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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-2/grammar/term_weight.ma".
17 (* RELOCATION ***************************************************************)
19 inductive lift: term → nat → nat → term → Prop ≝
20 | lift_sort : ∀k,d,e. lift (⋆k) d e (⋆k)
21 | lift_lref_lt: ∀i,d,e. i < d → lift (#i) d e (#i)
22 | lift_lref_ge: ∀i,d,e. d ≤ i → lift (#i) d e (#(i + e))
23 | lift_bind : ∀I,V1,V2,T1,T2,d,e.
24 lift V1 d e V2 → lift T1 (d + 1) e T2 →
25 lift (𝕓{I} V1. T1) d e (𝕓{I} V2. T2)
26 | lift_flat : ∀I,V1,V2,T1,T2,d,e.
27 lift V1 d e V2 → lift T1 d e T2 →
28 lift (𝕗{I} V1. T1) d e (𝕗{I} V2. T2)
31 interpretation "relocation" 'RLift T1 d e T2 = (lift T1 d e T2).
33 (* Basic properties *********************************************************)
35 lemma lift_lref_ge_minus: ∀d,e,i. d + e ≤ i → ↑[d, e] #(i - e) ≡ #i.
36 #d #e #i #H >(plus_minus_m_m i e) in ⊢ (? ? ? ? %) /3/
39 lemma lift_refl: ∀T,d. ↑[d, 0] T ≡ T.
42 | #i #d elim (lt_or_ge i d) /2/
47 lemma lift_total: ∀T1,d,e. ∃T2. ↑[d,e] T1 ≡ T2.
50 | #i #d #e elim (lt_or_ge i d) /3/
51 | * #I #V1 #T1 #IHV1 #IHT1 #d #e
52 elim (IHV1 d e) -IHV1 #V2 #HV12
53 [ elim (IHT1 (d+1) e) -IHT1 /3/
54 | elim (IHT1 d e) -IHT1 /3/
59 lemma lift_split: ∀d1,e2,T1,T2. ↑[d1, e2] T1 ≡ T2 → ∀d2,e1.
60 d1 ≤ d2 → d2 ≤ d1 + e1 → e1 ≤ e2 →
61 ∃∃T. ↑[d1, e1] T1 ≡ T & ↑[d2, e2 - e1] T ≡ T2.
62 #d1 #e2 #T1 #T2 #H elim H -H d1 e2 T1 T2
64 | #i #d1 #e2 #Hid1 #d2 #e1 #Hd12 #_ #_
65 lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2 /4/
66 | #i #d1 #e2 #Hid1 #d2 #e1 #_ #Hd21 #He12
67 lapply (transitive_le …(i+e1) Hd21 ?) /2/ -Hd21 #Hd21
68 <(arith_d1 i e2 e1) // /3/
69 | #I #V1 #V2 #T1 #T2 #d1 #e2 #_ #_ #IHV #IHT #d2 #e1 #Hd12 #Hd21 #He12
70 elim (IHV … Hd12 Hd21 He12) -IHV #V0 #HV0a #HV0b
71 elim (IHT (d2+1) … ? ? He12) /3 width = 5/
72 | #I #V1 #V2 #T1 #T2 #d1 #e2 #_ #_ #IHV #IHT #d2 #e1 #Hd12 #Hd21 #He12
73 elim (IHV … Hd12 Hd21 He12) -IHV #V0 #HV0a #HV0b
74 elim (IHT d2 … ? ? He12) /3 width = 5/
78 (* Basic forward lemmas *****************************************************)
80 lemma tw_lift: ∀d,e,T1,T2. ↑[d, e] T1 ≡ T2 → #T1 = #T2.
81 #d #e #T1 #T2 #H elim H -d e T1 T2; normalize //
84 (* Basic inversion lemmas ***************************************************)
86 lemma lift_inv_refl_aux: ∀d,e,T1,T2. ↑[d, e] T1 ≡ T2 → e = 0 → T1 = T2.
87 #d #e #T1 #T2 #H elim H -H d e T1 T2 /3/
90 lemma lift_inv_refl: ∀d,T1,T2. ↑[d, 0] T1 ≡ T2 → T1 = T2.
93 lemma lift_inv_sort1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀k. T1 = ⋆k → T2 = ⋆k.
94 #d #e #T1 #T2 * -d e T1 T2 //
95 [ #i #d #e #_ #k #H destruct
96 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
97 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
101 lemma lift_inv_sort1: ∀d,e,T2,k. ↑[d,e] ⋆k ≡ T2 → T2 = ⋆k.
104 lemma lift_inv_lref1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀i. T1 = #i →
105 (i < d ∧ T2 = #i) ∨ (d ≤ i ∧ T2 = #(i + e)).
106 #d #e #T1 #T2 * -d e T1 T2
107 [ #k #d #e #i #H destruct
108 | #j #d #e #Hj #i #Hi destruct /3/
109 | #j #d #e #Hj #i #Hi destruct /3/
110 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
111 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
115 lemma lift_inv_lref1: ∀d,e,T2,i. ↑[d,e] #i ≡ T2 →
116 (i < d ∧ T2 = #i) ∨ (d ≤ i ∧ T2 = #(i + e)).
119 lemma lift_inv_lref1_lt: ∀d,e,T2,i. ↑[d,e] #i ≡ T2 → i < d → T2 = #i.
120 #d #e #T2 #i #H elim (lift_inv_lref1 … H) -H * //
121 #Hdi #_ #Hid lapply (le_to_lt_to_lt … Hdi Hid) -Hdi Hid #Hdd
122 elim (lt_refl_false … Hdd)
125 lemma lift_inv_lref1_ge: ∀d,e,T2,i. ↑[d,e] #i ≡ T2 → d ≤ i → T2 = #(i + e).
126 #d #e #T2 #i #H elim (lift_inv_lref1 … H) -H * //
127 #Hid #_ #Hdi lapply (le_to_lt_to_lt … Hdi Hid) -Hdi Hid #Hdd
128 elim (lt_refl_false … Hdd)
131 lemma lift_inv_bind1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
132 ∀I,V1,U1. T1 = 𝕓{I} V1.U1 →
133 ∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 &
135 #d #e #T1 #T2 * -d e T1 T2
136 [ #k #d #e #I #V1 #U1 #H destruct
137 | #i #d #e #_ #I #V1 #U1 #H destruct
138 | #i #d #e #_ #I #V1 #U1 #H destruct
139 | #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V1 #U1 #H destruct /2 width=5/
140 | #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V1 #U1 #H destruct
144 lemma lift_inv_bind1: ∀d,e,T2,I,V1,U1. ↑[d,e] 𝕓{I} V1. U1 ≡ T2 →
145 ∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 &
149 lemma lift_inv_flat1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
150 ∀I,V1,U1. T1 = 𝕗{I} V1.U1 →
151 ∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d,e] U1 ≡ U2 &
153 #d #e #T1 #T2 * -d e T1 T2
154 [ #k #d #e #I #V1 #U1 #H destruct
155 | #i #d #e #_ #I #V1 #U1 #H destruct
156 | #i #d #e #_ #I #V1 #U1 #H destruct
157 | #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V1 #U1 #H destruct
158 | #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V1 #U1 #H destruct /2 width=5/
162 lemma lift_inv_flat1: ∀d,e,T2,I,V1,U1. ↑[d,e] 𝕗{I} V1. U1 ≡ T2 →
163 ∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d,e] U1 ≡ U2 &
167 lemma lift_inv_sort2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀k. T2 = ⋆k → T1 = ⋆k.
168 #d #e #T1 #T2 * -d e T1 T2 //
169 [ #i #d #e #_ #k #H destruct
170 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
171 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
175 lemma lift_inv_sort2: ∀d,e,T1,k. ↑[d,e] T1 ≡ ⋆k → T1 = ⋆k.
178 lemma lift_inv_lref2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀i. T2 = #i →
179 (i < d ∧ T1 = #i) ∨ (d + e ≤ i ∧ T1 = #(i - e)).
180 #d #e #T1 #T2 * -d e T1 T2
181 [ #k #d #e #i #H destruct
182 | #j #d #e #Hj #i #Hi destruct /3/
183 | #j #d #e #Hj #i #Hi destruct <minus_plus_m_m /4/
184 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
185 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
189 lemma lift_inv_lref2: ∀d,e,T1,i. ↑[d,e] T1 ≡ #i →
190 (i < d ∧ T1 = #i) ∨ (d + e ≤ i ∧ T1 = #(i - e)).
193 lemma lift_inv_lref2_lt: ∀d,e,T1,i. ↑[d,e] T1 ≡ #i → i < d → T1 = #i.
194 #d #e #T1 #i #H elim (lift_inv_lref2 … H) -H * //
195 #Hdi #_ #Hid lapply (le_to_lt_to_lt … Hdi Hid) -Hdi Hid #Hdd
196 elim (plus_lt_false … Hdd)
199 lemma lift_inv_lref2_ge: ∀d,e,T1,i. ↑[d,e] T1 ≡ #i → d + e ≤ i → T1 = #(i - e).
200 #d #e #T1 #i #H elim (lift_inv_lref2 … H) -H * //
201 #Hid #_ #Hdi lapply (le_to_lt_to_lt … Hdi Hid) -Hdi Hid #Hdd
202 elim (plus_lt_false … Hdd)
205 lemma lift_inv_bind2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
206 ∀I,V2,U2. T2 = 𝕓{I} V2.U2 →
207 ∃∃V1,U1. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 &
209 #d #e #T1 #T2 * -d e T1 T2
210 [ #k #d #e #I #V2 #U2 #H destruct
211 | #i #d #e #_ #I #V2 #U2 #H destruct
212 | #i #d #e #_ #I #V2 #U2 #H destruct
213 | #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V2 #U2 #H destruct /2 width=5/
214 | #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V2 #U2 #H destruct
218 lemma lift_inv_bind2: ∀d,e,T1,I,V2,U2. ↑[d,e] T1 ≡ 𝕓{I} V2. U2 →
219 ∃∃V1,U1. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 &
223 lemma lift_inv_flat2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
224 ∀I,V2,U2. T2 = 𝕗{I} V2.U2 →
225 ∃∃V1,U1. ↑[d,e] V1 ≡ V2 & ↑[d,e] U1 ≡ U2 &
227 #d #e #T1 #T2 * -d e T1 T2
228 [ #k #d #e #I #V2 #U2 #H destruct
229 | #i #d #e #_ #I #V2 #U2 #H destruct
230 | #i #d #e #_ #I #V2 #U2 #H destruct
231 | #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V2 #U2 #H destruct
232 | #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V2 #U2 #H destruct /2 width = 5/
236 lemma lift_inv_flat2: ∀d,e,T1,I,V2,U2. ↑[d,e] T1 ≡ 𝕗{I} V2. U2 →
237 ∃∃V1,U1. ↑[d,e] V1 ≡ V2 & ↑[d,e] U1 ≡ U2 &