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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/substitution/lift_neg.ma".
16 include "basic_2/substitution/drop_drop.ma".
17 include "basic_2/multiple/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:relation3 lenv term term): relation4 ynat term lenv lenv ≝
23 | llpx_sn_alt_r_intro: ∀L1,L2,T,l.
24 (∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) →
25 ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → I1 = I2 ∧ R K1 V1 V2
27 (∀I1,I2,K1,K2,V1,V2,i. l ≤ 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 l T L1 L2
32 (* Compact definition of llpx_sn_alt_r **************************************)
34 lemma llpx_sn_alt_r_intro_alt: ∀R,L1,L2,T,l. |L1| = |L2| →
35 (∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) →
36 ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 →
37 ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn_alt_r R 0 V1 K1 K2
38 ) → llpx_sn_alt_r R l T L1 L2.
39 #R #L1 #L2 #T #l #HL12 #IH @llpx_sn_alt_r_intro // -HL12
40 #I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #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,l. |L1| = |L2| → (
46 ∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) →
47 ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 →
48 ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn_alt_r R 0 V1 K1 K2 & S 0 V1 K1 K2
50 ∀L1,L2,T,l. llpx_sn_alt_r R l T L1 L2 → S l T L1 L2.
51 #R #S #IH #L1 #L2 #T #l #H elim H -L1 -L2 -T -l
52 #L1 #L2 #T #l #H1 #H2 #HL12 #IH2 @IH -IH // -HL12
53 #I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #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,l. llpx_sn_alt_r R l T L1 L2 →
59 ∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) →
60 ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 →
61 ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn_alt_r R 0 V1 K1 K2.
62 #R #L1 #L2 #T #l #H @(llpx_sn_alt_r_ind_alt … H) -L1 -L2 -T -l
63 #L1 #L2 #T #l #HL12 #IH @conj // -HL12
64 #I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #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,l. llpx_sn_alt_r R l (ⓕ{I}V.T) L1 L2 →
71 llpx_sn_alt_r R l V L1 L2 ∧ llpx_sn_alt_r R l T L1 L2.
72 #R #I #L1 #L2 #V #T #l #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 #Hli #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,l. llpx_sn_alt_r R l (ⓑ{a,I}V.T) L1 L2 →
80 llpx_sn_alt_r R l V L1 L2 ∧ llpx_sn_alt_r R (⫯l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
81 #R #a #I #L1 #L2 #V #T #l #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 #Hli #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 … Hli) -Hli * #Hli #Hi <yminus_SO2 in Hli; #Hli
87 lapply (drop_inv_drop1_lt … HLK1 ?) -HLK1 /2 width=1 by ylt_O/ #HLK1
88 lapply (drop_inv_drop1_lt … HLK2 ?) -HLK2 /2 width=1 by ylt_O/ #HLK2
89 elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 /3 width=9 by nlift_bind_dx, and3_intro/
93 (* Basic forward lemmas *****************************************************)
95 lemma llpx_sn_alt_r_fwd_length: ∀R,L1,L2,T,l. llpx_sn_alt_r R l T L1 L2 → |L1| = |L2|.
96 #R #L1 #L2 #T #l #H elim (llpx_sn_alt_r_inv_alt … H) -H //
99 lemma llpx_sn_alt_r_fwd_lref: ∀R,L1,L2,l,i. llpx_sn_alt_r R l (#i) L1 L2 →
100 ∨∨ |L1| ≤ i ∧ |L2| ≤ i
102 | ∃∃I,K1,K2,V1,V2. ⬇[i] L1 ≡ K1.ⓑ{I}V1 &
103 ⬇[i] L2 ≡ K2.ⓑ{I}V2 &
104 llpx_sn_alt_r R (yinj 0) V1 K1 K2 &
105 R K1 V1 V2 & l ≤ yinj i.
106 #R #L1 #L2 #l #i #H elim (llpx_sn_alt_r_inv_alt … H) -H
107 #HL12 #IH elim (lt_or_ge i (|L1|)) /3 width=1 by or3_intro0, conj/
108 elim (ylt_split i l) /3 width=1 by or3_intro1/
109 #Hli #HL1 elim (drop_O1_lt (Ⓕ) … HL1)
110 #I1 #K1 #V1 #HLK1 elim (drop_O1_lt (Ⓕ) L2 i) //
111 #I2 #K2 #V2 #HLK2 elim (IH … HLK1 HLK2) -IH
112 /3 width=9 by nlift_lref_be_SO, or3_intro2, ex5_5_intro/
115 (* Basic properties *********************************************************)
117 lemma llpx_sn_alt_r_sort: ∀R,L1,L2,l,s. |L1| = |L2| → llpx_sn_alt_r R l (⋆s) L1 L2.
118 #R #L1 #L2 #l #s #HL12 @llpx_sn_alt_r_intro_alt // -HL12
119 #I1 #I2 #K1 #K2 #V1 #V2 #i #_ #H elim (H (⋆s)) //
122 lemma llpx_sn_alt_r_gref: ∀R,L1,L2,l,p. |L1| = |L2| → llpx_sn_alt_r R l (§p) L1 L2.
123 #R #L1 #L2 #l #p #HL12 @llpx_sn_alt_r_intro_alt // -HL12
124 #I1 #I2 #K1 #K2 #V1 #V2 #i #_ #H elim (H (§p)) //
127 lemma llpx_sn_alt_r_skip: ∀R,L1,L2,l,i. |L1| = |L2| → yinj i < l → llpx_sn_alt_r R l (#i) L1 L2.
128 #R #L1 #L2 #l #i #HL12 #Hil @llpx_sn_alt_r_intro_alt // -HL12
129 #I1 #I2 #K1 #K2 #V1 #V2 #j #Hlj #H elim (H (#i)) -H
130 /4 width=3 by lift_lref_lt, ylt_yle_trans, ylt_inv_inj/
133 lemma llpx_sn_alt_r_free: ∀R,L1,L2,l,i. |L1| ≤ i → |L2| ≤ i → |L1| = |L2| →
134 llpx_sn_alt_r R l (#i) L1 L2.
135 #R #L1 #L2 #l #i #HL1 #_ #HL12 @llpx_sn_alt_r_intro_alt // -HL12
136 #I1 #I2 #K1 #K2 #V1 #V2 #j #_ #H #HLK1 elim (H (#(i-1))) -H
137 lapply (drop_fwd_length_lt2 … HLK1) -HLK1
138 /4 width=3 by lift_lref_ge_minus, yle_inj, transitive_le/
141 lemma llpx_sn_alt_r_lref: ∀R,I,L1,L2,K1,K2,V1,V2,l,i. l ≤ yinj i →
142 ⬇[i] L1 ≡ K1.ⓑ{I}V1 → ⬇[i] L2 ≡ K2.ⓑ{I}V2 →
143 llpx_sn_alt_r R 0 V1 K1 K2 → R K1 V1 V2 →
144 llpx_sn_alt_r R l (#i) L1 L2.
145 #R #I #L1 #L2 #K1 #K2 #V1 #V2 #l #i #Hli #HLK1 #HLK2 #HK12 #HV12 @llpx_sn_alt_r_intro_alt
146 [ lapply (llpx_sn_alt_r_fwd_length … HK12) -HK12 #HK12
147 @(drop_fwd_length_eq2 … HLK1 HLK2) normalize //
148 | #Z1 #Z2 #Y1 #Y2 #X1 #X2 #j #Hlj #H #HLY1 #HLY2
149 elim (lt_or_eq_or_gt i j) #Hij destruct
150 [ elim (H (#i)) -H /3 width=1 by lift_lref_lt, ylt_inj/
151 | lapply (drop_mono … HLY1 … HLK1) -HLY1 -HLK1 #H destruct
152 lapply (drop_mono … HLY2 … HLK2) -HLY2 -HLK2 #H destruct /2 width=1 by and3_intro/
153 | elim (H (#(i-1))) -H /3 width=1 by lift_lref_ge_minus, yle_inj/
158 lemma llpx_sn_alt_r_flat: ∀R,I,L1,L2,V,T,l.
159 llpx_sn_alt_r R l V L1 L2 → llpx_sn_alt_r R l T L1 L2 →
160 llpx_sn_alt_r R l (ⓕ{I}V.T) L1 L2.
161 #R #I #L1 #L2 #V #T #l #HV #HT
162 elim (llpx_sn_alt_r_inv_alt … HV) -HV #HL12 #IHV
163 elim (llpx_sn_alt_r_inv_alt … HT) -HT #_ #IHT
164 @llpx_sn_alt_r_intro_alt // -HL12
165 #I1 #I2 #K1 #K2 #V1 #V2 #i #Hli #HnVT #HLK1 #HLK2
166 elim (nlift_inv_flat … HnVT) -HnVT #H
167 [ elim (IHV … HLK1 … HLK2) -IHV /2 width=2 by and3_intro/
168 | elim (IHT … HLK1 … HLK2) -IHT /3 width=2 by and3_intro/
172 lemma llpx_sn_alt_r_bind: ∀R,a,I,L1,L2,V,T,l.
173 llpx_sn_alt_r R l V L1 L2 →
174 llpx_sn_alt_r R (⫯l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) →
175 llpx_sn_alt_r R l (ⓑ{a,I}V.T) L1 L2.
176 #R #a #I #L1 #L2 #V #T #l #HV #HT
177 elim (llpx_sn_alt_r_inv_alt … HV) -HV #HL12 #IHV
178 elim (llpx_sn_alt_r_inv_alt … HT) -HT #_ #IHT
179 @llpx_sn_alt_r_intro_alt // -HL12
180 #I1 #I2 #K1 #K2 #V1 #V2 #i #Hli #HnVT #HLK1 #HLK2
181 elim (nlift_inv_bind … HnVT) -HnVT #H
182 [ elim (IHV … HLK1 … HLK2) -IHV /2 width=2 by and3_intro/
183 | elim IHT -IHT /2 width=12 by drop_drop, yle_succ, and3_intro/
187 (* Main properties **********************************************************)
189 theorem llpx_sn_lpx_sn_alt_r: ∀R,L1,L2,T,l. llpx_sn R l T L1 L2 → llpx_sn_alt_r R l T L1 L2.
190 #R #L1 #L2 #T #l #H elim H -L1 -L2 -T -l
191 /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/
194 (* Main inversion lemmas ****************************************************)
196 theorem llpx_sn_alt_r_inv_lpx_sn: ∀R,T,L1,L2,l. llpx_sn_alt_r R l T L1 L2 → llpx_sn R l T L1 L2.
197 #R #T #L1 @(f2_ind … rfw … L1 T) -L1 -T #x #IH #L1 * *
198 [1,3: /3 width=4 by llpx_sn_alt_r_fwd_length, llpx_sn_gref, llpx_sn_sort/
199 | #i #Hx #L2 #l #H lapply (llpx_sn_alt_r_fwd_length … H)
200 #HL12 elim (llpx_sn_alt_r_fwd_lref … H) -H
201 [ * /2 width=1 by llpx_sn_free/
202 | /2 width=1 by llpx_sn_skip/
203 | * /4 width=9 by llpx_sn_lref, drop_fwd_rfw/
205 | #a #I #V #T #Hx #L2 #l #H elim (llpx_sn_alt_r_inv_bind … H) -H
206 /3 width=1 by llpx_sn_bind/
207 | #I #V #T #Hx #L2 #l #H elim (llpx_sn_alt_r_inv_flat … H) -H
208 /3 width=1 by llpx_sn_flat/
212 (* Alternative definition of llpx_sn (recursive) ****************************)
214 lemma llpx_sn_intro_alt_r: ∀R,L1,L2,T,l. |L1| = |L2| →
215 (∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) →
216 ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 →
217 ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn R 0 V1 K1 K2
218 ) → llpx_sn R l T L1 L2.
219 #R #L1 #L2 #T #l #HL12 #IH @llpx_sn_alt_r_inv_lpx_sn
220 @llpx_sn_alt_r_intro_alt // -HL12
221 #I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2
222 elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /3 width=1 by llpx_sn_lpx_sn_alt_r, and3_intro/
225 lemma llpx_sn_ind_alt_r: ∀R. ∀S:relation4 ynat term lenv lenv.
226 (∀L1,L2,T,l. |L1| = |L2| → (
227 ∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) →
228 ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 →
229 ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn R 0 V1 K1 K2 & S 0 V1 K1 K2
231 ∀L1,L2,T,l. llpx_sn R l T L1 L2 → S l T L1 L2.
232 #R #S #IH1 #L1 #L2 #T #l #H lapply (llpx_sn_lpx_sn_alt_r … H) -H
233 #H @(llpx_sn_alt_r_ind_alt … H) -L1 -L2 -T -l
234 #L1 #L2 #T #l #HL12 #IH2 @IH1 -IH1 // -HL12
235 #I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2
236 elim (IH2 … HnT HLK1 HLK2) -IH2 -HnT -HLK1 -HLK2 /3 width=1 by llpx_sn_alt_r_inv_lpx_sn, and4_intro/
239 lemma llpx_sn_inv_alt_r: ∀R,L1,L2,T,l. llpx_sn R l T L1 L2 →
241 ∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) →
242 ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 →
243 ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn R 0 V1 K1 K2.
244 #R #L1 #L2 #T #l #H lapply (llpx_sn_lpx_sn_alt_r … H) -H
245 #H elim (llpx_sn_alt_r_inv_alt … H) -H
247 #I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2
248 elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /3 width=1 by llpx_sn_alt_r_inv_lpx_sn, and3_intro/