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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/extlrsubeq_4.ma".
16 include "basic_2/relocation/ldrop.ma".
18 (* LOCAL ENVIRONMENT REFINEMENT FOR EXTENDED SUBSTITUTION *******************)
20 inductive lsuby: relation4 nat nat lenv lenv ≝
21 | lsuby_atom: ∀L,d,e. lsuby d e L (⋆)
22 | lsuby_zero: ∀I1,I2,L1,L2,V1,V2.
23 lsuby 0 0 L1 L2 → lsuby 0 0 (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2)
24 | lsuby_pair: ∀I1,I2,L1,L2,V,e. lsuby 0 e L1 L2 →
25 lsuby 0 (e + 1) (L1.ⓑ{I1}V) (L2.ⓑ{I2}V)
26 | lsuby_succ: ∀I1,I2,L1,L2,V1,V2,d,e.
27 lsuby d e L1 L2 → lsuby (d + 1) e (L1. ⓑ{I1}V1) (L2. ⓑ{I2} V2)
31 "local environment refinement (extended substitution)"
32 'ExtLRSubEq L1 d e L2 = (lsuby d e L1 L2).
34 (* Basic properties *********************************************************)
36 lemma lsuby_pair_lt: ∀I1,I2,L1,L2,V,e. L1 ⊑×[0, e-1] L2 → 0 < e →
37 L1.ⓑ{I1}V ⊑×[0, e] L2.ⓑ{I2}V.
38 #I1 #I2 #L1 #L2 #V #e #HL12 #He >(plus_minus_m_m e 1) /2 width=1 by lsuby_pair/
41 lemma lsuby_succ_lt: ∀I1,I2,L1,L2,V1,V2,d,e. L1 ⊑×[d-1, e] L2 → 0 < d →
42 L1.ⓑ{I1}V1 ⊑×[d, e] L2. ⓑ{I2}V2.
43 #I1 #I2 #L1 #L2 #V1 #V2 #d #e #HL12 #Hd >(plus_minus_m_m d 1) /2 width=1 by lsuby_succ/
46 lemma lsuby_refl: ∀L,d,e. L ⊑×[d, e] L.
48 #L #I #V #IHL #d @(nat_ind_plus … d) -d /2 width=1 by lsuby_succ/
49 #e @(nat_ind_plus … e) -e /2 width=2 by lsuby_pair, lsuby_zero/
52 lemma lsuby_length: ∀L1,L2. |L2| ≤ |L1| → L1 ⊑×[0, 0] L2.
54 [ #X #H lapply (le_n_O_to_eq … H) -H
55 #H lapply (length_inv_zero_sn … H) #H destruct /2 width=1 by lsuby_atom/
56 | #L1 #I1 #V1 #IHL1 * normalize
57 /4 width=2 by lsuby_zero, le_S_S_to_le/
61 (* Basic inversion lemmas ***************************************************)
63 fact lsuby_inv_atom1_aux: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 → L1 = ⋆ → L2 = ⋆.
64 #L1 #L2 #d #e * -L1 -L2 -d -e //
65 [ #I1 #I2 #L1 #L2 #V1 #V2 #_ #H destruct
66 | #I1 #I2 #L1 #L2 #V #e #_ #H destruct
67 | #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #H destruct
71 lemma lsuby_inv_atom1: ∀L2,d,e. ⋆ ⊑×[d, e] L2 → L2 = ⋆.
72 /2 width=5 by lsuby_inv_atom1_aux/ qed-.
74 fact lsuby_inv_zero1_aux: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 →
75 ∀J1,K1,W1. L1 = K1.ⓑ{J1}W1 → d = 0 → e = 0 →
77 ∃∃J2,K2,W2. K1 ⊑×[0, 0] K2 & L2 = K2.ⓑ{J2}W2.
78 #L1 #L2 #d #e * -L1 -L2 -d -e /2 width=1 by or_introl/
79 [ #I1 #I2 #L1 #L2 #V1 #V2 #HL12 #J1 #K1 #W1 #H #_ #_ destruct
80 /3 width=5 by ex2_3_intro, or_intror/
81 | #I1 #I2 #L1 #L2 #V #e #_ #J1 #K1 #W1 #_ #_
82 <plus_n_Sm #H destruct
83 | #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J1 #K1 #W1 #_
84 <plus_n_Sm #H destruct
88 lemma lsuby_inv_zero1: ∀I1,K1,L2,V1. K1.ⓑ{I1}V1 ⊑×[0, 0] L2 →
90 ∃∃I2,K2,V2. K1 ⊑×[0, 0] K2 & L2 = K2.ⓑ{I2}V2.
91 /2 width=9 by lsuby_inv_zero1_aux/ qed-.
93 fact lsuby_inv_pair1_aux: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 →
94 ∀J1,K1,W. L1 = K1.ⓑ{J1}W → d = 0 → 0 < e →
96 ∃∃J2,K2. K1 ⊑×[0, e-1] K2 & L2 = K2.ⓑ{J2}W.
97 #L1 #L2 #d #e * -L1 -L2 -d -e /2 width=1 by or_introl/
98 [ #I1 #I2 #L1 #L2 #V1 #V2 #_ #J1 #K1 #W #_ #_ #H
99 elim (lt_zero_false … H)
100 | #I1 #I2 #L1 #L2 #V #e #HL12 #J1 #K1 #W #H #_ #_ destruct
101 /3 width=4 by ex2_2_intro, or_intror/
102 | #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J1 #K1 #W #_
103 <plus_n_Sm #H destruct
107 lemma lsuby_inv_pair1: ∀I1,K1,L2,V,e. K1.ⓑ{I1}V ⊑×[0, e] L2 → 0 < e →
109 ∃∃I2,K2. K1 ⊑×[0, e-1] K2 & L2 = K2.ⓑ{I2}V.
110 /2 width=6 by lsuby_inv_pair1_aux/ qed-.
113 fact lsuby_inv_succ1_aux: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 →
114 ∀J1,K1,W1. L1 = K1.ⓑ{J1}W1 → 0 < d →
116 ∃∃J2,K2,W2. K1 ⊑×[d-1, e] K2 & L2 = K2.ⓑ{J2}W2.
117 #L1 #L2 #d #e * -L1 -L2 -d -e /2 width=1 by or_introl/
118 [ #I1 #I2 #L1 #L2 #V1 #V2 #_ #J1 #K1 #W1 #_ #H
119 elim (lt_zero_false … H)
120 | #I1 #I2 #L1 #L2 #V #e #_ #J1 #K1 #W1 #_ #H
121 elim (lt_zero_false … H)
122 | #I1 #I2 #L1 #L2 #V1 #V2 #d #e #HL12 #J1 #K1 #W1 #H #_ destruct
123 /3 width=5 by ex2_3_intro, or_intror/
127 lemma lsuby_inv_succ1: ∀I1,K1,L2,V1,d,e. K1.ⓑ{I1}V1 ⊑×[d, e] L2 → 0 < d →
129 ∃∃I2,K2,V2. K1 ⊑×[d - 1, e] K2 & L2 = K2.ⓑ{I2}V2.
130 /2 width=5 by lsuby_inv_succ1_aux/ qed-.
132 fact lsuby_inv_zero2_aux: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 →
133 ∀J2,K2,W2. L2 = K2.ⓑ{J2}W2 → d = 0 → e = 0 →
134 ∃∃J1,K1,W1. K1 ⊑×[0, 0] K2 & L1 = K1.ⓑ{J1}W1.
135 #L1 #L2 #d #e * -L1 -L2 -d -e
136 [ #L1 #d #e #J2 #K2 #W1 #H destruct
137 | #I1 #I2 #L1 #L2 #V1 #V2 #HL12 #J2 #K2 #W2 #H #_ #_ destruct
138 /2 width=5 by ex2_3_intro/
139 | #I1 #I2 #L1 #L2 #V #e #_ #J2 #K2 #W2 #_ #_
140 <plus_n_Sm #H destruct
141 | #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J2 #K2 #W2 #_
142 <plus_n_Sm #H destruct
146 lemma lsuby_inv_zero2: ∀I2,K2,L1,V2. L1 ⊑×[0, 0] K2.ⓑ{I2}V2 →
147 ∃∃I1,K1,V1. K1 ⊑×[0, 0] K2 & L1 = K1.ⓑ{I1}V1.
148 /2 width=9 by lsuby_inv_zero2_aux/ qed-.
150 fact lsuby_inv_pair2_aux: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 →
151 ∀J2,K2,W. L2 = K2.ⓑ{J2}W → d = 0 → 0 < e →
152 ∃∃J1,K1. K1 ⊑×[0, e-1] K2 & L1 = K1.ⓑ{J1}W.
153 #L1 #L2 #d #e * -L1 -L2 -d -e
154 [ #L1 #d #e #J2 #K2 #W #H destruct
155 | #I1 #I2 #L1 #L2 #V1 #V2 #_ #J2 #K2 #W #_ #_ #H
156 elim (lt_zero_false … H)
157 | #I1 #I2 #L1 #L2 #V #e #HL12 #J2 #K2 #W #H #_ #_ destruct
158 /2 width=4 by ex2_2_intro/
159 | #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J2 #K2 #W #_
160 <plus_n_Sm #H destruct
164 lemma lsuby_inv_pair2: ∀I2,K2,L1,V,e. L1 ⊑×[0, e] K2.ⓑ{I2}V → 0 < e →
165 ∃∃I1,K1. K1 ⊑×[0, e-1] K2 & L1 = K1.ⓑ{I1}V.
166 /2 width=6 by lsuby_inv_pair2_aux/ qed-.
168 fact lsuby_inv_succ2_aux: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 →
169 ∀J2,K2,W2. L2 = K2.ⓑ{J2}W2 → 0 < d →
170 ∃∃J1,K1,W1. K1 ⊑×[d-1, e] K2 & L1 = K1.ⓑ{J1}W1.
171 #L1 #L2 #d #e * -L1 -L2 -d -e
172 [ #L1 #d #e #J2 #K2 #W2 #H destruct
173 | #I1 #I2 #L1 #L2 #V1 #V2 #_ #J2 #K2 #W2 #_ #H
174 elim (lt_zero_false … H)
175 | #I1 #I2 #L1 #L2 #V #e #_ #J2 #K1 #W2 #_ #H
176 elim (lt_zero_false … H)
177 | #I1 #I2 #L1 #L2 #V1 #V2 #d #e #HL12 #J2 #K2 #W2 #H #_ destruct
178 /2 width=5 by ex2_3_intro/
182 lemma lsuby_inv_succ2: ∀I2,K2,L1,V2,d,e. L1 ⊑×[d, e] K2.ⓑ{I2}V2 → 0 < d →
183 ∃∃I1,K1,V1. K1 ⊑×[d-1, e] K2 & L1 = K1.ⓑ{I1}V1.
184 /2 width=5 by lsuby_inv_succ2_aux/ qed-.
186 (* Basic forward lemmas *****************************************************)
188 lemma lsuby_fwd_length: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 → |L2| ≤ |L1|.
189 #L1 #L2 #d #e #H elim H -L1 -L2 -d -e normalize /2 width=1 by le_S_S/
192 lemma lsuby_fwd_ldrop2_be: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 →
193 ∀I2,K2,W,i. ⇩[0, i] L2 ≡ K2.ⓑ{I2}W →
195 ∃∃I1,K1. K1 ⊑×[0, d+e-i-1] K2 & ⇩[0, i] L1 ≡ K1.ⓑ{I1}W.
196 #L1 #L2 #d #e #H elim H -L1 -L2 -d -e
197 [ #L1 #d #e #J2 #K2 #W #i #H
198 elim (ldrop_inv_atom1 … H) -H #H destruct
199 | #I1 #I2 #L1 #L2 #V1 #V2 #_ #_ #J2 #K2 #W #i #_ #_ #H
200 elim (lt_zero_false … H)
201 | #I1 #I2 #L1 #L2 #V #e #HL12 #IHL12 #J2 #K2 #W #i #H #_ #Hie
202 elim (ldrop_inv_O1_pair1 … H) -H * #Hi #HLK1
203 [ -IHL12 -Hie destruct normalize -I2
204 <minus_n_O <minus_plus_m_m /2 width=4 by ldrop_pair, ex2_2_intro/
206 elim (IHL12 … HLK1) -IHL12 -HLK1 // [2: /2 width=1 by lt_plus_to_minus/ ] -Hie normalize
207 >minus_minus_comm >arith_b1 /3 width=4 by ldrop_ldrop_lt, ex2_2_intro/
209 | #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #IHL12 #J2 #K2 #W #i #H #Hdi >plus_plus_comm_23 #Hide
210 elim (le_inv_plus_l … Hdi) #_ #Hi
211 lapply (ldrop_inv_ldrop1_lt … H ?) -H // #HLK1
212 elim (IHL12 … HLK1) -IHL12 -HLK1
213 [2,3: /2 width=1 by lt_plus_to_minus, monotonic_pred/ ] -Hdi -Hide
214 >minus_minus_comm >arith_b1 /3 width=4 by ldrop_ldrop_lt, ex2_2_intro/