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 (* The basic inversion lemmas ***********************************************)
38 lemma lift_inv_sort1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀k. T1 = ⋆k → T2 = ⋆k.
39 #d #e #T1 #T2 #H elim H -H d e T1 T2 //
40 [ #i #d #e #_ #k #H destruct
41 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #k #H destruct
42 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #k #H destruct
46 lemma lift_inv_sort1: ∀d,e,T2,k. ↑[d,e] ⋆k ≡ T2 → T2 = ⋆k.
47 #d #e #T2 #k #H lapply (lift_inv_sort1_aux … H) /2/
50 lemma lift_inv_lref1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀i. T1 = #i →
51 (i < d ∧ T2 = #i) ∨ (d ≤ i ∧ T2 = #(i + e)).
52 #d #e #T1 #T2 #H elim H -H d e T1 T2
53 [ #k #d #e #i #H destruct
54 | #j #d #e #Hj #i #Hi destruct /3/
55 | #j #d #e #Hj #i #Hi destruct /3/
56 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #i #H destruct
57 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #i #H destruct
61 lemma lift_inv_lref1: ∀d,e,T2,i. ↑[d,e] #i ≡ T2 →
62 (i < d ∧ T2 = #i) ∨ (d ≤ i ∧ T2 = #(i + e)).
63 #d #e #T2 #i #H lapply (lift_inv_lref1_aux … H) /2/
66 lemma lift_inv_bind1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
67 ∀I,V1,U1. T1 = 𝕓{I} V1.U1 →
68 ∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 &
70 #d #e #T1 #T2 #H elim H -H d e T1 T2
71 [ #k #d #e #I #V1 #U1 #H destruct
72 | #i #d #e #_ #I #V1 #U1 #H destruct
73 | #i #d #e #_ #I #V1 #U1 #H destruct
74 | #J #W1 #W2 #T1 #T2 #d #e #HW #HT #_ #_ #I #V1 #U1 #H destruct /2 width=5/
75 | #J #W1 #W2 #T1 #T2 #d #e #HW #HT #_ #_ #I #V1 #U1 #H destruct
79 lemma lift_inv_bind1: ∀d,e,T2,I,V1,U1. ↑[d,e] 𝕓{I} V1. U1 ≡ T2 →
80 ∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 &
82 #d #e #T2 #I #V1 #U1 #H lapply (lift_inv_bind1_aux … H) /2/
85 lemma lift_inv_flat1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
86 ∀I,V1,U1. T1 = 𝕗{I} V1.U1 →
87 ∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d,e] U1 ≡ U2 &
89 #d #e #T1 #T2 #H elim H -H d e T1 T2
90 [ #k #d #e #I #V1 #U1 #H destruct
91 | #i #d #e #_ #I #V1 #U1 #H destruct
92 | #i #d #e #_ #I #V1 #U1 #H destruct
93 | #J #W1 #W2 #T1 #T2 #d #e #HW #HT #_ #_ #I #V1 #U1 #H destruct
94 | #J #W1 #W2 #T1 #T2 #d #e #HW #HT #_ #_ #I #V1 #U1 #H destruct /2 width=5/
98 lemma lift_inv_flat1: ∀d,e,T2,I,V1,U1. ↑[d,e] 𝕗{I} V1. U1 ≡ T2 →
99 ∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d,e] U1 ≡ U2 &
101 #d #e #T2 #I #V1 #U1 #H lapply (lift_inv_flat1_aux … H) /2/
104 lemma lift_inv_sort2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀k. T2 = ⋆k → T1 = ⋆k.
105 #d #e #T1 #T2 #H elim H -H d e T1 T2 //
106 [ #i #d #e #_ #k #H destruct
107 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #k #H destruct
108 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #k #H destruct
112 lemma lift_inv_sort2: ∀d,e,T1,k. ↑[d,e] T1 ≡ ⋆k → T1 = ⋆k.
113 #d #e #T1 #k #H lapply (lift_inv_sort2_aux … H) /2/
116 lemma lift_inv_lref2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀i. T2 = #i →
117 (i < d ∧ T1 = #i) ∨ (d + e ≤ i ∧ T1 = #(i - e)).
118 #d #e #T1 #T2 #H elim H -H d e T1 T2
119 [ #k #d #e #i #H destruct
120 | #j #d #e #Hj #i #Hi destruct /3/
121 | #j #d #e #Hj #i #Hi destruct <minus_plus_m_m /4/
122 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #i #H destruct
123 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #i #H destruct
127 lemma lift_inv_lref2: ∀d,e,T1,i. ↑[d,e] T1 ≡ #i →
128 (i < d ∧ T1 = #i) ∨ (d + e ≤ i ∧ T1 = #(i - e)).
129 #d #e #T1 #i #H lapply (lift_inv_lref2_aux … H) /2/
132 lemma lift_inv_bind2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
133 ∀I,V2,U2. T2 = 𝕓{I} V2.U2 →
134 ∃∃V1,U1. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 &
136 #d #e #T1 #T2 #H elim H -H d e T1 T2
137 [ #k #d #e #I #V2 #U2 #H destruct
138 | #i #d #e #_ #I #V2 #U2 #H destruct
139 | #i #d #e #_ #I #V2 #U2 #H destruct
140 | #J #W1 #W2 #T1 #T2 #d #e #HW #HT #_ #_ #I #V2 #U2 #H destruct /2 width=5/
141 | #J #W1 #W2 #T1 #T2 #d #e #HW #HT #_ #_ #I #V2 #U2 #H destruct
145 lemma lift_inv_bind2: ∀d,e,T1,I,V2,U2. ↑[d,e] T1 ≡ 𝕓{I} V2. U2 →
146 ∃∃V1,U1. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 &
148 #d #e #T1 #I #V2 #U2 #H lapply (lift_inv_bind2_aux … H) /2/
151 lemma lift_inv_flat2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
152 ∀I,V2,U2. T2 = 𝕗{I} V2.U2 →
153 ∃∃V1,U1. ↑[d,e] V1 ≡ V2 & ↑[d,e] U1 ≡ U2 &
155 #d #e #T1 #T2 #H elim H -H d e T1 T2
156 [ #k #d #e #I #V2 #U2 #H destruct
157 | #i #d #e #_ #I #V2 #U2 #H destruct
158 | #i #d #e #_ #I #V2 #U2 #H destruct
159 | #J #W1 #W2 #T1 #T2 #d #e #HW #HT #_ #_ #I #V2 #U2 #H destruct
160 | #J #W1 #W2 #T1 #T2 #d #e #HW #HT #_ #_ #I #V2 #U2 #H destruct /2 width = 5/
164 lemma lift_inv_flat2: ∀d,e,T1,I,V2,U2. ↑[d,e] T1 ≡ 𝕗{I} V2. U2 →
165 ∃∃V1,U1. ↑[d,e] V1 ≡ V2 & ↑[d,e] U1 ≡ U2 &
167 #d #e #T1 #I #V2 #U2 #H lapply (lift_inv_flat2_aux … H) /2/