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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/xoa/ex_4_3.ma".
16 include "ground_2/xoa/ex_5_4.ma".
17 include "ground_2/xoa/or_3.ma".
18 include "ground_2/ynat/ynat_minus_sn.ma".
19 include "ground_2/ynat/ynat_minus_dx.ma".
20 include "basic_2A/notation/relations/psubst_6.ma".
21 include "basic_2A/grammar/genv.ma".
22 include "basic_2A/substitution/lsuby.ma".
24 (* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************)
27 inductive cpy: ynat → ynat → relation4 genv lenv term term ≝
28 | cpy_atom : ∀I,G,L,l,m. cpy l m G L (⓪{I}) (⓪{I})
29 | cpy_subst: ∀I,G,L,K,V,W,i,l,m. l ≤ yinj i → i < l+m →
30 ⬇[i] L ≡ K.ⓑ{I}V → ⬆[0, i+1] V ≡ W → cpy l m G L (#i) W
31 | cpy_bind : ∀a,I,G,L,V1,V2,T1,T2,l,m.
32 cpy l m G L V1 V2 → cpy (↑l) m G (L.ⓑ{I}V1) T1 T2 →
33 cpy l m G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
34 | cpy_flat : ∀I,G,L,V1,V2,T1,T2,l,m.
35 cpy l m G L V1 V2 → cpy l m G L T1 T2 →
36 cpy l m G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
39 interpretation "context-sensitive extended ordinary substritution (term)"
40 'PSubst G L T1 l m T2 = (cpy l m G L T1 T2).
42 (* Basic properties *********************************************************)
44 lemma lsuby_cpy_trans: ∀G,l,m. lsub_trans … (cpy l m G) (lsuby l m).
45 #G #l #m #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2 -l -m
47 | #I #G #L1 #K1 #V #W #i #l #m #Hli #Hilm #HLK1 #HVW #L2 #HL12
48 elim (lsuby_drop_trans_be … HL12 … HLK1) -HL12 -HLK1 /2 width=5 by cpy_subst/
49 | /4 width=1 by lsuby_succ, cpy_bind/
50 | /3 width=1 by cpy_flat/
54 lemma cpy_refl: ∀G,T,L,l,m. ⦃G, L⦄ ⊢ T ▶[l, m] T.
55 #G #T elim T -T // * /2 width=1 by cpy_bind, cpy_flat/
58 (* Basic_1: was: subst1_ex *)
59 lemma cpy_full: ∀I,G,K,V,T1,L,l. ⬇[l] L ≡ K.ⓑ{I}V →
60 ∃∃T2,T. ⦃G, L⦄ ⊢ T1 ▶[l, 1] T2 & ⬆[l, 1] T ≡ T2.
61 #I #G #K #V #T1 elim T1 -T1
63 /2 width=4 by lift_sort, lift_gref, ex2_2_intro/
64 elim (lt_or_eq_or_gt i l) #Hil
65 /3 width=4 by lift_lref_ge_minus, lift_lref_lt, ex2_2_intro/
67 elim (lift_total V 0 (i+1)) #W #HVW
68 elim (lift_split … HVW i i)
69 /4 width=5 by cpy_subst, ylt_inj, ex2_2_intro/
70 | * [ #a ] #J #W1 #U1 #IHW1 #IHU1 #L #l #HLK
71 elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
72 [ elim (IHU1 (L.ⓑ{J}W1) (l+1)) -IHU1
73 /3 width=9 by cpy_bind, drop_drop, lift_bind, ex2_2_intro/
74 | elim (IHU1 … HLK) -IHU1 -HLK
75 /3 width=8 by cpy_flat, lift_flat, ex2_2_intro/
80 lemma cpy_weak: ∀G,L,T1,T2,l1,m1. ⦃G, L⦄ ⊢ T1 ▶[l1, m1] T2 →
81 ∀l2,m2. l2 ≤ l1 → l1 + m1 ≤ l2 + m2 →
82 ⦃G, L⦄ ⊢ T1 ▶[l2, m2] T2.
83 #G #L #T1 #T2 #l1 #m1 #H elim H -G -L -T1 -T2 -l1 -m1 //
84 [ /3 width=5 by cpy_subst, ylt_yle_trans, yle_trans/
85 | /4 width=3 by cpy_bind, ylt_yle_trans, yle_succ/
86 | /3 width=1 by cpy_flat/
90 lemma cpy_weak_top: ∀G,L,T1,T2,l,m.
91 ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶[l, |L| - l] T2.
92 #G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m //
93 [ #I #G #L #K #V #W #i #l #m #Hli #_ #HLK #HVW
94 lapply (drop_fwd_length_lt2 … HLK) #Hil
95 lapply (ylt_inj … Hil) -Hil #Hil
96 lapply (yle_ylt_trans … Hil Hli) #Hl
97 @(cpy_subst … Hli … HLK HVW)
98 <yplus_minus_dx /2 width=1 by ylt_fwd_le/
99 | #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #HT12 #IHV normalize in match (|?|);
100 /2 width=1 by cpy_bind/
101 | /2 width=1 by cpy_flat/
105 lemma cpy_weak_full: ∀G,L,T1,T2,l,m.
106 ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶[0, |L|] T2.
107 #G #L #T1 #T2 #l #m #HT12
108 lapply (cpy_weak … HT12 0 (l + m) ? ?) -HT12
109 /2 width=2 by cpy_weak_top/
112 lemma cpy_split_up: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ∀i. i ≤ l + m →
113 ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[l, i-l] T & ⦃G, L⦄ ⊢ T ▶[i, l+m-i] T2.
114 #G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m
115 [ /2 width=3 by ex2_intro/
116 | #I #G #L #K #V #W #i #l #m #Hli #Hilm #HLK #HVW #j #Hjlm
117 elim (ylt_split i j) [ -Hilm -Hjlm | -Hli ]
118 /4 width=9 by cpy_subst, ylt_yle_trans, ex2_intro/
119 | #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #i #Hilm
120 elim (IHV12 i) -IHV12 // #V
121 elim (IHT12 (i+1)) -IHT12 /2 width=1 by yle_succ/ -Hilm
122 >yplus_SO2 >yplus_succ1 #T #HT1 #HT2
123 lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V) ?) -HT2
124 /3 width=5 by lsuby_succ, ex2_intro, cpy_bind/
125 | #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #i #Hilm
126 elim (IHV12 i) -IHV12 // elim (IHT12 i) -IHT12 // -Hilm
127 /3 width=5 by ex2_intro, cpy_flat/
131 lemma cpy_split_down: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ∀i. i ≤ l + m →
132 ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[i, l+m-i] T & ⦃G, L⦄ ⊢ T ▶[l, i-l] T2.
133 #G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m
134 [ /2 width=3 by ex2_intro/
135 | #I #G #L #K #V #W #i #l #m #Hli #Hilm #HLK #HVW #j #Hjlm
136 elim (ylt_split i j) [ -Hilm -Hjlm | -Hli ]
137 /4 width=9 by cpy_subst, ylt_yle_trans, ex2_intro/
138 | #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #i #Hilm
139 elim (IHV12 i) -IHV12 // #V
140 elim (IHT12 (i+1)) -IHT12 /2 width=1 by yle_succ/ -Hilm
141 >yplus_SO2 >yplus_succ1 #T #HT1 #HT2
142 lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V) ?) -HT2
143 /3 width=5 by lsuby_succ, ex2_intro, cpy_bind/
144 | #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #i #Hilm
145 elim (IHV12 i) -IHV12 // elim (IHT12 i) -IHT12 // -Hilm
146 /3 width=5 by ex2_intro, cpy_flat/
150 (* Basic forward lemmas *****************************************************)
152 lemma cpy_fwd_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2 →
153 ∀T1,l,m. ⬆[l, m] T1 ≡ U1 →
154 l ≤ lt → l + m ≤ lt + mt →
155 ∃∃T2. ⦃G, L⦄ ⊢ U1 ▶[l+m, lt+mt-(l+m)] U2 & ⬆[l, m] T2 ≡ U2.
156 #G #L #U1 #U2 #lt #mt #H elim H -G -L -U1 -U2 -lt -mt
157 [ * #i #G #L #lt #mt #T1 #l #m #H #_
158 [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by ex2_intro/
159 | elim (lift_inv_lref2 … H) -H * #Hil #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
160 | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by ex2_intro/
162 | #I #G #L #K #V #W #i #lt #mt #Hlti #Hilmt #HLK #HVW #T1 #l #m #H #Hllt #Hlmlmt
163 elim (lift_inv_lref2 … H) -H * #Hil #H destruct [ -V -Hilmt -Hlmlmt | -Hlti -Hllt ]
164 [ elim (ylt_yle_false … Hllt) -Hllt /3 width=3 by yle_ylt_trans, ylt_inj/
165 | elim (le_inv_plus_l … Hil) #Hlim #Hmi
166 elim (lift_split … HVW l (i-m+1) ? ? ?) [2,3,4: /2 width=1 by le_S_S, le_S/ ] -Hlim
167 #T2 #_ >plus_minus // <minus_minus /2 width=1 by le_S/ <minus_n_n <plus_n_O #H -Hmi
168 @(ex2_intro … H) -H @(cpy_subst … HLK HVW) /2 width=1 by yle_inj/ >ymax_pre_sn_comm // (**) (* explicit constructor *)
170 | #a #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #X #l #m #H #Hllt #Hlmlmt
171 elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
172 elim (IHW12 … HVW1) -V1 -IHW12 //
173 elim (IHU12 … HTU1) -T1 -IHU12 /2 width=1 by yle_succ/
174 <yplus_inj >yplus_SO2 >yplus_succ1 >yplus_succ1
175 /3 width=2 by cpy_bind, lift_bind, ex2_intro/
176 | #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #X #l #m #H #Hllt #Hlmlmt
177 elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
178 elim (IHW12 … HVW1) -V1 -IHW12 // elim (IHU12 … HTU1) -T1 -IHU12
179 /3 width=2 by cpy_flat, lift_flat, ex2_intro/
183 lemma cpy_fwd_tw: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ♯{T1} ≤ ♯{T2}.
184 #G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m normalize
185 /3 width=1 by monotonic_le_plus_l, le_plus/
188 (* Basic inversion lemmas ***************************************************)
190 fact cpy_inv_atom1_aux: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ∀J. T1 = ⓪{J} →
192 ∃∃I,K,V,i. l ≤ yinj i & i < l + m &
196 #G #L #T1 #T2 #l #m * -G -L -T1 -T2 -l -m
197 [ #I #G #L #l #m #J #H destruct /2 width=1 by or_introl/
198 | #I #G #L #K #V #T2 #i #l #m #Hli #Hilm #HLK #HVT2 #J #H destruct /3 width=9 by ex5_4_intro, or_intror/
199 | #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #J #H destruct
200 | #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #J #H destruct
204 lemma cpy_inv_atom1: ∀I,G,L,T2,l,m. ⦃G, L⦄ ⊢ ⓪{I} ▶[l, m] T2 →
206 ∃∃J,K,V,i. l ≤ yinj i & i < l + m &
210 /2 width=4 by cpy_inv_atom1_aux/ qed-.
212 (* Basic_1: was: subst1_gen_sort *)
213 lemma cpy_inv_sort1: ∀G,L,T2,k,l,m. ⦃G, L⦄ ⊢ ⋆k ▶[l, m] T2 → T2 = ⋆k.
214 #G #L #T2 #k #l #m #H
215 elim (cpy_inv_atom1 … H) -H //
216 * #I #K #V #i #_ #_ #_ #_ #H destruct
219 (* Basic_1: was: subst1_gen_lref *)
220 lemma cpy_inv_lref1: ∀G,L,T2,i,l,m. ⦃G, L⦄ ⊢ #i ▶[l, m] T2 →
222 ∃∃I,K,V. l ≤ i & i < l + m &
225 #G #L #T2 #i #l #m #H
226 elim (cpy_inv_atom1 … H) -H /2 width=1 by or_introl/
227 * #I #K #V #j #Hlj #Hjlm #HLK #HVT2 #H destruct /3 width=5 by ex4_3_intro, or_intror/
230 lemma cpy_inv_gref1: ∀G,L,T2,p,l,m. ⦃G, L⦄ ⊢ §p ▶[l, m] T2 → T2 = §p.
231 #G #L #T2 #p #l #m #H
232 elim (cpy_inv_atom1 … H) -H //
233 * #I #K #V #i #_ #_ #_ #_ #H destruct
236 fact cpy_inv_bind1_aux: ∀G,L,U1,U2,l,m. ⦃G, L⦄ ⊢ U1 ▶[l, m] U2 →
237 ∀a,I,V1,T1. U1 = ⓑ{a,I}V1.T1 →
238 ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[l, m] V2 &
239 ⦃G, L. ⓑ{I}V1⦄ ⊢ T1 ▶[↑l, m] T2 &
241 #G #L #U1 #U2 #l #m * -G -L -U1 -U2 -l -m
242 [ #I #G #L #l #m #b #J #W1 #U1 #H destruct
243 | #I #G #L #K #V #W #i #l #m #_ #_ #_ #_ #b #J #W1 #U1 #H destruct
244 | #a #I #G #L #V1 #V2 #T1 #T2 #l #m #HV12 #HT12 #b #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
245 | #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #b #J #W1 #U1 #H destruct
249 lemma cpy_inv_bind1: ∀a,I,G,L,V1,T1,U2,l,m. ⦃G, L⦄ ⊢ ⓑ{a,I} V1. T1 ▶[l, m] U2 →
250 ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[l, m] V2 &
251 ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶[↑l, m] T2 &
253 /2 width=3 by cpy_inv_bind1_aux/ qed-.
255 fact cpy_inv_flat1_aux: ∀G,L,U1,U2,l,m. ⦃G, L⦄ ⊢ U1 ▶[l, m] U2 →
256 ∀I,V1,T1. U1 = ⓕ{I}V1.T1 →
257 ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[l, m] V2 &
258 ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 &
260 #G #L #U1 #U2 #l #m * -G -L -U1 -U2 -l -m
261 [ #I #G #L #l #m #J #W1 #U1 #H destruct
262 | #I #G #L #K #V #W #i #l #m #_ #_ #_ #_ #J #W1 #U1 #H destruct
263 | #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #J #W1 #U1 #H destruct
264 | #I #G #L #V1 #V2 #T1 #T2 #l #m #HV12 #HT12 #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
268 lemma cpy_inv_flat1: ∀I,G,L,V1,T1,U2,l,m. ⦃G, L⦄ ⊢ ⓕ{I} V1. T1 ▶[l, m] U2 →
269 ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[l, m] V2 &
270 ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 &
272 /2 width=3 by cpy_inv_flat1_aux/ qed-.
275 fact cpy_inv_refl_O2_aux: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → m = 0 → T1 = T2.
276 #G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m
278 | #I #G #L #K #V #W #i #l #m #Hli #Hilm #_ #_ #H destruct
279 elim (ylt_yle_false … Hli) -Hli //
280 | /3 width=1 by eq_f2/
281 | /3 width=1 by eq_f2/
285 lemma cpy_inv_refl_O2: ∀G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 ▶[l, 0] T2 → T1 = T2.
286 /2 width=6 by cpy_inv_refl_O2_aux/ qed-.
288 (* Basic_1: was: subst1_gen_lift_eq *)
289 lemma cpy_inv_lift1_eq: ∀G,T1,U1,l,m. ⬆[l, m] T1 ≡ U1 →
290 ∀L,U2. ⦃G, L⦄ ⊢ U1 ▶[l, m] U2 → U1 = U2.
291 #G #T1 #U1 #l #m #HTU1 #L #U2 #HU12 elim (cpy_fwd_up … HU12 … HTU1) -HU12 -HTU1
292 /2 width=4 by cpy_inv_refl_O2/
295 (* Basic_1: removed theorems 25:
296 subst0_gen_sort subst0_gen_lref subst0_gen_head subst0_gen_lift_lt
297 subst0_gen_lift_false subst0_gen_lift_ge subst0_refl subst0_trans
298 subst0_lift_lt subst0_lift_ge subst0_lift_ge_S subst0_lift_ge_s
299 subst0_subst0 subst0_subst0_back subst0_weight_le subst0_weight_lt
300 subst0_confluence_neq subst0_confluence_eq subst0_tlt_head
301 subst0_confluence_lift subst0_tlt
302 subst1_head subst1_gen_head subst1_lift_S subst1_confluence_lift