Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate ordinal.agda @ 51:83b13f1f4f42
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Mon, 27 May 2019 15:00:45 +0900 |
| parents | e584686a1307 |
| children | cd9cf8b09610 |
| rev | line source |
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| 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 | |
| 24 | 25 data _d<_ {n : Level} : {lx ly : Nat} → OrdinalD {n} lx → OrdinalD {n} ly → Set n where |
| 26 Φ< : {lx : Nat} → {x : OrdinalD {n} lx} → Φ lx d< OSuc lx x | |
| 27 s< : {lx : Nat} → {x y : OrdinalD {n} lx} → x d< y → OSuc lx x d< OSuc lx y | |
| 41 | 28 ℵΦ< : {lx : Nat} → Φ (Suc lx) d< (ℵ lx) |
| 34 | 29 ℵ< : {lx : Nat} → {x : OrdinalD {n} (Suc lx) } → ¬ℵ x → OSuc (Suc lx) x d< (ℵ lx) |
| 30 ℵs< : {lx : Nat} → (ℵ lx) d< OSuc (Suc lx) (ℵ lx) | |
| 35 | 31 ℵss< : {lx : Nat} → {x : OrdinalD {n} (Suc lx) } → (ℵ lx) d< x → (ℵ lx) d< OSuc (Suc lx) x |
| 17 | 32 |
| 33 open Ordinal | |
| 34 | |
| 27 | 35 _o<_ : {n : Level} ( x y : Ordinal ) → Set n |
| 17 | 36 _o<_ x y = (lv x < lv y ) ∨ ( ord x d< ord y ) |
| 3 | 37 |
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equalitu and internal parametorisity
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38 o<-subst : {n : Level } {Z X z x : Ordinal {n}} → Z o< X → Z ≡ z → X ≡ x → z o< x |
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39 o<-subst df refl refl = df |
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40 |
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41 open import Data.Nat.Properties |
| 6 | 42 open import Data.Empty |
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problem on Ordinal ( OSuc ℵ )
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43 open import Data.Unit using ( ⊤ ) |
| 6 | 44 open import Relation.Nullary |
| 45 | |
| 46 open import Relation.Binary | |
| 47 open import Relation.Binary.Core | |
| 48 | |
| 24 | 49 o∅ : {n : Level} → Ordinal {n} |
| 50 o∅ = record { lv = Zero ; ord = Φ Zero } | |
| 21 | 51 |
| 34 | 52 s<refl : {n : Level } {lx : Nat } { x : OrdinalD {n} lx } → x d< OSuc lx x |
| 53 s<refl {n} {lv} {Φ lv} = Φ< | |
| 54 s<refl {n} {lv} {OSuc lv x} = s< s<refl | |
| 55 s<refl {n} {Suc lv} {ℵ lv} = ℵs< | |
| 56 | |
| 39 | 57 open import Relation.Binary.HeterogeneousEquality using (_≅_;refl) |
| 58 | |
| 59 ordinal-cong : {n : Level} {x y : Ordinal {n}} → | |
| 60 lv x ≡ lv y → ord x ≅ ord y → x ≡ y | |
| 61 ordinal-cong refl refl = refl | |
| 21 | 62 |
| 46 | 63 ordinal-lv : {n : Level} {x y : Ordinal {n}} → x ≡ y → lv x ≡ lv y |
| 64 ordinal-lv refl = refl | |
| 65 | |
| 66 ordinal-d : {n : Level} {x y : Ordinal {n}} → x ≡ y → ord x ≅ ord y | |
| 67 ordinal-d refl = refl | |
| 68 | |
| 24 | 69 ≡→¬d< : {n : Level} → {lv : Nat} → {x : OrdinalD {n} lv } → x d< x → ⊥ |
| 70 ≡→¬d< {n} {lx} {OSuc lx y} (s< t) = ≡→¬d< t | |
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71 |
| 24 | 72 trio<> : {n : Level} → {lx : Nat} {x : OrdinalD {n} lx } { y : OrdinalD lx } → y d< x → x d< y → ⊥ |
| 34 | 73 trio<> {n} {lx} {.(OSuc lx _)} {.(OSuc lx _)} (s< s) (s< t) = trio<> s t |
| 74 trio<> {_} {.(Suc _)} {.(OSuc (Suc _) (ℵ _))} {.(ℵ _)} ℵs< (ℵ< {_} {.(ℵ _)} ()) | |
| 75 trio<> {_} {.(Suc _)} {.(ℵ _)} {.(OSuc (Suc _) (ℵ _))} (ℵ< ()) ℵs< | |
| 35 | 76 trio<> {n} {lx} {.(OSuc lx _)} {.(Φ lx)} Φ< () |
| 77 trio<> {n} {.(Suc _)} {.(ℵ _)} {.(Φ (Suc _))} ℵΦ< () | |
| 78 trio<> {n} {.(Suc _)} {.(ℵ _)} {.(OSuc (Suc _) (Φ (Suc _)))} (ℵ< ¬ℵΦ) (ℵss< ()) | |
| 79 trio<> {n} {.(Suc _)} {.(ℵ _)} {.(OSuc (Suc _) (OSuc (Suc _) _))} (ℵ< (¬ℵs x)) (ℵss< x<y) = trio<> (ℵ< x) x<y | |
| 80 trio<> {n} {.(Suc _)} {.(OSuc (Suc _) (Φ (Suc _)))} {.(ℵ _)} (ℵss< ()) (ℵ< ¬ℵΦ) | |
| 81 trio<> {n} {.(Suc _)} {.(OSuc (Suc _) (OSuc (Suc _) _))} {.(ℵ _)} (ℵss< y<x) (ℵ< (¬ℵs x)) = trio<> y<x (ℵ< x) | |
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82 |
| 24 | 83 trio<≡ : {n : Level} → {lx : Nat} {x : OrdinalD {n} lx } { y : OrdinalD lx } → x ≡ y → x d< y → ⊥ |
| 17 | 84 trio<≡ refl = ≡→¬d< |
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85 |
| 24 | 86 trio>≡ : {n : Level} → {lx : Nat} {x : OrdinalD {n} lx } { y : OrdinalD lx } → x ≡ y → y d< x → ⊥ |
| 17 | 87 trio>≡ refl = ≡→¬d< |
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88 |
| 24 | 89 triO : {n : Level} → {lx ly : Nat} → OrdinalD {n} lx → OrdinalD {n} ly → Tri (lx < ly) ( lx ≡ ly ) ( lx > ly ) |
| 90 triO {n} {lx} {ly} x y = <-cmp lx ly | |
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91 |
| 35 | 92 fin : {n : Level} → {lx : Nat} → {y : OrdinalD {n} (Suc lx) } → y d< (ℵ lx) → ¬ℵ y |
| 93 fin {_} {_} {Φ (Suc _)} ℵΦ< = ¬ℵΦ | |
| 94 fin {_} {_} {OSuc (Suc _) _} (ℵ< x) = ¬ℵs x | |
| 95 | |
| 24 | 96 triOrdd : {n : Level} → {lx : Nat} → Trichotomous _≡_ ( _d<_ {n} {lx} {lx} ) |
| 97 triOrdd {_} {lv} (Φ lv) (Φ lv) = tri≈ ≡→¬d< refl ≡→¬d< | |
| 98 triOrdd {_} {Suc lv} (ℵ lv) (ℵ lv) = tri≈ ≡→¬d< refl ≡→¬d< | |
| 99 triOrdd {_} {lv} (Φ lv) (OSuc lv y) = tri< Φ< (λ ()) ( λ lt → trio<> lt Φ< ) | |
| 41 | 100 triOrdd {_} {.(Suc lv)} (Φ (Suc lv)) (ℵ lv) = tri< ℵΦ< (λ ()) ( λ lt → trio<> lt ℵΦ<) |
| 101 triOrdd {_} {Suc lv} (ℵ lv) (Φ (Suc lv)) = tri> ( λ lt → trio<> lt ℵΦ< ) (λ ()) ℵΦ< | |
| 35 | 102 triOrdd {_} {Suc lv} (ℵ lv) (OSuc (Suc lv) y ) with triOrdd (ℵ lv) y |
| 103 triOrdd {_} {Suc lv} (ℵ lv) (OSuc .(Suc lv) y) | tri< a ¬b ¬c = tri< (ℵss< a) (λ ()) (trio<> (ℵss< a) ) | |
| 104 triOrdd {_} {Suc lv} (ℵ lv) (OSuc .(Suc lv) y) | tri≈ ¬a refl ¬c = tri< ℵs< (λ ()) ( λ lt → trio<> lt ℵs< ) | |
| 105 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 | 106 triOrdd {_} {lv} (OSuc lv x) (Φ lv) = tri> (λ lt → trio<> lt Φ<) (λ ()) Φ< |
| 35 | 107 triOrdd {_} {.(Suc lv)} (OSuc (Suc lv) x) (ℵ lv) with triOrdd x (ℵ lv) |
| 108 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 ))) | |
| 109 triOrdd {_} {.(Suc lv)} (OSuc (Suc lv) x) (ℵ lv) | tri≈ ¬a refl ¬c = tri> (λ lt → trio<> lt ℵs< ) (λ ()) ℵs< | |
| 110 triOrdd {_} {.(Suc lv)} (OSuc (Suc lv) x) (ℵ lv) | tri> ¬a ¬b c = tri> (λ lt → trio<> lt (ℵss< c )) (λ ()) ( ℵss< c ) | |
| 24 | 111 triOrdd {_} {lv} (OSuc lv x) (OSuc lv y) with triOrdd x y |
| 112 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) ) | |
| 113 triOrdd {_} {lv} (OSuc lv x) (OSuc lv x) | tri≈ ¬a refl ¬c = tri≈ ≡→¬d< refl ≡→¬d< | |
| 114 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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115 |
| 24 | 116 d<→lv : {n : Level} {x y : Ordinal {n}} → ord x d< ord y → lv x ≡ lv y |
| 17 | 117 d<→lv Φ< = refl |
| 118 d<→lv (s< lt) = refl | |
| 119 d<→lv ℵΦ< = refl | |
| 34 | 120 d<→lv (ℵ< _) = refl |
| 121 d<→lv ℵs< = refl | |
| 35 | 122 d<→lv (ℵss< _) = refl |
| 123 | |
| 124 xsyℵ : {n : Level} {lx : Nat} {x y : OrdinalD {n} lx } → x d< y → ¬ℵ y → ¬ℵ x | |
| 125 xsyℵ {_} {_} {Φ lv₁} {y} x<y t = ¬ℵΦ | |
| 126 xsyℵ {_} {_} {OSuc lv₁ x} {OSuc lv₁ y} (s< x<y) (¬ℵs t) = ¬ℵs ( xsyℵ x<y t) | |
| 127 xsyℵ {_} {_} {OSuc .(Suc _) x} {.(ℵ _)} (ℵ< x₁) () | |
| 128 xsyℵ {_} {_} {ℵ lv₁} {.(OSuc (Suc lv₁) (ℵ lv₁))} ℵs< (¬ℵs t) = t | |
| 129 xsyℵ (ℵss< ()) (¬ℵs ¬ℵΦ) | |
| 130 xsyℵ (ℵss< x<y) (¬ℵs t) = xsyℵ x<y t | |
| 16 | 131 |
| 24 | 132 orddtrans : {n : Level} {lx : Nat} {x y z : OrdinalD {n} lx } → x d< y → y d< z → x d< z |
| 133 orddtrans {_} {lx} {.(Φ lx)} {.(OSuc lx _)} {.(OSuc lx _)} Φ< (s< y<z) = Φ< | |
| 41 | 134 orddtrans {_} {Suc lx} {Φ (Suc lx)} {OSuc (Suc lx) y} {ℵ lx} Φ< (ℵ< _) = ℵΦ< |
| 24 | 135 orddtrans {_} {lx} {.(OSuc lx _)} {.(OSuc lx _)} {.(OSuc lx _)} (s< x<y) (s< y<z) = s< ( orddtrans x<y y<z ) |
| 34 | 136 orddtrans {_} {Suc lx} {.(OSuc (Suc lx) _)} {.(OSuc (Suc lx) (Φ (Suc lx)))} {.(ℵ lx)} (s< ()) (ℵ< ¬ℵΦ) |
| 137 orddtrans ℵs< (ℵ< ()) | |
| 35 | 138 orddtrans {n} {Suc lx} {OSuc (Suc lx) x} {OSuc (Suc ly) y} {ℵ _} (s< x<y) (ℵ< t) = ℵ< ( xsyℵ x<y t ) |
| 139 orddtrans {n} {.(Suc _)} {.(Φ (Suc _))} {.(ℵ _)} {.(OSuc (Suc _) (ℵ _))} ℵΦ< ℵs< = Φ< | |
| 41 | 140 orddtrans {n} {.(Suc _)} {OSuc (Suc _) .(Φ (Suc _))} {.(ℵ _)} {OSuc (Suc _) (ℵ k)} (ℵ< ¬ℵΦ) ℵs< = s< ℵΦ< |
| 35 | 141 orddtrans {n} {.(Suc _)} {OSuc (Suc lv) (OSuc (Suc _) x)} {ℵ lv} {.(OSuc (Suc _) (ℵ _))} (ℵ< (¬ℵs t)) ℵs< = s< ( ℵ< t ) |
| 142 orddtrans {n} {.(Suc lv)} {ℵ lv} {OSuc .(Suc lv) (ℵ lv)} {OSuc .(Suc lv) .(OSuc (Suc lv) (ℵ lv))} ℵs< (s< ℵs<) = ℵss< ℵs< | |
| 143 orddtrans ℵΦ< (ℵss< y<z) = Φ< | |
| 41 | 144 orddtrans (ℵ< {lx} {Φ .(Suc lx)} nxx) (ℵss< {_} {k} y<z) = s< (orddtrans ℵΦ< y<z) |
| 35 | 145 orddtrans (ℵ< {lx} {OSuc .(Suc lx) xx} (¬ℵs nxx)) (ℵss< y<z) = s< (orddtrans (ℵ< nxx) y<z) |
| 146 orddtrans (ℵ< {.lv₁} {ℵ lv₁} ()) (ℵss< y<z) | |
| 147 orddtrans (ℵss< x<y) (s< y<z) = ℵss< ( orddtrans x<y y<z ) | |
| 148 orddtrans (ℵss< ()) (ℵ< ¬ℵΦ) | |
| 149 orddtrans (ℵss< ℵs<) (ℵ< (¬ℵs ())) | |
| 150 orddtrans (ℵss< (ℵss< x<y)) (ℵ< (¬ℵs x)) = orddtrans (ℵss< x<y) ( ℵ< x ) | |
| 151 orddtrans {n} {Suc lx} {x} {y} {z} ℵs< (s< (ℵss< {lx} {ss} y<z)) = ℵss< ( ℵss< y<z ) | |
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152 |
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153 max : (x y : Nat) → Nat |
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154 max Zero Zero = Zero |
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155 max Zero (Suc x) = (Suc x) |
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156 max (Suc x) Zero = (Suc x) |
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157 max (Suc x) (Suc y) = Suc ( max x y ) |
| 3 | 158 |
| 24 | 159 maxαd : {n : Level} → { lx : Nat } → OrdinalD {n} lx → OrdinalD lx → OrdinalD lx |
| 17 | 160 maxαd x y with triOrdd x y |
| 161 maxαd x y | tri< a ¬b ¬c = y | |
| 162 maxαd x y | tri≈ ¬a b ¬c = x | |
| 163 maxαd x y | tri> ¬a ¬b c = x | |
| 6 | 164 |
| 24 | 165 maxα : {n : Level} → Ordinal {n} → Ordinal → Ordinal |
| 17 | 166 maxα x y with <-cmp (lv x) (lv y) |
| 167 maxα x y | tri< a ¬b ¬c = x | |
| 168 maxα x y | tri> ¬a ¬b c = y | |
| 169 maxα x y | tri≈ ¬a refl ¬c = record { lv = lv x ; ord = maxαd (ord x) (ord y) } | |
| 7 | 170 |
| 24 | 171 _o≤_ : {n : Level} → Ordinal → Ordinal → Set (suc n) |
| 23 | 172 a o≤ b = (a ≡ b) ∨ ( a o< b ) |
| 173 | |
| 27 | 174 ordtrans : {n : Level} {x y z : Ordinal {n} } → x o< y → y o< z → x o< z |
| 175 ordtrans {n} {x} {y} {z} (case1 x₁) (case1 x₂) = case1 ( <-trans x₁ x₂ ) | |
| 176 ordtrans {n} {x} {y} {z} (case1 x₁) (case2 x₂) with d<→lv x₂ | |
| 177 ... | refl = case1 x₁ | |
| 178 ordtrans {n} {x} {y} {z} (case2 x₁) (case1 x₂) with d<→lv x₁ | |
| 179 ... | refl = case1 x₂ | |
| 180 ordtrans {n} {x} {y} {z} (case2 x₁) (case2 x₂) with d<→lv x₁ | d<→lv x₂ | |
| 181 ... | refl | refl = case2 ( orddtrans x₁ x₂ ) | |
| 182 | |
| 183 | |
| 24 | 184 trio< : {n : Level } → Trichotomous {suc n} _≡_ _o<_ |
| 23 | 185 trio< a b with <-cmp (lv a) (lv b) |
| 24 | 186 trio< a b | tri< a₁ ¬b ¬c = tri< (case1 a₁) (λ refl → ¬b (cong ( λ x → lv x ) refl ) ) lemma1 where |
| 187 lemma1 : ¬ (Suc (lv b) ≤ lv a) ∨ (ord b d< ord a) | |
| 188 lemma1 (case1 x) = ¬c x | |
| 189 lemma1 (case2 x) with d<→lv x | |
| 190 lemma1 (case2 x) | refl = ¬b refl | |
| 191 trio< a b | tri> ¬a ¬b c = tri> lemma1 (λ refl → ¬b (cong ( λ x → lv x ) refl ) ) (case1 c) where | |
| 192 lemma1 : ¬ (Suc (lv a) ≤ lv b) ∨ (ord a d< ord b) | |
| 193 lemma1 (case1 x) = ¬a x | |
| 194 lemma1 (case2 x) with d<→lv x | |
| 195 lemma1 (case2 x) | refl = ¬b refl | |
| 23 | 196 trio< a b | tri≈ ¬a refl ¬c with triOrdd ( ord a ) ( ord b ) |
| 24 | 197 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 |
| 198 lemma1 : (record { lv = _ ; ord = x }) ≡ b → x ≡ ord b | |
| 199 lemma1 refl = refl | |
| 200 lemma2 : ¬ (Suc (lv b) ≤ lv b) ∨ (ord b d< x) | |
| 201 lemma2 (case1 x) = ¬a x | |
| 202 lemma2 (case2 x) = trio<> x a | |
| 203 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 | |
| 204 lemma1 : (record { lv = _ ; ord = x }) ≡ b → x ≡ ord b | |
| 205 lemma1 refl = refl | |
| 206 lemma2 : ¬ (Suc (lv b) ≤ lv b) ∨ (x d< ord b) | |
| 207 lemma2 (case1 x) = ¬a x | |
| 208 lemma2 (case2 x) = trio<> x c | |
| 209 trio< record { lv = .(lv b) ; ord = x } b | tri≈ ¬a refl ¬c | tri≈ ¬a₁ refl ¬c₁ = tri≈ lemma1 refl lemma1 where | |
| 210 lemma1 : ¬ (Suc (lv b) ≤ lv b) ∨ (ord b d< ord b) | |
| 211 lemma1 (case1 x) = ¬a x | |
| 212 lemma1 (case2 x) = ≡→¬d< x | |
| 23 | 213 |
| 24 | 214 OrdTrans : {n : Level} → Transitive {suc n} _o≤_ |
| 16 | 215 OrdTrans (case1 refl) (case1 refl) = case1 refl |
| 216 OrdTrans (case1 refl) (case2 lt2) = case2 lt2 | |
| 217 OrdTrans (case2 lt1) (case1 refl) = case2 lt1 | |
| 17 | 218 OrdTrans (case2 (case1 x)) (case2 (case1 y)) = case2 (case1 ( <-trans x y ) ) |
| 219 OrdTrans (case2 (case1 x)) (case2 (case2 y)) with d<→lv y | |
| 220 OrdTrans (case2 (case1 x)) (case2 (case2 y)) | refl = case2 (case1 x ) | |
| 221 OrdTrans (case2 (case2 x)) (case2 (case1 y)) with d<→lv x | |
| 222 OrdTrans (case2 (case2 x)) (case2 (case1 y)) | refl = case2 (case1 y) | |
| 223 OrdTrans (case2 (case2 x)) (case2 (case2 y)) with d<→lv x | d<→lv y | |
| 224 OrdTrans (case2 (case2 x)) (case2 (case2 y)) | refl | refl = case2 (case2 (orddtrans x y )) | |
| 16 | 225 |
| 24 | 226 OrdPreorder : {n : Level} → Preorder (suc n) (suc n) (suc n) |
| 227 OrdPreorder {n} = record { Carrier = Ordinal | |
| 16 | 228 ; _≈_ = _≡_ |
| 23 | 229 ; _∼_ = _o≤_ |
| 16 | 230 ; isPreorder = record { |
| 231 isEquivalence = record { refl = refl ; sym = sym ; trans = trans } | |
| 232 ; reflexive = case1 | |
| 24 | 233 ; trans = OrdTrans |
| 16 | 234 } |
| 235 } | |
| 236 | |
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problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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237 TransFinite : {n : Level} → { ψ : Ordinal {n} → Set n } |
| 22 | 238 → ( ∀ (lx : Nat ) → ψ ( record { lv = Suc lx ; ord = ℵ lx } )) |
| 24 | 239 → ( ∀ (lx : Nat ) → ψ ( record { lv = lx ; ord = Φ lx } ) ) |
| 240 → ( ∀ (lx : Nat ) → (x : OrdinalD lx ) → ψ ( record { lv = lx ; ord = x } ) → ψ ( record { lv = lx ; ord = OSuc lx x } ) ) | |
| 22 | 241 → ∀ (x : Ordinal) → ψ x |
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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242 TransFinite caseℵ caseΦ caseOSuc record { lv = lv ; ord = Φ lv } = caseΦ lv |
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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243 TransFinite caseℵ caseΦ caseOSuc record { lv = lv ; ord = OSuc lv ord₁ } = caseOSuc lv ord₁ |
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problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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244 ( TransFinite caseℵ caseΦ caseOSuc (record { lv = lv ; ord = ord₁ } )) |
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problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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245 TransFinite caseℵ caseΦ caseOSuc record { lv = Suc lv₁ ; ord = ℵ lv₁ } = caseℵ lv₁ |
| 22 | 246 |