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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 "ground_2/ynat/ynat_max.ma".
16 include "basic_2/notation/relations/extpsubst_6.ma".
17 include "basic_2/grammar/genv.ma".
18 include "basic_2/grammar/cl_shift.ma".
19 include "basic_2/relocation/ldrop_append.ma".
20 include "basic_2/relocation/lsuby.ma".
22 (* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************)
25 inductive cpy: ynat → ynat → relation4 genv lenv term term ≝
26 | cpy_atom : ∀I,G,L,d,e. cpy d e G L (⓪{I}) (⓪{I})
27 | cpy_subst: ∀I,G,L,K,V,W,i,d,e. d ≤ yinj i → i < d+e →
28 ⇩[0, i] L ≡ K.ⓑ{I}V → ⇧[0, i+1] V ≡ W → cpy d e G L (#i) W
29 | cpy_bind : ∀a,I,G,L,V1,V2,T1,T2,d,e.
30 cpy d e G L V1 V2 → cpy (⫯d) e G (L.ⓑ{I}V2) T1 T2 →
31 cpy d e G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
32 | cpy_flat : ∀I,G,L,V1,V2,T1,T2,d,e.
33 cpy d e G L V1 V2 → cpy d e G L T1 T2 →
34 cpy d e G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
37 interpretation "context-sensitive extended ordinary substritution (term)"
38 'ExtPSubst G L T1 d e T2 = (cpy d e G L T1 T2).
40 (* Basic properties *********************************************************)
42 lemma lsuby_cpy_trans: ∀G,d,e. lsub_trans … (cpy d e G) (lsuby d e).
43 #G #d #e #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2 -d -e
45 | #I #G #L1 #K1 #V #W #i #d #e #Hdi #Hide #HLK1 #HVW #L2 #HL12
46 elim (lsuby_fwd_ldrop2_be … HL12 … HLK1) -HL12 -HLK1 /2 width=5 by cpy_subst/
47 | /4 width=1 by lsuby_succ, cpy_bind/
48 | /3 width=1 by cpy_flat/
52 lemma cpy_refl: ∀G,T,L,d,e. ⦃G, L⦄ ⊢ T ▶×[d, e] T.
53 #G #T elim T -T // * /2 width=1 by cpy_bind, cpy_flat/
56 lemma cpy_full: ∀I,G,K,V,T1,L,d. ⇩[0, d] L ≡ K.ⓑ{I}V →
57 ∃∃T2,T. ⦃G, L⦄ ⊢ T1 ▶×[d, 1] T2 & ⇧[d, 1] T ≡ T2.
58 #I #G #K #V #T1 elim T1 -T1
60 /2 width=4 by lift_sort, lift_gref, ex2_2_intro/
61 elim (lt_or_eq_or_gt i d) #Hid
62 /3 width=4 by lift_lref_ge_minus, lift_lref_lt, ex2_2_intro/
64 elim (lift_total V 0 (i+1)) #W #HVW
65 elim (lift_split … HVW i i)
66 /4 width=5 by cpy_subst, ylt_inj, ex2_2_intro/
67 | * [ #a ] #J #W1 #U1 #IHW1 #IHU1 #L #d #HLK
68 elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
69 [ elim (IHU1 (L.ⓑ{J}W2) (d+1)) -IHU1
70 /3 width=9 by cpy_bind, ldrop_ldrop, lift_bind, ex2_2_intro/
71 | elim (IHU1 … HLK) -IHU1 -HLK
72 /3 width=8 by cpy_flat, lift_flat, ex2_2_intro/
77 lemma cpy_weak: ∀G,L,T1,T2,d1,e1. ⦃G, L⦄ ⊢ T1 ▶×[d1, e1] T2 →
78 ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 →
79 ⦃G, L⦄ ⊢ T1 ▶×[d2, e2] T2.
80 #G #L #T1 #T2 #d1 #e1 #H elim H -G -L -T1 -T2 -d1 -e1 //
81 [ /3 width=5 by cpy_subst, ylt_yle_trans, yle_trans/
82 | /4 width=3 by cpy_bind, ylt_yle_trans, yle_succ/
83 | /3 width=1 by cpy_flat/
87 lemma cpy_weak_top: ∀G,L,T1,T2,d,e.
88 ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶×[d, |L| - d] T2.
89 #G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e //
90 [ #I #G #L #K #V #W #i #d #e #Hdi #_ #HLK #HVW
91 lapply (ldrop_fwd_length_lt2 … HLK)
92 /4 width=5 by cpy_subst, ylt_yle_trans, ylt_inj/
93 | #a #I #G #L #V1 #V2 normalize in match (|L.ⓑ{I}V2|); (**) (* |?| does not work *)
94 /2 width=1 by cpy_bind/
95 | /2 width=1 by cpy_flat/
99 lemma cpy_weak_full: ∀G,L,T1,T2,d,e.
100 ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶×[0, |L|] T2.
101 #G #L #T1 #T2 #d #e #HT12
102 lapply (cpy_weak … HT12 0 (d + e) ? ?) -HT12
103 /2 width=2 by cpy_weak_top/
106 lemma cpy_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶×[dt, et] U2 →
107 ∀T1,d,e. ⇧[d, e] T1 ≡ U1 →
108 d ≤ dt → d + e ≤ dt + et →
109 ∃∃T2. ⦃G, L⦄ ⊢ U1 ▶×[d+e, dt+et-(d+e)] U2 & ⇧[d, e] T2 ≡ U2.
110 #G #L #U1 #U2 #dt #et #H elim H -G -L -U1 -U2 -dt -et
111 [ * #i #G #L #dt #et #T1 #d #e #H #_
112 [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by ex2_intro/
113 | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
114 | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by ex2_intro/
116 | #I #G #L #K #V #W #i #dt #et #Hdti #Hidet #HLK #HVW #T1 #d #e #H #Hddt #Hdedet
117 elim (lift_inv_lref2 … H) -H * #Hid #H destruct [ -V -Hidet -Hdedet | -Hdti -Hddt ]
118 [ elim (ylt_yle_false … Hddt) -Hddt /3 width=3 by yle_ylt_trans, ylt_inj/
119 | elim (le_inv_plus_l … Hid) #Hdie #Hei
120 elim (lift_split … HVW d (i-e+1) ? ? ?) [2,3,4: /2 width=1 by le_S_S, le_S/ ] -Hdie
121 #T2 #_ >plus_minus // <minus_minus /2 width=1 by le_S/ <minus_n_n <plus_n_O #H -Hei
122 @(ex2_intro … H) -H @(cpy_subst … HLK HVW) /2 width=1 by yle_inj/ >ymax_pre_sn_comm // (**) (* explicit constructor *)
124 | #a #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #X #d #e #H #Hddt #Hdedet
125 elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
126 elim (IHW12 … HVW1) -V1 -IHW12 //
127 elim (IHU12 … HTU1) -T1 -IHU12 /2 width=1 by yle_succ/
128 <yplus_inj >yplus_SO2 >yplus_succ1 >yplus_succ1
129 /3 width=2 by cpy_bind, lift_bind, ex2_intro/
130 | #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #X #d #e #H #Hddt #Hdedet
131 elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
132 elim (IHW12 … HVW1) -V1 -IHW12 // elim (IHU12 … HTU1) -T1 -IHU12
133 /3 width=2 by cpy_flat, lift_flat, ex2_intro/
137 lemma cpy_split_up: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 → ∀i. i ≤ d + e →
138 ∃∃T. ⦃G, L⦄ ⊢ T1 ▶×[d, i-d] T & ⦃G, L⦄ ⊢ T ▶×[i, d+e-i] T2.
139 #G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e
140 [ /2 width=3 by ex2_intro/
141 | #I #G #L #K #V #W #i #d #e #Hdi #Hide #HLK #HVW #j #Hjde
142 elim (ylt_split i j) [ -Hide -Hjde | -Hdi ]
143 /4 width=9 by cpy_subst, ylt_yle_trans, ex2_intro/
144 | #a #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hide
145 elim (IHV12 i) -IHV12 // #V
146 elim (IHT12 (i+1)) -IHT12 /2 width=1 by yle_succ/ -Hide
147 >yplus_SO2 >yplus_succ1 #T #HT1 #HT2
148 lapply (lsuby_cpy_trans … HT1 (L.ⓑ{I}V) ?) -HT1
149 /3 width=5 by lsuby_succ, ex2_intro, cpy_bind/
150 | #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hide
151 elim (IHV12 i) -IHV12 // elim (IHT12 i) -IHT12 // -Hide
152 /3 width=5 by ex2_intro, cpy_flat/
156 lemma cpy_split_down: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 → ∀i. i ≤ d + e →
157 ∃∃T. ⦃G, L⦄ ⊢ T1 ▶×[i, d+e-i] T & ⦃G, L⦄ ⊢ T ▶×[d, i-d] T2.
158 #G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e
159 [ /2 width=3 by ex2_intro/
160 | #I #G #L #K #V #W #i #d #e #Hdi #Hide #HLK #HVW #j #Hjde
161 elim (ylt_split i j) [ -Hide -Hjde | -Hdi ]
162 /4 width=9 by cpy_subst, ylt_yle_trans, ex2_intro/
163 | #a #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hide
164 elim (IHV12 i) -IHV12 // #V
165 elim (IHT12 (i+1)) -IHT12 /2 width=1 by yle_succ/ -Hide
166 >yplus_SO2 >yplus_succ1 #T #HT1 #HT2
167 lapply (lsuby_cpy_trans … HT1 (L. ⓑ{I} V) ?) -HT1
168 /3 width=5 by lsuby_succ, ex2_intro, cpy_bind/
169 | #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hide
170 elim (IHV12 i) -IHV12 // elim (IHT12 i) -IHT12 // -Hide
171 /3 width=5 by ex2_intro, cpy_flat/
175 lemma cpy_append: ∀G,d,e. l_appendable_sn … (cpy d e G).
176 #G #d #e #K #T1 #T2 #H elim H -G -K -T1 -T2 -d -e
177 /2 width=1 by cpy_atom, cpy_bind, cpy_flat/
178 #I #G #K #K0 #V #W #i #d #e #Hdi #Hide #HK0 #HVW #L
179 lapply (ldrop_fwd_length_lt2 … HK0) #H
180 @(cpy_subst I … (L@@K0) … HVW) // (**) (* /4/ does not work *)
181 @(ldrop_O1_append_sn_le … HK0) /2 width=2 by lt_to_le/
184 (* Basic inversion lemmas ***************************************************)
186 fact cpy_inv_atom1_aux: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 → ∀J. T1 = ⓪{J} →
188 ∃∃I,K,V,i. d ≤ yinj i & i < d + e &
189 ⇩[O, i] L ≡ K.ⓑ{I}V &
192 #G #L #T1 #T2 #d #e * -G -L -T1 -T2 -d -e
193 [ #I #G #L #d #e #J #H destruct /2 width=1 by or_introl/
194 | #I #G #L #K #V #T2 #i #d #e #Hdi #Hide #HLK #HVT2 #J #H destruct /3 width=9 by ex5_4_intro, or_intror/
195 | #a #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #J #H destruct
196 | #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #J #H destruct
200 lemma cpy_inv_atom1: ∀I,G,L,T2,d,e. ⦃G, L⦄ ⊢ ⓪{I} ▶×[d, e] T2 →
202 ∃∃J,K,V,i. d ≤ yinj i & i < d + e &
203 ⇩[O, i] L ≡ K.ⓑ{J}V &
206 /2 width=4 by cpy_inv_atom1_aux/ qed-.
208 lemma cpy_inv_sort1: ∀G,L,T2,k,d,e. ⦃G, L⦄ ⊢ ⋆k ▶×[d, e] T2 → T2 = ⋆k.
209 #G #L #T2 #k #d #e #H
210 elim (cpy_inv_atom1 … H) -H //
211 * #I #K #V #i #_ #_ #_ #_ #H destruct
214 lemma cpy_inv_lref1: ∀G,L,T2,i,d,e. ⦃G, L⦄ ⊢ #i ▶×[d, e] T2 →
216 ∃∃I,K,V. d ≤ i & i < d + e &
217 ⇩[O, i] L ≡ K.ⓑ{I}V &
219 #G #L #T2 #i #d #e #H
220 elim (cpy_inv_atom1 … H) -H /2 width=1 by or_introl/
221 * #I #K #V #j #Hdj #Hjde #HLK #HVT2 #H destruct /3 width=5 by ex4_3_intro, or_intror/
224 lemma cpy_inv_gref1: ∀G,L,T2,p,d,e. ⦃G, L⦄ ⊢ §p ▶×[d, e] T2 → T2 = §p.
225 #G #L #T2 #p #d #e #H
226 elim (cpy_inv_atom1 … H) -H //
227 * #I #K #V #i #_ #_ #_ #_ #H destruct
230 fact cpy_inv_bind1_aux: ∀G,L,U1,U2,d,e. ⦃G, L⦄ ⊢ U1 ▶×[d, e] U2 →
231 ∀a,I,V1,T1. U1 = ⓑ{a,I}V1.T1 →
232 ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶×[d, e] V2 &
233 ⦃G, L. ⓑ{I}V2⦄ ⊢ T1 ▶×[⫯d, e] T2 &
235 #G #L #U1 #U2 #d #e * -G -L -U1 -U2 -d -e
236 [ #I #G #L #d #e #b #J #W1 #U1 #H destruct
237 | #I #G #L #K #V #W #i #d #e #_ #_ #_ #_ #b #J #W1 #U1 #H destruct
238 | #a #I #G #L #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #b #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
239 | #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #b #J #W1 #U1 #H destruct
243 lemma cpy_inv_bind1: ∀a,I,G,L,V1,T1,U2,d,e. ⦃G, L⦄ ⊢ ⓑ{a,I} V1. T1 ▶×[d, e] U2 →
244 ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶×[d, e] V2 &
245 ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ▶×[⫯d, e] T2 &
247 /2 width=3 by cpy_inv_bind1_aux/ qed-.
249 fact cpy_inv_flat1_aux: ∀G,L,U1,U2,d,e. ⦃G, L⦄ ⊢ U1 ▶×[d, e] U2 →
250 ∀I,V1,T1. U1 = ⓕ{I}V1.T1 →
251 ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶×[d, e] V2 &
252 ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 &
254 #G #L #U1 #U2 #d #e * -G -L -U1 -U2 -d -e
255 [ #I #G #L #d #e #J #W1 #U1 #H destruct
256 | #I #G #L #K #V #W #i #d #e #_ #_ #_ #_ #J #W1 #U1 #H destruct
257 | #a #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #J #W1 #U1 #H destruct
258 | #I #G #L #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
262 lemma cpy_inv_flat1: ∀I,G,L,V1,T1,U2,d,e. ⦃G, L⦄ ⊢ ⓕ{I} V1. T1 ▶×[d, e] U2 →
263 ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶×[d, e] V2 &
264 ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 &
266 /2 width=3 by cpy_inv_flat1_aux/ qed-.
269 fact cpy_inv_refl_O2_aux: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 → e = 0 → T1 = T2.
270 #G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e
272 | #I #G #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #H destruct
273 elim (ylt_yle_false … Hdi) -Hdi //
274 | /3 width=1 by eq_f2/
275 | /3 width=1 by eq_f2/
279 lemma cpy_inv_refl_O2: ∀G,L,T1,T2,d. ⦃G, L⦄ ⊢ T1 ▶×[d, 0] T2 → T1 = T2.
280 /2 width=6 by cpy_inv_refl_O2_aux/ qed-.
282 lemma cpy_inv_lift1_eq: ∀G,T1,U1,d,e. ⇧[d, e] T1 ≡ U1 →
283 ∀L,U2. ⦃G, L⦄ ⊢ U1 ▶×[d, e] U2 → U1 = U2.
284 #G #T1 #U1 #d #e #HTU1 #L #U2 #HU12 elim (cpy_up … HU12 … HTU1) -HU12 -HTU1
285 /2 width=4 by cpy_inv_refl_O2/
288 (* Basic forward lemmas *****************************************************)
290 lemma cpy_fwd_tw: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 → ♯{T1} ≤ ♯{T2}.
291 #G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e normalize
292 /3 width=1 by monotonic_le_plus_l, le_plus/
295 lemma cpy_fwd_shift1: ∀G,L1,L,T1,T,d,e. ⦃G, L⦄ ⊢ L1 @@ T1 ▶×[d, e] T →
296 ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
297 #G #L1 @(lenv_ind_dx … L1) -L1 normalize
298 [ #L #T1 #T #d #e #HT1
299 @(ex2_2_intro … (⋆)) // (**) (* explicit constructor *)
300 | #I #L1 #V1 #IH #L #T1 #X #d #e
301 >shift_append_assoc normalize #H
302 elim (cpy_inv_bind1 … H) -H
303 #V0 #T0 #_ #HT10 #H destruct
304 elim (IH … HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct
305 >append_length >HL12 -HL12
306 @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] (**) (* explicit constructor *)
307 /2 width=3 by trans_eq/