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15 include "basic_2/notation/relations/psubsteval_6.ma".
16 include "basic_2/relocation/cny.ma".
17 include "basic_2/substitution/cpys.ma".
19 (* EVALUATION FOR CONTEXT-SENSITIVE EXTENDED SUBSTITUTION ON TERMS **********)
21 definition cpye: ynat → ynat → relation4 genv lenv term term ≝
22 λd,e,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 ∧ ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃T2⦄.
24 interpretation "evaluation for context-sensitive extended substitution (term)"
25 'PSubstEval G L T1 T2 d e = (cpye d e G L T1 T2).
27 (* Basic_properties *********************************************************)
29 lemma cpye_sort: ∀G,L,d,e,k. ⦃G, L⦄ ⊢ ⋆k ▶*[d, e] 𝐍⦃⋆k⦄.
30 /3 width=5 by cny_sort, conj/ qed.
32 lemma cpye_free: ∀G,L,d,e,i. |L| ≤ i → ⦃G, L⦄ ⊢ #i ▶*[d, e] 𝐍⦃#i⦄.
33 /3 width=6 by cny_lref_free, conj/ qed.
35 lemma cpye_top: ∀G,L,d,e,i. d + e ≤ yinj i → ⦃G, L⦄ ⊢ #i ▶*[d, e] 𝐍⦃#i⦄.
36 /3 width=6 by cny_lref_top, conj/ qed.
38 lemma cpye_skip: ∀G,L,d,e,i. yinj i < d → ⦃G, L⦄ ⊢ #i ▶*[d, e] 𝐍⦃#i⦄.
39 /3 width=6 by cny_lref_skip, conj/ qed.
41 lemma cpye_gref: ∀G,L,d,e,p. ⦃G, L⦄ ⊢ §p ▶*[d, e] 𝐍⦃§p⦄.
42 /3 width=5 by cny_gref, conj/ qed.
44 lemma cpye_bind: ∀G,L,V1,V2,d,e. ⦃G, L⦄ ⊢ V1 ▶*[d, e] 𝐍⦃V2⦄ →
45 ∀I,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶*[⫯d, e] 𝐍⦃T2⦄ →
46 ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ▶*[d, e] 𝐍⦃ⓑ{a,I}V2.T2⦄.
47 #G #L #V1 #V2 #d #e * #HV12 #HV2 #I #T1 #T2 *
48 /5 width=8 by cpys_bind, cny_bind, lsuby_cny_conf, lsuby_succ, conj/
51 lemma cpye_flat: ∀G,L,V1,V2,d,e. ⦃G, L⦄ ⊢ V1 ▶*[d, e] 𝐍⦃V2⦄ →
52 ∀T1,T2. ⦃G, L⦄ ⊢ T1 ▶*[d, e] 𝐍⦃T2⦄ →
53 ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ▶*[d, e] 𝐍⦃ⓕ{I}V2.T2⦄.
54 #G #L #V1 #V2 #d #e * #HV12 #HV2 #T1 #T2 *
55 /3 width=7 by cpys_flat, cny_flat, conj/
58 (* Basic inversion lemmas ***************************************************)