Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate OD.agda @ 315:35e1214fa093
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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| date | Fri, 03 Jul 2020 16:50:19 +0900 |
| parents | 6b09b5af9fcd |
| children | c030a9655e79 |
| 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 -- | |
| 309 | 61 -- We also assumes HODs are isomorphic to Ordinals, which is ususally proved by Goedel number tricks. |
| 62 -- ODs have an ovbious maximum, but Ordinals are not. So HOD cannot be a maximum OD. | |
| 290 | 63 -- |
| 64 -- In classical Set Theory, sup is defined by Uion. Since we are working on constructive logic, | |
| 65 -- we need explict assumption on sup. | |
| 309 | 66 -- In order to allow sup on od→ord HOD, solutions of a HOD have to have a maximu. |
| 67 -- | |
| 68 -- ==→o≡ is necessary to prove axiom of extensionality. | |
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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 |
| 304 | 80 odmax : Ordinal |
| 308 | 81 <odmax : {y : Ordinal} → def od y → y o< odmax |
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82 |
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83 open HOD |
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84 |
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85 record ODAxiom : Set (suc n) where |
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86 field |
| 304 | 87 -- HOD is isomorphic to Ordinal (by means of Goedel number) |
| 303 | 88 od→ord : HOD → Ordinal |
| 89 ord→od : Ordinal → HOD | |
| 90 c<→o< : {x y : HOD } → def (od y) ( od→ord x ) → od→ord x o< od→ord y | |
| 312 | 91 ⊆→o≤ : {y z : HOD } → ({x : Ordinal} → def (od y) x → def (od z) x ) → od→ord y o< osuc (od→ord z) |
| 303 | 92 oiso : {x : HOD } → ord→od ( od→ord x ) ≡ x |
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93 diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x |
| 303 | 94 ==→o≡ : { x y : HOD } → (od x == od y) → x ≡ y |
| 306 | 95 sup-o : (A : HOD) → (( x : Ordinal ) → def (od A) x → Ordinal ) → Ordinal |
| 96 sup-o< : (A : HOD) → { ψ : ( x : Ordinal ) → def (od A) x → Ordinal } → ∀ {x : Ordinal } → (lt : def (od A) x ) → ψ x lt o< sup-o A ψ | |
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97 |
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98 postulate odAxiom : ODAxiom |
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99 open ODAxiom odAxiom |
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100 |
| 303 | 101 -- maxod : {x : OD} → od→ord x o< od→ord Ords |
| 102 -- maxod {x} = c<→o< OneObj | |
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103 |
| 304 | 104 -- we have not this contradiction |
| 303 | 105 -- bad-bad : ⊥ |
| 304 | 106 -- bad-bad = osuc-< <-osuc (c<→o< { record { od = record { def = λ x → One }; <odmax = {!!} } } OneObj) |
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107 |
| 179 | 108 -- Ordinal in OD ( and ZFSet ) Transitive Set |
| 303 | 109 Ord : ( a : Ordinal ) → HOD |
| 304 | 110 Ord a = record { od = record { def = λ y → y o< a } ; odmax = a ; <odmax = lemma } where |
| 111 lemma : {x : Ordinal} → x o< a → x o< a | |
| 112 lemma {x} lt = lt | |
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113 |
| 303 | 114 od∅ : HOD |
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115 od∅ = Ord o∅ |
| 40 | 116 |
| 303 | 117 odef : HOD → Ordinal → Set n |
| 118 odef A x = def ( od A ) x | |
| 123 | 119 |
| 303 | 120 o<→c<→HOD=Ord : ( {x y : Ordinal } → x o< y → odef (ord→od y) x ) → {x : HOD } → x ≡ Ord (od→ord x) |
| 121 o<→c<→HOD=Ord o<→c< {x} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where | |
| 122 lemma1 : {y : Ordinal} → odef x y → odef (Ord (od→ord x)) y | |
| 123 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)) | |
| 124 lemma2 : {y : Ordinal} → odef (Ord (od→ord x)) y → odef x y | |
| 125 lemma2 {y} lt = subst (λ k → odef k y ) oiso (o<→c< {y} {od→ord x} lt ) | |
| 95 | 126 |
| 303 | 127 _∋_ : ( a x : HOD ) → Set n |
| 128 _∋_ a x = odef a ( od→ord x ) | |
| 129 | |
| 130 _c<_ : ( x a : HOD ) → Set n | |
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131 x c< a = a ∋ x |
| 103 | 132 |
| 303 | 133 cseq : {n : Level} → HOD → HOD |
| 308 | 134 cseq x = record { od = record { def = λ y → odef x (osuc y) } ; odmax = osuc (odmax x) ; <odmax = lemma } where |
| 135 lemma : {y : Ordinal} → def (od x) (osuc y) → y o< osuc (odmax x) | |
| 136 lemma {y} lt = ordtrans <-osuc (ordtrans (<odmax x lt) <-osuc ) | |
| 95 | 137 |
| 303 | 138 odef-subst : {Z : HOD } {X : Ordinal }{z : HOD } {x : Ordinal }→ odef Z X → Z ≡ z → X ≡ x → odef z x |
| 139 odef-subst df refl refl = df | |
| 95 | 140 |
| 303 | 141 otrans : {n : Level} {a x y : Ordinal } → odef (Ord a) x → odef (Ord x) y → odef (Ord a) y |
| 187 | 142 otrans x<a y<x = ordtrans y<x x<a |
| 123 | 143 |
| 303 | 144 odef→o< : {X : HOD } → {x : Ordinal } → odef X x → x o< od→ord X |
| 145 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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146 |
| 51 | 147 -- avoiding lv != Zero error |
| 303 | 148 orefl : { x : HOD } → { y : Ordinal } → od→ord x ≡ y → od→ord x ≡ y |
| 51 | 149 orefl refl = refl |
| 150 | |
| 303 | 151 ==-iso : { x y : HOD } → od (ord→od (od→ord x)) == od (ord→od (od→ord y)) → od x == od y |
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152 ==-iso {x} {y} eq = record { |
| 303 | 153 eq→ = λ d → lemma ( eq→ eq (odef-subst d (sym oiso) refl )) ; |
| 154 eq← = λ d → lemma ( eq← eq (odef-subst d (sym oiso) refl )) } | |
| 51 | 155 where |
| 303 | 156 lemma : {x : HOD } {z : Ordinal } → odef (ord→od (od→ord x)) z → odef x z |
| 157 lemma {x} {z} d = odef-subst d oiso refl | |
| 51 | 158 |
| 303 | 159 =-iso : {x y : HOD } → (od x == od y) ≡ (od (ord→od (od→ord x)) == od y) |
| 160 =-iso {_} {y} = cong ( λ k → od k == od y ) (sym oiso) | |
| 57 | 161 |
| 303 | 162 ord→== : { x y : HOD } → od→ord x ≡ od→ord y → od x == od y |
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163 ord→== {x} {y} eq = ==-iso (lemma (od→ord x) (od→ord y) (orefl eq)) where |
| 303 | 164 lemma : ( ox oy : Ordinal ) → ox ≡ oy → od (ord→od ox) == od (ord→od oy) |
| 271 | 165 lemma ox ox refl = ==-refl |
| 51 | 166 |
| 303 | 167 o≡→== : { x y : Ordinal } → x ≡ y → od (ord→od x) == od (ord→od y) |
| 271 | 168 o≡→== {x} {.x} refl = ==-refl |
| 51 | 169 |
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170 o∅≡od∅ : ord→od (o∅ ) ≡ od∅ |
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171 o∅≡od∅ = ==→o≡ lemma where |
| 303 | 172 lemma0 : {x : Ordinal} → odef (ord→od o∅) x → odef od∅ x |
| 173 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 | |
| 174 lemma1 : {x : Ordinal} → odef od∅ x → odef (ord→od o∅) x | |
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175 lemma1 {x} lt = ⊥-elim (¬x<0 lt) |
| 303 | 176 lemma : od (ord→od o∅) == od od∅ |
| 150 | 177 lemma = record { eq→ = lemma0 ; eq← = lemma1 } |
| 178 | |
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179 ord-od∅ : od→ord (od∅ ) ≡ o∅ |
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180 ord-od∅ = sym ( subst (λ k → k ≡ od→ord (od∅ ) ) diso (cong ( λ k → od→ord k ) o∅≡od∅ ) ) |
| 80 | 181 |
| 303 | 182 ∅0 : record { def = λ x → Lift n ⊥ } == od od∅ |
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183 eq→ ∅0 {w} (lift ()) |
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184 eq← ∅0 {w} lt = lift (¬x<0 lt) |
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185 |
| 303 | 186 ∅< : { x y : HOD } → odef x (od→ord y ) → ¬ ( od x == od od∅ ) |
| 271 | 187 ∅< {x} {y} d eq with eq→ (==-trans eq (==-sym ∅0) ) d |
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188 ∅< {x} {y} d eq | lift () |
| 57 | 189 |
| 303 | 190 ∅6 : { x : HOD } → ¬ ( x ∋ x ) -- no Russel paradox |
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191 ∅6 {x} x∋x = o<¬≡ refl ( c<→o< {x} {x} x∋x ) |
| 51 | 192 |
| 303 | 193 odef-iso : {A B : HOD } {x y : Ordinal } → x ≡ y → (odef A y → odef B y) → odef A x → odef B x |
| 194 odef-iso refl t = t | |
| 76 | 195 |
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196 is-o∅ : ( x : Ordinal ) → Dec ( x ≡ o∅ ) |
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197 is-o∅ x with trio< x o∅ |
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198 is-o∅ x | tri< a ¬b ¬c = no ¬b |
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199 is-o∅ x | tri≈ ¬a b ¬c = yes b |
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200 is-o∅ x | tri> ¬a ¬b c = no ¬b |
| 57 | 201 |
| 303 | 202 _,_ : HOD → HOD → HOD |
| 308 | 203 x , y = record { od = record { def = λ t → (t ≡ od→ord x ) ∨ ( t ≡ od→ord y ) } ; odmax = omax (od→ord x) (od→ord y) ; <odmax = lemma } where |
| 204 lemma : {t : Ordinal} → (t ≡ od→ord x) ∨ (t ≡ od→ord y) → t o< omax (od→ord x) (od→ord y) | |
| 205 lemma {t} (case1 refl) = omax-x _ _ | |
| 206 lemma {t} (case2 refl) = omax-y _ _ | |
| 207 | |
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208 |
| 79 | 209 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) |
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210 -- postulate f-extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality n (suc n) |
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211 |
| 303 | 212 in-codomain : (X : HOD ) → ( ψ : HOD → HOD ) → HOD |
| 304 | 213 in-codomain X ψ = record { od = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( odef X y ∧ ( x ≡ od→ord (ψ (ord→od y ))))) } ; odmax = {!!} ; <odmax = {!!} } |
| 141 | 214 |
| 303 | 215 -- Power Set of X ( or constructible by λ y → odef X (od→ord y ) |
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216 |
| 303 | 217 ZFSubset : (A x : HOD ) → HOD |
| 308 | 218 ZFSubset A x = record { od = record { def = λ y → odef A y ∧ odef x y } ; odmax = omin (odmax A) (odmax x) ; <odmax = lemma } where -- roughly x = A → Set |
| 219 lemma : {y : Ordinal} → def (od A) y ∧ def (od x) y → y o< omin (odmax A) (odmax x) | |
| 220 lemma {y} and = min1 (<odmax A (proj1 and)) (<odmax x (proj2 and)) | |
| 221 | |
| 303 | 222 record _⊆_ ( A B : HOD ) : Set (suc n) where |
| 271 | 223 field |
| 303 | 224 incl : { x : HOD } → A ∋ x → B ∋ x |
| 271 | 225 |
| 226 open _⊆_ | |
| 190 | 227 infixr 220 _⊆_ |
| 228 | |
| 303 | 229 subset-lemma : {A x : HOD } → ( {y : HOD } → x ∋ y → ZFSubset A x ∋ y ) ⇔ ( x ⊆ A ) |
| 271 | 230 subset-lemma {A} {x} = record { |
| 231 proj1 = λ lt → record { incl = λ x∋z → proj1 (lt x∋z) } | |
| 232 ; proj2 = λ x⊆A lt → record { proj1 = incl x⊆A lt ; proj2 = lt } | |
| 190 | 233 } |
| 234 | |
| 312 | 235 power< : {A x : HOD } → x ⊆ A → Ord (osuc (od→ord A)) ∋ x |
| 236 power< {A} {x} x⊆A = ⊆→o≤ (λ {y} x∋y → subst (λ k → def (od A) k) diso (lemma y x∋y ) ) where | |
| 237 lemma : (y : Ordinal) → def (od x) y → def (od A) (od→ord (ord→od y)) | |
| 238 lemma y x∋y = incl x⊆A (subst (λ k → def (od x) k ) (sym diso) x∋y ) | |
| 239 | |
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240 open import Data.Unit |
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241 |
| 303 | 242 ε-induction : { ψ : HOD → Set (suc n)} |
| 243 → ( {x : HOD } → ({ y : HOD } → x ∋ y → ψ y ) → ψ x ) | |
| 244 → (x : HOD ) → ψ x | |
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245 ε-induction {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (osuc (od→ord x)) <-osuc ) where |
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246 induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (ord→od oy)) → ψ (ord→od ox) |
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247 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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248 ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (ord→od oy) |
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249 ε-induction-ord ox {oy} lt = TransFinite {λ oy → ψ (ord→od oy)} induction oy |
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250 |
| 303 | 251 HOD→ZF : ZF |
| 252 HOD→ZF = record { | |
| 253 ZFSet = HOD | |
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254 ; _∋_ = _∋_ |
| 303 | 255 ; _≈_ = _=h=_ |
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256 ; ∅ = od∅ |
| 28 | 257 ; _,_ = _,_ |
| 258 ; Union = Union | |
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259 ; Power = Power |
| 28 | 260 ; Select = Select |
| 261 ; Replace = Replace | |
| 161 | 262 ; infinite = infinite |
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263 ; isZF = isZF |
| 28 | 264 } where |
| 303 | 265 ZFSet = HOD -- is less than Ords because of maxod |
| 266 Select : (X : HOD ) → ((x : HOD ) → Set n ) → HOD | |
| 308 | 267 Select X ψ = record { od = record { def = λ x → ( odef X x ∧ ψ ( ord→od x )) } ; odmax = odmax X ; <odmax = λ y → <odmax X (proj1 y) } |
| 310 | 268 Replace : HOD → (HOD → HOD) → HOD |
| 311 | 269 Replace X ψ = record { od = record { def = λ x → (x o< sup-o X (λ y X∋y → od→ord (ψ (ord→od y)))) ∧ odef (in-codomain X ψ) x } ; odmax = {!!} ; <odmax = {!!} } -- ( λ x → od→ord (ψ x)) |
| 144 | 270 _∩_ : ( A B : ZFSet ) → ZFSet |
| 309 | 271 A ∩ B = record { od = record { def = λ x → odef A x ∧ odef B x } ; odmax = omin (odmax A) (odmax B) ; <odmax = λ y → min1 (<odmax A (proj1 y)) (<odmax B (proj2 y))} |
| 303 | 272 Union : HOD → HOD |
| 304 | 273 Union U = record { od = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((odef U u) ∧ (odef (ord→od u) x))) } ; odmax = {!!} ; <odmax = {!!} } |
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274 _∈_ : ( A B : ZFSet ) → Set n |
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275 A ∈ B = B ∋ A |
| 312 | 276 |
| 277 OPwr : (A : HOD ) → HOD | |
| 278 OPwr A = Ord ( sup-o (Ord (osuc (od→ord A))) ( λ x A∋x → od→ord ( ZFSubset A (ord→od x)) ) ) | |
| 279 | |
| 303 | 280 Power : HOD → HOD |
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281 Power A = Replace (OPwr (Ord (od→ord A))) ( λ x → A ∩ x ) |
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282 -- ｛_｝ : ZFSet → ZFSet |
| 287 | 283 -- ｛ x ｝ = ( x , x ) -- it works but we don't use |
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284 |
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285 data infinite-d : ( x : Ordinal ) → Set n where |
| 161 | 286 iφ : infinite-d o∅ |
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287 isuc : {x : Ordinal } → infinite-d x → |
| 161 | 288 infinite-d (od→ord ( Union (ord→od x , (ord→od x , ord→od x ) ) )) |
| 289 | |
| 303 | 290 infinite : HOD |
| 304 | 291 infinite = record { od = record { def = λ x → infinite-d x } ; odmax = {!!} ; <odmax = {!!} } |
| 303 | 292 |
| 293 _=h=_ : (x y : HOD) → Set n | |
| 294 x =h= y = od x == od y | |
| 161 | 295 |
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296 infixr 200 _∈_ |
| 96 | 297 -- infixr 230 _∩_ _∪_ |
| 303 | 298 isZF : IsZF (HOD ) _∋_ _=h=_ od∅ _,_ Union Power Select Replace infinite |
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299 isZF = record { |
| 271 | 300 isEquivalence = record { refl = ==-refl ; sym = ==-sym; trans = ==-trans } |
| 247 | 301 ; pair→ = pair→ |
| 302 ; pair← = pair← | |
| 72 | 303 ; union→ = union→ |
| 304 ; union← = union← | |
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305 ; empty = empty |
| 129 | 306 ; power→ = power→ |
| 76 | 307 ; power← = power← |
| 186 | 308 ; extensionality = λ {A} {B} {w} → extensionality {A} {B} {w} |
| 274 | 309 ; ε-induction = ε-induction |
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310 ; infinity∅ = infinity∅ |
| 160 | 311 ; infinity = infinity |
| 116 | 312 ; selection = λ {X} {ψ} {y} → selection {X} {ψ} {y} |
| 135 | 313 ; replacement← = replacement← |
| 314 ; replacement→ = replacement→ | |
| 274 | 315 -- ; choice-func = choice-func |
| 316 -- ; choice = choice | |
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317 } where |
| 129 | 318 |
| 303 | 319 pair→ : ( x y t : ZFSet ) → (x , y) ∋ t → ( t =h= x ) ∨ ( t =h= y ) |
| 320 pair→ x y t (case1 t≡x ) = case1 (subst₂ (λ j k → j =h= k ) oiso oiso (o≡→== t≡x )) | |
| 321 pair→ x y t (case2 t≡y ) = case2 (subst₂ (λ j k → j =h= k ) oiso oiso (o≡→== t≡y )) | |
| 247 | 322 |
| 303 | 323 pair← : ( x y t : ZFSet ) → ( t =h= x ) ∨ ( t =h= y ) → (x , y) ∋ t |
| 324 pair← x y t (case1 t=h=x) = case1 (cong (λ k → od→ord k ) (==→o≡ t=h=x)) | |
| 325 pair← x y t (case2 t=h=y) = case2 (cong (λ k → od→ord k ) (==→o≡ t=h=y)) | |
| 247 | 326 |
| 303 | 327 empty : (x : HOD ) → ¬ (od∅ ∋ x) |
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328 empty x = ¬x<0 |
| 129 | 329 |
| 271 | 330 o<→c< : {x y : Ordinal } → x o< y → (Ord x) ⊆ (Ord y) |
| 331 o<→c< lt = record { incl = λ z → ordtrans z lt } | |
| 155 | 332 |
| 271 | 333 ⊆→o< : {x y : Ordinal } → (Ord x) ⊆ (Ord y) → x o< osuc y |
| 155 | 334 ⊆→o< {x} {y} lt with trio< x y |
| 335 ⊆→o< {x} {y} lt | tri< a ¬b ¬c = ordtrans a <-osuc | |
| 336 ⊆→o< {x} {y} lt | tri≈ ¬a b ¬c = subst ( λ k → k o< osuc y) (sym b) <-osuc | |
| 271 | 337 ⊆→o< {x} {y} lt | tri> ¬a ¬b c with (incl lt) (o<-subst c (sym diso) refl ) |
| 155 | 338 ... | ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt diso refl )) |
| 151 | 339 |
| 303 | 340 union→ : (X z u : HOD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z |
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341 union→ X z u xx not = ⊥-elim ( not (od→ord u) ( record { proj1 = proj1 xx |
| 303 | 342 ; proj2 = subst ( λ k → odef k (od→ord z)) (sym oiso) (proj2 xx) } )) |
| 343 union← : (X z : HOD) (X∋z : Union X ∋ z) → ¬ ( (u : HOD ) → ¬ ((X ∋ u) ∧ (u ∋ z ))) | |
| 258 | 344 union← X z UX∋z = FExists _ lemma UX∋z where |
| 303 | 345 lemma : {y : Ordinal} → odef X y ∧ odef (ord→od y) (od→ord z) → ¬ ((u : HOD) → ¬ (X ∋ u) ∧ (u ∋ z)) |
| 346 lemma {y} xx not = not (ord→od y) record { proj1 = subst ( λ k → odef X k ) (sym diso) (proj1 xx ) ; proj2 = proj2 xx } | |
| 144 | 347 |
| 303 | 348 ψiso : {ψ : HOD → Set n} {x y : HOD } → ψ x → x ≡ y → ψ y |
| 144 | 349 ψiso {ψ} t refl = t |
| 303 | 350 selection : {ψ : HOD → Set n} {X y : HOD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y) |
| 144 | 351 selection {ψ} {X} {y} = record { |
| 352 proj1 = λ cond → record { proj1 = proj1 cond ; proj2 = ψiso {ψ} (proj2 cond) (sym oiso) } | |
| 353 ; proj2 = λ select → record { proj1 = proj1 select ; proj2 = ψiso {ψ} (proj2 select) oiso } | |
| 354 } | |
| 311 | 355 sup-c< : (ψ : HOD → HOD) → {X x : HOD} → X ∋ x → od→ord (ψ x) o< (sup-o X (λ y X∋y → od→ord (ψ (ord→od y)))) |
| 356 sup-c< ψ {X} {x} lt = subst (λ k → od→ord (ψ k) o< _ ) oiso (sup-o< X lt ) | |
| 303 | 357 replacement← : {ψ : HOD → HOD} (X x : HOD) → X ∋ x → Replace X ψ ∋ ψ x |
| 311 | 358 replacement← {ψ} X x lt = record { proj1 = sup-c< ψ {X} {x} lt ; proj2 = lemma } where |
| 303 | 359 lemma : odef (in-codomain X ψ) (od→ord (ψ x)) |
| 150 | 360 lemma not = ⊥-elim ( not ( od→ord x ) (record { proj1 = lt ; proj2 = cong (λ k → od→ord (ψ k)) (sym oiso)} )) |
| 303 | 361 replacement→ : {ψ : HOD → HOD} (X x : HOD) → (lt : Replace X ψ ∋ x) → ¬ ( (y : HOD) → ¬ (x =h= ψ y)) |
| 150 | 362 replacement→ {ψ} X x lt = contra-position lemma (lemma2 (proj2 lt)) where |
| 303 | 363 lemma2 : ¬ ((y : Ordinal) → ¬ odef X y ∧ ((od→ord x) ≡ od→ord (ψ (ord→od y)))) |
| 364 → ¬ ((y : Ordinal) → ¬ odef X y ∧ (ord→od (od→ord x) =h= ψ (ord→od y))) | |
| 144 | 365 lemma2 not not2 = not ( λ y d → not2 y (record { proj1 = proj1 d ; proj2 = lemma3 (proj2 d)})) where |
| 303 | 366 lemma3 : {y : Ordinal } → (od→ord x ≡ od→ord (ψ (ord→od y))) → (ord→od (od→ord x) =h= ψ (ord→od y)) |
| 367 lemma3 {y} eq = subst (λ k → ord→od (od→ord x) =h= k ) oiso (o≡→== eq ) | |
| 368 lemma : ( (y : HOD) → ¬ (x =h= ψ y)) → ( (y : Ordinal) → ¬ odef X y ∧ (ord→od (od→ord x) =h= ψ (ord→od y)) ) | |
| 369 lemma not y not2 = not (ord→od y) (subst (λ k → k =h= ψ (ord→od y)) oiso ( proj2 not2 )) | |
| 144 | 370 |
| 371 --- | |
| 372 --- Power Set | |
| 373 --- | |
| 303 | 374 --- First consider ordinals in HOD |
| 100 | 375 --- |
| 303 | 376 --- ZFSubset A x = record { def = λ y → odef A y ∧ odef x y } subset of A |
| 100 | 377 -- |
| 378 -- | |
| 303 | 379 ∩-≡ : { a b : HOD } → ({x : HOD } → (a ∋ x → b ∋ x)) → a =h= ( b ∩ a ) |
| 142 | 380 ∩-≡ {a} {b} inc = record { |
| 381 eq→ = λ {x} x<a → record { proj2 = x<a ; | |
| 303 | 382 proj1 = odef-subst {_} {_} {b} {x} (inc (odef-subst {_} {_} {a} {_} x<a refl (sym diso) )) refl diso } ; |
| 142 | 383 eq← = λ {x} x<a∩b → proj2 x<a∩b } |
| 100 | 384 -- |
| 258 | 385 -- Transitive Set case |
| 303 | 386 -- 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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387 -- OPwr (Ord a) is a sup of ZFSubset (Ord a) t, so OPwr (Ord a) ∋ t |
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388 -- OPwr A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) ) |
| 100 | 389 -- |
| 303 | 390 ord-power← : (a : Ordinal ) (t : HOD) → ({x : HOD} → (t ∋ x → (Ord a) ∋ x)) → OPwr (Ord a) ∋ t |
| 391 ord-power← a t t→A = odef-subst {_} {_} {OPwr (Ord a)} {od→ord t} | |
| 127 | 392 lemma refl (lemma1 lemma-eq )where |
| 303 | 393 lemma-eq : ZFSubset (Ord a) t =h= t |
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394 eq→ lemma-eq {z} w = proj2 w |
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395 eq← lemma-eq {z} w = record { proj2 = w ; |
| 303 | 396 proj1 = odef-subst {_} {_} {(Ord a)} {z} |
| 397 ( t→A (odef-subst {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso } | |
| 398 lemma1 : {a : Ordinal } { t : HOD } | |
| 399 → (eq : ZFSubset (Ord a) t =h= t) → od→ord (ZFSubset (Ord a) (ord→od (od→ord t))) ≡ od→ord t | |
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400 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 )) |
| 312 | 401 lemma2 : (od→ord t) o< (osuc (od→ord (Ord a))) |
| 402 lemma2 = ⊆→o≤ {t} {Ord a} (λ {x} x<t → subst (λ k → def (od (Ord a)) k) diso (t→A (subst (λ k → def (od t) k) (sym diso) x<t))) | |
| 403 lemma : od→ord (ZFSubset (Ord a) (ord→od (od→ord t)) ) o< sup-o (Ord (osuc (od→ord (Ord a)))) (λ x lt → od→ord (ZFSubset (Ord a) (ord→od x))) | |
| 311 | 404 lemma = sup-o< _ lemma2 |
| 129 | 405 |
| 144 | 406 -- |
| 303 | 407 -- Every set in HOD is a subset of Ordinals, so make OPwr (Ord (od→ord A)) first |
| 258 | 408 -- then replace of all elements of the Power set by A ∩ y |
| 144 | 409 -- |
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410 -- Power A = Replace (OPwr (Ord (od→ord A))) ( λ y → A ∩ y ) |
| 166 | 411 |
| 412 -- we have oly double negation form because of the replacement axiom | |
| 413 -- | |
| 303 | 414 power→ : ( A t : HOD) → Power A ∋ t → {x : HOD} → t ∋ x → ¬ ¬ (A ∋ x) |
| 258 | 415 power→ A t P∋t {x} t∋x = FExists _ lemma5 lemma4 where |
| 142 | 416 a = od→ord A |
| 303 | 417 lemma2 : ¬ ( (y : HOD) → ¬ (t =h= (A ∩ y))) |
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418 lemma2 = replacement→ (OPwr (Ord (od→ord A))) t P∋t |
| 303 | 419 lemma3 : (y : HOD) → t =h= ( A ∩ y ) → ¬ ¬ (A ∋ x) |
| 166 | 420 lemma3 y eq not = not (proj1 (eq→ eq t∋x)) |
| 303 | 421 lemma4 : ¬ ((y : Ordinal) → ¬ (t =h= (A ∩ ord→od y))) |
| 422 lemma4 not = lemma2 ( λ y not1 → not (od→ord y) (subst (λ k → t =h= ( A ∩ k )) (sym oiso) not1 )) | |
| 423 lemma5 : {y : Ordinal} → t =h= (A ∩ ord→od y) → ¬ ¬ (odef A (od→ord x)) | |
| 166 | 424 lemma5 {y} eq not = (lemma3 (ord→od y) eq) not |
| 425 | |
| 303 | 426 power← : (A t : HOD) → ({x : HOD} → (t ∋ x → A ∋ x)) → Power A ∋ t |
| 311 | 427 power← A t t→A = record { proj1 = lemma1 ; proj2 = lemma2 } where |
| 142 | 428 a = od→ord A |
| 303 | 429 lemma0 : {x : HOD} → t ∋ x → Ord a ∋ x |
| 142 | 430 lemma0 {x} t∋x = c<→o< (t→A t∋x) |
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431 lemma3 : OPwr (Ord a) ∋ t |
| 142 | 432 lemma3 = ord-power← a t lemma0 |
| 152 | 433 lemma4 : (A ∩ ord→od (od→ord t)) ≡ t |
| 434 lemma4 = let open ≡-Reasoning in begin | |
| 435 A ∩ ord→od (od→ord t) | |
| 436 ≡⟨ cong (λ k → A ∩ k) oiso ⟩ | |
| 437 A ∩ t | |
| 438 ≡⟨ sym (==→o≡ ( ∩-≡ t→A )) ⟩ | |
| 439 t | |
| 440 ∎ | |
| 312 | 441 --- (od→ord t) o< (sup-o (Ord (osuc (od→ord (Ord (od→ord A))))) (λ x A∋x → od→ord (ZFSubset (Ord (od→ord A)) (ord→od x)))) |
| 314 | 442 sup1 = sup-o (Ord (osuc (od→ord (Ord (od→ord A))))) (λ x A∋x → od→ord (ZFSubset (Ord (od→ord A)) (ord→od x))) |
| 313 | 443 lemma9 : def (od (Ord (Ordinals.osuc O (od→ord (Ord (od→ord A)))))) (od→ord (Ord (od→ord A))) |
| 444 lemma9 = <-osuc | |
| 315 | 445 lemmab : od→ord (ZFSubset (Ord (od→ord A)) (ord→od (od→ord (Ord (od→ord A) )))) o< sup1 |
| 446 lemmab = sup-o< (Ord (osuc (od→ord (Ord (od→ord A))))) lemma9 | |
| 447 lemmad : Ord (osuc (od→ord A)) ∋ t | |
| 448 lemmad = subst (λ k → k o< osuc (od→ord A)) {!!} {!!} -- ( ⊆→o≤ {ord→od (od→ord t)} {ord→od (od→ord (Ord (od→ord t)))} (λ {x} lt → {!!} ) ) | |
| 449 lemmac : ZFSubset (Ord (od→ord A)) (ord→od (od→ord (Ord (od→ord A) ))) =h= Ord (od→ord A) | |
| 450 lemmac = record { eq→ = {!!} ; eq← = {!!} } | |
| 451 lemmae : od→ord (ZFSubset (Ord (od→ord A)) (ord→od (od→ord (Ord (od→ord A))))) ≡ od→ord (Ord (od→ord A)) | |
| 452 lemmae = cong (λ k → od→ord k ) ( ==→o≡ lemmac) | |
| 311 | 453 lemma7 : def (od (OPwr (Ord (od→ord A)))) (od→ord t) |
| 315 | 454 lemma7 with osuc-≡< lemmad |
| 455 lemma7 | case2 lt = ordtrans (c<→o< lt) (subst (λ k → k o< sup1) lemmae lemmab ) | |
| 456 lemma7 | case1 eq with osuc-≡< lemmad | |
| 457 lemma7 | case1 eq | case1 eq1 = subst (λ k → k o< sup1) (trans lemmae {!!}) lemmab -- od→ord (Ord (od→ord A)) ≡ od→ord t | |
| 458 lemma7 | case1 eq | case2 lt = ordtrans {!!} (subst (λ k → k o< sup1) lemmae lemmab ) | |
| 310 | 459 lemma1 : od→ord t o< sup-o (OPwr (Ord (od→ord A))) (λ x lt → od→ord (A ∩ (ord→od x))) |
| 460 lemma1 = subst (λ k → od→ord k o< sup-o (OPwr (Ord (od→ord A))) (λ x lt → od→ord (A ∩ (ord→od x)))) | |
| 311 | 461 lemma4 (sup-o< (OPwr (Ord (od→ord A))) lemma7 ) |
| 303 | 462 lemma2 : odef (in-codomain (OPwr (Ord (od→ord A))) (_∩_ A)) (od→ord t) |
| 151 | 463 lemma2 not = ⊥-elim ( not (od→ord t) (record { proj1 = lemma3 ; proj2 = lemma6 }) ) where |
| 464 lemma6 : od→ord t ≡ od→ord (A ∩ ord→od (od→ord t)) | |
| 303 | 465 lemma6 = cong ( λ k → od→ord k ) (==→o≡ (subst (λ k → t =h= (A ∩ k)) (sym oiso) ( ∩-≡ t→A ))) |
| 142 | 466 |
| 311 | 467 |
| 271 | 468 ord⊆power : (a : Ordinal) → (Ord (osuc a)) ⊆ (Power (Ord a)) |
| 469 ord⊆power a = record { incl = λ {x} lt → power← (Ord a) x (lemma lt) } where | |
| 303 | 470 lemma : {x y : HOD} → od→ord x o< osuc a → x ∋ y → Ord a ∋ y |
| 271 | 471 lemma lt y<x with osuc-≡< lt |
| 472 lemma lt y<x | case1 refl = c<→o< y<x | |
| 473 lemma lt y<x | case2 x<a = ordtrans (c<→o< y<x) x<a | |
| 262 | 474 |
| 276 | 475 continuum-hyphotheis : (a : Ordinal) → Set (suc n) |
| 476 continuum-hyphotheis a = Power (Ord a) ⊆ Ord (osuc a) | |
| 129 | 477 |
| 303 | 478 extensionality0 : {A B : HOD } → ((z : HOD) → (A ∋ z) ⇔ (B ∋ z)) → A =h= B |
| 479 eq→ (extensionality0 {A} {B} eq ) {x} d = odef-iso {A} {B} (sym diso) (proj1 (eq (ord→od x))) d | |
| 480 eq← (extensionality0 {A} {B} eq ) {x} d = odef-iso {B} {A} (sym diso) (proj2 (eq (ord→od x))) d | |
| 186 | 481 |
| 303 | 482 extensionality : {A B w : HOD } → ((z : HOD ) → (A ∋ z) ⇔ (B ∋ z)) → (w ∋ A) ⇔ (w ∋ B) |
| 186 | 483 proj1 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) ( ==→o≡ (extensionality0 {A} {B} eq) ) d |
| 484 proj2 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) (sym ( ==→o≡ (extensionality0 {A} {B} eq) )) d | |
| 129 | 485 |
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486 infinity∅ : infinite ∋ od∅ |
| 303 | 487 infinity∅ = odef-subst {_} {_} {infinite} {od→ord (od∅ )} iφ refl lemma where |
| 161 | 488 lemma : o∅ ≡ od→ord od∅ |
| 489 lemma = let open ≡-Reasoning in begin | |
| 490 o∅ | |
| 491 ≡⟨ sym diso ⟩ | |
| 492 od→ord ( ord→od o∅ ) | |
| 493 ≡⟨ cong ( λ k → od→ord k ) o∅≡od∅ ⟩ | |
| 494 od→ord od∅ | |
| 495 ∎ | |
| 303 | 496 infinity : (x : HOD) → infinite ∋ x → infinite ∋ Union (x , (x , x )) |
| 497 infinity x lt = odef-subst {_} {_} {infinite} {od→ord (Union (x , (x , x )))} ( isuc {od→ord x} lt ) refl lemma where | |
| 161 | 498 lemma : od→ord (Union (ord→od (od→ord x) , (ord→od (od→ord x) , ord→od (od→ord x)))) |
| 499 ≡ od→ord (Union (x , (x , x))) | |
| 500 lemma = cong (λ k → od→ord (Union ( k , ( k , k ) ))) oiso | |
| 501 | |
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502 |
| 303 | 503 Union = ZF.Union HOD→ZF |
| 504 Power = ZF.Power HOD→ZF | |
| 505 Select = ZF.Select HOD→ZF | |
| 506 Replace = ZF.Replace HOD→ZF | |
| 507 isZF = ZF.isZF HOD→ZF |