Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate OD.agda @ 276:6f10c47e4e7a
separate choice
fix sup-o
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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| date | Sat, 09 May 2020 09:02:52 +0900 |
| parents | 29a85a427ed2 |
| children | d9d3654baee1 |
| rev | line source |
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| 16 | 1 open import Level |
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2 open import Ordinals |
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3 module OD {n : Level } (O : Ordinals {n} ) where |
| 3 | 4 |
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5 open import zf |
| 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 | |
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14 open import logic |
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15 open import nat |
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16 |
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17 open inOrdinal O |
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18 |
| 27 | 19 -- Ordinal Definable Set |
| 11 | 20 |
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21 record OD : Set (suc n ) where |
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22 field |
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23 def : (x : Ordinal ) → Set n |
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24 |
| 141 | 25 open OD |
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26 |
| 120 | 27 open _∧_ |
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28 open _∨_ |
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29 open Bool |
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30 |
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31 record _==_ ( a b : OD ) : Set n where |
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32 field |
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33 eq→ : ∀ { x : Ordinal } → def a x → def b x |
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34 eq← : ∀ { x : Ordinal } → def b x → def a x |
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35 |
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36 id : {A : Set n} → A → A |
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37 id x = x |
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38 |
| 271 | 39 ==-refl : { x : OD } → x == x |
| 40 ==-refl {x} = record { eq→ = id ; eq← = id } | |
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41 |
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42 open _==_ |
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43 |
| 271 | 44 ==-trans : { x y z : OD } → x == y → y == z → x == z |
| 45 ==-trans x=y y=z = record { eq→ = λ {m} t → eq→ y=z (eq→ x=y t) ; eq← = λ {m} t → eq← x=y (eq← y=z t) } | |
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46 |
| 271 | 47 ==-sym : { x y : OD } → x == y → y == x |
| 48 ==-sym x=y = record { eq→ = λ {m} t → eq← x=y t ; eq← = λ {m} t → eq→ x=y t } | |
| 49 | |
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50 |
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51 ⇔→== : { x y : OD } → ( {z : Ordinal } → def x z ⇔ def y z) → x == y |
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52 eq→ ( ⇔→== {x} {y} eq ) {z} m = proj1 eq m |
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53 eq← ( ⇔→== {x} {y} eq ) {z} m = proj2 eq m |
| 120 | 54 |
| 179 | 55 -- Ordinal in OD ( and ZFSet ) Transitive Set |
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56 Ord : ( a : Ordinal ) → OD |
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57 Ord a = record { def = λ y → y o< a } |
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58 |
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59 od∅ : OD |
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60 od∅ = Ord o∅ |
| 40 | 61 |
| 258 | 62 -- next assumptions are our axiom |
| 63 -- it defines a subset of OD, which is called HOD, usually defined as | |
| 64 -- HOD = { x | TC x ⊆ OD } | |
| 65 -- where TC x is a transitive clusure of x, i.e. Union of all elemnts of all subset of x | |
| 66 | |
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67 postulate |
| 141 | 68 -- OD can be iso to a subset of Ordinal ( by means of Godel Set ) |
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69 od→ord : OD → Ordinal |
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70 ord→od : Ordinal → OD |
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71 c<→o< : {x y : OD } → def y ( od→ord x ) → od→ord x o< od→ord y |
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72 oiso : {x : OD } → ord→od ( od→ord x ) ≡ x |
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73 diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x |
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74 ==→o≡ : { x y : OD } → (x == y) → x ≡ y |
| 258 | 75 -- next assumption causes ∀ x ∋ ∅ . It menas only an ordinal is allowed as OD |
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76 -- o<→c< : {n : Level} {x y : Ordinal } → x o< y → def (ord→od y) x |
| 159 | 77 -- ord→od x ≡ Ord x results the same |
| 276 | 78 -- supermum as Replacement Axiom ( corresponding Ordinal of OD has maximum ) |
| 79 sup-o : ( OD → Ordinal ) → Ordinal | |
| 80 sup-o< : { ψ : OD → Ordinal } → ∀ {x : OD } → ψ x o< sup-o ψ | |
| 111 | 81 -- contra-position of mimimulity of supermum required in Power Set Axiom |
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82 -- sup-x : {n : Level } → ( Ordinal → Ordinal ) → Ordinal |
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83 -- sup-lb : {n : Level } → { ψ : Ordinal → Ordinal } → {z : Ordinal } → z o< sup-o ψ → z o< osuc (ψ (sup-x ψ)) |
| 276 | 84 |
| 85 data One : Set n where | |
| 86 OneObj : One | |
| 87 | |
| 88 Ords : OD | |
| 89 Ords = record { def = λ x → One } | |
| 90 | |
| 91 maxod : {x : OD} → od→ord x o< od→ord Ords | |
| 92 maxod {x} = c<→o< OneObj | |
| 258 | 93 |
| 94 o<→c<→OD=Ord : ( {x y : Ordinal } → x o< y → def (ord→od y) x ) → {x : OD } → x ≡ Ord (od→ord x) | |
| 95 o<→c<→OD=Ord o<→c< {x} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where | |
| 96 lemma1 : {y : Ordinal} → def x y → def (Ord (od→ord x)) y | |
| 97 lemma1 {y} lt = subst ( λ k → k o< od→ord x ) diso (c<→o< {ord→od y} {x} (subst (λ k → def x k ) (sym diso) lt)) | |
| 98 lemma2 : {y : Ordinal} → def (Ord (od→ord x)) y → def x y | |
| 99 lemma2 {y} lt = subst (λ k → def k y ) oiso (o<→c< {y} {od→ord x} lt ) | |
| 123 | 100 |
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101 _∋_ : ( a x : OD ) → Set n |
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102 _∋_ a x = def a ( od→ord x ) |
| 95 | 103 |
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104 _c<_ : ( x a : OD ) → Set n |
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105 x c< a = a ∋ x |
| 103 | 106 |
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107 cseq : {n : Level} → OD → OD |
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108 cseq x = record { def = λ y → def x (osuc y) } where |
| 113 | 109 |
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110 def-subst : {Z : OD } {X : Ordinal }{z : OD } {x : Ordinal }→ def Z X → Z ≡ z → X ≡ x → def z x |
| 95 | 111 def-subst df refl refl = df |
| 112 | |
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113 sup-od : ( OD → OD ) → OD |
| 276 | 114 sup-od ψ = Ord ( sup-o ( λ x → od→ord (ψ x)) ) |
| 95 | 115 |
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116 sup-c< : ( ψ : OD → OD ) → ∀ {x : OD } → def ( sup-od ψ ) (od→ord ( ψ x )) |
| 276 | 117 sup-c< ψ {x} = def-subst {_} {_} {Ord ( sup-o ( λ x → od→ord (ψ x)) )} {od→ord ( ψ x )} |
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118 lemma refl (cong ( λ k → od→ord (ψ k) ) oiso) where |
| 276 | 119 lemma : od→ord (ψ (ord→od (od→ord x))) o< sup-o (λ x → od→ord (ψ x)) |
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120 lemma = subst₂ (λ j k → j o< k ) refl diso (o<-subst (sup-o< ) refl (sym diso) ) |
| 28 | 121 |
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122 otrans : {n : Level} {a x y : Ordinal } → def (Ord a) x → def (Ord x) y → def (Ord a) y |
| 187 | 123 otrans x<a y<x = ordtrans y<x x<a |
| 123 | 124 |
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125 def→o< : {X : OD } → {x : Ordinal } → def X x → x o< od→ord X |
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126 def→o< {X} {x} lt = o<-subst {_} {_} {x} {od→ord X} ( c<→o< ( def-subst {X} {x} lt (sym oiso) (sym diso) )) diso diso |
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127 |
| 258 | 128 |
| 51 | 129 -- avoiding lv != Zero error |
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130 orefl : { x : OD } → { y : Ordinal } → od→ord x ≡ y → od→ord x ≡ y |
| 51 | 131 orefl refl = refl |
| 132 | |
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133 ==-iso : { x y : OD } → ord→od (od→ord x) == ord→od (od→ord y) → x == y |
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134 ==-iso {x} {y} eq = record { |
| 51 | 135 eq→ = λ d → lemma ( eq→ eq (def-subst d (sym oiso) refl )) ; |
| 136 eq← = λ d → lemma ( eq← eq (def-subst d (sym oiso) refl )) } | |
| 137 where | |
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138 lemma : {x : OD } {z : Ordinal } → def (ord→od (od→ord x)) z → def x z |
| 51 | 139 lemma {x} {z} d = def-subst d oiso refl |
| 140 | |
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141 =-iso : {x y : OD } → (x == y) ≡ (ord→od (od→ord x) == y) |
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142 =-iso {_} {y} = cong ( λ k → k == y ) (sym oiso) |
| 57 | 143 |
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144 ord→== : { x y : OD } → od→ord x ≡ od→ord y → x == y |
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145 ord→== {x} {y} eq = ==-iso (lemma (od→ord x) (od→ord y) (orefl eq)) where |
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146 lemma : ( ox oy : Ordinal ) → ox ≡ oy → (ord→od ox) == (ord→od oy) |
| 271 | 147 lemma ox ox refl = ==-refl |
| 51 | 148 |
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149 o≡→== : { x y : Ordinal } → x ≡ y → ord→od x == ord→od y |
| 271 | 150 o≡→== {x} {.x} refl = ==-refl |
| 51 | 151 |
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152 o∅≡od∅ : ord→od (o∅ ) ≡ od∅ |
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153 o∅≡od∅ = ==→o≡ lemma where |
| 150 | 154 lemma0 : {x : Ordinal} → def (ord→od o∅) x → def od∅ x |
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155 lemma0 {x} lt = o<-subst (c<→o< {ord→od x} {ord→od o∅} (def-subst {ord→od o∅} {x} lt refl (sym diso)) ) diso diso |
| 150 | 156 lemma1 : {x : Ordinal} → def od∅ x → def (ord→od o∅) x |
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157 lemma1 {x} lt = ⊥-elim (¬x<0 lt) |
| 150 | 158 lemma : ord→od o∅ == od∅ |
| 159 lemma = record { eq→ = lemma0 ; eq← = lemma1 } | |
| 160 | |
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161 ord-od∅ : od→ord (od∅ ) ≡ o∅ |
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162 ord-od∅ = sym ( subst (λ k → k ≡ od→ord (od∅ ) ) diso (cong ( λ k → od→ord k ) o∅≡od∅ ) ) |
| 80 | 163 |
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164 ∅0 : record { def = λ x → Lift n ⊥ } == od∅ |
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165 eq→ ∅0 {w} (lift ()) |
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166 eq← ∅0 {w} lt = lift (¬x<0 lt) |
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167 |
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168 ∅< : { x y : OD } → def x (od→ord y ) → ¬ ( x == od∅ ) |
| 271 | 169 ∅< {x} {y} d eq with eq→ (==-trans eq (==-sym ∅0) ) d |
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170 ∅< {x} {y} d eq | lift () |
| 57 | 171 |
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172 ∅6 : { x : OD } → ¬ ( x ∋ x ) -- no Russel paradox |
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173 ∅6 {x} x∋x = o<¬≡ refl ( c<→o< {x} {x} x∋x ) |
| 51 | 174 |
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175 def-iso : {A B : OD } {x y : Ordinal } → x ≡ y → (def A y → def B y) → def A x → def B x |
| 76 | 176 def-iso refl t = t |
| 177 | |
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178 is-o∅ : ( x : Ordinal ) → Dec ( x ≡ o∅ ) |
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179 is-o∅ x with trio< x o∅ |
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180 is-o∅ x | tri< a ¬b ¬c = no ¬b |
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181 is-o∅ x | tri≈ ¬a b ¬c = yes b |
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182 is-o∅ x | tri> ¬a ¬b c = no ¬b |
| 57 | 183 |
| 254 | 184 _,_ : OD → OD → OD |
| 185 x , y = record { def = λ t → (t ≡ od→ord x ) ∨ ( t ≡ od→ord y ) } -- Ord (omax (od→ord x) (od→ord y)) | |
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186 |
| 79 | 187 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) |
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188 -- postulate f-extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality n (suc n) |
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189 |
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190 in-codomain : (X : OD ) → ( ψ : OD → OD ) → OD |
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191 in-codomain X ψ = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( def X y ∧ ( x ≡ od→ord (ψ (ord→od y ))))) } |
| 141 | 192 |
| 96 | 193 -- Power Set of X ( or constructible by λ y → def X (od→ord y ) |
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97
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194 |
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195 ZFSubset : (A x : OD ) → OD |
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191
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196 ZFSubset A x = record { def = λ y → def A y ∧ def x y } -- roughly x = A → Set |
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197 |
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198 Def : (A : OD ) → OD |
| 276 | 199 Def A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A x) ) ) |
| 190 | 200 |
| 271 | 201 -- _⊆_ : ( A B : OD ) → ∀{ x : OD } → Set n |
| 202 -- _⊆_ A B {x} = A ∋ x → B ∋ x | |
| 203 | |
| 204 record _⊆_ ( A B : OD ) : Set (suc n) where | |
| 205 field | |
| 206 incl : { x : OD } → A ∋ x → B ∋ x | |
| 207 | |
| 208 open _⊆_ | |
| 190 | 209 |
| 210 infixr 220 _⊆_ | |
| 211 | |
| 271 | 212 subset-lemma : {A x : OD } → ( {y : OD } → x ∋ y → ZFSubset A x ∋ y ) ⇔ ( x ⊆ A ) |
| 213 subset-lemma {A} {x} = record { | |
| 214 proj1 = λ lt → record { incl = λ x∋z → proj1 (lt x∋z) } | |
| 215 ; proj2 = λ x⊆A lt → record { proj1 = incl x⊆A lt ; proj2 = lt } | |
| 190 | 216 } |
| 217 | |
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261
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218 open import Data.Unit |
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219 |
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220 ε-induction : { ψ : OD → Set (suc n)} |
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221 → ( {x : OD } → ({ y : OD } → x ∋ y → ψ y ) → ψ x ) |
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222 → (x : OD ) → ψ x |
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223 ε-induction {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (osuc (od→ord x)) <-osuc ) where |
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224 induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (ord→od oy)) → ψ (ord→od ox) |
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225 induction ox prev = ind ( λ {y} lt → subst (λ k → ψ k ) oiso (prev (od→ord y) (o<-subst (c<→o< lt) refl diso ))) |
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226 ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (ord→od oy) |
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227 ε-induction-ord ox {oy} lt = TransFinite {λ oy → ψ (ord→od oy)} induction oy |
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228 |
| 262 | 229 -- minimal-2 : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → (y : OD ) → ¬ ( def (minimal x ne) (od→ord y)) ∧ (def x (od→ord y) ) |
| 230 -- minimal-2 x ne y = contra-position ( ε-induction (λ {z} ind → {!!} ) x ) ( λ p → {!!} ) | |
| 231 | |
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232 OD→ZF : ZF |
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233 OD→ZF = record { |
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234 ZFSet = OD |
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235 ; _∋_ = _∋_ |
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236 ; _≈_ = _==_ |
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237 ; ∅ = od∅ |
| 28 | 238 ; _,_ = _,_ |
| 239 ; Union = Union | |
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240 ; Power = Power |
| 28 | 241 ; Select = Select |
| 242 ; Replace = Replace | |
| 161 | 243 ; infinite = infinite |
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244 ; isZF = isZF |
| 28 | 245 } where |
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246 ZFSet = OD |
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247 Select : (X : OD ) → ((x : OD ) → Set n ) → OD |
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248 Select X ψ = record { def = λ x → ( def X x ∧ ψ ( ord→od x )) } |
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249 Replace : OD → (OD → OD ) → OD |
| 276 | 250 Replace X ψ = record { def = λ x → (x o< sup-o ( λ x → od→ord (ψ x))) ∧ def (in-codomain X ψ) x } |
| 144 | 251 _∩_ : ( A B : ZFSet ) → ZFSet |
| 145 | 252 A ∩ B = record { def = λ x → def A x ∧ def B x } |
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253 Union : OD → OD |
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254 Union U = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((def U u) ∧ (def (ord→od u) x))) } |
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255 _∈_ : ( A B : ZFSet ) → Set n |
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256 A ∈ B = B ∋ A |
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257 Power : OD → OD |
| 129 | 258 Power A = Replace (Def (Ord (od→ord A))) ( λ x → A ∩ x ) |
| 103 | 259 ｛_｝ : ZFSet → ZFSet |
| 260 ｛ x ｝ = ( x , x ) | |
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261 |
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262 data infinite-d : ( x : Ordinal ) → Set n where |
| 161 | 263 iφ : infinite-d o∅ |
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264 isuc : {x : Ordinal } → infinite-d x → |
| 161 | 265 infinite-d (od→ord ( Union (ord→od x , (ord→od x , ord→od x ) ) )) |
| 266 | |
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267 infinite : OD |
| 161 | 268 infinite = record { def = λ x → infinite-d x } |
| 269 | |
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270 infixr 200 _∈_ |
| 96 | 271 -- infixr 230 _∩_ _∪_ |
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272 isZF : IsZF (OD ) _∋_ _==_ od∅ _,_ Union Power Select Replace infinite |
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273 isZF = record { |
| 271 | 274 isEquivalence = record { refl = ==-refl ; sym = ==-sym; trans = ==-trans } |
| 247 | 275 ; pair→ = pair→ |
| 276 ; pair← = pair← | |
| 72 | 277 ; union→ = union→ |
| 278 ; union← = union← | |
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279 ; empty = empty |
| 129 | 280 ; power→ = power→ |
| 76 | 281 ; power← = power← |
| 186 | 282 ; extensionality = λ {A} {B} {w} → extensionality {A} {B} {w} |
| 274 | 283 ; ε-induction = ε-induction |
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284 ; infinity∅ = infinity∅ |
| 160 | 285 ; infinity = infinity |
| 116 | 286 ; selection = λ {X} {ψ} {y} → selection {X} {ψ} {y} |
| 135 | 287 ; replacement← = replacement← |
| 288 ; replacement→ = replacement→ | |
| 274 | 289 -- ; choice-func = choice-func |
| 290 -- ; choice = choice | |
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291 } where |
| 129 | 292 |
| 247 | 293 pair→ : ( x y t : ZFSet ) → (x , y) ∋ t → ( t == x ) ∨ ( t == y ) |
| 294 pair→ x y t (case1 t≡x ) = case1 (subst₂ (λ j k → j == k ) oiso oiso (o≡→== t≡x )) | |
| 295 pair→ x y t (case2 t≡y ) = case2 (subst₂ (λ j k → j == k ) oiso oiso (o≡→== t≡y )) | |
| 296 | |
| 297 pair← : ( x y t : ZFSet ) → ( t == x ) ∨ ( t == y ) → (x , y) ∋ t | |
| 298 pair← x y t (case1 t==x) = case1 (cong (λ k → od→ord k ) (==→o≡ t==x)) | |
| 299 pair← x y t (case2 t==y) = case2 (cong (λ k → od→ord k ) (==→o≡ t==y)) | |
| 300 | |
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301 empty : (x : OD ) → ¬ (od∅ ∋ x) |
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302 empty x = ¬x<0 |
| 129 | 303 |
| 271 | 304 o<→c< : {x y : Ordinal } → x o< y → (Ord x) ⊆ (Ord y) |
| 305 o<→c< lt = record { incl = λ z → ordtrans z lt } | |
| 155 | 306 |
| 271 | 307 ⊆→o< : {x y : Ordinal } → (Ord x) ⊆ (Ord y) → x o< osuc y |
| 155 | 308 ⊆→o< {x} {y} lt with trio< x y |
| 309 ⊆→o< {x} {y} lt | tri< a ¬b ¬c = ordtrans a <-osuc | |
| 310 ⊆→o< {x} {y} lt | tri≈ ¬a b ¬c = subst ( λ k → k o< osuc y) (sym b) <-osuc | |
| 271 | 311 ⊆→o< {x} {y} lt | tri> ¬a ¬b c with (incl lt) (o<-subst c (sym diso) refl ) |
| 155 | 312 ... | ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt diso refl )) |
| 151 | 313 |
| 144 | 314 union→ : (X z u : OD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z |
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315 union→ X z u xx not = ⊥-elim ( not (od→ord u) ( record { proj1 = proj1 xx |
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316 ; proj2 = subst ( λ k → def k (od→ord z)) (sym oiso) (proj2 xx) } )) |
| 159 | 317 union← : (X z : OD) (X∋z : Union X ∋ z) → ¬ ( (u : OD ) → ¬ ((X ∋ u) ∧ (u ∋ z ))) |
| 258 | 318 union← X z UX∋z = FExists _ lemma UX∋z where |
| 165 | 319 lemma : {y : Ordinal} → def X y ∧ def (ord→od y) (od→ord z) → ¬ ((u : OD) → ¬ (X ∋ u) ∧ (u ∋ z)) |
| 320 lemma {y} xx not = not (ord→od y) record { proj1 = subst ( λ k → def X k ) (sym diso) (proj1 xx ) ; proj2 = proj2 xx } | |
| 144 | 321 |
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322 ψiso : {ψ : OD → Set n} {x y : OD } → ψ x → x ≡ y → ψ y |
| 144 | 323 ψiso {ψ} t refl = t |
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324 selection : {ψ : OD → Set n} {X y : OD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y) |
| 144 | 325 selection {ψ} {X} {y} = record { |
| 326 proj1 = λ cond → record { proj1 = proj1 cond ; proj2 = ψiso {ψ} (proj2 cond) (sym oiso) } | |
| 327 ; proj2 = λ select → record { proj1 = proj1 select ; proj2 = ψiso {ψ} (proj2 select) oiso } | |
| 328 } | |
| 329 replacement← : {ψ : OD → OD} (X x : OD) → X ∋ x → Replace X ψ ∋ ψ x | |
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330 replacement← {ψ} X x lt = record { proj1 = sup-c< ψ {x} ; proj2 = lemma } where |
| 144 | 331 lemma : def (in-codomain X ψ) (od→ord (ψ x)) |
| 150 | 332 lemma not = ⊥-elim ( not ( od→ord x ) (record { proj1 = lt ; proj2 = cong (λ k → od→ord (ψ k)) (sym oiso)} )) |
| 144 | 333 replacement→ : {ψ : OD → OD} (X x : OD) → (lt : Replace X ψ ∋ x) → ¬ ( (y : OD) → ¬ (x == ψ y)) |
| 150 | 334 replacement→ {ψ} X x lt = contra-position lemma (lemma2 (proj2 lt)) where |
| 335 lemma2 : ¬ ((y : Ordinal) → ¬ def X y ∧ ((od→ord x) ≡ od→ord (ψ (ord→od y)))) | |
| 336 → ¬ ((y : Ordinal) → ¬ def X y ∧ (ord→od (od→ord x) == ψ (ord→od y))) | |
| 144 | 337 lemma2 not not2 = not ( λ y d → not2 y (record { proj1 = proj1 d ; proj2 = lemma3 (proj2 d)})) where |
| 150 | 338 lemma3 : {y : Ordinal } → (od→ord x ≡ od→ord (ψ (ord→od y))) → (ord→od (od→ord x) == ψ (ord→od y)) |
| 144 | 339 lemma3 {y} eq = subst (λ k → ord→od (od→ord x) == k ) oiso (o≡→== eq ) |
| 150 | 340 lemma : ( (y : OD) → ¬ (x == ψ y)) → ( (y : Ordinal) → ¬ def X y ∧ (ord→od (od→ord x) == ψ (ord→od y)) ) |
| 341 lemma not y not2 = not (ord→od y) (subst (λ k → k == ψ (ord→od y)) oiso ( proj2 not2 )) | |
| 144 | 342 |
| 343 --- | |
| 344 --- Power Set | |
| 345 --- | |
| 346 --- First consider ordinals in OD | |
| 100 | 347 --- |
| 348 --- ZFSubset A x = record { def = λ y → def A y ∧ def x y } subset of A | |
| 349 -- | |
| 350 -- | |
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351 ∩-≡ : { a b : OD } → ({x : OD } → (a ∋ x → b ∋ x)) → a == ( b ∩ a ) |
| 142 | 352 ∩-≡ {a} {b} inc = record { |
| 353 eq→ = λ {x} x<a → record { proj2 = x<a ; | |
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354 proj1 = def-subst {_} {_} {b} {x} (inc (def-subst {_} {_} {a} {_} x<a refl (sym diso) )) refl diso } ; |
| 142 | 355 eq← = λ {x} x<a∩b → proj2 x<a∩b } |
| 100 | 356 -- |
| 258 | 357 -- Transitive Set case |
| 358 -- we have t ∋ x → Ord a ∋ x means t is a subset of Ord a, that is ZFSubset (Ord a) t == t | |
| 359 -- Def (Ord a) is a sup of ZFSubset (Ord a) t, so Def (Ord a) ∋ t | |
| 360 -- Def A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) ) | |
| 100 | 361 -- |
| 141 | 362 ord-power← : (a : Ordinal ) (t : OD) → ({x : OD} → (t ∋ x → (Ord a) ∋ x)) → Def (Ord a) ∋ t |
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363 ord-power← a t t→A = def-subst {_} {_} {Def (Ord a)} {od→ord t} |
| 127 | 364 lemma refl (lemma1 lemma-eq )where |
| 129 | 365 lemma-eq : ZFSubset (Ord a) t == t |
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366 eq→ lemma-eq {z} w = proj2 w |
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367 eq← lemma-eq {z} w = record { proj2 = w ; |
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368 proj1 = def-subst {_} {_} {(Ord a)} {z} |
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369 ( t→A (def-subst {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso } |
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370 lemma1 : {a : Ordinal } { t : OD } |
| 129 | 371 → (eq : ZFSubset (Ord a) t == t) → od→ord (ZFSubset (Ord a) (ord→od (od→ord t))) ≡ od→ord t |
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372 lemma1 {a} {t} eq = subst (λ k → od→ord (ZFSubset (Ord a) k) ≡ od→ord t ) (sym oiso) (cong (λ k → od→ord k ) (==→o≡ eq )) |
| 276 | 373 lemma : od→ord (ZFSubset (Ord a) (ord→od (od→ord t)) ) o< sup-o (λ x → od→ord (ZFSubset (Ord a) x)) |
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374 lemma = sup-o< |
| 129 | 375 |
| 144 | 376 -- |
| 258 | 377 -- Every set in OD is a subset of Ordinals, so make Def (Ord (od→ord A)) first |
| 378 -- then replace of all elements of the Power set by A ∩ y | |
| 144 | 379 -- |
| 142 | 380 -- Power A = Replace (Def (Ord (od→ord A))) ( λ y → A ∩ y ) |
| 166 | 381 |
| 382 -- we have oly double negation form because of the replacement axiom | |
| 383 -- | |
| 384 power→ : ( A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → ¬ ¬ (A ∋ x) | |
| 258 | 385 power→ A t P∋t {x} t∋x = FExists _ lemma5 lemma4 where |
| 142 | 386 a = od→ord A |
| 387 lemma2 : ¬ ( (y : OD) → ¬ (t == (A ∩ y))) | |
| 388 lemma2 = replacement→ (Def (Ord (od→ord A))) t P∋t | |
| 166 | 389 lemma3 : (y : OD) → t == ( A ∩ y ) → ¬ ¬ (A ∋ x) |
| 390 lemma3 y eq not = not (proj1 (eq→ eq t∋x)) | |
| 142 | 391 lemma4 : ¬ ((y : Ordinal) → ¬ (t == (A ∩ ord→od y))) |
| 392 lemma4 not = lemma2 ( λ y not1 → not (od→ord y) (subst (λ k → t == ( A ∩ k )) (sym oiso) not1 )) | |
| 166 | 393 lemma5 : {y : Ordinal} → t == (A ∩ ord→od y) → ¬ ¬ (def A (od→ord x)) |
| 394 lemma5 {y} eq not = (lemma3 (ord→od y) eq) not | |
| 395 | |
| 142 | 396 power← : (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t |
| 397 power← A t t→A = record { proj1 = lemma1 ; proj2 = lemma2 } where | |
| 398 a = od→ord A | |
| 399 lemma0 : {x : OD} → t ∋ x → Ord a ∋ x | |
| 400 lemma0 {x} t∋x = c<→o< (t→A t∋x) | |
| 401 lemma3 : Def (Ord a) ∋ t | |
| 402 lemma3 = ord-power← a t lemma0 | |
| 152 | 403 lemma4 : (A ∩ ord→od (od→ord t)) ≡ t |
| 404 lemma4 = let open ≡-Reasoning in begin | |
| 405 A ∩ ord→od (od→ord t) | |
| 406 ≡⟨ cong (λ k → A ∩ k) oiso ⟩ | |
| 407 A ∩ t | |
| 408 ≡⟨ sym (==→o≡ ( ∩-≡ t→A )) ⟩ | |
| 409 t | |
| 410 ∎ | |
| 276 | 411 lemma1 : od→ord t o< sup-o (λ x → od→ord (A ∩ x)) |
| 412 lemma1 = subst (λ k → od→ord k o< sup-o (λ x → od→ord (A ∩ x))) | |
| 413 lemma4 (sup-o< {λ x → od→ord (A ∩ x)} ) | |
| 142 | 414 lemma2 : def (in-codomain (Def (Ord (od→ord A))) (_∩_ A)) (od→ord t) |
| 151 | 415 lemma2 not = ⊥-elim ( not (od→ord t) (record { proj1 = lemma3 ; proj2 = lemma6 }) ) where |
| 416 lemma6 : od→ord t ≡ od→ord (A ∩ ord→od (od→ord t)) | |
| 417 lemma6 = cong ( λ k → od→ord k ) (==→o≡ (subst (λ k → t == (A ∩ k)) (sym oiso) ( ∩-≡ t→A ))) | |
| 142 | 418 |
| 271 | 419 ord⊆power : (a : Ordinal) → (Ord (osuc a)) ⊆ (Power (Ord a)) |
| 420 ord⊆power a = record { incl = λ {x} lt → power← (Ord a) x (lemma lt) } where | |
| 421 lemma : {x y : OD} → od→ord x o< osuc a → x ∋ y → Ord a ∋ y | |
| 422 lemma lt y<x with osuc-≡< lt | |
| 423 lemma lt y<x | case1 refl = c<→o< y<x | |
| 424 lemma lt y<x | case2 x<a = ordtrans (c<→o< y<x) x<a | |
| 262 | 425 |
| 276 | 426 continuum-hyphotheis : (a : Ordinal) → Set (suc n) |
| 427 continuum-hyphotheis a = Power (Ord a) ⊆ Ord (osuc a) | |
| 129 | 428 |
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429 extensionality0 : {A B : OD } → ((z : OD) → (A ∋ z) ⇔ (B ∋ z)) → A == B |
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430 eq→ (extensionality0 {A} {B} eq ) {x} d = def-iso {A} {B} (sym diso) (proj1 (eq (ord→od x))) d |
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431 eq← (extensionality0 {A} {B} eq ) {x} d = def-iso {B} {A} (sym diso) (proj2 (eq (ord→od x))) d |
| 186 | 432 |
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433 extensionality : {A B w : OD } → ((z : OD ) → (A ∋ z) ⇔ (B ∋ z)) → (w ∋ A) ⇔ (w ∋ B) |
| 186 | 434 proj1 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) ( ==→o≡ (extensionality0 {A} {B} eq) ) d |
| 435 proj2 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) (sym ( ==→o≡ (extensionality0 {A} {B} eq) )) d | |
| 129 | 436 |
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437 infinity∅ : infinite ∋ od∅ |
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438 infinity∅ = def-subst {_} {_} {infinite} {od→ord (od∅ )} iφ refl lemma where |
| 161 | 439 lemma : o∅ ≡ od→ord od∅ |
| 440 lemma = let open ≡-Reasoning in begin | |
| 441 o∅ | |
| 442 ≡⟨ sym diso ⟩ | |
| 443 od→ord ( ord→od o∅ ) | |
| 444 ≡⟨ cong ( λ k → od→ord k ) o∅≡od∅ ⟩ | |
| 445 od→ord od∅ | |
| 446 ∎ | |
| 447 infinity : (x : OD) → infinite ∋ x → infinite ∋ Union (x , (x , x )) | |
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448 infinity x lt = def-subst {_} {_} {infinite} {od→ord (Union (x , (x , x )))} ( isuc {od→ord x} lt ) refl lemma where |
| 161 | 449 lemma : od→ord (Union (ord→od (od→ord x) , (ord→od (od→ord x) , ord→od (od→ord x)))) |
| 450 ≡ od→ord (Union (x , (x , x))) | |
| 451 lemma = cong (λ k → od→ord (Union ( k , ( k , k ) ))) oiso | |
| 452 | |
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453 |
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454 Union = ZF.Union OD→ZF |
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455 Power = ZF.Power OD→ZF |
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456 Select = ZF.Select OD→ZF |
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457 Replace = ZF.Replace OD→ZF |
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458 isZF = ZF.isZF OD→ZF |