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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_plus.ma".
16 include "basic_2/notation/relations/extlrsubeq_4.ma".
17 include "basic_2/relocation/ldrop.ma".
19 (* LOCAL ENVIRONMENT REFINEMENT FOR EXTENDED SUBSTITUTION *******************)
21 inductive lsuby: relation4 ynat ynat lenv lenv ≝
22 | lsuby_atom: ∀L,d,e. lsuby d e L (⋆)
23 | lsuby_zero: ∀I1,I2,L1,L2,V1,V2.
24 lsuby 0 0 L1 L2 → lsuby 0 0 (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2)
25 | lsuby_pair: ∀I1,I2,L1,L2,V,e. lsuby 0 e L1 L2 →
26 lsuby 0 (⫯e) (L1.ⓑ{I1}V) (L2.ⓑ{I2}V)
27 | lsuby_succ: ∀I1,I2,L1,L2,V1,V2,d,e.
28 lsuby d e L1 L2 → lsuby (⫯d) e (L1. ⓑ{I1}V1) (L2. ⓑ{I2} V2)
32 "local environment refinement (extended substitution)"
33 'ExtLRSubEq L1 d e L2 = (lsuby d e L1 L2).
35 (* Basic properties *********************************************************)
37 lemma lsuby_pair_lt: ∀I1,I2,L1,L2,V,e. L1 ⊑×[0, ⫰e] L2 → 0 < e →
38 L1.ⓑ{I1}V ⊑×[0, e] L2.ⓑ{I2}V.
39 #I1 #I2 #L1 #L2 #V #e #HL12 #He <(ylt_inv_O1 … He) /2 width=1 by lsuby_pair/
42 lemma lsuby_succ_lt: ∀I1,I2,L1,L2,V1,V2,d,e. L1 ⊑×[⫰d, e] L2 → 0 < d →
43 L1.ⓑ{I1}V1 ⊑×[d, e] L2. ⓑ{I2}V2.
44 #I1 #I2 #L1 #L2 #V1 #V2 #d #e #HL12 #Hd <(ylt_inv_O1 … Hd) /2 width=1 by lsuby_succ/
47 lemma lsuby_refl: ∀L,d,e. L ⊑×[d, e] L.
49 #L #I #V #IHL #d elim (ynat_cases … d) [| * #x ]
50 #Hd destruct /2 width=1 by lsuby_succ/
51 #e elim (ynat_cases … e) [| * #x ]
52 #He destruct /2 width=1 by lsuby_zero, lsuby_pair/
55 lemma lsuby_length: ∀L1,L2. |L2| ≤ |L1| → L1 ⊑×[yinj 0, yinj 0] L2.
57 [ #X #H lapply (le_n_O_to_eq … H) -H
58 #H lapply (length_inv_zero_sn … H) #H destruct /2 width=1 by lsuby_atom/
59 | #L1 #I1 #V1 #IHL1 * normalize
60 /4 width=2 by lsuby_zero, le_S_S_to_le/
64 lemma lsuby_sym: ∀d,e,L1,L2. L1 ⊑×[d, e] L2 → |L1| = |L2| → L2 ⊑×[d, e] L1.
65 #d #e #L1 #L2 #H elim H -d -e -L1 -L2
66 [ #L1 #d #e #H >(length_inv_zero_dx … H) -L1 //
67 | /2 width=1 by lsuby_length/
68 | #I1 #I2 #L1 #L2 #V #e #_ #IHL12 #H lapply (injective_plus_l … H)
69 /3 width=1 by lsuby_pair/
70 | #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #IHL12 #H lapply (injective_plus_l … H)
71 /3 width=1 by lsuby_succ/
75 (* Basic inversion lemmas ***************************************************)
77 fact lsuby_inv_atom1_aux: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 → L1 = ⋆ → L2 = ⋆.
78 #L1 #L2 #d #e * -L1 -L2 -d -e //
79 [ #I1 #I2 #L1 #L2 #V1 #V2 #_ #H destruct
80 | #I1 #I2 #L1 #L2 #V #e #_ #H destruct
81 | #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #H destruct
85 lemma lsuby_inv_atom1: ∀L2,d,e. ⋆ ⊑×[d, e] L2 → L2 = ⋆.
86 /2 width=5 by lsuby_inv_atom1_aux/ qed-.
88 fact lsuby_inv_zero1_aux: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 →
89 ∀J1,K1,W1. L1 = K1.ⓑ{J1}W1 → d = 0 → e = 0 →
91 ∃∃J2,K2,W2. K1 ⊑×[0, 0] K2 & L2 = K2.ⓑ{J2}W2.
92 #L1 #L2 #d #e * -L1 -L2 -d -e /2 width=1 by or_introl/
93 [ #I1 #I2 #L1 #L2 #V1 #V2 #HL12 #J1 #K1 #W1 #H #_ #_ destruct
94 /3 width=5 by ex2_3_intro, or_intror/
95 | #I1 #I2 #L1 #L2 #V #e #_ #J1 #K1 #W1 #_ #_ #H
96 elim (ysucc_inv_O_dx … H)
97 | #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J1 #K1 #W1 #_ #H
98 elim (ysucc_inv_O_dx … H)
102 lemma lsuby_inv_zero1: ∀I1,K1,L2,V1. K1.ⓑ{I1}V1 ⊑×[0, 0] L2 →
104 ∃∃I2,K2,V2. K1 ⊑×[0, 0] K2 & L2 = K2.ⓑ{I2}V2.
105 /2 width=9 by lsuby_inv_zero1_aux/ qed-.
107 fact lsuby_inv_pair1_aux: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 →
108 ∀J1,K1,W. L1 = K1.ⓑ{J1}W → d = 0 → 0 < e →
110 ∃∃J2,K2. K1 ⊑×[0, ⫰e] K2 & L2 = K2.ⓑ{J2}W.
111 #L1 #L2 #d #e * -L1 -L2 -d -e /2 width=1 by or_introl/
112 [ #I1 #I2 #L1 #L2 #V1 #V2 #_ #J1 #K1 #W #_ #_ #H
113 elim (ylt_yle_false … H) //
114 | #I1 #I2 #L1 #L2 #V #e #HL12 #J1 #K1 #W #H #_ #_ destruct
115 /3 width=4 by ex2_2_intro, or_intror/
116 | #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J1 #K1 #W #_ #H
117 elim (ysucc_inv_O_dx … H)
121 lemma lsuby_inv_pair1: ∀I1,K1,L2,V,e. K1.ⓑ{I1}V ⊑×[0, e] L2 → 0 < e →
123 ∃∃I2,K2. K1 ⊑×[0, ⫰e] K2 & L2 = K2.ⓑ{I2}V.
124 /2 width=6 by lsuby_inv_pair1_aux/ qed-.
126 fact lsuby_inv_succ1_aux: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 →
127 ∀J1,K1,W1. L1 = K1.ⓑ{J1}W1 → 0 < d →
129 ∃∃J2,K2,W2. K1 ⊑×[⫰d, e] K2 & L2 = K2.ⓑ{J2}W2.
130 #L1 #L2 #d #e * -L1 -L2 -d -e /2 width=1 by or_introl/
131 [ #I1 #I2 #L1 #L2 #V1 #V2 #_ #J1 #K1 #W1 #_ #H
132 elim (ylt_yle_false … H) //
133 | #I1 #I2 #L1 #L2 #V #e #_ #J1 #K1 #W1 #_ #H
134 elim (ylt_yle_false … H) //
135 | #I1 #I2 #L1 #L2 #V1 #V2 #d #e #HL12 #J1 #K1 #W1 #H #_ destruct
136 /3 width=5 by ex2_3_intro, or_intror/
140 lemma lsuby_inv_succ1: ∀I1,K1,L2,V1,d,e. K1.ⓑ{I1}V1 ⊑×[d, e] L2 → 0 < d →
142 ∃∃I2,K2,V2. K1 ⊑×[⫰d, e] K2 & L2 = K2.ⓑ{I2}V2.
143 /2 width=5 by lsuby_inv_succ1_aux/ qed-.
145 fact lsuby_inv_zero2_aux: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 →
146 ∀J2,K2,W2. L2 = K2.ⓑ{J2}W2 → d = 0 → e = 0 →
147 ∃∃J1,K1,W1. K1 ⊑×[0, 0] K2 & L1 = K1.ⓑ{J1}W1.
148 #L1 #L2 #d #e * -L1 -L2 -d -e
149 [ #L1 #d #e #J2 #K2 #W1 #H destruct
150 | #I1 #I2 #L1 #L2 #V1 #V2 #HL12 #J2 #K2 #W2 #H #_ #_ destruct
151 /2 width=5 by ex2_3_intro/
152 | #I1 #I2 #L1 #L2 #V #e #_ #J2 #K2 #W2 #_ #_ #H
153 elim (ysucc_inv_O_dx … H)
154 | #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J2 #K2 #W2 #_ #H
155 elim (ysucc_inv_O_dx … H)
159 lemma lsuby_inv_zero2: ∀I2,K2,L1,V2. L1 ⊑×[0, 0] K2.ⓑ{I2}V2 →
160 ∃∃I1,K1,V1. K1 ⊑×[0, 0] K2 & L1 = K1.ⓑ{I1}V1.
161 /2 width=9 by lsuby_inv_zero2_aux/ qed-.
163 fact lsuby_inv_pair2_aux: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 →
164 ∀J2,K2,W. L2 = K2.ⓑ{J2}W → d = 0 → 0 < e →
165 ∃∃J1,K1. K1 ⊑×[0, ⫰e] K2 & L1 = K1.ⓑ{J1}W.
166 #L1 #L2 #d #e * -L1 -L2 -d -e
167 [ #L1 #d #e #J2 #K2 #W #H destruct
168 | #I1 #I2 #L1 #L2 #V1 #V2 #_ #J2 #K2 #W #_ #_ #H
169 elim (ylt_yle_false … H) //
170 | #I1 #I2 #L1 #L2 #V #e #HL12 #J2 #K2 #W #H #_ #_ destruct
171 /2 width=4 by ex2_2_intro/
172 | #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J2 #K2 #W #_ #H
173 elim (ysucc_inv_O_dx … H)
177 lemma lsuby_inv_pair2: ∀I2,K2,L1,V,e. L1 ⊑×[0, e] K2.ⓑ{I2}V → 0 < e →
178 ∃∃I1,K1. K1 ⊑×[0, ⫰e] K2 & L1 = K1.ⓑ{I1}V.
179 /2 width=6 by lsuby_inv_pair2_aux/ qed-.
181 fact lsuby_inv_succ2_aux: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 →
182 ∀J2,K2,W2. L2 = K2.ⓑ{J2}W2 → 0 < d →
183 ∃∃J1,K1,W1. K1 ⊑×[⫰d, e] K2 & L1 = K1.ⓑ{J1}W1.
184 #L1 #L2 #d #e * -L1 -L2 -d -e
185 [ #L1 #d #e #J2 #K2 #W2 #H destruct
186 | #I1 #I2 #L1 #L2 #V1 #V2 #_ #J2 #K2 #W2 #_ #H
187 elim (ylt_yle_false … H) //
188 | #I1 #I2 #L1 #L2 #V #e #_ #J2 #K1 #W2 #_ #H
189 elim (ylt_yle_false … H) //
190 | #I1 #I2 #L1 #L2 #V1 #V2 #d #e #HL12 #J2 #K2 #W2 #H #_ destruct
191 /2 width=5 by ex2_3_intro/
195 lemma lsuby_inv_succ2: ∀I2,K2,L1,V2,d,e. L1 ⊑×[d, e] K2.ⓑ{I2}V2 → 0 < d →
196 ∃∃I1,K1,V1. K1 ⊑×[⫰d, e] K2 & L1 = K1.ⓑ{I1}V1.
197 /2 width=5 by lsuby_inv_succ2_aux/ qed-.
199 (* Basic forward lemmas *****************************************************)
201 lemma lsuby_fwd_length: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 → |L2| ≤ |L1|.
202 #L1 #L2 #d #e #H elim H -L1 -L2 -d -e normalize /2 width=1 by le_S_S/
205 lemma lsuby_fwd_ldrop2_be: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 →
206 ∀I2,K2,W,i. ⇩[0, i] L2 ≡ K2.ⓑ{I2}W →
208 ∃∃I1,K1. K1 ⊑×[0, ⫰(d+e-i)] K2 & ⇩[0, i] L1 ≡ K1.ⓑ{I1}W.
209 #L1 #L2 #d #e #H elim H -L1 -L2 -d -e
210 [ #L1 #d #e #J2 #K2 #W #i #H
211 elim (ldrop_inv_atom1 … H) -H #H destruct
212 | #I1 #I2 #L1 #L2 #V1 #V2 #_ #_ #J2 #K2 #W #i #_ #_ #H
213 elim (ylt_yle_false … H) //
214 | #I1 #I2 #L1 #L2 #V #e #HL12 #IHL12 #J2 #K2 #W #i #H #_ >yplus_O1
215 elim (ldrop_inv_O1_pair1 … H) -H * #Hi #HLK1 [ -IHL12 | -HL12 ]
216 [ #_ destruct -I2 >ypred_succ
217 /2 width=4 by ldrop_pair, ex2_2_intro/
218 | lapply (ylt_inv_O1 i ?) /2 width=1 by ylt_inj/
219 #H <H -H #H lapply (ylt_inv_succ … H) -H
220 #Hie elim (IHL12 … HLK1) -IHL12 -HLK1 // -Hie
221 >yminus_succ <yminus_inj /3 width=4 by ldrop_ldrop_lt, ex2_2_intro/
223 | #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #IHL12 #J2 #K2 #W #i #HLK2 #Hdi
224 elim (yle_inv_succ1 … Hdi) -Hdi
225 #Hdi #Hi <Hi >yplus_succ1 #H lapply (ylt_inv_succ … H) -H
226 #Hide lapply (ldrop_inv_ldrop1_lt … HLK2 ?) -HLK2 /2 width=1 by ylt_O/
227 #HLK1 elim (IHL12 … HLK1) -IHL12 -HLK1 <yminus_inj >yminus_SO2
228 /4 width=4 by ylt_O, ldrop_ldrop_lt, ex2_2_intro/