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/relocation/ldrop_ldrop.ma".
16 include "basic_2/relocation/cpy.ma".
18 (* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************)
20 (* Relocation properties ****************************************************)
22 lemma cpy_lift_le: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶×[dt, et] T2 →
23 ∀L,U1,U2,d,e. ⇩[d, e] L ≡ K →
24 ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 →
25 dt + et ≤ d → ⦃G, L⦄ ⊢ U1 ▶×[dt, et] U2.
26 #G #K #T1 #T2 #dt #et #H elim H -G -K -T1 -T2 -dt -et
27 [ #I #G #K #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
28 >(lift_mono … H1 … H2) -H1 -H2 //
29 | #I #G #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #d #e #HLK #H #HWU2 #Hdetd
30 lapply (lt_to_le_to_lt … Hidet … Hdetd) -Hdetd #Hid
31 lapply (lift_inv_lref1_lt … H … Hid) -H #H destruct
32 elim (lift_trans_ge … HVW … HWU2) -W // <minus_plus #W #HVW #HWU2
33 elim (ldrop_trans_le … HLK … HKV) -K /2 width=2 by lt_to_le/ #X #HLK #H
34 elim (ldrop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K #Y #_ #HVY
35 >(lift_mono … HVY … HVW) -Y -HVW #H destruct /2 width=5 by cpy_subst/
36 | #a #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd
37 elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
38 elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
39 /4 width=6 by cpy_bind, ldrop_skip, le_S_S/
40 | #G #I #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd
41 elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
42 elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
43 /3 width=6 by cpy_flat/
47 lemma cpy_lift_be: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶×[dt, et] T2 →
48 ∀L,U1,U2,d,e. ⇩[d, e] L ≡ K →
49 ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 →
50 dt ≤ d → d ≤ dt + et → ⦃G, L⦄ ⊢ U1 ▶×[dt, et + e] U2.
51 #G #K #T1 #T2 #dt #et #H elim H -G -K -T1 -T2 -dt -et
52 [ #I #G #K #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_ #_
53 >(lift_mono … H1 … H2) -H1 -H2 //
54 | #I #G #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #d #e #HLK #H #HWU2 #Hdtd #_
55 elim (lift_inv_lref1 … H) -H * #Hid #H destruct
57 lapply (lt_to_le_to_lt … (dt+et+e) Hidet ?) // -Hidet #Hidete
58 elim (lift_trans_ge … HVW … HWU2) -W // <minus_plus #W #HVW #HWU2
59 elim (ldrop_trans_le … HLK … HKV) -K /2 width=2 by lt_to_le/ #X #HLK #H
60 elim (ldrop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K #Y #_ #HVY
61 >(lift_mono … HVY … HVW) -V #H destruct /2 width=5 by cpy_subst/
63 lapply (transitive_le … Hdtd Hid) -Hdtd #Hdti
64 lapply (lift_trans_be … HVW … HWU2 ? ?) -W /2 width=1 by le_S/ >plus_plus_comm_23 #HVU2
65 lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hid /3 width=5 by cpy_subst, lt_minus_to_plus_r, transitive_le/
67 | #a #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdtd #Hddet
68 elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
69 elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
70 /4 width=6 by cpy_bind, ldrop_skip, le_S_S/ (**) (* auto a bit slow *)
71 | #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd
72 elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
73 elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
74 /3 width=6 by cpy_flat/
78 lemma cpy_lift_ge: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶×[dt, et] T2 →
79 ∀L,U1,U2,d,e. ⇩[d, e] L ≡ K →
80 ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 →
81 d ≤ dt → ⦃G, L⦄ ⊢ U1 ▶×[dt + e, et] U2.
82 #G #K #T1 #T2 #dt #et #H elim H -G -K -T1 -T2 -dt -et
83 [ #I #G #K #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
84 >(lift_mono … H1 … H2) -H1 -H2 //
85 | #I #G #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #d #e #HLK #H #HWU2 #Hddt
86 lapply (transitive_le … Hddt … Hdti) -Hddt #Hid
87 lapply (lift_inv_lref1_ge … H … Hid) -H #H destruct
88 lapply (lift_trans_be … HVW … HWU2 ? ?) -W /2 width=1 by le_S/ >plus_plus_comm_23 #HVU2
89 lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hid /3 width=5 by cpy_subst, lt_minus_to_plus_r, monotonic_le_plus_l/
90 | #a #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hddt
91 elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
92 elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
93 /4 width=5 by cpy_bind, ldrop_skip, le_minus_to_plus/
94 | #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hddt
95 elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
96 elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
97 /3 width=5 by cpy_flat/
101 lemma cpy_inv_lift1_le: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶×[dt, et] U2 →
102 ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
104 ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶×[dt, et] T2 & ⇧[d, e] T2 ≡ U2.
105 #G #L #U1 #U2 #dt #et #H elim H -G -L -U1 -U2 -dt -et
106 [ * #G #L #i #dt #et #K #d #e #_ #T1 #H #_
107 [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by cpy_atom, lift_sort, ex2_intro/
108 | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3 by cpy_atom, lift_lref_ge_minus, lift_lref_lt, ex2_intro/
109 | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by cpy_atom, lift_gref, ex2_intro/
111 | #I #G #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdetd
112 lapply (lt_to_le_to_lt … Hidet … Hdetd) -Hdetd #Hid
113 lapply (lift_inv_lref2_lt … H … Hid) -H #H destruct
114 elim (ldrop_conf_lt … HLK … HLKV ?) -L // #L #U #HKL #_ #HUV
115 elim (lift_trans_le … HUV … HVW ?) -V // >minus_plus <plus_minus_m_m // -Hid /3 width=5 by cpy_subst, ex2_intro/
116 | #a #I #G #L #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
117 elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
118 elim (IHV12 … HLK … HWV1) -V1 // #W2 #HW12 #HWV2
119 elim (IHU12 … HTU1) -IHU12 -HTU1
120 /3 width=5 by cpy_bind, ldrop_skip, lift_bind, le_S_S, ex2_intro/
121 | #I #G #L #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
122 elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
123 elim (IHV12 … HLK … HWV1) -V1 //
124 elim (IHU12 … HLK … HTU1) -U1 -HLK
125 /3 width=5 by cpy_flat, lift_flat, ex2_intro/
129 lemma cpy_inv_lift1_be: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶×[dt, et] U2 →
130 ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
131 dt ≤ d → d + e ≤ dt + et →
132 ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶×[dt, et - e] T2 & ⇧[d, e] T2 ≡ U2.
133 #G #L #U1 #U2 #dt #et #H elim H -G -L -U1 -U2 -dt -et
134 [ * #G #L #i #dt #et #K #d #e #_ #T1 #H #_
135 [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by cpy_atom, lift_sort, ex2_intro/
136 | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3 by cpy_atom, lift_lref_ge_minus, lift_lref_lt, ex2_intro/
137 | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by cpy_atom, lift_gref, ex2_intro/
139 | #I #G #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdtd #Hdedet
140 lapply (le_fwd_plus_plus_ge … Hdtd … Hdedet) #Heet
141 elim (lift_inv_lref2 … H) -H * #Hid #H destruct
143 lapply (lt_to_le_to_lt … (dt + (et - e)) Hid ?) [ <le_plus_minus /2 width=1 by le_plus_to_minus_r/ ] -Hdedet #Hidete
144 elim (ldrop_conf_lt … HLK … HLKV) -L // #L #U #HKL #_ #HUV
145 elim (lift_trans_le … HUV … HVW) -V // >minus_plus <plus_minus_m_m // -Hid /3 width=5 by cpy_subst, ex2_intro/
147 lapply (transitive_le … (i - e) Hdtd ?) /2 width=1 by le_plus_to_minus_r/ -Hdtd #Hdtie
148 elim (le_inv_plus_l … Hid) #Hdie #Hei
149 lapply (ldrop_conf_ge … HLK … HLKV ?) -L // #HKV
150 elim (lift_split … HVW d (i-e+1)) -HVW [2,3,4: /2 width=1 by le_S_S, le_S/ ] -Hid -Hdie
151 #V1 #HV1 >plus_minus // <minus_minus /2 width=1 by le_S/ <minus_n_n <plus_n_O #H
152 @(ex2_intro … H) @(cpy_subst … Hdtie … HKV HV1) (**) (* explicit constructor *)
153 >commutative_plus >plus_minus /2 width=1 by monotonic_lt_minus_l/
155 | #a #I #G #L #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdtd #Hdedet
156 elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
157 elim (IHV12 … HLK … HWV1) -V1 // #W2 #HW12 #HWV2
158 elim (IHU12 … HTU1) -U1
159 [5: /2 width=2 by ldrop_skip/ |2: skip
160 |3: >plus_plus_comm_23 >(plus_plus_comm_23 dt) /2 width=1 by le_S_S/
161 |4: /2 width=1 by le_S_S/
163 /3 width=5 by cpy_bind, lift_bind, ex2_intro/
164 | #I #G #L #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdtd #Hdedet
165 elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
166 elim (IHV12 … HLK … HWV1) -V1 //
167 elim (IHU12 … HLK … HTU1) -U1 -HLK //
168 /3 width=5 by cpy_flat, lift_flat, ex2_intro/
172 lemma cpy_inv_lift1_ge: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶×[dt, et] U2 →
173 ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
175 ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶×[dt - e, et] T2 & ⇧[d, e] T2 ≡ U2.
176 #G #L #U1 #U2 #dt #et #H elim H -G -L -U1 -U2 -dt -et
177 [ * #G #L #i #dt #et #K #d #e #_ #T1 #H #_
178 [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by cpy_atom, lift_sort, ex2_intro/
179 | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3 by cpy_atom, lift_lref_ge_minus, lift_lref_lt, ex2_intro/
180 | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by cpy_atom, lift_gref, ex2_intro/
182 | #I #G #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdedt
183 lapply (transitive_le … Hdedt … Hdti) #Hdei
184 elim (le_inv_plus_l … Hdedt) -Hdedt #_ #Hedt
185 elim (le_inv_plus_l … Hdei) #Hdie #Hei
186 lapply (lift_inv_lref2_ge … H … Hdei) -H #H destruct
187 lapply (ldrop_conf_ge … HLK … HLKV ?) -L // #HKV
188 elim (lift_split … HVW d (i-e+1)) -HVW [2,3,4: /2 width=1 by le_S_S, le_S/ ] -Hdei -Hdie
189 #V0 #HV10 >plus_minus // <minus_minus /2 width=1 by le_S/ <minus_n_n <plus_n_O #H
190 @(ex2_intro … H) @(cpy_subst … HKV HV10) /2 width=1 by monotonic_le_minus_l2/ (**) (* explicit constructor *)
191 >plus_minus /2 width=1 by monotonic_lt_minus_l/
192 | #a #I #G #L #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
193 elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
194 elim (le_inv_plus_l … Hdetd) #_ #Hedt
195 elim (IHV12 … HLK … HWV1) -V1 // #W2 #HW12 #HWV2
196 elim (IHU12 … HTU1) -U1 [4: @ldrop_skip // |2: skip |3: /2 width=1 by le_S_S/ ]
197 <plus_minus /3 width=5 by cpy_bind, lift_bind, ex2_intro/
198 | #I #G #L #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
199 elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
200 elim (IHV12 … HLK … HWV1) -V1 //
201 elim (IHU12 … HLK … HTU1) -U1 -HLK /3 width=5 by cpy_flat, lift_flat, ex2_intro/
205 lemma cpy_inv_lift1_eq: ∀G,L,U1,U2,d,e.
206 ⦃G, L⦄ ⊢ U1 ▶×[d, e] U2 → ∀T1. ⇧[d, e] T1 ≡ U1 → U1 = U2.
207 #G #L #U1 #U2 #d #e #H elim H -G -L -U1 -U2 -d -e
209 | #I #G #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #T1 #H
210 elim (lift_inv_lref2 … H) -H * #H
211 [ lapply (le_to_lt_to_lt … Hdi … H) -Hdi -H #H
212 elim (lt_refl_false … H)
213 | lapply (lt_to_le_to_lt … Hide … H) -Hide -H #H
214 elim (lt_refl_false … H)
216 | #a #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
217 elim (lift_inv_bind2 … HX) -HX #V #T #HV1 #HT1 #H destruct
219 | #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
220 elim (lift_inv_flat2 … HX) -HX #V #T #HV1 #HT1 #H destruct
225 lemma cpy_inv_lift1_ge_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶×[dt, et] U2 →
226 ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
227 d ≤ dt → dt ≤ d + e → d + e ≤ dt + et →
228 ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶×[d, dt + et - (d + e)] T2 & ⇧[d, e] T2 ≡ U2.
229 #G #L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet
230 elim (cpy_split_up … HU12 (d + e)) -HU12 // -Hdedet #U #HU1 #HU2
231 lapply (cpy_weak … HU1 d e ? ?) -HU1 // [ >commutative_plus /2 width=1 by le_minus_to_plus_r/ ] -Hddt -Hdtde #HU1
232 lapply (cpy_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct
233 elim (cpy_inv_lift1_ge … HU2 … HLK … HTU1) -U -L // <minus_plus_m_m /2 width=3 by ex2_intro/
236 lemma cpy_inv_lift1_be_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶×[dt, et] U2 →
237 ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
238 dt ≤ d → dt + et ≤ d + e →
239 ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶×[dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2.
240 #G #L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hdtd #Hdetde
241 lapply (cpy_weak … HU12 dt (d + e - dt) ? ?) -HU12 /2 width=3 by transitive_le, le_n/ -Hdetde #HU12
242 elim (cpy_inv_lift1_be … HU12 … HLK … HTU1) -U1 -L /2 width=3 by ex2_intro/
245 lemma cpy_inv_lift1_le_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶×[dt, et] U2 →
246 ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
247 dt ≤ d → d ≤ dt + et → dt + et ≤ d + e →
248 ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶×[dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2.
249 #G #L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hdtd #Hddet #Hdetde
250 elim (cpy_split_up … HU12 d) -HU12 // #U #HU1 #HU2
251 elim (cpy_inv_lift1_le … HU1 … HLK … HTU1) -U1 [2: >commutative_plus /2 width=1 by le_minus_to_plus_r/ ] -Hdtd #T #HT1 #HTU
252 lapply (cpy_weak … HU2 d e ? ?) -HU2 // [ >commutative_plus <plus_minus_m_m // ] -Hddet -Hdetde #HU2
253 lapply (cpy_inv_lift1_eq … HU2 … HTU) -L #H destruct /2 width=3 by ex2_intro/