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/language/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_con2 : ∀I,V1,V2,T1,T2,d,e.
21 lift V1 d e V2 → lift T1 (d + 1) e T2 →
22 lift (♭I V1. T1) d e (♭I V2. T2)
25 interpretation "relocation" 'RLift T1 d e T2 = (lift T1 d e T2).
27 (* The basic properties *****************************************************)
29 lemma lift_lref_ge_minus: ∀d,e,i. d + e ≤ i → ↑[d,e] #(i - e) ≡ #i.
30 #d #e #i #H >(plus_minus_m_m i e) in ⊢ (? ? ? ? %) /3/
33 (* The basic inversion lemmas ***********************************************)
35 lemma lift_inv_sort2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀k. T2 = ⋆k → T1 = ⋆k.
36 #d #e #T1 #T2 #H elim H -H d e T1 T2 //
37 [ #i #d #e #_ #k #H destruct
38 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #k #H destruct
42 lemma lift_inv_sort2: ∀d,e,T1,k. ↑[d,e] T1 ≡ ⋆k → T1 = ⋆k.
43 #d #e #T1 #k #H lapply (lift_inv_sort2_aux … H) /2/
46 lemma lift_inv_lref2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀i. T2 = #i →
47 (i < d ∧ T1 = #i) ∨ (d + e ≤ i ∧ T1 = #(i - e)).
48 #d #e #T1 #T2 #H elim H -H d e T1 T2
49 [ #k #d #e #i #H destruct
50 | #j #d #e #Hj #i #Hi destruct /3/
51 | #j #d #e #Hj #i #Hi destruct <minus_plus_m_m /4/
52 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #i #H destruct
56 lemma lift_inv_lref2: ∀d,e,T1,i. ↑[d,e] T1 ≡ #i →
57 (i < d ∧ T1 = #i) ∨ (d + e ≤ i ∧ T1 = #(i - e)).
58 #d #e #T1 #i #H lapply (lift_inv_lref2_aux … H) /2/
61 lemma lift_inv_con22_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
62 ∀I,V2,U2. T2 = ♭I V2.U2 →
63 ∃V1,U1. ↑[d,e] V1 ≡ V2 ∧ ↑[d+1,e] U1 ≡ U2 ∧
65 #d #e #T1 #T2 #H elim H -H d e T1 T2
66 [ #k #d #e #I #V2 #U2 #H destruct
67 | #i #d #e #_ #I #V2 #U2 #H destruct
68 | #i #d #e #_ #I #V2 #U2 #H destruct
69 | #J #W1 #W2 #T1 #T2 #d #e #HW #HT #_ #_ #I #V2 #U2 #H destruct /5/
72 lemma lift_inv_con22: ∀d,e,T1,I,V2,U2. ↑[d,e] T1 ≡ ♭I V2. U2 →
73 ∃V1,U1. ↑[d,e] V1 ≡ V2 ∧ ↑[d+1,e] U1 ≡ U2 ∧
75 #d #e #T1 #I #V2 #U2 #H lapply (lift_inv_con22_aux … H) /2/
78 (* the main properies *******************************************************)
80 theorem lift_trans_rev: ∀d1,e1,T1,T. ↑[d1,e1] T1 ≡ T →
81 ∀d2,e2,T2. ↑[d2 + e1, e2] T2 ≡ T →
83 ∃T0. ↑[d1, e1] T0 ≡ T2 ∧ ↑[d2, e2] T0 ≡ T1.
84 #d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T
85 [ #k #d1 #e1 #d2 #e2 #T2 #Hk #Hd12
86 lapply (lift_inv_sort2 … Hk) -Hk #Hk destruct -T2 /3/
87 | #i #d1 #e1 #Hid1 #d2 #e2 #T2 #Hi #Hd12
88 lapply (lift_inv_lref2 … Hi) -Hi #H
89 elim H -H #H elim H -H #Hid2 #Hi (**) (* decompose*)
91 [ -Hid2 /5/ (**) (* /4/ *)
92 | elim (lt_false d1 ?)
93 @(le_to_lt_to_lt … Hd12) -Hd12 @(le_to_lt_to_lt … Hid1) -Hid1 /2/
95 | #i #d1 #e1 #Hid1 #d2 #e2 #T2 #Hi #Hd12
96 lapply (lift_inv_lref2 … Hi) -Hi #H
97 elim H -H #H elim H -H #Hid2 #Hi (**) (* decompose*)
99 [ -Hd12; lapply (lt_plus_to_lt_l … Hid2) -Hid2 #Hid2 /4/
100 | -Hid1; lapply (arith1 … Hid2) -Hid2 #Hid2
101 @(ex_intro … #(i - e2)) @conj
102 [ >le_plus_minus_comm [ @lift_lref_ge @(transitive_le … Hd12) -Hd12 /2/ | -Hd12 /2/ ]
103 | -Hd12 >(plus_minus_m_m i e2) in ⊢ (? ? ? ? %) /3/
106 | #I #W1 #W #U1 #U #d1 #e1 #_ #_ #IHW #IHU #d2 #e2 #T2 #H #Hd12
107 lapply (lift_inv_con22 … H) -H #H
108 elim H -H #W2 #H elim H -H #U2 #H elim H -H #H elim H -H #HW2 #HU2 #H (**) (* decompose*)
110 elim (IHW … HW2 ?) // -IHW HW2 #W0 #H
111 elim H -H #HW2 #HW1 (**) (* decompose*)
112 >plus_plus_comm_23 in HU2 #HU2 elim (IHU … HU2 ?) /2/ -IHU HU2 Hd12 #U0 #H
113 elim H -H #HU2 #HU1 (**) (* decompose*)
114 @(ex_intro … ♭I W0. U0) /3/ (**) (* /4/ *)
118 theorem lift_free: ∀d1,e2,T1,T2. ↑[d1, e2] T1 ≡ T2 → ∀d2,e1.
119 d1 ≤ d2 → d2 ≤ d1 + e1 → e1 ≤ e2 →
120 ∃T. ↑[d1, e1] T1 ≡ T ∧ ↑[d2, e2 - e1] T ≡ T2.
121 #d1 #e2 #T1 #T2 #H elim H -H d1 e2 T1 T2
123 | #i #d1 #e2 #Hid1 #d2 #e1 #Hd12 #_ #_
124 lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2 /4/
125 | #i #d1 #e2 #Hid1 #d2 #e1 #_ #Hd21 #He12
126 lapply (transitive_le …(i+e1) Hd21 ?) /2/ -Hd21 #Hd21
127 <(plus_plus_minus_m_m e1 e2 i) //
128 @(ex_intro … #(i+e1)) /3/ (**) (* /4/ *)
129 | #I #V1 #V2 #T1 #T2 #d1 #e2 #_ #_ #IHV #IHT #d2 #e1 #Hd12 #Hd21 #He12
130 elim (IHV … Hd12 Hd21 He12) -IHV #V0 #H
131 elim H -H #HV0a #HV0b (**) (* decompose *)
132 elim (IHT (d2+1) … ? ? He12) /2/ -IHT Hd12 Hd21 He12 #T0 #H
133 elim H -H #HT0a #HT0b (**) (* decompose *)
134 @(ex_intro … ♭I V0.T0) /3/ (**) (* /4/ *)