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15 include "basic_2/unfold/delift_lift.ma".
17 (* INVERSE BASIC TERM RELOCATION *******************************************)
19 (* alternative definition of inverse basic term relocation *)
20 inductive delifta: nat → nat → lenv → relation term ≝
21 | delifta_sort : ∀L,d,e,k. delifta d e L (⋆k) (⋆k)
22 | delifta_lref_lt: ∀L,d,e,i. i < d → delifta d e L (#i) (#i)
23 | delifta_lref_be: ∀L,K,V1,V2,W2,i,d,e. d ≤ i → i < d + e →
24 ⇩[0, i] L ≡ K. ⓓV1 → delifta 0 (d + e - i - 1) K V1 V2 →
25 ⇧[0, d] V2 ≡ W2 → delifta d e L (#i) W2
26 | delifta_lref_ge: ∀L,d,e,i. d + e ≤ i → delifta d e L (#i) (#(i - e))
27 | delifta_gref : ∀L,d,e,p. delifta d e L (§p) (§p)
28 | delifta_bind : ∀L,a,I,V1,V2,T1,T2,d,e.
29 delifta d e L V1 V2 → delifta (d + 1) e (L. ⓑ{I} V2) T1 T2 →
30 delifta d e L (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2)
31 | delifta_flat : ∀L,I,V1,V2,T1,T2,d,e.
32 delifta d e L V1 V2 → delifta d e L T1 T2 →
33 delifta d e L (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
36 interpretation "inverse basic relocation (term) alternative"
37 'TSubstAlt L T1 d e T2 = (delifta d e L T1 T2).
39 (* Basic properties *********************************************************)
41 lemma delifta_lsubs_trans: ∀L1,T1,T2,d,e. L1 ⊢ ▼▼*[d, e] T1 ≡ T2 →
42 ∀L2. L2 ≼ [d, e] L1 → L2 ⊢ ▼▼*[d, e] T1 ≡ T2.
43 #L1 #T1 #T2 #d #e #H elim H -L1 -T1 -T2 -d -e // /2 width=1/
44 [ #L1 #K1 #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
45 elim (ldrop_lsubs_ldrop2_abbr … HL12 … HLK1 ? ?) -HL12 -HLK1 // /3 width=6/
51 lemma delift_delifta: ∀L,T1,T2,d,e. L ⊢ ▼*[d, e] T1 ≡ T2 → L ⊢ ▼▼*[d, e] T1 ≡ T2.
52 #L #T1 @(f2_ind … fw … L T1) -L -T1 #n #IH #L *
53 [ * #i #Hn #T2 #d #e #H destruct
54 [ >(delift_inv_sort1 … H) -H //
55 | elim (delift_inv_lref1 … H) -H * /2 width=1/
56 #K #V1 #V2 #Hdi #Hide #HLK #HV12 #HVT2
57 lapply (ldrop_pair2_fwd_fw … HLK) #H
58 lapply (IH … HV12) // -H /2 width=6/
59 | >(delift_inv_gref1 … H) -H //
61 | * [ #a ] #I #V1 #T1 #Hn #X #d #e #H
62 [ elim (delift_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
63 lapply (delift_lsubs_trans … HT12 (L.ⓑ{I}V1) ?) -HT12 /2 width=1/ #HT12
64 lapply (IH … HV12) -HV12 // #HV12
65 lapply (IH … HT12) -IH -HT12 /2 width=1/ #HT12
66 lapply (delifta_lsubs_trans … HT12 (L.ⓑ{I}V2) ?) -HT12 /2 width=1/
67 | elim (delift_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
68 lapply (IH … HV12) -HV12 //
69 lapply (IH … HT12) -IH -HT12 // /2 width=1/
74 (* Basic inversion lemmas ***************************************************)
76 lemma delifta_delift: ∀L,T1,T2,d,e. L ⊢ ▼▼*[d, e] T1 ≡ T2 → L ⊢ ▼*[d, e] T1 ≡ T2.
77 #L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e // /2 width=1/ /2 width=6/
80 lemma delift_ind_alt: ∀R:ℕ→ℕ→lenv→relation term.
81 (∀L,d,e,k. R d e L (⋆k) (⋆k)) →
82 (∀L,d,e,i. i < d → R d e L (#i) (#i)) →
83 (∀L,K,V1,V2,W2,i,d,e. d ≤ i → i < d + e →
84 ⇩[O, i] L ≡ K.ⓓV1 → K ⊢ ▼*[O, d + e - i - 1] V1 ≡ V2 →
85 ⇧[O, d] V2 ≡ W2 → R O (d+e-i-1) K V1 V2 → R d e L (#i) W2
87 (∀L,d,e,i. d + e ≤ i → R d e L (#i) (#(i - e))) →
88 (∀L,d,e,p. R d e L (§p) (§p)) →
89 (∀L,a,I,V1,V2,T1,T2,d,e. L ⊢ ▼*[d, e] V1 ≡ V2 →
90 L.ⓑ{I}V2 ⊢ ▼*[d + 1, e] T1 ≡ T2 → R d e L V1 V2 →
91 R (d+1) e (L.ⓑ{I}V2) T1 T2 → R d e L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
93 (∀L,I,V1,V2,T1,T2,d,e. L ⊢ ▼*[d, e] V1 ≡ V2 →
94 L⊢ ▼*[d, e] T1 ≡ T2 → R d e L V1 V2 →
95 R d e L T1 T2 → R d e L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
97 ∀d,e,L,T1,T2. L ⊢ ▼*[d, e] T1 ≡ T2 → R d e L T1 T2.
98 #R #H1 #H2 #H3 #H4 #H5 #H6 #H7 #d #e #L #T1 #T2 #H elim (delift_delifta … H) -L -T1 -T2 -d -e
99 // /2 width=1 by delifta_delift/ /3 width=1 by delifta_delift/ /3 width=7 by delifta_delift/