Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate ordinal.agda @ 30:3b0fdb95618e
problem on Ordinal ( OSuc ℵ )
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Tue, 21 May 2019 00:30:01 +0900 |
| parents | fce60b99dc55 |
| children | 97ff3f7933fe 2b853472cb24 |
| rev | line source |
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| 16 | 1 open import Level |
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posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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2 module ordinal where |
| 3 | 3 |
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4 open import zf |
| 3 | 5 |
| 23 | 6 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) |
| 3 | 7 |
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8 open import Relation.Binary.PropositionalEquality |
| 3 | 9 |
| 24 | 10 data OrdinalD {n : Level} : (lv : Nat) → Set n where |
| 11 Φ : (lv : Nat) → OrdinalD lv | |
| 12 OSuc : (lv : Nat) → OrdinalD {n} lv → OrdinalD lv | |
| 17 | 13 ℵ_ : (lv : Nat) → OrdinalD (Suc lv) |
| 3 | 14 |
| 24 | 15 record Ordinal {n : Level} : Set n where |
| 16 | 16 field |
| 17 lv : Nat | |
| 24 | 18 ord : OrdinalD {n} lv |
| 16 | 19 |
| 24 | 20 data _d<_ {n : Level} : {lx ly : Nat} → OrdinalD {n} lx → OrdinalD {n} ly → Set n where |
| 21 Φ< : {lx : Nat} → {x : OrdinalD {n} lx} → Φ lx d< OSuc lx x | |
| 22 s< : {lx : Nat} → {x y : OrdinalD {n} lx} → x d< y → OSuc lx x d< OSuc lx y | |
| 23 ℵΦ< : {lx : Nat} → {x : OrdinalD {n} (Suc lx) } → Φ (Suc lx) d< (ℵ lx) | |
| 24 ℵ< : {lx : Nat} → {x : OrdinalD {n} (Suc lx) } → OSuc (Suc lx) x d< (ℵ lx) | |
| 17 | 25 |
| 26 open Ordinal | |
| 27 | |
| 27 | 28 _o<_ : {n : Level} ( x y : Ordinal ) → Set n |
| 17 | 29 _o<_ x y = (lv x < lv y ) ∨ ( ord x d< ord y ) |
| 3 | 30 |
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31 open import Data.Nat.Properties |
| 6 | 32 open import Data.Empty |
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problem on Ordinal ( OSuc ℵ )
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33 open import Data.Unit using ( ⊤ ) |
| 6 | 34 open import Relation.Nullary |
| 35 | |
| 36 open import Relation.Binary | |
| 37 open import Relation.Binary.Core | |
| 38 | |
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problem on Ordinal ( OSuc ℵ )
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39 ¬ℵ : {n : Level} {lx : Nat } ( x : OrdinalD {n} lx ) → Set |
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problem on Ordinal ( OSuc ℵ )
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40 ¬ℵ (Φ lv₁) = ⊤ |
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problem on Ordinal ( OSuc ℵ )
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41 ¬ℵ (OSuc lv₁ x) = ¬ℵ x |
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problem on Ordinal ( OSuc ℵ )
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42 ¬ℵ (ℵ lv₁) = ⊥ |
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problem on Ordinal ( OSuc ℵ )
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43 |
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problem on Ordinal ( OSuc ℵ )
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44 s<refl : {n : Level } {lx : Nat } { x : OrdinalD {n} lx } → ¬ℵ x → x d< OSuc lx x |
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problem on Ordinal ( OSuc ℵ )
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45 s<refl {n} {.lv₁} {Φ lv₁} t = Φ< |
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problem on Ordinal ( OSuc ℵ )
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46 s<refl {n} {.lv₁} {OSuc lv₁ x} t = s< (s<refl t) |
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problem on Ordinal ( OSuc ℵ )
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47 s<refl {n} {.(Suc lv₁)} {ℵ lv₁} () |
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problem on Ordinal ( OSuc ℵ )
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48 |
| 24 | 49 o∅ : {n : Level} → Ordinal {n} |
| 50 o∅ = record { lv = Zero ; ord = Φ Zero } | |
| 21 | 51 |
| 52 | |
| 24 | 53 ≡→¬d< : {n : Level} → {lv : Nat} → {x : OrdinalD {n} lv } → x d< x → ⊥ |
| 54 ≡→¬d< {n} {lx} {OSuc lx y} (s< t) = ≡→¬d< t | |
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55 |
| 24 | 56 trio<> : {n : Level} → {lx : Nat} {x : OrdinalD {n} lx } { y : OrdinalD lx } → y d< x → x d< y → ⊥ |
| 57 trio<> {n} {lx} {.(OSuc lx _)} {.(OSuc lx _)} (s< s) (s< t) = | |
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58 trio<> s t |
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59 |
| 24 | 60 trio<≡ : {n : Level} → {lx : Nat} {x : OrdinalD {n} lx } { y : OrdinalD lx } → x ≡ y → x d< y → ⊥ |
| 17 | 61 trio<≡ refl = ≡→¬d< |
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62 |
| 24 | 63 trio>≡ : {n : Level} → {lx : Nat} {x : OrdinalD {n} lx } { y : OrdinalD lx } → x ≡ y → y d< x → ⊥ |
| 17 | 64 trio>≡ refl = ≡→¬d< |
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try to fix axiom of replacement
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65 |
| 24 | 66 triO : {n : Level} → {lx ly : Nat} → OrdinalD {n} lx → OrdinalD {n} ly → Tri (lx < ly) ( lx ≡ ly ) ( lx > ly ) |
| 67 triO {n} {lx} {ly} x y = <-cmp lx ly | |
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68 |
| 24 | 69 triOrdd : {n : Level} → {lx : Nat} → Trichotomous _≡_ ( _d<_ {n} {lx} {lx} ) |
| 70 triOrdd {_} {lv} (Φ lv) (Φ lv) = tri≈ ≡→¬d< refl ≡→¬d< | |
| 71 triOrdd {_} {Suc lv} (ℵ lv) (ℵ lv) = tri≈ ≡→¬d< refl ≡→¬d< | |
| 72 triOrdd {_} {lv} (Φ lv) (OSuc lv y) = tri< Φ< (λ ()) ( λ lt → trio<> lt Φ< ) | |
| 73 triOrdd {_} {.(Suc lv)} (Φ (Suc lv)) (ℵ lv) = tri< (ℵΦ< {_} {lv} {Φ (Suc lv)} ) (λ ()) ( λ lt → trio<> lt ((ℵΦ< {_} {lv} {Φ (Suc lv)} )) ) | |
| 74 triOrdd {_} {Suc lv} (ℵ lv) (Φ (Suc lv)) = tri> ( λ lt → trio<> lt (ℵΦ< {_} {lv} {Φ (Suc lv)} ) ) (λ ()) (ℵΦ< {_} {lv} {Φ (Suc lv)} ) | |
| 75 triOrdd {_} {Suc lv} (ℵ lv) (OSuc (Suc lv) y) = tri> ( λ lt → trio<> lt (ℵ< {_} {lv} {y} ) ) (λ ()) (ℵ< {_} {lv} {y} ) | |
| 76 triOrdd {_} {lv} (OSuc lv x) (Φ lv) = tri> (λ lt → trio<> lt Φ<) (λ ()) Φ< | |
| 77 triOrdd {_} {.(Suc lv)} (OSuc (Suc lv) x) (ℵ lv) = tri< ℵ< (λ ()) (λ lt → trio<> lt ℵ< ) | |
| 78 triOrdd {_} {lv} (OSuc lv x) (OSuc lv y) with triOrdd x y | |
| 79 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) ) | |
| 80 triOrdd {_} {lv} (OSuc lv x) (OSuc lv x) | tri≈ ¬a refl ¬c = tri≈ ≡→¬d< refl ≡→¬d< | |
| 81 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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82 |
| 24 | 83 d<→lv : {n : Level} {x y : Ordinal {n}} → ord x d< ord y → lv x ≡ lv y |
| 17 | 84 d<→lv Φ< = refl |
| 85 d<→lv (s< lt) = refl | |
| 86 d<→lv ℵΦ< = refl | |
| 87 d<→lv ℵ< = refl | |
| 16 | 88 |
| 24 | 89 orddtrans : {n : Level} {lx : Nat} {x y z : OrdinalD {n} lx } → x d< y → y d< z → x d< z |
| 90 orddtrans {_} {lx} {.(Φ lx)} {.(OSuc lx _)} {.(OSuc lx _)} Φ< (s< y<z) = Φ< | |
| 91 orddtrans {_} {Suc lx} {Φ (Suc lx)} {OSuc (Suc lx) y} {ℵ lx} Φ< ℵ< = ℵΦ< {_} {lx} {y} | |
| 92 orddtrans {_} {lx} {.(OSuc lx _)} {.(OSuc lx _)} {.(OSuc lx _)} (s< x<y) (s< y<z) = s< ( orddtrans x<y y<z ) | |
| 93 orddtrans {_} {Suc lx} {.(OSuc (Suc lx) _)} {.(OSuc (Suc lx) _)} {.(ℵ _)} (s< x<y) ℵ< = ℵ< | |
| 94 orddtrans {_} {Suc lx} {Φ (Suc lx)} {.(ℵ _)} {z} ℵΦ< () | |
| 95 orddtrans {_} {Suc lx} {OSuc (Suc lx) _} {.(ℵ _)} {z} ℵ< () | |
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96 |
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97 max : (x y : Nat) → Nat |
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98 max Zero Zero = Zero |
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99 max Zero (Suc x) = (Suc x) |
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100 max (Suc x) Zero = (Suc x) |
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101 max (Suc x) (Suc y) = Suc ( max x y ) |
| 3 | 102 |
| 24 | 103 maxαd : {n : Level} → { lx : Nat } → OrdinalD {n} lx → OrdinalD lx → OrdinalD lx |
| 17 | 104 maxαd x y with triOrdd x y |
| 105 maxαd x y | tri< a ¬b ¬c = y | |
| 106 maxαd x y | tri≈ ¬a b ¬c = x | |
| 107 maxαd x y | tri> ¬a ¬b c = x | |
| 6 | 108 |
| 24 | 109 maxα : {n : Level} → Ordinal {n} → Ordinal → Ordinal |
| 17 | 110 maxα x y with <-cmp (lv x) (lv y) |
| 111 maxα x y | tri< a ¬b ¬c = x | |
| 112 maxα x y | tri> ¬a ¬b c = y | |
| 113 maxα x y | tri≈ ¬a refl ¬c = record { lv = lv x ; ord = maxαd (ord x) (ord y) } | |
| 7 | 114 |
| 24 | 115 _o≤_ : {n : Level} → Ordinal → Ordinal → Set (suc n) |
| 23 | 116 a o≤ b = (a ≡ b) ∨ ( a o< b ) |
| 117 | |
| 27 | 118 ordtrans : {n : Level} {x y z : Ordinal {n} } → x o< y → y o< z → x o< z |
| 119 ordtrans {n} {x} {y} {z} (case1 x₁) (case1 x₂) = case1 ( <-trans x₁ x₂ ) | |
| 120 ordtrans {n} {x} {y} {z} (case1 x₁) (case2 x₂) with d<→lv x₂ | |
| 121 ... | refl = case1 x₁ | |
| 122 ordtrans {n} {x} {y} {z} (case2 x₁) (case1 x₂) with d<→lv x₁ | |
| 123 ... | refl = case1 x₂ | |
| 124 ordtrans {n} {x} {y} {z} (case2 x₁) (case2 x₂) with d<→lv x₁ | d<→lv x₂ | |
| 125 ... | refl | refl = case2 ( orddtrans x₁ x₂ ) | |
| 126 | |
| 127 | |
| 24 | 128 trio< : {n : Level } → Trichotomous {suc n} _≡_ _o<_ |
| 23 | 129 trio< a b with <-cmp (lv a) (lv b) |
| 24 | 130 trio< a b | tri< a₁ ¬b ¬c = tri< (case1 a₁) (λ refl → ¬b (cong ( λ x → lv x ) refl ) ) lemma1 where |
| 131 lemma1 : ¬ (Suc (lv b) ≤ lv a) ∨ (ord b d< ord a) | |
| 132 lemma1 (case1 x) = ¬c x | |
| 133 lemma1 (case2 x) with d<→lv x | |
| 134 lemma1 (case2 x) | refl = ¬b refl | |
| 135 trio< a b | tri> ¬a ¬b c = tri> lemma1 (λ refl → ¬b (cong ( λ x → lv x ) refl ) ) (case1 c) where | |
| 136 lemma1 : ¬ (Suc (lv a) ≤ lv b) ∨ (ord a d< ord b) | |
| 137 lemma1 (case1 x) = ¬a x | |
| 138 lemma1 (case2 x) with d<→lv x | |
| 139 lemma1 (case2 x) | refl = ¬b refl | |
| 23 | 140 trio< a b | tri≈ ¬a refl ¬c with triOrdd ( ord a ) ( ord b ) |
| 24 | 141 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 |
| 142 lemma1 : (record { lv = _ ; ord = x }) ≡ b → x ≡ ord b | |
| 143 lemma1 refl = refl | |
| 144 lemma2 : ¬ (Suc (lv b) ≤ lv b) ∨ (ord b d< x) | |
| 145 lemma2 (case1 x) = ¬a x | |
| 146 lemma2 (case2 x) = trio<> x a | |
| 147 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 | |
| 148 lemma1 : (record { lv = _ ; ord = x }) ≡ b → x ≡ ord b | |
| 149 lemma1 refl = refl | |
| 150 lemma2 : ¬ (Suc (lv b) ≤ lv b) ∨ (x d< ord b) | |
| 151 lemma2 (case1 x) = ¬a x | |
| 152 lemma2 (case2 x) = trio<> x c | |
| 153 trio< record { lv = .(lv b) ; ord = x } b | tri≈ ¬a refl ¬c | tri≈ ¬a₁ refl ¬c₁ = tri≈ lemma1 refl lemma1 where | |
| 154 lemma1 : ¬ (Suc (lv b) ≤ lv b) ∨ (ord b d< ord b) | |
| 155 lemma1 (case1 x) = ¬a x | |
| 156 lemma1 (case2 x) = ≡→¬d< x | |
| 23 | 157 |
| 24 | 158 OrdTrans : {n : Level} → Transitive {suc n} _o≤_ |
| 16 | 159 OrdTrans (case1 refl) (case1 refl) = case1 refl |
| 160 OrdTrans (case1 refl) (case2 lt2) = case2 lt2 | |
| 161 OrdTrans (case2 lt1) (case1 refl) = case2 lt1 | |
| 17 | 162 OrdTrans (case2 (case1 x)) (case2 (case1 y)) = case2 (case1 ( <-trans x y ) ) |
| 163 OrdTrans (case2 (case1 x)) (case2 (case2 y)) with d<→lv y | |
| 164 OrdTrans (case2 (case1 x)) (case2 (case2 y)) | refl = case2 (case1 x ) | |
| 165 OrdTrans (case2 (case2 x)) (case2 (case1 y)) with d<→lv x | |
| 166 OrdTrans (case2 (case2 x)) (case2 (case1 y)) | refl = case2 (case1 y) | |
| 167 OrdTrans (case2 (case2 x)) (case2 (case2 y)) with d<→lv x | d<→lv y | |
| 168 OrdTrans (case2 (case2 x)) (case2 (case2 y)) | refl | refl = case2 (case2 (orddtrans x y )) | |
| 16 | 169 |
| 24 | 170 OrdPreorder : {n : Level} → Preorder (suc n) (suc n) (suc n) |
| 171 OrdPreorder {n} = record { Carrier = Ordinal | |
| 16 | 172 ; _≈_ = _≡_ |
| 23 | 173 ; _∼_ = _o≤_ |
| 16 | 174 ; isPreorder = record { |
| 175 isEquivalence = record { refl = refl ; sym = sym ; trans = trans } | |
| 176 ; reflexive = case1 | |
| 24 | 177 ; trans = OrdTrans |
| 16 | 178 } |
| 179 } | |
| 180 | |
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181 TransFinite : {n : Level} → { ψ : Ordinal {n} → Set n } |
| 22 | 182 → ( ∀ (lx : Nat ) → ψ ( record { lv = Suc lx ; ord = ℵ lx } )) |
| 24 | 183 → ( ∀ (lx : Nat ) → ψ ( record { lv = lx ; ord = Φ lx } ) ) |
| 184 → ( ∀ (lx : Nat ) → (x : OrdinalD lx ) → ψ ( record { lv = lx ; ord = x } ) → ψ ( record { lv = lx ; ord = OSuc lx x } ) ) | |
| 22 | 185 → ∀ (x : Ordinal) → ψ x |
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problem on Ordinal ( OSuc ℵ )
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186 TransFinite caseℵ caseΦ caseOSuc record { lv = lv ; ord = Φ lv } = caseΦ lv |
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187 TransFinite caseℵ caseΦ caseOSuc record { lv = lv ; ord = OSuc lv ord₁ } = caseOSuc lv ord₁ |
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188 ( TransFinite caseℵ caseΦ caseOSuc (record { lv = lv ; ord = ord₁ } )) |
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189 TransFinite caseℵ caseΦ caseOSuc record { lv = Suc lv₁ ; ord = ℵ lv₁ } = caseℵ lv₁ |
| 22 | 190 |