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_2A/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_le_plus_l/
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 (lt_to_le_to_lt … Hil1 Hl12) -Hl12 #Hil2
49 lapply (lift_inv_lref2_lt … Hi ?) -Hi /3 width=3 by lift_lref_lt, lt_plus_to_minus_r, lt_to_le_to_lt, ex2_intro/
50 | #i #l1 #m1 #Hil1 #l2 #m2 #T2 #Hi #Hl12
51 elim (lift_inv_lref2 … Hi) -Hi * #Hil2 #H destruct
52 [ -Hl12 lapply (lt_plus_to_lt_l … Hil2) -Hil2 #Hil2 /3 width=3 by lift_lref_lt, lift_lref_ge, ex2_intro/
53 | -Hil1 >plus_plus_comm_23 in Hil2; #H lapply (le_plus_to_le_r … H) -H #H
54 elim (le_inv_plus_l … H) -H #Hilm2 #Hm2i
55 lapply (transitive_le … Hl12 Hilm2) -Hl12 #Hl12
56 >le_plus_minus_comm // >(plus_minus_m_m i m2) in ⊢ (? ? ? %);
57 /4 width=3 by lift_lref_ge, ex2_intro/
59 | #p #l1 #m1 #l2 #m2 #T2 #Hk #Hl12
60 lapply (lift_inv_gref2 … Hk) -Hk #Hk destruct /3 width=3 by lift_gref, ex2_intro/
61 | #a #I #W1 #W #U1 #U #l1 #m1 #_ #_ #IHW #IHU #l2 #m2 #T2 #H #Hl12
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) /3 width=5 by lift_bind, le_S_S, ex2_intro/
65 | #I #W1 #W #U1 #U #l1 #m1 #_ #_ #IHW #IHU #l2 #m2 #T2 #H #Hl12
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 by lift_flat, ex2_intro/
72 (* Note: apparently this was missing in basic_1 *)
73 theorem lift_div_be: ∀l1,m1,T1,T. ⬆[l1, m1] T1 ≡ T →
74 ∀m,m2,T2. ⬆[l1 + m, m2] T2 ≡ T →
75 m ≤ m1 → m1 ≤ m + m2 →
76 ∃∃T0. ⬆[l1, m] T0 ≡ T2 & ⬆[l1, m + m2 - m1] T0 ≡ T1.
77 #l1 #m1 #T1 #T #H elim H -l1 -m1 -T1 -T
78 [ #k #l1 #m1 #m #m2 #T2 #H >(lift_inv_sort2 … H) -H /2 width=3 by lift_sort, ex2_intro/
79 | #i #l1 #m1 #Hil1 #m #m2 #T2 #H #Hm1 #Hm1m2
80 >(lift_inv_lref2_lt … H) -H /3 width=3 by lift_lref_lt, lt_plus_to_minus_r, lt_to_le_to_lt, ex2_intro/
81 | #i #l1 #m1 #Hil1 #m #m2 #T2 #H #Hm1 #Hm1m2
82 elim (lt_or_ge (i+m1) (l1+m+m2)) #Him1l1m2
83 [ elim (lift_inv_lref2_be … H) -H /2 width=1 by le_plus/
84 | >(lift_inv_lref2_ge … H ?) -H //
85 lapply (le_plus_to_minus … Him1l1m2) #Hl1m21i
86 elim (le_inv_plus_l … Him1l1m2) -Him1l1m2 #Hl1m12 #Hm2im1
87 @ex2_intro [2: /2 width=1 by lift_lref_ge_minus/ | skip ] -Hl1m12
88 @lift_lref_ge_minus_eq [ >plus_minus_associative // | /2 width=1 by minus_le_minus_minus_comm/ ]
90 | #p #l1 #m1 #m #m2 #T2 #H >(lift_inv_gref2 … H) -H /2 width=3 by lift_gref, ex2_intro/
91 | #a #I #V1 #V #T1 #T #l1 #m1 #_ #_ #IHV1 #IHT1 #m #m2 #X #H #Hm1 #Hm1m2
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 /3 width=5 by lift_bind, ex2_intro/
95 | #I #V1 #V #T1 #T #l1 #m1 #_ #_ #IHV1 #IHT1 #m #m2 #X #H #Hm1 #Hm1m2
96 elim (lift_inv_flat2 … H) -H #V2 #T2 #HV2 #HT2 #H destruct
97 elim (IHV1 … HV2) -V //
98 elim (IHT1 … HT2) -T /3 width=5 by lift_flat, ex2_intro/
102 theorem lift_mono: ∀l,m,T,U1. ⬆[l,m] T ≡ U1 → ∀U2. ⬆[l,m] T ≡ U2 → U1 = U2.
103 #l #m #T #U1 #H elim H -l -m -T -U1
105 lapply (lift_inv_sort1 … HX) -HX //
106 | #i #l #m #Hil #X #HX
107 lapply (lift_inv_lref1_lt … HX ?) -HX //
108 | #i #l #m #Hli #X #HX
109 lapply (lift_inv_lref1_ge … HX ?) -HX //
111 lapply (lift_inv_gref1 … HX) -HX //
112 | #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #X #HX
113 elim (lift_inv_bind1 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1 by eq_f2/
114 | #I #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #X #HX
115 elim (lift_inv_flat1 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1 by eq_f2/
119 (* Basic_1: was: lift_free (left to right) *)
120 theorem lift_trans_be: ∀l1,m1,T1,T. ⬆[l1, m1] T1 ≡ T →
121 ∀l2,m2,T2. ⬆[l2, m2] T ≡ T2 →
122 l1 ≤ l2 → l2 ≤ l1 + m1 → ⬆[l1, m1 + m2] T1 ≡ T2.
123 #l1 #m1 #T1 #T #H elim H -l1 -m1 -T1 -T
124 [ #k #l1 #m1 #l2 #m2 #T2 #HT2 #_ #_
125 >(lift_inv_sort1 … HT2) -HT2 //
126 | #i #l1 #m1 #Hil1 #l2 #m2 #T2 #HT2 #Hl12 #_
127 lapply (lt_to_le_to_lt … Hil1 Hl12) -Hl12 #Hil2
128 lapply (lift_inv_lref1_lt … HT2 Hil2) /2 width=1 by lift_lref_lt/
129 | #i #l1 #m1 #Hil1 #l2 #m2 #T2 #HT2 #_ #Hl21
130 lapply (lift_inv_lref1_ge … HT2 ?) -HT2
131 [ @(transitive_le … Hl21 ?) -Hl21 /2 width=1 by monotonic_le_plus_l/
132 | -Hl21 /2 width=1 by lift_lref_ge/
134 | #p #l1 #m1 #l2 #m2 #T2 #HT2 #_ #_
135 >(lift_inv_gref1 … HT2) -HT2 //
136 | #a #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hl12 #Hl21
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 by lift_bind, le_S_S/ (**) (* full auto a bit slow *)
140 | #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hl12 #Hl21
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 by lift_flat/ (**) (* full auto a bit slow *)
147 (* Basic_1: was: lift_d (right to left) *)
148 theorem lift_trans_le: ∀l1,m1,T1,T. ⬆[l1, m1] T1 ≡ T →
149 ∀l2,m2,T2. ⬆[l2, m2] T ≡ T2 → l2 ≤ l1 →
150 ∃∃T0. ⬆[l2, m2] T1 ≡ T0 & ⬆[l1 + m2, m1] T0 ≡ T2.
151 #l1 #m1 #T1 #T #H elim H -l1 -m1 -T1 -T
152 [ #k #l1 #m1 #l2 #m2 #X #HX #_
153 >(lift_inv_sort1 … HX) -HX /2 width=3 by lift_sort, ex2_intro/
154 | #i #l1 #m1 #Hil1 #l2 #m2 #X #HX #_
155 lapply (lt_to_le_to_lt … (l1+m2) Hil1 ?) // #Him2
156 elim (lift_inv_lref1 … HX) -HX * #Hil2 #HX destruct /4 width=3 by lift_lref_ge_minus, lift_lref_lt, lt_minus_to_plus, monotonic_le_plus_l, ex2_intro/
157 | #i #l1 #m1 #Hil1 #l2 #m2 #X #HX #Hl21
158 lapply (transitive_le … Hl21 Hil1) -Hl21 #Hil2
159 lapply (lift_inv_lref1_ge … HX ?) -HX /2 width=3 by transitive_le/ #HX destruct
160 >plus_plus_comm_23 /4 width=3 by lift_lref_ge_minus, lift_lref_ge, monotonic_le_plus_l, ex2_intro/
161 | #p #l1 #m1 #l2 #m2 #X #HX #_
162 >(lift_inv_gref1 … HX) -HX /2 width=3 by lift_gref, ex2_intro/
163 | #a #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hl21
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 /3 width=5 by lift_bind, le_S_S, ex2_intro/
167 | #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hl21
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 by lift_flat, ex2_intro/
174 (* Basic_1: was: lift_d (left to right) *)
175 theorem lift_trans_ge: ∀l1,m1,T1,T. ⬆[l1, m1] T1 ≡ T →
176 ∀l2,m2,T2. ⬆[l2, m2] T ≡ T2 → l1 + m1 ≤ l2 →
177 ∃∃T0. ⬆[l2 - m1, m2] T1 ≡ T0 & ⬆[l1, m1] T0 ≡ T2.
178 #l1 #m1 #T1 #T #H elim H -l1 -m1 -T1 -T
179 [ #k #l1 #m1 #l2 #m2 #X #HX #_
180 >(lift_inv_sort1 … HX) -HX /2 width=3 by lift_sort, ex2_intro/
181 | #i #l1 #m1 #Hil1 #l2 #m2 #X #HX #Hlml
182 lapply (lt_to_le_to_lt … (l1+m1) Hil1 ?) // #Hil1m
183 lapply (lt_to_le_to_lt … (l2-m1) Hil1 ?) /2 width=1 by le_plus_to_minus_r/ #Hil2m
184 lapply (lt_to_le_to_lt … Hil1m Hlml) -Hil1m -Hlml #Hil2
185 lapply (lift_inv_lref1_lt … HX ?) -HX // #HX destruct /3 width=3 by lift_lref_lt, ex2_intro/
186 | #i #l1 #m1 #Hil1 #l2 #m2 #X #HX #_
187 elim (lift_inv_lref1 … HX) -HX * #Himl #HX destruct /4 width=3 by lift_lref_lt, lift_lref_ge, monotonic_le_minus_l, lt_plus_to_minus_r, transitive_le, ex2_intro/
188 | #p #l1 #m1 #l2 #m2 #X #HX #_
189 >(lift_inv_gref1 … HX) -HX /2 width=3 by lift_gref, ex2_intro/
190 | #a #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hlml
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 by le_S_S/ #T
194 <plus_minus /3 width=5 by lift_bind, le_plus_to_minus_r, le_plus_b, ex2_intro/
195 | #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hlml
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 by lift_flat, ex2_intro/
202 (* Advanced properties ******************************************************)
204 lemma lift_conf_O1: ∀T,T1,l1,m1. ⬆[l1, m1] T ≡ T1 → ∀T2,m2. ⬆[0, m2] T ≡ T2 →
205 ∃∃T0. ⬆[0, m2] T1 ≡ T0 & ⬆[l1 + m2, m1] T2 ≡ T0.
206 #T #T1 #l1 #m1 #HT1 #T2 #m2 #HT2
207 elim (lift_total T1 0 m2) #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 by ex2_intro/
212 lemma lift_conf_be: ∀T,T1,l,m1. ⬆[l, m1] T ≡ T1 → ∀T2,m2. ⬆[l, m2] T ≡ T2 →
213 m1 ≤ m2 → ⬆[l + m1, m2 - m1] T1 ≡ T2.
214 #T #T1 #l #m1 #HT1 #T2 #m2 #HT2 #Hm12
215 elim (lift_split … HT2 (l+m1) m1) -HT2 // #X #H
216 >(lift_mono … H … HT1) -T //