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 (* 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/
31 lapply (lift_inv_gref2 … HX) -HX //
32 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
33 elim (lift_inv_bind2 … HX) -HX #V #T #HV1 #HT1 #HX destruct -X /3/
34 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
35 elim (lift_inv_flat2 … HX) -HX #V #T #HV1 #HT1 #HX destruct -X /3/
39 (* Basic_1: was: lift_gen_lift *)
40 theorem lift_div_le: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
41 ∀d2,e2,T2. ↑[d2 + e1, e2] T2 ≡ T →
43 ∃∃T0. ↑[d1, e1] T0 ≡ T2 & ↑[d2, e2] T0 ≡ T1.
44 #d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T
45 [ #k #d1 #e1 #d2 #e2 #T2 #Hk #Hd12
46 lapply (lift_inv_sort2 … Hk) -Hk #Hk destruct -T2 /3/
47 | #i #d1 #e1 #Hid1 #d2 #e2 #T2 #Hi #Hd12
48 lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2
49 lapply (lift_inv_lref2_lt … Hi ?) -Hi /2/ /3/
50 | #i #d1 #e1 #Hid1 #d2 #e2 #T2 #Hi #Hd12
51 elim (lift_inv_lref2 … Hi) -Hi * #Hid2 #H destruct -T2
52 [ -Hd12; lapply (lt_plus_to_lt_l … Hid2) -Hid2 #Hid2 /3/
53 | -Hid1; lapply (arith1 … Hid2) -Hid2 #Hid2
54 @(ex2_1_intro … #(i - e2))
55 [ >le_plus_minus_comm [ @lift_lref_ge @(transitive_le … Hd12) -Hd12 /2/ | -Hd12 /2/ ]
56 | -Hd12 >(plus_minus_m_m i e2) in ⊢ (? ? ? ? %) /2/ /3/
59 | #p #d1 #e1 #d2 #e2 #T2 #Hk #Hd12
60 lapply (lift_inv_gref2 … Hk) -Hk #Hk destruct -T2 /3/
61 | #I #W1 #W #U1 #U #d1 #e1 #_ #_ #IHW #IHU #d2 #e2 #T2 #H #Hd12
62 lapply (lift_inv_bind2 … H) -H * #W2 #U2 #HW2 #HU2 #H destruct -T2;
63 elim (IHW … HW2 ?) // -IHW HW2 #W0 #HW2 #HW1
64 >plus_plus_comm_23 in HU2 #HU2 elim (IHU … HU2 ?) /2/ /3 width = 5/
65 | #I #W1 #W #U1 #U #d1 #e1 #_ #_ #IHW #IHU #d2 #e2 #T2 #H #Hd12
66 lapply (lift_inv_flat2 … H) -H * #W2 #U2 #HW2 #HU2 #H destruct -T2;
67 elim (IHW … HW2 ?) // -IHW HW2 #W0 #HW2 #HW1
68 elim (IHU … HU2 ?) // /3 width = 5/
72 theorem lift_mono: ∀d,e,T,U1. ↑[d,e] T ≡ U1 → ∀U2. ↑[d,e] T ≡ U2 → U1 = U2.
73 #d #e #T #U1 #H elim H -H d e T U1
75 lapply (lift_inv_sort1 … HX) -HX //
76 | #i #d #e #Hid #X #HX
77 lapply (lift_inv_lref1_lt … HX ?) -HX //
78 | #i #d #e #Hdi #X #HX
79 lapply (lift_inv_lref1_ge … HX ?) -HX //
81 lapply (lift_inv_gref1 … HX) -HX //
82 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
83 elim (lift_inv_bind1 … HX) -HX #V #T #HV1 #HT1 #HX destruct -X /3/
84 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
85 elim (lift_inv_flat1 … HX) -HX #V #T #HV1 #HT1 #HX destruct -X /3/
89 (* Basic_1: was: lift_free (left to right) *)
90 theorem lift_trans_be: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
91 ∀d2,e2,T2. ↑[d2, e2] T ≡ T2 →
92 d1 ≤ d2 → d2 ≤ d1 + e1 → ↑[d1, e1 + e2] T1 ≡ T2.
93 #d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T
94 [ #k #d1 #e1 #d2 #e2 #T2 #HT2 #_ #_
95 >(lift_inv_sort1 … HT2) -HT2 //
96 | #i #d1 #e1 #Hid1 #d2 #e2 #T2 #HT2 #Hd12 #_
97 lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2
98 lapply (lift_inv_lref1_lt … HT2 Hid2) /2/
99 | #i #d1 #e1 #Hid1 #d2 #e2 #T2 #HT2 #_ #Hd21
100 lapply (lift_inv_lref1_ge … HT2 ?) -HT2
101 [ @(transitive_le … Hd21 ?) -Hd21 /2/
104 | #p #d1 #e1 #d2 #e2 #T2 #HT2 #_ #_
105 >(lift_inv_gref1 … HT2) -HT2 //
106 | #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd12 #Hd21
107 elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
108 lapply (IHV12 … HV20 ? ?) // -IHV12 HV20 #HV10
109 lapply (IHT12 … HT20 ? ?) /2/
110 | #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd12 #Hd21
111 elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
112 lapply (IHV12 … HV20 ? ?) // -IHV12 HV20 #HV10
113 lapply (IHT12 … HT20 ? ?) // /2/
117 (* Basic_1: was: lift_d (right to left) *)
118 theorem lift_trans_le: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
119 ∀d2,e2,T2. ↑[d2, e2] T ≡ T2 → d2 ≤ d1 →
120 ∃∃T0. ↑[d2, e2] T1 ≡ T0 & ↑[d1 + e2, e1] T0 ≡ T2.
121 #d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T
122 [ #k #d1 #e1 #d2 #e2 #X #HX #_
123 >(lift_inv_sort1 … HX) -HX /2/
124 | #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #_
125 lapply (lt_to_le_to_lt … (d1+e2) Hid1 ?) // #Hie2
126 elim (lift_inv_lref1 … HX) -HX * #Hid2 #HX destruct -X /3/ /4/
127 | #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #Hd21
128 lapply (transitive_le … Hd21 Hid1) -Hd21 #Hid2
129 lapply (lift_inv_lref1_ge … HX ?) -HX /2/ #HX destruct -X;
130 >plus_plus_comm_23 /4/
131 | #p #d1 #e1 #d2 #e2 #X #HX #_
132 >(lift_inv_gref1 … HX) -HX /2/
133 | #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd21
134 elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
135 elim (IHV12 … HV20 ?) -IHV12 HV20 //
136 elim (IHT12 … HT20 ?) -IHT12 HT20 /2/ /3 width=5/
137 | #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd21
138 elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
139 elim (IHV12 … HV20 ?) -IHV12 HV20 //
140 elim (IHT12 … HT20 ?) -IHT12 HT20 // /3 width=5/
144 (* Basic_1: was: lift_d (left to right) *)
145 theorem lift_trans_ge: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
146 ∀d2,e2,T2. ↑[d2, e2] T ≡ T2 → d1 + e1 ≤ d2 →
147 ∃∃T0. ↑[d2 - e1, e2] T1 ≡ T0 & ↑[d1, e1] T0 ≡ T2.
148 #d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T
149 [ #k #d1 #e1 #d2 #e2 #X #HX #_
150 >(lift_inv_sort1 … HX) -HX /2/
151 | #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #Hded
152 lapply (lt_to_le_to_lt … (d1+e1) Hid1 ?) // #Hid1e
153 lapply (lt_to_le_to_lt … (d2-e1) Hid1 ?) /2/ #Hid2e
154 lapply (lt_to_le_to_lt … Hid1e Hded) -Hid1e Hded #Hid2
155 lapply (lift_inv_lref1_lt … HX ?) -HX // #HX destruct -X /3/
156 | #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #_
157 elim (lift_inv_lref1 … HX) -HX * #Hied #HX destruct -X;
158 [ /4/ | >plus_plus_comm_23 /4/ ]
159 | #p #d1 #e1 #d2 #e2 #X #HX #_
160 >(lift_inv_gref1 … HX) -HX /2/
161 | #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hded
162 elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
163 elim (IHV12 … HV20 ?) -IHV12 HV20 //
164 elim (IHT12 … HT20 ?) -IHT12 HT20 /2/ #T
165 <plus_minus /2/ /3 width=5/
166 | #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hded
167 elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
168 elim (IHV12 … HV20 ?) -IHV12 HV20 //
169 elim (IHT12 … HT20 ?) -IHT12 HT20 // /3 width=5/ (**) (* just /3 width=5/ crashes *)