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 "Basic-2/substitution/lift.ma".
17 (* RELOCATION ***************************************************************)
19 (* Main properies ***********************************************************)
21 theorem 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 theorem 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 theorem lift_mono: ∀d,e,T,U1. ↑[d,e] T ≡ U1 → ∀U2. ↑[d,e] T ≡ U2 → U1 = U2.
67 #d #e #T #U1 #H elim H -H d e T U1
69 lapply (lift_inv_sort1 … HX) -HX //
70 | #i #d #e #Hid #X #HX
71 lapply (lift_inv_lref1_lt … HX ?) -HX //
72 | #i #d #e #Hdi #X #HX
73 lapply (lift_inv_lref1_ge … HX ?) -HX //
74 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
75 elim (lift_inv_bind1 … HX) -HX #V #T #HV1 #HT1 #HX destruct -X /3/
76 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
77 elim (lift_inv_flat1 … HX) -HX #V #T #HV1 #HT1 #HX destruct -X /3/
81 theorem lift_trans_be: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
82 ∀d2,e2,T2. ↑[d2, e2] T ≡ T2 →
83 d1 ≤ d2 → d2 ≤ d1 + e1 → ↑[d1, e1 + e2] T1 ≡ T2.
84 #d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T
85 [ #k #d1 #e1 #d2 #e2 #T2 #HT2 #_ #_
86 >(lift_inv_sort1 … HT2) -HT2 //
87 | #i #d1 #e1 #Hid1 #d2 #e2 #T2 #HT2 #Hd12 #_
88 lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2
89 lapply (lift_inv_lref1_lt … HT2 Hid2) /2/
90 | #i #d1 #e1 #Hid1 #d2 #e2 #T2 #HT2 #_ #Hd21
91 lapply (lift_inv_lref1_ge … HT2 ?) -HT2
92 [ @(transitive_le … Hd21 ?) -Hd21 /2/
95 | #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd12 #Hd21
96 elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
97 lapply (IHV12 … HV20 ? ?) // -IHV12 HV20 #HV10
98 lapply (IHT12 … HT20 ? ?) /2/
99 | #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd12 #Hd21
100 elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
101 lapply (IHV12 … HV20 ? ?) // -IHV12 HV20 #HV10
102 lapply (IHT12 … HT20 ? ?) /2/
106 theorem lift_trans_le: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
107 ∀d2,e2,T2. ↑[d2, e2] T ≡ T2 → d2 ≤ d1 →
108 ∃∃T0. ↑[d2, e2] T1 ≡ T0 & ↑[d1 + e2, e1] T0 ≡ T2.
109 #d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T
110 [ #k #d1 #e1 #d2 #e2 #X #HX #_
111 >(lift_inv_sort1 … HX) -HX /2/
112 | #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #_
113 lapply (lt_to_le_to_lt … (d1+e2) Hid1 ?) // #Hie2
114 elim (lift_inv_lref1 … HX) -HX * #Hid2 #HX destruct -X /4/
115 | #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #Hd21
116 lapply (transitive_le … Hd21 Hid1) -Hd21 #Hid2
117 lapply (lift_inv_lref1_ge … HX ?) -HX /2/ #HX destruct -X;
118 >plus_plus_comm_23 /4/
119 | #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd21
120 elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
121 elim (IHV12 … HV20 ?) -IHV12 HV20 //
122 elim (IHT12 … HT20 ?) -IHT12 HT20 /3 width=5/
123 | #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd21
124 elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
125 elim (IHV12 … HV20 ?) -IHV12 HV20 //
126 elim (IHT12 … HT20 ?) -IHT12 HT20 /3 width=5/
130 theorem lift_trans_ge: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
131 ∀d2,e2,T2. ↑[d2, e2] T ≡ T2 → d1 + e1 ≤ d2 →
132 ∃∃T0. ↑[d2 - e1, e2] T1 ≡ T0 & ↑[d1, e1] T0 ≡ T2.
133 #d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T
134 [ #k #d1 #e1 #d2 #e2 #X #HX #_
135 >(lift_inv_sort1 … HX) -HX /2/
136 | #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #Hded
137 lapply (lt_to_le_to_lt … (d1+e1) Hid1 ?) // #Hid1e
138 lapply (lt_to_le_to_lt … (d2-e1) Hid1 ?) /2/ #Hid2e
139 lapply (lt_to_le_to_lt … Hid1e Hded) -Hid1e Hded #Hid2
140 lapply (lift_inv_lref1_lt … HX ?) -HX // #HX destruct -X /3/
141 | #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #_
142 elim (lift_inv_lref1 … HX) -HX * #Hied #HX destruct -X;
143 [2: >plus_plus_comm_23] /4/
144 | #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hded
145 elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
146 elim (IHV12 … HV20 ?) -IHV12 HV20 //
147 elim (IHT12 … HT20 ?) -IHT12 HT20 /2/ #T
148 <plus_minus /3 width=5/
149 | #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hded
150 elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
151 elim (IHV12 … HV20 ?) -IHV12 HV20 //
152 elim (IHT12 … HT20 ?) -IHT12 HT20 /3 width=5/