1 (**************************************************************************)
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 *)
13 (**************************************************************************)
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 inversion lemmas ****************************************************)
34 lemma llpx_sn_alt_inv_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_inv_flat: ∀R,I,L1,L2,V,T,d. llpx_sn_alt R d (ⓕ{I}V.T) L1 L2 →
46 llpx_sn_alt R d V L1 L2 ∧ llpx_sn_alt R d T L1 L2.
47 #R #I #L1 #L2 #V #T #d #H elim (llpx_sn_alt_inv_gen … H) -H
48 #HL12 #IH @conj @llpx_sn_alt_intro // -HL12
49 #I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #H #HLK1 #HLK2
50 elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2
51 /3 width=8 by nlift_flat_sn, nlift_flat_dx, conj/
54 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 →
55 llpx_sn_alt R d V L1 L2 ∧ llpx_sn_alt R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
56 #R #a #I #L1 #L2 #V #T #d #H elim (llpx_sn_alt_inv_gen … H) -H
57 #HL12 #IH @conj @llpx_sn_alt_intro [3,6: normalize // ] -HL12
58 #I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #H #HLK1 #HLK2
59 [1,2: elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2
60 /3 width=9 by nlift_bind_sn, conj/
61 |3,4: lapply (yle_inv_succ1 … Hdi) -Hdi * #Hdi #Hi
62 lapply (ldrop_inv_drop1_lt … HLK1 ?) -HLK1 /2 width=1 by ylt_O/ #HLK1
63 lapply (ldrop_inv_drop1_lt … HLK2 ?) -HLK2 /2 width=1 by ylt_O/ #HLK2
64 elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2
65 [1,3,4,6: /2 width=1 by conj/ ]
66 @nlift_bind_dx <plus_minus_m_m /2 width=2 by ylt_O/
70 (* Basic forward lemmas ******************************************************)
72 lemma llpx_sn_alt_fwd_length: ∀R,L1,L2,T,d. llpx_sn_alt R d T L1 L2 → |L1| = |L2|.
73 #R #L1 #L2 #T #d * -L1 -L2 -T -d //
76 lemma llpx_sn_alt_fwd_lref: ∀R,L1,L2,d,i. llpx_sn_alt R d (#i) L1 L2 →
77 ∨∨ |L1| ≤ i ∧ |L2| ≤ i
79 | ∃∃I,K1,K2,V1,V2. ⇩[i] L1 ≡ K1.ⓑ{I}V1 &
81 llpx_sn_alt R (yinj 0) V1 K1 K2 &
82 R K1 V1 V2 & d ≤ yinj i.
83 #R #L1 #L2 #d #i #H elim (llpx_sn_alt_inv_gen … H) -H
84 #HL12 #IH elim (lt_or_ge i (|L1|)) /3 width=1 by or3_intro0, conj/
85 elim (ylt_split i d) /3 width=1 by or3_intro1/
86 #Hdi #HL1 elim (ldrop_O1_lt … HL1)
87 #I1 #K1 #V1 #HLK1 elim (ldrop_O1_lt L2 i) //
88 #I2 #K2 #V2 #HLK2 elim (IH … HLK1 HLK2) -IH
89 /3 width=9 by nlift_lref_be_SO, or3_intro2, ex5_5_intro/
92 (* Basic properties **********************************************************)
94 lemma llpx_sn_alt_intro_alt: ∀R,L1,L2,T,d. |L1| = |L2| →
95 (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
96 ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
97 ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn_alt R 0 V1 K1 K2
98 ) → llpx_sn_alt R d T L1 L2.
99 #R #L1 #L2 #T #d #HL12 #IH @llpx_sn_alt_intro // -HL12
100 #I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
101 elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by conj/
104 lemma llpx_sn_alt_sort: ∀R,L1,L2,d,k. |L1| = |L2| → llpx_sn_alt R d (⋆k) L1 L2.
105 #R #L1 #L2 #d #k #HL12 @llpx_sn_alt_intro_alt // -HL12
106 #I1 #I2 #K1 #K2 #V1 #V2 #i #_ #H elim (H (⋆k)) //
109 lemma llpx_sn_alt_gref: ∀R,L1,L2,d,p. |L1| = |L2| → llpx_sn_alt R d (§p) L1 L2.
110 #R #L1 #L2 #d #p #HL12 @llpx_sn_alt_intro_alt // -HL12
111 #I1 #I2 #K1 #K2 #V1 #V2 #i #_ #H elim (H (§p)) //
114 lemma llpx_sn_alt_skip: ∀R,L1,L2,d,i. |L1| = |L2| → yinj i < d → llpx_sn_alt R d (#i) L1 L2.
115 #R #L1 #L2 #d #i #HL12 #Hid @llpx_sn_alt_intro_alt // -HL12
116 #I1 #I2 #K1 #K2 #V1 #V2 #j #Hdj #H elim (H (#i)) -H
117 /4 width=3 by lift_lref_lt, ylt_yle_trans, ylt_inv_inj/
120 lemma llpx_sn_alt_free: ∀R,L1,L2,d,i. |L1| ≤ i → |L2| ≤ i → |L1| = |L2| →
121 llpx_sn_alt R d (#i) L1 L2.
122 #R #L1 #L2 #d #i #HL1 #_ #HL12 @llpx_sn_alt_intro_alt // -HL12
123 #I1 #I2 #K1 #K2 #V1 #V2 #j #_ #H #HLK1 elim (H (#(i-1))) -H
124 lapply (ldrop_fwd_length_lt2 … HLK1) -HLK1
125 /3 width=3 by lift_lref_ge_minus, lt_to_le_to_lt/
128 lemma llpx_sn_alt_lref: ∀R,I,L1,L2,K1,K2,V1,V2,d,i. d ≤ yinj i →
129 ⇩[i] L1 ≡ K1.ⓑ{I}V1 → ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
130 llpx_sn_alt R 0 V1 K1 K2 → R K1 V1 V2 →
131 llpx_sn_alt R d (#i) L1 L2.
132 #R #I #L1 #L2 #K1 #K2 #V1 #V2 #d #i #Hdi #HLK1 #HLK2 #HK12 #HV12 @llpx_sn_alt_intro_alt
133 [ lapply (llpx_sn_alt_fwd_length … HK12) -HK12 #HK12
134 @(ldrop_fwd_length_eq2 … HLK1 HLK2) normalize //
135 | #Z1 #Z2 #Y1 #Y2 #X1 #X2 #j #Hdj #H #HLY1 #HLY2
136 elim (lt_or_eq_or_gt i j) #Hij destruct
137 [ elim (H (#i)) -H /2 width=1 by lift_lref_lt/
138 | lapply (ldrop_mono … HLY1 … HLK1) -HLY1 -HLK1 #H destruct
139 lapply (ldrop_mono … HLY2 … HLK2) -HLY2 -HLK2 #H destruct /2 width=1 by and3_intro/
140 | elim (H (#(i-1))) -H /2 width=1 by lift_lref_ge_minus/
145 lemma llpx_sn_alt_flat: ∀R,I,L1,L2,V,T,d.
146 llpx_sn_alt R d V L1 L2 → llpx_sn_alt R d T L1 L2 →
147 llpx_sn_alt R d (ⓕ{I}V.T) L1 L2.
148 #R #I #L1 #L2 #V #T #d #HV #HT
149 elim (llpx_sn_alt_inv_gen … HV) -HV #HL12 #IHV
150 elim (llpx_sn_alt_inv_gen … HT) -HT #_ #IHT
151 @llpx_sn_alt_intro_alt // -HL12
152 #I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #HnVT #HLK1 #HLK2
153 elim (nlift_inv_flat … HnVT) -HnVT #H
154 [ elim (IHV … HLK1 … HLK2) -IHV /2 width=2 by and3_intro/
155 | elim (IHT … HLK1 … HLK2) -IHT /3 width=2 by and3_intro/
159 lemma llpx_sn_alt_bind: ∀R,a,I,L1,L2,V,T,d.
160 llpx_sn_alt R d V L1 L2 →
161 llpx_sn_alt R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) →
162 llpx_sn_alt R d (ⓑ{a,I}V.T) L1 L2.
163 #R #a #I #L1 #L2 #V #T #d #HV #HT
164 elim (llpx_sn_alt_inv_gen … HV) -HV #HL12 #IHV
165 elim (llpx_sn_alt_inv_gen … HT) -HT #_ #IHT
166 @llpx_sn_alt_intro_alt // -HL12
167 #I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #HnVT #HLK1 #HLK2
168 elim (nlift_inv_bind … HnVT) -HnVT #H
169 [ elim (IHV … HLK1 … HLK2) -IHV /2 width=2 by and3_intro/
170 | elim IHT -IHT /2 width=12 by ldrop_drop, yle_succ, and3_intro/
174 (* Main properties **********************************************************)
176 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.
177 #R #L1 #L2 #T #d #H elim H -L1 -L2 -T -d
178 /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/
181 (* Main inversion lemmas ****************************************************)
183 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.
184 #R #T #L1 @(f2_ind … rfw … L1 T) -L1 -T #n #IH #L1 * *
185 [1,3: /3 width=4 by llpx_sn_alt_fwd_length, llpx_sn_gref, llpx_sn_sort/
186 | #i #Hn #L2 #d #H lapply (llpx_sn_alt_fwd_length … H)
187 #HL12 elim (llpx_sn_alt_fwd_lref … H) -H
188 [ * /2 width=1 by llpx_sn_free/
189 | /2 width=1 by llpx_sn_skip/
190 | * /4 width=9 by llpx_sn_lref, ldrop_fwd_rfw/
192 | #a #I #V #T #Hn #L2 #d #H elim (llpx_sn_alt_inv_bind … H) -H
193 /3 width=1 by llpx_sn_bind/
194 | #I #V #T #Hn #L2 #d #H elim (llpx_sn_alt_inv_flat … H) -H
195 /3 width=1 by llpx_sn_flat/
199 (* Advanced properties ******************************************************)
201 lemma llpx_sn_intro_alt: ∀R,L1,L2,T,d. |L1| = |L2| →
202 (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
203 ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
204 ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn R 0 V1 K1 K2
205 ) → llpx_sn R d T L1 L2.
206 #R #L1 #L2 #T #d #HL12 #IH @llpx_sn_alt_inv_lpx_sn
207 @llpx_sn_alt_intro_alt // -HL12
208 #I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
209 elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /3 width=1 by llpx_sn_lpx_sn_alt, and3_intro/
212 (* Advanced inversion lemmas ************************************************)
214 lemma llpx_sn_inv_gen: ∀R,L1,L2,T,d. llpx_sn R d T 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 K1 V1 V2 & llpx_sn R 0 V1 K1 K2.
219 #R #L1 #L2 #T #d #H lapply (llpx_sn_lpx_sn_alt … H) -H
220 #H elim (llpx_sn_alt_inv_gen … H) -H
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_alt_inv_lpx_sn, and3_intro/