Mercurial > hg > Members > kono > Proof > ZF-in-agda
changeset 169:acbcbd98d588
trans finite on ε-induction
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Fri, 19 Jul 2019 05:12:08 +0900 |
| parents | b25a4eca06a6 |
| children | c96f28c3c387 |
| files | HOD.agda |
| diffstat | 1 files changed, 12 insertions(+), 12 deletions(-) [+] |
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--- a/HOD.agda Fri Jul 19 03:27:58 2019 +0900 +++ b/HOD.agda Fri Jul 19 05:12:08 2019 +0900 @@ -474,16 +474,16 @@ → (x : OD {suc n} ) → ψ x ε-induction {n} {m} {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (osuc (od→ord x)) <-osuc) where ε-induction-ord : ( ox : Ordinal {suc n} ) {oy : Ordinal {suc n} } → oy o< ox → ψ (ord→od oy) - ε-induction-ord record { lv = Zero ; ord = (Φ 0) } (case1 ()) - ε-induction-ord record { lv = Zero ; ord = (Φ 0) } (case2 ()) - ε-induction-ord record { lv = lx ; ord = (OSuc lx ox) } {oy} y<x = - ind {ord→od oy} ( λ {y} lt → subst (λ k → ψ k ) oiso (ε-induction-ord (record { lv = lx ; ord = ox} ) (lemma y lt ))) where - lemma : (y : OD) → ord→od oy ∋ y → od→ord y o< record { lv = lx ; ord = ox } - lemma y lt with osuc-≡< y<x - lemma y lt | case1 refl = o<-subst (c<→o< lt) refl diso - lemma y lt | case2 lt1 = ordtrans (o<-subst (c<→o< lt) refl diso) lt1 - ε-induction-ord record { lv = (Suc lx) ; ord = (Φ (Suc lx)) } {record { lv = ly ; ord = OSuc ly oy }} (case1 (s≤s x)) = - ind {ord→od record { lv = ly ; ord = OSuc ly oy }} ( - λ {y} lt → subst ( λ k → ψ k ) oiso ( ε-induction-ord record { lv = lx ; ord = (Φ lx) } {od→ord y} {!!})) where - ε-induction-ord record { lv = (Suc lx) ; ord = (Φ (Suc lx)) } {record { lv = ly ; ord = Φ ly }} (case1 (s≤s x)) = {!!} + ε-induction-ord ox = TransFinite {suc n} {suc n ⊔ m} {λ z → {oy : Ordinal {suc n} } → oy o< z → ψ (ord→od oy) } lemma1 lemma2 ox where + lemma1 : (lx : Nat) {oy : Ordinal} → oy o< record { lv = lx ; ord = Φ lx } → ψ (ord→od oy) + lemma1 Zero {oy} (case1 ()) + lemma1 Zero {oy} (case2 ()) + lemma1 (Suc lx) {record { lv = Zero ; ord = Φ 0 }} (case1 (s≤s z≤n)) = {!!} + lemma1 (Suc lx) {record { lv = Zero ; ord = OSuc 0 oy }} (case1 (s≤s z≤n)) = {!!} + lemma1 (Suc (Suc lx)) {record { lv = Suc ly ; ord = Φ (Suc ly) }} (case1 (s≤s (s≤s x))) = {!!} + lemma1 (Suc (Suc lx)) {record { lv = Suc ly ; ord = OSuc (Suc ly) oy }} (case1 (s≤s (s≤s x))) = {!!} + lemma2 : (lx : Nat) (x₁ : OrdinalD lx) → + ({oy : Ordinal} → oy o< record { lv = lx ; ord = x₁ } → ψ (ord→od oy)) → + {oy : Ordinal} → oy o< record { lv = lx ; ord = OSuc lx x₁ } → ψ (ord→od oy) + lemma2 lx x1 p lt = ind ( λ {y} lty → subst (λ k → ψ k) oiso (p {!!} ))