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7 (* ||T|| The HELM team. *)
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15 notation "hvbox( L ⊢ break ▼ ▼ * [ term 46 d , break term 46 e ] break term 46 T1 ≡ break term 46 T2 )"
16 non associative with precedence 45
17 for @{ 'TSubstAlt $L $T1 $d $e $T2 }.
19 include "basic_2/unfold/delift_lift.ma".
21 (* INVERSE BASIC TERM RELOCATION *******************************************)
23 (* alternative definition of inverse basic term relocation *)
24 inductive delifta: nat → nat → lenv → relation term ≝
25 | delifta_sort : ∀L,d,e,k. delifta d e L (⋆k) (⋆k)
26 | delifta_lref_lt: ∀L,d,e,i. i < d → delifta d e L (#i) (#i)
27 | delifta_lref_be: ∀L,K,V1,V2,W2,i,d,e. d ≤ i → i < d + e →
28 ⇩[0, i] L ≡ K. ⓓV1 → delifta 0 (d + e - i - 1) K V1 V2 →
29 ⇧[0, d] V2 ≡ W2 → delifta d e L (#i) W2
30 | delifta_lref_ge: ∀L,d,e,i. d + e ≤ i → delifta d e L (#i) (#(i - e))
31 | delifta_gref : ∀L,d,e,p. delifta d e L (§p) (§p)
32 | delifta_bind : ∀L,a,I,V1,V2,T1,T2,d,e.
33 delifta d e L V1 V2 → delifta (d + 1) e (L. ⓑ{I} V2) T1 T2 →
34 delifta d e L (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2)
35 | delifta_flat : ∀L,I,V1,V2,T1,T2,d,e.
36 delifta d e L V1 V2 → delifta d e L T1 T2 →
37 delifta d e L (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
40 interpretation "inverse basic relocation (term) alternative"
41 'TSubstAlt L T1 d e T2 = (delifta d e L T1 T2).
43 (* Basic properties *********************************************************)
45 lemma delifta_lsubr_trans: ∀L1,T1,T2,d,e. L1 ⊢ ▼▼*[d, e] T1 ≡ T2 →
46 ∀L2. L2 ⊑ [d, e] L1 → L2 ⊢ ▼▼*[d, e] T1 ≡ T2.
47 #L1 #T1 #T2 #d #e #H elim H -L1 -T1 -T2 -d -e // /2 width=1/
48 [ #L1 #K1 #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
49 elim (ldrop_lsubr_ldrop2_abbr … HL12 … HLK1 ? ?) -HL12 -HLK1 // /3 width=6/
55 lemma delift_delifta: ∀L,T1,T2,d,e. L ⊢ ▼*[d, e] T1 ≡ T2 → L ⊢ ▼▼*[d, e] T1 ≡ T2.
56 #L #T1 @(f2_ind … fw … L T1) -L -T1 #n #IH #L *
57 [ * #i #Hn #T2 #d #e #H destruct
58 [ >(delift_inv_sort1 … H) -H //
59 | elim (delift_inv_lref1 … H) -H * /2 width=1/
60 #K #V1 #V2 #Hdi #Hide #HLK #HV12 #HVT2
61 lapply (ldrop_pair2_fwd_fw … HLK) #H
62 lapply (IH … HV12) // -H /2 width=6/
63 | >(delift_inv_gref1 … H) -H //
65 | * [ #a ] #I #V1 #T1 #Hn #X #d #e #H
66 [ elim (delift_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
67 lapply (delift_lsubr_trans … HT12 (L.ⓑ{I}V1) ?) -HT12 /2 width=1/ #HT12
68 lapply (IH … HV12) -HV12 // #HV12
69 lapply (IH … HT12) -IH -HT12 /2 width=1/ #HT12
70 lapply (delifta_lsubr_trans … HT12 (L.ⓑ{I}V2) ?) -HT12 /2 width=1/
71 | elim (delift_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
72 lapply (IH … HV12) -HV12 //
73 lapply (IH … HT12) -IH -HT12 // /2 width=1/
78 (* Basic inversion lemmas ***************************************************)
80 lemma delifta_delift: ∀L,T1,T2,d,e. L ⊢ ▼▼*[d, e] T1 ≡ T2 → L ⊢ ▼*[d, e] T1 ≡ T2.
81 #L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e // /2 width=1/ /2 width=6/
84 lemma delift_ind_alt: ∀R:ℕ→ℕ→lenv→relation term.
85 (∀L,d,e,k. R d e L (⋆k) (⋆k)) →
86 (∀L,d,e,i. i < d → R d e L (#i) (#i)) →
87 (∀L,K,V1,V2,W2,i,d,e. d ≤ i → i < d + e →
88 ⇩[O, i] L ≡ K.ⓓV1 → K ⊢ ▼*[O, d + e - i - 1] V1 ≡ V2 →
89 ⇧[O, d] V2 ≡ W2 → R O (d+e-i-1) K V1 V2 → R d e L (#i) W2
91 (∀L,d,e,i. d + e ≤ i → R d e L (#i) (#(i - e))) →
92 (∀L,d,e,p. R d e L (§p) (§p)) →
93 (∀L,a,I,V1,V2,T1,T2,d,e. L ⊢ ▼*[d, e] V1 ≡ V2 →
94 L.ⓑ{I}V2 ⊢ ▼*[d + 1, e] T1 ≡ T2 → R d e L V1 V2 →
95 R (d+1) e (L.ⓑ{I}V2) T1 T2 → R d e L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
97 (∀L,I,V1,V2,T1,T2,d,e. L ⊢ ▼*[d, e] V1 ≡ V2 →
98 L⊢ ▼*[d, e] T1 ≡ T2 → R d e L V1 V2 →
99 R d e L T1 T2 → R d e L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
101 ∀d,e,L,T1,T2. L ⊢ ▼*[d, e] T1 ≡ T2 → R d e L T1 T2.
102 #R #H1 #H2 #H3 #H4 #H5 #H6 #H7 #d #e #L #T1 #T2 #H elim (delift_delifta … H) -L -T1 -T2 -d -e
103 // /2 width=1 by delifta_delift/ /3 width=1 by delifta_delift/ /3 width=7 by delifta_delift/