Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate OD.agda @ 399:03a4e1b8f3fb
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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| date | Mon, 27 Jul 2020 19:58:46 +0900 |
| parents | fbe1a49876ad |
| children | 48ea49494fd1 |
| rev | line source |
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| 364 | 1 {-# OPTIONS --allow-unsolved-metas #-} |
| 16 | 2 open import Level |
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3 open import Ordinals |
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4 module OD {n : Level } (O : Ordinals {n} ) where |
| 3 | 5 |
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6 open import zf |
| 23 | 7 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) |
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8 open import Relation.Binary.PropositionalEquality |
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9 open import Data.Nat.Properties |
| 6 | 10 open import Data.Empty |
| 11 open import Relation.Nullary | |
| 12 open import Relation.Binary | |
| 13 open import Relation.Binary.Core | |
| 14 | |
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15 open import logic |
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16 open import nat |
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17 |
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18 open inOrdinal O |
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19 |
| 27 | 20 -- Ordinal Definable Set |
| 11 | 21 |
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22 record OD : Set (suc n ) where |
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23 field |
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24 def : (x : Ordinal ) → Set n |
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25 |
| 141 | 26 open OD |
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27 |
| 120 | 28 open _∧_ |
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29 open _∨_ |
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30 open Bool |
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31 |
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32 record _==_ ( a b : OD ) : Set n where |
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33 field |
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34 eq→ : ∀ { x : Ordinal } → def a x → def b x |
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35 eq← : ∀ { x : Ordinal } → def b x → def a x |
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36 |
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37 id : {A : Set n} → A → A |
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38 id x = x |
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39 |
| 271 | 40 ==-refl : { x : OD } → x == x |
| 41 ==-refl {x} = record { eq→ = id ; eq← = id } | |
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42 |
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43 open _==_ |
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44 |
| 271 | 45 ==-trans : { x y z : OD } → x == y → y == z → x == z |
| 46 ==-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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47 |
| 271 | 48 ==-sym : { x y : OD } → x == y → y == x |
| 49 ==-sym x=y = record { eq→ = λ {m} t → eq← x=y t ; eq← = λ {m} t → eq→ x=y t } | |
| 50 | |
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51 |
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52 ⇔→== : { x y : OD } → ( {z : Ordinal } → def x z ⇔ def y z) → x == y |
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53 eq→ ( ⇔→== {x} {y} eq ) {z} m = proj1 eq m |
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54 eq← ( ⇔→== {x} {y} eq ) {z} m = proj2 eq m |
| 120 | 55 |
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56 -- next assumptions are our axiom |
| 322 | 57 -- |
| 58 -- OD is an equation on Ordinals, so it contains Ordinals. If these Ordinals have one-to-one | |
| 59 -- correspondence to the OD then the OD looks like a ZF Set. | |
| 60 -- | |
| 61 -- If all ZF Set have supreme upper bound, the solutions of OD have to be bounded, i.e. | |
| 62 -- bbounded ODs are ZF Set. Unbounded ODs are classes. | |
| 63 -- | |
| 290 | 64 -- In classical Set Theory, HOD is used, as a subset of OD, |
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65 -- HOD = { x | TC x ⊆ OD } |
| 290 | 66 -- where TC x is a transitive clusure of x, i.e. Union of all elemnts of all subset of x. |
| 322 | 67 -- This is not possible because we don't have V yet. So we assumes HODs are bounded OD. |
| 290 | 68 -- |
| 309 | 69 -- We also assumes HODs are isomorphic to Ordinals, which is ususally proved by Goedel number tricks. |
| 322 | 70 -- There two contraints on the HOD order, one is ∋, the other one is ⊂. |
| 71 -- ODs have an ovbious maximum, but Ordinals are not, but HOD has no maximum because there is an aribtrary | |
| 72 -- bound on each HOD. | |
| 290 | 73 -- |
| 322 | 74 -- In classical Set Theory, sup is defined by Uion, since we are working on constructive logic, |
| 290 | 75 -- we need explict assumption on sup. |
| 309 | 76 -- |
| 77 -- ==→o≡ is necessary to prove axiom of extensionality. | |
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78 |
| 303 | 79 -- Ordinals in OD , the maximum |
| 80 Ords : OD | |
| 81 Ords = record { def = λ x → One } | |
| 82 | |
| 83 record HOD : Set (suc n) where | |
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84 field |
| 303 | 85 od : OD |
| 304 | 86 odmax : Ordinal |
| 308 | 87 <odmax : {y : Ordinal} → def od y → y o< odmax |
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88 |
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89 open HOD |
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90 |
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91 record ODAxiom : Set (suc n) where |
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92 field |
| 304 | 93 -- HOD is isomorphic to Ordinal (by means of Goedel number) |
| 303 | 94 od→ord : HOD → Ordinal |
| 95 ord→od : Ordinal → HOD | |
| 96 c<→o< : {x y : HOD } → def (od y) ( od→ord x ) → od→ord x o< od→ord y | |
| 335 | 97 ⊆→o≤ : {y z : HOD } → ({x : Ordinal} → def (od y) x → def (od z) x ) → od→ord y o< osuc (od→ord z) |
| 303 | 98 oiso : {x : HOD } → ord→od ( od→ord x ) ≡ x |
| 322 | 99 diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x |
| 335 | 100 ==→o≡ : {x y : HOD } → (od x == od y) → x ≡ y |
| 101 sup-o : (A : HOD) → ( ( x : Ordinal ) → def (od A) x → Ordinal ) → Ordinal | |
| 306 | 102 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 |
| 367 | 103 -- possible order restriction |
| 104 ho< : {x : HOD} → od→ord x o< next (odmax x) | |
| 105 | |
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106 |
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107 postulate odAxiom : ODAxiom |
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108 open ODAxiom odAxiom |
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109 |
| 363 | 110 -- odmax minimality |
| 111 -- | |
| 112 -- since we have ==→o≡ , so odmax have to be unique. We should have odmaxmin in HOD. | |
| 113 -- We can calculate the minimum using sup but it is tedius. | |
| 114 -- Only Select has non minimum odmax. | |
| 115 -- We have the same problem on 'def' itself, but we leave it. | |
| 116 | |
| 117 odmaxmin : Set (suc n) | |
| 118 odmaxmin = (y : HOD) (z : Ordinal) → ((x : Ordinal)→ def (od y) x → x o< z) → odmax y o< osuc z | |
| 119 | |
| 335 | 120 -- OD ⇔ Ordinal leads a contradiction, so we need bounded HOD |
| 121 ¬OD-order : ( od→ord : OD → Ordinal ) → ( ord→od : Ordinal → OD ) → ( { x y : OD } → def y ( od→ord x ) → od→ord x o< od→ord y) → ⊥ | |
| 122 ¬OD-order od→ord ord→od c<→o< = osuc-< <-osuc (c<→o< {Ords} OneObj ) | |
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123 |
| 335 | 124 -- Open supreme upper bound leads a contradition, so we use domain restriction on sup |
| 125 ¬open-sup : ( sup-o : (Ordinal → Ordinal ) → Ordinal) → ((ψ : Ordinal → Ordinal ) → (x : Ordinal) → ψ x o< sup-o ψ ) → ⊥ | |
| 126 ¬open-sup sup-o sup-o< = o<> <-osuc (sup-o< next-ord (sup-o next-ord)) where | |
| 127 next-ord : Ordinal → Ordinal | |
| 128 next-ord x = osuc x | |
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129 |
| 179 | 130 -- Ordinal in OD ( and ZFSet ) Transitive Set |
| 303 | 131 Ord : ( a : Ordinal ) → HOD |
| 304 | 132 Ord a = record { od = record { def = λ y → y o< a } ; odmax = a ; <odmax = lemma } where |
| 133 lemma : {x : Ordinal} → x o< a → x o< a | |
| 134 lemma {x} lt = lt | |
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135 |
| 303 | 136 od∅ : HOD |
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137 od∅ = Ord o∅ |
| 40 | 138 |
| 303 | 139 odef : HOD → Ordinal → Set n |
| 140 odef A x = def ( od A ) x | |
| 123 | 141 |
| 303 | 142 _∋_ : ( a x : HOD ) → Set n |
| 143 _∋_ a x = odef a ( od→ord x ) | |
| 144 | |
| 145 _c<_ : ( x a : HOD ) → Set n | |
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146 x c< a = a ∋ x |
| 103 | 147 |
| 396 | 148 d→∋ : ( a : HOD ) { x : Ordinal} → odef a x → a ∋ (ord→od x) |
| 149 d→∋ a lt = subst (λ k → odef a k ) (sym diso) lt | |
| 150 | |
| 361 | 151 cseq : HOD → HOD |
| 308 | 152 cseq x = record { od = record { def = λ y → odef x (osuc y) } ; odmax = osuc (odmax x) ; <odmax = lemma } where |
| 153 lemma : {y : Ordinal} → def (od x) (osuc y) → y o< osuc (odmax x) | |
| 154 lemma {y} lt = ordtrans <-osuc (ordtrans (<odmax x lt) <-osuc ) | |
| 95 | 155 |
| 303 | 156 odef-subst : {Z : HOD } {X : Ordinal }{z : HOD } {x : Ordinal }→ odef Z X → Z ≡ z → X ≡ x → odef z x |
| 157 odef-subst df refl refl = df | |
| 95 | 158 |
| 361 | 159 otrans : {a x y : Ordinal } → odef (Ord a) x → odef (Ord x) y → odef (Ord a) y |
| 187 | 160 otrans x<a y<x = ordtrans y<x x<a |
| 123 | 161 |
| 303 | 162 odef→o< : {X : HOD } → {x : Ordinal } → odef X x → x o< od→ord X |
| 163 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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164 |
| 388 | 165 odefo→o< : {X y : Ordinal } → odef (ord→od X) y → y o< X |
| 166 odefo→o< {X} {y} lt = subst₂ (λ j k → j o< k ) diso diso ( c<→o< (subst (λ k → odef (ord→od X) k ) (sym diso ) lt )) | |
| 167 | |
| 396 | 168 -- If we have reverse of c<→o<, everything becomes Ordinal |
| 169 o<→c<→HOD=Ord : ( o<→c< : {x y : Ordinal } → x o< y → odef (ord→od y) x ) → {x : HOD } → x ≡ Ord (od→ord x) | |
| 170 o<→c<→HOD=Ord o<→c< {x} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where | |
| 171 lemma1 : {y : Ordinal} → odef x y → odef (Ord (od→ord x)) y | |
| 172 lemma1 {y} lt = subst ( λ k → k o< od→ord x ) diso (c<→o< {ord→od y} {x} (d→∋ x lt)) | |
| 173 lemma2 : {y : Ordinal} → odef (Ord (od→ord x)) y → odef x y | |
| 174 lemma2 {y} lt = subst (λ k → odef k y ) oiso (o<→c< {y} {od→ord x} lt ) | |
| 175 | |
| 51 | 176 -- avoiding lv != Zero error |
| 303 | 177 orefl : { x : HOD } → { y : Ordinal } → od→ord x ≡ y → od→ord x ≡ y |
| 51 | 178 orefl refl = refl |
| 179 | |
| 303 | 180 ==-iso : { x y : HOD } → od (ord→od (od→ord x)) == od (ord→od (od→ord y)) → od x == od y |
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181 ==-iso {x} {y} eq = record { |
| 303 | 182 eq→ = λ d → lemma ( eq→ eq (odef-subst d (sym oiso) refl )) ; |
| 183 eq← = λ d → lemma ( eq← eq (odef-subst d (sym oiso) refl )) } | |
| 51 | 184 where |
| 303 | 185 lemma : {x : HOD } {z : Ordinal } → odef (ord→od (od→ord x)) z → odef x z |
| 186 lemma {x} {z} d = odef-subst d oiso refl | |
| 51 | 187 |
| 303 | 188 =-iso : {x y : HOD } → (od x == od y) ≡ (od (ord→od (od→ord x)) == od y) |
| 189 =-iso {_} {y} = cong ( λ k → od k == od y ) (sym oiso) | |
| 57 | 190 |
| 303 | 191 ord→== : { x y : HOD } → od→ord x ≡ od→ord y → od x == od y |
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192 ord→== {x} {y} eq = ==-iso (lemma (od→ord x) (od→ord y) (orefl eq)) where |
| 303 | 193 lemma : ( ox oy : Ordinal ) → ox ≡ oy → od (ord→od ox) == od (ord→od oy) |
| 271 | 194 lemma ox ox refl = ==-refl |
| 51 | 195 |
| 303 | 196 o≡→== : { x y : Ordinal } → x ≡ y → od (ord→od x) == od (ord→od y) |
| 271 | 197 o≡→== {x} {.x} refl = ==-refl |
| 51 | 198 |
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199 o∅≡od∅ : ord→od (o∅ ) ≡ od∅ |
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200 o∅≡od∅ = ==→o≡ lemma where |
| 303 | 201 lemma0 : {x : Ordinal} → odef (ord→od o∅) x → odef od∅ x |
| 202 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 | |
| 203 lemma1 : {x : Ordinal} → odef od∅ x → odef (ord→od o∅) x | |
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204 lemma1 {x} lt = ⊥-elim (¬x<0 lt) |
| 303 | 205 lemma : od (ord→od o∅) == od od∅ |
| 150 | 206 lemma = record { eq→ = lemma0 ; eq← = lemma1 } |
| 207 | |
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208 ord-od∅ : od→ord (od∅ ) ≡ o∅ |
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209 ord-od∅ = sym ( subst (λ k → k ≡ od→ord (od∅ ) ) diso (cong ( λ k → od→ord k ) o∅≡od∅ ) ) |
| 80 | 210 |
| 303 | 211 ∅0 : record { def = λ x → Lift n ⊥ } == od od∅ |
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212 eq→ ∅0 {w} (lift ()) |
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213 eq← ∅0 {w} lt = lift (¬x<0 lt) |
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214 |
| 303 | 215 ∅< : { x y : HOD } → odef x (od→ord y ) → ¬ ( od x == od od∅ ) |
| 271 | 216 ∅< {x} {y} d eq with eq→ (==-trans eq (==-sym ∅0) ) d |
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217 ∅< {x} {y} d eq | lift () |
| 57 | 218 |
| 303 | 219 ∅6 : { x : HOD } → ¬ ( x ∋ x ) -- no Russel paradox |
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220 ∅6 {x} x∋x = o<¬≡ refl ( c<→o< {x} {x} x∋x ) |
| 51 | 221 |
| 303 | 222 odef-iso : {A B : HOD } {x y : Ordinal } → x ≡ y → (odef A y → odef B y) → odef A x → odef B x |
| 223 odef-iso refl t = t | |
| 76 | 224 |
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225 is-o∅ : ( x : Ordinal ) → Dec ( x ≡ o∅ ) |
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226 is-o∅ x with trio< x o∅ |
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227 is-o∅ x | tri< a ¬b ¬c = no ¬b |
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228 is-o∅ x | tri≈ ¬a b ¬c = yes b |
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229 is-o∅ x | tri> ¬a ¬b c = no ¬b |
| 57 | 230 |
| 335 | 231 -- the pair |
| 338 | 232 _,_ : HOD → HOD → HOD |
| 308 | 233 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 |
| 234 lemma : {t : Ordinal} → (t ≡ od→ord x) ∨ (t ≡ od→ord y) → t o< omax (od→ord x) (od→ord y) | |
| 235 lemma {t} (case1 refl) = omax-x _ _ | |
| 236 lemma {t} (case2 refl) = omax-y _ _ | |
| 237 | |
| 343 | 238 pair-xx<xy : {x y : HOD} → od→ord (x , x) o< osuc (od→ord (x , y) ) |
| 239 pair-xx<xy {x} {y} = ⊆→o≤ lemma where | |
| 240 lemma : {z : Ordinal} → def (od (x , x)) z → def (od (x , y)) z | |
| 241 lemma {z} (case1 refl) = case1 refl | |
| 242 lemma {z} (case2 refl) = case1 refl | |
| 243 | |
| 393 | 244 pair-<xy : {x y : HOD} → {n : Ordinal} → od→ord x o< next n → od→ord y o< next n → od→ord (x , y) o< next n |
| 245 pair-<xy {x} {y} {o} x<nn y<nn with trio< (od→ord x) (od→ord y) | inspect (omax (od→ord x)) (od→ord y) | |
| 246 ... | tri< a ¬b ¬c | record { eq = eq1 } = next< (subst (λ k → k o< next o ) (sym eq1) (osuc<nx y<nn)) ho< | |
| 247 ... | tri> ¬a ¬b c | record { eq = eq1 } = next< (subst (λ k → k o< next o ) (sym eq1) (osuc<nx x<nn)) ho< | |
| 248 ... | tri≈ ¬a b ¬c | record { eq = eq1 } = next< (subst (λ k → k o< next o ) (omax≡ _ _ b) (subst (λ k → osuc k o< next o) b (osuc<nx x<nn))) ho< | |
| 249 | |
| 339 | 250 -- another form of infinite |
| 343 | 251 -- pair-ord< : {x : Ordinal } → Set n |
| 252 pair-ord< : {x : HOD } → ( {y : HOD } → od→ord y o< next (odmax y) ) → od→ord ( x , x ) o< next (od→ord x) | |
| 253 pair-ord< {x} ho< = subst (λ k → od→ord (x , x) o< k ) lemmab0 lemmab1 where | |
| 254 lemmab0 : next (odmax (x , x)) ≡ next (od→ord x) | |
| 255 lemmab0 = trans (cong (λ k → next k) (omxx _)) (sym nexto≡) | |
| 256 lemmab1 : od→ord (x , x) o< next ( odmax (x , x)) | |
| 257 lemmab1 = ho< | |
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258 |
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259 pair<y : {x y : HOD } → y ∋ x → od→ord (x , x) o< osuc (od→ord y) |
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260 pair<y {x} {y} y∋x = ⊆→o≤ lemma where |
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261 lemma : {z : Ordinal} → def (od (x , x)) z → def (od y) z |
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262 lemma (case1 refl) = y∋x |
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263 lemma (case2 refl) = y∋x |
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264 |
| 361 | 265 -- another possible restriction. We reqest no minimality on odmax, so it may arbitrary larger. |
| 266 odmax<od→ord : { x y : HOD } → x ∋ y → Set n | |
| 267 odmax<od→ord {x} {y} x∋y = odmax x o< od→ord x | |
| 268 | |
| 376 | 269 in-codomain : (X : HOD ) → ( ψ : HOD → HOD ) → OD |
| 270 in-codomain X ψ = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( odef X y ∧ ( x ≡ od→ord (ψ (ord→od y ))))) } | |
| 141 | 271 |
| 360 | 272 _∩_ : ( A B : HOD ) → HOD |
| 273 A ∩ B = record { od = record { def = λ x → odef A x ∧ odef B x } | |
| 274 ; odmax = omin (odmax A) (odmax B) ; <odmax = λ y → min1 (<odmax A (proj1 y)) (<odmax B (proj2 y))} | |
| 308 | 275 |
| 303 | 276 record _⊆_ ( A B : HOD ) : Set (suc n) where |
| 271 | 277 field |
| 303 | 278 incl : { x : HOD } → A ∋ x → B ∋ x |
| 271 | 279 |
| 280 open _⊆_ | |
| 190 | 281 infixr 220 _⊆_ |
| 282 | |
| 387 | 283 trans-⊆ : { A B C : HOD} → A ⊆ B → B ⊆ C → A ⊆ C |
| 284 trans-⊆ A⊆B B⊆C = record { incl = λ x → incl B⊆C (incl A⊆B x) } | |
| 285 | |
| 286 refl-⊆ : {A : HOD} → A ⊆ A | |
| 287 refl-⊆ {A} = record { incl = λ x → x } | |
| 288 | |
| 335 | 289 od⊆→o≤ : {x y : HOD } → x ⊆ y → od→ord x o< osuc (od→ord y) |
| 396 | 290 od⊆→o≤ {x} {y} lt = ⊆→o≤ {x} {y} (λ {z} x>z → subst (λ k → def (od y) k ) diso (incl lt (d→∋ x x>z))) |
| 335 | 291 |
| 292 -- if we have od→ord (x , x) ≡ osuc (od→ord x), ⊆→o≤ → c<→o< | |
| 338 | 293 ⊆→o≤→c<→o< : ({x : HOD} → od→ord (x , x) ≡ osuc (od→ord x) ) |
| 294 → ({y z : HOD } → ({x : Ordinal} → def (od y) x → def (od z) x ) → od→ord y o< osuc (od→ord z) ) | |
| 295 → {x y : HOD } → def (od y) ( od→ord x ) → od→ord x o< od→ord y | |
| 296 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x with trio< (od→ord x) (od→ord y) | |
| 297 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x | tri< a ¬b ¬c = a | |
| 298 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x | tri≈ ¬a b ¬c = ⊥-elim ( o<¬≡ (peq {x}) (pair<y (subst (λ k → k ∋ x) (sym ( ==→o≡ {x} {y} (ord→== b))) y∋x ))) | |
| 299 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x | tri> ¬a ¬b c = | |
| 300 ⊥-elim ( o<> (⊆→o≤ {x , x} {y} y⊆x,x ) lemma1 ) where | |
| 301 lemma : {z : Ordinal} → (z ≡ od→ord x) ∨ (z ≡ od→ord x) → od→ord x ≡ z | |
| 302 lemma (case1 refl) = refl | |
| 303 lemma (case2 refl) = refl | |
| 304 y⊆x,x : {z : Ordinals.ord O} → def (od (x , x)) z → def (od y) z | |
| 305 y⊆x,x {z} lt = subst (λ k → def (od y) k ) (lemma lt) y∋x | |
| 306 lemma1 : osuc (od→ord y) o< od→ord (x , x) | |
| 307 lemma1 = subst (λ k → osuc (od→ord y) o< k ) (sym (peq {x})) (osucc c ) | |
| 335 | 308 |
| 360 | 309 subset-lemma : {A x : HOD } → ( {y : HOD } → x ∋ y → (A ∩ x ) ∋ y ) ⇔ ( x ⊆ A ) |
| 271 | 310 subset-lemma {A} {x} = record { |
| 311 proj1 = λ lt → record { incl = λ x∋z → proj1 (lt x∋z) } | |
| 312 ; proj2 = λ x⊆A lt → record { proj1 = incl x⊆A lt ; proj2 = lt } | |
| 190 | 313 } |
| 314 | |
| 312 | 315 power< : {A x : HOD } → x ⊆ A → Ord (osuc (od→ord A)) ∋ x |
| 316 power< {A} {x} x⊆A = ⊆→o≤ (λ {y} x∋y → subst (λ k → def (od A) k) diso (lemma y x∋y ) ) where | |
| 317 lemma : (y : Ordinal) → def (od x) y → def (od A) (od→ord (ord→od y)) | |
| 396 | 318 lemma y x∋y = incl x⊆A (d→∋ x x∋y) |
| 312 | 319 |
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320 open import Data.Unit |
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321 |
| 324 | 322 ε-induction : { ψ : HOD → Set n} |
| 303 | 323 → ( {x : HOD } → ({ y : HOD } → x ∋ y → ψ y ) → ψ x ) |
| 324 → (x : HOD ) → ψ x | |
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325 ε-induction {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (osuc (od→ord x)) <-osuc ) where |
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326 induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (ord→od oy)) → ψ (ord→od ox) |
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327 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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328 ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (ord→od oy) |
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329 ε-induction-ord ox {oy} lt = TransFinite {λ oy → ψ (ord→od oy)} induction oy |
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330 |
| 388 | 331 -- level trick (what'a shame) for LEM / minimal |
| 332 ε-induction1 : { ψ : HOD → Set (suc n)} | |
| 333 → ( {x : HOD } → ({ y : HOD } → x ∋ y → ψ y ) → ψ x ) | |
| 334 → (x : HOD ) → ψ x | |
| 335 ε-induction1 {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (osuc (od→ord x)) <-osuc ) where | |
| 336 induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (ord→od oy)) → ψ (ord→od ox) | |
| 337 induction ox prev = ind ( λ {y} lt → subst (λ k → ψ k ) oiso (prev (od→ord y) (o<-subst (c<→o< lt) refl diso ))) | |
| 338 ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (ord→od oy) | |
| 339 ε-induction-ord ox {oy} lt = TransFinite1 {λ oy → ψ (ord→od oy)} induction oy | |
| 330 | 340 |
| 376 | 341 Select : (X : HOD ) → ((x : HOD ) → Set n ) → HOD |
| 342 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) } | |
| 377 | 343 |
| 376 | 344 Replace : HOD → (HOD → HOD) → HOD |
| 345 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 } | |
| 365 | 346 ; odmax = rmax ; <odmax = rmax<} where |
| 347 rmax : Ordinal | |
| 376 | 348 rmax = sup-o X (λ y X∋y → od→ord (ψ (ord→od y))) |
| 365 | 349 rmax< : {y : Ordinal} → (y o< rmax) ∧ def (in-codomain X ψ) y → y o< rmax |
| 350 rmax< lt = proj1 lt | |
| 377 | 351 |
| 378 | 352 -- |
| 353 -- If we have LEM, Replace' is equivalent to Replace | |
| 354 -- | |
| 377 | 355 in-codomain' : (X : HOD ) → ((x : HOD) → X ∋ x → HOD) → OD |
| 356 in-codomain' X ψ = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( odef X y ∧ ((lt : odef X y) → x ≡ od→ord (ψ (ord→od y ) (d→∋ X lt) )))) } | |
| 357 Replace' : (X : HOD) → ((x : HOD) → X ∋ x → HOD) → HOD | |
| 358 Replace' X ψ = record { od = record { def = λ x → (x o< sup-o X (λ y X∋y → od→ord (ψ (ord→od y) (d→∋ X X∋y) ))) ∧ def (in-codomain' X ψ) x } | |
| 359 ; odmax = rmax ; <odmax = rmax< } where | |
| 360 rmax : Ordinal | |
| 361 rmax = sup-o X (λ y X∋y → od→ord (ψ (ord→od y) (d→∋ X X∋y))) | |
| 362 rmax< : {y : Ordinal} → (y o< rmax) ∧ def (in-codomain' X ψ) y → y o< rmax | |
| 363 rmax< lt = proj1 lt | |
| 364 | |
| 365 | 365 Union : HOD → HOD |
| 366 Union U = record { od = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((odef U u) ∧ (odef (ord→od u) x))) } | |
| 367 ; odmax = osuc (od→ord U) ; <odmax = umax< } where | |
| 368 umax< : {y : Ordinal} → ¬ ((u : Ordinal) → ¬ def (od U) u ∧ def (od (ord→od u)) y) → y o< osuc (od→ord U) | |
| 369 umax< {y} not = lemma (FExists _ lemma1 not ) where | |
| 370 lemma0 : {x : Ordinal} → def (od (ord→od x)) y → y o< x | |
| 396 | 371 lemma0 {x} x<y = subst₂ (λ j k → j o< k ) diso diso (c<→o< (d→∋ (ord→od x) x<y )) |
| 365 | 372 lemma2 : {x : Ordinal} → def (od U) x → x o< od→ord U |
| 396 | 373 lemma2 {x} x<U = subst (λ k → k o< od→ord U ) diso (c<→o< (d→∋ U x<U)) |
| 365 | 374 lemma1 : {x : Ordinal} → def (od U) x ∧ def (od (ord→od x)) y → ¬ (od→ord U o< y) |
| 375 lemma1 {x} lt u<y = o<> u<y (ordtrans (lemma0 (proj2 lt)) (lemma2 (proj1 lt)) ) | |
| 376 lemma : ¬ ((od→ord U) o< y ) → y o< osuc (od→ord U) | |
| 377 lemma not with trio< y (od→ord U) | |
| 378 lemma not | tri< a ¬b ¬c = ordtrans a <-osuc | |
| 379 lemma not | tri≈ ¬a refl ¬c = <-osuc | |
| 380 lemma not | tri> ¬a ¬b c = ⊥-elim (not c) | |
| 381 _∈_ : ( A B : HOD ) → Set n | |
| 382 A ∈ B = B ∋ A | |
| 383 | |
| 384 OPwr : (A : HOD ) → HOD | |
| 385 OPwr A = Ord ( sup-o (Ord (osuc (od→ord A))) ( λ x A∋x → od→ord ( A ∩ (ord→od x)) ) ) | |
| 386 | |
| 387 Power : HOD → HOD | |
| 376 | 388 Power A = Replace (OPwr (Ord (od→ord A))) ( λ x → A ∩ x ) |
| 365 | 389 -- ｛_｝ : ZFSet → ZFSet |
| 390 -- ｛ x ｝ = ( x , x ) -- better to use (x , x) directly | |
| 391 | |
| 392 data infinite-d : ( x : Ordinal ) → Set n where | |
| 393 iφ : infinite-d o∅ | |
| 394 isuc : {x : Ordinal } → infinite-d x → | |
| 395 infinite-d (od→ord ( Union (ord→od x , (ord→od x , ord→od x ) ) )) | |
| 396 | |
| 397 -- ω can be diverged in our case, since we have no restriction on the corresponding ordinal of a pair. | |
| 398 -- We simply assumes infinite-d y has a maximum. | |
| 399 -- | |
| 400 -- This means that many of OD may not be HODs because of the od→ord mapping divergence. | |
| 401 -- We should have some axioms to prevent this such as od→ord x o< next (odmax x). | |
| 402 -- | |
| 367 | 403 -- postulate |
| 404 -- ωmax : Ordinal | |
| 405 -- <ωmax : {y : Ordinal} → infinite-d y → y o< ωmax | |
| 406 -- | |
| 407 -- infinite : HOD | |
| 408 -- infinite = record { od = record { def = λ x → infinite-d x } ; odmax = ωmax ; <odmax = <ωmax } | |
| 365 | 409 |
| 410 infinite : HOD | |
| 367 | 411 infinite = record { od = record { def = λ x → infinite-d x } ; odmax = next o∅ ; <odmax = lemma } where |
| 365 | 412 u : (y : Ordinal ) → HOD |
| 413 u y = Union (ord→od y , (ord→od y , ord→od y)) | |
| 414 -- next< : {x y z : Ordinal} → x o< next z → y o< next x → y o< next z | |
| 415 lemma8 : {y : Ordinal} → od→ord (ord→od y , ord→od y) o< next (odmax (ord→od y , ord→od y)) | |
| 416 lemma8 = ho< | |
| 417 --- (x,y) < next (omax x y) < next (osuc y) = next y | |
| 418 lemmaa : {x y : HOD} → od→ord x o< od→ord y → od→ord (x , y) o< next (od→ord y) | |
| 419 lemmaa {x} {y} x<y = subst (λ k → od→ord (x , y) o< k ) (sym nexto≡) (subst (λ k → od→ord (x , y) o< next k ) (sym (omax< _ _ x<y)) ho< ) | |
| 420 lemma81 : {y : Ordinal} → od→ord (ord→od y , ord→od y) o< next (od→ord (ord→od y)) | |
| 421 lemma81 {y} = nexto=n (subst (λ k → od→ord (ord→od y , ord→od y) o< k ) (cong (λ k → next k) (omxx _)) lemma8) | |
| 422 lemma9 : {y : Ordinal} → od→ord (ord→od y , (ord→od y , ord→od y)) o< next (od→ord (ord→od y , ord→od y)) | |
| 423 lemma9 = lemmaa (c<→o< (case1 refl)) | |
| 424 lemma71 : {y : Ordinal} → od→ord (ord→od y , (ord→od y , ord→od y)) o< next (od→ord (ord→od y)) | |
| 425 lemma71 = next< lemma81 lemma9 | |
| 426 lemma1 : {y : Ordinal} → od→ord (u y) o< next (osuc (od→ord (ord→od y , (ord→od y , ord→od y)))) | |
| 427 lemma1 = ho< | |
| 428 --- main recursion | |
| 429 lemma : {y : Ordinal} → infinite-d y → y o< next o∅ | |
| 430 lemma {o∅} iφ = x<nx | |
| 431 lemma (isuc {y} x) = next< (lemma x) (next< (subst (λ k → od→ord (ord→od y , (ord→od y , ord→od y)) o< next k) diso lemma71 ) (nexto=n lemma1)) | |
| 432 | |
| 393 | 433 ω<next-o∅ : {y : Ordinal} → infinite-d y → y o< next o∅ |
| 434 ω<next-o∅ {y} lt = <odmax infinite lt | |
| 365 | 435 |
| 436 nat→ω : Nat → HOD | |
| 437 nat→ω Zero = od∅ | |
| 438 nat→ω (Suc y) = Union (nat→ω y , (nat→ω y , nat→ω y)) | |
| 439 | |
| 440 ω→nat : (n : HOD) → infinite ∋ n → Nat | |
| 441 ω→nat n = lemma where | |
| 442 lemma : {y : Ordinal} → infinite-d y → Nat | |
| 443 lemma iφ = Zero | |
| 444 lemma (isuc lt) = Suc (lemma lt) | |
| 445 | |
| 446 ω∋nat→ω : {n : Nat} → def (od infinite) (od→ord (nat→ω n)) | |
| 367 | 447 ω∋nat→ω {Zero} = subst (λ k → def (od infinite) k) (sym ord-od∅) iφ |
| 448 ω∋nat→ω {Suc n} = subst (λ k → def (od infinite) k) lemma (isuc ( ω∋nat→ω {n})) where | |
| 449 lemma : od→ord (Union (ord→od (od→ord (nat→ω n)) , (ord→od (od→ord (nat→ω n)) , ord→od (od→ord (nat→ω n))))) ≡ od→ord (nat→ω (Suc n)) | |
| 450 lemma = subst (λ k → od→ord (Union (k , ( k , k ))) ≡ od→ord (nat→ω (Suc n))) (sym oiso) refl | |
| 365 | 451 |
| 452 _=h=_ : (x y : HOD) → Set n | |
| 453 x =h= y = od x == od y | |
| 454 | |
| 396 | 455 open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) |
| 397 | 456 postulate f-extensionality : { n m : Level} → HE.Extensionality n m |
| 396 | 457 |
| 397 | 458 |
| 459 ord∋eq : {A i : HOD } → { f : (x : HOD ) → A ∋ x → Set (suc n)} | |
| 396 | 460 → ( ( x : Ordinal ) → ( lx : odef A x ) → f (ord→od x) (d→∋ A lx) ) |
| 461 → ( lt : A ∋ i ) → f i lt | |
| 398 | 462 ord∋eq {A} {i} {f} of lt = lemma oiso lt where |
| 399 | 463 lemma2 : (j : Ordinal) → (ltj : odef A j ) |
| 464 → (i : HOD) → (lti : odef A (od→ord i) ) | |
| 465 → (ord→od j ≡ i ) → d→∋ A ltj ≅ lti | |
| 466 lemma2 j _ i _ refl = HE.refl | |
| 467 lemma1 : (j : Ordinal) → (ltj : odef A j ) | |
| 468 → (i : HOD) → (lti : odef A (od→ord i) ) | |
| 469 → (ord→od j ≡ i ) → d→∋ A ltj ≅ lti | |
| 470 → f (ord→od j) (d→∋ A ltj) ≡ f i lti | |
| 471 lemma1 j _ i _ refl HE.refl = refl | |
| 472 lemma0 : (j : Ordinal) → (lt : odef A j ) → f (ord→od j) (d→∋ A lt) | |
| 473 lemma0 j lt = of j lt | |
| 398 | 474 lemma : {i : HOD } → {j : Ordinal} → ord→od j ≡ i → (lti : odef A (od→ord i)) → f i lti |
| 475 lemma {i} {j} refl lt = subst (λ k → f (ord→od j) k ) {!!} ( of j (subst (λ k → odef A k ) diso lt )) | |
| 396 | 476 |
| 477 nat→ω-iso : {i : HOD} → (lt : infinite ∋ i ) → nat→ω ( ω→nat i lt ) ≡ i | |
| 397 | 478 nat→ω-iso {i} lt = ord∋eq {infinite} {i} (λ x lx → lemma lx ) lt where |
| 396 | 479 lemma : {x : Ordinal} → (lx : infinite-d x ) → nat→ω ( ω→nat (ord→od x) (d→∋ infinite lx) ) ≡ ord→od x |
| 399 | 480 lemma {o∅} iφ = begin |
| 481 nat→ω (ω→nat (ord→od o∅) (d→∋ infinite iφ)) | |
| 482 ≡⟨ {!!} ⟩ | |
| 483 nat→ω Zero | |
| 484 ≡⟨ sym o∅≡od∅ ⟩ | |
| 485 ord→od o∅ | |
| 486 ∎ where open ≡-Reasoning | |
| 487 lemma {x1} (isuc lx) = {!!} | |
| 488 | |
| 396 | 489 |
| 365 | 490 infixr 200 _∈_ |
| 491 -- infixr 230 _∩_ _∪_ | |
| 492 | |
| 493 pair→ : ( x y t : HOD ) → (x , y) ∋ t → ( t =h= x ) ∨ ( t =h= y ) | |
| 494 pair→ x y t (case1 t≡x ) = case1 (subst₂ (λ j k → j =h= k ) oiso oiso (o≡→== t≡x )) | |
| 495 pair→ x y t (case2 t≡y ) = case2 (subst₂ (λ j k → j =h= k ) oiso oiso (o≡→== t≡y )) | |
| 496 | |
| 497 pair← : ( x y t : HOD ) → ( t =h= x ) ∨ ( t =h= y ) → (x , y) ∋ t | |
| 498 pair← x y t (case1 t=h=x) = case1 (cong (λ k → od→ord k ) (==→o≡ t=h=x)) | |
| 499 pair← x y t (case2 t=h=y) = case2 (cong (λ k → od→ord k ) (==→o≡ t=h=y)) | |
| 500 | |
| 501 empty : (x : HOD ) → ¬ (od∅ ∋ x) | |
| 502 empty x = ¬x<0 | |
| 503 | |
| 504 o<→c< : {x y : Ordinal } → x o< y → (Ord x) ⊆ (Ord y) | |
| 505 o<→c< lt = record { incl = λ z → ordtrans z lt } | |
| 506 | |
| 507 ⊆→o< : {x y : Ordinal } → (Ord x) ⊆ (Ord y) → x o< osuc y | |
| 508 ⊆→o< {x} {y} lt with trio< x y | |
| 509 ⊆→o< {x} {y} lt | tri< a ¬b ¬c = ordtrans a <-osuc | |
| 510 ⊆→o< {x} {y} lt | tri≈ ¬a b ¬c = subst ( λ k → k o< osuc y) (sym b) <-osuc | |
| 511 ⊆→o< {x} {y} lt | tri> ¬a ¬b c with (incl lt) (o<-subst c (sym diso) refl ) | |
| 512 ... | ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt diso refl )) | |
| 513 | |
| 514 union→ : (X z u : HOD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z | |
| 515 union→ X z u xx not = ⊥-elim ( not (od→ord u) ( record { proj1 = proj1 xx | |
| 516 ; proj2 = subst ( λ k → odef k (od→ord z)) (sym oiso) (proj2 xx) } )) | |
| 517 union← : (X z : HOD) (X∋z : Union X ∋ z) → ¬ ( (u : HOD ) → ¬ ((X ∋ u) ∧ (u ∋ z ))) | |
| 518 union← X z UX∋z = FExists _ lemma UX∋z where | |
| 519 lemma : {y : Ordinal} → odef X y ∧ odef (ord→od y) (od→ord z) → ¬ ((u : HOD) → ¬ (X ∋ u) ∧ (u ∋ z)) | |
| 396 | 520 lemma {y} xx not = not (ord→od y) record { proj1 = d→∋ X (proj1 xx) ; proj2 = proj2 xx } |
| 365 | 521 |
| 376 | 522 ψiso : {ψ : HOD → Set n} {x y : HOD } → ψ x → x ≡ y → ψ y |
| 523 ψiso {ψ} t refl = t | |
| 524 selection : {ψ : HOD → Set n} {X y : HOD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y) | |
| 525 selection {ψ} {X} {y} = record { | |
| 526 proj1 = λ cond → record { proj1 = proj1 cond ; proj2 = ψiso {ψ} (proj2 cond) (sym oiso) } | |
| 527 ; proj2 = λ select → record { proj1 = proj1 select ; proj2 = ψiso {ψ} (proj2 select) oiso } | |
| 528 } | |
| 377 | 529 |
| 395 | 530 selection-in-domain : {ψ : HOD → Set n} {X y : HOD} → Select X ψ ∋ y → X ∋ y |
| 531 selection-in-domain {ψ} {X} {y} lt = proj1 ((proj2 (selection {ψ} {X} )) lt) | |
| 532 | |
| 376 | 533 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)))) |
| 534 sup-c< ψ {X} {x} lt = subst (λ k → od→ord (ψ k) o< _ ) oiso (sup-o< X lt ) | |
| 535 replacement← : {ψ : HOD → HOD} (X x : HOD) → X ∋ x → Replace X ψ ∋ ψ x | |
| 536 replacement← {ψ} X x lt = record { proj1 = sup-c< ψ {X} {x} lt ; proj2 = lemma } where | |
| 537 lemma : def (in-codomain X ψ) (od→ord (ψ x)) | |
| 538 lemma not = ⊥-elim ( not ( od→ord x ) (record { proj1 = lt ; proj2 = cong (λ k → od→ord (ψ k)) (sym oiso)} )) | |
| 539 replacement→ : {ψ : HOD → HOD} (X x : HOD) → (lt : Replace X ψ ∋ x) → ¬ ( (y : HOD) → ¬ (x =h= ψ y)) | |
| 540 replacement→ {ψ} X x lt = contra-position lemma (lemma2 (proj2 lt)) where | |
| 541 lemma2 : ¬ ((y : Ordinal) → ¬ odef X y ∧ ((od→ord x) ≡ od→ord (ψ (ord→od y)))) | |
| 542 → ¬ ((y : Ordinal) → ¬ odef X y ∧ (ord→od (od→ord x) =h= ψ (ord→od y))) | |
| 365 | 543 lemma2 not not2 = not ( λ y d → not2 y (record { proj1 = proj1 d ; proj2 = lemma3 (proj2 d)})) where |
| 376 | 544 lemma3 : {y : Ordinal } → (od→ord x ≡ od→ord (ψ (ord→od y))) → (ord→od (od→ord x) =h= ψ (ord→od y)) |
| 365 | 545 lemma3 {y} eq = subst (λ k → ord→od (od→ord x) =h= k ) oiso (o≡→== eq ) |
| 376 | 546 lemma : ( (y : HOD) → ¬ (x =h= ψ y)) → ( (y : Ordinal) → ¬ odef X y ∧ (ord→od (od→ord x) =h= ψ (ord→od y)) ) |
| 547 lemma not y not2 = not (ord→od y) (subst (λ k → k =h= ψ (ord→od y)) oiso ( proj2 not2 )) | |
| 365 | 548 |
| 549 --- | |
| 550 --- Power Set | |
| 551 --- | |
| 552 --- First consider ordinals in HOD | |
| 553 --- | |
| 554 --- A ∩ x = record { def = λ y → odef A y ∧ odef x y } subset of A | |
| 555 -- | |
| 556 -- | |
| 557 ∩-≡ : { a b : HOD } → ({x : HOD } → (a ∋ x → b ∋ x)) → a =h= ( b ∩ a ) | |
| 558 ∩-≡ {a} {b} inc = record { | |
| 559 eq→ = λ {x} x<a → record { proj2 = x<a ; | |
| 396 | 560 proj1 = odef-subst {_} {_} {b} {x} (inc (d→∋ a x<a)) refl diso } ; |
| 365 | 561 eq← = λ {x} x<a∩b → proj2 x<a∩b } |
| 562 -- | |
| 563 -- Transitive Set case | |
| 564 -- we have t ∋ x → Ord a ∋ x means t is a subset of Ord a, that is (Ord a) ∩ t =h= t | |
| 565 -- OPwr (Ord a) is a sup of (Ord a) ∩ t, so OPwr (Ord a) ∋ t | |
| 566 -- OPwr A = Ord ( sup-o ( λ x → od→ord ( A ∩ (ord→od x )) ) ) | |
| 567 -- | |
| 568 ord-power← : (a : Ordinal ) (t : HOD) → ({x : HOD} → (t ∋ x → (Ord a) ∋ x)) → OPwr (Ord a) ∋ t | |
| 569 ord-power← a t t→A = odef-subst {_} {_} {OPwr (Ord a)} {od→ord t} | |
| 570 lemma refl (lemma1 lemma-eq )where | |
| 571 lemma-eq : ((Ord a) ∩ t) =h= t | |
| 572 eq→ lemma-eq {z} w = proj2 w | |
| 573 eq← lemma-eq {z} w = record { proj2 = w ; | |
| 574 proj1 = odef-subst {_} {_} {(Ord a)} {z} | |
| 396 | 575 ( t→A (d→∋ t w)) refl diso } |
| 365 | 576 lemma1 : {a : Ordinal } { t : HOD } |
| 577 → (eq : ((Ord a) ∩ t) =h= t) → od→ord ((Ord a) ∩ (ord→od (od→ord t))) ≡ od→ord t | |
| 578 lemma1 {a} {t} eq = subst (λ k → od→ord ((Ord a) ∩ k) ≡ od→ord t ) (sym oiso) (cong (λ k → od→ord k ) (==→o≡ eq )) | |
| 579 lemma2 : (od→ord t) o< (osuc (od→ord (Ord a))) | |
| 396 | 580 lemma2 = ⊆→o≤ {t} {Ord a} (λ {x} x<t → subst (λ k → def (od (Ord a)) k) diso (t→A (d→∋ t x<t))) |
| 365 | 581 lemma : od→ord ((Ord a) ∩ (ord→od (od→ord t)) ) o< sup-o (Ord (osuc (od→ord (Ord a)))) (λ x lt → od→ord ((Ord a) ∩ (ord→od x))) |
| 582 lemma = sup-o< _ lemma2 | |
| 583 | |
| 584 -- | |
| 585 -- Every set in HOD is a subset of Ordinals, so make OPwr (Ord (od→ord A)) first | |
| 586 -- then replace of all elements of the Power set by A ∩ y | |
| 587 -- | |
| 588 -- Power A = Replace (OPwr (Ord (od→ord A))) ( λ y → A ∩ y ) | |
| 589 | |
| 590 -- we have oly double negation form because of the replacement axiom | |
| 591 -- | |
| 592 power→ : ( A t : HOD) → Power A ∋ t → {x : HOD} → t ∋ x → ¬ ¬ (A ∋ x) | |
| 593 power→ A t P∋t {x} t∋x = FExists _ lemma5 lemma4 where | |
| 594 a = od→ord A | |
| 595 lemma2 : ¬ ( (y : HOD) → ¬ (t =h= (A ∩ y))) | |
| 376 | 596 lemma2 = replacement→ {λ x → A ∩ x} (OPwr (Ord (od→ord A))) t P∋t |
| 365 | 597 lemma3 : (y : HOD) → t =h= ( A ∩ y ) → ¬ ¬ (A ∋ x) |
| 598 lemma3 y eq not = not (proj1 (eq→ eq t∋x)) | |
| 599 lemma4 : ¬ ((y : Ordinal) → ¬ (t =h= (A ∩ ord→od y))) | |
| 600 lemma4 not = lemma2 ( λ y not1 → not (od→ord y) (subst (λ k → t =h= ( A ∩ k )) (sym oiso) not1 )) | |
| 601 lemma5 : {y : Ordinal} → t =h= (A ∩ ord→od y) → ¬ ¬ (odef A (od→ord x)) | |
| 602 lemma5 {y} eq not = (lemma3 (ord→od y) eq) not | |
| 603 | |
| 604 power← : (A t : HOD) → ({x : HOD} → (t ∋ x → A ∋ x)) → Power A ∋ t | |
| 605 power← A t t→A = record { proj1 = lemma1 ; proj2 = lemma2 } where | |
| 606 a = od→ord A | |
| 607 lemma0 : {x : HOD} → t ∋ x → Ord a ∋ x | |
| 608 lemma0 {x} t∋x = c<→o< (t→A t∋x) | |
| 609 lemma3 : OPwr (Ord a) ∋ t | |
| 610 lemma3 = ord-power← a t lemma0 | |
| 611 lemma4 : (A ∩ ord→od (od→ord t)) ≡ t | |
| 612 lemma4 = let open ≡-Reasoning in begin | |
| 613 A ∩ ord→od (od→ord t) | |
| 614 ≡⟨ cong (λ k → A ∩ k) oiso ⟩ | |
| 615 A ∩ t | |
| 616 ≡⟨ sym (==→o≡ ( ∩-≡ {t} {A} t→A )) ⟩ | |
| 617 t | |
| 618 ∎ | |
| 619 sup1 : Ordinal | |
| 620 sup1 = sup-o (Ord (osuc (od→ord (Ord (od→ord A))))) (λ x A∋x → od→ord ((Ord (od→ord A)) ∩ (ord→od x))) | |
| 621 lemma9 : def (od (Ord (Ordinals.osuc O (od→ord (Ord (od→ord A)))))) (od→ord (Ord (od→ord A))) | |
| 622 lemma9 = <-osuc | |
| 623 lemmab : od→ord ((Ord (od→ord A)) ∩ (ord→od (od→ord (Ord (od→ord A) )))) o< sup1 | |
| 624 lemmab = sup-o< (Ord (osuc (od→ord (Ord (od→ord A))))) lemma9 | |
| 625 lemmad : Ord (osuc (od→ord A)) ∋ t | |
| 396 | 626 lemmad = ⊆→o≤ (λ {x} lt → subst (λ k → def (od A) k ) diso (t→A (d→∋ t lt))) |
| 365 | 627 lemmac : ((Ord (od→ord A)) ∩ (ord→od (od→ord (Ord (od→ord A) )))) =h= Ord (od→ord A) |
| 628 lemmac = record { eq→ = lemmaf ; eq← = lemmag } where | |
| 629 lemmaf : {x : Ordinal} → def (od ((Ord (od→ord A)) ∩ (ord→od (od→ord (Ord (od→ord A)))))) x → def (od (Ord (od→ord A))) x | |
| 630 lemmaf {x} lt = proj1 lt | |
| 631 lemmag : {x : Ordinal} → def (od (Ord (od→ord A))) x → def (od ((Ord (od→ord A)) ∩ (ord→od (od→ord (Ord (od→ord A)))))) x | |
| 632 lemmag {x} lt = record { proj1 = lt ; proj2 = subst (λ k → def (od k) x) (sym oiso) lt } | |
| 633 lemmae : od→ord ((Ord (od→ord A)) ∩ (ord→od (od→ord (Ord (od→ord A))))) ≡ od→ord (Ord (od→ord A)) | |
| 634 lemmae = cong (λ k → od→ord k ) ( ==→o≡ lemmac) | |
| 635 lemma7 : def (od (OPwr (Ord (od→ord A)))) (od→ord t) | |
| 636 lemma7 with osuc-≡< lemmad | |
| 637 lemma7 | case2 lt = ordtrans (c<→o< lt) (subst (λ k → k o< sup1) lemmae lemmab ) | |
| 638 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 | |
| 639 lemmah : {x : Ordinal } → def (od (ord→od (od→ord t))) x → def (od (ord→od (od→ord (Ord (od→ord t))))) x | |
| 640 lemmah {x} lt = subst (λ k → def (od k) x ) (sym oiso) (subst (λ k → k o< (od→ord t)) | |
| 641 diso | |
| 642 (c<→o< (subst₂ (λ j k → def (od j) k) oiso (sym diso) lt ))) | |
| 643 lemma7 | case1 eq | case1 eq1 = subst (λ k → k o< sup1) (trans lemmae lemmai) lemmab where | |
| 644 lemmai : od→ord (Ord (od→ord A)) ≡ od→ord t | |
| 645 lemmai = let open ≡-Reasoning in begin | |
| 646 od→ord (Ord (od→ord A)) | |
| 647 ≡⟨ sym (cong (λ k → od→ord (Ord k)) eq) ⟩ | |
| 648 od→ord (Ord (od→ord t)) | |
| 649 ≡⟨ sym diso ⟩ | |
| 650 od→ord (ord→od (od→ord (Ord (od→ord t)))) | |
| 651 ≡⟨ sym eq1 ⟩ | |
| 652 od→ord (ord→od (od→ord t)) | |
| 653 ≡⟨ diso ⟩ | |
| 654 od→ord t | |
| 655 ∎ | |
| 656 lemma7 | case1 eq | case2 lt = ordtrans lemmaj (subst (λ k → k o< sup1) lemmae lemmab ) where | |
| 657 lemmak : od→ord (ord→od (od→ord (Ord (od→ord t)))) ≡ od→ord (Ord (od→ord A)) | |
| 658 lemmak = let open ≡-Reasoning in begin | |
| 659 od→ord (ord→od (od→ord (Ord (od→ord t)))) | |
| 660 ≡⟨ diso ⟩ | |
| 661 od→ord (Ord (od→ord t)) | |
| 662 ≡⟨ cong (λ k → od→ord (Ord k)) eq ⟩ | |
| 663 od→ord (Ord (od→ord A)) | |
| 664 ∎ | |
| 665 lemmaj : od→ord t o< od→ord (Ord (od→ord A)) | |
| 666 lemmaj = subst₂ (λ j k → j o< k ) diso lemmak lt | |
| 667 lemma1 : od→ord t o< sup-o (OPwr (Ord (od→ord A))) (λ x lt → od→ord (A ∩ (ord→od x))) | |
| 668 lemma1 = subst (λ k → od→ord k o< sup-o (OPwr (Ord (od→ord A))) (λ x lt → od→ord (A ∩ (ord→od x)))) | |
| 669 lemma4 (sup-o< (OPwr (Ord (od→ord A))) lemma7 ) | |
| 376 | 670 lemma2 : def (in-codomain (OPwr (Ord (od→ord A))) (_∩_ A)) (od→ord t) |
| 671 lemma2 not = ⊥-elim ( not (od→ord t) (record { proj1 = lemma3 ; proj2 = lemma6 }) ) where | |
| 365 | 672 lemma6 : od→ord t ≡ od→ord (A ∩ ord→od (od→ord t)) |
| 673 lemma6 = cong ( λ k → od→ord k ) (==→o≡ (subst (λ k → t =h= (A ∩ k)) (sym oiso) ( ∩-≡ {t} {A} t→A ))) | |
| 674 | |
| 675 | |
| 676 ord⊆power : (a : Ordinal) → (Ord (osuc a)) ⊆ (Power (Ord a)) | |
| 677 ord⊆power a = record { incl = λ {x} lt → power← (Ord a) x (lemma lt) } where | |
| 678 lemma : {x y : HOD} → od→ord x o< osuc a → x ∋ y → Ord a ∋ y | |
| 679 lemma lt y<x with osuc-≡< lt | |
| 680 lemma lt y<x | case1 refl = c<→o< y<x | |
| 681 lemma lt y<x | case2 x<a = ordtrans (c<→o< y<x) x<a | |
| 682 | |
| 683 continuum-hyphotheis : (a : Ordinal) → Set (suc n) | |
| 684 continuum-hyphotheis a = Power (Ord a) ⊆ Ord (osuc a) | |
| 685 | |
| 686 extensionality0 : {A B : HOD } → ((z : HOD) → (A ∋ z) ⇔ (B ∋ z)) → A =h= B | |
| 687 eq→ (extensionality0 {A} {B} eq ) {x} d = odef-iso {A} {B} (sym diso) (proj1 (eq (ord→od x))) d | |
| 688 eq← (extensionality0 {A} {B} eq ) {x} d = odef-iso {B} {A} (sym diso) (proj2 (eq (ord→od x))) d | |
| 689 | |
| 690 extensionality : {A B w : HOD } → ((z : HOD ) → (A ∋ z) ⇔ (B ∋ z)) → (w ∋ A) ⇔ (w ∋ B) | |
| 691 proj1 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) ( ==→o≡ (extensionality0 {A} {B} eq) ) d | |
| 692 proj2 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) (sym ( ==→o≡ (extensionality0 {A} {B} eq) )) d | |
| 693 | |
| 694 infinity∅ : infinite ∋ od∅ | |
| 695 infinity∅ = odef-subst {_} {_} {infinite} {od→ord (od∅ )} iφ refl lemma where | |
| 696 lemma : o∅ ≡ od→ord od∅ | |
| 697 lemma = let open ≡-Reasoning in begin | |
| 698 o∅ | |
| 699 ≡⟨ sym diso ⟩ | |
| 700 od→ord ( ord→od o∅ ) | |
| 701 ≡⟨ cong ( λ k → od→ord k ) o∅≡od∅ ⟩ | |
| 702 od→ord od∅ | |
| 703 ∎ | |
| 704 infinity : (x : HOD) → infinite ∋ x → infinite ∋ Union (x , (x , x )) | |
| 705 infinity x lt = odef-subst {_} {_} {infinite} {od→ord (Union (x , (x , x )))} ( isuc {od→ord x} lt ) refl lemma where | |
| 706 lemma : od→ord (Union (ord→od (od→ord x) , (ord→od (od→ord x) , ord→od (od→ord x)))) | |
| 707 ≡ od→ord (Union (x , (x , x))) | |
| 708 lemma = cong (λ k → od→ord (Union ( k , ( k , k ) ))) oiso | |
| 709 | |
| 710 isZF : IsZF (HOD ) _∋_ _=h=_ od∅ _,_ Union Power Select Replace infinite | |
| 711 isZF = record { | |
| 712 isEquivalence = record { refl = ==-refl ; sym = ==-sym; trans = ==-trans } | |
| 713 ; pair→ = pair→ | |
| 714 ; pair← = pair← | |
| 715 ; union→ = union→ | |
| 716 ; union← = union← | |
| 717 ; empty = empty | |
| 718 ; power→ = power→ | |
| 719 ; power← = power← | |
| 720 ; extensionality = λ {A} {B} {w} → extensionality {A} {B} {w} | |
| 721 ; ε-induction = ε-induction | |
| 722 ; infinity∅ = infinity∅ | |
| 723 ; infinity = infinity | |
| 376 | 724 ; selection = λ {X} {ψ} {y} → selection {X} {ψ} {y} |
| 725 ; replacement← = replacement← | |
| 726 ; replacement→ = λ {ψ} → replacement→ {ψ} | |
| 365 | 727 -- ; choice-func = choice-func |
| 728 -- ; choice = choice | |
| 729 } | |
| 730 | |
| 303 | 731 HOD→ZF : ZF |
| 732 HOD→ZF = record { | |
| 733 ZFSet = HOD | |
|
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equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
42
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changeset
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734 ; _∋_ = _∋_ |
| 365 | 735 ; _≈_ = _=h=_ |
|
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fce60b99dc55
posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
28
diff
changeset
|
736 ; ∅ = od∅ |
| 28 | 737 ; _,_ = _,_ |
| 365 | 738 ; Union = Union |
| 739 ; Power = Power | |
| 740 ; Select = Select | |
| 741 ; Replace = Replace | |
| 742 ; infinite = infinite | |
| 743 ; isZF = isZF | |
| 744 } | |
| 745 | |
| 144 | 746 |