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 (**************************************************************************)
17 (* RELOCATION ***************************************************************)
19 (* Policy: level metavariables : d, e
20 height metavariables: h, k
22 (* Note: indexes start at zero *)
23 let rec lift h d M on M ≝ match M with
24 [ VRef i ⇒ #(tri … i d i (i + h) (i + h))
25 | Abst A ⇒ 𝛌. (lift h (d+1) A)
26 | Appl B A ⇒ @(lift h d B). (lift h d A)
29 interpretation "relocation" 'Lift h d M = (lift h d M).
31 notation "hvbox( ↑ [ term 46 d , break term 46 h ] break term 46 M )"
32 non associative with precedence 46
33 for @{ 'Lift $h $d $M }.
35 notation > "hvbox( ↑ [ term 46 h ] break term 46 M )"
36 non associative with precedence 46
37 for @{ 'Lift $h 0 $M }.
39 notation > "hvbox( ↑ term 46 M )"
40 non associative with precedence 46
41 for @{ 'Lift 1 0 $M }.
43 lemma lift_vref_lt: ∀d,h,i. i < d → ↑[d, h] #i = #i.
47 lemma lift_vref_ge: ∀d,h,i. d ≤ i → ↑[d, h] #i = #(i+h).
48 #d #h #i #H elim (le_to_or_lt_eq … H) -H
49 normalize // /3 width=1/
52 lemma lift_vref_pred: ∀d,i. d < i → ↑[d, 1] #(i-1) = #i.
53 #d #i #Hdi >lift_vref_ge /2 width=1/
54 <plus_minus_m_m // /2 width=2/
57 lemma lift_id: ∀M,d. ↑[d, 0] M = M.
59 [ #i #d elim (lt_or_ge i d) /2 width=1/
65 lemma lift_inv_vref_lt: ∀j,d. j < d → ∀h,M. ↑[d, h] M = #j → M = #j.
66 #j #d #Hjd #h * normalize
67 [ #i elim (lt_or_eq_or_gt i d) #Hid
68 [ >(tri_lt ???? … Hid) -Hid -Hjd //
69 | #H destruct >tri_eq in Hjd; #H
70 elim (plus_lt_false … H)
71 | >(tri_gt ???? … Hid)
72 lapply (transitive_lt … Hjd Hid) -d #H #H0 destruct
73 elim (plus_lt_false … H)
80 lemma lift_inv_vref_ge: ∀j,d. d ≤ j → ∀h,M. ↑[d, h] M = #j →
81 d + h ≤ j ∧ M = #(j-h).
82 #j #d #Hdj #h * normalize
83 [ #i elim (lt_or_eq_or_gt i d) #Hid
84 [ >(tri_lt ???? … Hid) #H destruct
85 lapply (le_to_lt_to_lt … Hdj Hid) -Hdj -Hid #H
86 elim (lt_refl_false … H)
87 | #H -Hdj destruct /2 width=1/
88 | >(tri_gt ???? … Hid) #H -Hdj destruct /4 width=1/
95 lemma lift_inv_vref_be: ∀j,d,h. d ≤ j → j < d + h → ∀M. ↑[d, h] M = #j → ⊥.
96 #j #d #h #Hdj #Hjdh #M #H elim (lift_inv_vref_ge … H) -H // -Hdj #Hdhj #_ -M
97 lapply (lt_to_le_to_lt … Hjdh Hdhj) -d -h #H
98 elim (lt_refl_false … H)
101 lemma lift_inv_vref_ge_plus: ∀j,d,h. d + h ≤ j →
102 ∀M. ↑[d, h] M = #j → M = #(j-h).
103 #j #d #h #Hdhj #M #H elim (lift_inv_vref_ge … H) -H // -M /2 width=2/
106 lemma lift_inv_abst: ∀C,d,h,M. ↑[d, h] M = 𝛌.C →
107 ∃∃A. ↑[d+1, h] A = C & M = 𝛌.A.
110 | #A #H destruct /2 width=3/
115 lemma lift_inv_appl: ∀D,C,d,h,M. ↑[d, h] M = @D.C →
116 ∃∃B,A. ↑[d, h] B = D & ↑[d, h] A = C & M = @B.A.
117 #D #C #d #h * normalize
120 | #B #A #H destruct /2 width=5/
124 theorem lift_lift_le: ∀h1,h2,M,d1,d2. d2 ≤ d1 →
125 ↑[d2, h2] ↑[d1, h1] M = ↑[d1 + h2, h1] ↑[d2, h2] M.
127 [ #i #d1 #d2 #Hd21 elim (lt_or_ge i d2) #Hid2
128 [ lapply (lt_to_le_to_lt … Hid2 Hd21) -Hd21 #Hid1
129 >(lift_vref_lt … Hid1) >(lift_vref_lt … Hid2)
130 >lift_vref_lt // /2 width=1/
131 | >(lift_vref_ge … Hid2) elim (lt_or_ge i d1) #Hid1
132 [ >(lift_vref_lt … Hid1) >(lift_vref_ge … Hid2)
133 >lift_vref_lt // -d2 /2 width=1/
134 | >(lift_vref_ge … Hid1) >lift_vref_ge /2 width=1/
135 >lift_vref_ge // /2 width=1/
138 | normalize #A #IHA #d1 #d2 #Hd21 >IHA // /2 width=1/
139 | normalize #B #A #IHB #IHA #d1 #d2 #Hd21 >IHB >IHA //
143 theorem lift_lift_be: ∀h1,h2,M,d1,d2. d1 ≤ d2 → d2 ≤ d1 + h1 →
144 ↑[d2, h2] ↑[d1, h1] M = ↑[d1, h1 + h2] M.
146 [ #i #d1 #d2 #Hd12 #Hd21 elim (lt_or_ge i d1) #Hid1
147 [ lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 -Hd21 #Hid2
148 >(lift_vref_lt … Hid1) >(lift_vref_lt … Hid1) /2 width=1/
149 | lapply (transitive_le … (i+h1) Hd21 ?) -Hd21 -Hd12 /2 width=1/ #Hd2
150 >(lift_vref_ge … Hid1) >(lift_vref_ge … Hid1) /2 width=1/
152 | normalize #A #IHA #d1 #d2 #Hd12 #Hd21 >IHA // /2 width=1/
153 | normalize #B #A #IHB #IHA #d1 #d2 #Hd12 #Hd21 >IHB >IHA //
157 theorem lift_lift_ge: ∀h1,h2,M,d1,d2. d1 + h1 ≤ d2 →
158 ↑[d2, h2] ↑[d1, h1] M = ↑[d1, h1] ↑[d2 - h1, h2] M.
159 #h1 #h2 #M #d1 #d2 #Hd12
160 >(lift_lift_le h2 h1) /2 width=1/ <plus_minus_m_m // /2 width=2/
163 (* Note: this is "∀h,d. injective … (lift h d)" *)
164 theorem lift_inj: ∀h,M1,M2,d. ↑[d, h] M2 = ↑[d, h] M1 → M2 = M1.
166 [ #i #M2 #d #H elim (lt_or_ge i d) #Hid
167 [ >(lift_vref_lt … Hid) in H; #H
168 >(lift_inv_vref_lt … Hid … H) -M2 -d -h //
169 | >(lift_vref_ge … Hid) in H; #H
170 >(lift_inv_vref_ge_plus … H) -M2 // /2 width=1/
172 | normalize #A1 #IHA1 #M2 #d #H
173 elim (lift_inv_abst … H) -H #A2 #HA12 #H destruct
174 >(IHA1 … HA12) -IHA1 -A2 //
175 | normalize #B1 #A1 #IHB1 #IHA1 #M2 #d #H
176 elim (lift_inv_appl … H) -H #B2 #A2 #HB12 #HA12 #H destruct
177 >(IHB1 … HB12) -IHB1 -B2 >(IHA1 … HA12) -IHA1 -A2 //
181 theorem lift_inv_lift_le: ∀h1,h2,M1,M2,d1,d2. d2 ≤ d1 →
182 ↑[d2, h2] M2 = ↑[d1 + h2, h1] M1 →
183 ∃∃M. ↑[d1, h1] M = M2 & ↑[d2, h2] M = M1.
184 #h1 #h2 #M1 elim M1 -M1
185 [ #i #M2 #d1 #d2 #Hd21 elim (lt_or_ge i (d1+h2)) #Hid1
186 [ >(lift_vref_lt … Hid1) elim (lt_or_ge i d2) #Hid2 #H
187 [ lapply (lt_to_le_to_lt … Hid2 Hd21) -Hd21 -Hid1 #Hid1
188 >(lift_inv_vref_lt … Hid2 … H) -M2 /3 width=3/
189 | elim (lift_inv_vref_ge … H) -H -Hd21 // -Hid2 #Hdh2i #H destruct
190 elim (le_inv_plus_l … Hdh2i) -Hdh2i #Hd2i #Hh2i
191 @(ex2_intro … (#(i-h2))) [ /4 width=1/ ] -Hid1
192 >lift_vref_ge // -Hd2i /3 width=1/ (**) (* auto: needs some help here *)
194 | elim (le_inv_plus_l … Hid1) #Hd1i #Hh2i
195 lapply (transitive_le (d2+h2) … Hid1) /2 width=1/ -Hd21 #Hdh2i
196 elim (le_inv_plus_l … Hdh2i) #Hd2i #_
197 >(lift_vref_ge … Hid1) #H -Hid1
198 >(lift_inv_vref_ge_plus … H) -H /2 width=3/ -Hdh2i
199 @(ex2_intro … (#(i-h2))) (**) (* auto: needs some help here *)
200 [ >lift_vref_ge // -Hd1i /3 width=1/
201 | >lift_vref_ge // -Hd2i -Hd1i /3 width=1/
204 | normalize #A1 #IHA1 #M2 #d1 #d2 #Hd21 #H
205 elim (lift_inv_abst … H) -H >plus_plus_comm_23 #A2 #HA12 #H destruct
206 elim (IHA1 … HA12) -IHA1 -HA12 /2 width=1/ -Hd21 #A #HA2 #HA1
207 @(ex2_intro … (𝛌.A)) normalize //
208 | normalize #B1 #A1 #IHB1 #IHA1 #M2 #d1 #d2 #Hd21 #H
209 elim (lift_inv_appl … H) -H #B2 #A2 #HB12 #HA12 #H destruct
210 elim (IHB1 … HB12) -IHB1 -HB12 // #B #HB2 #HB1
211 elim (IHA1 … HA12) -IHA1 -HA12 // -Hd21 #A #HA2 #HA1
212 @(ex2_intro … (@B.A)) normalize //
216 theorem lift_inv_lift_be: ∀h1,h2,M1,M2,d1,d2. d1 ≤ d2 → d2 ≤ d1 + h1 →
217 ↑[d2, h2] M2 = ↑[d1, h1 + h2] M1 → ↑[d1, h1] M1 = M2.
218 #h1 #h2 #M1 elim M1 -M1
219 [ #i #M2 #d1 #d2 #Hd12 #Hd21 elim (lt_or_ge i d1) #Hid1
220 [ lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 -Hd21 #Hid2
221 >(lift_vref_lt … Hid1) #H >(lift_inv_vref_lt … Hid2 … H) -h2 -M2 -d2 /2 width=1/
222 | lapply (transitive_le … (i+h1) Hd21 ?) -Hd12 -Hd21 /2 width=1/ #Hd2
223 >(lift_vref_ge … Hid1) #H >(lift_inv_vref_ge_plus … H) -M2 /2 width=1/
225 | normalize #A1 #IHA1 #M2 #d1 #d2 #Hd12 #Hd21 #H
226 elim (lift_inv_abst … H) -H #A #HA12 #H destruct
227 >(IHA1 … HA12) -IHA1 -HA12 // /2 width=1/
228 | normalize #B1 #A1 #IHB1 #IHA1 #M2 #d1 #d2 #Hd12 #Hd21 #H
229 elim (lift_inv_appl … H) -H #B #A #HB12 #HA12 #H destruct
230 >(IHB1 … HB12) -IHB1 -HB12 // >(IHA1 … HA12) -IHA1 -HA12 //
234 theorem lift_inv_lift_ge: ∀h1,h2,M1,M2,d1,d2. d1 + h1 ≤ d2 →
235 ↑[d2, h2] M2 = ↑[d1, h1] M1 →
236 ∃∃M. ↑[d1, h1] M = M2 & ↑[d2 - h1, h2] M = M1.
237 #h1 #h2 #M1 #M2 #d1 #d2 #Hd12 #H
238 elim (le_inv_plus_l … Hd12) -Hd12 #Hd12 #Hh1d2
239 lapply (sym_eq term … H) -H >(plus_minus_m_m … Hh1d2) in ⊢ (???%→?); -Hh1d2 #H
240 elim (lift_inv_lift_le … Hd12 … H) -H -Hd12 /2 width=3/
243 definition liftable: predicate (relation term) ≝ λR.
244 ∀h,M1,M2. R M1 M2 → ∀d. R (↑[d, h] M1) (↑[d, h] M2).
246 definition deliftable_sn: predicate (relation term) ≝ λR.
247 ∀h,N1,N2. R N1 N2 → ∀d,M1. ↑[d, h] M1 = N1 →
248 ∃∃M2. R M1 M2 & ↑[d, h] M2 = N2.
250 lemma star_liftable: ∀R. liftable R → liftable (star … R).
251 #R #HR #h #M1 #M2 #H elim H -M2 // /3 width=3/
254 lemma star_deliftable_sn: ∀R. deliftable_sn R → deliftable_sn (star … R).
255 #R #HR #h #N1 #N2 #H elim H -N2 /2 width=3/
256 #N #N2 #_ #HN2 #IHN1 #d #M1 #HMN1
257 elim (IHN1 … HMN1) -N1 #M #HM1 #HMN
258 elim (HR … HN2 … HMN) -N /3 width=3/
261 lemma lstar_liftable: ∀T,R. (∀t. liftable (R t)) →
262 ∀l. liftable (lstar T … R l).
263 #T #R #HR #l #h #M1 #M2 #H
264 @(lstar_ind_l ????????? H) -l -M1 // /3 width=3/
267 lemma lstar_deliftable_sn: ∀T,R. (∀t. deliftable_sn (R t)) →
268 ∀l. deliftable_sn (lstar T … R l).
269 #T #R #HR #l #h #N1 #N2 #H
270 @(lstar_ind_l ????????? H) -l -N1 /2 width=3/
271 #t #l #N1 #N #HN1 #_ #IHN2 #d #M1 #HMN1
272 elim (HR … HN1 … HMN1) -N1 #M #HM1 #HMN
273 elim (IHN2 … HMN) -N /3 width=3/