Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate OD.agda @ 322:a9d380378efd
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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| date | Fri, 03 Jul 2020 22:54:45 +0900 |
| parents | a81824502ebd |
| children | e228e96965f0 |
| 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 |
| 322 | 56 -- |
| 57 -- OD is an equation on Ordinals, so it contains Ordinals. If these Ordinals have one-to-one | |
| 58 -- correspondence to the OD then the OD looks like a ZF Set. | |
| 59 -- | |
| 60 -- If all ZF Set have supreme upper bound, the solutions of OD have to be bounded, i.e. | |
| 61 -- bbounded ODs are ZF Set. Unbounded ODs are classes. | |
| 62 -- | |
| 290 | 63 -- In classical Set Theory, HOD is used, as a subset of OD, |
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64 -- HOD = { x | TC x ⊆ OD } |
| 290 | 65 -- where TC x is a transitive clusure of x, i.e. Union of all elemnts of all subset of x. |
| 322 | 66 -- This is not possible because we don't have V yet. So we assumes HODs are bounded OD. |
| 290 | 67 -- |
| 309 | 68 -- We also assumes HODs are isomorphic to Ordinals, which is ususally proved by Goedel number tricks. |
| 322 | 69 -- There two contraints on the HOD order, one is ∋, the other one is ⊂. |
| 70 -- ODs have an ovbious maximum, but Ordinals are not, but HOD has no maximum because there is an aribtrary | |
| 71 -- bound on each HOD. | |
| 290 | 72 -- |
| 322 | 73 -- In classical Set Theory, sup is defined by Uion, since we are working on constructive logic, |
| 290 | 74 -- we need explict assumption on sup. |
| 309 | 75 -- |
| 76 -- ==→o≡ is necessary to prove axiom of extensionality. | |
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77 |
| 303 | 78 data One : Set n where |
| 79 OneObj : One | |
| 80 | |
| 81 -- Ordinals in OD , the maximum | |
| 82 Ords : OD | |
| 83 Ords = record { def = λ x → One } | |
| 84 | |
| 85 record HOD : Set (suc n) where | |
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86 field |
| 303 | 87 od : OD |
| 304 | 88 odmax : Ordinal |
| 308 | 89 <odmax : {y : Ordinal} → def od y → y o< odmax |
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90 |
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91 open HOD |
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92 |
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93 record ODAxiom : Set (suc n) where |
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94 field |
| 304 | 95 -- HOD is isomorphic to Ordinal (by means of Goedel number) |
| 303 | 96 od→ord : HOD → Ordinal |
| 97 ord→od : Ordinal → HOD | |
| 98 c<→o< : {x y : HOD } → def (od y) ( od→ord x ) → od→ord x o< od→ord y | |
| 312 | 99 ⊆→o≤ : {y z : HOD } → ({x : Ordinal} → def (od y) x → def (od z) x ) → od→ord y o< osuc (od→ord z) |
| 303 | 100 oiso : {x : HOD } → ord→od ( od→ord x ) ≡ x |
| 322 | 101 diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x |
| 102 ==→o≡ : { x y : HOD } → (od x == od y) → x ≡ y | |
| 306 | 103 sup-o : (A : HOD) → (( x : Ordinal ) → def (od A) x → Ordinal ) → Ordinal |
| 104 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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105 |
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106 postulate odAxiom : ODAxiom |
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107 open ODAxiom odAxiom |
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108 |
| 303 | 109 -- maxod : {x : OD} → od→ord x o< od→ord Ords |
| 110 -- maxod {x} = c<→o< OneObj | |
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111 |
| 304 | 112 -- we have not this contradiction |
| 303 | 113 -- bad-bad : ⊥ |
| 304 | 114 -- bad-bad = osuc-< <-osuc (c<→o< { record { od = record { def = λ x → One }; <odmax = {!!} } } OneObj) |
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115 |
| 179 | 116 -- Ordinal in OD ( and ZFSet ) Transitive Set |
| 303 | 117 Ord : ( a : Ordinal ) → HOD |
| 304 | 118 Ord a = record { od = record { def = λ y → y o< a } ; odmax = a ; <odmax = lemma } where |
| 119 lemma : {x : Ordinal} → x o< a → x o< a | |
| 120 lemma {x} lt = lt | |
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121 |
| 303 | 122 od∅ : HOD |
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123 od∅ = Ord o∅ |
| 40 | 124 |
| 303 | 125 odef : HOD → Ordinal → Set n |
| 126 odef A x = def ( od A ) x | |
| 123 | 127 |
| 303 | 128 o<→c<→HOD=Ord : ( {x y : Ordinal } → x o< y → odef (ord→od y) x ) → {x : HOD } → x ≡ Ord (od→ord x) |
| 129 o<→c<→HOD=Ord o<→c< {x} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where | |
| 130 lemma1 : {y : Ordinal} → odef x y → odef (Ord (od→ord x)) y | |
| 131 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)) | |
| 132 lemma2 : {y : Ordinal} → odef (Ord (od→ord x)) y → odef x y | |
| 133 lemma2 {y} lt = subst (λ k → odef k y ) oiso (o<→c< {y} {od→ord x} lt ) | |
| 95 | 134 |
| 303 | 135 _∋_ : ( a x : HOD ) → Set n |
| 136 _∋_ a x = odef a ( od→ord x ) | |
| 137 | |
| 138 _c<_ : ( x a : HOD ) → Set n | |
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139 x c< a = a ∋ x |
| 103 | 140 |
| 303 | 141 cseq : {n : Level} → HOD → HOD |
| 308 | 142 cseq x = record { od = record { def = λ y → odef x (osuc y) } ; odmax = osuc (odmax x) ; <odmax = lemma } where |
| 143 lemma : {y : Ordinal} → def (od x) (osuc y) → y o< osuc (odmax x) | |
| 144 lemma {y} lt = ordtrans <-osuc (ordtrans (<odmax x lt) <-osuc ) | |
| 95 | 145 |
| 303 | 146 odef-subst : {Z : HOD } {X : Ordinal }{z : HOD } {x : Ordinal }→ odef Z X → Z ≡ z → X ≡ x → odef z x |
| 147 odef-subst df refl refl = df | |
| 95 | 148 |
| 303 | 149 otrans : {n : Level} {a x y : Ordinal } → odef (Ord a) x → odef (Ord x) y → odef (Ord a) y |
| 187 | 150 otrans x<a y<x = ordtrans y<x x<a |
| 123 | 151 |
| 303 | 152 odef→o< : {X : HOD } → {x : Ordinal } → odef X x → x o< od→ord X |
| 153 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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154 |
| 51 | 155 -- avoiding lv != Zero error |
| 303 | 156 orefl : { x : HOD } → { y : Ordinal } → od→ord x ≡ y → od→ord x ≡ y |
| 51 | 157 orefl refl = refl |
| 158 | |
| 303 | 159 ==-iso : { x y : HOD } → od (ord→od (od→ord x)) == od (ord→od (od→ord y)) → od x == od y |
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160 ==-iso {x} {y} eq = record { |
| 303 | 161 eq→ = λ d → lemma ( eq→ eq (odef-subst d (sym oiso) refl )) ; |
| 162 eq← = λ d → lemma ( eq← eq (odef-subst d (sym oiso) refl )) } | |
| 51 | 163 where |
| 303 | 164 lemma : {x : HOD } {z : Ordinal } → odef (ord→od (od→ord x)) z → odef x z |
| 165 lemma {x} {z} d = odef-subst d oiso refl | |
| 51 | 166 |
| 303 | 167 =-iso : {x y : HOD } → (od x == od y) ≡ (od (ord→od (od→ord x)) == od y) |
| 168 =-iso {_} {y} = cong ( λ k → od k == od y ) (sym oiso) | |
| 57 | 169 |
| 303 | 170 ord→== : { x y : HOD } → od→ord x ≡ od→ord y → od x == od y |
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171 ord→== {x} {y} eq = ==-iso (lemma (od→ord x) (od→ord y) (orefl eq)) where |
| 303 | 172 lemma : ( ox oy : Ordinal ) → ox ≡ oy → od (ord→od ox) == od (ord→od oy) |
| 271 | 173 lemma ox ox refl = ==-refl |
| 51 | 174 |
| 303 | 175 o≡→== : { x y : Ordinal } → x ≡ y → od (ord→od x) == od (ord→od y) |
| 271 | 176 o≡→== {x} {.x} refl = ==-refl |
| 51 | 177 |
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178 o∅≡od∅ : ord→od (o∅ ) ≡ od∅ |
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179 o∅≡od∅ = ==→o≡ lemma where |
| 303 | 180 lemma0 : {x : Ordinal} → odef (ord→od o∅) x → odef od∅ x |
| 181 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 | |
| 182 lemma1 : {x : Ordinal} → odef od∅ x → odef (ord→od o∅) x | |
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183 lemma1 {x} lt = ⊥-elim (¬x<0 lt) |
| 303 | 184 lemma : od (ord→od o∅) == od od∅ |
| 150 | 185 lemma = record { eq→ = lemma0 ; eq← = lemma1 } |
| 186 | |
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187 ord-od∅ : od→ord (od∅ ) ≡ o∅ |
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188 ord-od∅ = sym ( subst (λ k → k ≡ od→ord (od∅ ) ) diso (cong ( λ k → od→ord k ) o∅≡od∅ ) ) |
| 80 | 189 |
| 303 | 190 ∅0 : record { def = λ x → Lift n ⊥ } == od od∅ |
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191 eq→ ∅0 {w} (lift ()) |
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192 eq← ∅0 {w} lt = lift (¬x<0 lt) |
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193 |
| 303 | 194 ∅< : { x y : HOD } → odef x (od→ord y ) → ¬ ( od x == od od∅ ) |
| 271 | 195 ∅< {x} {y} d eq with eq→ (==-trans eq (==-sym ∅0) ) d |
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196 ∅< {x} {y} d eq | lift () |
| 57 | 197 |
| 303 | 198 ∅6 : { x : HOD } → ¬ ( x ∋ x ) -- no Russel paradox |
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199 ∅6 {x} x∋x = o<¬≡ refl ( c<→o< {x} {x} x∋x ) |
| 51 | 200 |
| 303 | 201 odef-iso : {A B : HOD } {x y : Ordinal } → x ≡ y → (odef A y → odef B y) → odef A x → odef B x |
| 202 odef-iso refl t = t | |
| 76 | 203 |
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204 is-o∅ : ( x : Ordinal ) → Dec ( x ≡ o∅ ) |
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205 is-o∅ x with trio< x o∅ |
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206 is-o∅ x | tri< a ¬b ¬c = no ¬b |
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207 is-o∅ x | tri≈ ¬a b ¬c = yes b |
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208 is-o∅ x | tri> ¬a ¬b c = no ¬b |
| 57 | 209 |
| 303 | 210 _,_ : HOD → HOD → HOD |
| 308 | 211 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 |
| 212 lemma : {t : Ordinal} → (t ≡ od→ord x) ∨ (t ≡ od→ord y) → t o< omax (od→ord x) (od→ord y) | |
| 213 lemma {t} (case1 refl) = omax-x _ _ | |
| 214 lemma {t} (case2 refl) = omax-y _ _ | |
| 215 | |
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216 |
| 79 | 217 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) |
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218 -- postulate f-extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality n (suc n) |
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219 |
| 318 | 220 in-codomain : (X : HOD ) → ( ψ : HOD → HOD ) → OD |
| 221 in-codomain X ψ = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( odef X y ∧ ( x ≡ od→ord (ψ (ord→od y ))))) } | |
| 141 | 222 |
| 303 | 223 -- Power Set of X ( or constructible by λ y → odef X (od→ord y ) |
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224 |
| 303 | 225 ZFSubset : (A x : HOD ) → HOD |
| 308 | 226 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 |
| 227 lemma : {y : Ordinal} → def (od A) y ∧ def (od x) y → y o< omin (odmax A) (odmax x) | |
| 228 lemma {y} and = min1 (<odmax A (proj1 and)) (<odmax x (proj2 and)) | |
| 229 | |
| 303 | 230 record _⊆_ ( A B : HOD ) : Set (suc n) where |
| 271 | 231 field |
| 303 | 232 incl : { x : HOD } → A ∋ x → B ∋ x |
| 271 | 233 |
| 234 open _⊆_ | |
| 190 | 235 infixr 220 _⊆_ |
| 236 | |
| 303 | 237 subset-lemma : {A x : HOD } → ( {y : HOD } → x ∋ y → ZFSubset A x ∋ y ) ⇔ ( x ⊆ A ) |
| 271 | 238 subset-lemma {A} {x} = record { |
| 239 proj1 = λ lt → record { incl = λ x∋z → proj1 (lt x∋z) } | |
| 240 ; proj2 = λ x⊆A lt → record { proj1 = incl x⊆A lt ; proj2 = lt } | |
| 190 | 241 } |
| 242 | |
| 312 | 243 power< : {A x : HOD } → x ⊆ A → Ord (osuc (od→ord A)) ∋ x |
| 244 power< {A} {x} x⊆A = ⊆→o≤ (λ {y} x∋y → subst (λ k → def (od A) k) diso (lemma y x∋y ) ) where | |
| 245 lemma : (y : Ordinal) → def (od x) y → def (od A) (od→ord (ord→od y)) | |
| 246 lemma y x∋y = incl x⊆A (subst (λ k → def (od x) k ) (sym diso) x∋y ) | |
| 247 | |
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248 open import Data.Unit |
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249 |
| 303 | 250 ε-induction : { ψ : HOD → Set (suc n)} |
| 251 → ( {x : HOD } → ({ y : HOD } → x ∋ y → ψ y ) → ψ x ) | |
| 252 → (x : HOD ) → ψ x | |
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253 ε-induction {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (osuc (od→ord x)) <-osuc ) where |
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254 induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (ord→od oy)) → ψ (ord→od ox) |
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255 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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256 ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (ord→od oy) |
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257 ε-induction-ord ox {oy} lt = TransFinite {λ oy → ψ (ord→od oy)} induction oy |
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258 |
| 303 | 259 HOD→ZF : ZF |
| 260 HOD→ZF = record { | |
| 261 ZFSet = HOD | |
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262 ; _∋_ = _∋_ |
| 303 | 263 ; _≈_ = _=h=_ |
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264 ; ∅ = od∅ |
| 28 | 265 ; _,_ = _,_ |
| 266 ; Union = Union | |
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267 ; Power = Power |
| 28 | 268 ; Select = Select |
| 269 ; Replace = Replace | |
| 161 | 270 ; infinite = infinite |
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271 ; isZF = isZF |
| 28 | 272 } where |
| 303 | 273 ZFSet = HOD -- is less than Ords because of maxod |
| 274 Select : (X : HOD ) → ((x : HOD ) → Set n ) → HOD | |
| 308 | 275 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 | 276 Replace : HOD → (HOD → HOD) → HOD |
| 318 | 277 Replace X ψ = record { od = record { def = λ x → (x o< sup-o X (λ y X∋y → od→ord (ψ (ord→od y)))) ∧ def (in-codomain X ψ) x } |
| 278 ; odmax = rmax ; <odmax = rmax<} where | |
| 279 rmax : Ordinal | |
| 280 rmax = sup-o X (λ y X∋y → od→ord (ψ (ord→od y))) | |
| 281 rmax< : {y : Ordinal} → (y o< rmax) ∧ def (in-codomain X ψ) y → y o< rmax | |
| 282 rmax< lt = proj1 lt | |
| 144 | 283 _∩_ : ( A B : ZFSet ) → ZFSet |
| 318 | 284 A ∩ B = record { od = record { def = λ x → odef A x ∧ odef B x } |
| 285 ; odmax = omin (odmax A) (odmax B) ; <odmax = λ y → min1 (<odmax A (proj1 y)) (<odmax B (proj2 y))} | |
| 303 | 286 Union : HOD → HOD |
| 318 | 287 Union U = record { od = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((odef U u) ∧ (odef (ord→od u) x))) } |
| 288 ; odmax = osuc (od→ord U) ; <odmax = umax< } where | |
| 289 umax< : {y : Ordinal} → ¬ ((u : Ordinal) → ¬ def (od U) u ∧ def (od (ord→od u)) y) → y o< osuc (od→ord U) | |
| 319 | 290 umax< {y} not = lemma (FExists _ lemma1 not ) where |
| 291 lemma0 : {x : Ordinal} → def (od (ord→od x)) y → y o< x | |
| 292 lemma0 {x} x<y = subst₂ (λ j k → j o< k ) diso diso (c<→o< (subst (λ k → def (od (ord→od x)) k) (sym diso) x<y)) | |
| 293 lemma2 : {x : Ordinal} → def (od U) x → x o< od→ord U | |
| 294 lemma2 {x} x<U = subst (λ k → k o< od→ord U ) diso (c<→o< (subst (λ k → def (od U) k) (sym diso) x<U)) | |
| 295 lemma1 : {x : Ordinal} → def (od U) x ∧ def (od (ord→od x)) y → ¬ (od→ord U o< y) | |
| 296 lemma1 {x} lt u<y = o<> u<y (ordtrans (lemma0 (proj2 lt)) (lemma2 (proj1 lt)) ) | |
| 297 lemma : ¬ ((od→ord U) o< y ) → y o< osuc (od→ord U) | |
| 298 lemma not with trio< y (od→ord U) | |
| 299 lemma not | tri< a ¬b ¬c = ordtrans a <-osuc | |
| 300 lemma not | tri≈ ¬a refl ¬c = <-osuc | |
| 301 lemma not | tri> ¬a ¬b c = ⊥-elim (not c) | |
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302 _∈_ : ( A B : ZFSet ) → Set n |
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303 A ∈ B = B ∋ A |
| 312 | 304 |
| 305 OPwr : (A : HOD ) → HOD | |
| 306 OPwr A = Ord ( sup-o (Ord (osuc (od→ord A))) ( λ x A∋x → od→ord ( ZFSubset A (ord→od x)) ) ) | |
| 307 | |
| 303 | 308 Power : HOD → HOD |
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309 Power A = Replace (OPwr (Ord (od→ord A))) ( λ x → A ∩ x ) |
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310 -- ｛_｝ : ZFSet → ZFSet |
| 287 | 311 -- ｛ x ｝ = ( x , x ) -- it works but we don't use |
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312 |
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313 data infinite-d : ( x : Ordinal ) → Set n where |
| 161 | 314 iφ : infinite-d o∅ |
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315 isuc : {x : Ordinal } → infinite-d x → |
| 161 | 316 infinite-d (od→ord ( Union (ord→od x , (ord→od x , ord→od x ) ) )) |
| 317 | |
| 303 | 318 infinite : HOD |
| 320 | 319 infinite = record { od = record { def = λ x → infinite-d x } ; odmax = next o∅ ; <odmax = lemma } where |
| 320 lemma : {y : Ordinal} → infinite-d y → y o< next o∅ | |
| 321 lemma {o∅} iφ = proj1 next-limit | |
| 321 | 322 lemma (isuc {x} i) = lemma1 where -- proj2 next-limit ? ( lemma i ) |
| 323 lemma1 : od→ord (Union (ord→od x , (ord→od x , ord→od x))) o< next o∅ | |
| 322 | 324 lemma1 = {!!} |
| 321 | 325 |
| 320 | 326 |
| 303 | 327 |
| 328 _=h=_ : (x y : HOD) → Set n | |
| 329 x =h= y = od x == od y | |
| 161 | 330 |
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331 infixr 200 _∈_ |
| 96 | 332 -- infixr 230 _∩_ _∪_ |
| 303 | 333 isZF : IsZF (HOD ) _∋_ _=h=_ od∅ _,_ Union Power Select Replace infinite |
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334 isZF = record { |
| 271 | 335 isEquivalence = record { refl = ==-refl ; sym = ==-sym; trans = ==-trans } |
| 247 | 336 ; pair→ = pair→ |
| 337 ; pair← = pair← | |
| 72 | 338 ; union→ = union→ |
| 339 ; union← = union← | |
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340 ; empty = empty |
| 129 | 341 ; power→ = power→ |
| 76 | 342 ; power← = power← |
| 186 | 343 ; extensionality = λ {A} {B} {w} → extensionality {A} {B} {w} |
| 274 | 344 ; ε-induction = ε-induction |
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345 ; infinity∅ = infinity∅ |
| 160 | 346 ; infinity = infinity |
| 116 | 347 ; selection = λ {X} {ψ} {y} → selection {X} {ψ} {y} |
| 135 | 348 ; replacement← = replacement← |
| 317 | 349 ; replacement→ = λ {ψ} → replacement→ {ψ} |
| 274 | 350 -- ; choice-func = choice-func |
| 351 -- ; choice = choice | |
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352 } where |
| 129 | 353 |
| 303 | 354 pair→ : ( x y t : ZFSet ) → (x , y) ∋ t → ( t =h= x ) ∨ ( t =h= y ) |
| 355 pair→ x y t (case1 t≡x ) = case1 (subst₂ (λ j k → j =h= k ) oiso oiso (o≡→== t≡x )) | |
| 356 pair→ x y t (case2 t≡y ) = case2 (subst₂ (λ j k → j =h= k ) oiso oiso (o≡→== t≡y )) | |
| 247 | 357 |
| 303 | 358 pair← : ( x y t : ZFSet ) → ( t =h= x ) ∨ ( t =h= y ) → (x , y) ∋ t |
| 359 pair← x y t (case1 t=h=x) = case1 (cong (λ k → od→ord k ) (==→o≡ t=h=x)) | |
| 360 pair← x y t (case2 t=h=y) = case2 (cong (λ k → od→ord k ) (==→o≡ t=h=y)) | |
| 247 | 361 |
| 303 | 362 empty : (x : HOD ) → ¬ (od∅ ∋ x) |
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363 empty x = ¬x<0 |
| 129 | 364 |
| 271 | 365 o<→c< : {x y : Ordinal } → x o< y → (Ord x) ⊆ (Ord y) |
| 366 o<→c< lt = record { incl = λ z → ordtrans z lt } | |
| 155 | 367 |
| 271 | 368 ⊆→o< : {x y : Ordinal } → (Ord x) ⊆ (Ord y) → x o< osuc y |
| 155 | 369 ⊆→o< {x} {y} lt with trio< x y |
| 370 ⊆→o< {x} {y} lt | tri< a ¬b ¬c = ordtrans a <-osuc | |
| 371 ⊆→o< {x} {y} lt | tri≈ ¬a b ¬c = subst ( λ k → k o< osuc y) (sym b) <-osuc | |
| 271 | 372 ⊆→o< {x} {y} lt | tri> ¬a ¬b c with (incl lt) (o<-subst c (sym diso) refl ) |
| 155 | 373 ... | ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt diso refl )) |
| 151 | 374 |
| 303 | 375 union→ : (X z u : HOD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z |
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376 union→ X z u xx not = ⊥-elim ( not (od→ord u) ( record { proj1 = proj1 xx |
| 303 | 377 ; proj2 = subst ( λ k → odef k (od→ord z)) (sym oiso) (proj2 xx) } )) |
| 378 union← : (X z : HOD) (X∋z : Union X ∋ z) → ¬ ( (u : HOD ) → ¬ ((X ∋ u) ∧ (u ∋ z ))) | |
| 258 | 379 union← X z UX∋z = FExists _ lemma UX∋z where |
| 303 | 380 lemma : {y : Ordinal} → odef X y ∧ odef (ord→od y) (od→ord z) → ¬ ((u : HOD) → ¬ (X ∋ u) ∧ (u ∋ z)) |
| 381 lemma {y} xx not = not (ord→od y) record { proj1 = subst ( λ k → odef X k ) (sym diso) (proj1 xx ) ; proj2 = proj2 xx } | |
| 144 | 382 |
| 303 | 383 ψiso : {ψ : HOD → Set n} {x y : HOD } → ψ x → x ≡ y → ψ y |
| 144 | 384 ψiso {ψ} t refl = t |
| 303 | 385 selection : {ψ : HOD → Set n} {X y : HOD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y) |
| 144 | 386 selection {ψ} {X} {y} = record { |
| 387 proj1 = λ cond → record { proj1 = proj1 cond ; proj2 = ψiso {ψ} (proj2 cond) (sym oiso) } | |
| 388 ; proj2 = λ select → record { proj1 = proj1 select ; proj2 = ψiso {ψ} (proj2 select) oiso } | |
| 389 } | |
| 311 | 390 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)))) |
| 391 sup-c< ψ {X} {x} lt = subst (λ k → od→ord (ψ k) o< _ ) oiso (sup-o< X lt ) | |
| 303 | 392 replacement← : {ψ : HOD → HOD} (X x : HOD) → X ∋ x → Replace X ψ ∋ ψ x |
| 311 | 393 replacement← {ψ} X x lt = record { proj1 = sup-c< ψ {X} {x} lt ; proj2 = lemma } where |
| 318 | 394 lemma : def (in-codomain X ψ) (od→ord (ψ x)) |
| 150 | 395 lemma not = ⊥-elim ( not ( od→ord x ) (record { proj1 = lt ; proj2 = cong (λ k → od→ord (ψ k)) (sym oiso)} )) |
| 303 | 396 replacement→ : {ψ : HOD → HOD} (X x : HOD) → (lt : Replace X ψ ∋ x) → ¬ ( (y : HOD) → ¬ (x =h= ψ y)) |
| 150 | 397 replacement→ {ψ} X x lt = contra-position lemma (lemma2 (proj2 lt)) where |
| 303 | 398 lemma2 : ¬ ((y : Ordinal) → ¬ odef X y ∧ ((od→ord x) ≡ od→ord (ψ (ord→od y)))) |
| 399 → ¬ ((y : Ordinal) → ¬ odef X y ∧ (ord→od (od→ord x) =h= ψ (ord→od y))) | |
| 144 | 400 lemma2 not not2 = not ( λ y d → not2 y (record { proj1 = proj1 d ; proj2 = lemma3 (proj2 d)})) where |
| 303 | 401 lemma3 : {y : Ordinal } → (od→ord x ≡ od→ord (ψ (ord→od y))) → (ord→od (od→ord x) =h= ψ (ord→od y)) |
| 402 lemma3 {y} eq = subst (λ k → ord→od (od→ord x) =h= k ) oiso (o≡→== eq ) | |
| 403 lemma : ( (y : HOD) → ¬ (x =h= ψ y)) → ( (y : Ordinal) → ¬ odef X y ∧ (ord→od (od→ord x) =h= ψ (ord→od y)) ) | |
| 404 lemma not y not2 = not (ord→od y) (subst (λ k → k =h= ψ (ord→od y)) oiso ( proj2 not2 )) | |
| 144 | 405 |
| 406 --- | |
| 407 --- Power Set | |
| 408 --- | |
| 303 | 409 --- First consider ordinals in HOD |
| 100 | 410 --- |
| 303 | 411 --- ZFSubset A x = record { def = λ y → odef A y ∧ odef x y } subset of A |
| 100 | 412 -- |
| 413 -- | |
| 303 | 414 ∩-≡ : { a b : HOD } → ({x : HOD } → (a ∋ x → b ∋ x)) → a =h= ( b ∩ a ) |
| 142 | 415 ∩-≡ {a} {b} inc = record { |
| 416 eq→ = λ {x} x<a → record { proj2 = x<a ; | |
| 303 | 417 proj1 = odef-subst {_} {_} {b} {x} (inc (odef-subst {_} {_} {a} {_} x<a refl (sym diso) )) refl diso } ; |
| 142 | 418 eq← = λ {x} x<a∩b → proj2 x<a∩b } |
| 100 | 419 -- |
| 258 | 420 -- Transitive Set case |
| 303 | 421 -- 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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422 -- OPwr (Ord a) is a sup of ZFSubset (Ord a) t, so OPwr (Ord a) ∋ t |
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423 -- OPwr A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) ) |
| 100 | 424 -- |
| 303 | 425 ord-power← : (a : Ordinal ) (t : HOD) → ({x : HOD} → (t ∋ x → (Ord a) ∋ x)) → OPwr (Ord a) ∋ t |
| 426 ord-power← a t t→A = odef-subst {_} {_} {OPwr (Ord a)} {od→ord t} | |
| 127 | 427 lemma refl (lemma1 lemma-eq )where |
| 303 | 428 lemma-eq : ZFSubset (Ord a) t =h= t |
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429 eq→ lemma-eq {z} w = proj2 w |
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430 eq← lemma-eq {z} w = record { proj2 = w ; |
| 303 | 431 proj1 = odef-subst {_} {_} {(Ord a)} {z} |
| 432 ( t→A (odef-subst {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso } | |
| 433 lemma1 : {a : Ordinal } { t : HOD } | |
| 434 → (eq : ZFSubset (Ord a) t =h= t) → od→ord (ZFSubset (Ord a) (ord→od (od→ord t))) ≡ od→ord t | |
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435 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 | 436 lemma2 : (od→ord t) o< (osuc (od→ord (Ord a))) |
| 437 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))) | |
| 438 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 | 439 lemma = sup-o< _ lemma2 |
| 129 | 440 |
| 144 | 441 -- |
| 303 | 442 -- Every set in HOD is a subset of Ordinals, so make OPwr (Ord (od→ord A)) first |
| 258 | 443 -- then replace of all elements of the Power set by A ∩ y |
| 144 | 444 -- |
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445 -- Power A = Replace (OPwr (Ord (od→ord A))) ( λ y → A ∩ y ) |
| 166 | 446 |
| 447 -- we have oly double negation form because of the replacement axiom | |
| 448 -- | |
| 303 | 449 power→ : ( A t : HOD) → Power A ∋ t → {x : HOD} → t ∋ x → ¬ ¬ (A ∋ x) |
| 258 | 450 power→ A t P∋t {x} t∋x = FExists _ lemma5 lemma4 where |
| 142 | 451 a = od→ord A |
| 303 | 452 lemma2 : ¬ ( (y : HOD) → ¬ (t =h= (A ∩ y))) |
| 317 | 453 lemma2 = replacement→ {λ x → A ∩ x} (OPwr (Ord (od→ord A))) t P∋t |
| 303 | 454 lemma3 : (y : HOD) → t =h= ( A ∩ y ) → ¬ ¬ (A ∋ x) |
| 166 | 455 lemma3 y eq not = not (proj1 (eq→ eq t∋x)) |
| 303 | 456 lemma4 : ¬ ((y : Ordinal) → ¬ (t =h= (A ∩ ord→od y))) |
| 457 lemma4 not = lemma2 ( λ y not1 → not (od→ord y) (subst (λ k → t =h= ( A ∩ k )) (sym oiso) not1 )) | |
| 458 lemma5 : {y : Ordinal} → t =h= (A ∩ ord→od y) → ¬ ¬ (odef A (od→ord x)) | |
| 166 | 459 lemma5 {y} eq not = (lemma3 (ord→od y) eq) not |
| 460 | |
| 303 | 461 power← : (A t : HOD) → ({x : HOD} → (t ∋ x → A ∋ x)) → Power A ∋ t |
| 311 | 462 power← A t t→A = record { proj1 = lemma1 ; proj2 = lemma2 } where |
| 142 | 463 a = od→ord A |
| 303 | 464 lemma0 : {x : HOD} → t ∋ x → Ord a ∋ x |
| 142 | 465 lemma0 {x} t∋x = c<→o< (t→A t∋x) |
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466 lemma3 : OPwr (Ord a) ∋ t |
| 142 | 467 lemma3 = ord-power← a t lemma0 |
| 152 | 468 lemma4 : (A ∩ ord→od (od→ord t)) ≡ t |
| 469 lemma4 = let open ≡-Reasoning in begin | |
| 470 A ∩ ord→od (od→ord t) | |
| 471 ≡⟨ cong (λ k → A ∩ k) oiso ⟩ | |
| 472 A ∩ t | |
| 317 | 473 ≡⟨ sym (==→o≡ ( ∩-≡ {t} {A} t→A )) ⟩ |
| 152 | 474 t |
| 475 ∎ | |
| 317 | 476 sup1 : Ordinal |
| 314 | 477 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 | 478 lemma9 : def (od (Ord (Ordinals.osuc O (od→ord (Ord (od→ord A)))))) (od→ord (Ord (od→ord A))) |
| 479 lemma9 = <-osuc | |
| 315 | 480 lemmab : od→ord (ZFSubset (Ord (od→ord A)) (ord→od (od→ord (Ord (od→ord A) )))) o< sup1 |
| 481 lemmab = sup-o< (Ord (osuc (od→ord (Ord (od→ord A))))) lemma9 | |
| 482 lemmad : Ord (osuc (od→ord A)) ∋ t | |
| 317 | 483 lemmad = ⊆→o≤ (λ {x} lt → subst (λ k → def (od A) k ) diso (t→A (subst (λ k → def (od t) k ) (sym diso) lt))) |
| 315 | 484 lemmac : ZFSubset (Ord (od→ord A)) (ord→od (od→ord (Ord (od→ord A) ))) =h= Ord (od→ord A) |
| 317 | 485 lemmac = record { eq→ = lemmaf ; eq← = lemmag } where |
| 486 lemmaf : {x : Ordinal} → def (od (ZFSubset (Ord (od→ord A)) (ord→od (od→ord (Ord (od→ord A)))))) x → def (od (Ord (od→ord A))) x | |
| 487 lemmaf {x} lt = proj1 lt | |
| 488 lemmag : {x : Ordinal} → def (od (Ord (od→ord A))) x → def (od (ZFSubset (Ord (od→ord A)) (ord→od (od→ord (Ord (od→ord A)))))) x | |
| 489 lemmag {x} lt = record { proj1 = lt ; proj2 = subst (λ k → def (od k) x) (sym oiso) lt } | |
| 315 | 490 lemmae : od→ord (ZFSubset (Ord (od→ord A)) (ord→od (od→ord (Ord (od→ord A))))) ≡ od→ord (Ord (od→ord A)) |
| 491 lemmae = cong (λ k → od→ord k ) ( ==→o≡ lemmac) | |
| 311 | 492 lemma7 : def (od (OPwr (Ord (od→ord A)))) (od→ord t) |
| 315 | 493 lemma7 with osuc-≡< lemmad |
| 494 lemma7 | case2 lt = ordtrans (c<→o< lt) (subst (λ k → k o< sup1) lemmae lemmab ) | |
| 317 | 495 lemma7 | case1 eq with osuc-≡< (⊆→o≤ {ord→od (od→ord t)} {ord→od (od→ord (Ord (od→ord t)))} (λ {x} lt → lemmah lt )) where |
| 496 lemmah : {x : Ordinal } → def (od (ord→od (od→ord t))) x → def (od (ord→od (od→ord (Ord (od→ord t))))) x | |
| 497 lemmah {x} lt = subst (λ k → def (od k) x ) (sym oiso) (subst (λ k → k o< (od→ord t)) | |
| 498 diso | |
| 499 (c<→o< (subst₂ (λ j k → def (od j) k) oiso (sym diso) lt ))) | |
| 500 lemma7 | case1 eq | case1 eq1 = subst (λ k → k o< sup1) (trans lemmae lemmai) lemmab where | |
| 501 lemmai : od→ord (Ord (od→ord A)) ≡ od→ord t | |
| 502 lemmai = let open ≡-Reasoning in begin | |
| 503 od→ord (Ord (od→ord A)) | |
| 504 ≡⟨ sym (cong (λ k → od→ord (Ord k)) eq) ⟩ | |
| 505 od→ord (Ord (od→ord t)) | |
| 506 ≡⟨ sym diso ⟩ | |
| 507 od→ord (ord→od (od→ord (Ord (od→ord t)))) | |
| 508 ≡⟨ sym eq1 ⟩ | |
| 509 od→ord (ord→od (od→ord t)) | |
| 510 ≡⟨ diso ⟩ | |
| 511 od→ord t | |
| 512 ∎ | |
| 513 lemma7 | case1 eq | case2 lt = ordtrans lemmaj (subst (λ k → k o< sup1) lemmae lemmab ) where | |
| 514 lemmak : od→ord (ord→od (od→ord (Ord (od→ord t)))) ≡ od→ord (Ord (od→ord A)) | |
| 515 lemmak = let open ≡-Reasoning in begin | |
| 516 od→ord (ord→od (od→ord (Ord (od→ord t)))) | |
| 517 ≡⟨ diso ⟩ | |
| 518 od→ord (Ord (od→ord t)) | |
| 519 ≡⟨ cong (λ k → od→ord (Ord k)) eq ⟩ | |
| 520 od→ord (Ord (od→ord A)) | |
| 521 ∎ | |
| 522 lemmaj : od→ord t o< od→ord (Ord (od→ord A)) | |
| 523 lemmaj = subst₂ (λ j k → j o< k ) diso lemmak lt | |
| 310 | 524 lemma1 : od→ord t o< sup-o (OPwr (Ord (od→ord A))) (λ x lt → od→ord (A ∩ (ord→od x))) |
| 525 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 | 526 lemma4 (sup-o< (OPwr (Ord (od→ord A))) lemma7 ) |
| 318 | 527 lemma2 : def (in-codomain (OPwr (Ord (od→ord A))) (_∩_ A)) (od→ord t) |
| 151 | 528 lemma2 not = ⊥-elim ( not (od→ord t) (record { proj1 = lemma3 ; proj2 = lemma6 }) ) where |
| 529 lemma6 : od→ord t ≡ od→ord (A ∩ ord→od (od→ord t)) | |
| 317 | 530 lemma6 = cong ( λ k → od→ord k ) (==→o≡ (subst (λ k → t =h= (A ∩ k)) (sym oiso) ( ∩-≡ {t} {A} t→A ))) |
| 142 | 531 |
| 311 | 532 |
| 271 | 533 ord⊆power : (a : Ordinal) → (Ord (osuc a)) ⊆ (Power (Ord a)) |
| 534 ord⊆power a = record { incl = λ {x} lt → power← (Ord a) x (lemma lt) } where | |
| 303 | 535 lemma : {x y : HOD} → od→ord x o< osuc a → x ∋ y → Ord a ∋ y |
| 271 | 536 lemma lt y<x with osuc-≡< lt |
| 537 lemma lt y<x | case1 refl = c<→o< y<x | |
| 538 lemma lt y<x | case2 x<a = ordtrans (c<→o< y<x) x<a | |
| 262 | 539 |
| 276 | 540 continuum-hyphotheis : (a : Ordinal) → Set (suc n) |
| 541 continuum-hyphotheis a = Power (Ord a) ⊆ Ord (osuc a) | |
| 129 | 542 |
| 303 | 543 extensionality0 : {A B : HOD } → ((z : HOD) → (A ∋ z) ⇔ (B ∋ z)) → A =h= B |
| 544 eq→ (extensionality0 {A} {B} eq ) {x} d = odef-iso {A} {B} (sym diso) (proj1 (eq (ord→od x))) d | |
| 545 eq← (extensionality0 {A} {B} eq ) {x} d = odef-iso {B} {A} (sym diso) (proj2 (eq (ord→od x))) d | |
| 186 | 546 |
| 303 | 547 extensionality : {A B w : HOD } → ((z : HOD ) → (A ∋ z) ⇔ (B ∋ z)) → (w ∋ A) ⇔ (w ∋ B) |
| 186 | 548 proj1 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) ( ==→o≡ (extensionality0 {A} {B} eq) ) d |
| 549 proj2 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) (sym ( ==→o≡ (extensionality0 {A} {B} eq) )) d | |
| 129 | 550 |
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551 infinity∅ : infinite ∋ od∅ |
| 303 | 552 infinity∅ = odef-subst {_} {_} {infinite} {od→ord (od∅ )} iφ refl lemma where |
| 161 | 553 lemma : o∅ ≡ od→ord od∅ |
| 554 lemma = let open ≡-Reasoning in begin | |
| 555 o∅ | |
| 556 ≡⟨ sym diso ⟩ | |
| 557 od→ord ( ord→od o∅ ) | |
| 558 ≡⟨ cong ( λ k → od→ord k ) o∅≡od∅ ⟩ | |
| 559 od→ord od∅ | |
| 560 ∎ | |
| 303 | 561 infinity : (x : HOD) → infinite ∋ x → infinite ∋ Union (x , (x , x )) |
| 562 infinity x lt = odef-subst {_} {_} {infinite} {od→ord (Union (x , (x , x )))} ( isuc {od→ord x} lt ) refl lemma where | |
| 161 | 563 lemma : od→ord (Union (ord→od (od→ord x) , (ord→od (od→ord x) , ord→od (od→ord x)))) |
| 564 ≡ od→ord (Union (x , (x , x))) | |
| 565 lemma = cong (λ k → od→ord (Union ( k , ( k , k ) ))) oiso | |
| 566 | |
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| 303 | 568 Union = ZF.Union HOD→ZF |
| 569 Power = ZF.Power HOD→ZF | |
| 570 Select = ZF.Select HOD→ZF | |
| 571 Replace = ZF.Replace HOD→ZF | |
| 572 isZF = ZF.isZF HOD→ZF |