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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/relocation/lift_neg.ma".
16 include "basic_2/relocation/ldrop_ldrop.ma".
17 include "basic_2/substitution/llpx_sn.ma".
19 (* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
21 (* alternative definition of llpx_sn (recursive) *)
22 inductive llpx_sn_alt_r (R:relation4 bind2 lenv term term): relation4 ynat term lenv lenv ≝
23 | llpx_sn_alt_r_intro: ∀L1,L2,T,d.
24 (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
25 ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 → I1 = I2 ∧ R I1 K1 V1 V2
27 (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
28 ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 → llpx_sn_alt_r R 0 V1 K1 K2
29 ) → |L1| = |L2| → llpx_sn_alt_r R d T L1 L2
32 (* Compact definition of llpx_sn_alt_r **************************************)
34 lemma llpx_sn_alt_r_intro_alt: ∀R,L1,L2,T,d. |L1| = |L2| →
35 (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
36 ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
37 ∧∧ I1 = I2 & R I1 K1 V1 V2 & llpx_sn_alt_r R 0 V1 K1 K2
38 ) → llpx_sn_alt_r R d T L1 L2.
39 #R #L1 #L2 #T #d #HL12 #IH @llpx_sn_alt_r_intro // -HL12
40 #I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
41 elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by conj/
44 lemma llpx_sn_alt_r_ind_alt: ∀R. ∀S:relation4 ynat term lenv lenv.
45 (∀L1,L2,T,d. |L1| = |L2| → (
46 ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
47 ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
48 ∧∧ I1 = I2 & R I1 K1 V1 V2 & llpx_sn_alt_r R 0 V1 K1 K2 & S 0 V1 K1 K2
50 ∀L1,L2,T,d. llpx_sn_alt_r R d T L1 L2 → S d T L1 L2.
51 #R #S #IH #L1 #L2 #T #d #H elim H -L1 -L2 -T -d
52 #L1 #L2 #T #d #H1 #H2 #HL12 #IH2 @IH -IH // -HL12
53 #I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
54 elim (H1 … HnT HLK1 HLK2) -H1 /4 width=8 by and4_intro/
57 lemma llpx_sn_alt_r_inv_alt: ∀R,L1,L2,T,d. llpx_sn_alt_r R d T L1 L2 →
59 ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
60 ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
61 ∧∧ I1 = I2 & R I1 K1 V1 V2 & llpx_sn_alt_r R 0 V1 K1 K2.
62 #R #L1 #L2 #T #d #H @(llpx_sn_alt_r_ind_alt … H) -L1 -L2 -T -d
63 #L1 #L2 #T #d #HL12 #IH @conj // -HL12
64 #I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
65 elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by and3_intro/
68 (* Basic inversion lemmas ***************************************************)
70 lemma llpx_sn_alt_r_inv_flat: ∀R,I,L1,L2,V,T,d. llpx_sn_alt_r R d (ⓕ{I}V.T) L1 L2 →
71 llpx_sn_alt_r R d V L1 L2 ∧ llpx_sn_alt_r R d T L1 L2.
72 #R #I #L1 #L2 #V #T #d #H elim (llpx_sn_alt_r_inv_alt … H) -H
73 #HL12 #IH @conj @llpx_sn_alt_r_intro_alt // -HL12
74 #I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #H #HLK1 #HLK2
75 elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 //
76 /3 width=8 by nlift_flat_sn, nlift_flat_dx, and3_intro/
79 lemma llpx_sn_alt_r_inv_bind: ∀R,a,I,L1,L2,V,T,d. llpx_sn_alt_r R d (ⓑ{a,I}V.T) L1 L2 →
80 llpx_sn_alt_r R d V L1 L2 ∧ llpx_sn_alt_r R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
81 #R #a #I #L1 #L2 #V #T #d #H elim (llpx_sn_alt_r_inv_alt … H) -H
82 #HL12 #IH @conj @llpx_sn_alt_r_intro_alt [1,3: normalize // ] -HL12
83 #I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #H #HLK1 #HLK2
84 [ elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2
85 /3 width=9 by nlift_bind_sn, and3_intro/
86 | lapply (yle_inv_succ1 … Hdi) -Hdi * #Hdi #Hi
87 lapply (ldrop_inv_drop1_lt … HLK1 ?) -HLK1 /2 width=1 by ylt_O/ #HLK1
88 lapply (ldrop_inv_drop1_lt … HLK2 ?) -HLK2 /2 width=1 by ylt_O/ #HLK2
89 elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 /2 width=1 by and3_intro/
90 @nlift_bind_dx <plus_minus_m_m /2 width=2 by ylt_O/
94 (* Basic forward lemmas *****************************************************)
96 lemma llpx_sn_alt_r_fwd_length: ∀R,L1,L2,T,d. llpx_sn_alt_r R d T L1 L2 → |L1| = |L2|.
97 #R #L1 #L2 #T #d #H elim (llpx_sn_alt_r_inv_alt … H) -H //
100 lemma llpx_sn_alt_r_fwd_lref: ∀R,L1,L2,d,i. llpx_sn_alt_r R d (#i) L1 L2 →
101 ∨∨ |L1| ≤ i ∧ |L2| ≤ i
103 | ∃∃I,K1,K2,V1,V2. ⇩[i] L1 ≡ K1.ⓑ{I}V1 &
104 ⇩[i] L2 ≡ K2.ⓑ{I}V2 &
105 llpx_sn_alt_r R (yinj 0) V1 K1 K2 &
106 R I K1 V1 V2 & d ≤ yinj i.
107 #R #L1 #L2 #d #i #H elim (llpx_sn_alt_r_inv_alt … H) -H
108 #HL12 #IH elim (lt_or_ge i (|L1|)) /3 width=1 by or3_intro0, conj/
109 elim (ylt_split i d) /3 width=1 by or3_intro1/
110 #Hdi #HL1 elim (ldrop_O1_lt (Ⓕ) … HL1)
111 #I1 #K1 #V1 #HLK1 elim (ldrop_O1_lt (Ⓕ) L2 i) //
112 #I2 #K2 #V2 #HLK2 elim (IH … HLK1 HLK2) -IH
113 /3 width=9 by nlift_lref_be_SO, or3_intro2, ex5_5_intro/
116 (* Basic properties *********************************************************)
118 lemma llpx_sn_alt_r_sort: ∀R,L1,L2,d,k. |L1| = |L2| → llpx_sn_alt_r R d (⋆k) L1 L2.
119 #R #L1 #L2 #d #k #HL12 @llpx_sn_alt_r_intro_alt // -HL12
120 #I1 #I2 #K1 #K2 #V1 #V2 #i #_ #H elim (H (⋆k)) //
123 lemma llpx_sn_alt_r_gref: ∀R,L1,L2,d,p. |L1| = |L2| → llpx_sn_alt_r R d (§p) L1 L2.
124 #R #L1 #L2 #d #p #HL12 @llpx_sn_alt_r_intro_alt // -HL12
125 #I1 #I2 #K1 #K2 #V1 #V2 #i #_ #H elim (H (§p)) //
128 lemma llpx_sn_alt_r_skip: ∀R,L1,L2,d,i. |L1| = |L2| → yinj i < d → llpx_sn_alt_r R d (#i) L1 L2.
129 #R #L1 #L2 #d #i #HL12 #Hid @llpx_sn_alt_r_intro_alt // -HL12
130 #I1 #I2 #K1 #K2 #V1 #V2 #j #Hdj #H elim (H (#i)) -H
131 /4 width=3 by lift_lref_lt, ylt_yle_trans, ylt_inv_inj/
134 lemma llpx_sn_alt_r_free: ∀R,L1,L2,d,i. |L1| ≤ i → |L2| ≤ i → |L1| = |L2| →
135 llpx_sn_alt_r R d (#i) L1 L2.
136 #R #L1 #L2 #d #i #HL1 #_ #HL12 @llpx_sn_alt_r_intro_alt // -HL12
137 #I1 #I2 #K1 #K2 #V1 #V2 #j #_ #H #HLK1 elim (H (#(i-1))) -H
138 lapply (ldrop_fwd_length_lt2 … HLK1) -HLK1
139 /3 width=3 by lift_lref_ge_minus, lt_to_le_to_lt/
142 lemma llpx_sn_alt_r_lref: ∀R,I,L1,L2,K1,K2,V1,V2,d,i. d ≤ yinj i →
143 ⇩[i] L1 ≡ K1.ⓑ{I}V1 → ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
144 llpx_sn_alt_r R 0 V1 K1 K2 → R I K1 V1 V2 →
145 llpx_sn_alt_r R d (#i) L1 L2.
146 #R #I #L1 #L2 #K1 #K2 #V1 #V2 #d #i #Hdi #HLK1 #HLK2 #HK12 #HV12 @llpx_sn_alt_r_intro_alt
147 [ lapply (llpx_sn_alt_r_fwd_length … HK12) -HK12 #HK12
148 @(ldrop_fwd_length_eq2 … HLK1 HLK2) normalize //
149 | #Z1 #Z2 #Y1 #Y2 #X1 #X2 #j #Hdj #H #HLY1 #HLY2
150 elim (lt_or_eq_or_gt i j) #Hij destruct
151 [ elim (H (#i)) -H /2 width=1 by lift_lref_lt/
152 | lapply (ldrop_mono … HLY1 … HLK1) -HLY1 -HLK1 #H destruct
153 lapply (ldrop_mono … HLY2 … HLK2) -HLY2 -HLK2 #H destruct /2 width=1 by and3_intro/
154 | elim (H (#(i-1))) -H /2 width=1 by lift_lref_ge_minus/
159 lemma llpx_sn_alt_r_flat: ∀R,I,L1,L2,V,T,d.
160 llpx_sn_alt_r R d V L1 L2 → llpx_sn_alt_r R d T L1 L2 →
161 llpx_sn_alt_r R d (ⓕ{I}V.T) L1 L2.
162 #R #I #L1 #L2 #V #T #d #HV #HT
163 elim (llpx_sn_alt_r_inv_alt … HV) -HV #HL12 #IHV
164 elim (llpx_sn_alt_r_inv_alt … HT) -HT #_ #IHT
165 @llpx_sn_alt_r_intro_alt // -HL12
166 #I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #HnVT #HLK1 #HLK2
167 elim (nlift_inv_flat … HnVT) -HnVT #H
168 [ elim (IHV … HLK1 … HLK2) -IHV /2 width=2 by and3_intro/
169 | elim (IHT … HLK1 … HLK2) -IHT /3 width=2 by and3_intro/
173 lemma llpx_sn_alt_r_bind: ∀R,a,I,L1,L2,V,T,d.
174 llpx_sn_alt_r R d V L1 L2 →
175 llpx_sn_alt_r R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) →
176 llpx_sn_alt_r R d (ⓑ{a,I}V.T) L1 L2.
177 #R #a #I #L1 #L2 #V #T #d #HV #HT
178 elim (llpx_sn_alt_r_inv_alt … HV) -HV #HL12 #IHV
179 elim (llpx_sn_alt_r_inv_alt … HT) -HT #_ #IHT
180 @llpx_sn_alt_r_intro_alt // -HL12
181 #I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #HnVT #HLK1 #HLK2
182 elim (nlift_inv_bind … HnVT) -HnVT #H
183 [ elim (IHV … HLK1 … HLK2) -IHV /2 width=2 by and3_intro/
184 | elim IHT -IHT /2 width=12 by ldrop_drop, yle_succ, and3_intro/
188 (* Main properties **********************************************************)
190 theorem llpx_sn_lpx_sn_alt_r: ∀R,L1,L2,T,d. llpx_sn R d T L1 L2 → llpx_sn_alt_r R d T L1 L2.
191 #R #L1 #L2 #T #d #H elim H -L1 -L2 -T -d
192 /2 width=9 by llpx_sn_alt_r_sort, llpx_sn_alt_r_gref, llpx_sn_alt_r_skip, llpx_sn_alt_r_free, llpx_sn_alt_r_lref, llpx_sn_alt_r_flat, llpx_sn_alt_r_bind/
195 (* Main inversion lemmas ****************************************************)
197 theorem llpx_sn_alt_r_inv_lpx_sn: ∀R,T,L1,L2,d. llpx_sn_alt_r R d T L1 L2 → llpx_sn R d T L1 L2.
198 #R #T #L1 @(f2_ind … rfw … L1 T) -L1 -T #n #IH #L1 * *
199 [1,3: /3 width=4 by llpx_sn_alt_r_fwd_length, llpx_sn_gref, llpx_sn_sort/
200 | #i #Hn #L2 #d #H lapply (llpx_sn_alt_r_fwd_length … H)
201 #HL12 elim (llpx_sn_alt_r_fwd_lref … H) -H
202 [ * /2 width=1 by llpx_sn_free/
203 | /2 width=1 by llpx_sn_skip/
204 | * /4 width=9 by llpx_sn_lref, ldrop_fwd_rfw/
206 | #a #I #V #T #Hn #L2 #d #H elim (llpx_sn_alt_r_inv_bind … H) -H
207 /3 width=1 by llpx_sn_bind/
208 | #I #V #T #Hn #L2 #d #H elim (llpx_sn_alt_r_inv_flat … H) -H
209 /3 width=1 by llpx_sn_flat/
213 (* Alternative definition of llpx_sn (recursive) ****************************)
215 lemma llpx_sn_intro_alt_r: ∀R,L1,L2,T,d. |L1| = |L2| →
216 (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
217 ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
218 ∧∧ I1 = I2 & R I1 K1 V1 V2 & llpx_sn R 0 V1 K1 K2
219 ) → llpx_sn R d T L1 L2.
220 #R #L1 #L2 #T #d #HL12 #IH @llpx_sn_alt_r_inv_lpx_sn
221 @llpx_sn_alt_r_intro_alt // -HL12
222 #I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
223 elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /3 width=1 by llpx_sn_lpx_sn_alt_r, and3_intro/
226 lemma llpx_sn_ind_alt_r: ∀R. ∀S:relation4 ynat term lenv lenv.
227 (∀L1,L2,T,d. |L1| = |L2| → (
228 ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
229 ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
230 ∧∧ I1 = I2 & R I1 K1 V1 V2 & llpx_sn R 0 V1 K1 K2 & S 0 V1 K1 K2
232 ∀L1,L2,T,d. llpx_sn R d T L1 L2 → S d T L1 L2.
233 #R #S #IH1 #L1 #L2 #T #d #H lapply (llpx_sn_lpx_sn_alt_r … H) -H
234 #H @(llpx_sn_alt_r_ind_alt … H) -L1 -L2 -T -d
235 #L1 #L2 #T #d #HL12 #IH2 @IH1 -IH1 // -HL12
236 #I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
237 elim (IH2 … HnT HLK1 HLK2) -IH2 -HnT -HLK1 -HLK2 /3 width=1 by llpx_sn_alt_r_inv_lpx_sn, and4_intro/
240 lemma llpx_sn_inv_alt_r: ∀R,L1,L2,T,d. llpx_sn R d T L1 L2 →
242 ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
243 ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
244 ∧∧ I1 = I2 & R I1 K1 V1 V2 & llpx_sn R 0 V1 K1 K2.
245 #R #L1 #L2 #T #d #H lapply (llpx_sn_lpx_sn_alt_r … H) -H
246 #H elim (llpx_sn_alt_r_inv_alt … H) -H
248 #I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
249 elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /3 width=1 by llpx_sn_alt_r_inv_lpx_sn, and3_intro/