Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate OD.agda @ 303:7963b76df6e1
¬odmax based HOD
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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| date | Mon, 29 Jun 2020 17:56:06 +0900 |
| parents | 304c271b3d47 |
| children | 2c111bfcc89a |
| 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 |
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55 -- next assumptions are our axiom |
| 290 | 56 -- In classical Set Theory, HOD is used, as a subset of OD, |
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57 -- HOD = { x | TC x ⊆ OD } |
| 290 | 58 -- where TC x is a transitive clusure of x, i.e. Union of all elemnts of all subset of x. |
| 59 -- This is not possible because we don't have V yet. | |
| 60 -- We simply assume V=OD here. | |
| 61 -- | |
| 62 -- We also assumes ODs are isomorphic to Ordinals, which is ususally proved by Goedel number tricks. | |
| 63 -- ODs have an ovbious maximum, but Ordinals are not. This means, od→ord is not an on-to mapping. | |
| 64 -- | |
| 65 -- ==→o≡ is necessary to prove axiom of extensionality. | |
| 66 -- | |
| 67 -- In classical Set Theory, sup is defined by Uion. Since we are working on constructive logic, | |
| 68 -- we need explict assumption on sup. | |
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69 |
| 303 | 70 data One : Set n where |
| 71 OneObj : One | |
| 72 | |
| 73 -- Ordinals in OD , the maximum | |
| 74 Ords : OD | |
| 75 Ords = record { def = λ x → One } | |
| 76 | |
| 77 record HOD : Set (suc n) where | |
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78 field |
| 303 | 79 od : OD |
| 80 ¬odmax : ¬ (od ≡ Ords) | |
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81 |
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82 record OrdinalSubset (maxordinal : Ordinal) : Set (suc n) where |
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83 field |
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84 os→ : (x : Ordinal) → x o< maxordinal → Ordinal |
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85 os← : Ordinal → Ordinal |
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86 os←limit : (x : Ordinal) → os← x o< maxordinal |
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87 os-iso← : {x : Ordinal} → os→ (os← x) (os←limit x) ≡ x |
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88 os-iso→ : {x : Ordinal} → (lt : x o< maxordinal ) → os← (os→ x lt) ≡ x |
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89 |
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90 open HOD |
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91 |
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92 -- HOD→OD : {x : Ordinal} → HOD x → OD |
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93 -- HOD→OD hod = record { def = hdef {!!} } |
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94 |
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95 record ODAxiom : Set (suc n) where |
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96 -- OD can be iso to a subset of Ordinal ( by means of Godel Set ) |
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97 field |
| 303 | 98 od→ord : HOD → Ordinal |
| 99 ord→od : Ordinal → HOD | |
| 100 c<→o< : {x y : HOD } → def (od y) ( od→ord x ) → od→ord x o< od→ord y | |
| 101 oiso : {x : HOD } → ord→od ( od→ord x ) ≡ x | |
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102 diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x |
| 303 | 103 ==→o≡ : { x y : HOD } → (od x == od y) → x ≡ y |
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104 -- supermum as Replacement Axiom ( corresponding Ordinal of OD has maximum ) |
| 303 | 105 sup-od : ( HOD → HOD ) → HOD |
| 106 sup-c< : ( ψ : HOD → HOD ) → ∀ {x : HOD } → def (od ( sup-od ψ )) (od→ord ( ψ x )) | |
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107 -- contra-position of mimimulity of supermum required in Power Set Axiom which we don't use |
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108 -- sup-x : {n : Level } → ( OD → Ordinal ) → Ordinal |
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109 -- sup-lb : {n : Level } → { ψ : OD → Ordinal } → {z : Ordinal } → z o< sup-o ψ → z o< osuc (ψ (sup-x ψ)) |
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110 |
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111 postulate odAxiom : ODAxiom |
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112 open ODAxiom odAxiom |
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113 |
| 303 | 114 -- maxod : {x : OD} → od→ord x o< od→ord Ords |
| 115 -- maxod {x} = c<→o< OneObj | |
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116 |
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117 -- we have to avoid this contradiction |
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118 |
| 303 | 119 -- bad-bad : ⊥ |
| 120 -- bad-bad = osuc-< <-osuc (c<→o< { record { def = λ x → One }} OneObj) | |
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121 |
| 179 | 122 -- Ordinal in OD ( and ZFSet ) Transitive Set |
| 303 | 123 Ord : ( a : Ordinal ) → HOD |
| 124 Ord a = record { od = record { def = λ y → y o< a } ; ¬odmax = ? } | |
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125 |
| 303 | 126 od∅ : HOD |
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127 od∅ = Ord o∅ |
| 40 | 128 |
| 303 | 129 sup-o : ( HOD → Ordinal ) → Ordinal |
| 130 sup-o = ? | |
| 131 sup-o< : { ψ : HOD → Ordinal } → ∀ {x : HOD } → ψ x o< sup-o ψ | |
| 132 sup-o< = ? | |
| 258 | 133 |
| 303 | 134 odef : HOD → Ordinal → Set n |
| 135 odef A x = def ( od A ) x | |
| 123 | 136 |
| 303 | 137 o<→c<→HOD=Ord : ( {x y : Ordinal } → x o< y → odef (ord→od y) x ) → {x : HOD } → x ≡ Ord (od→ord x) |
| 138 o<→c<→HOD=Ord o<→c< {x} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where | |
| 139 lemma1 : {y : Ordinal} → odef x y → odef (Ord (od→ord x)) y | |
| 140 lemma1 {y} lt = subst ( λ k → k o< od→ord x ) diso (c<→o< {ord→od y} {x} (subst (λ k → odef x k ) (sym diso) lt)) | |
| 141 lemma2 : {y : Ordinal} → odef (Ord (od→ord x)) y → odef x y | |
| 142 lemma2 {y} lt = subst (λ k → odef k y ) oiso (o<→c< {y} {od→ord x} lt ) | |
| 95 | 143 |
| 303 | 144 _∋_ : ( a x : HOD ) → Set n |
| 145 _∋_ a x = odef a ( od→ord x ) | |
| 146 | |
| 147 _c<_ : ( x a : HOD ) → Set n | |
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148 x c< a = a ∋ x |
| 103 | 149 |
| 303 | 150 cseq : {n : Level} → HOD → HOD |
| 151 cseq x = record { od = record { def = λ y → odef x (osuc y) } ; ¬odmax = ? } where | |
| 95 | 152 |
| 303 | 153 odef-subst : {Z : HOD } {X : Ordinal }{z : HOD } {x : Ordinal }→ odef Z X → Z ≡ z → X ≡ x → odef z x |
| 154 odef-subst df refl refl = df | |
| 95 | 155 |
| 303 | 156 otrans : {n : Level} {a x y : Ordinal } → odef (Ord a) x → odef (Ord x) y → odef (Ord a) y |
| 187 | 157 otrans x<a y<x = ordtrans y<x x<a |
| 123 | 158 |
| 303 | 159 odef→o< : {X : HOD } → {x : Ordinal } → odef X x → x o< od→ord X |
| 160 odef→o< {X} {x} lt = o<-subst {_} {_} {x} {od→ord X} ( c<→o< ( odef-subst {X} {x} lt (sym oiso) (sym diso) )) diso diso | |
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161 |
| 258 | 162 |
| 51 | 163 -- avoiding lv != Zero error |
| 303 | 164 orefl : { x : HOD } → { y : Ordinal } → od→ord x ≡ y → od→ord x ≡ y |
| 51 | 165 orefl refl = refl |
| 166 | |
| 303 | 167 ==-iso : { x y : HOD } → od (ord→od (od→ord x)) == od (ord→od (od→ord y)) → od x == od y |
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168 ==-iso {x} {y} eq = record { |
| 303 | 169 eq→ = λ d → lemma ( eq→ eq (odef-subst d (sym oiso) refl )) ; |
| 170 eq← = λ d → lemma ( eq← eq (odef-subst d (sym oiso) refl )) } | |
| 51 | 171 where |
| 303 | 172 lemma : {x : HOD } {z : Ordinal } → odef (ord→od (od→ord x)) z → odef x z |
| 173 lemma {x} {z} d = odef-subst d oiso refl | |
| 51 | 174 |
| 303 | 175 =-iso : {x y : HOD } → (od x == od y) ≡ (od (ord→od (od→ord x)) == od y) |
| 176 =-iso {_} {y} = cong ( λ k → od k == od y ) (sym oiso) | |
| 57 | 177 |
| 303 | 178 ord→== : { x y : HOD } → od→ord x ≡ od→ord y → od x == od y |
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179 ord→== {x} {y} eq = ==-iso (lemma (od→ord x) (od→ord y) (orefl eq)) where |
| 303 | 180 lemma : ( ox oy : Ordinal ) → ox ≡ oy → od (ord→od ox) == od (ord→od oy) |
| 271 | 181 lemma ox ox refl = ==-refl |
| 51 | 182 |
| 303 | 183 o≡→== : { x y : Ordinal } → x ≡ y → od (ord→od x) == od (ord→od y) |
| 271 | 184 o≡→== {x} {.x} refl = ==-refl |
| 51 | 185 |
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186 o∅≡od∅ : ord→od (o∅ ) ≡ od∅ |
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187 o∅≡od∅ = ==→o≡ lemma where |
| 303 | 188 lemma0 : {x : Ordinal} → odef (ord→od o∅) x → odef od∅ x |
| 189 lemma0 {x} lt = o<-subst (c<→o< {ord→od x} {ord→od o∅} (odef-subst {ord→od o∅} {x} lt refl (sym diso)) ) diso diso | |
| 190 lemma1 : {x : Ordinal} → odef od∅ x → odef (ord→od o∅) x | |
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191 lemma1 {x} lt = ⊥-elim (¬x<0 lt) |
| 303 | 192 lemma : od (ord→od o∅) == od od∅ |
| 150 | 193 lemma = record { eq→ = lemma0 ; eq← = lemma1 } |
| 194 | |
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195 ord-od∅ : od→ord (od∅ ) ≡ o∅ |
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196 ord-od∅ = sym ( subst (λ k → k ≡ od→ord (od∅ ) ) diso (cong ( λ k → od→ord k ) o∅≡od∅ ) ) |
| 80 | 197 |
| 303 | 198 ∅0 : record { def = λ x → Lift n ⊥ } == od od∅ |
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199 eq→ ∅0 {w} (lift ()) |
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200 eq← ∅0 {w} lt = lift (¬x<0 lt) |
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201 |
| 303 | 202 ∅< : { x y : HOD } → odef x (od→ord y ) → ¬ ( od x == od od∅ ) |
| 271 | 203 ∅< {x} {y} d eq with eq→ (==-trans eq (==-sym ∅0) ) d |
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204 ∅< {x} {y} d eq | lift () |
| 57 | 205 |
| 303 | 206 ∅6 : { x : HOD } → ¬ ( x ∋ x ) -- no Russel paradox |
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207 ∅6 {x} x∋x = o<¬≡ refl ( c<→o< {x} {x} x∋x ) |
| 51 | 208 |
| 303 | 209 odef-iso : {A B : HOD } {x y : Ordinal } → x ≡ y → (odef A y → odef B y) → odef A x → odef B x |
| 210 odef-iso refl t = t | |
| 76 | 211 |
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212 is-o∅ : ( x : Ordinal ) → Dec ( x ≡ o∅ ) |
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213 is-o∅ x with trio< x o∅ |
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214 is-o∅ x | tri< a ¬b ¬c = no ¬b |
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215 is-o∅ x | tri≈ ¬a b ¬c = yes b |
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216 is-o∅ x | tri> ¬a ¬b c = no ¬b |
| 57 | 217 |
| 303 | 218 _,_ : HOD → HOD → HOD |
| 219 x , y = record { od = record { def = λ t → (t ≡ od→ord x ) ∨ ( t ≡ od→ord y ) } ; ¬odmax = ? } -- Ord (omax (od→ord x) (od→ord y)) | |
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220 |
| 79 | 221 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) |
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222 -- postulate f-extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality n (suc n) |
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223 |
| 303 | 224 in-codomain : (X : HOD ) → ( ψ : HOD → HOD ) → HOD |
| 225 in-codomain X ψ = record { od = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( odef X y ∧ ( x ≡ od→ord (ψ (ord→od y ))))) } ; ¬odmax = ? } | |
| 141 | 226 |
| 303 | 227 -- Power Set of X ( or constructible by λ y → odef X (od→ord y ) |
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228 |
| 303 | 229 ZFSubset : (A x : HOD ) → HOD |
| 230 ZFSubset A x = record { od = record { def = λ y → odef A y ∧ odef x y } ; ¬odmax = ? } -- roughly x = A → Set | |
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231 |
| 303 | 232 OPwr : (A : HOD ) → HOD |
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233 OPwr A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A x) ) ) |
| 190 | 234 |
| 303 | 235 -- _⊆_ : ( A B : HOD ) → ∀{ x : HOD } → Set n |
| 271 | 236 -- _⊆_ A B {x} = A ∋ x → B ∋ x |
| 237 | |
| 303 | 238 record _⊆_ ( A B : HOD ) : Set (suc n) where |
| 271 | 239 field |
| 303 | 240 incl : { x : HOD } → A ∋ x → B ∋ x |
| 271 | 241 |
| 242 open _⊆_ | |
| 190 | 243 |
| 244 infixr 220 _⊆_ | |
| 245 | |
| 303 | 246 subset-lemma : {A x : HOD } → ( {y : HOD } → x ∋ y → ZFSubset A x ∋ y ) ⇔ ( x ⊆ A ) |
| 271 | 247 subset-lemma {A} {x} = record { |
| 248 proj1 = λ lt → record { incl = λ x∋z → proj1 (lt x∋z) } | |
| 249 ; proj2 = λ x⊆A lt → record { proj1 = incl x⊆A lt ; proj2 = lt } | |
| 190 | 250 } |
| 251 | |
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252 open import Data.Unit |
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253 |
| 303 | 254 ε-induction : { ψ : HOD → Set (suc n)} |
| 255 → ( {x : HOD } → ({ y : HOD } → x ∋ y → ψ y ) → ψ x ) | |
| 256 → (x : HOD ) → ψ x | |
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257 ε-induction {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (osuc (od→ord x)) <-osuc ) where |
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258 induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (ord→od oy)) → ψ (ord→od ox) |
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259 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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260 ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (ord→od oy) |
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261 ε-induction-ord ox {oy} lt = TransFinite {λ oy → ψ (ord→od oy)} induction oy |
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262 |
| 303 | 263 -- minimal-2 : (x : HOD ) → ( ne : ¬ (x == od∅ ) ) → (y : HOD ) → ¬ ( odef (minimal x ne) (od→ord y)) ∧ (odef x (od→ord y) ) |
| 262 | 264 -- minimal-2 x ne y = contra-position ( ε-induction (λ {z} ind → {!!} ) x ) ( λ p → {!!} ) |
| 265 | |
| 303 | 266 HOD→ZF : ZF |
| 267 HOD→ZF = record { | |
| 268 ZFSet = HOD | |
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269 ; _∋_ = _∋_ |
| 303 | 270 ; _≈_ = _=h=_ |
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271 ; ∅ = od∅ |
| 28 | 272 ; _,_ = _,_ |
| 273 ; Union = Union | |
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274 ; Power = Power |
| 28 | 275 ; Select = Select |
| 276 ; Replace = Replace | |
| 161 | 277 ; infinite = infinite |
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278 ; isZF = isZF |
| 28 | 279 } where |
| 303 | 280 ZFSet = HOD -- is less than Ords because of maxod |
| 281 Select : (X : HOD ) → ((x : HOD ) → Set n ) → HOD | |
| 282 Select X ψ = record { od = record { def = λ x → ( odef X x ∧ ψ ( ord→od x )) } ; ¬odmax = ? } | |
| 283 Replace : HOD → (HOD → HOD ) → HOD | |
| 284 Replace X ψ = record { od = record { def = λ x → (x o< sup-o ( λ x → od→ord (ψ x))) ∧ odef (in-codomain X ψ) x } ; ¬odmax = ? } | |
| 144 | 285 _∩_ : ( A B : ZFSet ) → ZFSet |
| 303 | 286 A ∩ B = record { od = record { def = λ x → odef A x ∧ odef B x } ; ¬odmax = ? } |
| 287 Union : HOD → HOD | |
| 288 Union U = record { od = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((odef U u) ∧ (odef (ord→od u) x))) } ; ¬odmax = ? } | |
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289 _∈_ : ( A B : ZFSet ) → Set n |
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290 A ∈ B = B ∋ A |
| 303 | 291 Power : HOD → HOD |
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292 Power A = Replace (OPwr (Ord (od→ord A))) ( λ x → A ∩ x ) |
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293 -- ｛_｝ : ZFSet → ZFSet |
| 287 | 294 -- ｛ x ｝ = ( x , x ) -- it works but we don't use |
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295 |
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296 data infinite-d : ( x : Ordinal ) → Set n where |
| 161 | 297 iφ : infinite-d o∅ |
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298 isuc : {x : Ordinal } → infinite-d x → |
| 161 | 299 infinite-d (od→ord ( Union (ord→od x , (ord→od x , ord→od x ) ) )) |
| 300 | |
| 303 | 301 infinite : HOD |
| 302 infinite = record { od = record { def = λ x → infinite-d x } ; ¬odmax = ? } | |
| 303 | |
| 304 _=h=_ : (x y : HOD) → Set n | |
| 305 x =h= y = od x == od y | |
| 161 | 306 |
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307 infixr 200 _∈_ |
| 96 | 308 -- infixr 230 _∩_ _∪_ |
| 303 | 309 isZF : IsZF (HOD ) _∋_ _=h=_ od∅ _,_ Union Power Select Replace infinite |
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310 isZF = record { |
| 271 | 311 isEquivalence = record { refl = ==-refl ; sym = ==-sym; trans = ==-trans } |
| 247 | 312 ; pair→ = pair→ |
| 313 ; pair← = pair← | |
| 72 | 314 ; union→ = union→ |
| 315 ; union← = union← | |
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316 ; empty = empty |
| 129 | 317 ; power→ = power→ |
| 76 | 318 ; power← = power← |
| 186 | 319 ; extensionality = λ {A} {B} {w} → extensionality {A} {B} {w} |
| 274 | 320 ; ε-induction = ε-induction |
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321 ; infinity∅ = infinity∅ |
| 160 | 322 ; infinity = infinity |
| 116 | 323 ; selection = λ {X} {ψ} {y} → selection {X} {ψ} {y} |
| 135 | 324 ; replacement← = replacement← |
| 325 ; replacement→ = replacement→ | |
| 274 | 326 -- ; choice-func = choice-func |
| 327 -- ; choice = choice | |
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328 } where |
| 129 | 329 |
| 303 | 330 pair→ : ( x y t : ZFSet ) → (x , y) ∋ t → ( t =h= x ) ∨ ( t =h= y ) |
| 331 pair→ x y t (case1 t≡x ) = case1 (subst₂ (λ j k → j =h= k ) oiso oiso (o≡→== t≡x )) | |
| 332 pair→ x y t (case2 t≡y ) = case2 (subst₂ (λ j k → j =h= k ) oiso oiso (o≡→== t≡y )) | |
| 247 | 333 |
| 303 | 334 pair← : ( x y t : ZFSet ) → ( t =h= x ) ∨ ( t =h= y ) → (x , y) ∋ t |
| 335 pair← x y t (case1 t=h=x) = case1 (cong (λ k → od→ord k ) (==→o≡ t=h=x)) | |
| 336 pair← x y t (case2 t=h=y) = case2 (cong (λ k → od→ord k ) (==→o≡ t=h=y)) | |
| 247 | 337 |
| 303 | 338 empty : (x : HOD ) → ¬ (od∅ ∋ x) |
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339 empty x = ¬x<0 |
| 129 | 340 |
| 271 | 341 o<→c< : {x y : Ordinal } → x o< y → (Ord x) ⊆ (Ord y) |
| 342 o<→c< lt = record { incl = λ z → ordtrans z lt } | |
| 155 | 343 |
| 271 | 344 ⊆→o< : {x y : Ordinal } → (Ord x) ⊆ (Ord y) → x o< osuc y |
| 155 | 345 ⊆→o< {x} {y} lt with trio< x y |
| 346 ⊆→o< {x} {y} lt | tri< a ¬b ¬c = ordtrans a <-osuc | |
| 347 ⊆→o< {x} {y} lt | tri≈ ¬a b ¬c = subst ( λ k → k o< osuc y) (sym b) <-osuc | |
| 271 | 348 ⊆→o< {x} {y} lt | tri> ¬a ¬b c with (incl lt) (o<-subst c (sym diso) refl ) |
| 155 | 349 ... | ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt diso refl )) |
| 151 | 350 |
| 303 | 351 union→ : (X z u : HOD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z |
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352 union→ X z u xx not = ⊥-elim ( not (od→ord u) ( record { proj1 = proj1 xx |
| 303 | 353 ; proj2 = subst ( λ k → odef k (od→ord z)) (sym oiso) (proj2 xx) } )) |
| 354 union← : (X z : HOD) (X∋z : Union X ∋ z) → ¬ ( (u : HOD ) → ¬ ((X ∋ u) ∧ (u ∋ z ))) | |
| 258 | 355 union← X z UX∋z = FExists _ lemma UX∋z where |
| 303 | 356 lemma : {y : Ordinal} → odef X y ∧ odef (ord→od y) (od→ord z) → ¬ ((u : HOD) → ¬ (X ∋ u) ∧ (u ∋ z)) |
| 357 lemma {y} xx not = not (ord→od y) record { proj1 = subst ( λ k → odef X k ) (sym diso) (proj1 xx ) ; proj2 = proj2 xx } | |
| 144 | 358 |
| 303 | 359 ψiso : {ψ : HOD → Set n} {x y : HOD } → ψ x → x ≡ y → ψ y |
| 144 | 360 ψiso {ψ} t refl = t |
| 303 | 361 selection : {ψ : HOD → Set n} {X y : HOD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y) |
| 144 | 362 selection {ψ} {X} {y} = record { |
| 363 proj1 = λ cond → record { proj1 = proj1 cond ; proj2 = ψiso {ψ} (proj2 cond) (sym oiso) } | |
| 364 ; proj2 = λ select → record { proj1 = proj1 select ; proj2 = ψiso {ψ} (proj2 select) oiso } | |
| 365 } | |
| 303 | 366 replacement← : {ψ : HOD → HOD} (X x : HOD) → X ∋ x → Replace X ψ ∋ ψ x |
| 367 replacement← {ψ} X x lt = record { proj1 = ? ; proj2 = lemma } where -- sup-c< ψ {x} | |
| 368 lemma : odef (in-codomain X ψ) (od→ord (ψ x)) | |
| 150 | 369 lemma not = ⊥-elim ( not ( od→ord x ) (record { proj1 = lt ; proj2 = cong (λ k → od→ord (ψ k)) (sym oiso)} )) |
| 303 | 370 replacement→ : {ψ : HOD → HOD} (X x : HOD) → (lt : Replace X ψ ∋ x) → ¬ ( (y : HOD) → ¬ (x =h= ψ y)) |
| 150 | 371 replacement→ {ψ} X x lt = contra-position lemma (lemma2 (proj2 lt)) where |
| 303 | 372 lemma2 : ¬ ((y : Ordinal) → ¬ odef X y ∧ ((od→ord x) ≡ od→ord (ψ (ord→od y)))) |
| 373 → ¬ ((y : Ordinal) → ¬ odef X y ∧ (ord→od (od→ord x) =h= ψ (ord→od y))) | |
| 144 | 374 lemma2 not not2 = not ( λ y d → not2 y (record { proj1 = proj1 d ; proj2 = lemma3 (proj2 d)})) where |
| 303 | 375 lemma3 : {y : Ordinal } → (od→ord x ≡ od→ord (ψ (ord→od y))) → (ord→od (od→ord x) =h= ψ (ord→od y)) |
| 376 lemma3 {y} eq = subst (λ k → ord→od (od→ord x) =h= k ) oiso (o≡→== eq ) | |
| 377 lemma : ( (y : HOD) → ¬ (x =h= ψ y)) → ( (y : Ordinal) → ¬ odef X y ∧ (ord→od (od→ord x) =h= ψ (ord→od y)) ) | |
| 378 lemma not y not2 = not (ord→od y) (subst (λ k → k =h= ψ (ord→od y)) oiso ( proj2 not2 )) | |
| 144 | 379 |
| 380 --- | |
| 381 --- Power Set | |
| 382 --- | |
| 303 | 383 --- First consider ordinals in HOD |
| 100 | 384 --- |
| 303 | 385 --- ZFSubset A x = record { def = λ y → odef A y ∧ odef x y } subset of A |
| 100 | 386 -- |
| 387 -- | |
| 303 | 388 ∩-≡ : { a b : HOD } → ({x : HOD } → (a ∋ x → b ∋ x)) → a =h= ( b ∩ a ) |
| 142 | 389 ∩-≡ {a} {b} inc = record { |
| 390 eq→ = λ {x} x<a → record { proj2 = x<a ; | |
| 303 | 391 proj1 = odef-subst {_} {_} {b} {x} (inc (odef-subst {_} {_} {a} {_} x<a refl (sym diso) )) refl diso } ; |
| 142 | 392 eq← = λ {x} x<a∩b → proj2 x<a∩b } |
| 100 | 393 -- |
| 258 | 394 -- Transitive Set case |
| 303 | 395 -- we have t ∋ x → Ord a ∋ x means t is a subset of Ord a, that is ZFSubset (Ord a) t =h= t |
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396 -- OPwr (Ord a) is a sup of ZFSubset (Ord a) t, so OPwr (Ord a) ∋ t |
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397 -- OPwr A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) ) |
| 100 | 398 -- |
| 303 | 399 ord-power← : (a : Ordinal ) (t : HOD) → ({x : HOD} → (t ∋ x → (Ord a) ∋ x)) → OPwr (Ord a) ∋ t |
| 400 ord-power← a t t→A = odef-subst {_} {_} {OPwr (Ord a)} {od→ord t} | |
| 127 | 401 lemma refl (lemma1 lemma-eq )where |
| 303 | 402 lemma-eq : ZFSubset (Ord a) t =h= t |
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403 eq→ lemma-eq {z} w = proj2 w |
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404 eq← lemma-eq {z} w = record { proj2 = w ; |
| 303 | 405 proj1 = odef-subst {_} {_} {(Ord a)} {z} |
| 406 ( t→A (odef-subst {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso } | |
| 407 lemma1 : {a : Ordinal } { t : HOD } | |
| 408 → (eq : ZFSubset (Ord a) t =h= t) → od→ord (ZFSubset (Ord a) (ord→od (od→ord t))) ≡ od→ord t | |
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409 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 | 410 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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411 lemma = sup-o< |
| 129 | 412 |
| 144 | 413 -- |
| 303 | 414 -- Every set in HOD is a subset of Ordinals, so make OPwr (Ord (od→ord A)) first |
| 258 | 415 -- then replace of all elements of the Power set by A ∩ y |
| 144 | 416 -- |
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417 -- Power A = Replace (OPwr (Ord (od→ord A))) ( λ y → A ∩ y ) |
| 166 | 418 |
| 419 -- we have oly double negation form because of the replacement axiom | |
| 420 -- | |
| 303 | 421 power→ : ( A t : HOD) → Power A ∋ t → {x : HOD} → t ∋ x → ¬ ¬ (A ∋ x) |
| 258 | 422 power→ A t P∋t {x} t∋x = FExists _ lemma5 lemma4 where |
| 142 | 423 a = od→ord A |
| 303 | 424 lemma2 : ¬ ( (y : HOD) → ¬ (t =h= (A ∩ y))) |
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425 lemma2 = replacement→ (OPwr (Ord (od→ord A))) t P∋t |
| 303 | 426 lemma3 : (y : HOD) → t =h= ( A ∩ y ) → ¬ ¬ (A ∋ x) |
| 166 | 427 lemma3 y eq not = not (proj1 (eq→ eq t∋x)) |
| 303 | 428 lemma4 : ¬ ((y : Ordinal) → ¬ (t =h= (A ∩ ord→od y))) |
| 429 lemma4 not = lemma2 ( λ y not1 → not (od→ord y) (subst (λ k → t =h= ( A ∩ k )) (sym oiso) not1 )) | |
| 430 lemma5 : {y : Ordinal} → t =h= (A ∩ ord→od y) → ¬ ¬ (odef A (od→ord x)) | |
| 166 | 431 lemma5 {y} eq not = (lemma3 (ord→od y) eq) not |
| 432 | |
| 303 | 433 power← : (A t : HOD) → ({x : HOD} → (t ∋ x → A ∋ x)) → Power A ∋ t |
| 142 | 434 power← A t t→A = record { proj1 = lemma1 ; proj2 = lemma2 } where |
| 435 a = od→ord A | |
| 303 | 436 lemma0 : {x : HOD} → t ∋ x → Ord a ∋ x |
| 142 | 437 lemma0 {x} t∋x = c<→o< (t→A t∋x) |
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438 lemma3 : OPwr (Ord a) ∋ t |
| 142 | 439 lemma3 = ord-power← a t lemma0 |
| 152 | 440 lemma4 : (A ∩ ord→od (od→ord t)) ≡ t |
| 441 lemma4 = let open ≡-Reasoning in begin | |
| 442 A ∩ ord→od (od→ord t) | |
| 443 ≡⟨ cong (λ k → A ∩ k) oiso ⟩ | |
| 444 A ∩ t | |
| 445 ≡⟨ sym (==→o≡ ( ∩-≡ t→A )) ⟩ | |
| 446 t | |
| 447 ∎ | |
| 276 | 448 lemma1 : od→ord t o< sup-o (λ x → od→ord (A ∩ x)) |
| 449 lemma1 = subst (λ k → od→ord k o< sup-o (λ x → od→ord (A ∩ x))) | |
| 450 lemma4 (sup-o< {λ x → od→ord (A ∩ x)} ) | |
| 303 | 451 lemma2 : odef (in-codomain (OPwr (Ord (od→ord A))) (_∩_ A)) (od→ord t) |
| 151 | 452 lemma2 not = ⊥-elim ( not (od→ord t) (record { proj1 = lemma3 ; proj2 = lemma6 }) ) where |
| 453 lemma6 : od→ord t ≡ od→ord (A ∩ ord→od (od→ord t)) | |
| 303 | 454 lemma6 = cong ( λ k → od→ord k ) (==→o≡ (subst (λ k → t =h= (A ∩ k)) (sym oiso) ( ∩-≡ t→A ))) |
| 142 | 455 |
| 271 | 456 ord⊆power : (a : Ordinal) → (Ord (osuc a)) ⊆ (Power (Ord a)) |
| 457 ord⊆power a = record { incl = λ {x} lt → power← (Ord a) x (lemma lt) } where | |
| 303 | 458 lemma : {x y : HOD} → od→ord x o< osuc a → x ∋ y → Ord a ∋ y |
| 271 | 459 lemma lt y<x with osuc-≡< lt |
| 460 lemma lt y<x | case1 refl = c<→o< y<x | |
| 461 lemma lt y<x | case2 x<a = ordtrans (c<→o< y<x) x<a | |
| 262 | 462 |
| 276 | 463 continuum-hyphotheis : (a : Ordinal) → Set (suc n) |
| 464 continuum-hyphotheis a = Power (Ord a) ⊆ Ord (osuc a) | |
| 129 | 465 |
| 303 | 466 extensionality0 : {A B : HOD } → ((z : HOD) → (A ∋ z) ⇔ (B ∋ z)) → A =h= B |
| 467 eq→ (extensionality0 {A} {B} eq ) {x} d = odef-iso {A} {B} (sym diso) (proj1 (eq (ord→od x))) d | |
| 468 eq← (extensionality0 {A} {B} eq ) {x} d = odef-iso {B} {A} (sym diso) (proj2 (eq (ord→od x))) d | |
| 186 | 469 |
| 303 | 470 extensionality : {A B w : HOD } → ((z : HOD ) → (A ∋ z) ⇔ (B ∋ z)) → (w ∋ A) ⇔ (w ∋ B) |
| 186 | 471 proj1 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) ( ==→o≡ (extensionality0 {A} {B} eq) ) d |
| 472 proj2 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) (sym ( ==→o≡ (extensionality0 {A} {B} eq) )) d | |
| 129 | 473 |
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474 infinity∅ : infinite ∋ od∅ |
| 303 | 475 infinity∅ = odef-subst {_} {_} {infinite} {od→ord (od∅ )} iφ refl lemma where |
| 161 | 476 lemma : o∅ ≡ od→ord od∅ |
| 477 lemma = let open ≡-Reasoning in begin | |
| 478 o∅ | |
| 479 ≡⟨ sym diso ⟩ | |
| 480 od→ord ( ord→od o∅ ) | |
| 481 ≡⟨ cong ( λ k → od→ord k ) o∅≡od∅ ⟩ | |
| 482 od→ord od∅ | |
| 483 ∎ | |
| 303 | 484 infinity : (x : HOD) → infinite ∋ x → infinite ∋ Union (x , (x , x )) |
| 485 infinity x lt = odef-subst {_} {_} {infinite} {od→ord (Union (x , (x , x )))} ( isuc {od→ord x} lt ) refl lemma where | |
| 161 | 486 lemma : od→ord (Union (ord→od (od→ord x) , (ord→od (od→ord x) , ord→od (od→ord x)))) |
| 487 ≡ od→ord (Union (x , (x , x))) | |
| 488 lemma = cong (λ k → od→ord (Union ( k , ( k , k ) ))) oiso | |
| 489 | |
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490 |
| 303 | 491 Union = ZF.Union HOD→ZF |
| 492 Power = ZF.Power HOD→ZF | |
| 493 Select = ZF.Select HOD→ZF | |
| 494 Replace = ZF.Replace HOD→ZF | |
| 495 isZF = ZF.isZF HOD→ZF |