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 (**************************************************************************)
15 include "lambda-delta/substitution/lift_defs.ma".
17 (* RELOCATION ***************************************************************)
19 (* Main properies ***********************************************************)
21 lemma lift_inj: ∀d,e,T1,U. ↑[d,e] T1 ≡ U → ∀T2. ↑[d,e] T2 ≡ U → T1 = T2.
22 #d #e #T1 #U #H elim H -H d e T1 U
24 lapply (lift_inv_sort2 … HX) -HX //
25 | #i #d #e #Hid #X #HX
26 lapply (lift_inv_lref2_lt … HX ?) -HX //
27 | #i #d #e #Hdi #X #HX
28 lapply (lift_inv_lref2_ge … HX ?) -HX /2/
29 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
30 elim (lift_inv_bind2 … HX) -HX #V #T #HV1 #HT1 #HX destruct -X /3/
31 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
32 elim (lift_inv_flat2 … HX) -HX #V #T #HV1 #HT1 #HX destruct -X /3/
36 lemma lift_div_le: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
37 ∀d2,e2,T2. ↑[d2 + e1, e2] T2 ≡ T →
39 ∃∃T0. ↑[d1, e1] T0 ≡ T2 & ↑[d2, e2] T0 ≡ T1.
40 #d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T
41 [ #k #d1 #e1 #d2 #e2 #T2 #Hk #Hd12
42 lapply (lift_inv_sort2 … Hk) -Hk #Hk destruct -T2 /3/
43 | #i #d1 #e1 #Hid1 #d2 #e2 #T2 #Hi #Hd12
44 lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2
45 lapply (lift_inv_lref2_lt … Hi ?) -Hi /3/
46 | #i #d1 #e1 #Hid1 #d2 #e2 #T2 #Hi #Hd12
47 elim (lift_inv_lref2 … Hi) -Hi * #Hid2 #H destruct -T2
48 [ -Hd12; lapply (lt_plus_to_lt_l … Hid2) -Hid2 #Hid2 /3/
49 | -Hid1; lapply (arith1 … Hid2) -Hid2 #Hid2
50 @(ex2_1_intro … #(i - e2))
51 [ >le_plus_minus_comm [ @lift_lref_ge @(transitive_le … Hd12) -Hd12 /2/ | -Hd12 /2/ ]
52 | -Hd12 >(plus_minus_m_m i e2) in ⊢ (? ? ? ? %) /3/
55 | #I #W1 #W #U1 #U #d1 #e1 #_ #_ #IHW #IHU #d2 #e2 #T2 #H #Hd12
56 lapply (lift_inv_bind2 … H) -H * #W2 #U2 #HW2 #HU2 #H destruct -T2;
57 elim (IHW … HW2 ?) // -IHW HW2 #W0 #HW2 #HW1
58 >plus_plus_comm_23 in HU2 #HU2 elim (IHU … HU2 ?) /3 width = 5/
59 | #I #W1 #W #U1 #U #d1 #e1 #_ #_ #IHW #IHU #d2 #e2 #T2 #H #Hd12
60 lapply (lift_inv_flat2 … H) -H * #W2 #U2 #HW2 #HU2 #H destruct -T2;
61 elim (IHW … HW2 ?) // -IHW HW2 #W0 #HW2 #HW1
62 elim (IHU … HU2 ?) /3 width = 5/
66 lemma lift_free: ∀d1,e2,T1,T2. ↑[d1, e2] T1 ≡ T2 → ∀d2,e1.
67 d1 ≤ d2 → d2 ≤ d1 + e1 → e1 ≤ e2 →
68 ∃∃T. ↑[d1, e1] T1 ≡ T & ↑[d2, e2 - e1] T ≡ T2.
69 #d1 #e2 #T1 #T2 #H elim H -H d1 e2 T1 T2
71 | #i #d1 #e2 #Hid1 #d2 #e1 #Hd12 #_ #_
72 lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2 /4/
73 | #i #d1 #e2 #Hid1 #d2 #e1 #_ #Hd21 #He12
74 lapply (transitive_le …(i+e1) Hd21 ?) /2/ -Hd21 #Hd21
75 <(arith_d1 i e2 e1) // /3/
76 | #I #V1 #V2 #T1 #T2 #d1 #e2 #_ #_ #IHV #IHT #d2 #e1 #Hd12 #Hd21 #He12
77 elim (IHV … Hd12 Hd21 He12) -IHV #V0 #HV0a #HV0b
78 elim (IHT (d2+1) … ? ? He12) /3 width = 5/
79 | #I #V1 #V2 #T1 #T2 #d1 #e2 #_ #_ #IHV #IHT #d2 #e1 #Hd12 #Hd21 #He12
80 elim (IHV … Hd12 Hd21 He12) -IHV #V0 #HV0a #HV0b
81 elim (IHT d2 … ? ? He12) /3 width = 5/
85 lemma lift_mono: ∀d,e,T,U1. ↑[d,e] T ≡ U1 → ∀U2. ↑[d,e] T ≡ U2 → U1 = U2.
86 #d #e #T #U1 #H elim H -H d e T U1
88 lapply (lift_inv_sort1 … HX) -HX //
89 | #i #d #e #Hid #X #HX
90 lapply (lift_inv_lref1_lt … HX ?) -HX //
91 | #i #d #e #Hdi #X #HX
92 lapply (lift_inv_lref1_ge … HX ?) -HX //
93 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
94 elim (lift_inv_bind1 … HX) -HX #V #T #HV1 #HT1 #HX destruct -X /3/
95 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
96 elim (lift_inv_flat1 … HX) -HX #V #T #HV1 #HT1 #HX destruct -X /3/
100 lemma lift_trans_be: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
101 ∀d2,e2,T2. ↑[d2, e2] T ≡ T2 →
102 d1 ≤ d2 → d2 ≤ d1 + e1 → ↑[d1, e1 + e2] T1 ≡ T2.
103 #d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T
104 [ #k #d1 #e1 #d2 #e2 #T2 #HT2 #_ #_
105 >(lift_inv_sort1 … HT2) -HT2 //
106 | #i #d1 #e1 #Hid1 #d2 #e2 #T2 #HT2 #Hd12 #_
107 lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2
108 lapply (lift_inv_lref1_lt … HT2 Hid2) /2/
109 | #i #d1 #e1 #Hid1 #d2 #e2 #T2 #HT2 #_ #Hd21
110 lapply (lift_inv_lref1_ge … HT2 ?) -HT2
111 [ @(transitive_le … Hd21 ?) -Hd21 /2/
114 | #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd12 #Hd21
115 elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
116 lapply (IHV12 … HV20 ? ?) // -IHV12 HV20 #HV10
117 lapply (IHT12 … HT20 ? ?) /2/
118 | #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd12 #Hd21
119 elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
120 lapply (IHV12 … HV20 ? ?) // -IHV12 HV20 #HV10
121 lapply (IHT12 … HT20 ? ?) /2/
125 lemma lift_trans_le: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
126 ∀d2,e2,T2. ↑[d2, e2] T ≡ T2 → d2 ≤ d1 →
127 ∃∃T0. ↑[d2, e2] T1 ≡ T0 & ↑[d1 + e2, e1] T0 ≡ T2.
128 #d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T
129 [ #k #d1 #e1 #d2 #e2 #X #HX #_
130 >(lift_inv_sort1 … HX) -HX /2/
131 | #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #_
132 lapply (lt_to_le_to_lt … (d1+e2) Hid1 ?) // #Hie2
133 elim (lift_inv_lref1 … HX) -HX * #Hid2 #HX destruct -X /4/
134 | #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #Hd21
135 lapply (transitive_le … Hd21 Hid1) -Hd21 #Hid2
136 lapply (lift_inv_lref1_ge … HX ?) -HX /2/ #HX destruct -X;
137 >plus_plus_comm_23 /4/
138 | #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd21
139 elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
140 elim (IHV12 … HV20 ?) -IHV12 HV20 //
141 elim (IHT12 … HT20 ?) -IHT12 HT20 /3 width=5/
142 | #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd21
143 elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
144 elim (IHV12 … HV20 ?) -IHV12 HV20 //
145 elim (IHT12 … HT20 ?) -IHT12 HT20 /3 width=5/
149 lemma lift_trans_ge: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
150 ∀d2,e2,T2. ↑[d2, e2] T ≡ T2 → d1 + e1 ≤ d2 →
151 ∃∃T0. ↑[d2 - e1, e2] T1 ≡ T0 & ↑[d1, e1] T0 ≡ T2.
152 #d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T
153 [ #k #d1 #e1 #d2 #e2 #X #HX #_
154 >(lift_inv_sort1 … HX) -HX /2/
155 | #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #Hded
156 lapply (lt_to_le_to_lt … (d1+e1) Hid1 ?) // #Hid1e
157 lapply (lt_to_le_to_lt … (d2-e1) Hid1 ?) /2/ #Hid2e
158 lapply (lt_to_le_to_lt … Hid1e Hded) -Hid1e Hded #Hid2
159 lapply (lift_inv_lref1_lt … HX ?) -HX // #HX destruct -X /3/
160 | #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #_
161 elim (lift_inv_lref1 … HX) -HX * #Hied #HX destruct -X;
162 [2: >plus_plus_comm_23] /4/
163 | #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hded
164 elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
165 elim (IHV12 … HV20 ?) -IHV12 HV20 //
166 elim (IHT12 … HT20 ?) -IHT12 HT20 /2/ #T
167 <plus_minus /3 width=5/
168 | #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hded
169 elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
170 elim (IHV12 … HV20 ?) -IHV12 HV20 //
171 elim (IHT12 … HT20 ?) -IHT12 HT20 /3 width=5/