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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 *)
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15 include "basic_2/relocation/lift_neg.ma".
16 include "basic_2/relocation/ldrop_ldrop.ma".
17 include "basic_2/relocation/llpx_sn.ma".
19 (* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
21 (* alternative definition of llpx_sn_alt *)
22 inductive llpx_sn_alt (R:relation3 lenv term term): relation4 ynat term lenv lenv ≝
23 | llpx_sn_alt_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 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 0 V1 K1 K2
29 ) → |L1| = |L2| → llpx_sn_alt R d T L1 L2
32 (* Basic forward lemmas ******************************************************)
34 lemma llpx_sn_alt_fwd_gen: ∀R,L1,L2,T,d. llpx_sn_alt R d T L1 L2 →
36 ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
37 ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
38 ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn_alt R 0 V1 K1 K2.
39 #R #L1 #L2 #T #d * -L1 -L2 -T -d
40 #L1 #L2 #T #d #IH1 #IH2 #HL12 @conj //
41 #I1 #I2 #K1 #K2 #HLK1 #HLK2 #i #Hid #HnT #HLK1 #HLK2
42 elim (IH1 … HnT HLK1 HLK2) -IH1 /4 width=8 by and3_intro/
45 lemma llpx_sn_alt_fwd_length: ∀R,L1,L2,T,d. llpx_sn_alt R d T L1 L2 → |L1| = |L2|.
46 #R #L1 #L2 #T #d * -L1 -L2 -T -d //
49 fact llpx_sn_alt_fwd_lref_aux: ∀R,L1,L2,X,d. llpx_sn_alt R d X L1 L2 → ∀i. X = #i →
50 ∨∨ |L1| ≤ i ∧ |L2| ≤ i
52 | ∃∃I,K1,K2,V1,V2. ⇩[i] L1 ≡ K1.ⓑ{I}V1 &
54 llpx_sn_alt R (yinj 0) V1 K1 K2 &
55 R K1 V1 V2 & d ≤ yinj i.
56 #R #L1 #L2 #X #d * -L1 -L2 -X -d
57 #L1 #L2 #X #d #H1X #H2X #HL12 #i #H destruct
58 elim (lt_or_ge i (|L1|)) /3 width=1 by or3_intro0, conj/
59 elim (ylt_split i d) /3 width=1 by or3_intro1/
60 #Hdi #HL1 elim (ldrop_O1_lt … HL1) #I1 #K1 #V1 #HLK1
61 elim (ldrop_O1_lt L2 i) // #I2 #K2 #V2 #HLK2
62 elim (H1X … HLK1 HLK2) -H1X /2 width=3 by nlift_lref_be_SO/ #H #HV12 destruct
63 lapply (H2X … HLK1 HLK2) -H2X /2 width=3 by nlift_lref_be_SO/
64 /3 width=9 by or3_intro2, ex5_5_intro/
67 lemma llpx_sn_alt_fwd_lref: ∀R,L1,L2,d,i. llpx_sn_alt R d (#i) L1 L2 →
68 ∨∨ |L1| ≤ i ∧ |L2| ≤ i
70 | ∃∃I,K1,K2,V1,V2. ⇩[i] L1 ≡ K1.ⓑ{I}V1 &
72 llpx_sn_alt R (yinj 0) V1 K1 K2 &
73 R K1 V1 V2 & d ≤ yinj i.
74 /2 width=3 by llpx_sn_alt_fwd_lref_aux/ qed-.
76 (* Basic inversion lemmas ****************************************************)
78 fact llpx_sn_alt_inv_flat_aux: ∀R,L1,L2,X,d. llpx_sn_alt R d X L1 L2 →
80 llpx_sn_alt R d V L1 L2 ∧ llpx_sn_alt R d T L1 L2.
81 #R #L1 #L2 #X #d * -L1 -L2 -X -d
82 #L1 #L2 #X #d #H1X #H2X #HL12
84 @conj @llpx_sn_alt_intro // -HL12
85 /4 width=8 by nlift_flat_sn, nlift_flat_dx/
88 lemma llpx_sn_alt_inv_flat: ∀R,I,L1,L2,V,T,d. llpx_sn_alt R d (ⓕ{I}V.T) L1 L2 →
89 llpx_sn_alt R d V L1 L2 ∧ llpx_sn_alt R d T L1 L2.
90 /2 width=4 by llpx_sn_alt_inv_flat_aux/ qed-.
92 fact llpx_sn_alt_inv_bind_aux: ∀R,L1,L2,X,d. llpx_sn_alt R d X L1 L2 →
93 ∀a,I,V,T. X = ⓑ{a,I}V.T →
94 llpx_sn_alt R d V L1 L2 ∧ llpx_sn_alt R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
95 #R #L1 #L2 #X #d * -L1 -L2 -X -d
96 #L1 #L2 #X #d #H1X #H2X #HL12
97 #a #I #V #T #H destruct
98 @conj @llpx_sn_alt_intro [3,6: normalize /2 width=1 by eq_f2/ ] -HL12
99 #I1 #I2 #K1 #K2 #W1 #W2 #i #Hdi #H #HLK1 #HLK2
100 [1,2: /4 width=9 by nlift_bind_sn/ ]
101 lapply (yle_inv_succ1 … Hdi) -Hdi * #Hdi #Hi
102 lapply (ldrop_inv_drop1_lt … HLK1 ?) -HLK1 /2 width=1 by ylt_O/ #HLK1
103 lapply (ldrop_inv_drop1_lt … HLK2 ?) -HLK2 /2 width=1 by ylt_O/ #HLK2
104 [ @(H1X … HLK1 HLK2) | @(H2X … HLK1 HLK2) ] // -I1 -I2 -L1 -L2 -K1 -K2 -W1 -W2
105 @nlift_bind_dx <plus_minus_m_m /2 width=2 by ylt_O/
108 lemma llpx_sn_alt_inv_bind: ∀R,a,I,L1,L2,V,T,d. llpx_sn_alt R d (ⓑ{a,I}V.T) L1 L2 →
109 llpx_sn_alt R d V L1 L2 ∧ llpx_sn_alt R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
110 /2 width=4 by llpx_sn_alt_inv_bind_aux/ qed-.
112 (* Basic properties **********************************************************)
114 lemma llpx_sn_alt_intro_alt: ∀R,L1,L2,T,d. |L1| = |L2| →
115 (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
116 ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
117 ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn_alt R 0 V1 K1 K2
118 ) → llpx_sn_alt R d T L1 L2.
119 #R #L1 #L2 #T #d #HL12 #IH @llpx_sn_alt_intro // -HL12
120 #I1 #I2 #K1 #K2 #HLK1 #HLK2 #i #Hid #HnT #HLK1 #HLK2
121 elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by conj/
124 lemma llpx_sn_alt_sort: ∀R,L1,L2,d,k. |L1| = |L2| → llpx_sn_alt R d (⋆k) L1 L2.
125 #R #L1 #L2 #d #k #HL12 @llpx_sn_alt_intro // -HL12
126 #I1 #I2 #K1 #K2 #V1 #V2 #i #_ #H elim (H (⋆k)) //
129 lemma llpx_sn_alt_gref: ∀R,L1,L2,d,p. |L1| = |L2| → llpx_sn_alt R d (§p) L1 L2.
130 #R #L1 #L2 #d #p #HL12 @llpx_sn_alt_intro // -HL12
131 #I1 #I2 #K1 #K2 #V1 #V2 #i #_ #H elim (H (§p)) //
134 lemma llpx_sn_alt_skip: ∀R,L1,L2,d,i. |L1| = |L2| → yinj i < d → llpx_sn_alt R d (#i) L1 L2.
135 #R #L1 #L2 #d #i #HL12 #Hid @llpx_sn_alt_intro // -HL12
136 #I1 #I2 #K1 #K2 #V1 #V2 #j #Hdj #H elim (H (#i)) -H
137 /4 width=3 by lift_lref_lt, ylt_yle_trans, ylt_inv_inj/
140 lemma llpx_sn_alt_free: ∀R,L1,L2,d,i. |L1| ≤ i → |L2| ≤ i → |L1| = |L2| →
141 llpx_sn_alt R d (#i) L1 L2.
142 #R #L1 #L2 #d #i #HL1 #_ #HL12 @llpx_sn_alt_intro // -HL12
143 #I1 #I2 #K1 #K2 #V1 #V2 #j #_ #H #HLK1 elim (H (#(i-1))) -H
144 lapply (ldrop_fwd_length_lt2 … HLK1) -HLK1
145 /3 width=3 by lift_lref_ge_minus, lt_to_le_to_lt/
148 lemma llpx_sn_alt_lref: ∀R,I,L1,L2,K1,K2,V1,V2,d,i. d ≤ yinj i →
149 ⇩[i] L1 ≡ K1.ⓑ{I}V1 → ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
150 llpx_sn_alt R 0 V1 K1 K2 → R K1 V1 V2 →
151 llpx_sn_alt R d (#i) L1 L2.
152 #R #I #L1 #L2 #K1 #K2 #V1 #V2 #d #i #Hdi #HLK1 #HLK2 #HK12 #HV12 @llpx_sn_alt_intro
153 [1,2: #Z1 #Z2 #Y1 #Y2 #X1 #X2 #j #Hdj #H #HLY1 #HLY2
154 elim (lt_or_eq_or_gt i j) #Hij destruct
155 [1,4: elim (H (#i)) -H /2 width=1 by lift_lref_lt/
156 |2,5: lapply (ldrop_mono … HLY1 … HLK1) -HLY1 -HLK1 #H destruct
157 lapply (ldrop_mono … HLY2 … HLK2) -HLY2 -HLK2 #H destruct /2 width=1 by conj/
158 |3,6: elim (H (#(i-1))) -H /2 width=1 by lift_lref_ge_minus/
160 | lapply (llpx_sn_alt_fwd_length … HK12) -HK12 #HK12
161 @(ldrop_fwd_length_eq2 … HLK1 HLK2) normalize /2 width=1 by eq_f2/
165 fact llpx_sn_alt_flat_aux: ∀R,I,L1,L2,V,d. llpx_sn_alt R d V L1 L2 →
166 ∀Y1,Y2,T,m. llpx_sn_alt R m T Y1 Y2 →
167 Y1 = L1 → Y2 = L2 → m = d →
168 llpx_sn_alt R d (ⓕ{I}V.T) L1 L2.
169 #R #I #L1 #L2 #V #d * -L1 -L2 -V -d #L1 #L2 #V #d #H1V #H2V #HL12
170 #Y1 #Y2 #T #m * -Y1 -Y2 -T -m #Y1 #Y2 #T #m #H1T #H2T #_
171 #HT1 #HY2 #Hm destruct
172 @llpx_sn_alt_intro // -HL12
173 #J1 #J2 #K1 #K2 #W1 #W2 #i #Hdi #HnVT #HLK1 #HLK2
174 elim (nlift_inv_flat … HnVT) -HnVT /3 width=8 by/
177 lemma llpx_sn_alt_flat: ∀R,I,L1,L2,V,T,d.
178 llpx_sn_alt R d V L1 L2 → llpx_sn_alt R d T L1 L2 →
179 llpx_sn_alt R d (ⓕ{I}V.T) L1 L2.
180 /2 width=7 by llpx_sn_alt_flat_aux/ qed.
182 fact llpx_sn_alt_bind_aux: ∀R,a,I,L1,L2,V,d. llpx_sn_alt R d V L1 L2 →
183 ∀Y1,Y2,T,m. llpx_sn_alt R m T Y1 Y2 →
184 Y1 = L1.ⓑ{I}V → Y2 = L2.ⓑ{I}V → m = ⫯d →
185 llpx_sn_alt R d (ⓑ{a,I}V.T) L1 L2.
186 #R #a #I #L1 #L2 #V #d * -L1 -L2 -V -d #L1 #L2 #V #d #H1V #H2V #HL12
187 #Y1 #Y2 #T #m * -Y1 -Y2 -T -m #Y1 #Y2 #T #m #H1T #H2T #_
188 #HT1 #HY2 #Hm destruct
189 @llpx_sn_alt_intro // -HL12
190 #J1 #J2 #K1 #K2 #W1 #W2 #i #Hdi #HnVT #HLK1 #HLK2
191 elim (nlift_inv_bind … HnVT) -HnVT /3 width=8 by ldrop_drop, yle_succ/
194 lemma llpx_sn_alt_bind: ∀R,a,I,L1,L2,V,T,d.
195 llpx_sn_alt R d V L1 L2 →
196 llpx_sn_alt R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) →
197 llpx_sn_alt R d (ⓑ{a,I}V.T) L1 L2.
198 /2 width=7 by llpx_sn_alt_bind_aux/ qed.
200 (* Main properties **********************************************************)
202 theorem llpx_sn_lpx_sn_alt: ∀R,L1,L2,T,d. llpx_sn R d T L1 L2 → llpx_sn_alt R d T L1 L2.
203 #R #L1 #L2 #T #d #H elim H -L1 -L2 -T -d
204 /2 width=9 by llpx_sn_alt_sort, llpx_sn_alt_gref, llpx_sn_alt_skip, llpx_sn_alt_free, llpx_sn_alt_lref, llpx_sn_alt_flat, llpx_sn_alt_bind/
207 (* Main inversion lemmas ****************************************************)
209 theorem llpx_sn_alt_inv_lpx_sn: ∀R,T,L1,L2,d. llpx_sn_alt R d T L1 L2 → llpx_sn R d T L1 L2.
210 #R #T #L1 @(f2_ind … rfw … L1 T) -L1 -T #n #IH #L1 * *
211 [1,3: /3 width=4 by llpx_sn_alt_fwd_length, llpx_sn_gref, llpx_sn_sort/
212 | #i #Hn #L2 #d #H lapply (llpx_sn_alt_fwd_length … H)
213 #HL12 elim (llpx_sn_alt_fwd_lref … H) -H
214 [ * /2 width=1 by llpx_sn_free/
215 | /2 width=1 by llpx_sn_skip/
216 | * /4 width=9 by llpx_sn_lref, ldrop_fwd_rfw/
218 | #a #I #V #T #Hn #L2 #d #H elim (llpx_sn_alt_inv_bind … H) -H
219 /3 width=1 by llpx_sn_bind/
220 | #I #V #T #Hn #L2 #d #H elim (llpx_sn_alt_inv_flat … H) -H
221 /3 width=1 by llpx_sn_flat/
225 (* Advanced properties ******************************************************)
227 lemma llpx_sn_intro_alt: ∀R,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 K1 V1 V2 & llpx_sn R 0 V1 K1 K2
231 ) → llpx_sn R d T L1 L2.
232 #R #L1 #L2 #T #d #HL12 #IH @llpx_sn_alt_inv_lpx_sn
233 @llpx_sn_alt_intro_alt // -HL12
234 #I1 #I2 #K1 #K2 #HLK1 #HLK2 #i #Hid #HnT #HLK1 #HLK2
235 elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /3 width=1 by llpx_sn_lpx_sn_alt, and3_intro/
238 (* Advanced forward lemmas lemmas *******************************************)
240 lemma llpx_sn_fwd_alt: ∀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 K1 V1 V2 & llpx_sn R 0 V1 K1 K2.
245 #R #L1 #L2 #T #d #H lapply (llpx_sn_lpx_sn_alt … H) -H
246 #H elim (llpx_sn_alt_fwd_gen … H) -H
248 #I1 #I2 #K1 #K2 #HLK1 #HLK2 #i #Hid #HnT #HLK1 #HLK2
249 elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /3 width=1 by llpx_sn_alt_inv_lpx_sn, and3_intro/