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 properties ***********************************************************)
21 (* Basic_1: was: lift_inj *)
22 theorem lift_inj: ∀l,m,T1,U. ⬆[l, m] T1 ≡ U → ∀T2. ⬆[l, m] T2 ≡ U → T1 = T2.
23 #l #m #T1 #U #H elim H -l -m -T1 -U
25 lapply (lift_inv_sort2 … HX) -HX //
26 | #i #l #m #Hil #X #HX
27 lapply (lift_inv_lref2_lt … HX ?) -HX //
28 | #i #l #m #Hli #X #HX
29 lapply (lift_inv_lref2_ge … HX ?) -HX /2 width=1 by monotonic_yle_plus_dx/
31 lapply (lift_inv_gref2 … HX) -HX //
32 | #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #X #HX
33 elim (lift_inv_bind2 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1 by eq_f2/
34 | #I #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #X #HX
35 elim (lift_inv_flat2 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1 by eq_f2/
39 (* Basic_1: was: lift_gen_lift *)
40 theorem lift_div_le: ∀l1,m1,T1,T. ⬆[l1, m1] T1 ≡ T →
41 ∀l2,m2,T2. ⬆[l2 + m1, m2] T2 ≡ T →
43 ∃∃T0. ⬆[l1, m1] T0 ≡ T2 & ⬆[l2, m2] T0 ≡ T1.
44 #l1 #m1 #T1 #T #H elim H -l1 -m1 -T1 -T
45 [ #k #l1 #m1 #l2 #m2 #T2 #Hk #Hl12
46 lapply (lift_inv_sort2 … Hk) -Hk #Hk destruct /3 width=3 by lift_sort, ex2_intro/
47 | #i #l1 #m1 #Hil1 #l2 #m2 #T2 #Hi #Hl12
48 lapply (ylt_yle_trans … Hl12 Hil1) -Hl12 #Hil2
49 lapply (lift_inv_lref2_lt … Hi ?) -Hi /3 width=3 by lift_lref_lt, ylt_plus_dx1_trans, ex2_intro/
50 | #i #l1 #m1 #Hil1 #l2 #m2 #T2 #Hi #Hl12
51 elim (lift_inv_lref2 … Hi) -Hi * <yplus_inj #Hil2 #H destruct
52 [ -Hl12 lapply (ylt_inv_monotonic_plus_dx … Hil2) -Hil2 #Hil2 /3 width=3 by lift_lref_lt, lift_lref_ge, ex2_intro/
53 | -Hil1 >yplus_comm_23 in Hil2; #H lapply ( yle_inv_monotonic_plus_dx … H) -H #H
54 elim (yle_inv_plus_inj2 … H) -H >yminus_inj #Hl2im2 #H
55 lapply (yle_inv_inj … H) -H #Hm2i
56 lapply (yle_trans … Hl12 … Hl2im2) -Hl12 #Hl1im2
57 >le_plus_minus_comm // >(plus_minus_m_m i m2) in ⊢ (? ? ? %);
58 /3 width=3 by lift_lref_ge, ex2_intro/
60 | #p #l1 #m1 #l2 #m2 #T2 #Hk #Hl12
61 lapply (lift_inv_gref2 … Hk) -Hk #Hk destruct /3 width=3 by lift_gref, ex2_intro/
62 | #a #I #W1 #W #U1 #U #l1 #m1 #_ #_ #IHW #IHU #l2 #m2 #T2 #H #Hl12
63 lapply (lift_inv_bind2 … H) -H * #W2 #U2 #HW2 #HU2 #H destruct
64 elim (IHW … HW2) // -IHW -HW2 #W0 #HW2 #HW1
65 <yplus_succ1 in HU2; #HU2 elim (IHU … HU2) /3 width=5 by yle_succ, lift_bind, ex2_intro/
66 | #I #W1 #W #U1 #U #l1 #m1 #_ #_ #IHW #IHU #l2 #m2 #T2 #H #Hl12
67 lapply (lift_inv_flat2 … H) -H * #W2 #U2 #HW2 #HU2 #H destruct
68 elim (IHW … HW2) // -IHW -HW2 #W0 #HW2 #HW1
69 elim (IHU … HU2) /3 width=5 by lift_flat, ex2_intro/
73 (* Note: apparently this was missing in basic_1 *)
74 theorem lift_div_be: ∀l1,m1,T1,T. ⬆[l1, m1] T1 ≡ T →
75 ∀m,m2,T2. ⬆[l1 + yinj m, m2] T2 ≡ T →
76 m ≤ m1 → m1 ≤ m + m2 →
77 ∃∃T0. ⬆[l1, m] T0 ≡ T2 & ⬆[l1, m + m2 - m1] T0 ≡ T1.
78 #l1 #m1 #T1 #T #H elim H -l1 -m1 -T1 -T
79 [ #k #l1 #m1 #m #m2 #T2 #H >(lift_inv_sort2 … H) -H /2 width=3 by lift_sort, ex2_intro/
80 | #i #l1 #m1 #Hil1 #m #m2 #T2 #H #Hm1 #Hm1m2
81 >(lift_inv_lref2_lt … H) -H /3 width=3 by ylt_plus_dx1_trans, lift_lref_lt, ex2_intro/
82 | #i #l1 #m1 #Hil1 #m #m2 #T2 #H #Hm1 #Hm1m2
83 elim (ylt_split (i+m1) (l1+m+m2)) #H0
84 [ elim (lift_inv_lref2_be … H) -H /3 width=2 by monotonic_yle_plus, yle_inj/
85 | >(lift_inv_lref2_ge … H ?) -H //
86 lapply (yle_plus2_to_minus_inj2 … H0) #Hl1m21i
87 elim (yle_inv_plus_inj2 … H0) -H0 #Hl1m12 #Hm2im1
88 @ex2_intro [2: /2 width=1 by lift_lref_ge_minus/ | skip ]
89 @lift_lref_ge_minus_eq
90 [ <yminus_inj <yplus_inj >yplus_minus_assoc_inj /2 width=1 by yle_inj/
91 | /2 width=1 by minus_le_minus_minus_comm/
94 | #p #l1 #m1 #m #m2 #T2 #H >(lift_inv_gref2 … H) -H /2 width=3 by lift_gref, ex2_intro/
95 | #a #I #V1 #V #T1 #T #l1 #m1 #_ #_ #IHV1 #IHT1 #m #m2 #X #H #Hm1 #Hm1m2
96 elim (lift_inv_bind2 … H) -H #V2 #T2 #HV2 <yplus_succ1 #HT2 #H destruct
97 elim (IHV1 … HV2) -V //
98 elim (IHT1 … HT2) -T /3 width=5 by lift_bind, ex2_intro/
99 | #I #V1 #V #T1 #T #l1 #m1 #_ #_ #IHV1 #IHT1 #m #m2 #X #H #Hm1 #Hm1m2
100 elim (lift_inv_flat2 … H) -H #V2 #T2 #HV2 #HT2 #H destruct
101 elim (IHV1 … HV2) -V //
102 elim (IHT1 … HT2) -T /3 width=5 by lift_flat, ex2_intro/
106 theorem lift_mono: ∀l,m,T,U1. ⬆[l,m] T ≡ U1 → ∀U2. ⬆[l,m] T ≡ U2 → U1 = U2.
107 #l #m #T #U1 #H elim H -l -m -T -U1
109 lapply (lift_inv_sort1 … HX) -HX //
110 | #i #l #m #Hil #X #HX
111 lapply (lift_inv_lref1_lt … HX ?) -HX //
112 | #i #l #m #Hli #X #HX
113 lapply (lift_inv_lref1_ge … HX ?) -HX //
115 lapply (lift_inv_gref1 … HX) -HX //
116 | #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #X #HX
117 elim (lift_inv_bind1 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1 by eq_f2/
118 | #I #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #X #HX
119 elim (lift_inv_flat1 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1 by eq_f2/
123 (* Basic_1: was: lift_free (left to right) *)
124 theorem lift_trans_be: ∀l1,m1,T1,T. ⬆[l1, m1] T1 ≡ T →
125 ∀l2,m2,T2. ⬆[l2, m2] T ≡ T2 →
126 l1 ≤ l2 → l2 ≤ l1 + m1 → ⬆[l1, m1 + m2] T1 ≡ T2.
127 #l1 #m1 #T1 #T #H elim H -l1 -m1 -T1 -T
128 [ #k #l1 #m1 #l2 #m2 #T2 #HT2 #_ #_
129 >(lift_inv_sort1 … HT2) -HT2 //
130 | #i #l1 #m1 #Hil1 #l2 #m2 #T2 #HT2 #Hl12 #_
131 lapply (ylt_yle_trans … Hl12 Hil1) -Hl12 #Hil2
132 lapply (lift_inv_lref1_lt … HT2 Hil2) /2 width=1 by lift_lref_lt/
133 | #i #l1 #m1 #Hil1 #l2 #m2 #T2 #HT2 #_ #Hl21
134 lapply (lift_inv_lref1_ge … HT2 ?) -HT2
135 [ @(yle_trans … Hl21) -Hl21 /2 width=1 by monotonic_yle_plus_dx/
136 | -Hl21 /2 width=1 by lift_lref_ge/
138 | #p #l1 #m1 #l2 #m2 #T2 #HT2 #_ #_
139 >(lift_inv_gref1 … HT2) -HT2 //
140 | #a #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hl12 #Hl21
141 elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
142 lapply (IHV12 … HV20 ? ?) // -IHV12 -HV20 #HV10
143 lapply (IHT12 … HT20 ? ?) /2 width=1 by lift_bind, yle_succ/ (**) (* full auto a bit slow *)
144 | #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hl12 #Hl21
145 elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
146 lapply (IHV12 … HV20 ? ?) // -IHV12 -HV20 #HV10
147 lapply (IHT12 … HT20 ? ?) /2 width=1 by lift_flat/ (**) (* full auto a bit slow *)
151 (* Basic_1: was: lift_d (right to left) *)
152 theorem lift_trans_le: ∀l1,m1,T1,T. ⬆[l1, m1] T1 ≡ T →
153 ∀l2,m2,T2. ⬆[l2, m2] T ≡ T2 → l2 ≤ l1 →
154 ∃∃T0. ⬆[l2, m2] T1 ≡ T0 & ⬆[l1 + m2, m1] T0 ≡ T2.
155 #l1 #m1 #T1 #T #H elim H -l1 -m1 -T1 -T
156 [ #k #l1 #m1 #l2 #m2 #X #HX #_
157 >(lift_inv_sort1 … HX) -HX /2 width=3 by lift_sort, ex2_intro/
158 | #i #l1 #m1 #Hil1 #l2 #m2 #X #HX #_
159 lapply (ylt_yle_trans … (l1+m2) ? Hil1) // #Him2
160 elim (lift_inv_lref1 … HX) -HX * #Hil2 #HX destruct
161 /4 width=3 by monotonic_ylt_plus_dx, monotonic_yle_plus_dx, lift_lref_ge_minus, lift_lref_lt, ex2_intro/
162 | #i #l1 #m1 #Hil1 #l2 #m2 #X #HX #Hl21
163 lapply (yle_trans … Hl21 … Hil1) -Hl21 #Hil2
164 lapply (lift_inv_lref1_ge … HX ?) -HX /2 width=3 by yle_plus_dx1_trans/ #HX destruct
165 >plus_plus_comm_23 /4 width=3 by monotonic_yle_plus_dx, lift_lref_ge_minus, lift_lref_ge, ex2_intro/
166 | #p #l1 #m1 #l2 #m2 #X #HX #_
167 >(lift_inv_gref1 … HX) -HX /2 width=3 by lift_gref, ex2_intro/
168 | #a #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hl21
169 elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
170 elim (IHV12 … HV20) -IHV12 -HV20 //
171 elim (IHT12 … HT20) -IHT12 -HT20 /3 width=5 by lift_bind, yle_succ, ex2_intro/
172 | #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hl21
173 elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
174 elim (IHV12 … HV20) -IHV12 -HV20 //
175 elim (IHT12 … HT20) -IHT12 -HT20 /3 width=5 by lift_flat, ex2_intro/
179 (* Basic_1: was: lift_d (left to right) *)
180 theorem lift_trans_ge: ∀l1,m1,T1,T. ⬆[l1, m1] T1 ≡ T →
181 ∀l2,m2,T2. ⬆[l2, m2] T ≡ T2 → l1 + m1 ≤ l2 →
182 ∃∃T0. ⬆[l2 - m1, m2] T1 ≡ T0 & ⬆[l1, m1] T0 ≡ T2.
183 #l1 #m1 #T1 #T #H elim H -l1 -m1 -T1 -T
184 [ #k #l1 #m1 #l2 #m2 #X #HX #_
185 >(lift_inv_sort1 … HX) -HX /2 width=3 by lift_sort, ex2_intro/
186 | #i #l1 #m1 #Hil1 #l2 #m2 #X #HX #Hlml
187 lapply (ylt_yle_trans … (l1+m1) ? Hil1) // #Hil1m
188 lapply (ylt_yle_trans … (l2-m1) ? Hil1) /2 width=1 by yle_plus1_to_minus_inj2/ #Hil2m
189 lapply (ylt_yle_trans … Hlml Hil1m) -Hil1m -Hlml #Hil2
190 lapply (lift_inv_lref1_lt … HX ?) -HX // #HX destruct /3 width=3 by lift_lref_lt, ex2_intro/
191 | #i #l1 #m1 #Hil1 #l2 #m2 #X #HX #_
192 elim (lift_inv_lref1 … HX) -HX * <yplus_inj #Himl #HX destruct
193 [ /4 width=3 by lift_lref_lt, lift_lref_ge, ylt_plus1_to_minus_inj2, ex2_intro/
194 | /4 width=3 by lift_lref_ge, yle_plus_dx1_trans, monotonic_yle_minus_dx, ex2_intro/
196 | #p #l1 #m1 #l2 #m2 #X #HX #_
197 >(lift_inv_gref1 … HX) -HX /2 width=3 by lift_gref, ex2_intro/
198 | #a #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hlml
199 elim (yle_inv_plus_inj2 … Hlml) #Hlm #Hml
200 elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
201 elim (IHV12 … HV20) -IHV12 -HV20 //
202 elim (IHT12 … HT20) -IHT12 -HT20 /2 width=1 by yle_succ/ -Hlml
203 #T >yminus_succ1_inj /3 width=5 by lift_bind, ex2_intro/
204 | #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hlml
205 elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
206 elim (IHV12 … HV20) -IHV12 -HV20 //
207 elim (IHT12 … HT20) -IHT12 -HT20 /3 width=5 by lift_flat, ex2_intro/
211 (* Advanced properties ******************************************************)
213 lemma lift_conf_O1: ∀T,T1,l1,m1. ⬆[l1, m1] T ≡ T1 → ∀T2,m2. ⬆[0, m2] T ≡ T2 →
214 ∃∃T0. ⬆[0, m2] T1 ≡ T0 & ⬆[l1 + m2, m1] T2 ≡ T0.
215 #T #T1 #l1 #m1 #HT1 #T2 #m2 #HT2
216 elim (lift_total T1 0 m2) #T0 #HT10
217 elim (lift_trans_le … HT1 … HT10) -HT1 // #X #HTX #HT20
218 lapply (lift_mono … HTX … HT2) -T #H destruct /2 width=3 by ex2_intro/
221 lemma lift_conf_be: ∀T,T1,l,m1. ⬆[l, m1] T ≡ T1 → ∀T2,m2. ⬆[l, m2] T ≡ T2 →
222 m1 ≤ m2 → ⬆[l + yinj m1, m2 - m1] T1 ≡ T2.
223 #T #T1 #l #m1 #HT1 #T2 #m2 #HT2 #Hm12
224 elim (lift_split … HT2 (l+m1) m1) -HT2 // #X #H
225 >(lift_mono … H … HT1) -T //