Mercurial > hg > Members > kono > Proof > ZF-in-agda

comparison ordinal.agda @ 73:dd430a95610f

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fix ordinal
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sat, 01 Jun 2019 18:17:24 +0900
parents f39f1a90d154
children 819da8c08f05
comparison
equal deleted inserted replaced
72:f39f1a90d154 73:dd430a95610f
40 40
41 o<-subst : {n : Level } {Z X z x : Ordinal {n}} → Z o< X → Z ≡ z → X ≡ x → z o< x 41 o<-subst : {n : Level } {Z X z x : Ordinal {n}} → Z o< X → Z ≡ z → X ≡ x → z o< x
42 o<-subst df refl refl = df 42 o<-subst df refl refl = df
43 43
44 osuc : {n : Level} ( x : Ordinal {n} ) → Ordinal {n} 44 osuc : {n : Level} ( x : Ordinal {n} ) → Ordinal {n}
45 osuc record { lv = lx ; ord = (Φ lv) } = record { lv = lx ; ord = OSuc lx (Φ lv) } 45 osuc record { lv = 0 ; ord = (Φ lv) } = record { lv = 0 ; ord = OSuc 0 (Φ lv) }
46 osuc record { lv = Suc lx ; ord = (Φ (Suc lv)) } = record { lv = Suc lx ; ord = ℵ lv }
46 osuc record { lv = lx ; ord = (OSuc lx ox ) } = record { lv = lx ; ord = OSuc lx (OSuc lx ox) } 47 osuc record { lv = lx ; ord = (OSuc lx ox ) } = record { lv = lx ; ord = OSuc lx (OSuc lx ox) }
47 osuc record { lv = Suc lx ; ord = ℵ lx } = record { lv = Suc lx ; ord = OSuc (Suc lx) (ℵ lx) } 48 osuc record { lv = Suc lx ; ord = ℵ lx } = record { lv = Suc lx ; ord = OSuc (Suc lx) (ℵ lx) }
48 49
49 open import Data.Nat.Properties 50 open import Data.Nat.Properties
50 open import Data.Empty 51 open import Data.Empty
61 s<refl {n} {lv} {Φ lv} = Φ< 62 s<refl {n} {lv} {Φ lv} = Φ<
62 s<refl {n} {lv} {OSuc lv x} = s< s<refl 63 s<refl {n} {lv} {OSuc lv x} = s< s<refl
63 s<refl {n} {Suc lv} {ℵ lv} = ℵs< 64 s<refl {n} {Suc lv} {ℵ lv} = ℵs<
64 65
65 <-osuc : {n : Level} { x : Ordinal {n} } → x o< osuc x 66 <-osuc : {n : Level} { x : Ordinal {n} } → x o< osuc x
66 <-osuc {n} { record { lv = lv ; ord = (Φ .(lv)) } } = case2 Φ< 67 <-osuc {n} { record { lv = 0 ; ord = Φ 0 } } = case2 Φ<
67 <-osuc {n} { record { lv = lv ; ord = (OSuc lv ox ) } } = case2 ( s< s<refl ) 68 <-osuc {n} { record { lv = (Suc lv) ; ord = Φ (Suc lv) } } = case2 ℵΦ<
69 <-osuc {n} {record { lv = Zero ; ord = OSuc .0 ox }} = case2 ( s< s<refl )
70 <-osuc {n} {record { lv = Suc lv₁ ; ord = OSuc .(Suc lv₁) ox }} = case2 ( s< s<refl )
68 <-osuc {n} { record { lv = .(Suc lv₁) ; ord = (ℵ lv₁) } } = case2 ℵs< 71 <-osuc {n} { record { lv = .(Suc lv₁) ; ord = (ℵ lv₁) } } = case2 ℵs<
72
73 osuc-lveq : {n : Level} { x : Ordinal {n} } → lv x ≡ lv ( osuc x )
74 osuc-lveq {n} {record { lv = 0 ; ord = Φ 0 }} = refl
75 osuc-lveq {n} {record { lv = Suc lv ; ord = Φ (Suc lv) }} = refl
76 osuc-lveq {n} {record { lv = Zero ; ord = OSuc .0 ord₁ }} = refl
77 osuc-lveq {n} {record { lv = Suc lv₁ ; ord = OSuc .(Suc lv₁) ord₁ }} = refl
78 osuc-lveq {n} {record { lv = .(Suc lv₁) ; ord = ℵ lv₁ }} = refl
79
80 nat-<> : { x y : Nat } → x < y → y < x → ⊥
81 nat-<> (s≤s x<y) (s≤s y<x) = nat-<> x<y y<x
82
83 osuc-< : {n : Level} { x y : Ordinal {n} } → y o< osuc x → x o< y → ⊥
84 osuc-< {n} {record { lv = .0 ; ord = Φ .0 }} {record { lv = .(Suc _) ; ord = ord }} (case1 ()) (case1 (s≤s z≤n))
85 osuc-< {n} {record { lv = .0 ; ord = OSuc .0 ord₁ }} {record { lv = .(Suc _) ; ord = ord }} (case1 ()) (case1 (s≤s z≤n))
86 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 }}
87 osuc-< {n} {record { lv = Zero ; ord = Φ .0 }} {record { lv = ly ; ord = yo }} (case1 lt1) (case1 lt2) | eq = nat-<> lt1 lt2
88 osuc-< {n} {record { lv = Suc lx ; ord = Φ .(Suc lx) }} {record { lv = ly ; ord = yo }} (case1 lt1) (case1 lt2) | eq = nat-<> lt1 lt2
89 osuc-< {n} {record { lv = Zero ; ord = OSuc .0 xo }} {record { lv = ly ; ord = yo }} (case1 lt1) (case1 lt2) | eq = nat-<> lt1 lt2
90 osuc-< {n} {record { lv = Suc lx ; ord = OSuc .(Suc lx) xo }} {record { lv = ly ; ord = yo }} (case1 lt1) (case1 lt2) | eq = nat-<> lt1 lt2
91 osuc-< {n} {record { lv = .(Suc lv₁) ; ord = ℵ lv₁ }} {record { lv = ly ; ord = yo }} (case1 lt1) (case1 lt2) | refl = nat-<> lt1 lt2
92 osuc-< {n} {x} {y} (case1 x₁) (case2 x₂) = {!!}
93 osuc-< {n} {x} {y} (case2 x₁) (case1 x₂) = {!!}
94 osuc-< {n} {x} {y} (case2 x₁) (case2 x₂) = {!!}
69 95
70 open import Relation.Binary.HeterogeneousEquality using (_≅_;refl) 96 open import Relation.Binary.HeterogeneousEquality using (_≅_;refl)
71 97
72 ordinal-cong : {n : Level} {x y : Ordinal {n}} → 98 ordinal-cong : {n : Level} {x y : Ordinal {n}} →
73 lv x ≡ lv y → ord x ≅ ord y → x ≡ y 99 lv x ≡ lv y → ord x ≅ ord y → x ≡ y