Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate ordinal.agda @ 34:c9ad0d97ce41
fix oridinal
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Wed, 22 May 2019 11:52:49 +0900 |
| parents | 2b853472cb24 |
| children | 88b77cecaeba |
| 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 | |
| 28 ℵΦ< : {lx : Nat} → {x : OrdinalD {n} (Suc lx) } → 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) | |
| 17 | 31 |
| 32 open Ordinal | |
| 33 | |
| 27 | 34 _o<_ : {n : Level} ( x y : Ordinal ) → Set n |
| 17 | 35 _o<_ x y = (lv x < lv y ) ∨ ( ord x d< ord y ) |
| 3 | 36 |
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37 open import Data.Nat.Properties |
| 6 | 38 open import Data.Empty |
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problem on Ordinal ( OSuc ℵ )
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39 open import Data.Unit using ( ⊤ ) |
| 6 | 40 open import Relation.Nullary |
| 41 | |
| 42 open import Relation.Binary | |
| 43 open import Relation.Binary.Core | |
| 44 | |
| 24 | 45 o∅ : {n : Level} → Ordinal {n} |
| 46 o∅ = record { lv = Zero ; ord = Φ Zero } | |
| 21 | 47 |
| 34 | 48 s<refl : {n : Level } {lx : Nat } { x : OrdinalD {n} lx } → x d< OSuc lx x |
| 49 s<refl {n} {lv} {Φ lv} = Φ< | |
| 50 s<refl {n} {lv} {OSuc lv x} = s< s<refl | |
| 51 s<refl {n} {Suc lv} {ℵ lv} = ℵs< | |
| 52 | |
| 21 | 53 |
| 24 | 54 ≡→¬d< : {n : Level} → {lv : Nat} → {x : OrdinalD {n} lv } → x d< x → ⊥ |
| 55 ≡→¬d< {n} {lx} {OSuc lx y} (s< t) = ≡→¬d< t | |
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56 |
| 24 | 57 trio<> : {n : Level} → {lx : Nat} {x : OrdinalD {n} lx } { y : OrdinalD lx } → y d< x → x d< y → ⊥ |
| 34 | 58 trio<> {n} {lx} {.(OSuc lx _)} {.(OSuc lx _)} (s< s) (s< t) = trio<> s t |
| 59 trio<> {_} {.(Suc _)} {.(OSuc (Suc _) (ℵ _))} {.(ℵ _)} ℵs< (ℵ< {_} {.(ℵ _)} ()) | |
| 60 trio<> {_} {.(Suc _)} {.(ℵ _)} {.(OSuc (Suc _) (ℵ _))} (ℵ< ()) ℵs< | |
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61 |
| 24 | 62 trio<≡ : {n : Level} → {lx : Nat} {x : OrdinalD {n} lx } { y : OrdinalD lx } → x ≡ y → x d< y → ⊥ |
| 17 | 63 trio<≡ refl = ≡→¬d< |
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64 |
| 24 | 65 trio>≡ : {n : Level} → {lx : Nat} {x : OrdinalD {n} lx } { y : OrdinalD lx } → x ≡ y → y d< x → ⊥ |
| 17 | 66 trio>≡ refl = ≡→¬d< |
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67 |
| 24 | 68 triO : {n : Level} → {lx ly : Nat} → OrdinalD {n} lx → OrdinalD {n} ly → Tri (lx < ly) ( lx ≡ ly ) ( lx > ly ) |
| 69 triO {n} {lx} {ly} x y = <-cmp lx ly | |
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70 |
| 24 | 71 triOrdd : {n : Level} → {lx : Nat} → Trichotomous _≡_ ( _d<_ {n} {lx} {lx} ) |
| 72 triOrdd {_} {lv} (Φ lv) (Φ lv) = tri≈ ≡→¬d< refl ≡→¬d< | |
| 73 triOrdd {_} {Suc lv} (ℵ lv) (ℵ lv) = tri≈ ≡→¬d< refl ≡→¬d< | |
| 74 triOrdd {_} {lv} (Φ lv) (OSuc lv y) = tri< Φ< (λ ()) ( λ lt → trio<> lt Φ< ) | |
| 75 triOrdd {_} {.(Suc lv)} (Φ (Suc lv)) (ℵ lv) = tri< (ℵΦ< {_} {lv} {Φ (Suc lv)} ) (λ ()) ( λ lt → trio<> lt ((ℵΦ< {_} {lv} {Φ (Suc lv)} )) ) | |
| 76 triOrdd {_} {Suc lv} (ℵ lv) (Φ (Suc lv)) = tri> ( λ lt → trio<> lt (ℵΦ< {_} {lv} {Φ (Suc lv)} ) ) (λ ()) (ℵΦ< {_} {lv} {Φ (Suc lv)} ) | |
| 34 | 77 triOrdd {_} {Suc lv} (ℵ lv) (OSuc (Suc lv) y) = tri> ( λ lt → trio<> lt (ℵ< {_} {lv} {y} {!!}) ) (λ ()) (ℵ< {_} {lv} {y} {!!}) |
| 24 | 78 triOrdd {_} {lv} (OSuc lv x) (Φ lv) = tri> (λ lt → trio<> lt Φ<) (λ ()) Φ< |
| 34 | 79 triOrdd {_} {.(Suc lv)} (OSuc (Suc lv) x) (ℵ lv) = tri< (ℵ< {!!}) (λ ()) (λ lt → trio<> lt (ℵ< {!!}) ) |
| 24 | 80 triOrdd {_} {lv} (OSuc lv x) (OSuc lv y) with triOrdd x y |
| 81 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) ) | |
| 82 triOrdd {_} {lv} (OSuc lv x) (OSuc lv x) | tri≈ ¬a refl ¬c = tri≈ ≡→¬d< refl ≡→¬d< | |
| 83 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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84 |
| 24 | 85 d<→lv : {n : Level} {x y : Ordinal {n}} → ord x d< ord y → lv x ≡ lv y |
| 17 | 86 d<→lv Φ< = refl |
| 87 d<→lv (s< lt) = refl | |
| 88 d<→lv ℵΦ< = refl | |
| 34 | 89 d<→lv (ℵ< _) = refl |
| 90 d<→lv ℵs< = refl | |
| 16 | 91 |
| 24 | 92 orddtrans : {n : Level} {lx : Nat} {x y z : OrdinalD {n} lx } → x d< y → y d< z → x d< z |
| 93 orddtrans {_} {lx} {.(Φ lx)} {.(OSuc lx _)} {.(OSuc lx _)} Φ< (s< y<z) = Φ< | |
| 34 | 94 orddtrans {_} {Suc lx} {Φ (Suc lx)} {OSuc (Suc lx) y} {ℵ lx} Φ< (ℵ< _) = ℵΦ< {_} {lx} {y} |
| 24 | 95 orddtrans {_} {lx} {.(OSuc lx _)} {.(OSuc lx _)} {.(OSuc lx _)} (s< x<y) (s< y<z) = s< ( orddtrans x<y y<z ) |
| 34 | 96 orddtrans {_} {Suc lx} {.(OSuc (Suc lx) _)} {.(OSuc (Suc lx) (Φ (Suc lx)))} {.(ℵ lx)} (s< ()) (ℵ< ¬ℵΦ) |
| 97 orddtrans {_} {Suc lx} {OSuc (Suc lx) x} {OSuc (Suc lx) (OSuc (Suc lx) y)} {.(ℵ lx)} (s< x<y) (ℵ< (¬ℵs nℵ)) = ℵ< lemma where | |
| 98 lemma : ¬ℵ x | |
| 99 lemma = {!!} | |
| 100 orddtrans ℵΦ< ℵs< = {!!} | |
| 101 orddtrans (ℵ< ¬ℵΦ) ℵs< = {!!} | |
| 102 orddtrans (ℵ< (¬ℵs nℵ)) ℵs< = {!!} | |
| 103 orddtrans ℵs< (s< ℵs<) = {!!} | |
| 104 orddtrans ℵs< (ℵ< ()) | |
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105 |
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106 max : (x y : Nat) → Nat |
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107 max Zero Zero = Zero |
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108 max Zero (Suc x) = (Suc x) |
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109 max (Suc x) Zero = (Suc x) |
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110 max (Suc x) (Suc y) = Suc ( max x y ) |
| 3 | 111 |
| 24 | 112 maxαd : {n : Level} → { lx : Nat } → OrdinalD {n} lx → OrdinalD lx → OrdinalD lx |
| 17 | 113 maxαd x y with triOrdd x y |
| 114 maxαd x y | tri< a ¬b ¬c = y | |
| 115 maxαd x y | tri≈ ¬a b ¬c = x | |
| 116 maxαd x y | tri> ¬a ¬b c = x | |
| 6 | 117 |
| 24 | 118 maxα : {n : Level} → Ordinal {n} → Ordinal → Ordinal |
| 17 | 119 maxα x y with <-cmp (lv x) (lv y) |
| 120 maxα x y | tri< a ¬b ¬c = x | |
| 121 maxα x y | tri> ¬a ¬b c = y | |
| 122 maxα x y | tri≈ ¬a refl ¬c = record { lv = lv x ; ord = maxαd (ord x) (ord y) } | |
| 7 | 123 |
| 24 | 124 _o≤_ : {n : Level} → Ordinal → Ordinal → Set (suc n) |
| 23 | 125 a o≤ b = (a ≡ b) ∨ ( a o< b ) |
| 126 | |
| 27 | 127 ordtrans : {n : Level} {x y z : Ordinal {n} } → x o< y → y o< z → x o< z |
| 128 ordtrans {n} {x} {y} {z} (case1 x₁) (case1 x₂) = case1 ( <-trans x₁ x₂ ) | |
| 129 ordtrans {n} {x} {y} {z} (case1 x₁) (case2 x₂) with d<→lv x₂ | |
| 130 ... | refl = case1 x₁ | |
| 131 ordtrans {n} {x} {y} {z} (case2 x₁) (case1 x₂) with d<→lv x₁ | |
| 132 ... | refl = case1 x₂ | |
| 133 ordtrans {n} {x} {y} {z} (case2 x₁) (case2 x₂) with d<→lv x₁ | d<→lv x₂ | |
| 134 ... | refl | refl = case2 ( orddtrans x₁ x₂ ) | |
| 135 | |
| 136 | |
| 24 | 137 trio< : {n : Level } → Trichotomous {suc n} _≡_ _o<_ |
| 23 | 138 trio< a b with <-cmp (lv a) (lv b) |
| 24 | 139 trio< a b | tri< a₁ ¬b ¬c = tri< (case1 a₁) (λ refl → ¬b (cong ( λ x → lv x ) refl ) ) lemma1 where |
| 140 lemma1 : ¬ (Suc (lv b) ≤ lv a) ∨ (ord b d< ord a) | |
| 141 lemma1 (case1 x) = ¬c x | |
| 142 lemma1 (case2 x) with d<→lv x | |
| 143 lemma1 (case2 x) | refl = ¬b refl | |
| 144 trio< a b | tri> ¬a ¬b c = tri> lemma1 (λ refl → ¬b (cong ( λ x → lv x ) refl ) ) (case1 c) where | |
| 145 lemma1 : ¬ (Suc (lv a) ≤ lv b) ∨ (ord a d< ord b) | |
| 146 lemma1 (case1 x) = ¬a x | |
| 147 lemma1 (case2 x) with d<→lv x | |
| 148 lemma1 (case2 x) | refl = ¬b refl | |
| 23 | 149 trio< a b | tri≈ ¬a refl ¬c with triOrdd ( ord a ) ( ord b ) |
| 24 | 150 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 |
| 151 lemma1 : (record { lv = _ ; ord = x }) ≡ b → x ≡ ord b | |
| 152 lemma1 refl = refl | |
| 153 lemma2 : ¬ (Suc (lv b) ≤ lv b) ∨ (ord b d< x) | |
| 154 lemma2 (case1 x) = ¬a x | |
| 155 lemma2 (case2 x) = trio<> x a | |
| 156 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 | |
| 157 lemma1 : (record { lv = _ ; ord = x }) ≡ b → x ≡ ord b | |
| 158 lemma1 refl = refl | |
| 159 lemma2 : ¬ (Suc (lv b) ≤ lv b) ∨ (x d< ord b) | |
| 160 lemma2 (case1 x) = ¬a x | |
| 161 lemma2 (case2 x) = trio<> x c | |
| 162 trio< record { lv = .(lv b) ; ord = x } b | tri≈ ¬a refl ¬c | tri≈ ¬a₁ refl ¬c₁ = tri≈ lemma1 refl lemma1 where | |
| 163 lemma1 : ¬ (Suc (lv b) ≤ lv b) ∨ (ord b d< ord b) | |
| 164 lemma1 (case1 x) = ¬a x | |
| 165 lemma1 (case2 x) = ≡→¬d< x | |
| 23 | 166 |
| 24 | 167 OrdTrans : {n : Level} → Transitive {suc n} _o≤_ |
| 16 | 168 OrdTrans (case1 refl) (case1 refl) = case1 refl |
| 169 OrdTrans (case1 refl) (case2 lt2) = case2 lt2 | |
| 170 OrdTrans (case2 lt1) (case1 refl) = case2 lt1 | |
| 17 | 171 OrdTrans (case2 (case1 x)) (case2 (case1 y)) = case2 (case1 ( <-trans x y ) ) |
| 172 OrdTrans (case2 (case1 x)) (case2 (case2 y)) with d<→lv y | |
| 173 OrdTrans (case2 (case1 x)) (case2 (case2 y)) | refl = case2 (case1 x ) | |
| 174 OrdTrans (case2 (case2 x)) (case2 (case1 y)) with d<→lv x | |
| 175 OrdTrans (case2 (case2 x)) (case2 (case1 y)) | refl = case2 (case1 y) | |
| 176 OrdTrans (case2 (case2 x)) (case2 (case2 y)) with d<→lv x | d<→lv y | |
| 177 OrdTrans (case2 (case2 x)) (case2 (case2 y)) | refl | refl = case2 (case2 (orddtrans x y )) | |
| 16 | 178 |
| 24 | 179 OrdPreorder : {n : Level} → Preorder (suc n) (suc n) (suc n) |
| 180 OrdPreorder {n} = record { Carrier = Ordinal | |
| 16 | 181 ; _≈_ = _≡_ |
| 23 | 182 ; _∼_ = _o≤_ |
| 16 | 183 ; isPreorder = record { |
| 184 isEquivalence = record { refl = refl ; sym = sym ; trans = trans } | |
| 185 ; reflexive = case1 | |
| 24 | 186 ; trans = OrdTrans |
| 16 | 187 } |
| 188 } | |
| 189 | |
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190 TransFinite : {n : Level} → { ψ : Ordinal {n} → Set n } |
| 22 | 191 → ( ∀ (lx : Nat ) → ψ ( record { lv = Suc lx ; ord = ℵ lx } )) |
| 24 | 192 → ( ∀ (lx : Nat ) → ψ ( record { lv = lx ; ord = Φ lx } ) ) |
| 193 → ( ∀ (lx : Nat ) → (x : OrdinalD lx ) → ψ ( record { lv = lx ; ord = x } ) → ψ ( record { lv = lx ; ord = OSuc lx x } ) ) | |
| 22 | 194 → ∀ (x : Ordinal) → ψ x |
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195 TransFinite caseℵ caseΦ caseOSuc record { lv = lv ; ord = Φ lv } = caseΦ lv |
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196 TransFinite caseℵ caseΦ caseOSuc record { lv = lv ; ord = OSuc lv ord₁ } = caseOSuc lv ord₁ |
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197 ( TransFinite caseℵ caseΦ caseOSuc (record { lv = lv ; ord = ord₁ } )) |
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198 TransFinite caseℵ caseΦ caseOSuc record { lv = Suc lv₁ ; ord = ℵ lv₁ } = caseℵ lv₁ |
| 22 | 199 |