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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 *)
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15 include "basic_2/substitution/ldrop_ldrop.ma".
16 include "basic_2/substitution/cpy.ma".
18 (* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************)
20 (* Properties on relocation *************************************************)
22 (* Basic_1: was: subst1_lift_lt *)
23 lemma cpy_lift_le: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶[dt, et] T2 →
24 ∀L,U1,U2,s,d,e. ⇩[s, d, e] L ≡ K →
25 ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 →
26 dt + et ≤ d → ⦃G, L⦄ ⊢ U1 ▶[dt, et] U2.
27 #G #K #T1 #T2 #dt #et #H elim H -G -K -T1 -T2 -dt -et
28 [ #I #G #K #dt #et #L #U1 #U2 #s #d #e #_ #H1 #H2 #_
29 >(lift_mono … H1 … H2) -H1 -H2 //
30 | #I #G #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #s #d #e #HLK #H #HWU2 #Hdetd
31 lapply (ylt_yle_trans … Hdetd … Hidet) -Hdetd #Hid
32 lapply (ylt_inv_inj … Hid) -Hid #Hid
33 lapply (lift_inv_lref1_lt … H … Hid) -H #H destruct
34 elim (lift_trans_ge … HVW … HWU2) -W // <minus_plus #W #HVW #HWU2
35 elim (ldrop_trans_le … HLK … HKV) -K /2 width=2 by lt_to_le/ #X #HLK #H
36 elim (ldrop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K #Y #_ #HVY
37 >(lift_mono … HVY … HVW) -Y -HVW #H destruct /2 width=5 by cpy_subst/
38 | #a #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #d #e #HLK #H1 #H2 #Hdetd
39 elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
40 elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
41 /4 width=7 by cpy_bind, ldrop_skip, yle_succ/
42 | #G #I #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #d #e #HLK #H1 #H2 #Hdetd
43 elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
44 elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
45 /3 width=7 by cpy_flat/
49 lemma cpy_lift_be: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶[dt, et] T2 →
50 ∀L,U1,U2,s,d,e. ⇩[s, d, e] L ≡ K →
51 ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 →
52 dt ≤ d → d ≤ dt + et → ⦃G, L⦄ ⊢ U1 ▶[dt, et + e] U2.
53 #G #K #T1 #T2 #dt #et #H elim H -G -K -T1 -T2 -dt -et
54 [ #I #G #K #dt #et #L #U1 #U2 #s #d #e #_ #H1 #H2 #_ #_
55 >(lift_mono … H1 … H2) -H1 -H2 //
56 | #I #G #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #s #d #e #HLK #H #HWU2 #Hdtd #_
57 elim (lift_inv_lref1 … H) -H * #Hid #H destruct
59 lapply (ylt_yle_trans … (dt+et+e) … Hidet) // -Hidet #Hidete
60 elim (lift_trans_ge … HVW … HWU2) -W // <minus_plus #W #HVW #HWU2
61 elim (ldrop_trans_le … HLK … HKV) -K /2 width=2 by lt_to_le/ #X #HLK #H
62 elim (ldrop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K #Y #_ #HVY
63 >(lift_mono … HVY … HVW) -V #H destruct /2 width=5 by cpy_subst/
65 elim (yle_inv_inj2 … Hdtd) -Hdtd #dtt #Hdtd #H destruct
66 lapply (transitive_le … Hdtd Hid) -Hdtd #Hdti
67 lapply (lift_trans_be … HVW … HWU2 ? ?) -W /2 width=1 by le_S/ >plus_plus_comm_23 #HVU2
68 lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hid
69 /4 width=5 by cpy_subst, ldrop_inv_gen, monotonic_ylt_plus_dx, yle_plus_dx1_trans, yle_inj/
71 | #a #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #d #e #HLK #H1 #H2 #Hdtd #Hddet
72 elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
73 elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
74 /4 width=7 by cpy_bind, ldrop_skip, yle_succ/
75 | #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #d #e #HLK #H1 #H2 #Hdetd
76 elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
77 elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
78 /3 width=7 by cpy_flat/
82 (* Basic_1: was: subst1_lift_ge *)
83 lemma cpy_lift_ge: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶[dt, et] T2 →
84 ∀L,U1,U2,s,d,e. ⇩[s, d, e] L ≡ K →
85 ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 →
86 d ≤ dt → ⦃G, L⦄ ⊢ U1 ▶[dt+e, et] U2.
87 #G #K #T1 #T2 #dt #et #H elim H -G -K -T1 -T2 -dt -et
88 [ #I #G #K #dt #et #L #U1 #U2 #s #d #e #_ #H1 #H2 #_
89 >(lift_mono … H1 … H2) -H1 -H2 //
90 | #I #G #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #s #d #e #HLK #H #HWU2 #Hddt
91 lapply (yle_trans … Hddt … Hdti) -Hddt #Hid
92 elim (yle_inv_inj2 … Hid) -Hid #dd #Hddi #H0 destruct
93 lapply (lift_inv_lref1_ge … H … Hddi) -H #H destruct
94 lapply (lift_trans_be … HVW … HWU2 ? ?) -W /2 width=1 by le_S/ >plus_plus_comm_23 #HVU2
95 lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hddi
96 /3 width=5 by cpy_subst, ldrop_inv_gen, monotonic_ylt_plus_dx, monotonic_yle_plus_dx/
97 | #a #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #d #e #HLK #H1 #H2 #Hddt
98 elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
99 elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
100 /4 width=6 by cpy_bind, ldrop_skip, yle_succ/
101 | #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #d #e #HLK #H1 #H2 #Hddt
102 elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
103 elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
104 /3 width=6 by cpy_flat/
108 (* Inversion lemmas on relocation *******************************************)
110 (* Basic_1: was: subst1_gen_lift_lt *)
111 lemma cpy_inv_lift1_le: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶[dt, et] U2 →
112 ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
114 ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[dt, et] T2 & ⇧[d, e] T2 ≡ U2.
115 #G #L #U1 #U2 #dt #et #H elim H -G -L -U1 -U2 -dt -et
116 [ * #i #G #L #dt #et #K #s #d #e #_ #T1 #H #_
117 [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by ex2_intro/
118 | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
119 | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by ex2_intro/
121 | #I #G #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #s #d #e #HLK #T1 #H #Hdetd
122 lapply (ylt_yle_trans … Hdetd … Hidet) -Hdetd #Hid
123 lapply (ylt_inv_inj … Hid) -Hid #Hid
124 lapply (lift_inv_lref2_lt … H … Hid) -H #H destruct
125 elim (ldrop_conf_lt … HLK … HLKV) -L // #L #U #HKL #_ #HUV
126 elim (lift_trans_le … HUV … HVW) -V // >minus_plus <plus_minus_m_m // -Hid /3 width=5 by cpy_subst, ex2_intro/
127 | #a #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #K #s #d #e #HLK #X #H #Hdetd
128 elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
129 elim (IHW12 … HLK … HVW1) -IHW12 // #V2 #HV12 #HVW2
130 elim (IHU12 … HTU1) -IHU12 -HTU1
131 /3 width=6 by cpy_bind, yle_succ, ldrop_skip, lift_bind, ex2_intro/
132 | #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #K #s #d #e #HLK #X #H #Hdetd
133 elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
134 elim (IHW12 … HLK … HVW1) -W1 //
135 elim (IHU12 … HLK … HTU1) -U1 -HLK
136 /3 width=5 by cpy_flat, lift_flat, ex2_intro/
140 lemma cpy_inv_lift1_be: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶[dt, et] U2 →
141 ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
142 dt ≤ d → yinj d + e ≤ dt + et →
143 ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[dt, et-e] T2 & ⇧[d, e] T2 ≡ U2.
144 #G #L #U1 #U2 #dt #et #H elim H -G -L -U1 -U2 -dt -et
145 [ * #i #G #L #dt #et #K #s #d #e #_ #T1 #H #_ #_
146 [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by ex2_intro/
147 | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
148 | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by ex2_intro/
150 | #I #G #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #s #d #e #HLK #T1 #H #Hdtd #Hdedet
151 lapply (yle_fwd_plus_ge_inj … Hdtd Hdedet) #Heet
152 elim (lift_inv_lref2 … H) -H * #Hid #H destruct [ -Hdtd -Hidet | -Hdti -Hdedet ]
153 [ lapply (ylt_yle_trans i d (dt+(et-e)) ? ?) /2 width=1 by ylt_inj/
154 [ >yplus_minus_assoc_inj /2 width=1 by yle_plus1_to_minus_inj2/ ] -Hdedet #Hidete
155 elim (ldrop_conf_lt … HLK … HLKV) -L // #L #U #HKL #_ #HUV
156 elim (lift_trans_le … HUV … HVW) -V // >minus_plus <plus_minus_m_m // -Hid
157 /3 width=5 by cpy_subst, ex2_intro/
158 | elim (le_inv_plus_l … Hid) #Hdie #Hei
159 lapply (yle_trans … Hdtd (i-e) ?) /2 width=1 by yle_inj/ -Hdtd #Hdtie
160 lapply (ldrop_conf_ge … HLK … HLKV ?) -L // #HKV
161 elim (lift_split … HVW d (i-e+1)) -HVW [2,3,4: /2 width=1 by le_S_S, le_S/ ] -Hid -Hdie
162 #V1 #HV1 >plus_minus // <minus_minus /2 width=1 by le_S/ <minus_n_n <plus_n_O #H
163 @(ex2_intro … H) @(cpy_subst … HKV HV1) // (**) (* explicit constructor *)
164 >yplus_minus_assoc_inj /3 width=1 by monotonic_ylt_minus_dx, yle_inj/
166 | #a #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #K #s #d #e #HLK #X #H #Hdtd #Hdedet
167 elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
168 elim (IHW12 … HLK … HVW1) -IHW12 // #V2 #HV12 #HVW2
169 elim (IHU12 … HTU1) -U1
170 /3 width=6 by cpy_bind, ldrop_skip, lift_bind, yle_succ, ex2_intro/
171 | #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #K #s #d #e #HLK #X #H #Hdtd #Hdedet
172 elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
173 elim (IHW12 … HLK … HVW1) -W1 //
174 elim (IHU12 … HLK … HTU1) -U1 -HLK //
175 /3 width=5 by cpy_flat, lift_flat, ex2_intro/
179 (* Basic_1: was: subst1_gen_lift_ge *)
180 lemma cpy_inv_lift1_ge: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶[dt, et] U2 →
181 ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
183 ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[dt-e, et] T2 & ⇧[d, e] T2 ≡ U2.
184 #G #L #U1 #U2 #dt #et #H elim H -G -L -U1 -U2 -dt -et
185 [ * #i #G #L #dt #et #K #s #d #e #_ #T1 #H #_
186 [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by ex2_intro/
187 | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
188 | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by ex2_intro/
190 | #I #G #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #s #d #e #HLK #T1 #H #Hdedt
191 lapply (yle_trans … Hdedt … Hdti) #Hdei
192 elim (yle_inv_plus_inj2 … Hdedt) -Hdedt #_ #Hedt
193 elim (yle_inv_plus_inj2 … Hdei) #Hdie #Hei
194 lapply (lift_inv_lref2_ge … H ?) -H /2 width=1 by yle_inv_inj/ #H destruct
195 lapply (ldrop_conf_ge … HLK … HLKV ?) -L /2 width=1 by yle_inv_inj/ #HKV
196 elim (lift_split … HVW d (i-e+1)) -HVW [2,3,4: /3 width=1 by yle_inv_inj, le_S_S, le_S/ ] -Hdei -Hdie
197 #V0 #HV10 >plus_minus /2 width=1 by yle_inv_inj/ <minus_minus /3 width=1 by yle_inv_inj, le_S/ <minus_n_n <plus_n_O #H
198 @(ex2_intro … H) @(cpy_subst … HKV HV10) (**) (* explicit constructor *)
199 [ /2 width=1 by monotonic_yle_minus_dx/
200 | <yplus_minus_comm_inj /2 width=1 by monotonic_ylt_minus_dx/
202 | #a #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #K #s #d #e #HLK #X #H #Hdetd
203 elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
204 elim (yle_inv_plus_inj2 … Hdetd) #_ #Hedt
205 elim (IHW12 … HLK … HVW1) -IHW12 // #V2 #HV12 #HVW2
206 elim (IHU12 … HTU1) -U1 [4: @ldrop_skip // |2,5: skip |3: /2 width=1 by yle_succ/ ]
207 >yminus_succ1_inj /3 width=5 by cpy_bind, lift_bind, ex2_intro/
208 | #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #K #s #d #e #HLK #X #H #Hdetd
209 elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
210 elim (IHW12 … HLK … HVW1) -W1 //
211 elim (IHU12 … HLK … HTU1) -U1 -HLK /3 width=5 by cpy_flat, lift_flat, ex2_intro/
215 (* Advancd inversion lemmas on relocation ***********************************)
217 lemma cpy_inv_lift1_ge_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶[dt, et] U2 →
218 ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
219 d ≤ dt → dt ≤ yinj d + e → yinj d + e ≤ dt + et →
220 ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[d, dt + et - (yinj d + e)] T2 & ⇧[d, e] T2 ≡ U2.
221 #G #L #U1 #U2 #dt #et #HU12 #K #s #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet
222 elim (cpy_split_up … HU12 (d + e)) -HU12 // -Hdedet #U #HU1 #HU2
223 lapply (cpy_weak … HU1 d e ? ?) -HU1 // [ >ymax_pre_sn_comm // ] -Hddt -Hdtde #HU1
224 lapply (cpy_inv_lift1_eq … HTU1 … HU1) -HU1 #HU1 destruct
225 elim (cpy_inv_lift1_ge … HU2 … HLK … HTU1) -U -L /2 width=3 by ex2_intro/
228 lemma cpy_inv_lift1_be_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶[dt, et] U2 →
229 ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
230 dt ≤ d → dt + et ≤ yinj d + e →
231 ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[dt, d-dt] T2 & ⇧[d, e] T2 ≡ U2.
232 #G #L #U1 #U2 #dt #et #HU12 #K #s #d #e #HLK #T1 #HTU1 #Hdtd #Hdetde
233 lapply (cpy_weak … HU12 dt (d+e-dt) ? ?) -HU12 //
234 [ >ymax_pre_sn_comm /2 width=1 by yle_plus_dx1_trans/ ] -Hdetde #HU12
235 elim (cpy_inv_lift1_be … HU12 … HLK … HTU1) -U1 -L /2 width=3 by ex2_intro/
238 lemma cpy_inv_lift1_le_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶[dt, et] U2 →
239 ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
240 dt ≤ d → d ≤ dt + et → dt + et ≤ yinj d + e →
241 ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2.
242 #G #L #U1 #U2 #dt #et #HU12 #K #s #d #e #HLK #T1 #HTU1 #Hdtd #Hddet #Hdetde
243 elim (cpy_split_up … HU12 d) -HU12 // #U #HU1 #HU2
244 elim (cpy_inv_lift1_le … HU1 … HLK … HTU1) -U1
245 [2: >ymax_pre_sn_comm // ] -Hdtd #T #HT1 #HTU
246 lapply (cpy_weak … HU2 d e ? ?) -HU2 //
247 [ >ymax_pre_sn_comm // ] -Hddet -Hdetde #HU2
248 lapply (cpy_inv_lift1_eq … HTU … HU2) -L #H destruct /2 width=3 by ex2_intro/