Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate HOD.agda @ 116:47541e86c6ac
axiom of selection
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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| date | Wed, 26 Jun 2019 08:05:58 +0900 |
| parents | 277c2f3b8acb |
| children | a4c97390d312 |
| rev | line source |
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| 16 | 1 open import Level |
| 112 | 2 module HOD where |
| 3 | 3 |
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4 open import zf |
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5 open import ordinal |
| 23 | 6 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) |
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7 open import Relation.Binary.PropositionalEquality |
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8 open import Data.Nat.Properties |
| 6 | 9 open import Data.Empty |
| 10 open import Relation.Nullary | |
| 11 open import Relation.Binary | |
| 12 open import Relation.Binary.Core | |
| 13 | |
| 27 | 14 -- Ordinal Definable Set |
| 11 | 15 |
| 112 | 16 record HOD {n : Level} : Set (suc n) where |
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17 field |
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18 def : (x : Ordinal {n} ) → Set n |
| 114 | 19 otrans : {x : Ordinal {n} } → def x → { y : Ordinal {n} } → y o< x → def y |
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20 |
| 112 | 21 open HOD |
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22 open import Data.Unit |
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23 |
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24 open Ordinal |
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25 |
| 112 | 26 record _==_ {n : Level} ( a b : HOD {n} ) : Set n where |
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27 field |
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28 eq→ : ∀ { x : Ordinal {n} } → def a x → def b x |
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29 eq← : ∀ { x : Ordinal {n} } → def b x → def a x |
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30 |
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31 id : {n : Level} {A : Set n} → A → A |
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32 id x = x |
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33 |
| 112 | 34 eq-refl : {n : Level} { x : HOD {n} } → x == x |
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35 eq-refl {n} {x} = record { eq→ = id ; eq← = id } |
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36 |
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37 open _==_ |
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38 |
| 112 | 39 eq-sym : {n : Level} { x y : HOD {n} } → x == y → y == x |
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40 eq-sym eq = record { eq→ = eq← eq ; eq← = eq→ eq } |
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41 |
| 112 | 42 eq-trans : {n : Level} { x y z : HOD {n} } → x == y → y == z → x == z |
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43 eq-trans x=y y=z = record { eq→ = λ t → eq→ y=z ( eq→ x=y t) ; eq← = λ t → eq← x=y ( eq← y=z t) } |
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44 |
| 112 | 45 -- Ordinal in HOD ( and ZFSet ) |
| 46 Ord : { n : Level } → ( a : Ordinal {n} ) → HOD {n} | |
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47 Ord {n} a = record { def = λ y → y o< a ; otrans = lemma } where |
| 114 | 48 lemma : {x : Ordinal} → x o< a → {y : Ordinal} → y o< x → y o< a |
| 49 lemma {x} x<a {y} y<x = ordtrans {n} {y} {x} {a} y<x x<a | |
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50 |
| 112 | 51 od∅ : {n : Level} → HOD {n} |
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52 od∅ {n} = Ord o∅ |
| 40 | 53 |
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54 postulate |
| 112 | 55 -- HOD can be iso to a subset of Ordinal ( by means of Godel Set ) |
| 56 od→ord : {n : Level} → HOD {n} → Ordinal {n} | |
| 113 | 57 ord→od : {n : Level} → Ordinal {n} → HOD {n} |
| 58 oiso : {n : Level} {x : HOD {n}} → ord→od ( od→ord x ) ≡ x | |
| 59 diso : {n : Level} {x : Ordinal {n}} → od→ord ( ord→od x ) ≡ x | |
| 112 | 60 c<→o< : {n : Level} {x y : HOD {n} } → def y ( od→ord x ) → od→ord x o< od→ord y |
| 116 | 61 ord-Ord :{n : Level} {x : Ordinal {n}} → x ≡ od→ord (Ord x) |
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62 -- next assumption causes ∀ x ∋ ∅ . It menas only an ordinal becomes a set |
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63 -- o<→c< : {n : Level} {x y : Ordinal {n} } → x o< y → def (ord→od y) x |
| 100 | 64 -- supermum as Replacement Axiom |
| 95 | 65 sup-o : {n : Level } → ( Ordinal {n} → Ordinal {n}) → Ordinal {n} |
| 98 | 66 sup-o< : {n : Level } → { ψ : Ordinal {n} → Ordinal {n}} → ∀ {x : Ordinal {n}} → ψ x o< sup-o ψ |
| 111 | 67 -- contra-position of mimimulity of supermum required in Power Set Axiom |
| 98 | 68 sup-x : {n : Level } → ( Ordinal {n} → Ordinal {n}) → Ordinal {n} |
| 69 sup-lb : {n : Level } → { ψ : Ordinal {n} → Ordinal {n}} → {z : Ordinal {n}} → z o< sup-o ψ → z o< osuc (ψ (sup-x ψ)) | |
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70 -- sup-lb : {n : Level } → ( ψ : Ordinal {n} → Ordinal {n}) → ( ∀ {x : Ordinal {n}} → ψx o< z ) → z o< osuc ( sup-o ψ ) |
| 95 | 71 |
| 112 | 72 _∋_ : { n : Level } → ( a x : HOD {n} ) → Set n |
| 95 | 73 _∋_ {n} a x = def a ( od→ord x ) |
| 74 | |
| 112 | 75 _c<_ : { n : Level } → ( x a : HOD {n} ) → Set n |
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76 x c< a = a ∋ x |
| 103 | 77 |
| 112 | 78 _c≤_ : {n : Level} → HOD {n} → HOD {n} → Set (suc n) |
| 95 | 79 a c≤ b = (a ≡ b) ∨ ( b ∋ a ) |
| 80 | |
| 113 | 81 cseq : {n : Level} → HOD {n} → HOD {n} |
| 82 cseq x = record { def = λ y → osuc y o< (od→ord x) ; otrans = lemma } where | |
| 114 | 83 lemma : {ox : Ordinal} → osuc ox o< od→ord x → { oy : Ordinal}→ oy o< ox → osuc oy o< od→ord x |
| 84 lemma {ox} oox<Ox {oy} oy<ox = ordtrans (osucc oy<ox ) oox<Ox | |
| 113 | 85 |
| 112 | 86 def-subst : {n : Level } {Z : HOD {n}} {X : Ordinal {n} }{z : HOD {n}} {x : Ordinal {n} }→ def Z X → Z ≡ z → X ≡ x → def z x |
| 95 | 87 def-subst df refl refl = df |
| 88 | |
| 113 | 89 o<→c< : {n : Level} {x y : Ordinal {n} } → x o< y → Ord y ∋ Ord x |
| 90 o<→c< {n} {x} {y} lt = subst ( λ k → k o< y ) ord-Ord lt | |
| 91 | |
| 112 | 92 sup-od : {n : Level } → ( HOD {n} → HOD {n}) → HOD {n} |
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93 sup-od ψ = Ord ( sup-o ( λ x → od→ord (ψ (ord→od x ))) ) |
| 95 | 94 |
| 112 | 95 sup-c< : {n : Level } → ( ψ : HOD {n} → HOD {n}) → ∀ {x : HOD {n}} → def ( sup-od ψ ) (od→ord ( ψ x )) |
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96 sup-c< {n} ψ {x} = def-subst {n} {_} {_} {Ord ( sup-o ( λ x → od→ord (ψ (ord→od x ))) )} {od→ord ( ψ x )} |
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97 lemma refl (cong ( λ k → od→ord (ψ k) ) oiso) where |
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98 lemma : od→ord (ψ (ord→od (od→ord x))) o< sup-o (λ x → od→ord (ψ (ord→od x))) |
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99 lemma = subst₂ (λ j k → j o< k ) refl diso (o<-subst sup-o< refl (sym diso) ) |
| 28 | 100 |
| 37 | 101 ∅3 : {n : Level} → { x : Ordinal {n}} → ( ∀(y : Ordinal {n}) → ¬ (y o< x ) ) → x ≡ o∅ {n} |
| 81 | 102 ∅3 {n} {x} = TransFinite {n} c2 c3 x where |
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103 c0 : Nat → Ordinal {n} → Set n |
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104 c0 lx x = (∀(y : Ordinal {n}) → ¬ (y o< x)) → x ≡ o∅ {n} |
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105 c2 : (lx : Nat) → c0 lx (record { lv = lx ; ord = Φ lx } ) |
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106 c2 Zero not = refl |
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107 c2 (Suc lx) not with not ( record { lv = lx ; ord = Φ lx } ) |
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108 ... | t with t (case1 ≤-refl ) |
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109 c2 (Suc lx) not | t | () |
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110 c3 : (lx : Nat) (x₁ : OrdinalD lx) → c0 lx (record { lv = lx ; ord = x₁ }) → c0 lx (record { lv = lx ; ord = OSuc lx x₁ }) |
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111 c3 lx (Φ .lx) d not with not ( record { lv = lx ; ord = Φ lx } ) |
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112 ... | t with t (case2 Φ< ) |
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113 c3 lx (Φ .lx) d not | t | () |
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114 c3 lx (OSuc .lx x₁) d not with not ( record { lv = lx ; ord = OSuc lx x₁ } ) |
| 34 | 115 ... | t with t (case2 (s< s<refl ) ) |
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116 c3 lx (OSuc .lx x₁) d not | t | () |
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117 |
| 112 | 118 transitive : {n : Level } { z y x : HOD {suc n} } → z ∋ y → y ∋ x → z ∋ x |
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119 transitive {n} {z} {y} {x} z∋y x∋y with ordtrans ( c<→o< {suc n} {x} {y} x∋y ) ( c<→o< {suc n} {y} {z} z∋y ) |
| 111 | 120 ... | t = otrans z z∋y (c<→o< {suc n} {x} {y} x∋y ) |
| 36 | 121 |
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122 record Minimumo {n : Level } (x : Ordinal {n}) : Set (suc n) where |
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123 field |
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124 mino : Ordinal {n} |
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125 min<x : mino o< x |
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126 |
| 57 | 127 ∅5 : {n : Level} → { x : Ordinal {n} } → ¬ ( x ≡ o∅ {n} ) → o∅ {n} o< x |
| 128 ∅5 {n} {record { lv = Zero ; ord = (Φ .0) }} not = ⊥-elim (not refl) | |
| 129 ∅5 {n} {record { lv = Zero ; ord = (OSuc .0 ord) }} not = case2 Φ< | |
| 130 ∅5 {n} {record { lv = (Suc lv) ; ord = ord }} not = case1 (s≤s z≤n) | |
| 37 | 131 |
| 46 | 132 ord-iso : {n : Level} {y : Ordinal {n} } → record { lv = lv (od→ord (ord→od y)) ; ord = ord (od→ord (ord→od y)) } ≡ record { lv = lv y ; ord = ord y } |
| 133 ord-iso = cong ( λ k → record { lv = lv k ; ord = ord k } ) diso | |
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134 |
| 51 | 135 -- avoiding lv != Zero error |
| 112 | 136 orefl : {n : Level} → { x : HOD {n} } → { y : Ordinal {n} } → od→ord x ≡ y → od→ord x ≡ y |
| 51 | 137 orefl refl = refl |
| 138 | |
| 112 | 139 ==-iso : {n : Level} → { x y : HOD {n} } → ord→od (od→ord x) == ord→od (od→ord y) → x == y |
| 51 | 140 ==-iso {n} {x} {y} eq = record { |
| 141 eq→ = λ d → lemma ( eq→ eq (def-subst d (sym oiso) refl )) ; | |
| 142 eq← = λ d → lemma ( eq← eq (def-subst d (sym oiso) refl )) } | |
| 143 where | |
| 112 | 144 lemma : {x : HOD {n} } {z : Ordinal {n}} → def (ord→od (od→ord x)) z → def x z |
| 51 | 145 lemma {x} {z} d = def-subst d oiso refl |
| 146 | |
| 112 | 147 =-iso : {n : Level } {x y : HOD {suc n} } → (x == y) ≡ (ord→od (od→ord x) == y) |
| 57 | 148 =-iso {_} {_} {y} = cong ( λ k → k == y ) (sym oiso) |
| 149 | |
| 112 | 150 ord→== : {n : Level} → { x y : HOD {n} } → od→ord x ≡ od→ord y → x == y |
| 51 | 151 ord→== {n} {x} {y} eq = ==-iso (lemma (od→ord x) (od→ord y) (orefl eq)) where |
| 152 lemma : ( ox oy : Ordinal {n} ) → ox ≡ oy → (ord→od ox) == (ord→od oy) | |
| 153 lemma ox ox refl = eq-refl | |
| 154 | |
| 155 o≡→== : {n : Level} → { x y : Ordinal {n} } → x ≡ y → ord→od x == ord→od y | |
| 156 o≡→== {n} {x} {.x} refl = eq-refl | |
| 157 | |
| 158 >→¬< : {x y : Nat } → (x < y ) → ¬ ( y < x ) | |
| 159 >→¬< (s≤s x<y) (s≤s y<x) = >→¬< x<y y<x | |
| 160 | |
| 112 | 161 c≤-refl : {n : Level} → ( x : HOD {n} ) → x c≤ x |
| 51 | 162 c≤-refl x = case1 refl |
| 163 | |
| 112 | 164 ∋→o< : {n : Level} → { a x : HOD {suc n} } → a ∋ x → od→ord x o< od→ord a |
| 91 | 165 ∋→o< {n} {a} {x} lt = t where |
| 166 t : (od→ord x) o< (od→ord a) | |
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167 t = c<→o< {suc n} {x} {a} lt |
| 91 | 168 |
| 80 | 169 o∅≡od∅ : {n : Level} → ord→od (o∅ {suc n}) ≡ od∅ {suc n} |
| 170 o∅≡od∅ {n} with trio< {n} (o∅ {suc n}) (od→ord (od∅ {suc n} )) | |
| 171 o∅≡od∅ {n} | tri< a ¬b ¬c = ⊥-elim (lemma a) where | |
| 172 lemma : o∅ {suc n } o< (od→ord (od∅ {suc n} )) → ⊥ | |
| 113 | 173 lemma lt with o<→c< lt |
| 174 lemma lt | t = o<¬≡ refl t | |
| 80 | 175 o∅≡od∅ {n} | tri≈ ¬a b ¬c = trans (cong (λ k → ord→od k ) b ) oiso |
| 176 o∅≡od∅ {n} | tri> ¬a ¬b c = ⊥-elim (¬x<0 c) | |
| 177 | |
| 112 | 178 o<→¬c> : {n : Level} → { x y : HOD {n} } → (od→ord x ) o< ( od→ord y) → ¬ (y c< x ) |
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179 o<→¬c> {n} {x} {y} olt clt = o<> olt (c<→o< clt ) where |
| 51 | 180 |
| 112 | 181 o≡→¬c< : {n : Level} → { x y : HOD {n} } → (od→ord x ) ≡ ( od→ord y) → ¬ x c< y |
| 111 | 182 o≡→¬c< {n} {x} {y} oeq lt = o<¬≡ (orefl oeq ) (c<→o< lt) |
| 54 | 183 |
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184 ∅0 : {n : Level} → record { def = λ x → Lift n ⊥ ; otrans = λ () } == od∅ {n} |
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185 eq→ ∅0 {w} (lift ()) |
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186 eq← ∅0 {w} (case1 ()) |
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187 eq← ∅0 {w} (case2 ()) |
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188 |
| 112 | 189 ∅< : {n : Level} → { x y : HOD {n} } → def x (od→ord y ) → ¬ ( x == od∅ {n} ) |
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190 ∅< {n} {x} {y} d eq with eq→ (eq-trans eq (eq-sym ∅0) ) d |
| 60 | 191 ∅< {n} {x} {y} d eq | lift () |
| 57 | 192 |
| 112 | 193 -- ∅6 : {n : Level} → { x : HOD {suc n} } → ¬ ( x ∋ x ) -- no Russel paradox |
| 111 | 194 -- ∅6 {n} {x} x∋x = c<> {n} {x} {x} x∋x x∋x |
| 51 | 195 |
| 112 | 196 def-iso : {n : Level} {A B : HOD {n}} {x y : Ordinal {n}} → x ≡ y → (def A y → def B y) → def A x → def B x |
| 76 | 197 def-iso refl t = t |
| 198 | |
| 57 | 199 is-o∅ : {n : Level} → ( x : Ordinal {suc n} ) → Dec ( x ≡ o∅ {suc n} ) |
| 200 is-o∅ {n} record { lv = Zero ; ord = (Φ .0) } = yes refl | |
| 201 is-o∅ {n} record { lv = Zero ; ord = (OSuc .0 ord₁) } = no ( λ () ) | |
| 202 is-o∅ {n} record { lv = (Suc lv₁) ; ord = ord } = no (λ()) | |
| 203 | |
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204 open _∧_ |
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205 |
| 79 | 206 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) |
| 94 | 207 -- postulate f-extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality (suc n) (suc (suc n)) |
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208 |
| 112 | 209 csuc : {n : Level} → HOD {suc n} → HOD {suc n} |
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210 csuc x = ord→od ( osuc ( od→ord x )) |
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211 |
| 96 | 212 -- Power Set of X ( or constructible by λ y → def X (od→ord y ) |
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213 |
| 112 | 214 ZFSubset : {n : Level} → (A x : HOD {suc n} ) → HOD {suc n} |
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215 ZFSubset A x = record { def = λ y → def A y ∧ def x y ; otrans = {!!} } |
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216 |
| 112 | 217 Def : {n : Level} → (A : HOD {suc n}) → HOD {suc n} |
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218 Def {n} A = ord→od ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) ) |
| 96 | 219 |
| 220 -- Constructible Set on α | |
| 112 | 221 L : {n : Level} → (α : Ordinal {suc n}) → HOD {suc n} |
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222 L {n} record { lv = Zero ; ord = (Φ .0) } = od∅ |
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223 L {n} record { lv = lx ; ord = (OSuc lv ox) } = Def ( L {n} ( record { lv = lx ; ord = ox } ) ) |
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224 L {n} record { lv = (Suc lx) ; ord = (Φ (Suc lx)) } = -- Union ( L α ) |
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225 record { def = λ y → osuc y o< (od→ord (L {n} (record { lv = lx ; ord = Φ lx }) )) ; otrans = {!!} } |
| 89 | 226 |
| 111 | 227 omega : { n : Level } → Ordinal {n} |
| 228 omega = record { lv = Suc Zero ; ord = Φ 1 } | |
| 229 | |
| 112 | 230 HOD→ZF : {n : Level} → ZF {suc (suc n)} {suc n} |
| 231 HOD→ZF {n} = record { | |
| 232 ZFSet = HOD {suc n} | |
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233 ; _∋_ = _∋_ |
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234 ; _≈_ = _==_ |
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235 ; ∅ = od∅ |
| 28 | 236 ; _,_ = _,_ |
| 237 ; Union = Union | |
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238 ; Power = Power |
| 28 | 239 ; Select = Select |
| 240 ; Replace = Replace | |
| 111 | 241 ; infinite = Ord omega |
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242 ; isZF = isZF |
| 28 | 243 } where |
| 112 | 244 Replace : HOD {suc n} → (HOD {suc n} → HOD {suc n} ) → HOD {suc n} |
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245 Replace X ψ = sup-od ψ |
| 115 | 246 Select : (X : HOD {suc n} ) → ((x : HOD {suc n} ) → Set (suc n) ) → HOD {suc n} |
| 116 | 247 Select X ψ = record { def = λ x → ((y : Ordinal {suc n} ) → X ∋ ord→od y → ψ (ord→od y)) ∧ (X ∋ ord→od x ) ; otrans = lemma } where |
| 248 lemma : {x : Ordinal} → ((y : Ordinal) → X ∋ ord→od y → ψ (ord→od y)) ∧ (X ∋ ord→od x) → | |
| 249 {y : Ordinal} → y o< x → ((y₁ : Ordinal) → X ∋ ord→od y₁ → ψ (ord→od y₁)) ∧ (X ∋ ord→od y) | |
| 115 | 250 lemma {x} select {y} y<x = record { proj1 = proj1 select ; proj2 = y<X } where |
| 251 y<X : X ∋ ord→od y | |
| 252 y<X = otrans X (proj2 select) (o<-subst y<x (sym diso) (sym diso) ) | |
| 112 | 253 _,_ : HOD {suc n} → HOD {suc n} → HOD {suc n} |
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254 x , y = Ord (omax (od→ord x) (od→ord y)) |
| 112 | 255 Union : HOD {suc n} → HOD {suc n} |
| 113 | 256 Union U = cseq U |
| 77 | 257 -- power : ∀ X ∃ A ∀ t ( t ∈ A ↔ ( ∀ {x} → t ∋ x → X ∋ x ) |
| 112 | 258 Power : HOD {suc n} → HOD {suc n} |
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259 Power A = Def A |
| 112 | 260 ZFSet = HOD {suc n} |
| 54 | 261 _∈_ : ( A B : ZFSet ) → Set (suc n) |
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262 A ∈ B = B ∋ A |
| 54 | 263 _⊆_ : ( A B : ZFSet ) → ∀{ x : ZFSet } → Set (suc n) |
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264 _⊆_ A B {x} = A ∋ x → B ∋ x |
| 103 | 265 _∩_ : ( A B : ZFSet ) → ZFSet |
| 115 | 266 A ∩ B = Select (A , B) ( λ x → ( A ∋ x ) ∧ (B ∋ x) ) |
| 96 | 267 -- _∪_ : ( A B : ZFSet ) → ZFSet |
| 268 -- A ∪ B = Select (A , B) ( λ x → (A ∋ x) ∨ ( B ∋ x ) ) | |
| 103 | 269 ｛_｝ : ZFSet → ZFSet |
| 270 ｛ x ｝ = ( x , x ) | |
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271 |
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272 infixr 200 _∈_ |
| 96 | 273 -- infixr 230 _∩_ _∪_ |
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274 infixr 220 _⊆_ |
| 112 | 275 isZF : IsZF (HOD {suc n}) _∋_ _==_ od∅ _,_ Union Power Select Replace (Ord omega) |
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276 isZF = record { |
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277 isEquivalence = record { refl = eq-refl ; sym = eq-sym; trans = eq-trans } |
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278 ; pair = pair |
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279 ; union-u = λ _ z _ → csuc z |
| 72 | 280 ; union→ = union→ |
| 281 ; union← = union← | |
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282 ; empty = empty |
| 76 | 283 ; power→ = power→ |
| 284 ; power← = power← | |
| 285 ; extensionality = extensionality | |
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286 ; minimul = minimul |
| 51 | 287 ; regularity = regularity |
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288 ; infinity∅ = infinity∅ |
| 93 | 289 ; infinity = λ _ → infinity |
| 116 | 290 ; selection = λ {X} {ψ} {y} → selection {X} {ψ} {y} |
| 93 | 291 ; replacement = replacement |
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292 } where |
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293 open _∧_ |
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294 open Minimumo |
| 112 | 295 pair : (A B : HOD {suc n} ) → ((A , B) ∋ A) ∧ ((A , B) ∋ B) |
| 87 | 296 proj1 (pair A B ) = omax-x {n} (od→ord A) (od→ord B) |
| 297 proj2 (pair A B ) = omax-y {n} (od→ord A) (od→ord B) | |
| 112 | 298 empty : (x : HOD {suc n} ) → ¬ (od∅ ∋ x) |
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299 empty x (case1 ()) |
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300 empty x (case2 ()) |
| 100 | 301 --- |
| 302 --- ZFSubset A x = record { def = λ y → def A y ∧ def x y } subset of A | |
| 303 --- Power X = ord→od ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) ) Power X is a sup of all subset of A | |
| 304 -- | |
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305 -- if Power A ∋ t, from a propertiy of minimum sup there is osuc ZFSubset A ∋ t |
| 100 | 306 -- then ZFSubset A ≡ t or ZFSubset A ∋ t. In the former case ZFSubset A ∋ x implies A ∋ x |
| 307 -- In case of later, ZFSubset A ∋ t and t ∋ x implies ZFSubset A ∋ x by transitivity | |
| 308 -- | |
| 112 | 309 power→ : (A t : HOD) → Power A ∋ t → {x : HOD} → t ∋ x → A ∋ x |
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310 power→ A t P∋t {x} t∋x = proj1 lemma-s where |
| 112 | 311 minsup : HOD |
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312 minsup = ZFSubset A ( ord→od ( sup-x (λ x → od→ord ( ZFSubset A (ord→od x))))) |
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313 lemma-t : csuc minsup ∋ t |
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314 lemma-t = {!!} -- o<→c< (o<-subst (sup-lb (o<-subst (c<→o< P∋t) refl diso )) refl refl ) |
| 98 | 315 lemma-s : ZFSubset A ( ord→od ( sup-x (λ x → od→ord ( ZFSubset A (ord→od x))))) ∋ x |
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316 lemma-s with osuc-≡< ( o<-subst (c<→o< lemma-t ) refl diso ) |
| 111 | 317 lemma-s | case1 eq = def-subst {!!} oiso refl |
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318 lemma-s | case2 lt = transitive {n} {minsup} {t} {x} (def-subst {!!} oiso refl ) t∋x |
| 100 | 319 -- |
| 320 -- we have t ∋ x → A ∋ x means t is a subset of A, that is ZFSubset A t == t | |
| 321 -- Power A is a sup of ZFSubset A t, so Power A ∋ t | |
| 322 -- | |
| 112 | 323 power← : (A t : HOD) → ({x : HOD} → (t ∋ x → A ∋ x)) → Power A ∋ t |
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324 power← A t t→A = def-subst {suc n} {_} {_} {Power A} {od→ord t} |
| 103 | 325 {!!} refl lemma1 where |
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326 lemma-eq : ZFSubset A t == t |
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327 eq→ lemma-eq {z} w = proj2 w |
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328 eq← lemma-eq {z} w = record { proj2 = w ; |
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329 proj1 = def-subst {suc n} {_} {_} {A} {z} ( t→A (def-subst {suc n} {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso } |
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330 lemma1 : od→ord (ZFSubset A (ord→od (od→ord t))) ≡ od→ord t |
| 111 | 331 lemma1 = subst (λ k → od→ord (ZFSubset A k) ≡ od→ord t ) (sym oiso) {!!} |
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332 lemma : od→ord (ZFSubset A (ord→od (od→ord t)) ) o< sup-o (λ x → od→ord (ZFSubset A (ord→od x))) |
| 98 | 333 lemma = sup-o< |
| 112 | 334 union-lemma-u : {X z : HOD {suc n}} → (U>z : Union X ∋ z ) → csuc z ∋ z |
| 72 | 335 union-lemma-u {X} {z} U>z = lemma <-osuc where |
| 336 lemma : {oz ooz : Ordinal {suc n}} → oz o< ooz → def (ord→od ooz) oz | |
| 103 | 337 lemma {oz} {ooz} lt = def-subst {suc n} {ord→od ooz} {!!} refl refl |
| 112 | 338 union→ : (X z u : HOD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z |
| 72 | 339 union→ X y u xx with trio< ( od→ord u ) ( osuc ( od→ord y )) |
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340 union→ X y u xx | tri< a ¬b ¬c with osuc-< a (c<→o< (proj2 xx)) |
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341 union→ X y u xx | tri< a ¬b ¬c | () |
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342 union→ X y u xx | tri≈ ¬a b ¬c = lemma b (c<→o< (proj1 xx )) where |
| 73 | 343 lemma : {oX ou ooy : Ordinal {suc n}} → ou ≡ ooy → ou o< oX → ooy o< oX |
| 344 lemma refl lt = lt | |
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345 union→ X y u xx | tri> ¬a ¬b c = ordtrans {suc n} {osuc ( od→ord y )} {od→ord u} {od→ord X} c ( c<→o< (proj1 xx )) |
| 112 | 346 union← : (X z : HOD) (X∋z : Union X ∋ z) → (X ∋ csuc z) ∧ (csuc z ∋ z ) |
| 103 | 347 union← X z X∋z = record { proj1 = def-subst {suc n} {_} {_} {X} {od→ord (csuc z )} {!!} oiso (sym diso) ; proj2 = union-lemma-u X∋z } |
| 112 | 348 ψiso : {ψ : HOD {suc n} → Set (suc n)} {x y : HOD {suc n}} → ψ x → x ≡ y → ψ y |
| 54 | 349 ψiso {ψ} t refl = t |
| 116 | 350 selection : {X : HOD } {ψ : (x : HOD ) → Set (suc n)} {y : HOD} → (((y₁ : HOD) → X ∋ y₁ → ψ y₁) ∧ (X ∋ y)) ⇔ (Select X ψ ∋ y) |
| 115 | 351 selection {X} {ψ} {y} = record { proj1 = λ s → record { |
| 116 | 352 proj1 = λ y1 y1<X → proj1 s (ord→od y1) y1<X ; proj2 = subst (λ k → def X k ) (sym diso) (proj2 s) } |
| 353 ; proj2 = λ s → record { proj1 = λ y1 dy1 → subst (λ k → ψ k ) oiso ((proj1 s) (od→ord y1) (def-subst {suc n} {_} {_} {X} {_} dy1 refl (sym diso ))) | |
| 354 ; proj2 = def-subst {suc n} {_} {_} {X} {od→ord y} (proj2 s ) refl diso } } where | |
| 112 | 355 replacement : {ψ : HOD → HOD} (X x : HOD) → Replace X ψ ∋ ψ x |
| 93 | 356 replacement {ψ} X x = sup-c< ψ {x} |
| 112 | 357 ∅-iso : {x : HOD} → ¬ (x == od∅) → ¬ ((ord→od (od→ord x)) == od∅) |
| 60 | 358 ∅-iso {x} neq = subst (λ k → ¬ k) (=-iso {n} ) neq |
| 112 | 359 minimul : (x : HOD {suc n} ) → ¬ (x == od∅ )→ HOD {suc n} |
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360 minimul x not = {!!} |
| 112 | 361 regularity : (x : HOD) (not : ¬ (x == od∅)) → |
| 115 | 362 (x ∋ minimul x not) ∧ (Select (minimul x not) (λ x₁ → (minimul x not ∋ x₁) ∧ (x ∋ x₁)) == od∅) |
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363 proj1 (regularity x not ) = {!!} |
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364 proj2 (regularity x not ) = record { eq→ = reg ; eq← = {!!} } where |
| 115 | 365 reg : {y : Ordinal} → def (Select (minimul x not) (λ x₂ → (minimul x not ∋ x₂) ∧ (x ∋ x₂))) y → def od∅ y |
| 111 | 366 reg {y} t = {!!} |
| 112 | 367 extensionality : {A B : HOD {suc n}} → ((z : HOD) → (A ∋ z) ⇔ (B ∋ z)) → A == B |
| 76 | 368 eq→ (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {A} {B} (sym diso) (proj1 (eq (ord→od x))) d |
| 369 eq← (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {B} {A} (sym diso) (proj2 (eq (ord→od x))) d | |
| 112 | 370 xx-union : {x : HOD {suc n}} → (x , x) ≡ record { def = λ z → z o< osuc (od→ord x) } |
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371 xx-union {x} = cong ( λ k → Ord k ) (omxx (od→ord x)) |
| 112 | 372 xxx-union : {x : HOD {suc n}} → (x , (x , x)) ≡ record { def = λ z → z o< osuc (osuc (od→ord x))} |
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373 xxx-union {x} = cong ( λ k → Ord k ) lemma where |
| 112 | 374 lemma1 : {x : HOD {suc n}} → od→ord x o< od→ord (x , x) |
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375 lemma1 {x} = c<→o< ( proj1 (pair x x ) ) |
| 112 | 376 lemma2 : {x : HOD {suc n}} → od→ord (x , x) ≡ osuc (od→ord x) |
| 111 | 377 lemma2 = trans ( cong ( λ k → od→ord k ) xx-union ) {!!} |
| 112 | 378 lemma : {x : HOD {suc n}} → omax (od→ord x) (od→ord (x , x)) ≡ osuc (osuc (od→ord x)) |
| 91 | 379 lemma {x} = trans ( sym ( omax< (od→ord x) (od→ord (x , x)) lemma1 ) ) ( cong ( λ k → osuc k ) lemma2 ) |
| 112 | 380 uxxx-union : {x : HOD {suc n}} → Union (x , (x , x)) ≡ record { def = λ z → osuc z o< osuc (osuc (od→ord x)) } |
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381 uxxx-union {x} = cong ( λ k → record { def = λ z → osuc z o< k ; otrans = {!!} } ) lemma where |
| 90 | 382 lemma : od→ord (x , (x , x)) ≡ osuc (osuc (od→ord x)) |
| 111 | 383 lemma = trans ( cong ( λ k → od→ord k ) xxx-union ) {!!} |
| 112 | 384 uxxx-2 : {x : HOD {suc n}} → record { def = λ z → osuc z o< osuc (osuc (od→ord x)) } == record { def = λ z → z o< osuc (od→ord x) } |
| 91 | 385 eq→ ( uxxx-2 {x} ) {m} lt = proj1 (osuc2 m (od→ord x)) lt |
| 386 eq← ( uxxx-2 {x} ) {m} lt = proj2 (osuc2 m (od→ord x)) lt | |
| 112 | 387 uxxx-ord : {x : HOD {suc n}} → od→ord (Union (x , (x , x))) ≡ osuc (od→ord x) |
| 111 | 388 uxxx-ord {x} = trans (cong (λ k → od→ord k ) uxxx-union) {!!} |
| 112 | 389 infinite : HOD {suc n} |
| 111 | 390 infinite = Ord omega |
| 391 infinity∅ : Ord omega ∋ od∅ {suc n} | |
| 392 infinity∅ = {!!} | |
| 112 | 393 infinity : (x : HOD) → infinite ∋ x → infinite ∋ Union (x , (x , x )) |
| 111 | 394 infinity x lt = {!!} where |
| 91 | 395 lemma : (ox : Ordinal {suc n} ) → ox o< omega → osuc ox o< omega |
| 396 lemma record { lv = Zero ; ord = (Φ .0) } (case1 (s≤s x)) = case1 (s≤s z≤n) | |
| 397 lemma record { lv = Zero ; ord = (OSuc .0 ord₁) } (case1 (s≤s x)) = case1 (s≤s z≤n) | |
| 398 lemma record { lv = (Suc lv₁) ; ord = (Φ .(Suc lv₁)) } (case1 (s≤s ())) | |
| 399 lemma record { lv = (Suc lv₁) ; ord = (OSuc .(Suc lv₁) ord₁) } (case1 (s≤s ())) | |
| 400 lemma record { lv = 1 ; ord = (Φ 1) } (case2 c2) with d<→lv c2 | |
| 401 lemma record { lv = (Suc Zero) ; ord = (Φ .1) } (case2 ()) | refl | |
| 103 | 402 -- ∀ X [ ∅ ∉ X → (∃ f : X → ⋃ X ) → ∀ A ∈ X ( f ( A ) ∈ A ) ] -- this form is no good since X is a transitive set |
| 403 -- ∀ z [ ∀ x ( x ∈ z → ¬ ( x ≈ ∅ ) ) ∧ ∀ x ∀ y ( x , y ∈ z ∧ ¬ ( x ≈ y ) → x ∩ y ≈ ∅ ) → ∃ u ∀ x ( x ∈ z → ∃ t ( u ∩ x) ≈ ｛ t ｝) ] | |
| 112 | 404 record Choice (z : HOD {suc n}) : Set (suc (suc n)) where |
| 103 | 405 field |
| 112 | 406 u : {x : HOD {suc n}} ( x∈z : x ∈ z ) → HOD {suc n} |
| 407 t : {x : HOD {suc n}} ( x∈z : x ∈ z ) → (x : HOD {suc n} ) → HOD {suc n} | |
| 408 choice : { x : HOD {suc n} } → ( x∈z : x ∈ z ) → ( u x∈z ∩ x) == ｛ t x∈z x ｝ | |
| 409 -- choice : {x : HOD {suc n}} ( x ∈ z → ¬ ( x ≈ ∅ ) ) → | |
| 410 -- axiom-of-choice : { X : HOD } → ( ¬x∅ : ¬ ( X == od∅ ) ) → { A : HOD } → (A∈X : A ∈ X ) → choice ¬x∅ A∈X ∈ A | |
| 103 | 411 -- axiom-of-choice {X} nx {A} lt = ¬∅=→∅∈ {!!} |
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