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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 (* Basic_1: was: lift_inj *)
22 theorem lift_inj: ∀d,e,T1,U. ↑[d,e] T1 ≡ U → ∀T2. ↑[d,e] T2 ≡ U → T1 = T2.
23 #d #e #T1 #U #H elim H -H d e T1 U
25 lapply (lift_inv_sort2 … HX) -HX //
26 | #i #d #e #Hid #X #HX
27 lapply (lift_inv_lref2_lt … HX ?) -HX //
28 | #i #d #e #Hdi #X #HX
29 lapply (lift_inv_lref2_ge … HX ?) -HX /2/
30 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
31 elim (lift_inv_bind2 … HX) -HX #V #T #HV1 #HT1 #HX destruct -X /3/
32 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
33 elim (lift_inv_flat2 … HX) -HX #V #T #HV1 #HT1 #HX destruct -X /3/
37 (* Basic_1: was: lift_gen_lift *)
38 theorem lift_div_le: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
39 ∀d2,e2,T2. ↑[d2 + e1, e2] T2 ≡ T →
41 ∃∃T0. ↑[d1, e1] T0 ≡ T2 & ↑[d2, e2] T0 ≡ T1.
42 #d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T
43 [ #k #d1 #e1 #d2 #e2 #T2 #Hk #Hd12
44 lapply (lift_inv_sort2 … Hk) -Hk #Hk destruct -T2 /3/
45 | #i #d1 #e1 #Hid1 #d2 #e2 #T2 #Hi #Hd12
46 lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2
47 lapply (lift_inv_lref2_lt … Hi ?) -Hi /3/
48 | #i #d1 #e1 #Hid1 #d2 #e2 #T2 #Hi #Hd12
49 elim (lift_inv_lref2 … Hi) -Hi * #Hid2 #H destruct -T2
50 [ -Hd12; lapply (lt_plus_to_lt_l … Hid2) -Hid2 #Hid2 /3/
51 | -Hid1; lapply (arith1 … Hid2) -Hid2 #Hid2
52 @(ex2_1_intro … #(i - e2))
53 [ >le_plus_minus_comm [ @lift_lref_ge @(transitive_le … Hd12) -Hd12 /2/ | -Hd12 /2/ ]
54 | -Hd12 >(plus_minus_m_m i e2) in ⊢ (? ? ? ? %) /3/
57 | #I #W1 #W #U1 #U #d1 #e1 #_ #_ #IHW #IHU #d2 #e2 #T2 #H #Hd12
58 lapply (lift_inv_bind2 … H) -H * #W2 #U2 #HW2 #HU2 #H destruct -T2;
59 elim (IHW … HW2 ?) // -IHW HW2 #W0 #HW2 #HW1
60 >plus_plus_comm_23 in HU2 #HU2 elim (IHU … HU2 ?) /3 width = 5/
61 | #I #W1 #W #U1 #U #d1 #e1 #_ #_ #IHW #IHU #d2 #e2 #T2 #H #Hd12
62 lapply (lift_inv_flat2 … H) -H * #W2 #U2 #HW2 #HU2 #H destruct -T2;
63 elim (IHW … HW2 ?) // -IHW HW2 #W0 #HW2 #HW1
64 elim (IHU … HU2 ?) /3 width = 5/
68 theorem lift_mono: ∀d,e,T,U1. ↑[d,e] T ≡ U1 → ∀U2. ↑[d,e] T ≡ U2 → U1 = U2.
69 #d #e #T #U1 #H elim H -H d e T U1
71 lapply (lift_inv_sort1 … HX) -HX //
72 | #i #d #e #Hid #X #HX
73 lapply (lift_inv_lref1_lt … HX ?) -HX //
74 | #i #d #e #Hdi #X #HX
75 lapply (lift_inv_lref1_ge … HX ?) -HX //
76 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
77 elim (lift_inv_bind1 … HX) -HX #V #T #HV1 #HT1 #HX destruct -X /3/
78 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
79 elim (lift_inv_flat1 … HX) -HX #V #T #HV1 #HT1 #HX destruct -X /3/
83 (* Basic_1: was: lift_free (left to right) *)
84 theorem lift_trans_be: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
85 ∀d2,e2,T2. ↑[d2, e2] T ≡ T2 →
86 d1 ≤ d2 → d2 ≤ d1 + e1 → ↑[d1, e1 + e2] T1 ≡ T2.
87 #d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T
88 [ #k #d1 #e1 #d2 #e2 #T2 #HT2 #_ #_
89 >(lift_inv_sort1 … HT2) -HT2 //
90 | #i #d1 #e1 #Hid1 #d2 #e2 #T2 #HT2 #Hd12 #_
91 lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2
92 lapply (lift_inv_lref1_lt … HT2 Hid2) /2/
93 | #i #d1 #e1 #Hid1 #d2 #e2 #T2 #HT2 #_ #Hd21
94 lapply (lift_inv_lref1_ge … HT2 ?) -HT2
95 [ @(transitive_le … Hd21 ?) -Hd21 /2/
98 | #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd12 #Hd21
99 elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
100 lapply (IHV12 … HV20 ? ?) // -IHV12 HV20 #HV10
101 lapply (IHT12 … HT20 ? ?) /2/
102 | #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd12 #Hd21
103 elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
104 lapply (IHV12 … HV20 ? ?) // -IHV12 HV20 #HV10
105 lapply (IHT12 … HT20 ? ?) /2/
109 (* Basic_1: was: lift_d (right to left) *)
110 theorem lift_trans_le: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
111 ∀d2,e2,T2. ↑[d2, e2] T ≡ T2 → d2 ≤ d1 →
112 ∃∃T0. ↑[d2, e2] T1 ≡ T0 & ↑[d1 + e2, e1] T0 ≡ T2.
113 #d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T
114 [ #k #d1 #e1 #d2 #e2 #X #HX #_
115 >(lift_inv_sort1 … HX) -HX /2/
116 | #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #_
117 lapply (lt_to_le_to_lt … (d1+e2) Hid1 ?) // #Hie2
118 elim (lift_inv_lref1 … HX) -HX * #Hid2 #HX destruct -X /4/
119 | #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #Hd21
120 lapply (transitive_le … Hd21 Hid1) -Hd21 #Hid2
121 lapply (lift_inv_lref1_ge … HX ?) -HX /2/ #HX destruct -X;
122 >plus_plus_comm_23 /4/
123 | #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd21
124 elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
125 elim (IHV12 … HV20 ?) -IHV12 HV20 //
126 elim (IHT12 … HT20 ?) -IHT12 HT20 /3 width=5/
127 | #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd21
128 elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
129 elim (IHV12 … HV20 ?) -IHV12 HV20 //
130 elim (IHT12 … HT20 ?) -IHT12 HT20 /3 width=5/
134 (* Basic_1: was: lift_d (left to right) *)
135 theorem lift_trans_ge: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
136 ∀d2,e2,T2. ↑[d2, e2] T ≡ T2 → d1 + e1 ≤ d2 →
137 ∃∃T0. ↑[d2 - e1, e2] T1 ≡ T0 & ↑[d1, e1] T0 ≡ T2.
138 #d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T
139 [ #k #d1 #e1 #d2 #e2 #X #HX #_
140 >(lift_inv_sort1 … HX) -HX /2/
141 | #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #Hded
142 lapply (lt_to_le_to_lt … (d1+e1) Hid1 ?) // #Hid1e
143 lapply (lt_to_le_to_lt … (d2-e1) Hid1 ?) /2/ #Hid2e
144 lapply (lt_to_le_to_lt … Hid1e Hded) -Hid1e Hded #Hid2
145 lapply (lift_inv_lref1_lt … HX ?) -HX // #HX destruct -X /3/
146 | #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #_
147 elim (lift_inv_lref1 … HX) -HX * #Hied #HX destruct -X;
148 [2: >plus_plus_comm_23] /4/
149 | #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hded
150 elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
151 elim (IHV12 … HV20 ?) -IHV12 HV20 //
152 elim (IHT12 … HT20 ?) -IHT12 HT20 /2/ #T
153 <plus_minus /3 width=5/
154 | #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hded
155 elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
156 elim (IHV12 … HV20 ?) -IHV12 HV20 //
157 elim (IHT12 … HT20 ?) -IHT12 HT20 /3 width=5/