Mercurial > hg > Members > kono > Proof > ZF-in-agda
diff ordinal.agda @ 73:dd430a95610f
fix ordinal
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sat, 01 Jun 2019 18:17:24 +0900 |
| parents | f39f1a90d154 |
| children | 819da8c08f05 |
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--- a/ordinal.agda Sat Jun 01 14:43:05 2019 +0900 +++ b/ordinal.agda Sat Jun 01 18:17:24 2019 +0900 @@ -42,7 +42,8 @@ o<-subst df refl refl = df osuc : {n : Level} ( x : Ordinal {n} ) → Ordinal {n} -osuc record { lv = lx ; ord = (Φ lv) } = record { lv = lx ; ord = OSuc lx (Φ lv) } +osuc record { lv = 0 ; ord = (Φ lv) } = record { lv = 0 ; ord = OSuc 0 (Φ lv) } +osuc record { lv = Suc lx ; ord = (Φ (Suc lv)) } = record { lv = Suc lx ; ord = ℵ lv } osuc record { lv = lx ; ord = (OSuc lx ox ) } = record { lv = lx ; ord = OSuc lx (OSuc lx ox) } osuc record { lv = Suc lx ; ord = ℵ lx } = record { lv = Suc lx ; ord = OSuc (Suc lx) (ℵ lx) } @@ -63,10 +64,35 @@ s<refl {n} {Suc lv} {ℵ lv} = ℵs< <-osuc : {n : Level} { x : Ordinal {n} } → x o< osuc x -<-osuc {n} { record { lv = lv ; ord = (Φ .(lv)) } } = case2 Φ< -<-osuc {n} { record { lv = lv ; ord = (OSuc lv ox ) } } = case2 ( s< s<refl ) +<-osuc {n} { record { lv = 0 ; ord = Φ 0 } } = case2 Φ< +<-osuc {n} { record { lv = (Suc lv) ; ord = Φ (Suc lv) } } = case2 ℵΦ< +<-osuc {n} {record { lv = Zero ; ord = OSuc .0 ox }} = case2 ( s< s<refl ) +<-osuc {n} {record { lv = Suc lv₁ ; ord = OSuc .(Suc lv₁) ox }} = case2 ( s< s<refl ) <-osuc {n} { record { lv = .(Suc lv₁) ; ord = (ℵ lv₁) } } = case2 ℵs< +osuc-lveq : {n : Level} { x : Ordinal {n} } → lv x ≡ lv ( osuc x ) +osuc-lveq {n} {record { lv = 0 ; ord = Φ 0 }} = refl +osuc-lveq {n} {record { lv = Suc lv ; ord = Φ (Suc lv) }} = refl +osuc-lveq {n} {record { lv = Zero ; ord = OSuc .0 ord₁ }} = refl +osuc-lveq {n} {record { lv = Suc lv₁ ; ord = OSuc .(Suc lv₁) ord₁ }} = refl +osuc-lveq {n} {record { lv = .(Suc lv₁) ; ord = ℵ lv₁ }} = refl + +nat-<> : { x y : Nat } → x < y → y < x → ⊥ +nat-<> (s≤s x<y) (s≤s y<x) = nat-<> x<y y<x + +osuc-< : {n : Level} { x y : Ordinal {n} } → y o< osuc x → x o< y → ⊥ +osuc-< {n} {record { lv = .0 ; ord = Φ .0 }} {record { lv = .(Suc _) ; ord = ord }} (case1 ()) (case1 (s≤s z≤n)) +osuc-< {n} {record { lv = .0 ; ord = OSuc .0 ord₁ }} {record { lv = .(Suc _) ; ord = ord }} (case1 ()) (case1 (s≤s z≤n)) +osuc-< {n} {record { lv = lx ; ord = xo }} {record { lv = ly ; ord = yo }} (case1 lt1) (case1 lt2) with osuc-lveq {n} {record { lv = lx ; ord = xo }} +osuc-< {n} {record { lv = Zero ; ord = Φ .0 }} {record { lv = ly ; ord = yo }} (case1 lt1) (case1 lt2) | eq = nat-<> lt1 lt2 +osuc-< {n} {record { lv = Suc lx ; ord = Φ .(Suc lx) }} {record { lv = ly ; ord = yo }} (case1 lt1) (case1 lt2) | eq = nat-<> lt1 lt2 +osuc-< {n} {record { lv = Zero ; ord = OSuc .0 xo }} {record { lv = ly ; ord = yo }} (case1 lt1) (case1 lt2) | eq = nat-<> lt1 lt2 +osuc-< {n} {record { lv = Suc lx ; ord = OSuc .(Suc lx) xo }} {record { lv = ly ; ord = yo }} (case1 lt1) (case1 lt2) | eq = nat-<> lt1 lt2 +osuc-< {n} {record { lv = .(Suc lv₁) ; ord = ℵ lv₁ }} {record { lv = ly ; ord = yo }} (case1 lt1) (case1 lt2) | refl = nat-<> lt1 lt2 +osuc-< {n} {x} {y} (case1 x₁) (case2 x₂) = {!!} +osuc-< {n} {x} {y} (case2 x₁) (case1 x₂) = {!!} +osuc-< {n} {x} {y} (case2 x₁) (case2 x₂) = {!!} + open import Relation.Binary.HeterogeneousEquality using (_≅_;refl) ordinal-cong : {n : Level} {x y : Ordinal {n}} →