 #_ #HU12 #H destruct /2 width=4/
265 | elim (lift_inv_flat2 … H) -H #W1 #U1 #_ #HU12 #H destruct /2 width=4/
270 (* Basic forward lemmas *****************************************************)
272 lemma lift_fwd_pair1: ∀I,T2,V1,U1,d,e. ⇧[d,e] ②{I}V1.U1 ≡ T2 →
273 ∃∃V2,U2. ⇧[d,e] V1 ≡ V2 & T2 = ②{I}V2.U2.
274 * [ #a ] #I #T2 #V1 #U1 #d #e #H
275 [ elim (lift_inv_bind1 … H) -H /2 width=4/
276 | elim (lift_inv_flat1 … H) -H /2 width=4/
280 lemma lift_fwd_pair2: ∀I,T1,V2,U2,d,e. ⇧[d,e] T1 ≡ ②{I}V2.U2 →
281 ∃∃V1,U1. ⇧[d,e] V1 ≡ V2 & T1 = ②{I}V1.U1.
282 * [ #a ] #I #T1 #V2 #U2 #d #e #H
283 [ elim (lift_inv_bind2 … H) -H /2 width=4/
284 | elim (lift_inv_flat2 … H) -H /2 width=4/
288 lemma lift_fwd_tw: ∀d,e,T1,T2. ⇧[d, e] T1 ≡ T2 → ♯{T1} = ♯{T2}.
289 #d #e #T1 #T2 #H elim H -d -e -T1 -T2 normalize //
292 lemma lift_simple_dx: ∀d,e,T1,T2. ⇧[d, e] T1 ≡ T2 → 𝐒⦃T1⦄ → 𝐒⦃T2⦄.
293 #d #e #T1 #T2 #H elim H -d -e -T1 -T2 //
294 #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #H
295 elim (simple_inv_bind … H)
298 lemma lift_simple_sn: ∀d,e,T1,T2. ⇧[d, e] T1 ≡ T2 → 𝐒⦃T2⦄ → 𝐒⦃T1⦄.
299 #d #e #T1 #T2 #H elim H -d -e -T1 -T2 //
300 #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #H
301 elim (simple_inv_bind … H)
304 (* Basic properties *********************************************************)
306 (* Basic_1: was: lift_lref_gt *)
307 lemma lift_lref_ge_minus: ∀d,e,i. d + e ≤ i → ⇧[d, e] #(i - e) ≡ #i.
308 #d #e #i #H >(plus_minus_m_m i e) in ⊢ (? ? ? ? %); /2 width=2/ /3 width=2/
311 lemma lift_lref_ge_minus_eq: ∀d,e,i,j. d + e ≤ i → j = i - e → ⇧[d, e] #j ≡ #i.
314 (* Basic_1: was: lift_r *)
315 lemma lift_refl: ∀T,d. ⇧[d, 0] T ≡ T.
317 [ * #i // #d elim (lt_or_ge i d) /2 width=1/
322 lemma lift_total: ∀T1,d,e. ∃T2. ⇧[d,e] T1 ≡ T2.
324 [ * #i /2 width=2/ #d #e elim (lt_or_ge i d) /3 width=2/
325 | * [ #a ] #I #V1 #T1 #IHV1 #IHT1 #d #e
326 elim (IHV1 d e) -IHV1 #V2 #HV12
327 [ elim (IHT1 (d+1) e) -IHT1 /3 width=2/
328 | elim (IHT1 d e) -IHT1 /3 width=2/
333 (* Basic_1: was: lift_free (right to left) *)
334 lemma lift_split: ∀d1,e2,T1,T2. ⇧[d1, e2] T1 ≡ T2 →
335 ∀d2,e1. d1 ≤ d2 → d2 ≤ d1 + e1 → e1 ≤ e2 →
336 ∃∃T. ⇧[d1, e1] T1 ≡ T & ⇧[d2, e2 - e1] T ≡ T2.
337 #d1 #e2 #T1 #T2 #H elim H -d1 -e2 -T1 -T2
339 | #i #d1 #e2 #Hid1 #d2 #e1 #Hd12 #_ #_
340 lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2 /4 width=3/
341 | #i #d1 #e2 #Hid1 #d2 #e1 #_ #Hd21 #He12
342 lapply (transitive_le … (i+e1) Hd21 ?) /2 width=1/ -Hd21 #Hd21
343 >(plus_minus_m_m e2 e1 ?) // /3 width=3/
345 | #a #I #V1 #V2 #T1 #T2 #d1 #e2 #_ #_ #IHV #IHT #d2 #e1 #Hd12 #Hd21 #He12
346 elim (IHV … Hd12 Hd21 He12) -IHV #V0 #HV0a #HV0b
347 elim (IHT (d2+1) … ? ? He12) /2 width=1/ /3 width=5/
348 | #I #V1 #V2 #T1 #T2 #d1 #e2 #_ #_ #IHV #IHT #d2 #e1 #Hd12 #Hd21 #He12
349 elim (IHV … Hd12 Hd21 He12) -IHV #V0 #HV0a #HV0b
350 elim (IHT d2 … ? ? He12) // /3 width=5/
354 (* Basic_1: was only: dnf_dec2 dnf_dec *)
355 lemma is_lift_dec: ∀T2,d,e. Decidable (∃T1. ⇧[d,e] T1 ≡ T2).
357 [ * [1,3: /3 width=2/ ] #i #d #e
358 elim (lt_dec i d) #Hid
360 | lapply (false_lt_to_le … Hid) -Hid #Hid
361 elim (lt_dec i (d + e)) #Hide
362 [ @or_intror * #T1 #H
363 elim (lift_inv_lref2_be … H Hid Hide)
364 | lapply (false_lt_to_le … Hide) -Hide /4 width=2/
367 | * [ #a ] #I #V2 #T2 #IHV2 #IHT2 #d #e
368 [ elim (IHV2 d e) -IHV2
369 [ * #V1 #HV12 elim (IHT2 (d+1) e) -IHT2
370 [ * #T1 #HT12 @or_introl /3 width=2/
371 | -V1 #HT2 @or_intror * #X #H
372 elim (lift_inv_bind2 … H) -H /3 width=2/
374 | -IHT2 #HV2 @or_intror * #X #H
375 elim (lift_inv_bind2 … H) -H /3 width=2/
377 | elim (IHV2 d e) -IHV2
378 [ * #V1 #HV12 elim (IHT2 d e) -IHT2
379 [ * #T1 #HT12 /4 width=2/
380 | -V1 #HT2 @or_intror * #X #H
381 elim (lift_inv_flat2 … H) -H /3 width=2/
383 | -IHT2 #HV2 @or_intror * #X #H
384 elim (lift_inv_flat2 … H) -H /3 width=2/
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