2 ||M|| This file is part of HELM, an Hypertextual, Electronic
3 ||A|| Library of Mathematics, developed at the Computer Science
4 ||T|| Department of the University of Bologna, Italy.
7 ||A|| This file is distributed under the terms of the
8 \ / GNU General Public License Version 2
10 V_______________________________________________________________ *)
12 include "lambda-delta/syntax/term.ma".
14 (* RELOCATION ***************************************************************)
16 inductive lift: term → nat → nat → term → Prop ≝
17 | lift_sort : ∀k,d,e. lift (⋆k) d e (⋆k)
18 | lift_lref_lt: ∀i,d,e. i < d → lift (#i) d e (#i)
19 | lift_lref_ge: ∀i,d,e. d ≤ i → lift (#i) d e (#(i + e))
20 | lift_bind : ∀I,V1,V2,T1,T2,d,e.
21 lift V1 d e V2 → lift T1 (d + 1) e T2 →
22 lift (𝕓{I} V1. T1) d e (𝕓{I} V2. T2)
23 | lift_flat : ∀I,V1,V2,T1,T2,d,e.
24 lift V1 d e V2 → lift T1 d e T2 →
25 lift (𝕗{I} V1. T1) d e (𝕗{I} V2. T2)
28 interpretation "relocation" 'RLift T1 d e T2 = (lift T1 d e T2).
30 (* The basic properties *****************************************************)
32 lemma lift_lref_ge_minus: ∀d,e,i. d + e ≤ i → ↑[d, e] #(i - e) ≡ #i.
33 #d #e #i #H >(plus_minus_m_m i e) in ⊢ (? ? ? ? %) /3/
36 lemma lift_refl: ∀T,d. ↑[d, 0] T ≡ T.
39 | #i #d elim (lt_or_ge i d) /2/
44 (* The basic inversion lemmas ***********************************************)
46 lemma lift_inv_refl_aux: ∀d,e,T1,T2. ↑[d, e] T1 ≡ T2 → e = 0 → T1 = T2.
47 #d #e #T1 #T2 #H elim H -H d e T1 T2 /3/
50 lemma lift_inv_refl: ∀d,T1,T2. ↑[d, 0] T1 ≡ T2 → T1 = T2.
53 lemma lift_inv_sort1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀k. T1 = ⋆k → T2 = ⋆k.
54 #d #e #T1 #T2 #H elim H -H d e T1 T2 //
55 [ #i #d #e #_ #k #H destruct
56 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #k #H destruct
57 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #k #H destruct
61 lemma lift_inv_sort1: ∀d,e,T2,k. ↑[d,e] ⋆k ≡ T2 → T2 = ⋆k.
64 lemma lift_inv_lref1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀i. T1 = #i →
65 (i < d ∧ T2 = #i) ∨ (d ≤ i ∧ T2 = #(i + e)).
66 #d #e #T1 #T2 #H elim H -H d e T1 T2
67 [ #k #d #e #i #H destruct
68 | #j #d #e #Hj #i #Hi destruct /3/
69 | #j #d #e #Hj #i #Hi destruct /3/
70 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #i #H destruct
71 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #i #H destruct
75 lemma lift_inv_lref1: ∀d,e,T2,i. ↑[d,e] #i ≡ T2 →
76 (i < d ∧ T2 = #i) ∨ (d ≤ i ∧ T2 = #(i + e)).
79 lemma lift_inv_lref1_lt: ∀d,e,T2,i. ↑[d,e] #i ≡ T2 → i < d → T2 = #i.
80 #d #e #T2 #i #H elim (lift_inv_lref1 … H) -H * //
81 #Hdi #_ #Hid lapply (le_to_lt_to_lt … Hdi Hid) -Hdi Hid #Hdd
85 lemma lift_inv_lref1_ge: ∀d,e,T2,i. ↑[d,e] #i ≡ T2 → d ≤ i → T2 = #(i + e).
86 #d #e #T2 #i #H elim (lift_inv_lref1 … H) -H * //
87 #Hid #_ #Hdi lapply (le_to_lt_to_lt … Hdi Hid) -Hdi Hid #Hdd
91 lemma lift_inv_bind1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
92 ∀I,V1,U1. T1 = 𝕓{I} V1.U1 →
93 ∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 &
95 #d #e #T1 #T2 #H elim H -H d e T1 T2
96 [ #k #d #e #I #V1 #U1 #H destruct
97 | #i #d #e #_ #I #V1 #U1 #H destruct
98 | #i #d #e #_ #I #V1 #U1 #H destruct
99 | #J #W1 #W2 #T1 #T2 #d #e #HW #HT #_ #_ #I #V1 #U1 #H destruct /2 width=5/
100 | #J #W1 #W2 #T1 #T2 #d #e #HW #HT #_ #_ #I #V1 #U1 #H destruct
104 lemma lift_inv_bind1: ∀d,e,T2,I,V1,U1. ↑[d,e] 𝕓{I} V1. U1 ≡ T2 →
105 ∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 &
109 lemma lift_inv_flat1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
110 ∀I,V1,U1. T1 = 𝕗{I} V1.U1 →
111 ∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d,e] U1 ≡ U2 &
113 #d #e #T1 #T2 #H elim H -H d e T1 T2
114 [ #k #d #e #I #V1 #U1 #H destruct
115 | #i #d #e #_ #I #V1 #U1 #H destruct
116 | #i #d #e #_ #I #V1 #U1 #H destruct
117 | #J #W1 #W2 #T1 #T2 #d #e #HW #HT #_ #_ #I #V1 #U1 #H destruct
118 | #J #W1 #W2 #T1 #T2 #d #e #HW #HT #_ #_ #I #V1 #U1 #H destruct /2 width=5/
122 lemma lift_inv_flat1: ∀d,e,T2,I,V1,U1. ↑[d,e] 𝕗{I} V1. U1 ≡ T2 →
123 ∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d,e] U1 ≡ U2 &
127 lemma lift_inv_sort2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀k. T2 = ⋆k → T1 = ⋆k.
128 #d #e #T1 #T2 #H elim H -H d e T1 T2 //
129 [ #i #d #e #_ #k #H destruct
130 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #k #H destruct
131 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #k #H destruct
135 lemma lift_inv_sort2: ∀d,e,T1,k. ↑[d,e] T1 ≡ ⋆k → T1 = ⋆k.
138 lemma lift_inv_lref2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀i. T2 = #i →
139 (i < d ∧ T1 = #i) ∨ (d + e ≤ i ∧ T1 = #(i - e)).
140 #d #e #T1 #T2 #H elim H -H d e T1 T2
141 [ #k #d #e #i #H destruct
142 | #j #d #e #Hj #i #Hi destruct /3/
143 | #j #d #e #Hj #i #Hi destruct <minus_plus_m_m /4/
144 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #i #H destruct
145 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #i #H destruct
149 lemma lift_inv_lref2: ∀d,e,T1,i. ↑[d,e] T1 ≡ #i →
150 (i < d ∧ T1 = #i) ∨ (d + e ≤ i ∧ T1 = #(i - e)).
153 lemma lift_inv_lref2_lt: ∀d,e,T1,i. ↑[d,e] T1 ≡ #i → i < d → T1 = #i.
154 #d #e #T1 #i #H elim (lift_inv_lref2 … H) -H * //
155 #Hdi #_ #Hid lapply (le_to_lt_to_lt … Hdi Hid) -Hdi Hid #Hdd
156 elim (plus_lt_false … Hdd)
159 lemma lift_inv_lref2_ge: ∀d,e,T1,i. ↑[d,e] T1 ≡ #i → d + e ≤ i → T1 = #(i - e).
160 #d #e #T1 #i #H elim (lift_inv_lref2 … H) -H * //
161 #Hid #_ #Hdi lapply (le_to_lt_to_lt … Hdi Hid) -Hdi Hid #Hdd
162 elim (plus_lt_false … Hdd)
165 lemma lift_inv_bind2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
166 ∀I,V2,U2. T2 = 𝕓{I} V2.U2 →
167 ∃∃V1,U1. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 &
169 #d #e #T1 #T2 #H elim H -H d e T1 T2
170 [ #k #d #e #I #V2 #U2 #H destruct
171 | #i #d #e #_ #I #V2 #U2 #H destruct
172 | #i #d #e #_ #I #V2 #U2 #H destruct
173 | #J #W1 #W2 #T1 #T2 #d #e #HW #HT #_ #_ #I #V2 #U2 #H destruct /2 width=5/
174 | #J #W1 #W2 #T1 #T2 #d #e #HW #HT #_ #_ #I #V2 #U2 #H destruct
178 lemma lift_inv_bind2: ∀d,e,T1,I,V2,U2. ↑[d,e] T1 ≡ 𝕓{I} V2. U2 →
179 ∃∃V1,U1. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 &
183 lemma lift_inv_flat2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
184 ∀I,V2,U2. T2 = 𝕗{I} V2.U2 →
185 ∃∃V1,U1. ↑[d,e] V1 ≡ V2 & ↑[d,e] U1 ≡ U2 &
187 #d #e #T1 #T2 #H elim H -H d e T1 T2
188 [ #k #d #e #I #V2 #U2 #H destruct
189 | #i #d #e #_ #I #V2 #U2 #H destruct
190 | #i #d #e #_ #I #V2 #U2 #H destruct
191 | #J #W1 #W2 #T1 #T2 #d #e #HW #HT #_ #_ #I #V2 #U2 #H destruct
192 | #J #W1 #W2 #T1 #T2 #d #e #HW #HT #_ #_ #I #V2 #U2 #H destruct /2 width = 5/
196 lemma lift_inv_flat2: ∀d,e,T1,I,V2,U2. ↑[d,e] T1 ≡ 𝕗{I} V2. U2 →
197 ∃∃V1,U1. ↑[d,e] V1 ≡ V2 & ↑[d,e] U1 ≡ U2 &