Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate ordinal.agda @ 74:819da8c08f05
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 |
| rev | line source |
|---|---|
| 34 | 1 {-# OPTIONS --allow-unsolved-metas #-} |
| 16 | 2 open import Level |
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posturate OD is isomorphic to Ordinal
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3 module ordinal where |
| 3 | 4 |
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5 open import zf |
| 3 | 6 |
| 23 | 7 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) |
| 3 | 8 |
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9 open import Relation.Binary.PropositionalEquality |
| 3 | 10 |
| 24 | 11 data OrdinalD {n : Level} : (lv : Nat) → Set n where |
| 12 Φ : (lv : Nat) → OrdinalD lv | |
| 13 OSuc : (lv : Nat) → OrdinalD {n} lv → OrdinalD lv | |
| 17 | 14 ℵ_ : (lv : Nat) → OrdinalD (Suc lv) |
| 3 | 15 |
| 24 | 16 record Ordinal {n : Level} : Set n where |
| 16 | 17 field |
| 18 lv : Nat | |
| 24 | 19 ord : OrdinalD {n} lv |
| 16 | 20 |
| 34 | 21 data ¬ℵ {n : Level} {lx : Nat } : ( x : OrdinalD {n} lx ) → Set where |
| 22 ¬ℵΦ : ¬ℵ (Φ lx) | |
| 23 ¬ℵs : {x : OrdinalD {n} lx } → ¬ℵ x → ¬ℵ (OSuc lx x) | |
| 24 | |
| 70 | 25 -- |
| 26 -- Φ (Suc lv) < ℵ lv < OSuc (Suc lv) (ℵ lv) < OSuc ... < OSuc (Suc lv) (Φ (Suc lv)) < OSuc ... < ℵ (Suc lv) | |
| 27 -- | |
| 24 | 28 data _d<_ {n : Level} : {lx ly : Nat} → OrdinalD {n} lx → OrdinalD {n} ly → Set n where |
| 29 Φ< : {lx : Nat} → {x : OrdinalD {n} lx} → Φ lx d< OSuc lx x | |
| 30 s< : {lx : Nat} → {x y : OrdinalD {n} lx} → x d< y → OSuc lx x d< OSuc lx y | |
| 41 | 31 ℵΦ< : {lx : Nat} → Φ (Suc lx) d< (ℵ lx) |
| 34 | 32 ℵ< : {lx : Nat} → {x : OrdinalD {n} (Suc lx) } → ¬ℵ x → OSuc (Suc lx) x d< (ℵ lx) |
| 33 ℵs< : {lx : Nat} → (ℵ lx) d< OSuc (Suc lx) (ℵ lx) | |
| 35 | 34 ℵss< : {lx : Nat} → {x : OrdinalD {n} (Suc lx) } → (ℵ lx) d< x → (ℵ lx) d< OSuc (Suc lx) x |
| 17 | 35 |
| 36 open Ordinal | |
| 37 | |
| 27 | 38 _o<_ : {n : Level} ( x y : Ordinal ) → Set n |
| 17 | 39 _o<_ x y = (lv x < lv y ) ∨ ( ord x d< ord y ) |
| 3 | 40 |
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equalitu and internal parametorisity
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41 o<-subst : {n : Level } {Z X z x : Ordinal {n}} → Z o< X → Z ≡ z → X ≡ x → z o< x |
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42 o<-subst df refl refl = df |
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43 |
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44 open import Data.Nat.Properties |
| 6 | 45 open import Data.Empty |
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problem on Ordinal ( OSuc ℵ )
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46 open import Data.Unit using ( ⊤ ) |
| 6 | 47 open import Relation.Nullary |
| 48 | |
| 49 open import Relation.Binary | |
| 50 open import Relation.Binary.Core | |
| 51 | |
| 24 | 52 o∅ : {n : Level} → Ordinal {n} |
| 53 o∅ = record { lv = Zero ; ord = Φ Zero } | |
| 21 | 54 |
| 34 | 55 s<refl : {n : Level } {lx : Nat } { x : OrdinalD {n} lx } → x d< OSuc lx x |
| 56 s<refl {n} {lv} {Φ lv} = Φ< | |
| 57 s<refl {n} {lv} {OSuc lv x} = s< s<refl | |
| 58 s<refl {n} {Suc lv} {ℵ lv} = ℵs< | |
| 59 | |
| 39 | 60 open import Relation.Binary.HeterogeneousEquality using (_≅_;refl) |
| 61 | |
| 62 ordinal-cong : {n : Level} {x y : Ordinal {n}} → | |
| 63 lv x ≡ lv y → ord x ≅ ord y → x ≡ y | |
| 64 ordinal-cong refl refl = refl | |
| 21 | 65 |
| 46 | 66 ordinal-lv : {n : Level} {x y : Ordinal {n}} → x ≡ y → lv x ≡ lv y |
| 67 ordinal-lv refl = refl | |
| 68 | |
| 69 ordinal-d : {n : Level} {x y : Ordinal {n}} → x ≡ y → ord x ≅ ord y | |
| 70 ordinal-d refl = refl | |
| 71 | |
| 24 | 72 ≡→¬d< : {n : Level} → {lv : Nat} → {x : OrdinalD {n} lv } → x d< x → ⊥ |
| 73 ≡→¬d< {n} {lx} {OSuc lx y} (s< t) = ≡→¬d< t | |
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74 |
| 24 | 75 trio<> : {n : Level} → {lx : Nat} {x : OrdinalD {n} lx } { y : OrdinalD lx } → y d< x → x d< y → ⊥ |
| 34 | 76 trio<> {n} {lx} {.(OSuc lx _)} {.(OSuc lx _)} (s< s) (s< t) = trio<> s t |
| 77 trio<> {_} {.(Suc _)} {.(OSuc (Suc _) (ℵ _))} {.(ℵ _)} ℵs< (ℵ< {_} {.(ℵ _)} ()) | |
| 78 trio<> {_} {.(Suc _)} {.(ℵ _)} {.(OSuc (Suc _) (ℵ _))} (ℵ< ()) ℵs< | |
| 35 | 79 trio<> {n} {lx} {.(OSuc lx _)} {.(Φ lx)} Φ< () |
| 80 trio<> {n} {.(Suc _)} {.(ℵ _)} {.(Φ (Suc _))} ℵΦ< () | |
| 81 trio<> {n} {.(Suc _)} {.(ℵ _)} {.(OSuc (Suc _) (Φ (Suc _)))} (ℵ< ¬ℵΦ) (ℵss< ()) | |
| 82 trio<> {n} {.(Suc _)} {.(ℵ _)} {.(OSuc (Suc _) (OSuc (Suc _) _))} (ℵ< (¬ℵs x)) (ℵss< x<y) = trio<> (ℵ< x) x<y | |
| 83 trio<> {n} {.(Suc _)} {.(OSuc (Suc _) (Φ (Suc _)))} {.(ℵ _)} (ℵss< ()) (ℵ< ¬ℵΦ) | |
| 84 trio<> {n} {.(Suc _)} {.(OSuc (Suc _) (OSuc (Suc _) _))} {.(ℵ _)} (ℵss< y<x) (ℵ< (¬ℵs x)) = trio<> y<x (ℵ< x) | |
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85 |
| 24 | 86 trio<≡ : {n : Level} → {lx : Nat} {x : OrdinalD {n} lx } { y : OrdinalD lx } → x ≡ y → x d< y → ⊥ |
| 17 | 87 trio<≡ refl = ≡→¬d< |
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88 |
| 24 | 89 trio>≡ : {n : Level} → {lx : Nat} {x : OrdinalD {n} lx } { y : OrdinalD lx } → x ≡ y → y d< x → ⊥ |
| 17 | 90 trio>≡ refl = ≡→¬d< |
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91 |
| 24 | 92 triO : {n : Level} → {lx ly : Nat} → OrdinalD {n} lx → OrdinalD {n} ly → Tri (lx < ly) ( lx ≡ ly ) ( lx > ly ) |
| 93 triO {n} {lx} {ly} x y = <-cmp lx ly | |
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94 |
| 35 | 95 fin : {n : Level} → {lx : Nat} → {y : OrdinalD {n} (Suc lx) } → y d< (ℵ lx) → ¬ℵ y |
| 96 fin {_} {_} {Φ (Suc _)} ℵΦ< = ¬ℵΦ | |
| 97 fin {_} {_} {OSuc (Suc _) _} (ℵ< x) = ¬ℵs x | |
| 98 | |
| 24 | 99 triOrdd : {n : Level} → {lx : Nat} → Trichotomous _≡_ ( _d<_ {n} {lx} {lx} ) |
| 100 triOrdd {_} {lv} (Φ lv) (Φ lv) = tri≈ ≡→¬d< refl ≡→¬d< | |
| 101 triOrdd {_} {Suc lv} (ℵ lv) (ℵ lv) = tri≈ ≡→¬d< refl ≡→¬d< | |
| 102 triOrdd {_} {lv} (Φ lv) (OSuc lv y) = tri< Φ< (λ ()) ( λ lt → trio<> lt Φ< ) | |
| 41 | 103 triOrdd {_} {.(Suc lv)} (Φ (Suc lv)) (ℵ lv) = tri< ℵΦ< (λ ()) ( λ lt → trio<> lt ℵΦ<) |
| 104 triOrdd {_} {Suc lv} (ℵ lv) (Φ (Suc lv)) = tri> ( λ lt → trio<> lt ℵΦ< ) (λ ()) ℵΦ< | |
| 35 | 105 triOrdd {_} {Suc lv} (ℵ lv) (OSuc (Suc lv) y ) with triOrdd (ℵ lv) y |
| 106 triOrdd {_} {Suc lv} (ℵ lv) (OSuc .(Suc lv) y) | tri< a ¬b ¬c = tri< (ℵss< a) (λ ()) (trio<> (ℵss< a) ) | |
| 107 triOrdd {_} {Suc lv} (ℵ lv) (OSuc .(Suc lv) y) | tri≈ ¬a refl ¬c = tri< ℵs< (λ ()) ( λ lt → trio<> lt ℵs< ) | |
| 108 triOrdd {_} {Suc lv} (ℵ lv) (OSuc .(Suc lv) y) | tri> ¬a ¬b c = tri> ( λ lt → trio<> lt ( ℵ< (fin c)) ) (λ ()) ( ℵ< (fin c) ) | |
| 24 | 109 triOrdd {_} {lv} (OSuc lv x) (Φ lv) = tri> (λ lt → trio<> lt Φ<) (λ ()) Φ< |
| 35 | 110 triOrdd {_} {.(Suc lv)} (OSuc (Suc lv) x) (ℵ lv) with triOrdd x (ℵ lv) |
| 111 triOrdd {_} {.(Suc lv)} (OSuc (Suc lv) x) (ℵ lv) | tri< a ¬b ¬c = tri< (ℵ< (fin a ) ) (λ ()) ( λ lt → trio<> lt (ℵ< (fin a ))) | |
| 112 triOrdd {_} {.(Suc lv)} (OSuc (Suc lv) x) (ℵ lv) | tri≈ ¬a refl ¬c = tri> (λ lt → trio<> lt ℵs< ) (λ ()) ℵs< | |
| 113 triOrdd {_} {.(Suc lv)} (OSuc (Suc lv) x) (ℵ lv) | tri> ¬a ¬b c = tri> (λ lt → trio<> lt (ℵss< c )) (λ ()) ( ℵss< c ) | |
| 24 | 114 triOrdd {_} {lv} (OSuc lv x) (OSuc lv y) with triOrdd x y |
| 115 triOrdd {_} {lv} (OSuc lv x) (OSuc lv y) | tri< a ¬b ¬c = tri< (s< a) (λ tx=ty → trio<≡ tx=ty (s< a) ) ( λ lt → trio<> lt (s< a) ) | |
| 116 triOrdd {_} {lv} (OSuc lv x) (OSuc lv x) | tri≈ ¬a refl ¬c = tri≈ ≡→¬d< refl ≡→¬d< | |
| 117 triOrdd {_} {lv} (OSuc lv x) (OSuc lv y) | tri> ¬a ¬b c = tri> ( λ lt → trio<> lt (s< c) ) (λ tx=ty → trio>≡ tx=ty (s< c) ) (s< c) | |
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118 |
| 24 | 119 d<→lv : {n : Level} {x y : Ordinal {n}} → ord x d< ord y → lv x ≡ lv y |
| 17 | 120 d<→lv Φ< = refl |
| 121 d<→lv (s< lt) = refl | |
| 122 d<→lv ℵΦ< = refl | |
| 34 | 123 d<→lv (ℵ< _) = refl |
| 124 d<→lv ℵs< = refl | |
| 35 | 125 d<→lv (ℵss< _) = refl |
| 126 | |
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127 osuc : {n : Level} ( x : Ordinal {n} ) → Ordinal {n} |
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128 osuc record { lv = 0 ; ord = (Φ lv) } = record { lv = 0 ; ord = OSuc 0 (Φ lv) } |
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129 osuc record { lv = Suc lx ; ord = (Φ (Suc lv)) } = record { lv = Suc lx ; ord = ℵ lv } |
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130 osuc record { lv = 0 ; ord = (OSuc 0 ox ) } = record { lv = 0 ; ord = OSuc 0 (OSuc 0 ox) } |
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131 osuc record { lv = Suc lx ; ord = (OSuc (Suc lx) ox ) } = record { lv = Suc lx ; ord = OSuc (Suc lx) (OSuc (Suc lx) ox) } |
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132 osuc record { lv = Suc lx ; ord = ℵ lx } = record { lv = Suc lx ; ord = OSuc (Suc lx) (ℵ lx) } |
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133 |
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134 <-osuc : {n : Level} { x : Ordinal {n} } → x o< osuc x |
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135 <-osuc {n} { record { lv = 0 ; ord = Φ 0 } } = case2 Φ< |
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136 <-osuc {n} { record { lv = (Suc lv) ; ord = Φ (Suc lv) } } = case2 ℵΦ< |
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137 <-osuc {n} {record { lv = Zero ; ord = OSuc .0 ox }} = case2 ( s< s<refl ) |
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138 <-osuc {n} {record { lv = Suc lv₁ ; ord = OSuc .(Suc lv₁) ox }} = case2 ( s< s<refl ) |
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139 <-osuc {n} { record { lv = .(Suc lv₁) ; ord = (ℵ lv₁) } } = case2 ℵs< |
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140 |
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141 osuc-lveq : {n : Level} { x : Ordinal {n} } → lv x ≡ lv ( osuc x ) |
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142 osuc-lveq {n} {record { lv = 0 ; ord = Φ 0 }} = refl |
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143 osuc-lveq {n} {record { lv = Suc lv ; ord = Φ (Suc lv) }} = refl |
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144 osuc-lveq {n} {record { lv = Zero ; ord = OSuc .0 ord₁ }} = refl |
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145 osuc-lveq {n} {record { lv = Suc lv₁ ; ord = OSuc .(Suc lv₁) ord₁ }} = refl |
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146 osuc-lveq {n} {record { lv = .(Suc lv₁) ; ord = ℵ lv₁ }} = refl |
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147 |
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148 nat-<> : { x y : Nat } → x < y → y < x → ⊥ |
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149 nat-<> (s≤s x<y) (s≤s y<x) = nat-<> x<y y<x |
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150 |
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151 nat-<≡ : { x : Nat } → x < x → ⊥ |
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152 nat-<≡ (s≤s lt) = nat-<≡ lt |
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153 |
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154 osuc-< : {n : Level} { x y : Ordinal {n} } → y o< osuc x → x o< y → ⊥ |
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155 osuc-< {n} {record { lv = .0 ; ord = Φ .0 }} {record { lv = .(Suc _) ; ord = ord }} (case1 ()) (case1 (s≤s z≤n)) |
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156 osuc-< {n} {record { lv = .0 ; ord = OSuc .0 ord₁ }} {record { lv = .(Suc _) ; ord = ord }} (case1 ()) (case1 (s≤s z≤n)) |
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157 osuc-< {n} {record { lv = Suc lx ; ord = Φ .(Suc lx) }} {record { lv = ly ; ord = yo }} (case1 lt1) (case1 lt2) = nat-<> lt1 lt2 |
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158 osuc-< {n} {record { lv = Suc lx ; ord = OSuc .(Suc lx) xo }} {record { lv = ly ; ord = yo }} (case1 lt1) (case1 lt2) = nat-<> lt1 lt2 |
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159 osuc-< {n} {record { lv = .(Suc lv₁) ; ord = ℵ lv₁ }} {record { lv = ly ; ord = yo }} (case1 lt1) (case1 lt2) = nat-<> lt1 lt2 |
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160 osuc-< {n} {x} {y} (case1 x₁) (case2 x₂) with d<→lv x₂ | osuc-lveq {n} {x} |
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161 ... | refl | eq = {!!} |
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162 osuc-< {n} {x} {y} (case2 x₁) (case1 x₂) with d<→lv x₁ | osuc-lveq {n} {x} |
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163 ... | refl | eq = {!!} |
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164 osuc-< {n} {record { lv = Zero ; ord = Φ .0 }} {record { lv = Zero ; ord = Φ .0 }} (case2 Φ<) (case2 ()) |
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165 osuc-< {n} {record { lv = Suc lv₁ ; ord = Φ .(Suc lv₁) }} {record { lv = Zero ; ord = Φ .0 }} (case2 ()) (case2 x₂) |
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166 osuc-< {n} {record { lv = lv₁ ; ord = Φ .lv₁ }} {record { lv = Suc lv₂ ; ord = Φ .(Suc lv₂) }} (case2 x₁) (case2 ()) |
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167 osuc-< {n} {record { lv = .0 ; ord = Φ .0 }} {record { lv = Zero ; ord = OSuc .0 ord₁ }} (case2 (s< ())) (case2 Φ<) |
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168 osuc-< {n} {record { lv = .1 ; ord = ℵ Zero }} {record { lv = .1 ; ord = ℵ Zero }} (case2 ℵs<) (case2 ()) |
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169 osuc-< {n} {record { lv = .1 ; ord = ℵ Zero }} {record { lv = .1 ; ord = ℵ Zero }} (case2 (ℵss< x₁)) (case2 ()) |
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170 osuc-< {n} {record { lv = .1 ; ord = ℵ Zero }} {record { lv = .(Suc (Suc lv₂)) ; ord = ℵ Suc lv₂ }} (case2 ()) (case2 x₂) |
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171 osuc-< {n} {record { lv = .(Suc (Suc lv₁)) ; ord = ℵ Suc lv₁ }} {record { lv = .(Suc (Suc lv₁)) ; ord = ℵ .(Suc lv₁) }} (case2 ℵs<) (case2 ()) |
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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 ()) |
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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<) = {!!} |
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174 osuc-< {n} {record { lv = Zero ; ord = (OSuc 0 ox) }} {record { lv = .0 ; ord = (OSuc 0 oy) }} (case2 (s< c1)) (case2 (s< c2)) = |
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175 osuc-< {n} {record { lv = Zero ; ord = ox }} {record { lv = 0 ; ord = oy }} (case2 {!!}) (case2 c2) |
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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)) = {!!} |
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177 osuc-< {n} {record { lv = .(Suc _) ; ord = .(OSuc (Suc _) _) }} {record { lv = .(Suc _) ; ord = .(ℵ _) }} (case2 (ℵss< c1)) (case2 (ℵ< x)) = {!!} |
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178 osuc-< {n} {record { lv = .(Suc _) ; ord = .(ℵ _) }} {record { lv = .(Suc _) ; ord = .(OSuc (Suc _) (ℵ _)) }} (case2 (s< c1)) (case2 ℵs<) = {!!} |
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179 osuc-< {n} {record { lv = .(Suc _) ; ord = .(ℵ _) }} {record { lv = .(Suc _) ; ord = .(OSuc (Suc _) _) }} (case2 (s< c1)) (case2 (ℵss< c2)) = {!!} |
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180 |
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181 |
| 35 | 182 xsyℵ : {n : Level} {lx : Nat} {x y : OrdinalD {n} lx } → x d< y → ¬ℵ y → ¬ℵ x |
| 183 xsyℵ {_} {_} {Φ lv₁} {y} x<y t = ¬ℵΦ | |
| 184 xsyℵ {_} {_} {OSuc lv₁ x} {OSuc lv₁ y} (s< x<y) (¬ℵs t) = ¬ℵs ( xsyℵ x<y t) | |
| 185 xsyℵ {_} {_} {OSuc .(Suc _) x} {.(ℵ _)} (ℵ< x₁) () | |
| 186 xsyℵ {_} {_} {ℵ lv₁} {.(OSuc (Suc lv₁) (ℵ lv₁))} ℵs< (¬ℵs t) = t | |
| 187 xsyℵ (ℵss< ()) (¬ℵs ¬ℵΦ) | |
| 188 xsyℵ (ℵss< x<y) (¬ℵs t) = xsyℵ x<y t | |
| 16 | 189 |
| 24 | 190 orddtrans : {n : Level} {lx : Nat} {x y z : OrdinalD {n} lx } → x d< y → y d< z → x d< z |
| 191 orddtrans {_} {lx} {.(Φ lx)} {.(OSuc lx _)} {.(OSuc lx _)} Φ< (s< y<z) = Φ< | |
| 41 | 192 orddtrans {_} {Suc lx} {Φ (Suc lx)} {OSuc (Suc lx) y} {ℵ lx} Φ< (ℵ< _) = ℵΦ< |
| 24 | 193 orddtrans {_} {lx} {.(OSuc lx _)} {.(OSuc lx _)} {.(OSuc lx _)} (s< x<y) (s< y<z) = s< ( orddtrans x<y y<z ) |
| 34 | 194 orddtrans {_} {Suc lx} {.(OSuc (Suc lx) _)} {.(OSuc (Suc lx) (Φ (Suc lx)))} {.(ℵ lx)} (s< ()) (ℵ< ¬ℵΦ) |
| 195 orddtrans ℵs< (ℵ< ()) | |
| 35 | 196 orddtrans {n} {Suc lx} {OSuc (Suc lx) x} {OSuc (Suc ly) y} {ℵ _} (s< x<y) (ℵ< t) = ℵ< ( xsyℵ x<y t ) |
| 197 orddtrans {n} {.(Suc _)} {.(Φ (Suc _))} {.(ℵ _)} {.(OSuc (Suc _) (ℵ _))} ℵΦ< ℵs< = Φ< | |
| 41 | 198 orddtrans {n} {.(Suc _)} {OSuc (Suc _) .(Φ (Suc _))} {.(ℵ _)} {OSuc (Suc _) (ℵ k)} (ℵ< ¬ℵΦ) ℵs< = s< ℵΦ< |
| 35 | 199 orddtrans {n} {.(Suc _)} {OSuc (Suc lv) (OSuc (Suc _) x)} {ℵ lv} {.(OSuc (Suc _) (ℵ _))} (ℵ< (¬ℵs t)) ℵs< = s< ( ℵ< t ) |
| 200 orddtrans {n} {.(Suc lv)} {ℵ lv} {OSuc .(Suc lv) (ℵ lv)} {OSuc .(Suc lv) .(OSuc (Suc lv) (ℵ lv))} ℵs< (s< ℵs<) = ℵss< ℵs< | |
| 201 orddtrans ℵΦ< (ℵss< y<z) = Φ< | |
| 41 | 202 orddtrans (ℵ< {lx} {Φ .(Suc lx)} nxx) (ℵss< {_} {k} y<z) = s< (orddtrans ℵΦ< y<z) |
| 35 | 203 orddtrans (ℵ< {lx} {OSuc .(Suc lx) xx} (¬ℵs nxx)) (ℵss< y<z) = s< (orddtrans (ℵ< nxx) y<z) |
| 204 orddtrans (ℵ< {.lv₁} {ℵ lv₁} ()) (ℵss< y<z) | |
| 205 orddtrans (ℵss< x<y) (s< y<z) = ℵss< ( orddtrans x<y y<z ) | |
| 206 orddtrans (ℵss< ()) (ℵ< ¬ℵΦ) | |
| 207 orddtrans (ℵss< ℵs<) (ℵ< (¬ℵs ())) | |
| 208 orddtrans (ℵss< (ℵss< x<y)) (ℵ< (¬ℵs x)) = orddtrans (ℵss< x<y) ( ℵ< x ) | |
| 209 orddtrans {n} {Suc lx} {x} {y} {z} ℵs< (s< (ℵss< {lx} {ss} y<z)) = ℵss< ( ℵss< y<z ) | |
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210 |
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211 max : (x y : Nat) → Nat |
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212 max Zero Zero = Zero |
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213 max Zero (Suc x) = (Suc x) |
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214 max (Suc x) Zero = (Suc x) |
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215 max (Suc x) (Suc y) = Suc ( max x y ) |
| 3 | 216 |
| 24 | 217 maxαd : {n : Level} → { lx : Nat } → OrdinalD {n} lx → OrdinalD lx → OrdinalD lx |
| 17 | 218 maxαd x y with triOrdd x y |
| 219 maxαd x y | tri< a ¬b ¬c = y | |
| 220 maxαd x y | tri≈ ¬a b ¬c = x | |
| 221 maxαd x y | tri> ¬a ¬b c = x | |
| 6 | 222 |
| 24 | 223 maxα : {n : Level} → Ordinal {n} → Ordinal → Ordinal |
| 17 | 224 maxα x y with <-cmp (lv x) (lv y) |
| 225 maxα x y | tri< a ¬b ¬c = x | |
| 226 maxα x y | tri> ¬a ¬b c = y | |
| 227 maxα x y | tri≈ ¬a refl ¬c = record { lv = lv x ; ord = maxαd (ord x) (ord y) } | |
| 7 | 228 |
| 24 | 229 _o≤_ : {n : Level} → Ordinal → Ordinal → Set (suc n) |
| 23 | 230 a o≤ b = (a ≡ b) ∨ ( a o< b ) |
| 231 | |
| 27 | 232 ordtrans : {n : Level} {x y z : Ordinal {n} } → x o< y → y o< z → x o< z |
| 233 ordtrans {n} {x} {y} {z} (case1 x₁) (case1 x₂) = case1 ( <-trans x₁ x₂ ) | |
| 234 ordtrans {n} {x} {y} {z} (case1 x₁) (case2 x₂) with d<→lv x₂ | |
| 235 ... | refl = case1 x₁ | |
| 236 ordtrans {n} {x} {y} {z} (case2 x₁) (case1 x₂) with d<→lv x₁ | |
| 237 ... | refl = case1 x₂ | |
| 238 ordtrans {n} {x} {y} {z} (case2 x₁) (case2 x₂) with d<→lv x₁ | d<→lv x₂ | |
| 239 ... | refl | refl = case2 ( orddtrans x₁ x₂ ) | |
| 240 | |
| 241 | |
| 24 | 242 trio< : {n : Level } → Trichotomous {suc n} _≡_ _o<_ |
| 23 | 243 trio< a b with <-cmp (lv a) (lv b) |
| 24 | 244 trio< a b | tri< a₁ ¬b ¬c = tri< (case1 a₁) (λ refl → ¬b (cong ( λ x → lv x ) refl ) ) lemma1 where |
| 245 lemma1 : ¬ (Suc (lv b) ≤ lv a) ∨ (ord b d< ord a) | |
| 246 lemma1 (case1 x) = ¬c x | |
| 247 lemma1 (case2 x) with d<→lv x | |
| 248 lemma1 (case2 x) | refl = ¬b refl | |
| 249 trio< a b | tri> ¬a ¬b c = tri> lemma1 (λ refl → ¬b (cong ( λ x → lv x ) refl ) ) (case1 c) where | |
| 250 lemma1 : ¬ (Suc (lv a) ≤ lv b) ∨ (ord a d< ord b) | |
| 251 lemma1 (case1 x) = ¬a x | |
| 252 lemma1 (case2 x) with d<→lv x | |
| 253 lemma1 (case2 x) | refl = ¬b refl | |
| 23 | 254 trio< a b | tri≈ ¬a refl ¬c with triOrdd ( ord a ) ( ord b ) |
| 24 | 255 trio< record { lv = .(lv b) ; ord = x } b | tri≈ ¬a refl ¬c | tri< a ¬b ¬c₁ = tri< (case2 a) (λ refl → ¬b (lemma1 refl )) lemma2 where |
| 256 lemma1 : (record { lv = _ ; ord = x }) ≡ b → x ≡ ord b | |
| 257 lemma1 refl = refl | |
| 258 lemma2 : ¬ (Suc (lv b) ≤ lv b) ∨ (ord b d< x) | |
| 259 lemma2 (case1 x) = ¬a x | |
| 260 lemma2 (case2 x) = trio<> x a | |
| 261 trio< record { lv = .(lv b) ; ord = x } b | tri≈ ¬a refl ¬c | tri> ¬a₁ ¬b c = tri> lemma2 (λ refl → ¬b (lemma1 refl )) (case2 c) where | |
| 262 lemma1 : (record { lv = _ ; ord = x }) ≡ b → x ≡ ord b | |
| 263 lemma1 refl = refl | |
| 264 lemma2 : ¬ (Suc (lv b) ≤ lv b) ∨ (x d< ord b) | |
| 265 lemma2 (case1 x) = ¬a x | |
| 266 lemma2 (case2 x) = trio<> x c | |
| 267 trio< record { lv = .(lv b) ; ord = x } b | tri≈ ¬a refl ¬c | tri≈ ¬a₁ refl ¬c₁ = tri≈ lemma1 refl lemma1 where | |
| 268 lemma1 : ¬ (Suc (lv b) ≤ lv b) ∨ (ord b d< ord b) | |
| 269 lemma1 (case1 x) = ¬a x | |
| 270 lemma1 (case2 x) = ≡→¬d< x | |
| 23 | 271 |
| 24 | 272 OrdTrans : {n : Level} → Transitive {suc n} _o≤_ |
| 16 | 273 OrdTrans (case1 refl) (case1 refl) = case1 refl |
| 274 OrdTrans (case1 refl) (case2 lt2) = case2 lt2 | |
| 275 OrdTrans (case2 lt1) (case1 refl) = case2 lt1 | |
| 17 | 276 OrdTrans (case2 (case1 x)) (case2 (case1 y)) = case2 (case1 ( <-trans x y ) ) |
| 277 OrdTrans (case2 (case1 x)) (case2 (case2 y)) with d<→lv y | |
| 278 OrdTrans (case2 (case1 x)) (case2 (case2 y)) | refl = case2 (case1 x ) | |
| 279 OrdTrans (case2 (case2 x)) (case2 (case1 y)) with d<→lv x | |
| 280 OrdTrans (case2 (case2 x)) (case2 (case1 y)) | refl = case2 (case1 y) | |
| 281 OrdTrans (case2 (case2 x)) (case2 (case2 y)) with d<→lv x | d<→lv y | |
| 282 OrdTrans (case2 (case2 x)) (case2 (case2 y)) | refl | refl = case2 (case2 (orddtrans x y )) | |
| 16 | 283 |
| 24 | 284 OrdPreorder : {n : Level} → Preorder (suc n) (suc n) (suc n) |
| 285 OrdPreorder {n} = record { Carrier = Ordinal | |
| 16 | 286 ; _≈_ = _≡_ |
| 23 | 287 ; _∼_ = _o≤_ |
| 16 | 288 ; isPreorder = record { |
| 289 isEquivalence = record { refl = refl ; sym = sym ; trans = trans } | |
| 290 ; reflexive = case1 | |
| 24 | 291 ; trans = OrdTrans |
| 16 | 292 } |
| 293 } | |
| 294 | |
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295 TransFinite : {n : Level} → { ψ : Ordinal {n} → Set n } |
| 22 | 296 → ( ∀ (lx : Nat ) → ψ ( record { lv = Suc lx ; ord = ℵ lx } )) |
| 24 | 297 → ( ∀ (lx : Nat ) → ψ ( record { lv = lx ; ord = Φ lx } ) ) |
| 298 → ( ∀ (lx : Nat ) → (x : OrdinalD lx ) → ψ ( record { lv = lx ; ord = x } ) → ψ ( record { lv = lx ; ord = OSuc lx x } ) ) | |
| 22 | 299 → ∀ (x : Ordinal) → ψ x |
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300 TransFinite caseℵ caseΦ caseOSuc record { lv = lv ; ord = Φ lv } = caseΦ lv |
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301 TransFinite caseℵ caseΦ caseOSuc record { lv = lv ; ord = OSuc lv ord₁ } = caseOSuc lv ord₁ |
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302 ( TransFinite caseℵ caseΦ caseOSuc (record { lv = lv ; ord = ord₁ } )) |
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303 TransFinite caseℵ caseΦ caseOSuc record { lv = Suc lv₁ ; ord = ℵ lv₁ } = caseℵ lv₁ |
| 22 | 304 |