Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate HOD.agda @ 140:312e27aa3cb5
remove otrans again. start over
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sun, 07 Jul 2019 23:02:47 +0900 |
| parents | 53077af367e9 |
| children | 21b2654985c4 |
| 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 |
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19 |
| 112 | 20 open HOD |
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21 open import Data.Unit |
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22 |
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23 open Ordinal |
| 120 | 24 open _∧_ |
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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 |
| 120 | 45 ⇔→== : {n : Level} { x y : HOD {suc n} } → ( {z : Ordinal {suc n}} → def x z ⇔ def y z) → x == y |
| 46 eq→ ( ⇔→== {n} {x} {y} eq ) {z} m = proj1 eq m | |
| 47 eq← ( ⇔→== {n} {x} {y} eq ) {z} m = proj2 eq m | |
| 48 | |
| 112 | 49 -- Ordinal in HOD ( and ZFSet ) |
| 50 Ord : { n : Level } → ( a : Ordinal {n} ) → HOD {n} | |
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51 Ord {n} a = record { def = λ y → y o< a } |
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52 |
| 112 | 53 od∅ : {n : Level} → HOD {n} |
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54 od∅ {n} = Ord o∅ |
| 40 | 55 |
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56 postulate |
| 112 | 57 -- HOD can be iso to a subset of Ordinal ( by means of Godel Set ) |
| 58 od→ord : {n : Level} → HOD {n} → Ordinal {n} | |
| 113 | 59 ord→od : {n : Level} → Ordinal {n} → HOD {n} |
| 129 | 60 c<→o< : {n : Level} {x y : HOD {n} } → def y ( od→ord x ) → od→ord x o< od→ord y |
| 113 | 61 oiso : {n : Level} {x : HOD {n}} → ord→od ( od→ord x ) ≡ x |
| 62 diso : {n : Level} {x : Ordinal {n}} → od→ord ( ord→od x ) ≡ x | |
| 116 | 63 ord-Ord :{n : Level} {x : Ordinal {n}} → x ≡ od→ord (Ord x) |
| 120 | 64 ==→o≡ : {n : Level} → { x y : HOD {suc n} } → (x == y) → x ≡ y |
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65 -- next assumption causes ∀ x ∋ ∅ . It menas only an ordinal becomes a set |
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66 -- o<→c< : {n : Level} {x y : Ordinal {n} } → x o< y → def (ord→od y) x |
| 100 | 67 -- supermum as Replacement Axiom |
| 95 | 68 sup-o : {n : Level } → ( Ordinal {n} → Ordinal {n}) → Ordinal {n} |
| 98 | 69 sup-o< : {n : Level } → { ψ : Ordinal {n} → Ordinal {n}} → ∀ {x : Ordinal {n}} → ψ x o< sup-o ψ |
| 111 | 70 -- contra-position of mimimulity of supermum required in Power Set Axiom |
| 98 | 71 sup-x : {n : Level } → ( Ordinal {n} → Ordinal {n}) → Ordinal {n} |
| 72 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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73 -- sup-lb : {n : Level } → ( ψ : Ordinal {n} → Ordinal {n}) → ( ∀ {x : Ordinal {n}} → ψx o< z ) → z o< osuc ( sup-o ψ ) |
| 117 | 74 minimul : {n : Level } → (x : HOD {suc n} ) → ¬ (x == od∅ )→ HOD {suc n} |
| 75 -- this should be ¬ (x == od∅ )→ ∃ ox → x ∋ Ord ox ( minimum of x ) | |
| 76 x∋minimul : {n : Level } → (x : HOD {suc n} ) → ( ne : ¬ (x == od∅ ) ) → def x ( od→ord ( minimul x ne ) ) | |
| 77 minimul-1 : {n : Level } → (x : HOD {suc n} ) → ( ne : ¬ (x == od∅ ) ) → (y : HOD {suc n}) → ¬ ( def (minimul x ne) (od→ord y)) ∧ (def x (od→ord y) ) | |
| 123 | 78 -- we should prove this in agda, but simply put here |
| 79 ===-≡ : {n : Level} { x y : HOD {suc n}} → x == y → x ≡ y | |
| 80 | |
| 81 Ord-ord : {n : Level } {ox : Ordinal {suc n}} → Ord ox ≡ ord→od ox | |
| 82 Ord-ord {n} {px} = trans (sym oiso) (cong ( λ k → ord→od k ) (sym ord-Ord)) | |
| 95 | 83 |
| 112 | 84 _∋_ : { n : Level } → ( a x : HOD {n} ) → Set n |
| 95 | 85 _∋_ {n} a x = def a ( od→ord x ) |
| 86 | |
| 112 | 87 _c<_ : { n : Level } → ( x a : HOD {n} ) → Set n |
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88 x c< a = a ∋ x |
| 103 | 89 |
| 112 | 90 _c≤_ : {n : Level} → HOD {n} → HOD {n} → Set (suc n) |
| 95 | 91 a c≤ b = (a ≡ b) ∨ ( b ∋ a ) |
| 92 | |
| 113 | 93 cseq : {n : Level} → HOD {n} → HOD {n} |
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94 cseq x = record { def = λ y → def x (osuc y) } where |
| 113 | 95 |
| 112 | 96 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 | 97 def-subst df refl refl = df |
| 98 | |
| 113 | 99 o<→c< : {n : Level} {x y : Ordinal {n} } → x o< y → Ord y ∋ Ord x |
| 100 o<→c< {n} {x} {y} lt = subst ( λ k → k o< y ) ord-Ord lt | |
| 101 | |
| 112 | 102 sup-od : {n : Level } → ( HOD {n} → HOD {n}) → HOD {n} |
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103 sup-od ψ = Ord ( sup-o ( λ x → od→ord (ψ (ord→od x ))) ) |
| 95 | 104 |
| 112 | 105 sup-c< : {n : Level } → ( ψ : HOD {n} → HOD {n}) → ∀ {x : HOD {n}} → def ( sup-od ψ ) (od→ord ( ψ x )) |
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106 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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107 lemma refl (cong ( λ k → od→ord (ψ k) ) oiso) where |
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108 lemma : od→ord (ψ (ord→od (od→ord x))) o< sup-o (λ x → od→ord (ψ (ord→od x))) |
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109 lemma = subst₂ (λ j k → j o< k ) refl diso (o<-subst sup-o< refl (sym diso) ) |
| 28 | 110 |
| 123 | 111 o<→o> : {n : Level} → { x y : Ordinal {n} } → (Ord x == Ord y) → x o< y → ⊥ |
| 112 o<→o> {n} {x} {y} record { eq→ = xy ; eq← = yx } (case1 lt) with o<-subst (yx (case1 lt)) ord-Ord refl | |
| 113 ... | oyx with o<¬≡ refl (c<→o< {n} {Ord x} oyx ) | |
| 114 ... | () | |
| 115 o<→o> {n} {x} {y} record { eq→ = xy ; eq← = yx } (case2 lt) with o<-subst (yx (case2 lt)) ord-Ord refl | |
| 116 ... | oyx with o<¬≡ refl (c<→o< {n} {Ord x} oyx ) | |
| 117 ... | () | |
| 118 | |
| 119 Ord==→≡ : {n : Level} { x y : Ordinal {suc n}} → Ord x == Ord y → x ≡ y | |
| 120 Ord==→≡ {n} {x} {y} eq with trio< x y | |
| 121 Ord==→≡ {n} {x} {y} eq | tri< a ¬b ¬c = ⊥-elim ( o<→o> eq a ) | |
| 122 Ord==→≡ {n} {x} {y} eq | tri≈ ¬a b ¬c = b | |
| 123 Ord==→≡ {n} {x} {y} eq | tri> ¬a ¬b c = ⊥-elim ( o<→o> (eq-sym eq) c ) | |
| 124 | |
| 125 | |
| 37 | 126 ∅3 : {n : Level} → { x : Ordinal {n}} → ( ∀(y : Ordinal {n}) → ¬ (y o< x ) ) → x ≡ o∅ {n} |
| 81 | 127 ∅3 {n} {x} = TransFinite {n} c2 c3 x where |
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128 c0 : Nat → Ordinal {n} → Set n |
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129 c0 lx x = (∀(y : Ordinal {n}) → ¬ (y o< x)) → x ≡ o∅ {n} |
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130 c2 : (lx : Nat) → c0 lx (record { lv = lx ; ord = Φ lx } ) |
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131 c2 Zero not = refl |
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132 c2 (Suc lx) not with not ( record { lv = lx ; ord = Φ lx } ) |
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133 ... | t with t (case1 ≤-refl ) |
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134 c2 (Suc lx) not | t | () |
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135 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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136 c3 lx (Φ .lx) d not with not ( record { lv = lx ; ord = Φ lx } ) |
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137 ... | t with t (case2 Φ< ) |
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138 c3 lx (Φ .lx) d not | t | () |
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139 c3 lx (OSuc .lx x₁) d not with not ( record { lv = lx ; ord = OSuc lx x₁ } ) |
| 34 | 140 ... | t with t (case2 (s< s<refl ) ) |
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141 c3 lx (OSuc .lx x₁) d not | t | () |
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142 |
| 57 | 143 ∅5 : {n : Level} → { x : Ordinal {n} } → ¬ ( x ≡ o∅ {n} ) → o∅ {n} o< x |
| 144 ∅5 {n} {record { lv = Zero ; ord = (Φ .0) }} not = ⊥-elim (not refl) | |
| 145 ∅5 {n} {record { lv = Zero ; ord = (OSuc .0 ord) }} not = case2 Φ< | |
| 146 ∅5 {n} {record { lv = (Suc lv) ; ord = ord }} not = case1 (s≤s z≤n) | |
| 37 | 147 |
| 46 | 148 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 } |
| 149 ord-iso = cong ( λ k → record { lv = lv k ; ord = ord k } ) diso | |
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150 |
| 51 | 151 -- avoiding lv != Zero error |
| 112 | 152 orefl : {n : Level} → { x : HOD {n} } → { y : Ordinal {n} } → od→ord x ≡ y → od→ord x ≡ y |
| 51 | 153 orefl refl = refl |
| 154 | |
| 112 | 155 ==-iso : {n : Level} → { x y : HOD {n} } → ord→od (od→ord x) == ord→od (od→ord y) → x == y |
| 51 | 156 ==-iso {n} {x} {y} eq = record { |
| 157 eq→ = λ d → lemma ( eq→ eq (def-subst d (sym oiso) refl )) ; | |
| 158 eq← = λ d → lemma ( eq← eq (def-subst d (sym oiso) refl )) } | |
| 159 where | |
| 112 | 160 lemma : {x : HOD {n} } {z : Ordinal {n}} → def (ord→od (od→ord x)) z → def x z |
| 51 | 161 lemma {x} {z} d = def-subst d oiso refl |
| 162 | |
| 112 | 163 =-iso : {n : Level } {x y : HOD {suc n} } → (x == y) ≡ (ord→od (od→ord x) == y) |
| 57 | 164 =-iso {_} {_} {y} = cong ( λ k → k == y ) (sym oiso) |
| 165 | |
| 112 | 166 ord→== : {n : Level} → { x y : HOD {n} } → od→ord x ≡ od→ord y → x == y |
| 51 | 167 ord→== {n} {x} {y} eq = ==-iso (lemma (od→ord x) (od→ord y) (orefl eq)) where |
| 168 lemma : ( ox oy : Ordinal {n} ) → ox ≡ oy → (ord→od ox) == (ord→od oy) | |
| 169 lemma ox ox refl = eq-refl | |
| 170 | |
| 171 o≡→== : {n : Level} → { x y : Ordinal {n} } → x ≡ y → ord→od x == ord→od y | |
| 172 o≡→== {n} {x} {.x} refl = eq-refl | |
| 173 | |
| 174 >→¬< : {x y : Nat } → (x < y ) → ¬ ( y < x ) | |
| 175 >→¬< (s≤s x<y) (s≤s y<x) = >→¬< x<y y<x | |
| 176 | |
| 112 | 177 c≤-refl : {n : Level} → ( x : HOD {n} ) → x c≤ x |
| 51 | 178 c≤-refl x = case1 refl |
| 179 | |
| 112 | 180 ∋→o< : {n : Level} → { a x : HOD {suc n} } → a ∋ x → od→ord x o< od→ord a |
| 91 | 181 ∋→o< {n} {a} {x} lt = t where |
| 182 t : (od→ord x) o< (od→ord a) | |
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183 t = c<→o< {suc n} {x} {a} lt |
| 91 | 184 |
| 80 | 185 o∅≡od∅ : {n : Level} → ord→od (o∅ {suc n}) ≡ od∅ {suc n} |
| 186 o∅≡od∅ {n} with trio< {n} (o∅ {suc n}) (od→ord (od∅ {suc n} )) | |
| 187 o∅≡od∅ {n} | tri< a ¬b ¬c = ⊥-elim (lemma a) where | |
| 188 lemma : o∅ {suc n } o< (od→ord (od∅ {suc n} )) → ⊥ | |
| 113 | 189 lemma lt with o<→c< lt |
| 190 lemma lt | t = o<¬≡ refl t | |
| 80 | 191 o∅≡od∅ {n} | tri≈ ¬a b ¬c = trans (cong (λ k → ord→od k ) b ) oiso |
| 192 o∅≡od∅ {n} | tri> ¬a ¬b c = ⊥-elim (¬x<0 c) | |
| 193 | |
| 112 | 194 o<→¬c> : {n : Level} → { x y : HOD {n} } → (od→ord x ) o< ( od→ord y) → ¬ (y c< x ) |
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195 o<→¬c> {n} {x} {y} olt clt = o<> olt (c<→o< clt ) where |
| 51 | 196 |
| 112 | 197 o≡→¬c< : {n : Level} → { x y : HOD {n} } → (od→ord x ) ≡ ( od→ord y) → ¬ x c< y |
| 111 | 198 o≡→¬c< {n} {x} {y} oeq lt = o<¬≡ (orefl oeq ) (c<→o< lt) |
| 54 | 199 |
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200 ∅0 : {n : Level} → record { def = λ x → Lift n ⊥ } == od∅ {n} |
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201 eq→ ∅0 {w} (lift ()) |
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202 eq← ∅0 {w} (case1 ()) |
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203 eq← ∅0 {w} (case2 ()) |
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204 |
| 112 | 205 ∅< : {n : Level} → { x y : HOD {n} } → def x (od→ord y ) → ¬ ( x == od∅ {n} ) |
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206 ∅< {n} {x} {y} d eq with eq→ (eq-trans eq (eq-sym ∅0) ) d |
| 60 | 207 ∅< {n} {x} {y} d eq | lift () |
| 57 | 208 |
| 120 | 209 ∅6 : {n : Level} → { x : HOD {suc n} } → ¬ ( x ∋ x ) -- no Russel paradox |
| 210 ∅6 {n} {x} x∋x = o<¬≡ refl ( c<→o< {suc n} {x} {x} x∋x ) | |
| 51 | 211 |
| 112 | 212 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 | 213 def-iso refl t = t |
| 214 | |
| 57 | 215 is-o∅ : {n : Level} → ( x : Ordinal {suc n} ) → Dec ( x ≡ o∅ {suc n} ) |
| 216 is-o∅ {n} record { lv = Zero ; ord = (Φ .0) } = yes refl | |
| 217 is-o∅ {n} record { lv = Zero ; ord = (OSuc .0 ord₁) } = no ( λ () ) | |
| 218 is-o∅ {n} record { lv = (Suc lv₁) ; ord = ord } = no (λ()) | |
| 219 | |
| 119 | 220 |
| 79 | 221 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) |
| 94 | 222 -- postulate f-extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality (suc n) (suc (suc n)) |
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223 |
| 112 | 224 csuc : {n : Level} → HOD {suc n} → HOD {suc n} |
| 122 | 225 csuc x = Ord ( osuc ( od→ord x )) |
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226 |
| 96 | 227 -- Power Set of X ( or constructible by λ y → def X (od→ord y ) |
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228 |
| 112 | 229 ZFSubset : {n : Level} → (A x : HOD {suc n} ) → HOD {suc n} |
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230 ZFSubset A x = record { def = λ y → def A y ∧ def x y } where |
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231 |
| 112 | 232 Def : {n : Level} → (A : HOD {suc n}) → HOD {suc n} |
| 121 | 233 Def {n} A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) ) |
| 96 | 234 |
| 129 | 235 OrdSubset : {n : Level} → (A x : Ordinal {suc n} ) → ZFSubset (Ord A) (Ord x) ≡ Ord ( minα A x ) |
| 236 OrdSubset {n} A x = ===-≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where | |
| 237 lemma1 : {y : Ordinal} → def (ZFSubset (Ord A) (Ord x)) y → def (Ord (minα A x)) y | |
| 238 lemma1 {y} s with trio< A x | |
| 239 lemma1 {y} s | tri< a ¬b ¬c = proj1 s | |
| 240 lemma1 {y} s | tri≈ ¬a refl ¬c = proj1 s | |
| 241 lemma1 {y} s | tri> ¬a ¬b c = proj2 s | |
| 242 lemma2 : {y : Ordinal} → def (Ord (minα A x)) y → def (ZFSubset (Ord A) (Ord x)) y | |
| 243 lemma2 {y} lt with trio< A x | |
| 244 lemma2 {y} lt | tri< a ¬b ¬c = record { proj1 = lt ; proj2 = ordtrans lt a } | |
| 245 lemma2 {y} lt | tri≈ ¬a refl ¬c = record { proj1 = lt ; proj2 = lt } | |
| 246 lemma2 {y} lt | tri> ¬a ¬b c = record { proj1 = ordtrans lt c ; proj2 = lt } | |
| 247 | |
| 96 | 248 -- Constructible Set on α |
| 122 | 249 -- Def X = record { def = λ y → ∀ (x : OD ) → y < X ∧ y < od→ord x } |
| 250 -- L (Φ 0) = Φ | |
| 251 -- L (OSuc lv n) = { Def ( L n ) } | |
| 252 -- L (Φ (Suc n)) = Union (L α) (α < Φ (Suc n) ) | |
| 112 | 253 L : {n : Level} → (α : Ordinal {suc n}) → HOD {suc n} |
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254 L {n} record { lv = Zero ; ord = (Φ .0) } = od∅ |
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255 L {n} record { lv = lx ; ord = (OSuc lv ox) } = Def ( L {n} ( record { lv = lx ; ord = ox } ) ) |
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256 L {n} record { lv = (Suc lx) ; ord = (Φ (Suc lx)) } = -- Union ( L α ) |
| 121 | 257 cseq ( Ord (od→ord (L {n} (record { lv = lx ; ord = Φ lx })))) |
| 89 | 258 |
| 123 | 259 L00 : {n : Level} → (ox : Ordinal {suc n}) → ox o< sup-o ( λ x → od→ord ( ZFSubset (Ord ox) (ord→od x ))) |
| 260 L00 {n} ox = o<-subst {suc n} {_} {_} {ox} {sup-o ( λ x → od→ord ( ZFSubset (Ord ox) (ord→od x )))} | |
| 261 (sup-o< {suc n} {λ x → od→ord ( ZFSubset (Ord ox) (ord→od x ))} {ox} ) (lemma0 ox) refl where | |
| 262 lemma1 : {n : Level } {ox z : Ordinal {suc n}} → ( def (Ord ox) z ∧ def (ord→od ox) z ) ⇔ def ( Ord ox ) z | |
| 263 lemma1 {n} {ox} {z} = record { proj1 = proj1 ; proj2 = λ t → record { proj1 = t ; proj2 = subst (λ k → def k z ) Ord-ord t }} | |
| 122 | 264 lemma0 : {n : Level} → (ox : Ordinal {suc n}) → od→ord (ZFSubset (Ord ox) (ord→od ox)) ≡ ox |
| 123 | 265 lemma0 {n} ox = trans (cong (λ k → od→ord k) (===-≡ (⇔→== lemma1) )) (sym ord-Ord) |
| 122 | 266 |
| 123 | 267 -- L0 : {n : Level} → (α : Ordinal {suc n}) → α o< β → L (osuc α) ∋ L α |
| 268 -- L1 : {n : Level} → (α β : Ordinal {suc n}) → α o< β → ∀ (x : HOD {suc n}) → L α ∋ x → L β ∋ x | |
| 122 | 269 |
| 111 | 270 omega : { n : Level } → Ordinal {n} |
| 271 omega = record { lv = Suc Zero ; ord = Φ 1 } | |
| 272 | |
| 112 | 273 HOD→ZF : {n : Level} → ZF {suc (suc n)} {suc n} |
| 274 HOD→ZF {n} = record { | |
| 275 ZFSet = HOD {suc n} | |
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276 ; _∋_ = _∋_ |
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277 ; _≈_ = _==_ |
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278 ; ∅ = od∅ |
| 28 | 279 ; _,_ = _,_ |
| 280 ; Union = Union | |
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281 ; Power = Power |
| 28 | 282 ; Select = Select |
| 283 ; Replace = Replace | |
| 111 | 284 ; infinite = Ord omega |
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285 ; isZF = isZF |
| 28 | 286 } where |
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287 Select : (X : HOD {suc n} ) → ((x : HOD {suc n} ) → Set (suc n) ) → HOD {suc n} |
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288 Select X ψ = record { def = λ x → ( def X x ∧ ψ ( ord→od x )) } |
| 112 | 289 Replace : HOD {suc n} → (HOD {suc n} → HOD {suc n} ) → HOD {suc n} |
| 139 | 290 Replace X ψ = Select ( Ord (sup-o ( λ x → od→ord (ψ (ord→od x ))))) ( λ x → ¬ (∀ (y : Ordinal ) → ¬ ( def X y ∧ ( x == ψ (Ord y) )))) |
| 112 | 291 _,_ : HOD {suc n} → HOD {suc n} → HOD {suc n} |
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292 x , y = Ord (omax (od→ord x) (od→ord y)) |
| 112 | 293 Union : HOD {suc n} → HOD {suc n} |
| 113 | 294 Union U = cseq U |
| 77 | 295 -- power : ∀ X ∃ A ∀ t ( t ∈ A ↔ ( ∀ {x} → t ∋ x → X ∋ x ) |
| 112 | 296 ZFSet = HOD {suc n} |
| 54 | 297 _∈_ : ( A B : ZFSet ) → Set (suc n) |
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298 A ∈ B = B ∋ A |
| 54 | 299 _⊆_ : ( A B : ZFSet ) → ∀{ x : ZFSet } → Set (suc n) |
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300 _⊆_ A B {x} = A ∋ x → B ∋ x |
| 103 | 301 _∩_ : ( A B : ZFSet ) → ZFSet |
| 115 | 302 A ∩ B = Select (A , B) ( λ x → ( A ∋ x ) ∧ (B ∋ x) ) |
| 129 | 303 Power : HOD {suc n} → HOD {suc n} |
| 304 Power A = Replace (Def (Ord (od→ord A))) ( λ x → A ∩ x ) | |
| 96 | 305 -- _∪_ : ( A B : ZFSet ) → ZFSet |
| 306 -- A ∪ B = Select (A , B) ( λ x → (A ∋ x) ∨ ( B ∋ x ) ) | |
| 103 | 307 ｛_｝ : ZFSet → ZFSet |
| 308 ｛ x ｝ = ( x , x ) | |
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309 |
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310 infixr 200 _∈_ |
| 96 | 311 -- infixr 230 _∩_ _∪_ |
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312 infixr 220 _⊆_ |
| 112 | 313 isZF : IsZF (HOD {suc n}) _∋_ _==_ od∅ _,_ Union Power Select Replace (Ord omega) |
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314 isZF = record { |
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315 isEquivalence = record { refl = eq-refl ; sym = eq-sym; trans = eq-trans } |
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316 ; pair = pair |
| 118 | 317 ; union-u = λ X z UX∋z → union-u {X} {z} UX∋z |
| 72 | 318 ; union→ = union→ |
| 319 ; union← = union← | |
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320 ; empty = empty |
| 129 | 321 ; power→ = power→ |
| 76 | 322 ; power← = power← |
| 323 ; extensionality = extensionality | |
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324 ; minimul = minimul |
| 51 | 325 ; regularity = regularity |
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326 ; infinity∅ = infinity∅ |
| 93 | 327 ; infinity = λ _ → infinity |
| 116 | 328 ; selection = λ {X} {ψ} {y} → selection {X} {ψ} {y} |
| 135 | 329 ; replacement← = replacement← |
| 330 ; replacement→ = replacement→ | |
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331 } where |
| 129 | 332 |
| 112 | 333 pair : (A B : HOD {suc n} ) → ((A , B) ∋ A) ∧ ((A , B) ∋ B) |
| 87 | 334 proj1 (pair A B ) = omax-x {n} (od→ord A) (od→ord B) |
| 335 proj2 (pair A B ) = omax-y {n} (od→ord A) (od→ord B) | |
| 129 | 336 |
| 112 | 337 empty : (x : HOD {suc n} ) → ¬ (od∅ ∋ x) |
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338 empty x (case1 ()) |
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339 empty x (case2 ()) |
| 129 | 340 |
| 100 | 341 --- |
| 342 --- ZFSubset A x = record { def = λ y → def A y ∧ def x y } subset of A | |
| 343 --- 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 | |
| 344 -- | |
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345 -- if Power A ∋ t, from a propertiy of minimum sup there is osuc ZFSubset A ∋ t |
| 100 | 346 -- then ZFSubset A ≡ t or ZFSubset A ∋ t. In the former case ZFSubset A ∋ x implies A ∋ x |
| 128 | 347 -- In case of later, ZFSubset A ∋ t and t ∋ x implies A ∋ x by transitivity |
| 100 | 348 -- |
| 129 | 349 POrd : {a : Ordinal } {t : HOD} → Def (Ord a) ∋ t → Def (Ord a) ∋ Ord (od→ord t) |
| 350 POrd {a} {t} P∋t = o<→c< P∋t | |
| 351 ord-power→ : (a : Ordinal ) ( t : HOD) → Def (Ord a) ∋ t → {x : HOD} → t ∋ x → Ord a ∋ x | |
| 352 ord-power→ a t P∋t {x} t∋x with osuc-≡< (sup-lb (POrd P∋t)) | |
| 127 | 353 ... | case1 eq = proj1 (def-subst (Ltx t∋x) (sym (subst₂ (λ j k → j ≡ k ) oiso oiso ( cong (λ k → ord→od k) (sym eq) ))) refl ) where |
| 354 Ltx : {n : Level} → {x t : HOD {suc n}} → t ∋ x → Ord (od→ord t) ∋ x | |
| 355 Ltx {n} {x} {t} lt = c<→o< lt | |
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356 ... | case2 lt = {!!} where |
| 129 | 357 sp = sup-x (λ x → od→ord ( ZFSubset (Ord a) (ord→od x))) |
| 112 | 358 minsup : HOD |
| 129 | 359 minsup = ZFSubset (Ord a) ( ord→od ( sup-x (λ x → od→ord ( ZFSubset (Ord a) (ord→od x))))) |
| 127 | 360 Ltx : {n : Level} → {x t : HOD {suc n}} → t ∋ x → Ord (od→ord t) ∋ x |
| 361 Ltx {n} {x} {t} lt = c<→o< lt | |
| 130 | 362 -- lemma1 hold because minsup is Ord (minα a sp) |
| 127 | 363 lemma1 : od→ord (Ord (od→ord t)) o< od→ord minsup → minsup ∋ Ord (od→ord t) |
| 129 | 364 lemma1 lt with OrdSubset a ( sup-x (λ x → od→ord ( ZFSubset (Ord a) (ord→od x)))) |
| 365 ... | eq with subst ( λ k → ZFSubset (Ord a) k ≡ Ord (minα a sp)) Ord-ord eq | |
| 366 ... | eq1 = def-subst {suc n} {_} {_} {minsup} {od→ord (Ord (od→ord t))} (o<→c< lt) lemma2 (sym ord-Ord) where | |
| 367 lemma2 : Ord (od→ord (ZFSubset (Ord a) (ord→od sp))) ≡ minsup | |
| 368 lemma2 = let open ≡-Reasoning in begin | |
| 369 Ord (od→ord (ZFSubset (Ord a) (ord→od sp))) | |
| 370 ≡⟨ cong (λ k → Ord (od→ord k)) eq1 ⟩ | |
| 371 Ord (od→ord (Ord (minα a sp))) | |
| 372 ≡⟨ cong (λ k → Ord (od→ord k)) Ord-ord ⟩ | |
| 373 Ord (od→ord (ord→od (minα a sp))) | |
| 374 ≡⟨ cong (λ k → Ord k) diso ⟩ | |
| 375 Ord (minα a sp) | |
| 376 ≡⟨ sym eq1 ⟩ | |
| 377 minsup | |
| 378 ∎ | |
| 100 | 379 -- |
| 380 -- we have t ∋ x → A ∋ x means t is a subset of A, that is ZFSubset A t == t | |
| 381 -- Power A is a sup of ZFSubset A t, so Power A ∋ t | |
| 382 -- | |
| 129 | 383 ord-power← : (a : Ordinal ) (t : HOD) → ({x : HOD} → (t ∋ x → (Ord a) ∋ x)) → Def (Ord a) ∋ t |
| 384 ord-power← a t t→A = def-subst {suc n} {_} {_} {Def (Ord a)} {od→ord t} | |
| 127 | 385 lemma refl (lemma1 lemma-eq )where |
| 129 | 386 lemma-eq : ZFSubset (Ord a) t == t |
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387 eq→ lemma-eq {z} w = proj2 w |
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388 eq← lemma-eq {z} w = record { proj2 = w ; |
| 129 | 389 proj1 = def-subst {suc n} {_} {_} {(Ord a)} {z} |
| 126 | 390 ( t→A (def-subst {suc n} {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso } |
| 129 | 391 lemma1 : {n : Level } {a : Ordinal {suc n}} { t : HOD {suc n}} |
| 392 → (eq : ZFSubset (Ord a) t == t) → od→ord (ZFSubset (Ord a) (ord→od (od→ord t))) ≡ od→ord t | |
| 393 lemma1 {n} {a} {t} eq = subst (λ k → od→ord (ZFSubset (Ord a) k) ≡ od→ord t ) (sym oiso) (cong (λ k → od→ord k ) (===-≡ eq )) | |
| 394 lemma : od→ord (ZFSubset (Ord a) (ord→od (od→ord t)) ) o< sup-o (λ x → od→ord (ZFSubset (Ord a) (ord→od x))) | |
| 98 | 395 lemma = sup-o< |
| 129 | 396 |
| 130 | 397 -- Power A = Replace (Def (Ord (od→ord A))) ( λ x → A ∩ x ) |
| 129 | 398 power→ : ( A t : HOD) → Power A ∋ t → {x : HOD} → t ∋ x → A ∋ x |
| 399 power→ = {!!} | |
| 400 power← : (A t : HOD) → ({x : HOD} → (t ∋ x → A ∋ x)) → Power A ∋ t | |
| 135 | 401 power← A t t→A = {!!} where |
| 130 | 402 a = od→ord A |
| 403 ψ : HOD → HOD | |
| 404 ψ y = Def (Ord a) ∩ y | |
| 129 | 405 |
| 118 | 406 union-u : {X z : HOD {suc n}} → (U>z : Union X ∋ z ) → HOD {suc n} |
| 407 union-u {X} {z} U>z = Ord ( osuc ( od→ord z ) ) | |
| 112 | 408 union→ : (X z u : HOD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z |
| 118 | 409 union→ X z u xx with trio< ( od→ord u ) ( osuc ( od→ord z )) |
| 410 union→ X z u xx | tri< a ¬b ¬c with osuc-< a (c<→o< (proj2 xx)) | |
| 411 union→ X z u xx | tri< a ¬b ¬c | () | |
| 122 | 412 union→ X z u xx | tri≈ ¬a b ¬c = def-subst {suc n} {_} {_} {X} {osuc (od→ord z)} (proj1 xx) refl b |
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413 union→ X z u xx | tri> ¬a ¬b c = {!!} |
| 118 | 414 union← : (X z : HOD) (X∋z : Union X ∋ z) → (X ∋ union-u {X} {z} X∋z ) ∧ (union-u {X} {z} X∋z ∋ z ) |
| 415 union← X z X∋z = record { proj1 = lemma ; proj2 = <-osuc } where | |
| 416 lemma : X ∋ union-u {X} {z} X∋z | |
| 417 lemma = def-subst {suc n} {_} {_} {X} {od→ord (Ord (osuc ( od→ord z )))} X∋z refl ord-Ord | |
| 129 | 418 |
| 138 | 419 -- ψiso : {ψ : HOD {suc n} → Set (suc n)} {x y : HOD {suc n}} → ψ x → x ≡ y → ψ y |
| 420 -- ψiso {ψ} t refl = t | |
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421 selection : {ψ : HOD → Set (suc n)} {X y : HOD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y) |
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422 selection {X} {ψ} {y} = {!!} |
| 139 | 423 replacement← : {ψ : HOD → HOD} (X x : HOD) → X ∋ x → Replace X ψ ∋ ψ x |
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424 replacement← {ψ} X x lt = {!!} |
| 139 | 425 replacement→ : {ψ : HOD → HOD} (X x : HOD) → (lt : Replace X ψ ∋ x) → ¬ ( (y : HOD) → ¬ (x == ψ y)) |
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426 replacement→ {ψ} X x lt = contra-position lemma {!!} where |
| 138 | 427 lemma : ( (y : HOD) → ¬ (x == ψ y)) → ( (y : Ordinal) → ¬ def X y ∧ (ord→od (od→ord x) == ψ (Ord y)) ) |
| 428 lemma not y not2 = not (ord→od y) (subst₂ ( λ k j → k == j ) oiso (cong (λ k → ψ k ) Ord-ord ) (proj2 not2 )) | |
| 129 | 429 |
| 112 | 430 ∅-iso : {x : HOD} → ¬ (x == od∅) → ¬ ((ord→od (od→ord x)) == od∅) |
| 60 | 431 ∅-iso {x} neq = subst (λ k → ¬ k) (=-iso {n} ) neq |
| 112 | 432 regularity : (x : HOD) (not : ¬ (x == od∅)) → |
| 115 | 433 (x ∋ minimul x not) ∧ (Select (minimul x not) (λ x₁ → (minimul x not ∋ x₁) ∧ (x ∋ x₁)) == od∅) |
| 117 | 434 proj1 (regularity x not ) = x∋minimul x not |
| 435 proj2 (regularity x not ) = record { eq→ = lemma1 ; eq← = λ {y} d → lemma2 {y} d } where | |
| 436 lemma1 : {x₁ : Ordinal} → def (Select (minimul x not) (λ x₂ → (minimul x not ∋ x₂) ∧ (x ∋ x₂))) x₁ → def od∅ x₁ | |
| 437 lemma1 {x₁} s = ⊥-elim ( minimul-1 x not (ord→od x₁) lemma3 ) where | |
| 438 lemma3 : def (minimul x not) (od→ord (ord→od x₁)) ∧ def x (od→ord (ord→od x₁)) | |
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439 lemma3 = {!!} |
| 117 | 440 lemma2 : {x₁ : Ordinal} → def od∅ x₁ → def (Select (minimul x not) (λ x₂ → (minimul x not ∋ x₂) ∧ (x ∋ x₂))) x₁ |
| 441 lemma2 {y} d = ⊥-elim (empty (ord→od y) (def-subst {suc n} {_} {_} {od∅} {od→ord (ord→od y)} d refl (sym diso) )) | |
| 129 | 442 |
| 112 | 443 extensionality : {A B : HOD {suc n}} → ((z : HOD) → (A ∋ z) ⇔ (B ∋ z)) → A == B |
| 76 | 444 eq→ (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {A} {B} (sym diso) (proj1 (eq (ord→od x))) d |
| 445 eq← (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {B} {A} (sym diso) (proj2 (eq (ord→od x))) d | |
| 129 | 446 |
| 119 | 447 open import Relation.Binary.PropositionalEquality |
| 448 uxxx-ord : {x : HOD {suc n}} → {y : Ordinal {suc n}} → def (Union (x , (x , x))) y ⇔ ( y o< osuc (od→ord x) ) | |
| 449 uxxx-ord {x} {y} = subst (λ k → k ⇔ ( y o< osuc (od→ord x) )) (sym lemma) ( osuc2 y (od→ord x)) where | |
| 450 lemma : {y : Ordinal {suc n}} → def (Union (x , (x , x))) y ≡ osuc y o< osuc (osuc (od→ord x)) | |
| 451 lemma {y} = let open ≡-Reasoning in begin | |
| 452 def (Union (x , (x , x))) y | |
| 453 ≡⟨⟩ | |
| 454 def ( Ord ( omax (od→ord x) (od→ord (Ord (omax (od→ord x) (od→ord x) )) ))) ( osuc y ) | |
| 455 ≡⟨⟩ | |
| 456 osuc y o< omax (od→ord x) (od→ord (Ord (omax (od→ord x) (od→ord x) )) ) | |
| 457 ≡⟨ cong (λ k → osuc y o< omax (od→ord x) k ) (sym ord-Ord) ⟩ | |
| 458 osuc y o< omax (od→ord x) (omax (od→ord x) (od→ord x) ) | |
| 459 ≡⟨ cong (λ k → osuc y o< k ) (omxxx (od→ord x) ) ⟩ | |
| 460 osuc y o< osuc (osuc (od→ord x)) | |
| 461 ∎ | |
| 112 | 462 infinite : HOD {suc n} |
| 111 | 463 infinite = Ord omega |
| 464 infinity∅ : Ord omega ∋ od∅ {suc n} | |
| 119 | 465 infinity∅ = o<-subst (case1 (s≤s z≤n) ) ord-Ord refl |
| 112 | 466 infinity : (x : HOD) → infinite ∋ x → infinite ∋ Union (x , (x , x )) |
| 120 | 467 infinity x lt = o<-subst ( lemma (od→ord x) lt ) eq refl where |
| 468 eq : osuc (od→ord x) ≡ od→ord (Union (x , (x , x))) | |
| 469 eq = let open ≡-Reasoning in begin | |
| 470 osuc (od→ord x) | |
| 471 ≡⟨ ord-Ord ⟩ | |
| 472 od→ord (Ord (osuc (od→ord x))) | |
| 473 ≡⟨ cong ( λ k → od→ord k ) ( sym (==→o≡ ( ⇔→== uxxx-ord ))) ⟩ | |
| 474 od→ord (Union (x , (x , x))) | |
| 475 ∎ | |
| 91 | 476 lemma : (ox : Ordinal {suc n} ) → ox o< omega → osuc ox o< omega |
| 477 lemma record { lv = Zero ; ord = (Φ .0) } (case1 (s≤s x)) = case1 (s≤s z≤n) | |
| 478 lemma record { lv = Zero ; ord = (OSuc .0 ord₁) } (case1 (s≤s x)) = case1 (s≤s z≤n) | |
| 479 lemma record { lv = (Suc lv₁) ; ord = (Φ .(Suc lv₁)) } (case1 (s≤s ())) | |
| 480 lemma record { lv = (Suc lv₁) ; ord = (OSuc .(Suc lv₁) ord₁) } (case1 (s≤s ())) | |
| 481 lemma record { lv = 1 ; ord = (Φ 1) } (case2 c2) with d<→lv c2 | |
| 482 lemma record { lv = (Suc Zero) ; ord = (Φ .1) } (case2 ()) | refl | |
| 103 | 483 -- ∀ X [ ∅ ∉ X → (∃ f : X → ⋃ X ) → ∀ A ∈ X ( f ( A ) ∈ A ) ] -- this form is no good since X is a transitive set |
| 484 -- ∀ z [ ∀ x ( x ∈ z → ¬ ( x ≈ ∅ ) ) ∧ ∀ x ∀ y ( x , y ∈ z ∧ ¬ ( x ≈ y ) → x ∩ y ≈ ∅ ) → ∃ u ∀ x ( x ∈ z → ∃ t ( u ∩ x) ≈ ｛ t ｝) ] | |
| 112 | 485 record Choice (z : HOD {suc n}) : Set (suc (suc n)) where |
| 103 | 486 field |
| 112 | 487 u : {x : HOD {suc n}} ( x∈z : x ∈ z ) → HOD {suc n} |
| 488 t : {x : HOD {suc n}} ( x∈z : x ∈ z ) → (x : HOD {suc n} ) → HOD {suc n} | |
| 489 choice : { x : HOD {suc n} } → ( x∈z : x ∈ z ) → ( u x∈z ∩ x) == ｛ t x∈z x ｝ | |
| 490 -- choice : {x : HOD {suc n}} ( x ∈ z → ¬ ( x ≈ ∅ ) ) → | |
| 491 -- axiom-of-choice : { X : HOD } → ( ¬x∅ : ¬ ( X == od∅ ) ) → { A : HOD } → (A∈X : A ∈ X ) → choice ¬x∅ A∈X ∈ A | |
| 103 | 492 -- axiom-of-choice {X} nx {A} lt = ¬∅=→∅∈ {!!} |
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