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 (* the main properies *******************************************************)
21 lemma lift_div_le: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
22 ∀d2,e2,T2. ↑[d2 + e1, e2] T2 ≡ T →
24 ∃∃T0. ↑[d1, e1] T0 ≡ T2 & ↑[d2, e2] T0 ≡ T1.
25 #d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T
26 [ #k #d1 #e1 #d2 #e2 #T2 #Hk #Hd12
27 lapply (lift_inv_sort2 … Hk) -Hk #Hk destruct -T2 /3/
28 | #i #d1 #e1 #Hid1 #d2 #e2 #T2 #Hi #Hd12
29 lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2
30 lapply (lift_inv_lref2_lt … Hi ?) -Hi /3/
31 | #i #d1 #e1 #Hid1 #d2 #e2 #T2 #Hi #Hd12
32 elim (lift_inv_lref2 … Hi) -Hi * #Hid2 #H destruct -T2
33 [ -Hd12; lapply (lt_plus_to_lt_l … Hid2) -Hid2 #Hid2 /3/
34 | -Hid1; lapply (arith1 … Hid2) -Hid2 #Hid2
35 @(ex2_1_intro … #(i - e2))
36 [ >le_plus_minus_comm [ @lift_lref_ge @(transitive_le … Hd12) -Hd12 /2/ | -Hd12 /2/ ]
37 | -Hd12 >(plus_minus_m_m i e2) in ⊢ (? ? ? ? %) /3/
40 | #I #W1 #W #U1 #U #d1 #e1 #_ #_ #IHW #IHU #d2 #e2 #T2 #H #Hd12
41 lapply (lift_inv_bind2 … H) -H * #W2 #U2 #HW2 #HU2 #H destruct -T2;
42 elim (IHW … HW2 ?) // -IHW HW2 #W0 #HW2 #HW1
43 >plus_plus_comm_23 in HU2 #HU2 elim (IHU … HU2 ?) /3 width = 5/
44 | #I #W1 #W #U1 #U #d1 #e1 #_ #_ #IHW #IHU #d2 #e2 #T2 #H #Hd12
45 lapply (lift_inv_flat2 … H) -H * #W2 #U2 #HW2 #HU2 #H destruct -T2;
46 elim (IHW … HW2 ?) // -IHW HW2 #W0 #HW2 #HW1
47 elim (IHU … HU2 ?) /3 width = 5/
51 lemma lift_free: ∀d1,e2,T1,T2. ↑[d1, e2] T1 ≡ T2 → ∀d2,e1.
52 d1 ≤ d2 → d2 ≤ d1 + e1 → e1 ≤ e2 →
53 ∃∃T. ↑[d1, e1] T1 ≡ T & ↑[d2, e2 - e1] T ≡ T2.
54 #d1 #e2 #T1 #T2 #H elim H -H d1 e2 T1 T2
56 | #i #d1 #e2 #Hid1 #d2 #e1 #Hd12 #_ #_
57 lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2 /4/
58 | #i #d1 #e2 #Hid1 #d2 #e1 #_ #Hd21 #He12
59 lapply (transitive_le …(i+e1) Hd21 ?) /2/ -Hd21 #Hd21
60 <(plus_plus_minus_m_m e1 e2 i) /3/
61 | #I #V1 #V2 #T1 #T2 #d1 #e2 #_ #_ #IHV #IHT #d2 #e1 #Hd12 #Hd21 #He12
62 elim (IHV … Hd12 Hd21 He12) -IHV #V0 #HV0a #HV0b
63 elim (IHT (d2+1) … ? ? He12) /3 width = 5/
64 | #I #V1 #V2 #T1 #T2 #d1 #e2 #_ #_ #IHV #IHT #d2 #e1 #Hd12 #Hd21 #He12
65 elim (IHV … Hd12 Hd21 He12) -IHV #V0 #HV0a #HV0b
66 elim (IHT d2 … ? ? He12) /3 width = 5/
70 lemma lift_trans_be: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
71 ∀d2,e2,T2. ↑[d2, e2] T ≡ T2 →
72 d1 ≤ d2 → d2 ≤ d1 + e1 → ↑[d1, e1 + e2] T1 ≡ T2.
73 #d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T
74 [ #k #d1 #e1 #d2 #e2 #T2 #HT2 #_ #_
75 >(lift_inv_sort1 … HT2) -HT2 //
76 | #i #d1 #e1 #Hid1 #d2 #e2 #T2 #HT2 #Hd12 #_
77 lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2
78 lapply (lift_inv_lref1_lt … HT2 Hid2) /2/
79 | #i #d1 #e1 #Hid1 #d2 #e2 #T2 #HT2 #_ #Hd21
80 lapply (lift_inv_lref1_ge … HT2 ?) -HT2
81 [ @(transitive_le … Hd21 ?) -Hd21 /2/
84 | #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd12 #Hd21
85 elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
86 lapply (IHV12 … HV20 ? ?) // -IHV12 HV20 #HV10
87 lapply (IHT12 … HT20 ? ?) /2/
88 | #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd12 #Hd21
89 elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
90 lapply (IHV12 … HV20 ? ?) // -IHV12 HV20 #HV10
91 lapply (IHT12 … HT20 ? ?) /2/
95 lemma lift_trans_le: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
96 ∀d2,e2,T2. ↑[d2, e2] T ≡ T2 → d2 ≤ d1 →
97 ∃∃T0. ↑[d2, e2] T1 ≡ T0 & ↑[d1 + e2, e1] T0 ≡ T2.
98 #d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T
99 [ #k #d1 #e1 #d2 #e2 #X #HX #_
100 >(lift_inv_sort1 … HX) -HX /2/
101 | #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #_
102 lapply (lt_to_le_to_lt … (d1+e2) Hid1 ?) // #Hie2
103 elim (lift_inv_lref1 … HX) -HX * #Hid2 #HX destruct -X /4/
104 | #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #Hd21
105 lapply (transitive_le … Hd21 Hid1) -Hd21 #Hid2
106 lapply (lift_inv_lref1_ge … HX ?) -HX /2/ #HX destruct -X;
107 >plus_plus_comm_23 /4/
108 | #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd21
109 elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
110 elim (IHV12 … HV20 ?) -IHV12 HV20 //
111 elim (IHT12 … HT20 ?) -IHT12 HT20 /3 width=5/
112 | #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd21
113 elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
114 elim (IHV12 … HV20 ?) -IHV12 HV20 //
115 elim (IHT12 … HT20 ?) -IHT12 HT20 /3 width=5/
119 lemma lift_trans_ge: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
120 ∀d2,e2,T2. ↑[d2, e2] T ≡ T2 → d1 + e1 ≤ d2 →
121 ∃∃T0. ↑[d2 - e1, e2] T1 ≡ T0 & ↑[d1, e1] T0 ≡ T2.
122 #d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T
123 [ #k #d1 #e1 #d2 #e2 #X #HX #_
124 >(lift_inv_sort1 … HX) -HX /2/
125 | #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #Hded
126 lapply (lt_to_le_to_lt … (d1+e1) Hid1 ?) // #Hid1e
127 lapply (lt_to_le_to_lt … (d2-e1) Hid1 ?) /2/ #Hid2e
128 lapply (lt_to_le_to_lt … Hid1e Hded) -Hid1e Hded #Hid2
129 lapply (lift_inv_lref1_lt … HX ?) -HX // #HX destruct -X /3/
130 | #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #_
131 elim (lift_inv_lref1 … HX) -HX * #Hied #HX destruct -X;
132 [2: >plus_plus_comm_23] /4/
133 | #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hded
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/ #T
137 <plus_minus /3 width=5/
138 | #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hded
139 elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
140 elim (IHV12 … HV20 ?) -IHV12 HV20 //
141 elim (IHT12 … HT20 ?) -IHT12 HT20 /3 width=5/