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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
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15 include "basic_2/notation/relations/psubst_6.ma".
16 include "basic_2/grammar/genv.ma".
17 include "basic_2/substitution/lsuby.ma".
19 (* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************)
22 inductive cpy: ynat → ynat → relation4 genv lenv term term ≝
23 | cpy_atom : ∀I,G,L,l,m. cpy l m G L (⓪{I}) (⓪{I})
24 | cpy_subst: ∀I,G,L,K,V,W,i,l,m. l ≤ i → i < l+m →
25 ⬇[i] L ≡ K.ⓑ{I}V → ⬆[0, ⫯i] V ≡ W → cpy l m G L (#i) W
26 | cpy_bind : ∀a,I,G,L,V1,V2,T1,T2,l,m.
27 cpy l m G L V1 V2 → cpy (⫯l) m G (L.ⓑ{I}V1) T1 T2 →
28 cpy l m G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
29 | cpy_flat : ∀I,G,L,V1,V2,T1,T2,l,m.
30 cpy l m G L V1 V2 → cpy l m G L T1 T2 →
31 cpy l m G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
34 interpretation "context-sensitive extended ordinary substritution (term)"
35 'PSubst G L T1 l m T2 = (cpy l m G L T1 T2).
37 (* Basic properties *********************************************************)
39 lemma lsuby_cpy_trans: ∀G,l,m. lsub_trans … (cpy l m G) (lsuby l m).
40 #G #l #m #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2 -l -m
42 | #I #G #L1 #K1 #V #W #i #l #m #Hli #Hilm #HLK1 #HVW #L2 #HL12
43 elim (ylt_inv_plus_dx … Hilm) #m0 #H0 #_
44 elim (lsuby_drop_trans_be … HL12 … HLK1 … H0) -HL12 -HLK1 -H0 /2 width=5 by cpy_subst/
45 | /4 width=1 by lsuby_succ, cpy_bind/
46 | /3 width=1 by cpy_flat/
50 lemma cpy_refl: ∀G,T,L,l,m. ⦃G, L⦄ ⊢ T ▶[l, m] T.
51 #G #T elim T -T // * /2 width=1 by cpy_bind, cpy_flat/
54 (* Basic_1: was: subst1_ex *)
55 lemma cpy_full: ∀I,G,K,V,T1,L,l. ⬇[l] L ≡ K.ⓑ{I}V →
56 ∃∃T2,T. ⦃G, L⦄ ⊢ T1 ▶[l, 1] T2 & ⬆[l, 1] T ≡ T2.
57 #I #G #K #V #T1 elim T1 -T1
59 /2 width=4 by lift_sort, lift_gref, ex2_2_intro/
60 elim (ylt_split_eq i l) /3 width=4 by lift_lref_pred, lift_lref_lt, ex2_2_intro/
61 #H destruct lapply (drop_fwd_Y2 … HLK) #Hi
62 elim (lift_total (⫯i) … V 0) /2 width=1 by ylt_succ/
63 #W #HVW elim (lift_split … HVW i i … 1)
64 /4 width=6 by cpy_subst, monotonic_ylt_plus_sn, ex2_2_intro/
65 | * [ #a ] #J #W1 #U1 #IHW1 #IHU1 #L #l #HLK
66 elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
67 [ elim (IHU1 (L.ⓑ{J}W1) (⫯l)) -IHU1
68 /3 width=9 by cpy_bind, drop_drop, lift_bind, ex2_2_intro/
69 | elim (IHU1 … HLK) -IHU1 -HLK
70 /3 width=8 by cpy_flat, lift_flat, ex2_2_intro/
75 lemma cpy_weak: ∀G,L,T1,T2,l1,m1. ⦃G, L⦄ ⊢ T1 ▶[l1, m1] T2 →
76 ∀l2,m2. l2 ≤ l1 → l1 + m1 ≤ l2 + m2 →
77 ⦃G, L⦄ ⊢ T1 ▶[l2, m2] T2.
78 #G #L #T1 #T2 #l1 #m1 #H elim H -G -L -T1 -T2 -l1 -m1 //
79 [ /3 width=5 by cpy_subst, ylt_yle_trans, yle_trans/
80 | /4 width=3 by cpy_bind, ylt_yle_trans, yle_succ/
81 | /3 width=1 by cpy_flat/
85 lemma cpy_weak_top: ∀G,L,T1,T2,l,m.
86 ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶[l, ∞] T2.
87 /2 width=5 by cpy_weak/ qed-.
89 lemma cpy_weak_full: ∀G,L,T1,T2,l,m.
90 ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶[0, ∞] T2.
91 /2 width=5 by cpy_weak/ qed-.
93 lemma cpy_split_up: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 →
94 ∀i,m2. i + m2 = l + m →
96 ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[l, m1] T & ⦃G, L⦄ ⊢ T ▶[i, m2] T2.
97 #G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m
98 [ /2 width=3 by ex2_intro/
99 | #I #G #L #K #V #W #i #l #m #Hli #Hilm #HLK #HVW #j #m2 #H2 #m1 #H1
100 elim (ylt_split i j) [ -Hilm -H2 | -Hli ]
101 /4 width=9 by cpy_subst, ylt_yle_trans, ex2_intro/
102 | #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #i #m2 #H2 #m1 #H1
103 elim (IHV12 … H2 … H1) -IHV12 #V
104 elim (IHT12 (⫯i) … m2 … m1) -IHT12 /2 width=1 by yle_succ/ -H2 -H1
105 #T #HT1 #HT2 lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V) ?) -HT2
106 /3 width=5 by lsuby_succ, ex2_intro, cpy_bind/
107 | #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #i #m2 #H2 #m1 #H1
108 elim (IHV12 … H2 … H1) -IHV12 elim (IHT12 … H2 … H1) -IHT12 -H2 -H1
109 /3 width=5 by ex2_intro, cpy_flat/
113 lemma cpy_split_down: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 →
114 ∀m1,m2. m = m1 + m2 →
115 ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[l+m2, m1] T & ⦃G, L⦄ ⊢ T ▶[l, m2] T2.
116 #G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m
117 [ /2 width=3 by ex2_intro/
118 | #I #G #L #K #V #W #i #l #m #Hli #Hilm #HLK #HVW #m1 #m2 #H destruct
119 elim (ylt_split i (l+m2)) [ -Hilm | -Hli ]
120 /3 width=9 by cpy_subst, ex2_intro/
121 | #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #m1 #m2 #H destruct
122 elim (IHV12 m1 m2) -IHV12 // #V
123 elim (IHT12 m1 m2) -IHT12 //
124 >yplus_succ1 #T #HT1 #HT2
125 lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V) ?) -HT2
126 /3 width=5 by lsuby_succ, ex2_intro, cpy_bind/
127 | #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #m1 #m2 #H destruct
128 elim (IHV12 m1 m2) -IHV12 // elim (IHT12 m1 m2) -IHT12 //
129 /3 width=5 by ex2_intro, cpy_flat/
133 (* Basic forward lemmas *****************************************************)
135 lemma cpy_fwd_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2 →
136 ∀T1,l,m. ⬆[l, m] T1 ≡ U1 →
137 l ≤ lt → l + m ≤ lt + mt →
138 ∃∃T2. ⦃G, L⦄ ⊢ U1 ▶[l+m, lt+mt-(l+m)] U2 & ⬆[l, m] T2 ≡ U2.
139 #G #L #U1 #U2 #lt #mt #H elim H -G -L -U1 -U2 -lt -mt
140 [ * #i #G #L #lt #mt #T1 #l #m #H #_
141 [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by ex2_intro/
142 | elim (lift_inv_lref2 … H) -H * #Hil #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
143 | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by ex2_intro/
145 | #I #G #L #K #V #W #i #lt #mt #Hlti #Hilmt #HLK #HVW #T1 #l #m #H #Hllt #Hlmlmt
146 elim (lift_inv_lref2 … H) -H * #Hil #H destruct [ -V -Hilmt -Hlmlmt | -Hlti -Hllt ]
147 [ elim (ylt_yle_false … Hllt) -Hllt /3 width=3 by yle_ylt_trans, ylt_inj/
148 | elim (yle_inv_plus_inj2 … Hil) #Hlim #Hmi
149 elim (lift_split … HVW l (⫯(i-m)) ? ? ?) [2,3,4: /2 width=1 by yle_succ_dx, le_S_S/ ] -Hlim
150 #T2 #_ >plus_minus /2 width=1 by yle_inv_inj/ <minus_minus /3 width=1 by le_S, yle_inv_inj/ <minus_n_n <plus_n_O #H -Hmi
151 @(ex2_intro … H) -H @(cpy_subst … HLK HVW) /2 width=1 by yle_inj/ >ymax_pre_sn_comm // (**) (* explicit constructor *)
153 | #a #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #X #l #m #H #Hllt #Hlmlmt
154 elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
155 elim (IHW12 … HVW1) -V1 -IHW12 //
156 elim (IHU12 … HTU1) -T1 -IHU12 /2 width=1 by yle_succ/
157 <yplus_inj >yplus_SO2 >yplus_succ1 >yplus_succ1
158 /3 width=2 by cpy_bind, lift_bind, ex2_intro/
159 | #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #X #l #m #H #Hllt #Hlmlmt
160 elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
161 elim (IHW12 … HVW1) -V1 -IHW12 // elim (IHU12 … HTU1) -T1 -IHU12
162 /3 width=2 by cpy_flat, lift_flat, ex2_intro/
166 lemma cpy_fwd_tw: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ♯{T1} ≤ ♯{T2}.
167 #G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m normalize
168 /3 width=1 by monotonic_le_plus_l, le_plus/
171 (* Basic inversion lemmas ***************************************************)
173 fact cpy_inv_atom1_aux: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ∀J. T1 = ⓪{J} →
175 ∃∃I,K,V,i. l ≤ yinj i & i < l + m &
179 #G #L #T1 #T2 #l #m * -G -L -T1 -T2 -l -m
180 [ #I #G #L #l #m #J #H destruct /2 width=1 by or_introl/
181 | #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/
182 | #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #J #H destruct
183 | #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #J #H destruct
187 lemma cpy_inv_atom1: ∀I,G,L,T2,l,m. ⦃G, L⦄ ⊢ ⓪{I} ▶[l, m] T2 →
189 ∃∃J,K,V,i. l ≤ yinj i & i < l + m &
193 /2 width=4 by cpy_inv_atom1_aux/ qed-.
195 (* Basic_1: was: subst1_gen_sort *)
196 lemma cpy_inv_sort1: ∀G,L,T2,k,l,m. ⦃G, L⦄ ⊢ ⋆k ▶[l, m] T2 → T2 = ⋆k.
197 #G #L #T2 #k #l #m #H
198 elim (cpy_inv_atom1 … H) -H //
199 * #I #K #V #i #_ #_ #_ #_ #H destruct
202 (* Basic_1: was: subst1_gen_lref *)
203 lemma cpy_inv_lref1: ∀G,L,T2,i,l,m. ⦃G, L⦄ ⊢ #i ▶[l, m] T2 →
205 ∃∃I,K,V. l ≤ i & i < l + m &
208 #G #L #T2 #i #l #m #H
209 elim (cpy_inv_atom1 … H) -H /2 width=1 by or_introl/
210 * #I #K #V #j #Hlj #Hjlm #HLK #HVT2 #H destruct /3 width=5 by ex4_3_intro, or_intror/
213 lemma cpy_inv_gref1: ∀G,L,T2,p,l,m. ⦃G, L⦄ ⊢ §p ▶[l, m] T2 → T2 = §p.
214 #G #L #T2 #p #l #m #H
215 elim (cpy_inv_atom1 … H) -H //
216 * #I #K #V #i #_ #_ #_ #_ #H destruct
219 fact cpy_inv_bind1_aux: ∀G,L,U1,U2,l,m. ⦃G, L⦄ ⊢ U1 ▶[l, m] U2 →
220 ∀a,I,V1,T1. U1 = ⓑ{a,I}V1.T1 →
221 ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[l, m] V2 &
222 ⦃G, L. ⓑ{I}V1⦄ ⊢ T1 ▶[⫯l, m] T2 &
224 #G #L #U1 #U2 #l #m * -G -L -U1 -U2 -l -m
225 [ #I #G #L #l #m #b #J #W1 #U1 #H destruct
226 | #I #G #L #K #V #W #i #l #m #_ #_ #_ #_ #b #J #W1 #U1 #H destruct
227 | #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/
228 | #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #b #J #W1 #U1 #H destruct
232 lemma cpy_inv_bind1: ∀a,I,G,L,V1,T1,U2,l,m. ⦃G, L⦄ ⊢ ⓑ{a,I} V1. T1 ▶[l, m] U2 →
233 ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[l, m] V2 &
234 ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶[⫯l, m] T2 &
236 /2 width=3 by cpy_inv_bind1_aux/ qed-.
238 fact cpy_inv_flat1_aux: ∀G,L,U1,U2,l,m. ⦃G, L⦄ ⊢ U1 ▶[l, m] U2 →
239 ∀I,V1,T1. U1 = ⓕ{I}V1.T1 →
240 ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[l, m] V2 &
241 ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 &
243 #G #L #U1 #U2 #l #m * -G -L -U1 -U2 -l -m
244 [ #I #G #L #l #m #J #W1 #U1 #H destruct
245 | #I #G #L #K #V #W #i #l #m #_ #_ #_ #_ #J #W1 #U1 #H destruct
246 | #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #J #W1 #U1 #H destruct
247 | #I #G #L #V1 #V2 #T1 #T2 #l #m #HV12 #HT12 #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
251 lemma cpy_inv_flat1: ∀I,G,L,V1,T1,U2,l,m. ⦃G, L⦄ ⊢ ⓕ{I} V1. T1 ▶[l, m] U2 →
252 ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[l, m] V2 &
253 ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 &
255 /2 width=3 by cpy_inv_flat1_aux/ qed-.
258 fact cpy_inv_refl_O2_aux: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → m = 0 → T1 = T2.
259 #G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m
261 | #I #G #L #K #V #W #i #l #m #Hli #Hilm #_ #_ #H destruct
262 elim (ylt_yle_false … Hli) -Hli //
263 | /3 width=1 by eq_f2/
264 | /3 width=1 by eq_f2/
268 lemma cpy_inv_refl_O2: ∀G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 ▶[l, 0] T2 → T1 = T2.
269 /2 width=6 by cpy_inv_refl_O2_aux/ qed-.
271 (* Basic_1: was: subst1_gen_lift_eq *)
272 lemma cpy_inv_lift1_eq: ∀G,T1,U1,l,m. ⬆[l, m] T1 ≡ U1 →
273 ∀L,U2. ⦃G, L⦄ ⊢ U1 ▶[l, m] U2 → U1 = U2.
274 #G #T1 #U1 #l #m #HTU1 #L #U2 #HU12 elim (cpy_fwd_up … HU12 … HTU1) -HU12 -HTU1
275 /2 width=4 by cpy_inv_refl_O2/
278 (* Basic_1: removed theorems 25:
279 subst0_gen_sort subst0_gen_lref subst0_gen_head subst0_gen_lift_lt
280 subst0_gen_lift_false subst0_gen_lift_ge subst0_refl subst0_trans
281 subst0_lift_lt subst0_lift_ge subst0_lift_ge_S subst0_lift_ge_s
282 subst0_subst0 subst0_subst0_back subst0_weight_le subst0_weight_lt
283 subst0_confluence_neq subst0_confluence_eq subst0_tlt_head
284 subst0_confluence_lift subst0_tlt
285 subst1_head subst1_gen_head subst1_lift_S subst1_confluence_lift