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 (* BASIC TERM 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 -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 width=1/
31 lapply (lift_inv_gref2 … HX) -HX //
32 | #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
33 elim (lift_inv_bind2 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1/
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 /3 width=1/
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 -d1 -e1 -T1 -T
45 [ #k #d1 #e1 #d2 #e2 #T2 #Hk #Hd12
46 lapply (lift_inv_sort2 … Hk) -Hk #Hk destruct /3 width=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 width=3/ /3 width=3/
50 | #i #d1 #e1 #Hid1 #d2 #e2 #T2 #Hi #Hd12
51 elim (lift_inv_lref2 … Hi) -Hi * #Hid2 #H destruct
52 [ -Hd12 lapply (lt_plus_to_lt_l … Hid2) -Hid2 #Hid2 /3 width=3/
53 | -Hid1 >plus_plus_comm_23 in Hid2; #H lapply (le_plus_to_le_r … H) -H #H
54 elim (le_inv_plus_l … H) -H #Hide2 #He2i
55 lapply (transitive_le … Hd12 Hide2) -Hd12 #Hd12
56 >le_plus_minus_comm // >(plus_minus_m_m i e2) in ⊢ (? ? ? %); // -He2i
59 | #p #d1 #e1 #d2 #e2 #T2 #Hk #Hd12
60 lapply (lift_inv_gref2 … Hk) -Hk #Hk destruct /3 width=3/
61 | #a #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
63 elim (IHW … HW2 ?) // -IHW -HW2 #W0 #HW2 #HW1
64 >plus_plus_comm_23 in HU2; #HU2 elim (IHU … HU2 ?) /2 width=1/ /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
67 elim (IHW … HW2 ?) // -IHW -HW2 #W0 #HW2 #HW1
68 elim (IHU … HU2 ?) // /3 width=5/
72 (* Note: apparently this was missing in basic_1 *)
73 theorem lift_div_be: ∀d1,e1,T1,T. ⇧[d1, e1] T1 ≡ T →
74 ∀e,e2,T2. ⇧[d1 + e, e2] T2 ≡ T →
75 e ≤ e1 → e1 ≤ e + e2 →
76 ∃∃T0. ⇧[d1, e] T0 ≡ T2 & ⇧[d1, e + e2 - e1] T0 ≡ T1.
77 #d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T
78 [ #k #d1 #e1 #e #e2 #T2 #H >(lift_inv_sort2 … H) -H /2 width=3/
79 | #i #d1 #e1 #Hid1 #e #e2 #T2 #H #He1 #He1e2
80 >(lift_inv_lref2_lt … H) -H [ /3 width=3/ | /2 width=3/ ]
81 | #i #d1 #e1 #Hid1 #e #e2 #T2 #H #He1 #He1e2
82 elim (lt_or_ge (i+e1) (d1+e+e2)) #Hie1d1e2
83 [ elim (lift_inv_lref2_be … H ? ?) -H // /2 width=1/
84 | >(lift_inv_lref2_ge … H ?) -H //
85 lapply (le_plus_to_minus … Hie1d1e2) #Hd1e21i
86 elim (le_inv_plus_l … Hie1d1e2) -Hie1d1e2 #Hd1e12 #He2ie1
87 @ex2_1_intro [2: /2 width=1/ | skip ] -Hd1e12
88 @lift_lref_ge_minus_eq [ >plus_minus_commutative // | /2 width=1/ ]
90 | #p #d1 #e1 #e #e2 #T2 #H >(lift_inv_gref2 … H) -H /2 width=3/
91 | #a #I #V1 #V #T1 #T #d1 #e1 #_ #_ #IHV1 #IHT1 #e #e2 #X #H #He1 #He1e2
92 elim (lift_inv_bind2 … H) -H #V2 #T2 #HV2 #HT2 #H destruct
93 elim (IHV1 … HV2 ? ?) -V // >plus_plus_comm_23 in HT2; #HT2
94 elim (IHT1 … HT2 ? ?) -T // -He1 -He1e2 /3 width=5/
95 | #I #V1 #V #T1 #T #d1 #e1 #_ #_ #IHV1 #IHT1 #e #e2 #X #H #He1 #He1e2
96 elim (lift_inv_flat2 … H) -H #V2 #T2 #HV2 #HT2 #H destruct
97 elim (IHV1 … HV2 ? ?) -V //
98 elim (IHT1 … HT2 ? ?) -T // -He1 -He1e2 /3 width=5/
102 theorem lift_mono: ∀d,e,T,U1. ⇧[d,e] T ≡ U1 → ∀U2. ⇧[d,e] T ≡ U2 → U1 = U2.
103 #d #e #T #U1 #H elim H -d -e -T -U1
105 lapply (lift_inv_sort1 … HX) -HX //
106 | #i #d #e #Hid #X #HX
107 lapply (lift_inv_lref1_lt … HX ?) -HX //
108 | #i #d #e #Hdi #X #HX
109 lapply (lift_inv_lref1_ge … HX ?) -HX //
111 lapply (lift_inv_gref1 … HX) -HX //
112 | #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
113 elim (lift_inv_bind1 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1/
114 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
115 elim (lift_inv_flat1 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1/
119 (* Basic_1: was: lift_free (left to right) *)
120 theorem lift_trans_be: ∀d1,e1,T1,T. ⇧[d1, e1] T1 ≡ T →
121 ∀d2,e2,T2. ⇧[d2, e2] T ≡ T2 →
122 d1 ≤ d2 → d2 ≤ d1 + e1 → ⇧[d1, e1 + e2] T1 ≡ T2.
123 #d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T
124 [ #k #d1 #e1 #d2 #e2 #T2 #HT2 #_ #_
125 >(lift_inv_sort1 … HT2) -HT2 //
126 | #i #d1 #e1 #Hid1 #d2 #e2 #T2 #HT2 #Hd12 #_
127 lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2
128 lapply (lift_inv_lref1_lt … HT2 Hid2) /2 width=1/
129 | #i #d1 #e1 #Hid1 #d2 #e2 #T2 #HT2 #_ #Hd21
130 lapply (lift_inv_lref1_ge … HT2 ?) -HT2
131 [ @(transitive_le … Hd21 ?) -Hd21 /2 width=1/
134 | #p #d1 #e1 #d2 #e2 #T2 #HT2 #_ #_
135 >(lift_inv_gref1 … HT2) -HT2 //
136 | #a #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd12 #Hd21
137 elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
138 lapply (IHV12 … HV20 ? ?) // -IHV12 -HV20 #HV10
139 lapply (IHT12 … HT20 ? ?) /2 width=1/
140 | #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd12 #Hd21
141 elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
142 lapply (IHV12 … HV20 ? ?) // -IHV12 -HV20 #HV10
143 lapply (IHT12 … HT20 ? ?) // /2 width=1/
147 (* Basic_1: was: lift_d (right to left) *)
148 theorem lift_trans_le: ∀d1,e1,T1,T. ⇧[d1, e1] T1 ≡ T →
149 ∀d2,e2,T2. ⇧[d2, e2] T ≡ T2 → d2 ≤ d1 →
150 ∃∃T0. ⇧[d2, e2] T1 ≡ T0 & ⇧[d1 + e2, e1] T0 ≡ T2.
151 #d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T
152 [ #k #d1 #e1 #d2 #e2 #X #HX #_
153 >(lift_inv_sort1 … HX) -HX /2 width=3/
154 | #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #_
155 lapply (lt_to_le_to_lt … (d1+e2) Hid1 ?) // #Hie2
156 elim (lift_inv_lref1 … HX) -HX * #Hid2 #HX destruct /3 width=3/ /4 width=3/
157 | #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #Hd21
158 lapply (transitive_le … Hd21 Hid1) -Hd21 #Hid2
159 lapply (lift_inv_lref1_ge … HX ?) -HX /2 width=3/ #HX destruct
160 >plus_plus_comm_23 /4 width=3/
161 | #p #d1 #e1 #d2 #e2 #X #HX #_
162 >(lift_inv_gref1 … HX) -HX /2 width=3/
163 | #a #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd21
164 elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
165 elim (IHV12 … HV20 ?) -IHV12 -HV20 //
166 elim (IHT12 … HT20 ?) -IHT12 -HT20 /2 width=1/ /3 width=5/
167 | #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd21
168 elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
169 elim (IHV12 … HV20 ?) -IHV12 -HV20 //
170 elim (IHT12 … HT20 ?) -IHT12 -HT20 // /3 width=5/
174 (* Basic_1: was: lift_d (left to right) *)
175 theorem lift_trans_ge: ∀d1,e1,T1,T. ⇧[d1, e1] T1 ≡ T →
176 ∀d2,e2,T2. ⇧[d2, e2] T ≡ T2 → d1 + e1 ≤ d2 →
177 ∃∃T0. ⇧[d2 - e1, e2] T1 ≡ T0 & ⇧[d1, e1] T0 ≡ T2.
178 #d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T
179 [ #k #d1 #e1 #d2 #e2 #X #HX #_
180 >(lift_inv_sort1 … HX) -HX /2 width=3/
181 | #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #Hded
182 lapply (lt_to_le_to_lt … (d1+e1) Hid1 ?) // #Hid1e
183 lapply (lt_to_le_to_lt … (d2-e1) Hid1 ?) /2 width=1/ #Hid2e
184 lapply (lt_to_le_to_lt … Hid1e Hded) -Hid1e -Hded #Hid2
185 lapply (lift_inv_lref1_lt … HX ?) -HX // #HX destruct /3 width=3/
186 | #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #_
187 elim (lift_inv_lref1 … HX) -HX * #Hied #HX destruct /4 width=3/
188 | #p #d1 #e1 #d2 #e2 #X #HX #_
189 >(lift_inv_gref1 … HX) -HX /2 width=3/
190 | #a #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hded
191 elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
192 elim (IHV12 … HV20 ?) -IHV12 -HV20 //
193 elim (IHT12 … HT20 ?) -IHT12 -HT20 /2 width=1/ #T
194 <plus_minus /2 width=2/ /3 width=5/
195 | #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hded
196 elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
197 elim (IHV12 … HV20 ?) -IHV12 -HV20 //
198 elim (IHT12 … HT20 ?) -IHT12 -HT20 // /3 width=5/
202 (* Advanced properties ******************************************************)
204 lemma lift_conf_O1: ∀T,T1,d1,e1. ⇧[d1, e1] T ≡ T1 → ∀T2,e2. ⇧[0, e2] T ≡ T2 →
205 ∃∃T0. ⇧[0, e2] T1 ≡ T0 & ⇧[d1 + e2, e1] T2 ≡ T0.
206 #T #T1 #d1 #e1 #HT1 #T2 #e2 #HT2
207 elim (lift_total T1 0 e2) #T0 #HT10
208 elim (lift_trans_le … HT1 … HT10 ?) -HT1 // #X #HTX #HT20
209 lapply (lift_mono … HTX … HT2) -T #H destruct /2 width=3/
212 lemma lift_conf_be: ∀T,T1,d,e1. ⇧[d, e1] T ≡ T1 → ∀T2,e2. ⇧[d, e2] T ≡ T2 →
213 e1 ≤ e2 → ⇧[d + e1, e2 - e1] T1 ≡ T2.
214 #T #T1 #d #e1 #HT1 #T2 #e2 #HT2 #He12
215 elim (lift_split … HT2 (d+e1) e1 ? ? ?) -HT2 // #X #H
216 >(lift_mono … H … HT1) -T //