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4 (* ||A|| A project by Andrea Asperti *)
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15 include "basic_2/notation/relations/psubststaralt_6.ma".
16 include "basic_2/substitution/cpys_lift.ma".
18 (* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
20 (* alternative definition of cpys *)
21 inductive cpysa: ynat → ynat → relation4 genv lenv term term ≝
22 | cpysa_atom : ∀I,G,L,d,e. cpysa d e G L (⓪{I}) (⓪{I})
23 | cpysa_subst: ∀I,G,L,K,V1,V2,W2,i,d,e. d ≤ yinj i → i < d+e →
24 ⇩[i] L ≡ K.ⓑ{I}V1 → cpysa 0 (⫰(d+e-i)) G K V1 V2 →
25 ⇧[0, i+1] V2 ≡ W2 → cpysa d e G L (#i) W2
26 | cpysa_bind : ∀a,I,G,L,V1,V2,T1,T2,d,e.
27 cpysa d e G L V1 V2 → cpysa (⫯d) e G (L.ⓑ{I}V1) T1 T2 →
28 cpysa d e G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
29 | cpysa_flat : ∀I,G,L,V1,V2,T1,T2,d,e.
30 cpysa d e G L V1 V2 → cpysa d e G L T1 T2 →
31 cpysa d e G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
35 "context-sensitive extended multiple substritution (term) alternative"
36 'PSubstStarAlt G L T1 d e T2 = (cpysa d e G L T1 T2).
38 (* Basic properties *********************************************************)
40 lemma lsuby_cpysa_trans: ∀G,d,e. lsub_trans … (cpysa d e G) (lsuby d e).
41 #G #d #e #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2 -d -e
43 | #I #G #L1 #K1 #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
44 elim (lsuby_ldrop_trans_be … HL12 … HLK1) -HL12 -HLK1 /3 width=7 by cpysa_subst/
45 | /4 width=1 by lsuby_succ, cpysa_bind/
46 | /3 width=1 by cpysa_flat/
50 lemma cpysa_refl: ∀G,T,L,d,e. ⦃G, L⦄ ⊢ T ▶▶*[d, e] T.
52 #I elim I -I /2 width=1 by cpysa_bind, cpysa_flat/
55 lemma cpysa_cpy_trans: ∀G,L,T1,T,d,e. ⦃G, L⦄ ⊢ T1 ▶▶*[d, e] T →
56 ∀T2. ⦃G, L⦄ ⊢ T ▶[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶▶*[d, e] T2.
57 #G #L #T1 #T #d #e #H elim H -G -L -T1 -T -d -e
58 [ #I #G #L #d #e #X #H
59 elim (cpy_inv_atom1 … H) -H // * /2 width=7 by cpysa_subst/
60 | #I #G #L #K #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK #_ #HVW2 #IHV12 #T2 #H
61 lapply (ldrop_fwd_drop2 … HLK) #H0LK
62 lapply (cpy_weak … H 0 (d+e) ? ?) -H // #H
63 elim (cpy_inv_lift1_be … H … H0LK … HVW2) -H -H0LK -HVW2
64 /3 width=7 by cpysa_subst, ylt_fwd_le_succ/
65 | #a #I #G #L #V1 #V #T1 #T #d #e #_ #_ #IHV1 #IHT1 #X #H
66 elim (cpy_inv_bind1 … H) -H #V2 #T2 #HV2 #HT2 #H destruct
67 /5 width=5 by cpysa_bind, lsuby_cpy_trans, lsuby_succ/
68 | #I #G #L #V1 #V #T1 #T #d #e #_ #_ #IHV1 #IHT1 #X #H
69 elim (cpy_inv_flat1 … H) -H #V2 #T2 #HV2 #HT2 #H destruct /3 width=1 by cpysa_flat/
73 lemma cpys_cpysa: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶▶*[d, e] T2.
74 /3 width=8 by cpysa_cpy_trans, cpys_ind/ qed.
76 (* Basic inversion lemmas ***************************************************)
78 lemma cpysa_inv_cpys: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶▶*[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2.
79 #G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e
80 /2 width=7 by cpys_subst, cpys_flat, cpys_bind, cpy_cpys/
83 (* Advanced eliminators *****************************************************)
85 lemma cpys_ind_alt: ∀R:ynat→ynat→relation4 genv lenv term term.
86 (∀I,G,L,d,e. R d e G L (⓪{I}) (⓪{I})) →
87 (∀I,G,L,K,V1,V2,W2,i,d,e. d ≤ yinj i → i < d + e →
88 ⇩[i] L ≡ K.ⓑ{I}V1 → ⦃G, K⦄ ⊢ V1 ▶*[O, ⫰(d+e-i)] V2 →
89 ⇧[O, i+1] V2 ≡ W2 → R O (⫰(d+e-i)) G K V1 V2 → R d e G L (#i) W2
91 (∀a,I,G,L,V1,V2,T1,T2,d,e. ⦃G, L⦄ ⊢ V1 ▶*[d, e] V2 →
92 ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶*[⫯d, e] T2 → R d e G L V1 V2 →
93 R (⫯d) e G (L.ⓑ{I}V1) T1 T2 → R d e G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
95 (∀I,G,L,V1,V2,T1,T2,d,e. ⦃G, L⦄ ⊢ V1 ▶*[d, e] V2 →
96 ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 → R d e G L V1 V2 →
97 R d e G L T1 T2 → R d e G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
99 ∀d,e,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 → R d e G L T1 T2.
100 #R #H1 #H2 #H3 #H4 #d #e #G #L #T1 #T2 #H elim (cpys_cpysa … H) -G -L -T1 -T2 -d -e
101 /3 width=8 by cpysa_inv_cpys/