Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate ordinal-definable.agda @ 111:1daa1d24348c
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author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Thu, 20 Jun 2019 13:18:18 +0900 |
parents | dab56d357fa3 |
children | c42352a7ee07 |
rev | line source |
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16 | 1 open import Level |
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posturate OD is isomorphic to Ordinal
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2 module ordinal-definable where |
3 | 3 |
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e11e95d5ddee
separete constructible set
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4 open import zf |
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5 open import ordinal |
3 | 6 |
23 | 7 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) |
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8 open import Relation.Binary.PropositionalEquality |
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9 open import Data.Nat.Properties |
6 | 10 open import Data.Empty |
11 open import Relation.Nullary | |
12 open import Relation.Binary | |
13 open import Relation.Binary.Core | |
14 | |
27 | 15 -- Ordinal Definable Set |
11 | 16 |
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17 record OD {n : Level} : Set (suc n) where |
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18 field |
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19 def : (x : Ordinal {n} ) → Set n |
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20 otrans : {x y : Ordinal {n} } → def x → y o< x → def y |
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21 |
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22 open OD |
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23 open import Data.Unit |
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24 |
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od→lv : {n : Level} → OD {n} → Nat
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25 open Ordinal |
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od→lv : {n : Level} → OD {n} → Nat
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26 |
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27 record _==_ {n : Level} ( a b : OD {n} ) : Set n where |
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28 field |
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29 eq→ : ∀ { x : Ordinal {n} } → def a x → def b x |
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30 eq← : ∀ { x : Ordinal {n} } → def b x → def a x |
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31 |
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32 id : {n : Level} {A : Set n} → A → A |
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33 id x = x |
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34 |
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35 eq-refl : {n : Level} { x : OD {n} } → x == x |
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36 eq-refl {n} {x} = record { eq→ = id ; eq← = id } |
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37 |
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38 open _==_ |
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39 |
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40 eq-sym : {n : Level} { x y : OD {n} } → x == y → y == x |
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41 eq-sym eq = record { eq→ = eq← eq ; eq← = eq→ eq } |
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42 |
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43 eq-trans : {n : Level} { x y z : OD {n} } → x == y → y == z → x == z |
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44 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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45 |
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46 -- Ordinal in OD ( and ZFSet ) |
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47 Ord : { n : Level } → ( a : Ordinal {n} ) → OD {n} |
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48 Ord {n} a = record { def = λ y → y o< a ; otrans = lemma } where |
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49 lemma : {x y : Ordinal} → x o< a → y o< x → y o< a |
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50 lemma {x} {y} x<a y<x = ordtrans {n} {y} {x} {a} y<x x<a |
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51 |
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52 -- od∅ : {n : Level} → OD {n} |
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53 -- od∅ {n} = record { def = λ _ → Lift n ⊥ } |
40 | 54 od∅ : {n : Level} → OD {n} |
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55 od∅ {n} = Ord o∅ |
40 | 56 |
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57 postulate |
100 | 58 -- OD can be iso to a subset of Ordinal ( by means of Godel Set ) |
95 | 59 od→ord : {n : Level} → OD {n} → Ordinal {n} |
60 ord→od : {n : Level} → Ordinal {n} → OD {n} | |
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61 c<→o< : {n : Level} {x y : OD {n} } → def y ( od→ord x ) → od→ord x o< od→ord y |
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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 |
95 | 64 oiso : {n : Level} {x : OD {n}} → ord→od ( od→ord x ) ≡ x |
65 diso : {n : Level} {x : Ordinal {n}} → od→ord ( ord→od x ) ≡ x | |
100 | 66 -- supermum as Replacement Axiom |
95 | 67 sup-o : {n : Level } → ( Ordinal {n} → Ordinal {n}) → Ordinal {n} |
98 | 68 sup-o< : {n : Level } → { ψ : Ordinal {n} → Ordinal {n}} → ∀ {x : Ordinal {n}} → ψ x o< sup-o ψ |
111 | 69 -- contra-position of mimimulity of supermum required in Power Set Axiom |
98 | 70 sup-x : {n : Level } → ( Ordinal {n} → Ordinal {n}) → Ordinal {n} |
71 sup-lb : {n : Level } → { ψ : Ordinal {n} → Ordinal {n}} → {z : Ordinal {n}} → z o< sup-o ψ → z o< osuc (ψ (sup-x ψ)) | |
109
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72 -- sup-lb : {n : Level } → ( ψ : Ordinal {n} → Ordinal {n}) → ( ∀ {x : Ordinal {n}} → ψx o< z ) → z o< osuc ( sup-o ψ ) |
95 | 73 |
74 _∋_ : { n : Level } → ( a x : OD {n} ) → Set n | |
75 _∋_ {n} a x = def a ( od→ord x ) | |
76 | |
77 _c<_ : { n : Level } → ( x a : OD {n} ) → Set n | |
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78 x c< a = a ∋ x |
103 | 79 |
80 postulate | |
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81 o<→c< : {n : Level} {x y : Ordinal {n} } → x o< y → Ord y ∋ Ord x |
95 | 82 |
83 _c≤_ : {n : Level} → OD {n} → OD {n} → Set (suc n) | |
84 a c≤ b = (a ≡ b) ∨ ( b ∋ a ) | |
85 | |
86 def-subst : {n : Level } {Z : OD {n}} {X : Ordinal {n} }{z : OD {n}} {x : Ordinal {n} }→ def Z X → Z ≡ z → X ≡ x → def z x | |
87 def-subst df refl refl = df | |
88 | |
89 sup-od : {n : Level } → ( OD {n} → OD {n}) → OD {n} | |
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90 sup-od ψ = Ord ( sup-o ( λ x → od→ord (ψ (ord→od x ))) ) |
95 | 91 |
92 sup-c< : {n : Level } → ( ψ : OD {n} → OD {n}) → ∀ {x : OD {n}} → def ( sup-od ψ ) (od→ord ( ψ x )) | |
109
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93 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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94 lemma refl (cong ( λ k → od→ord (ψ k) ) oiso) where |
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95 lemma : od→ord (ψ (ord→od (od→ord x))) o< sup-o (λ x → od→ord (ψ (ord→od x))) |
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96 lemma = subst₂ (λ j k → j o< k ) refl diso (o<-subst sup-o< refl (sym diso) ) |
28 | 97 |
37 | 98 ∅3 : {n : Level} → { x : Ordinal {n}} → ( ∀(y : Ordinal {n}) → ¬ (y o< x ) ) → x ≡ o∅ {n} |
81 | 99 ∅3 {n} {x} = TransFinite {n} c2 c3 x where |
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100 c0 : Nat → Ordinal {n} → Set n |
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101 c0 lx x = (∀(y : Ordinal {n}) → ¬ (y o< x)) → x ≡ o∅ {n} |
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102 c2 : (lx : Nat) → c0 lx (record { lv = lx ; ord = Φ lx } ) |
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103 c2 Zero not = refl |
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104 c2 (Suc lx) not with not ( record { lv = lx ; ord = Φ lx } ) |
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105 ... | t with t (case1 ≤-refl ) |
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106 c2 (Suc lx) not | t | () |
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107 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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108 c3 lx (Φ .lx) d not with not ( record { lv = lx ; ord = Φ lx } ) |
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109 ... | t with t (case2 Φ< ) |
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110 c3 lx (Φ .lx) d not | t | () |
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111 c3 lx (OSuc .lx x₁) d not with not ( record { lv = lx ; ord = OSuc lx x₁ } ) |
34 | 112 ... | t with t (case2 (s< s<refl ) ) |
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113 c3 lx (OSuc .lx x₁) d not | t | () |
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114 |
69 | 115 transitive : {n : Level } { z y x : OD {suc n} } → z ∋ y → y ∋ x → z ∋ x |
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116 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 | 117 ... | t = otrans z z∋y (c<→o< {suc n} {x} {y} x∋y ) |
36 | 118 |
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119 record Minimumo {n : Level } (x : Ordinal {n}) : Set (suc n) where |
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120 field |
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121 mino : Ordinal {n} |
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122 min<x : mino o< x |
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123 |
57 | 124 ∅5 : {n : Level} → { x : Ordinal {n} } → ¬ ( x ≡ o∅ {n} ) → o∅ {n} o< x |
125 ∅5 {n} {record { lv = Zero ; ord = (Φ .0) }} not = ⊥-elim (not refl) | |
126 ∅5 {n} {record { lv = Zero ; ord = (OSuc .0 ord) }} not = case2 Φ< | |
127 ∅5 {n} {record { lv = (Suc lv) ; ord = ord }} not = case1 (s≤s z≤n) | |
37 | 128 |
46 | 129 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 } |
130 ord-iso = cong ( λ k → record { lv = lv k ; ord = ord k } ) diso | |
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131 |
51 | 132 -- avoiding lv != Zero error |
133 orefl : {n : Level} → { x : OD {n} } → { y : Ordinal {n} } → od→ord x ≡ y → od→ord x ≡ y | |
134 orefl refl = refl | |
135 | |
136 ==-iso : {n : Level} → { x y : OD {n} } → ord→od (od→ord x) == ord→od (od→ord y) → x == y | |
137 ==-iso {n} {x} {y} eq = record { | |
138 eq→ = λ d → lemma ( eq→ eq (def-subst d (sym oiso) refl )) ; | |
139 eq← = λ d → lemma ( eq← eq (def-subst d (sym oiso) refl )) } | |
140 where | |
141 lemma : {x : OD {n} } {z : Ordinal {n}} → def (ord→od (od→ord x)) z → def x z | |
142 lemma {x} {z} d = def-subst d oiso refl | |
143 | |
57 | 144 =-iso : {n : Level } {x y : OD {suc n} } → (x == y) ≡ (ord→od (od→ord x) == y) |
145 =-iso {_} {_} {y} = cong ( λ k → k == y ) (sym oiso) | |
146 | |
51 | 147 ord→== : {n : Level} → { x y : OD {n} } → od→ord x ≡ od→ord y → x == y |
148 ord→== {n} {x} {y} eq = ==-iso (lemma (od→ord x) (od→ord y) (orefl eq)) where | |
149 lemma : ( ox oy : Ordinal {n} ) → ox ≡ oy → (ord→od ox) == (ord→od oy) | |
150 lemma ox ox refl = eq-refl | |
151 | |
152 o≡→== : {n : Level} → { x y : Ordinal {n} } → x ≡ y → ord→od x == ord→od y | |
153 o≡→== {n} {x} {.x} refl = eq-refl | |
154 | |
155 >→¬< : {x y : Nat } → (x < y ) → ¬ ( y < x ) | |
156 >→¬< (s≤s x<y) (s≤s y<x) = >→¬< x<y y<x | |
157 | |
158 c≤-refl : {n : Level} → ( x : OD {n} ) → x c≤ x | |
159 c≤-refl x = case1 refl | |
160 | |
91 | 161 ∋→o< : {n : Level} → { a x : OD {suc n} } → a ∋ x → od→ord x o< od→ord a |
162 ∋→o< {n} {a} {x} lt = t where | |
163 t : (od→ord x) o< (od→ord a) | |
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164 t = c<→o< {suc n} {x} {a} lt |
91 | 165 |
80 | 166 o∅≡od∅ : {n : Level} → ord→od (o∅ {suc n}) ≡ od∅ {suc n} |
167 o∅≡od∅ {n} with trio< {n} (o∅ {suc n}) (od→ord (od∅ {suc n} )) | |
168 o∅≡od∅ {n} | tri< a ¬b ¬c = ⊥-elim (lemma a) where | |
169 lemma : o∅ {suc n } o< (od→ord (od∅ {suc n} )) → ⊥ | |
103 | 170 lemma lt with def-subst {!!} oiso refl |
171 lemma lt | t = {!!} | |
80 | 172 o∅≡od∅ {n} | tri≈ ¬a b ¬c = trans (cong (λ k → ord→od k ) b ) oiso |
173 o∅≡od∅ {n} | tri> ¬a ¬b c = ⊥-elim (¬x<0 c) | |
174 | |
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175 ord-od∅ : {n : Level} → o∅ {suc n} ≡ od→ord (Ord (o∅ {suc n})) |
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176 ord-od∅ {n} with trio< {n} (o∅ {suc n}) (od→ord (Ord (o∅ {suc n}))) |
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177 ord-od∅ {n} | tri< a ¬b ¬c = ⊥-elim (lemma a) where |
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178 lemma : o∅ {suc n } o< (od→ord (Ord (o∅ {suc n}))) → ⊥ |
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179 lemma lt with o<→c< lt |
111 | 180 lemma lt | t = o<¬≡ refl t |
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181 ord-od∅ {n} | tri≈ ¬a b ¬c = b |
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182 ord-od∅ {n} | tri> ¬a ¬b c = ⊥-elim (¬x<0 c) |
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183 |
51 | 184 o<→¬c> : {n : Level} → { x y : OD {n} } → (od→ord x ) o< ( od→ord y) → ¬ (y c< x ) |
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185 o<→¬c> {n} {x} {y} olt clt = o<> olt (c<→o< clt ) where |
51 | 186 |
187 o≡→¬c< : {n : Level} → { x y : OD {n} } → (od→ord x ) ≡ ( od→ord y) → ¬ x c< y | |
111 | 188 o≡→¬c< {n} {x} {y} oeq lt = o<¬≡ (orefl oeq ) (c<→o< lt) |
54 | 189 |
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190 ∅0 : {n : Level} → record { def = λ x → Lift n ⊥ ; otrans = λ () } == od∅ {n} |
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191 eq→ ∅0 {w} (lift ()) |
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192 eq← ∅0 {w} (case1 ()) |
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193 eq← ∅0 {w} (case2 ()) |
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194 |
60 | 195 ∅< : {n : Level} → { x y : OD {n} } → def x (od→ord y ) → ¬ ( x == od∅ {n} ) |
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196 ∅< {n} {x} {y} d eq with eq→ (eq-trans eq (eq-sym ∅0) ) d |
60 | 197 ∅< {n} {x} {y} d eq | lift () |
57 | 198 |
111 | 199 -- ∅6 : {n : Level} → { x : OD {suc n} } → ¬ ( x ∋ x ) -- no Russel paradox |
200 -- ∅6 {n} {x} x∋x = c<> {n} {x} {x} x∋x x∋x | |
51 | 201 |
76 | 202 def-iso : {n : Level} {A B : OD {n}} {x y : Ordinal {n}} → x ≡ y → (def A y → def B y) → def A x → def B x |
203 def-iso refl t = t | |
204 | |
57 | 205 is-o∅ : {n : Level} → ( x : Ordinal {suc n} ) → Dec ( x ≡ o∅ {suc n} ) |
206 is-o∅ {n} record { lv = Zero ; ord = (Φ .0) } = yes refl | |
207 is-o∅ {n} record { lv = Zero ; ord = (OSuc .0 ord₁) } = no ( λ () ) | |
208 is-o∅ {n} record { lv = (Suc lv₁) ; ord = ord } = no (λ()) | |
209 | |
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210 open _∧_ |
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211 |
79 | 212 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) |
94 | 213 -- postulate f-extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality (suc n) (suc (suc n)) |
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214 |
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215 csuc : {n : Level} → OD {suc n} → OD {suc n} |
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216 csuc x = ord→od ( osuc ( od→ord x )) |
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217 |
96 | 218 -- Power Set of X ( or constructible by λ y → def X (od→ord y ) |
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219 |
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220 ZFSubset : {n : Level} → (A x : OD {suc n} ) → OD {suc n} |
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221 ZFSubset A x = record { def = λ y → def A y ∧ def x y ; otrans = {!!} } |
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222 |
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223 Def : {n : Level} → (A : OD {suc n}) → OD {suc n} |
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224 Def {n} A = ord→od ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) ) |
96 | 225 |
226 -- Constructible Set on α | |
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227 L : {n : Level} → (α : Ordinal {suc n}) → OD {suc n} |
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228 L {n} record { lv = Zero ; ord = (Φ .0) } = od∅ |
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229 L {n} record { lv = lx ; ord = (OSuc lv ox) } = Def ( L {n} ( record { lv = lx ; ord = ox } ) ) |
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230 L {n} record { lv = (Suc lx) ; ord = (Φ (Suc lx)) } = -- Union ( L α ) |
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231 record { def = λ y → osuc y o< (od→ord (L {n} (record { lv = lx ; ord = Φ lx }) )) ; otrans = {!!} } |
89 | 232 |
111 | 233 omega : { n : Level } → Ordinal {n} |
234 omega = record { lv = Suc Zero ; ord = Φ 1 } | |
235 | |
54 | 236 OD→ZF : {n : Level} → ZF {suc (suc n)} {suc n} |
40 | 237 OD→ZF {n} = record { |
54 | 238 ZFSet = OD {suc n} |
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239 ; _∋_ = _∋_ |
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240 ; _≈_ = _==_ |
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241 ; ∅ = od∅ |
28 | 242 ; _,_ = _,_ |
243 ; Union = Union | |
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244 ; Power = Power |
28 | 245 ; Select = Select |
246 ; Replace = Replace | |
111 | 247 ; infinite = Ord omega |
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248 ; isZF = isZF |
28 | 249 } where |
54 | 250 Replace : OD {suc n} → (OD {suc n} → OD {suc n} ) → OD {suc n} |
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251 Replace X ψ = sup-od ψ |
111 | 252 Select : (X : OD {suc n} ) → ((x : OD {suc n} ) → X ∋ x → Set (suc n) ) → OD {suc n} |
253 Select X ψ = record { def = λ x → ( (d : def X x ) → ψ (ord→od x) (subst (λ k → def X k ) (sym diso) d)) ; otrans = lemma } where | |
254 lemma : {x y : Ordinal} → ((d : def X x) → ψ (ord→od x) (subst (def X) (sym diso) d)) → | |
255 y o< x → (d : def X y) → ψ (ord→od y) (subst (def X) (sym diso) d) | |
256 lemma {x} {y} f y<x d = {!!} | |
54 | 257 _,_ : OD {suc n} → OD {suc n} → OD {suc n} |
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258 x , y = Ord (omax (od→ord x) (od→ord y)) |
54 | 259 Union : OD {suc n} → OD {suc n} |
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260 Union U = record { def = λ y → osuc y o< (od→ord U) ; otrans = {!!} } |
77 | 261 -- power : ∀ X ∃ A ∀ t ( t ∈ A ↔ ( ∀ {x} → t ∋ x → X ∋ x ) |
54 | 262 Power : OD {suc n} → OD {suc n} |
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263 Power A = Def A |
54 | 264 ZFSet = OD {suc n} |
265 _∈_ : ( A B : ZFSet ) → Set (suc n) | |
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266 A ∈ B = B ∋ A |
54 | 267 _⊆_ : ( A B : ZFSet ) → ∀{ x : ZFSet } → Set (suc n) |
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268 _⊆_ A B {x} = A ∋ x → B ∋ x |
103 | 269 _∩_ : ( A B : ZFSet ) → ZFSet |
111 | 270 A ∩ B = Select (A , B) ( λ x d → ( A ∋ x ) ∧ (B ∋ x) ) |
96 | 271 -- _∪_ : ( A B : ZFSet ) → ZFSet |
272 -- A ∪ B = Select (A , B) ( λ x → (A ∋ x) ∨ ( B ∋ x ) ) | |
103 | 273 {_} : ZFSet → ZFSet |
274 { x } = ( x , x ) | |
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275 |
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276 infixr 200 _∈_ |
96 | 277 -- infixr 230 _∩_ _∪_ |
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278 infixr 220 _⊆_ |
111 | 279 isZF : IsZF (OD {suc n}) _∋_ _==_ od∅ _,_ Union Power Select Replace (Ord omega) |
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280 isZF = record { |
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281 isEquivalence = record { refl = eq-refl ; sym = eq-sym; trans = eq-trans } |
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282 ; pair = pair |
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283 ; union-u = λ _ z _ → csuc z |
72 | 284 ; union→ = union→ |
285 ; union← = union← | |
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286 ; empty = empty |
76 | 287 ; power→ = power→ |
288 ; power← = power← | |
289 ; extensionality = extensionality | |
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290 ; minimul = minimul |
51 | 291 ; regularity = regularity |
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292 ; infinity∅ = infinity∅ |
93 | 293 ; infinity = λ _ → infinity |
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294 ; selection = λ {ψ} {X} {y} → selection {ψ} {X} {y} |
93 | 295 ; replacement = replacement |
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296 } where |
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297 open _∧_ |
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298 open Minimumo |
54 | 299 pair : (A B : OD {suc n} ) → ((A , B) ∋ A) ∧ ((A , B) ∋ B) |
87 | 300 proj1 (pair A B ) = omax-x {n} (od→ord A) (od→ord B) |
301 proj2 (pair A B ) = omax-y {n} (od→ord A) (od→ord B) | |
54 | 302 empty : (x : OD {suc n} ) → ¬ (od∅ ∋ x) |
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303 empty x (case1 ()) |
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304 empty x (case2 ()) |
100 | 305 --- |
306 --- ZFSubset A x = record { def = λ y → def A y ∧ def x y } subset of A | |
307 --- 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 | |
308 -- | |
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309 -- if Power A ∋ t, from a propertiy of minimum sup there is osuc ZFSubset A ∋ t |
100 | 310 -- then ZFSubset A ≡ t or ZFSubset A ∋ t. In the former case ZFSubset A ∋ x implies A ∋ x |
311 -- In case of later, ZFSubset A ∋ t and t ∋ x implies ZFSubset A ∋ x by transitivity | |
312 -- | |
76 | 313 power→ : (A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → A ∋ x |
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314 power→ A t P∋t {x} t∋x = proj1 lemma-s where |
98 | 315 minsup : OD |
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316 minsup = ZFSubset A ( ord→od ( sup-x (λ x → od→ord ( ZFSubset A (ord→od x))))) |
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317 lemma-t : csuc minsup ∋ t |
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318 lemma-t = {!!} -- o<→c< (o<-subst (sup-lb (o<-subst (c<→o< P∋t) refl diso )) refl refl ) |
98 | 319 lemma-s : ZFSubset A ( ord→od ( sup-x (λ x → od→ord ( ZFSubset A (ord→od x))))) ∋ x |
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320 lemma-s with osuc-≡< ( o<-subst (c<→o< lemma-t ) refl diso ) |
111 | 321 lemma-s | case1 eq = def-subst {!!} oiso refl |
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322 lemma-s | case2 lt = transitive {n} {minsup} {t} {x} (def-subst {!!} oiso refl ) t∋x |
100 | 323 -- |
324 -- we have t ∋ x → A ∋ x means t is a subset of A, that is ZFSubset A t == t | |
325 -- Power A is a sup of ZFSubset A t, so Power A ∋ t | |
326 -- | |
77 | 327 power← : (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t |
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328 power← A t t→A = def-subst {suc n} {_} {_} {Power A} {od→ord t} |
103 | 329 {!!} refl lemma1 where |
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330 lemma-eq : ZFSubset A t == t |
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331 eq→ lemma-eq {z} w = proj2 w |
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332 eq← lemma-eq {z} w = record { proj2 = w ; |
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333 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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334 lemma1 : od→ord (ZFSubset A (ord→od (od→ord t))) ≡ od→ord t |
111 | 335 lemma1 = subst (λ k → od→ord (ZFSubset A k) ≡ od→ord t ) (sym oiso) {!!} |
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336 lemma : od→ord (ZFSubset A (ord→od (od→ord t)) ) o< sup-o (λ x → od→ord (ZFSubset A (ord→od x))) |
98 | 337 lemma = sup-o< |
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338 union-lemma-u : {X z : OD {suc n}} → (U>z : Union X ∋ z ) → csuc z ∋ z |
72 | 339 union-lemma-u {X} {z} U>z = lemma <-osuc where |
340 lemma : {oz ooz : Ordinal {suc n}} → oz o< ooz → def (ord→od ooz) oz | |
103 | 341 lemma {oz} {ooz} lt = def-subst {suc n} {ord→od ooz} {!!} refl refl |
73 | 342 union→ : (X z u : OD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z |
72 | 343 union→ X y u xx with trio< ( od→ord u ) ( osuc ( od→ord y )) |
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344 union→ X y u xx | tri< a ¬b ¬c with osuc-< a (c<→o< (proj2 xx)) |
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345 union→ X y u xx | tri< a ¬b ¬c | () |
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346 union→ X y u xx | tri≈ ¬a b ¬c = lemma b (c<→o< (proj1 xx )) where |
73 | 347 lemma : {oX ou ooy : Ordinal {suc n}} → ou ≡ ooy → ou o< oX → ooy o< oX |
348 lemma refl lt = lt | |
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349 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 )) |
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350 union← : (X z : OD) (X∋z : Union X ∋ z) → (X ∋ csuc z) ∧ (csuc z ∋ z ) |
103 | 351 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 } |
54 | 352 ψiso : {ψ : OD {suc n} → Set (suc n)} {x y : OD {suc n}} → ψ x → x ≡ y → ψ y |
353 ψiso {ψ} t refl = t | |
111 | 354 selection : {X : OD } {ψ : (x : OD ) → x ∈ X → Set (suc n)} {y : OD} → ((d : X ∋ y ) → ψ y d ) ⇔ (Select X ψ ∋ y) |
355 selection {ψ} {X} {y} = {!!} | |
93 | 356 replacement : {ψ : OD → OD} (X x : OD) → Replace X ψ ∋ ψ x |
357 replacement {ψ} X x = sup-c< ψ {x} | |
60 | 358 ∅-iso : {x : OD} → ¬ (x == od∅) → ¬ ((ord→od (od→ord x)) == od∅) |
359 ∅-iso {x} neq = subst (λ k → ¬ k) (=-iso {n} ) neq | |
54 | 360 minimul : (x : OD {suc n} ) → ¬ (x == od∅ )→ OD {suc n} |
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361 minimul x not = {!!} |
57 | 362 regularity : (x : OD) (not : ¬ (x == od∅)) → |
111 | 363 (x ∋ minimul x not) ∧ (Select (minimul x not) (λ x₁ d → (minimul x not ∋ x₁) ∧ (x ∋ x₁)) == od∅) |
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364 proj1 (regularity x not ) = {!!} |
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365 proj2 (regularity x not ) = record { eq→ = reg ; eq← = {!!} } where |
111 | 366 reg : {y : Ordinal} → def (Select (minimul x not) (λ x₂ d → (minimul x not ∋ x₂) ∧ (x ∋ x₂))) y → def od∅ y |
367 reg {y} t = {!!} | |
76 | 368 extensionality : {A B : OD {suc n}} → ((z : OD) → (A ∋ z) ⇔ (B ∋ z)) → A == B |
369 eq→ (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {A} {B} (sym diso) (proj1 (eq (ord→od x))) d | |
370 eq← (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {B} {A} (sym diso) (proj2 (eq (ord→od x))) d | |
89 | 371 xx-union : {x : OD {suc n}} → (x , x) ≡ record { def = λ z → z o< osuc (od→ord x) } |
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372 xx-union {x} = cong ( λ k → Ord k ) (omxx (od→ord x)) |
89 | 373 xxx-union : {x : OD {suc n}} → (x , (x , x)) ≡ record { def = λ z → z o< osuc (osuc (od→ord x))} |
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374 xxx-union {x} = cong ( λ k → Ord k ) lemma where |
91 | 375 lemma1 : {x : OD {suc n}} → od→ord x o< od→ord (x , x) |
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376 lemma1 {x} = c<→o< ( proj1 (pair x x ) ) |
91 | 377 lemma2 : {x : OD {suc n}} → od→ord (x , x) ≡ osuc (od→ord x) |
111 | 378 lemma2 = trans ( cong ( λ k → od→ord k ) xx-union ) {!!} |
89 | 379 lemma : {x : OD {suc n}} → omax (od→ord x) (od→ord (x , x)) ≡ osuc (osuc (od→ord x)) |
91 | 380 lemma {x} = trans ( sym ( omax< (od→ord x) (od→ord (x , x)) lemma1 ) ) ( cong ( λ k → osuc k ) lemma2 ) |
90 | 381 uxxx-union : {x : OD {suc n}} → Union (x , (x , x)) ≡ record { def = λ z → osuc z o< osuc (osuc (od→ord x)) } |
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382 uxxx-union {x} = cong ( λ k → record { def = λ z → osuc z o< k ; otrans = {!!} } ) lemma where |
90 | 383 lemma : od→ord (x , (x , x)) ≡ osuc (osuc (od→ord x)) |
111 | 384 lemma = trans ( cong ( λ k → od→ord k ) xxx-union ) {!!} |
91 | 385 uxxx-2 : {x : OD {suc n}} → record { def = λ z → osuc z o< osuc (osuc (od→ord x)) } == record { def = λ z → z o< osuc (od→ord x) } |
386 eq→ ( uxxx-2 {x} ) {m} lt = proj1 (osuc2 m (od→ord x)) lt | |
387 eq← ( uxxx-2 {x} ) {m} lt = proj2 (osuc2 m (od→ord x)) lt | |
388 uxxx-ord : {x : OD {suc n}} → od→ord (Union (x , (x , x))) ≡ osuc (od→ord x) | |
111 | 389 uxxx-ord {x} = trans (cong (λ k → od→ord k ) uxxx-union) {!!} |
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390 infinite : OD {suc n} |
111 | 391 infinite = Ord omega |
392 infinity∅ : Ord omega ∋ od∅ {suc n} | |
393 infinity∅ = {!!} | |
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394 infinity : (x : OD) → infinite ∋ x → infinite ∋ Union (x , (x , x )) |
111 | 395 infinity x lt = {!!} where |
91 | 396 lemma : (ox : Ordinal {suc n} ) → ox o< omega → osuc ox o< omega |
397 lemma record { lv = Zero ; ord = (Φ .0) } (case1 (s≤s x)) = case1 (s≤s z≤n) | |
398 lemma record { lv = Zero ; ord = (OSuc .0 ord₁) } (case1 (s≤s x)) = case1 (s≤s z≤n) | |
399 lemma record { lv = (Suc lv₁) ; ord = (Φ .(Suc lv₁)) } (case1 (s≤s ())) | |
400 lemma record { lv = (Suc lv₁) ; ord = (OSuc .(Suc lv₁) ord₁) } (case1 (s≤s ())) | |
401 lemma record { lv = 1 ; ord = (Φ 1) } (case2 c2) with d<→lv c2 | |
402 lemma record { lv = (Suc Zero) ; ord = (Φ .1) } (case2 ()) | refl | |
103 | 403 -- ∀ X [ ∅ ∉ X → (∃ f : X → ⋃ X ) → ∀ A ∈ X ( f ( A ) ∈ A ) ] -- this form is no good since X is a transitive set |
404 -- ∀ z [ ∀ x ( x ∈ z → ¬ ( x ≈ ∅ ) ) ∧ ∀ x ∀ y ( x , y ∈ z ∧ ¬ ( x ≈ y ) → x ∩ y ≈ ∅ ) → ∃ u ∀ x ( x ∈ z → ∃ t ( u ∩ x) ≈ { t }) ] | |
405 record Choice (z : OD {suc n}) : Set (suc (suc n)) where | |
406 field | |
407 u : {x : OD {suc n}} ( x∈z : x ∈ z ) → OD {suc n} | |
408 t : {x : OD {suc n}} ( x∈z : x ∈ z ) → (x : OD {suc n} ) → OD {suc n} | |
409 choice : { x : OD {suc n} } → ( x∈z : x ∈ z ) → ( u x∈z ∩ x) == { t x∈z x } | |
410 -- choice : {x : OD {suc n}} ( x ∈ z → ¬ ( x ≈ ∅ ) ) → | |
411 -- axiom-of-choice : { X : OD } → ( ¬x∅ : ¬ ( X == od∅ ) ) → { A : OD } → (A∈X : A ∈ X ) → choice ¬x∅ A∈X ∈ A | |
412 -- axiom-of-choice {X} nx {A} lt = ¬∅=→∅∈ {!!} | |
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413 |