Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate ordinal.agda @ 81:96c932d0145d
simpler ordinal
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Tue, 04 Jun 2019 01:05:33 +0900 |
| parents | 714470702a8b |
| children | 4b2aff372b7c |
| 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⊔_ ) |
| 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 | |
| 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 |
| 70 | 20 -- |
| 21 -- Φ (Suc lv) < ℵ lv < OSuc (Suc lv) (ℵ lv) < OSuc ... < OSuc (Suc lv) (Φ (Suc lv)) < OSuc ... < ℵ (Suc lv) | |
| 22 -- | |
| 24 | 23 data _d<_ {n : Level} : {lx ly : Nat} → OrdinalD {n} lx → OrdinalD {n} ly → Set n where |
| 24 Φ< : {lx : Nat} → {x : OrdinalD {n} lx} → Φ lx d< OSuc lx x | |
| 25 s< : {lx : Nat} → {x y : OrdinalD {n} lx} → x d< y → OSuc lx x d< OSuc lx y | |
| 17 | 26 |
| 27 open Ordinal | |
| 28 | |
| 27 | 29 _o<_ : {n : Level} ( x y : Ordinal ) → Set n |
| 17 | 30 _o<_ x y = (lv x < lv y ) ∨ ( ord x d< ord y ) |
| 3 | 31 |
| 75 | 32 s<refl : {n : Level } {lx : Nat } { x : OrdinalD {n} lx } → x d< OSuc lx x |
| 33 s<refl {n} {lv} {Φ lv} = Φ< | |
| 34 s<refl {n} {lv} {OSuc lv x} = s< s<refl | |
| 35 | |
| 36 trio<> : {n : Level} → {lx : Nat} {x : OrdinalD {n} lx } { y : OrdinalD lx } → y d< x → x d< y → ⊥ | |
| 37 trio<> {n} {lx} {.(OSuc lx _)} {.(OSuc lx _)} (s< s) (s< t) = trio<> s t | |
| 38 trio<> {n} {lx} {.(OSuc lx _)} {.(Φ lx)} Φ< () | |
| 39 | |
| 40 d<→lv : {n : Level} {x y : Ordinal {n}} → ord x d< ord y → lv x ≡ lv y | |
| 41 d<→lv Φ< = refl | |
| 42 d<→lv (s< lt) = refl | |
| 43 | |
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equalitu and internal parametorisity
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44 o<-subst : {n : Level } {Z X z x : Ordinal {n}} → Z o< X → Z ≡ z → X ≡ x → z o< x |
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45 o<-subst df refl refl = df |
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46 |
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47 open import Data.Nat.Properties |
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problem on Ordinal ( OSuc ℵ )
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48 open import Data.Unit using ( ⊤ ) |
| 6 | 49 open import Relation.Nullary |
| 50 | |
| 51 open import Relation.Binary | |
| 52 open import Relation.Binary.Core | |
| 53 | |
| 24 | 54 o∅ : {n : Level} → Ordinal {n} |
| 55 o∅ = record { lv = Zero ; ord = Φ Zero } | |
| 21 | 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 } → x ≡ y → x d< y → ⊥ |
| 17 | 73 trio<≡ refl = ≡→¬d< |
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74 |
| 24 | 75 trio>≡ : {n : Level} → {lx : Nat} {x : OrdinalD {n} lx } { y : OrdinalD lx } → x ≡ y → y d< x → ⊥ |
| 17 | 76 trio>≡ refl = ≡→¬d< |
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77 |
| 24 | 78 triO : {n : Level} → {lx ly : Nat} → OrdinalD {n} lx → OrdinalD {n} ly → Tri (lx < ly) ( lx ≡ ly ) ( lx > ly ) |
| 79 triO {n} {lx} {ly} x y = <-cmp lx ly | |
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80 |
| 24 | 81 triOrdd : {n : Level} → {lx : Nat} → Trichotomous _≡_ ( _d<_ {n} {lx} {lx} ) |
| 82 triOrdd {_} {lv} (Φ lv) (Φ lv) = tri≈ ≡→¬d< refl ≡→¬d< | |
| 83 triOrdd {_} {lv} (Φ lv) (OSuc lv y) = tri< Φ< (λ ()) ( λ lt → trio<> lt Φ< ) | |
| 84 triOrdd {_} {lv} (OSuc lv x) (Φ lv) = tri> (λ lt → trio<> lt Φ<) (λ ()) Φ< | |
| 85 triOrdd {_} {lv} (OSuc lv x) (OSuc lv y) with triOrdd x y | |
| 86 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) ) | |
| 87 triOrdd {_} {lv} (OSuc lv x) (OSuc lv x) | tri≈ ¬a refl ¬c = tri≈ ≡→¬d< refl ≡→¬d< | |
| 88 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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89 |
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90 osuc : {n : Level} ( x : Ordinal {n} ) → Ordinal {n} |
| 75 | 91 osuc record { lv = lx ; ord = ox } = record { lv = lx ; ord = OSuc lx ox } |
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92 |
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93 <-osuc : {n : Level} { x : Ordinal {n} } → x o< osuc x |
| 75 | 94 <-osuc {n} {record { lv = lx ; ord = Φ .lx }} = case2 Φ< |
| 95 <-osuc {n} {record { lv = lx ; ord = OSuc .lx ox }} = case2 ( s< s<refl ) | |
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96 |
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97 osuc-lveq : {n : Level} { x : Ordinal {n} } → lv x ≡ lv ( osuc x ) |
| 75 | 98 osuc-lveq {n} = refl |
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99 |
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100 nat-<> : { x y : Nat } → x < y → y < x → ⊥ |
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101 nat-<> (s≤s x<y) (s≤s y<x) = nat-<> x<y y<x |
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102 |
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103 nat-<≡ : { x : Nat } → x < x → ⊥ |
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104 nat-<≡ (s≤s lt) = nat-<≡ lt |
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105 |
| 81 | 106 nat-≡< : { x y : Nat } → x ≡ y → x < y → ⊥ |
| 107 nat-≡< refl lt = nat-<≡ lt | |
| 108 | |
| 75 | 109 ¬a≤a : {la : Nat} → Suc la ≤ la → ⊥ |
| 110 ¬a≤a (s≤s lt) = ¬a≤a lt | |
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111 |
| 81 | 112 o<> : {n : Level} → {x y : Ordinal {n} } → y o< x → x o< y → ⊥ |
| 113 o<> {n} {x} {y} (case1 x₁) (case1 x₂) = nat-<> x₁ x₂ | |
| 114 o<> {n} {x} {y} (case1 x₁) (case2 x₂) = nat-≡< (sym (d<→lv x₂)) x₁ | |
| 115 o<> {n} {x} {y} (case2 x₁) (case1 x₂) = nat-≡< (sym (d<→lv x₁)) x₂ | |
| 116 o<> {n} {record { lv = lv₁ ; ord = .(OSuc lv₁ _) }} {record { lv = .lv₁ ; ord = .(Φ lv₁) }} (case2 Φ<) (case2 ()) | |
| 117 o<> {n} {record { lv = lv₁ ; ord = .(OSuc lv₁ _) }} {record { lv = .lv₁ ; ord = .(OSuc lv₁ _) }} (case2 (s< y<x)) (case2 (s< x<y)) = | |
| 118 o<> (case2 y<x) (case2 x<y) | |
| 16 | 119 |
| 24 | 120 orddtrans : {n : Level} {lx : Nat} {x y z : OrdinalD {n} lx } → x d< y → y d< z → x d< z |
| 121 orddtrans {_} {lx} {.(Φ lx)} {.(OSuc lx _)} {.(OSuc lx _)} Φ< (s< y<z) = Φ< | |
| 122 orddtrans {_} {lx} {.(OSuc lx _)} {.(OSuc lx _)} {.(OSuc lx _)} (s< x<y) (s< y<z) = s< ( orddtrans x<y y<z ) | |
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123 |
| 75 | 124 osuc-≡< : {n : Level} { a x : Ordinal {n} } → x o< osuc a → (x ≡ a ) ∨ (x o< a) |
| 125 osuc-≡< {n} {a} {x} (case1 lt) = case2 (case1 lt) | |
| 126 osuc-≡< {n} {record { lv = lv₁ ; ord = Φ .lv₁ }} {record { lv = .lv₁ ; ord = .(Φ lv₁) }} (case2 Φ<) = case1 refl | |
| 127 osuc-≡< {n} {record { lv = lv₁ ; ord = OSuc .lv₁ ord₁ }} {record { lv = .lv₁ ; ord = .(Φ lv₁) }} (case2 Φ<) = case2 (case2 Φ<) | |
| 128 osuc-≡< {n} {record { lv = lv₁ ; ord = Φ .lv₁ }} {record { lv = .lv₁ ; ord = .(OSuc lv₁ _) }} (case2 (s< ())) | |
| 129 osuc-≡< {n} {record { lv = la ; ord = OSuc la oa }} {record { lv = la ; ord = (OSuc la ox) }} (case2 (s< lt)) with | |
| 130 osuc-≡< {n} {record { lv = la ; ord = oa }} {record { lv = la ; ord = ox }} (case2 lt ) | |
| 131 ... | case1 refl = case1 refl | |
| 132 ... | case2 (case2 x) = case2 (case2( s< x) ) | |
| 133 ... | case2 (case1 x) = ⊥-elim (¬a≤a x) where | |
| 134 | |
| 135 osuc-< : {n : Level} { x y : Ordinal {n} } → y o< osuc x → x o< y → ⊥ | |
| 136 osuc-< {n} {x} {y} y<ox x<y with osuc-≡< y<ox | |
| 137 osuc-< {n} {x} {x} y<ox (case1 x₁) | case1 refl = ⊥-elim (¬a≤a x₁) | |
| 138 osuc-< {n} {x} {x} (case1 x₂) (case2 x₁) | case1 refl = ⊥-elim (¬a≤a x₂) | |
| 139 osuc-< {n} {x} {x} (case2 x₂) (case2 x₁) | case1 refl = ≡→¬d< x₁ | |
| 81 | 140 osuc-< {n} {x} {y} y<ox (case1 x₂) | case2 (case1 x₁) = nat-<> x₂ x₁ |
| 141 osuc-< {n} {x} {y} y<ox (case1 x₂) | case2 (case2 x₁) = nat-≡< (sym (d<→lv x₁)) x₂ | |
| 142 osuc-< {n} {x} {y} y<ox (case2 x<y) | case2 y<x = o<> (case2 x<y) y<x | |
| 75 | 143 |
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144 max : (x y : Nat) → Nat |
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145 max Zero Zero = Zero |
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146 max Zero (Suc x) = (Suc x) |
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147 max (Suc x) Zero = (Suc x) |
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148 max (Suc x) (Suc y) = Suc ( max x y ) |
| 3 | 149 |
| 24 | 150 maxαd : {n : Level} → { lx : Nat } → OrdinalD {n} lx → OrdinalD lx → OrdinalD lx |
| 17 | 151 maxαd x y with triOrdd x y |
| 152 maxαd x y | tri< a ¬b ¬c = y | |
| 153 maxαd x y | tri≈ ¬a b ¬c = x | |
| 154 maxαd x y | tri> ¬a ¬b c = x | |
| 6 | 155 |
| 24 | 156 maxα : {n : Level} → Ordinal {n} → Ordinal → Ordinal |
| 17 | 157 maxα x y with <-cmp (lv x) (lv y) |
| 158 maxα x y | tri< a ¬b ¬c = x | |
| 159 maxα x y | tri> ¬a ¬b c = y | |
| 160 maxα x y | tri≈ ¬a refl ¬c = record { lv = lv x ; ord = maxαd (ord x) (ord y) } | |
| 7 | 161 |
| 24 | 162 _o≤_ : {n : Level} → Ordinal → Ordinal → Set (suc n) |
| 23 | 163 a o≤ b = (a ≡ b) ∨ ( a o< b ) |
| 164 | |
| 27 | 165 ordtrans : {n : Level} {x y z : Ordinal {n} } → x o< y → y o< z → x o< z |
| 166 ordtrans {n} {x} {y} {z} (case1 x₁) (case1 x₂) = case1 ( <-trans x₁ x₂ ) | |
| 81 | 167 ordtrans {n} {x} {y} {z} (case1 x₁) (case2 x₂) with d<→lv x₂ |
| 27 | 168 ... | refl = case1 x₁ |
| 81 | 169 ordtrans {n} {x} {y} {z} (case2 x₁) (case1 x₂) with d<→lv x₁ |
| 27 | 170 ... | refl = case1 x₂ |
| 171 ordtrans {n} {x} {y} {z} (case2 x₁) (case2 x₂) with d<→lv x₁ | d<→lv x₂ | |
| 172 ... | refl | refl = case2 ( orddtrans x₁ x₂ ) | |
| 173 | |
| 24 | 174 trio< : {n : Level } → Trichotomous {suc n} _≡_ _o<_ |
| 23 | 175 trio< a b with <-cmp (lv a) (lv b) |
| 24 | 176 trio< a b | tri< a₁ ¬b ¬c = tri< (case1 a₁) (λ refl → ¬b (cong ( λ x → lv x ) refl ) ) lemma1 where |
| 177 lemma1 : ¬ (Suc (lv b) ≤ lv a) ∨ (ord b d< ord a) | |
| 178 lemma1 (case1 x) = ¬c x | |
| 81 | 179 lemma1 (case2 x) = ⊥-elim (nat-≡< (sym ( d<→lv x )) a₁ ) |
| 24 | 180 trio< a b | tri> ¬a ¬b c = tri> lemma1 (λ refl → ¬b (cong ( λ x → lv x ) refl ) ) (case1 c) where |
| 181 lemma1 : ¬ (Suc (lv a) ≤ lv b) ∨ (ord a d< ord b) | |
| 182 lemma1 (case1 x) = ¬a x | |
| 81 | 183 lemma1 (case2 x) = ⊥-elim (nat-≡< (sym ( d<→lv x )) c ) |
| 23 | 184 trio< a b | tri≈ ¬a refl ¬c with triOrdd ( ord a ) ( ord b ) |
| 24 | 185 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 |
| 186 lemma1 : (record { lv = _ ; ord = x }) ≡ b → x ≡ ord b | |
| 187 lemma1 refl = refl | |
| 188 lemma2 : ¬ (Suc (lv b) ≤ lv b) ∨ (ord b d< x) | |
| 189 lemma2 (case1 x) = ¬a x | |
| 190 lemma2 (case2 x) = trio<> x a | |
| 191 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 | |
| 192 lemma1 : (record { lv = _ ; ord = x }) ≡ b → x ≡ ord b | |
| 193 lemma1 refl = refl | |
| 194 lemma2 : ¬ (Suc (lv b) ≤ lv b) ∨ (x d< ord b) | |
| 195 lemma2 (case1 x) = ¬a x | |
| 196 lemma2 (case2 x) = trio<> x c | |
| 197 trio< record { lv = .(lv b) ; ord = x } b | tri≈ ¬a refl ¬c | tri≈ ¬a₁ refl ¬c₁ = tri≈ lemma1 refl lemma1 where | |
| 198 lemma1 : ¬ (Suc (lv b) ≤ lv b) ∨ (ord b d< ord b) | |
| 199 lemma1 (case1 x) = ¬a x | |
| 200 lemma1 (case2 x) = ≡→¬d< x | |
| 23 | 201 |
| 24 | 202 OrdTrans : {n : Level} → Transitive {suc n} _o≤_ |
| 16 | 203 OrdTrans (case1 refl) (case1 refl) = case1 refl |
| 204 OrdTrans (case1 refl) (case2 lt2) = case2 lt2 | |
| 205 OrdTrans (case2 lt1) (case1 refl) = case2 lt1 | |
| 81 | 206 OrdTrans (case2 x) (case2 y) = case2 (ordtrans x y) |
| 16 | 207 |
| 24 | 208 OrdPreorder : {n : Level} → Preorder (suc n) (suc n) (suc n) |
| 209 OrdPreorder {n} = record { Carrier = Ordinal | |
| 16 | 210 ; _≈_ = _≡_ |
| 23 | 211 ; _∼_ = _o≤_ |
| 16 | 212 ; isPreorder = record { |
| 213 isEquivalence = record { refl = refl ; sym = sym ; trans = trans } | |
| 214 ; reflexive = case1 | |
| 24 | 215 ; trans = OrdTrans |
| 16 | 216 } |
| 217 } | |
| 218 | |
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changeset
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219 TransFinite : {n : Level} → { ψ : Ordinal {n} → Set n } |
| 24 | 220 → ( ∀ (lx : Nat ) → ψ ( record { lv = lx ; ord = Φ lx } ) ) |
| 221 → ( ∀ (lx : Nat ) → (x : OrdinalD lx ) → ψ ( record { lv = lx ; ord = x } ) → ψ ( record { lv = lx ; ord = OSuc lx x } ) ) | |
| 22 | 222 → ∀ (x : Ordinal) → ψ x |
| 81 | 223 TransFinite caseΦ caseOSuc record { lv = lv ; ord = (Φ (lv)) } = caseΦ lv |
| 224 TransFinite caseΦ caseOSuc record { lv = lx ; ord = (OSuc lx ox) } = | |
| 225 caseOSuc lx ox (TransFinite caseΦ caseOSuc record { lv = lx ; ord = ox }) |