Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate ordinal.agda @ 75:714470702a8b
Union done
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sun, 02 Jun 2019 10:53:52 +0900 |
| parents | 819da8c08f05 |
| children | 96c932d0145d |
| rev | line source |
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| 34 | 1 {-# OPTIONS --allow-unsolved-metas #-} |
| 16 | 2 open import Level |
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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⊔_ ) |
| 75 | 8 open import Data.Empty |
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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 |
| 75 | 41 s<refl : {n : Level } {lx : Nat } { x : OrdinalD {n} lx } → x d< OSuc lx x |
| 42 s<refl {n} {lv} {Φ lv} = Φ< | |
| 43 s<refl {n} {lv} {OSuc lv x} = s< s<refl | |
| 44 s<refl {n} {Suc lv} {ℵ lv} = ℵs< | |
| 45 | |
| 46 trio<> : {n : Level} → {lx : Nat} {x : OrdinalD {n} lx } { y : OrdinalD lx } → y d< x → x d< y → ⊥ | |
| 47 trio<> {n} {lx} {.(OSuc lx _)} {.(OSuc lx _)} (s< s) (s< t) = trio<> s t | |
| 48 trio<> {_} {.(Suc _)} {.(OSuc (Suc _) (ℵ _))} {.(ℵ _)} ℵs< (ℵ< {_} {.(ℵ _)} ()) | |
| 49 trio<> {_} {.(Suc _)} {.(ℵ _)} {.(OSuc (Suc _) (ℵ _))} (ℵ< ()) ℵs< | |
| 50 trio<> {n} {lx} {.(OSuc lx _)} {.(Φ lx)} Φ< () | |
| 51 trio<> {n} {.(Suc _)} {.(ℵ _)} {.(Φ (Suc _))} ℵΦ< () | |
| 52 trio<> {n} {.(Suc _)} {.(ℵ _)} {.(OSuc (Suc _) (Φ (Suc _)))} (ℵ< ¬ℵΦ) (ℵss< ()) | |
| 53 trio<> {n} {.(Suc _)} {.(ℵ _)} {.(OSuc (Suc _) (OSuc (Suc _) _))} (ℵ< (¬ℵs x)) (ℵss< x<y) = trio<> (ℵ< x) x<y | |
| 54 trio<> {n} {.(Suc _)} {.(OSuc (Suc _) (Φ (Suc _)))} {.(ℵ _)} (ℵss< ()) (ℵ< ¬ℵΦ) | |
| 55 trio<> {n} {.(Suc _)} {.(OSuc (Suc _) (OSuc (Suc _) _))} {.(ℵ _)} (ℵss< y<x) (ℵ< (¬ℵs x)) = trio<> y<x (ℵ< x) | |
| 56 | |
| 57 d<→lv : {n : Level} {x y : Ordinal {n}} → ord x d< ord y → lv x ≡ lv y | |
| 58 d<→lv Φ< = refl | |
| 59 d<→lv (s< lt) = refl | |
| 60 d<→lv ℵΦ< = refl | |
| 61 d<→lv (ℵ< _) = refl | |
| 62 d<→lv ℵs< = refl | |
| 63 d<→lv (ℵss< _) = refl | |
| 64 | |
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65 o<-subst : {n : Level } {Z X z x : Ordinal {n}} → Z o< X → Z ≡ z → X ≡ x → z o< x |
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66 o<-subst df refl refl = df |
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67 |
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68 open import Data.Nat.Properties |
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problem on Ordinal ( OSuc ℵ )
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69 open import Data.Unit using ( ⊤ ) |
| 6 | 70 open import Relation.Nullary |
| 71 | |
| 72 open import Relation.Binary | |
| 73 open import Relation.Binary.Core | |
| 74 | |
| 24 | 75 o∅ : {n : Level} → Ordinal {n} |
| 76 o∅ = record { lv = Zero ; ord = Φ Zero } | |
| 21 | 77 |
| 39 | 78 open import Relation.Binary.HeterogeneousEquality using (_≅_;refl) |
| 79 | |
| 80 ordinal-cong : {n : Level} {x y : Ordinal {n}} → | |
| 81 lv x ≡ lv y → ord x ≅ ord y → x ≡ y | |
| 82 ordinal-cong refl refl = refl | |
| 21 | 83 |
| 46 | 84 ordinal-lv : {n : Level} {x y : Ordinal {n}} → x ≡ y → lv x ≡ lv y |
| 85 ordinal-lv refl = refl | |
| 86 | |
| 87 ordinal-d : {n : Level} {x y : Ordinal {n}} → x ≡ y → ord x ≅ ord y | |
| 88 ordinal-d refl = refl | |
| 89 | |
| 24 | 90 ≡→¬d< : {n : Level} → {lv : Nat} → {x : OrdinalD {n} lv } → x d< x → ⊥ |
| 91 ≡→¬d< {n} {lx} {OSuc lx y} (s< t) = ≡→¬d< t | |
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92 |
| 24 | 93 trio<≡ : {n : Level} → {lx : Nat} {x : OrdinalD {n} lx } { y : OrdinalD lx } → x ≡ y → x d< y → ⊥ |
| 17 | 94 trio<≡ refl = ≡→¬d< |
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95 |
| 24 | 96 trio>≡ : {n : Level} → {lx : Nat} {x : OrdinalD {n} lx } { y : OrdinalD lx } → x ≡ y → y d< x → ⊥ |
| 17 | 97 trio>≡ refl = ≡→¬d< |
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98 |
| 24 | 99 triO : {n : Level} → {lx ly : Nat} → OrdinalD {n} lx → OrdinalD {n} ly → Tri (lx < ly) ( lx ≡ ly ) ( lx > ly ) |
| 100 triO {n} {lx} {ly} x y = <-cmp lx ly | |
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101 |
| 35 | 102 fin : {n : Level} → {lx : Nat} → {y : OrdinalD {n} (Suc lx) } → y d< (ℵ lx) → ¬ℵ y |
| 103 fin {_} {_} {Φ (Suc _)} ℵΦ< = ¬ℵΦ | |
| 104 fin {_} {_} {OSuc (Suc _) _} (ℵ< x) = ¬ℵs x | |
| 105 | |
| 24 | 106 triOrdd : {n : Level} → {lx : Nat} → Trichotomous _≡_ ( _d<_ {n} {lx} {lx} ) |
| 107 triOrdd {_} {lv} (Φ lv) (Φ lv) = tri≈ ≡→¬d< refl ≡→¬d< | |
| 108 triOrdd {_} {Suc lv} (ℵ lv) (ℵ lv) = tri≈ ≡→¬d< refl ≡→¬d< | |
| 109 triOrdd {_} {lv} (Φ lv) (OSuc lv y) = tri< Φ< (λ ()) ( λ lt → trio<> lt Φ< ) | |
| 41 | 110 triOrdd {_} {.(Suc lv)} (Φ (Suc lv)) (ℵ lv) = tri< ℵΦ< (λ ()) ( λ lt → trio<> lt ℵΦ<) |
| 111 triOrdd {_} {Suc lv} (ℵ lv) (Φ (Suc lv)) = tri> ( λ lt → trio<> lt ℵΦ< ) (λ ()) ℵΦ< | |
| 35 | 112 triOrdd {_} {Suc lv} (ℵ lv) (OSuc (Suc lv) y ) with triOrdd (ℵ lv) y |
| 113 triOrdd {_} {Suc lv} (ℵ lv) (OSuc .(Suc lv) y) | tri< a ¬b ¬c = tri< (ℵss< a) (λ ()) (trio<> (ℵss< a) ) | |
| 114 triOrdd {_} {Suc lv} (ℵ lv) (OSuc .(Suc lv) y) | tri≈ ¬a refl ¬c = tri< ℵs< (λ ()) ( λ lt → trio<> lt ℵs< ) | |
| 115 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 | 116 triOrdd {_} {lv} (OSuc lv x) (Φ lv) = tri> (λ lt → trio<> lt Φ<) (λ ()) Φ< |
| 35 | 117 triOrdd {_} {.(Suc lv)} (OSuc (Suc lv) x) (ℵ lv) with triOrdd x (ℵ lv) |
| 118 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 ))) | |
| 119 triOrdd {_} {.(Suc lv)} (OSuc (Suc lv) x) (ℵ lv) | tri≈ ¬a refl ¬c = tri> (λ lt → trio<> lt ℵs< ) (λ ()) ℵs< | |
| 120 triOrdd {_} {.(Suc lv)} (OSuc (Suc lv) x) (ℵ lv) | tri> ¬a ¬b c = tri> (λ lt → trio<> lt (ℵss< c )) (λ ()) ( ℵss< c ) | |
| 24 | 121 triOrdd {_} {lv} (OSuc lv x) (OSuc lv y) with triOrdd x y |
| 122 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) ) | |
| 123 triOrdd {_} {lv} (OSuc lv x) (OSuc lv x) | tri≈ ¬a refl ¬c = tri≈ ≡→¬d< refl ≡→¬d< | |
| 124 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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126 osuc : {n : Level} ( x : Ordinal {n} ) → Ordinal {n} |
| 75 | 127 osuc record { lv = lx ; ord = ox } = record { lv = lx ; ord = OSuc lx ox } |
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128 |
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129 <-osuc : {n : Level} { x : Ordinal {n} } → x o< osuc x |
| 75 | 130 <-osuc {n} {record { lv = lx ; ord = Φ .lx }} = case2 Φ< |
| 131 <-osuc {n} {record { lv = lx ; ord = OSuc .lx ox }} = case2 ( s< s<refl ) | |
| 132 <-osuc {n} {record { lv = (Suc lx) ; ord = ℵ lx }} = case2 ℵs< | |
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133 |
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134 osuc-lveq : {n : Level} { x : Ordinal {n} } → lv x ≡ lv ( osuc x ) |
| 75 | 135 osuc-lveq {n} = refl |
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136 |
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137 nat-<> : { x y : Nat } → x < y → y < x → ⊥ |
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138 nat-<> (s≤s x<y) (s≤s y<x) = nat-<> x<y y<x |
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139 |
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140 nat-<≡ : { x : Nat } → x < x → ⊥ |
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141 nat-<≡ (s≤s lt) = nat-<≡ lt |
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142 |
| 75 | 143 ¬a≤a : {la : Nat} → Suc la ≤ la → ⊥ |
| 144 ¬a≤a (s≤s lt) = ¬a≤a lt | |
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145 |
| 35 | 146 xsyℵ : {n : Level} {lx : Nat} {x y : OrdinalD {n} lx } → x d< y → ¬ℵ y → ¬ℵ x |
| 147 xsyℵ {_} {_} {Φ lv₁} {y} x<y t = ¬ℵΦ | |
| 148 xsyℵ {_} {_} {OSuc lv₁ x} {OSuc lv₁ y} (s< x<y) (¬ℵs t) = ¬ℵs ( xsyℵ x<y t) | |
| 149 xsyℵ {_} {_} {OSuc .(Suc _) x} {.(ℵ _)} (ℵ< x₁) () | |
| 150 xsyℵ {_} {_} {ℵ lv₁} {.(OSuc (Suc lv₁) (ℵ lv₁))} ℵs< (¬ℵs t) = t | |
| 151 xsyℵ (ℵss< ()) (¬ℵs ¬ℵΦ) | |
| 152 xsyℵ (ℵss< x<y) (¬ℵs t) = xsyℵ x<y t | |
| 16 | 153 |
| 24 | 154 orddtrans : {n : Level} {lx : Nat} {x y z : OrdinalD {n} lx } → x d< y → y d< z → x d< z |
| 155 orddtrans {_} {lx} {.(Φ lx)} {.(OSuc lx _)} {.(OSuc lx _)} Φ< (s< y<z) = Φ< | |
| 41 | 156 orddtrans {_} {Suc lx} {Φ (Suc lx)} {OSuc (Suc lx) y} {ℵ lx} Φ< (ℵ< _) = ℵΦ< |
| 24 | 157 orddtrans {_} {lx} {.(OSuc lx _)} {.(OSuc lx _)} {.(OSuc lx _)} (s< x<y) (s< y<z) = s< ( orddtrans x<y y<z ) |
| 34 | 158 orddtrans {_} {Suc lx} {.(OSuc (Suc lx) _)} {.(OSuc (Suc lx) (Φ (Suc lx)))} {.(ℵ lx)} (s< ()) (ℵ< ¬ℵΦ) |
| 159 orddtrans ℵs< (ℵ< ()) | |
| 35 | 160 orddtrans {n} {Suc lx} {OSuc (Suc lx) x} {OSuc (Suc ly) y} {ℵ _} (s< x<y) (ℵ< t) = ℵ< ( xsyℵ x<y t ) |
| 161 orddtrans {n} {.(Suc _)} {.(Φ (Suc _))} {.(ℵ _)} {.(OSuc (Suc _) (ℵ _))} ℵΦ< ℵs< = Φ< | |
| 41 | 162 orddtrans {n} {.(Suc _)} {OSuc (Suc _) .(Φ (Suc _))} {.(ℵ _)} {OSuc (Suc _) (ℵ k)} (ℵ< ¬ℵΦ) ℵs< = s< ℵΦ< |
| 35 | 163 orddtrans {n} {.(Suc _)} {OSuc (Suc lv) (OSuc (Suc _) x)} {ℵ lv} {.(OSuc (Suc _) (ℵ _))} (ℵ< (¬ℵs t)) ℵs< = s< ( ℵ< t ) |
| 164 orddtrans {n} {.(Suc lv)} {ℵ lv} {OSuc .(Suc lv) (ℵ lv)} {OSuc .(Suc lv) .(OSuc (Suc lv) (ℵ lv))} ℵs< (s< ℵs<) = ℵss< ℵs< | |
| 165 orddtrans ℵΦ< (ℵss< y<z) = Φ< | |
| 41 | 166 orddtrans (ℵ< {lx} {Φ .(Suc lx)} nxx) (ℵss< {_} {k} y<z) = s< (orddtrans ℵΦ< y<z) |
| 35 | 167 orddtrans (ℵ< {lx} {OSuc .(Suc lx) xx} (¬ℵs nxx)) (ℵss< y<z) = s< (orddtrans (ℵ< nxx) y<z) |
| 168 orddtrans (ℵ< {.lv₁} {ℵ lv₁} ()) (ℵss< y<z) | |
| 169 orddtrans (ℵss< x<y) (s< y<z) = ℵss< ( orddtrans x<y y<z ) | |
| 170 orddtrans (ℵss< ()) (ℵ< ¬ℵΦ) | |
| 171 orddtrans (ℵss< ℵs<) (ℵ< (¬ℵs ())) | |
| 172 orddtrans (ℵss< (ℵss< x<y)) (ℵ< (¬ℵs x)) = orddtrans (ℵss< x<y) ( ℵ< x ) | |
| 173 orddtrans {n} {Suc lx} {x} {y} {z} ℵs< (s< (ℵss< {lx} {ss} y<z)) = ℵss< ( ℵss< y<z ) | |
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174 |
| 75 | 175 osuc-≡< : {n : Level} { a x : Ordinal {n} } → x o< osuc a → (x ≡ a ) ∨ (x o< a) |
| 176 osuc-≡< {n} {a} {x} (case1 lt) = case2 (case1 lt) | |
| 177 osuc-≡< {n} {record { lv = lv₁ ; ord = Φ .lv₁ }} {record { lv = .lv₁ ; ord = .(Φ lv₁) }} (case2 Φ<) = case1 refl | |
| 178 osuc-≡< {n} {record { lv = lv₁ ; ord = OSuc .lv₁ ord₁ }} {record { lv = .lv₁ ; ord = .(Φ lv₁) }} (case2 Φ<) = case2 (case2 Φ<) | |
| 179 osuc-≡< {n} {record { lv = .(Suc lv₁) ; ord = ℵ lv₁ }} {record { lv = .(Suc lv₁) ; ord = .(Φ (Suc lv₁)) }} (case2 Φ<) = case2 (case2 ℵΦ<) | |
| 180 osuc-≡< {n} {record { lv = lv₁ ; ord = Φ .lv₁ }} {record { lv = .lv₁ ; ord = .(OSuc lv₁ _) }} (case2 (s< ())) | |
| 181 osuc-≡< {n} {record { lv = la ; ord = OSuc la oa }} {record { lv = la ; ord = (OSuc la ox) }} (case2 (s< lt)) with | |
| 182 osuc-≡< {n} {record { lv = la ; ord = oa }} {record { lv = la ; ord = ox }} (case2 lt ) | |
| 183 ... | case1 refl = case1 refl | |
| 184 ... | case2 (case2 x) = case2 (case2( s< x) ) | |
| 185 ... | case2 (case1 x) = ⊥-elim (¬a≤a x) where | |
| 186 osuc-≡< {n} {record { lv = .(Suc lv₁) ; ord = ℵ lv₁ }} {record { lv = .(Suc lv₁) ; ord = .(OSuc (Suc lv₁) (Φ (Suc lv₁))) }} (case2 (s< ℵΦ<)) = case2 (case2 (ℵ< ¬ℵΦ )) | |
| 187 osuc-≡< {n} {record { lv = (Suc lx) ; ord = ℵ lx }} {record { lv = (Suc lx) ; ord = (OSuc (Suc lx) (OSuc (Suc lx) ox)) }} (case2 (s< (ℵ< x))) with | |
| 188 osuc-≡< {n} {record { lv = (Suc lx) ; ord = ℵ lx }} {record { lv = (Suc lx) ; ord = (OSuc (Suc lx) ox) }} (case2 (lemma (ℵ< x)) ) where | |
| 189 lemma : OSuc (Suc lx) ox d< (ℵ lx) → OSuc (Suc lx) ox d< ord (osuc (record { lv = Suc lx ; ord = ℵ lx })) | |
| 190 lemma lt = orddtrans lt s<refl | |
| 191 ... | case1 () | |
| 192 ... | case2 ttt = case2 ( case2 (ℵ< (¬ℵs x) )) | |
| 193 osuc-≡< {n} {record { lv = .(Suc _) ; ord = .(ℵ _) }} {record { lv = .(Suc _) ; ord = .(ℵ _) }} (case2 ℵs<) = case1 refl | |
| 194 osuc-≡< {n} {record { lv = .(Suc _) ; ord = Φ .(Suc _) }} {record { lv = .(Suc _) ; ord = .(ℵ _) }} (case2 (ℵss< lt)) = case2 (case2 lt) | |
| 195 osuc-≡< {n} {record { lv = .(Suc _) ; ord = OSuc .(Suc _) ord₁ }} {record { lv = .(Suc _) ; ord = .(ℵ _) }} (case2 (ℵss< lt)) = case2 (case2 lt) | |
| 196 osuc-≡< {n} {record { lv = .(Suc lv₁) ; ord = ℵ lv₁ }} {record { lv = .(Suc lv₁) ; ord = .(ℵ lv₁) }} (case2 (ℵss< lt)) = case1 refl | |
| 197 | |
| 198 osuc-< : {n : Level} { x y : Ordinal {n} } → y o< osuc x → x o< y → ⊥ | |
| 199 osuc-< {n} {x} {y} y<ox x<y with osuc-≡< y<ox | |
| 200 osuc-< {n} {x} {x} y<ox (case1 x₁) | case1 refl = ⊥-elim (¬a≤a x₁) | |
| 201 osuc-< {n} {x} {x} (case1 x₂) (case2 x₁) | case1 refl = ⊥-elim (¬a≤a x₂) | |
| 202 osuc-< {n} {x} {x} (case2 x₂) (case2 x₁) | case1 refl = ≡→¬d< x₁ | |
| 203 osuc-< {n} {x} {y} y<ox (case1 x₂) | case2 (case1 x₁) = nat-<> x₁ x₂ | |
| 204 osuc-< {n} {x} {y} y<ox (case2 x₂) | case2 (case1 x₁) with d<→lv x₂ | |
| 205 ... | refl = ⊥-elim (¬a≤a x₁) | |
| 206 osuc-< {n} {x} {y} y<ox (case1 x₁) | case2 (case2 y<x) with d<→lv y<x | |
| 207 ... | refl = ⊥-elim (¬a≤a x₁) | |
| 208 osuc-< {n} {x} {y} y<ox (case2 x<y) | case2 (case2 y<x) with d<→lv y<x | d<→lv x<y | |
| 209 ... | refl | refl = trio<> y<x x<y | |
| 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 |