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comparison ordinal.agda @ 74:819da8c08f05

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ordinal atomical successor?
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sat, 01 Jun 2019 19:19:40 +0900
parents dd430a95610f
children 714470702a8b
comparison
equal deleted inserted replaced
73:dd430a95610f 74:819da8c08f05
39 _o<_ x y = (lv x < lv y ) ∨ ( ord x d< ord y ) 39 _o<_ x y = (lv x < lv y ) ∨ ( ord x d< ord y )
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}
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 }
47 osuc record { lv = lx ; ord = (OSuc lx ox ) } = record { lv = lx ; ord = OSuc lx (OSuc lx ox) }
48 osuc record { lv = Suc lx ; ord = ℵ lx } = record { lv = Suc lx ; ord = OSuc (Suc lx) (ℵ lx) }
49
50 open import Data.Nat.Properties 44 open import Data.Nat.Properties
51 open import Data.Empty 45 open import Data.Empty
52 open import Data.Unit using ( ⊤ ) 46 open import Data.Unit using ( ⊤ )
53 open import Relation.Nullary 47 open import Relation.Nullary
54 48
60 54
61 s<refl : {n : Level } {lx : Nat } { x : OrdinalD {n} lx } → x d< OSuc lx x 55 s<refl : {n : Level } {lx : Nat } { x : OrdinalD {n} lx } → x d< OSuc lx x
62 s<refl {n} {lv} {Φ lv} = Φ< 56 s<refl {n} {lv} {Φ lv} = Φ<
63 s<refl {n} {lv} {OSuc lv x} = s< s<refl 57 s<refl {n} {lv} {OSuc lv x} = s< s<refl
64 s<refl {n} {Suc lv} {ℵ lv} = ℵs< 58 s<refl {n} {Suc lv} {ℵ lv} = ℵs<
65
66 <-osuc : {n : Level} { x : Ordinal {n} } → x o< osuc x
67 <-osuc {n} { record { lv = 0 ; ord = Φ 0 } } = case2 Φ<
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 )
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) = {!!}
95 59
96 open import Relation.Binary.HeterogeneousEquality using (_≅_;refl) 60 open import Relation.Binary.HeterogeneousEquality using (_≅_;refl)
97 61
98 ordinal-cong : {n : Level} {x y : Ordinal {n}} → 62 ordinal-cong : {n : Level} {x y : Ordinal {n}} →
99 lv x ≡ lv y → ord x ≅ ord y → x ≡ y 63 lv x ≡ lv y → ord x ≅ ord y → x ≡ y
157 d<→lv (s< lt) = refl 121 d<→lv (s< lt) = refl
158 d<→lv ℵΦ< = refl 122 d<→lv ℵΦ< = refl
159 d<→lv (ℵ< _) = refl 123 d<→lv (ℵ< _) = refl
160 d<→lv ℵs< = refl 124 d<→lv ℵs< = refl
161 d<→lv (ℵss< _) = refl 125 d<→lv (ℵss< _) = refl
126
127 osuc : {n : Level} ( x : Ordinal {n} ) → Ordinal {n}
128 osuc record { lv = 0 ; ord = (Φ lv) } = record { lv = 0 ; ord = OSuc 0 (Φ lv) }
129 osuc record { lv = Suc lx ; ord = (Φ (Suc lv)) } = record { lv = Suc lx ; ord = ℵ lv }
130 osuc record { lv = 0 ; ord = (OSuc 0 ox ) } = record { lv = 0 ; ord = OSuc 0 (OSuc 0 ox) }
131 osuc record { lv = Suc lx ; ord = (OSuc (Suc lx) ox ) } = record { lv = Suc lx ; ord = OSuc (Suc lx) (OSuc (Suc lx) ox) }
132 osuc record { lv = Suc lx ; ord = ℵ lx } = record { lv = Suc lx ; ord = OSuc (Suc lx) (ℵ lx) }
133
134 <-osuc : {n : Level} { x : Ordinal {n} } → x o< osuc x
135 <-osuc {n} { record { lv = 0 ; ord = Φ 0 } } = case2 Φ<
136 <-osuc {n} { record { lv = (Suc lv) ; ord = Φ (Suc lv) } } = case2 ℵΦ<
137 <-osuc {n} {record { lv = Zero ; ord = OSuc .0 ox }} = case2 ( s< s<refl )
138 <-osuc {n} {record { lv = Suc lv₁ ; ord = OSuc .(Suc lv₁) ox }} = case2 ( s< s<refl )
139 <-osuc {n} { record { lv = .(Suc lv₁) ; ord = (ℵ lv₁) } } = case2 ℵs<
140
141 osuc-lveq : {n : Level} { x : Ordinal {n} } → lv x ≡ lv ( osuc x )
142 osuc-lveq {n} {record { lv = 0 ; ord = Φ 0 }} = refl
143 osuc-lveq {n} {record { lv = Suc lv ; ord = Φ (Suc lv) }} = refl
144 osuc-lveq {n} {record { lv = Zero ; ord = OSuc .0 ord₁ }} = refl
145 osuc-lveq {n} {record { lv = Suc lv₁ ; ord = OSuc .(Suc lv₁) ord₁ }} = refl
146 osuc-lveq {n} {record { lv = .(Suc lv₁) ; ord = ℵ lv₁ }} = refl
147
148 nat-<> : { x y : Nat } → x < y → y < x → ⊥
149 nat-<> (s≤s x<y) (s≤s y<x) = nat-<> x<y y<x
150
151 nat-<≡ : { x : Nat } → x < x → ⊥
152 nat-<≡ (s≤s lt) = nat-<≡ lt
153
154 osuc-< : {n : Level} { x y : Ordinal {n} } → y o< osuc x → x o< y → ⊥
155 osuc-< {n} {record { lv = .0 ; ord = Φ .0 }} {record { lv = .(Suc _) ; ord = ord }} (case1 ()) (case1 (s≤s z≤n))
156 osuc-< {n} {record { lv = .0 ; ord = OSuc .0 ord₁ }} {record { lv = .(Suc _) ; ord = ord }} (case1 ()) (case1 (s≤s z≤n))
157 osuc-< {n} {record { lv = Suc lx ; ord = Φ .(Suc lx) }} {record { lv = ly ; ord = yo }} (case1 lt1) (case1 lt2) = nat-<> lt1 lt2
158 osuc-< {n} {record { lv = Suc lx ; ord = OSuc .(Suc lx) xo }} {record { lv = ly ; ord = yo }} (case1 lt1) (case1 lt2) = nat-<> lt1 lt2
159 osuc-< {n} {record { lv = .(Suc lv₁) ; ord = ℵ lv₁ }} {record { lv = ly ; ord = yo }} (case1 lt1) (case1 lt2) = nat-<> lt1 lt2
160 osuc-< {n} {x} {y} (case1 x₁) (case2 x₂) with d<→lv x₂ | osuc-lveq {n} {x}
161 ... | refl | eq = {!!}
162 osuc-< {n} {x} {y} (case2 x₁) (case1 x₂) with d<→lv x₁ | osuc-lveq {n} {x}
163 ... | refl | eq = {!!}
164 osuc-< {n} {record { lv = Zero ; ord = Φ .0 }} {record { lv = Zero ; ord = Φ .0 }} (case2 Φ<) (case2 ())
165 osuc-< {n} {record { lv = Suc lv₁ ; ord = Φ .(Suc lv₁) }} {record { lv = Zero ; ord = Φ .0 }} (case2 ()) (case2 x₂)
166 osuc-< {n} {record { lv = lv₁ ; ord = Φ .lv₁ }} {record { lv = Suc lv₂ ; ord = Φ .(Suc lv₂) }} (case2 x₁) (case2 ())
167 osuc-< {n} {record { lv = .0 ; ord = Φ .0 }} {record { lv = Zero ; ord = OSuc .0 ord₁ }} (case2 (s< ())) (case2 Φ<)
168 osuc-< {n} {record { lv = .1 ; ord = ℵ Zero }} {record { lv = .1 ; ord = ℵ Zero }} (case2 ℵs<) (case2 ())
169 osuc-< {n} {record { lv = .1 ; ord = ℵ Zero }} {record { lv = .1 ; ord = ℵ Zero }} (case2 (ℵss< x₁)) (case2 ())
170 osuc-< {n} {record { lv = .1 ; ord = ℵ Zero }} {record { lv = .(Suc (Suc lv₂)) ; ord = ℵ Suc lv₂ }} (case2 ()) (case2 x₂)
171 osuc-< {n} {record { lv = .(Suc (Suc lv₁)) ; ord = ℵ Suc lv₁ }} {record { lv = .(Suc (Suc lv₁)) ; ord = ℵ .(Suc lv₁) }} (case2 ℵs<) (case2 ())
172 osuc-< {n} {record { lv = .(Suc (Suc lv₁)) ; ord = ℵ Suc lv₁ }} {record { lv = .(Suc (Suc lv₁)) ; ord = ℵ .(Suc lv₁) }} (case2 (ℵss< x₁)) (case2 ())
173 osuc-< {n} {record { lv = Suc lv₁ ; ord = .(Φ (Suc lv₁)) }} {record { lv = .(Suc lv₁) ; ord = (OSuc (Suc lv₁) y) }} (case2 (ℵ< x)) (case2 Φ<) = {!!}
174 osuc-< {n} {record { lv = Zero ; ord = (OSuc 0 ox) }} {record { lv = .0 ; ord = (OSuc 0 oy) }} (case2 (s< c1)) (case2 (s< c2)) =
175 osuc-< {n} {record { lv = Zero ; ord = ox }} {record { lv = 0 ; ord = oy }} (case2 {!!}) (case2 c2)
176 osuc-< {n} {record { lv = Suc lv₁ ; ord = .(OSuc (Suc lv₁) _) }} {record { lv = .(Suc lv₁) ; ord = .(OSuc (Suc lv₁) _) }} (case2 (s< c1)) (case2 (s< c2)) = {!!}
177 osuc-< {n} {record { lv = .(Suc _) ; ord = .(OSuc (Suc _) _) }} {record { lv = .(Suc _) ; ord = .(ℵ _) }} (case2 (ℵss< c1)) (case2 (ℵ< x)) = {!!}
178 osuc-< {n} {record { lv = .(Suc _) ; ord = .(ℵ _) }} {record { lv = .(Suc _) ; ord = .(OSuc (Suc _) (ℵ _)) }} (case2 (s< c1)) (case2 ℵs<) = {!!}
179 osuc-< {n} {record { lv = .(Suc _) ; ord = .(ℵ _) }} {record { lv = .(Suc _) ; ord = .(OSuc (Suc _) _) }} (case2 (s< c1)) (case2 (ℵss< c2)) = {!!}
180
162 181
163 xsyℵ : {n : Level} {lx : Nat} {x y : OrdinalD {n} lx } → x d< y → ¬ℵ y → ¬ℵ x 182 xsyℵ : {n : Level} {lx : Nat} {x y : OrdinalD {n} lx } → x d< y → ¬ℵ y → ¬ℵ x
164 xsyℵ {_} {_} {Φ lv₁} {y} x<y t = ¬ℵΦ 183 xsyℵ {_} {_} {Φ lv₁} {y} x<y t = ¬ℵΦ
165 xsyℵ {_} {_} {OSuc lv₁ x} {OSuc lv₁ y} (s< x<y) (¬ℵs t) = ¬ℵs ( xsyℵ x<y t) 184 xsyℵ {_} {_} {OSuc lv₁ x} {OSuc lv₁ y} (s< x<y) (¬ℵs t) = ¬ℵs ( xsyℵ x<y t)
166 xsyℵ {_} {_} {OSuc .(Suc _) x} {.(ℵ _)} (ℵ< x₁) () 185 xsyℵ {_} {_} {OSuc .(Suc _) x} {.(ℵ _)} (ℵ< x₁) ()