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+ − 1 {-# OPTIONS --allow-unsolved-metas #-}
431
+ − 2 open import Level
+ − 3 open import Ordinals
+ − 4 module OD {n : Level } (O : Ordinals {n} ) where
+ − 5
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+ − 6 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ )
431
+ − 7 open import Relation.Binary.PropositionalEquality hiding ( [_] )
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+ − 8 open import Data.Nat.Properties
431
+ − 9 open import Data.Empty
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+ − 10 open import Data.Unit
431
+ − 11 open import Relation.Nullary
+ − 12 open import Relation.Binary hiding (_⇔_)
+ − 13 open import Relation.Binary.Core hiding (_⇔_)
+ − 14
+ − 15 open import logic
+ − 16 import OrdUtil
+ − 17 open import nat
+ − 18
+ − 19 open Ordinals.Ordinals O
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+ − 20 open Ordinals.IsOrdinals isOrdinal
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+ − 21 -- open Ordinals.IsNext isNext
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+ − 22 open OrdUtil O
+ − 23
+ − 24 -- Ordinal Definable Set
+ − 25
+ − 26 record OD : Set (suc n ) where
+ − 27 field
+ − 28 def : (x : Ordinal ) → Set n
+ − 29
+ − 30 open OD
+ − 31
+ − 32 open _∧_
+ − 33 open _∨_
+ − 34 open Bool
+ − 35
+ − 36 record _==_ ( a b : OD ) : Set n where
+ − 37 field
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+ − 38 eq→ : ∀ { x : Ordinal } → def a x → def b x
+ − 39 eq← : ∀ { x : Ordinal } → def b x → def a x
431
+ − 40
+ − 41 ==-refl : { x : OD } → x == x
+ − 42 ==-refl {x} = record { eq→ = λ x → x ; eq← = λ x → x }
+ − 43
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+ − 44 open _==_
431
+ − 45
+ − 46 ==-trans : { x y z : OD } → x == y → y == z → x == z
+ − 47 ==-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) }
+ − 48
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+ − 49 ==-sym : { x y : OD } → x == y → y == x
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+ − 50 ==-sym x=y = record { eq→ = λ {m} t → eq← x=y t ; eq← = λ {m} t → eq→ x=y t }
+ − 51
+ − 52
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+ − 53 ⇔→== : { x y : OD } → ( {z : Ordinal } → (def x z ⇔ def y z)) → x == y
+ − 54 eq→ ( ⇔→== {x} {y} eq ) {z} m = proj1 eq m
+ − 55 eq← ( ⇔→== {x} {y} eq ) {z} m = proj2 eq m
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+ − 56
+ − 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 --
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+ − 64 -- In classical Set Theory, HOD is used, as a subset of OD,
431
+ − 65 -- HOD = { x | TC x ⊆ OD }
+ − 66 -- where TC x is a transitive clusure of x, i.e. Union of all elemnts of all subset of x.
+ − 67 -- This is not possible because we don't have V yet. So we assumes HODs are bounded OD.
+ − 68 --
+ − 69 -- We also assumes HODs are isomorphic to Ordinals, which is ususally proved by Goedel number tricks.
+ − 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.
+ − 73 --
+ − 74 -- In classical Set Theory, sup is defined by Uion, since we are working on constructive logic,
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+ − 75 -- we need explict assumption on sup for unrestricted Replacement.
431
+ − 76 --
+ − 77 -- ==→o≡ is necessary to prove axiom of extensionality.
+ − 78
+ − 79 -- Ordinals in OD , the maximum
+ − 80 Ords : OD
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+ − 81 Ords = record { def = λ x → Lift n ⊤ }
431
+ − 82
+ − 83 record HOD : Set (suc n) where
+ − 84 field
+ − 85 od : OD
+ − 86 odmax : Ordinal
+ − 87 <odmax : {y : Ordinal} → def od y → y o< odmax
+ − 88
+ − 89 open HOD
+ − 90
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+ − 91 open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ )
+ − 92
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+ − 93 record ODAxiom : Set (suc n) where
431
+ − 94 field
+ − 95 -- HOD is isomorphic to Ordinal (by means of Goedel number)
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+ − 96 & : HOD → Ordinal
+ − 97 * : Ordinal → HOD
431
+ − 98 c<→o< : {x y : HOD } → def (od y) ( & x ) → & x o< & y
+ − 99 ⊆→o≤ : {y z : HOD } → ({x : Ordinal} → def (od y) x → def (od z) x ) → & y o< osuc (& z)
+ − 100 *iso : {x : HOD } → * ( & x ) ≡ x
+ − 101 &iso : {x : Ordinal } → & ( * x ) ≡ x
+ − 102 ==→o≡ : {x y : HOD } → (od x == od y) → x ≡ y
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+ − 103 ∋-irr : {X : HOD} {x : Ordinal } → (a b : def (od X) x) → a ≅ b
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+ − 104
431
+ − 105 postulate odAxiom : ODAxiom
+ − 106 open ODAxiom odAxiom
+ − 107
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+ − 108 -- possible order restriction (required in the axiom of Omega )
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+ − 109
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+ − 110 -- postulate odAxiom-ho< : ODAxiom-ho<
+ − 111 -- open ODAxiom-ho< odAxiom-ho<
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+ − 112
431
+ − 113 -- odmax minimality
+ − 114 --
+ − 115 -- since we have ==→o≡ , so odmax have to be unique. We should have odmaxmin in HOD.
+ − 116 -- We can calculate the minimum using sup but it is tedius.
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+ − 117 -- Only Select has non minimum odmax.
431
+ − 118 -- We have the same problem on 'def' itself, but we leave it.
+ − 119
+ − 120 odmaxmin : Set (suc n)
+ − 121 odmaxmin = (y : HOD) (z : Ordinal) → ((x : Ordinal)→ def (od y) x → x o< z) → odmax y o< osuc z
+ − 122
+ − 123 -- OD ⇔ Ordinal leads a contradiction, so we need bounded HOD
+ − 124 ¬OD-order : ( & : OD → Ordinal ) → ( * : Ordinal → OD ) → ( { x y : OD } → def y ( & x ) → & x o< & y) → ⊥
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+ − 125 ¬OD-order & * c<→o< = o≤> <-osuc (c<→o< {Ords} (lift tt) )
431
+ − 126
+ − 127 -- Ordinal in OD ( and ZFSet ) Transitive Set
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+ − 128 Ord : ( a : Ordinal ) → HOD
431
+ − 129 Ord a = record { od = record { def = λ y → y o< a } ; odmax = a ; <odmax = lemma } where
+ − 130 lemma : {x : Ordinal} → x o< a → x o< a
+ − 131 lemma {x} lt = lt
+ − 132
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+ − 133 od∅ : HOD
+ − 134 od∅ = Ord o∅
431
+ − 135
+ − 136 odef : HOD → Ordinal → Set n
+ − 137 odef A x = def ( od A ) x
+ − 138
+ − 139 _∋_ : ( a x : HOD ) → Set n
+ − 140 _∋_ a x = odef a ( & x )
+ − 141
+ − 142 -- _c<_ : ( x a : HOD ) → Set n
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+ − 143 -- x c< a = a ∋ x
431
+ − 144
+ − 145 d→∋ : ( a : HOD ) { x : Ordinal} → odef a x → a ∋ (* x)
+ − 146 d→∋ a lt = subst (λ k → odef a k ) (sym &iso) lt
+ − 147
+ − 148 -- odef-subst : {Z : HOD } {X : Ordinal }{z : HOD } {x : Ordinal }→ odef Z X → Z ≡ z → X ≡ x → odef z x
+ − 149 -- odef-subst df refl refl = df
+ − 150
+ − 151 otrans : {a x y : Ordinal } → odef (Ord a) x → odef (Ord x) y → odef (Ord a) y
+ − 152 otrans x<a y<x = ordtrans y<x x<a
+ − 153
+ − 154 -- If we have reverse of c<→o<, everything becomes Ordinal
+ − 155 ∈→c<→HOD=Ord : ( o<→c< : {x y : Ordinal } → x o< y → odef (* y) x ) → {x : HOD } → x ≡ Ord (& x)
+ − 156 ∈→c<→HOD=Ord o<→c< {x} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where
+ − 157 lemma1 : {y : Ordinal} → odef x y → odef (Ord (& x)) y
+ − 158 lemma1 {y} lt = subst ( λ k → k o< & x ) &iso (c<→o< {* y} {x} (d→∋ x lt))
+ − 159 lemma2 : {y : Ordinal} → odef (Ord (& x)) y → odef x y
+ − 160 lemma2 {y} lt = subst (λ k → odef k y ) *iso (o<→c< {y} {& x} lt )
+ − 161
+ − 162 -- avoiding lv != Zero error
+ − 163 orefl : { x : HOD } → { y : Ordinal } → & x ≡ y → & x ≡ y
+ − 164 orefl refl = refl
+ − 165
+ − 166 ==-iso : { x y : HOD } → od (* (& x)) == od (* (& y)) → od x == od y
+ − 167 ==-iso {x} {y} eq = record {
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+ − 168 eq→ = λ {z} d → lemma ( eq→ eq (subst (λ k → odef k z ) (sym *iso) d )) ;
+ − 169 eq← = λ {z} d → lemma ( eq← eq (subst (λ k → odef k z ) (sym *iso) d )) }
431
+ − 170 where
+ − 171 lemma : {x : HOD } {z : Ordinal } → odef (* (& x)) z → odef x z
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+ − 172 lemma {x} {z} d = subst (λ k → odef k z) (*iso) d
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+ − 173
+ − 174 =-iso : {x y : HOD } → (od x == od y) ≡ (od (* (& x)) == od y)
+ − 175 =-iso {_} {y} = cong ( λ k → od k == od y ) (sym *iso)
+ − 176
+ − 177 ord→== : { x y : HOD } → & x ≡ & y → od x == od y
+ − 178 ord→== {x} {y} eq = ==-iso (lemma (& x) (& y) (orefl eq)) where
+ − 179 lemma : ( ox oy : Ordinal ) → ox ≡ oy → od (* ox) == od (* oy)
+ − 180 lemma ox ox refl = ==-refl
+ − 181
+ − 182 o≡→== : { x y : Ordinal } → x ≡ y → od (* x) == od (* y)
+ − 183 o≡→== {x} {.x} refl = ==-refl
+ − 184
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+ − 185 *≡*→≡ : { x y : Ordinal } → * x ≡ * y → x ≡ y
+ − 186 *≡*→≡ eq = subst₂ (λ j k → j ≡ k ) &iso &iso ( cong (&) eq )
+ − 187
+ − 188 &≡&→≡ : { x y : HOD } → & x ≡ & y → x ≡ y
+ − 189 &≡&→≡ eq = subst₂ (λ j k → j ≡ k ) *iso *iso ( cong (*) eq )
+ − 190
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+ − 191 o∅≡od∅ : * (o∅ ) ≡ od∅
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+ − 192 o∅≡od∅ = ==→o≡ lemma where
+ − 193 lemma0 : {x : Ordinal} → odef (* o∅) x → odef od∅ x
+ − 194 lemma0 {x} lt with c<→o< {* x} {* o∅} (subst (λ k → odef (* o∅) k ) (sym &iso) lt)
+ − 195 ... | t = subst₂ (λ j k → j o< k ) &iso &iso t
+ − 196 lemma1 : {x : Ordinal} → odef od∅ x → odef (* o∅) x
+ − 197 lemma1 {x} lt = ⊥-elim (¬x<0 lt)
+ − 198 lemma : od (* o∅) == od od∅
+ − 199 lemma = record { eq→ = lemma0 ; eq← = lemma1 }
+ − 200
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+ − 201 ord-od∅ : & (od∅ ) ≡ o∅
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+ − 202 ord-od∅ = sym ( subst (λ k → k ≡ & (od∅ ) ) &iso (cong ( λ k → & k ) o∅≡od∅ ) )
+ − 203
+ − 204 ≡o∅→=od∅ : {x : HOD} → & x ≡ o∅ → od x == od od∅
+ − 205 ≡o∅→=od∅ {x} eq = record { eq→ = λ {y} lt → ⊥-elim ( ¬x<0 {y} (subst₂ (λ j k → j o< k ) &iso eq ( c<→o< {* y} {x} (d→∋ x lt))))
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+ − 206 ; eq← = λ {y} lt → ⊥-elim ( ¬x<0 lt )}
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+ − 207
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+ − 208 =od∅→≡o∅ : {x : HOD} → od x == od od∅ → & x ≡ o∅
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+ − 209 =od∅→≡o∅ {x} eq = trans (cong (λ k → & k ) (==→o≡ {x} {od∅} eq)) ord-od∅
+ − 210
448
+ − 211 ≡od∅→=od∅ : {x : HOD} → x ≡ od∅ → od x == od od∅
+ − 212 ≡od∅→=od∅ {x} eq = ≡o∅→=od∅ (subst (λ k → & x ≡ k ) ord-od∅ ( cong & eq ) )
+ − 213
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+ − 214 ∅0 : record { def = λ x → Lift n ⊥ } == od od∅
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+ − 215 eq→ ∅0 {w} (lift ())
+ − 216 eq← ∅0 {w} lt = lift (¬x<0 lt)
+ − 217
+ − 218 ∅< : { x y : HOD } → odef x (& y ) → ¬ ( od x == od od∅ )
+ − 219 ∅< {x} {y} d eq with eq→ (==-trans eq (==-sym ∅0) ) d
+ − 220 ∅< {x} {y} d eq | lift ()
450
+ − 221
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+ − 222 ¬x∋y→x≡od∅ : { x : HOD } → ({y : Ordinal } → ¬ odef x y ) → x ≡ od∅
+ − 223 ¬x∋y→x≡od∅ {x} nxy = ==→o≡ record { eq→ = λ {y} lt → ⊥-elim (nxy lt) ; eq← = λ {y} lt → ⊥-elim (¬x<0 lt) }
+ − 224
1148
+ − 225 0<P→ne : { x : HOD } → o∅ o< & x → ¬ ( od x == od od∅ )
+ − 226 0<P→ne {x} 0<x eq = ⊥-elim ( o<¬≡ (sym (=od∅→≡o∅ eq)) 0<x )
+ − 227
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+ − 228 ∈∅< : { x : HOD } {y : Ordinal } → odef x y → o∅ o< (& x)
+ − 229 ∈∅< {x} {y} d with trio< o∅ (& x)
+ − 230 ... | tri< a ¬b ¬c = a
+ − 231 ... | tri≈ ¬a b ¬c = ⊥-elim ( ∅< {x} {* y} (subst (λ k → odef x k ) (sym &iso) d ) ( ≡o∅→=od∅ (sym b) ) )
+ − 232 ... | tri> ¬a ¬b c = ⊥-elim ( ¬x<0 c )
+ − 233
431
+ − 234 ∅6 : { x : HOD } → ¬ ( x ∋ x ) -- no Russel paradox
+ − 235 ∅6 {x} x∋x = o<¬≡ refl ( c<→o< {x} {x} x∋x )
+ − 236
+ − 237 odef-iso : {A B : HOD } {x y : Ordinal } → x ≡ y → (odef A y → odef B y) → odef A x → odef B x
+ − 238 odef-iso refl t = t
+ − 239
+ − 240 is-o∅ : ( x : Ordinal ) → Dec ( x ≡ o∅ )
+ − 241 is-o∅ x with trio< x o∅
+ − 242 is-o∅ x | tri< a ¬b ¬c = no ¬b
+ − 243 is-o∅ x | tri≈ ¬a b ¬c = yes b
+ − 244 is-o∅ x | tri> ¬a ¬b c = no ¬b
+ − 245
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+ − 246 odef< : {b : Ordinal } { A : HOD } → odef A b → b o< & A
+ − 247 odef< {b} {A} ab = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) ab))
+ − 248
+ − 249 odef∧< : {A : HOD } {y : Ordinal} {n : Level } → {P : Set n} → odef A y ∧ P → y o< & A
+ − 250 odef∧< {A } {y} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p )))
+ − 251
431
+ − 252 -- the pair
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+ − 253 _,_ : HOD → HOD → HOD
431
+ − 254 x , y = record { od = record { def = λ t → (t ≡ & x ) ∨ ( t ≡ & y ) } ; odmax = omax (& x) (& y) ; <odmax = lemma } where
+ − 255 lemma : {t : Ordinal} → (t ≡ & x) ∨ (t ≡ & y) → t o< omax (& x) (& y)
+ − 256 lemma {t} (case1 refl) = omax-x _ _
+ − 257 lemma {t} (case2 refl) = omax-y _ _
+ − 258
+ − 259 pair<y : {x y : HOD } → y ∋ x → & (x , x) o< osuc (& y)
+ − 260 pair<y {x} {y} y∋x = ⊆→o≤ lemma where
+ − 261 lemma : {z : Ordinal} → def (od (x , x)) z → def (od y) z
+ − 262 lemma (case1 refl) = y∋x
+ − 263 lemma (case2 refl) = y∋x
+ − 264
688
+ − 265 -- another possible restriction. We require no minimality on odmax, so it may arbitrary larger.
431
+ − 266 odmax<& : { x y : HOD } → x ∋ y → Set n
+ − 267 odmax<& {x} {y} x∋y = odmax x o< & x
+ − 268
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+ − 269 in-codomain : (X : HOD ) → ( ψ : HOD → HOD ) → OD
431
+ − 270 in-codomain X ψ = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( odef X y ∧ ( x ≡ & (ψ (* y ))))) }
+ − 271
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+ − 272 _∩_ : ( A B : HOD ) → HOD
431
+ − 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))}
+ − 275
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+ − 276 _⊆_ : ( A B : HOD) → Set n
+ − 277 _⊆_ A B = { x : Ordinal } → odef A x → odef B x
431
+ − 278
+ − 279 infixr 220 _⊆_
+ − 280
+ − 281 -- if we have & (x , x) ≡ osuc (& x), ⊆→o≤ → c<→o<
+ − 282 ⊆→o≤→c<→o< : ({x : HOD} → & (x , x) ≡ osuc (& x) )
+ − 283 → ({y z : HOD } → ({x : Ordinal} → def (od y) x → def (od z) x ) → & y o< osuc (& z) )
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+ − 284 → {x y : HOD } → def (od y) ( & x ) → & x o< & y
431
+ − 285 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x with trio< (& x) (& y)
+ − 286 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x | tri< a ¬b ¬c = a
+ − 287 ⊆→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 )))
+ − 288 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x | tri> ¬a ¬b c =
+ − 289 ⊥-elim ( o<> (⊆→o≤ {x , x} {y} y⊆x,x ) lemma1 ) where
+ − 290 lemma : {z : Ordinal} → (z ≡ & x) ∨ (z ≡ & x) → & x ≡ z
+ − 291 lemma (case1 refl) = refl
+ − 292 lemma (case2 refl) = refl
+ − 293 y⊆x,x : {z : Ordinal} → def (od (x , x)) z → def (od y) z
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+ − 294 y⊆x,x {z} lt = subst (λ k → def (od y) k ) (lemma lt) y∋x
431
+ − 295 lemma1 : osuc (& y) o< & (x , x)
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+ − 296 lemma1 = subst (λ k → osuc (& y) o< k ) (sym (peq {x})) (osucc c )
431
+ − 297
+ − 298 ε-induction : { ψ : HOD → Set (suc n)}
+ − 299 → ( {x : HOD } → ({ y : HOD } → x ∋ y → ψ y ) → ψ x )
+ − 300 → (x : HOD ) → ψ x
+ − 301 ε-induction {ψ} ind x = subst (λ k → ψ k ) *iso (ε-induction-ord (osuc (& x)) <-osuc ) where
+ − 302 induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (* oy)) → ψ (* ox)
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+ − 303 induction ox prev = ind ( λ {y} lt → subst (λ k → ψ k ) *iso (prev (& y) (o<-subst (c<→o< lt) refl &iso )))
431
+ − 304 ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (* oy)
+ − 305 ε-induction-ord ox {oy} lt = TransFinite {λ oy → ψ (* oy)} induction oy
+ − 306
1109
+ − 307 ε-induction0 : { ψ : HOD → Set n}
+ − 308 → ( {x : HOD } → ({ y : HOD } → x ∋ y → ψ y ) → ψ x )
+ − 309 → (x : HOD ) → ψ x
+ − 310 ε-induction0 {ψ} ind x = subst (λ k → ψ k ) *iso (ε-induction-ord (osuc (& x)) <-osuc ) where
+ − 311 induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (* oy)) → ψ (* ox)
+ − 312 induction ox prev = ind ( λ {y} lt → subst (λ k → ψ k ) *iso (prev (& y) (o<-subst (c<→o< lt) refl &iso )))
+ − 313 ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (* oy)
+ − 314 ε-induction-ord ox {oy} lt = inOrdinal.TransFinite0 O {λ oy → ψ (* oy)} induction oy
+ − 315
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+ − 316 -- Open supreme upper bound leads a contradition, so we use domain restriction on sup
+ − 317 ¬open-sup : ( sup-o : (Ordinal → Ordinal ) → Ordinal) → ((ψ : Ordinal → Ordinal ) → (x : Ordinal) → ψ x o< sup-o ψ ) → ⊥
+ − 318 ¬open-sup sup-o sup-o< = o<> <-osuc (sup-o< next-ord (sup-o next-ord)) where
+ − 319 next-ord : Ordinal → Ordinal
+ − 320 next-ord x = osuc x
+ − 321
+ − 322 Select : (X : HOD ) → ((x : HOD ) → Set n ) → HOD
431
+ − 323 Select X ψ = record { od = record { def = λ x → ( odef X x ∧ ψ ( * x )) } ; odmax = odmax X ; <odmax = λ y → <odmax X (proj1 y) }
+ − 324
1095
+ − 325 _=h=_ : (x y : HOD) → Set n
+ − 326 x =h= y = od x == od y
+ − 327
+ − 328 record Own (A : HOD) (x : Ordinal) : Set n where
+ − 329 field
+ − 330 owner : Ordinal
+ − 331 ao : odef A owner
+ − 332 ox : odef (* owner) x
+ − 333
+ − 334 Union : HOD → HOD
+ − 335 Union U = record { od = record { def = λ x → Own U x } ; odmax = osuc (& U) ; <odmax = umax } where
+ − 336 umax : {y : Ordinal} → Own U y → y o< osuc (& U)
+ − 337 umax {y} uy = o<→≤ ( ordtrans (odef< (Own.ox uy)) (subst (λ k → k o< & U) (sym &iso) umax1) ) where
+ − 338 umax1 : Own.owner uy o< & U
+ − 339 umax1 = odef< (Own.ao uy)
+ − 340
+ − 341 union→ : (X z u : HOD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z
+ − 342 union→ X z u xx = record { owner = & u ; ao = proj1 xx ; ox = subst (λ k → odef k (& z)) (sym *iso) (proj2 xx) }
+ − 343 union← : (X z : HOD) (X∋z : Union X ∋ z) → ¬ ( (u : HOD ) → ¬ ((X ∋ u) ∧ (u ∋ z )))
+ − 344 union← X z UX∋z not = ⊥-elim ( not (* (Own.owner UX∋z)) ⟪ subst (λ k → odef X k) (sym &iso) ( Own.ao UX∋z) , Own.ox UX∋z ⟫ )
+ − 345
1303
+ − 346 --
+ − 347 --
+ − 348 --
+ − 349
1285
+ − 350 record RCod (COD : HOD) (ψ : HOD → HOD) : Set (suc n) where
+ − 351 field
+ − 352 ≤COD : ∀ {x : HOD } → ψ x ⊆ COD
+ − 353
1095
+ − 354 record Replaced (A : HOD) (ψ : Ordinal → Ordinal ) (x : Ordinal ) : Set n where
+ − 355 field
+ − 356 z : Ordinal
+ − 357 az : odef A z
+ − 358 x=ψz : x ≡ ψ z
+ − 359
1285
+ − 360 Replace : (D : HOD) → (ψ : HOD → HOD) → {C : HOD} → RCod C ψ → HOD
+ − 361 Replace X ψ {C} rc = record { od = record { def = λ x → Replaced X (λ z → & (ψ (* z))) x } ; odmax = osuc (& C)
+ − 362 ; <odmax = rmax< } where
+ − 363 rmax< : {y : Ordinal} → Replaced X (λ z → & (ψ (* z))) y → y o< osuc (& C)
+ − 364 rmax< {y} lt = subst (λ k → k o< osuc (& C)) r01 ( ⊆→o≤ (RCod.≤COD rc) ) where
1095
+ − 365 r01 : & (ψ ( * (Replaced.z lt ) )) ≡ y
+ − 366 r01 = sym (Replaced.x=ψz lt )
+ − 367
1285
+ − 368 replacement← : {ψ : HOD → HOD} (X x : HOD) → X ∋ x → {C : HOD} → (rc : RCod C ψ) → Replace X ψ rc ∋ ψ x
+ − 369 replacement← {ψ} X x lt {C} rc = record { z = & x ; az = lt ; x=ψz = cong (λ k → & (ψ k)) (sym *iso) }
+ − 370 replacement→ : {ψ : HOD → HOD} (X x : HOD) → {C : HOD} → (rc : RCod C ψ ) → (lt : Replace X ψ rc ∋ x)
+ − 371 → ¬ ( (y : HOD) → ¬ (x =h= ψ y))
+ − 372 replacement→ {ψ} X x {C} rc lt eq = eq (* (Replaced.z lt)) (ord→== (Replaced.x=ψz lt))
431
+ − 373
+ − 374 --
1091
+ − 375 -- If we have LEM, Replace' is equivalent to Replace
431
+ − 376 --
1095
+ − 377
1285
+ − 378 record RXCod (X COD : HOD) (ψ : (x : HOD) → X ∋ x → HOD) : Set (suc n) where
+ − 379 field
+ − 380 ≤COD : ∀ {x : HOD } → (lt : X ∋ x) → ψ x lt ⊆ COD
+ − 381
1095
+ − 382 record Replaced1 (A : HOD) (ψ : (x : Ordinal ) → odef A x → Ordinal ) (x : Ordinal ) : Set n where
+ − 383 field
+ − 384 z : Ordinal
+ − 385 az : odef A z
+ − 386 x=ψz : x ≡ ψ z az
431
+ − 387
1285
+ − 388 Replace' : (X : HOD) → (ψ : (x : HOD) → X ∋ x → HOD) → {C : HOD} → RXCod X C ψ → HOD
+ − 389 Replace' X ψ {C} rc = record { od = record { def = λ x → Replaced1 X (λ z xz → & (ψ (* z) (subst (λ k → odef X k) (sym &iso) xz) )) x } ; odmax = osuc (& C) ; <odmax = rmax< } where
+ − 390 rmax< : {y : Ordinal} → Replaced1 X (λ z xz → & (ψ (* z) (subst (λ k → odef X k) (sym &iso) xz) )) y → y o< osuc (& C)
+ − 391 rmax< {y} lt = subst (λ k → k o< osuc (& C)) r01 ( ⊆→o≤ (RXCod.≤COD rc (subst (λ k → odef X k) (sym &iso) (Replaced1.az lt) ))) where
1095
+ − 392 r01 : & (ψ ( * (Replaced1.z lt ) ) (subst (λ k → odef X k) (sym &iso) (Replaced1.az lt) )) ≡ y
+ − 393 r01 = sym (Replaced1.x=ψz lt )
+ − 394
1285
+ − 395 cod-conv : (X : HOD) → (ψ : (x : HOD) → X ∋ x → HOD) → {C : HOD} → (rc : RXCod X C ψ )
+ − 396 → RXCod (* (& X)) C (λ y xy → ψ y (subst (λ k → k ∋ y) *iso xy))
+ − 397 cod-conv X ψ {C} rc = record { ≤COD = λ {x} lt → RXCod.≤COD rc (subst (λ k → odef k (& x)) *iso lt) }
1218
+ − 398
1294
+ − 399 Replace'-iso : {X Y : HOD} → {fx : (x : HOD) → X ∋ x → HOD} {fy : (x : HOD) → Y ∋ x → HOD}
+ − 400 → {CX : HOD} → (rcx : RXCod X CX fx ) → {CY : HOD} → (rcy : RXCod Y CY fy )
+ − 401 → X ≡ Y → ( (x : HOD) → (xx : X ∋ x ) → (yy : Y ∋ x ) → fx _ xx ≡ fy _ yy )
+ − 402 → Replace' X fx rcx ≡ Replace' Y fy rcy
+ − 403 Replace'-iso {X} {X} {fx} {fy} _ _ refl eq = ==→o≡ record { eq→ = ri0 ; eq← = ri1 } where
+ − 404 ri0 : {x : Ordinal} → Replaced1 X (λ z xz → & (fx (* z) (subst (odef X) (sym &iso) xz))) x
+ − 405 → Replaced1 X (λ z xz → & (fy (* z) (subst (odef X) (sym &iso) xz))) x
+ − 406 ri0 {x} record { z = z ; az = az ; x=ψz = x=ψz } = record { z = z ; az = az ; x=ψz = trans x=ψz (cong (&) ( eq _ xz xz )) } where
+ − 407 xz : X ∋ * z
+ − 408 xz = subst (λ k → odef X k ) (sym &iso) az
+ − 409 ri1 : {x : Ordinal} → Replaced1 X (λ z xz → & (fy (* z) (subst (odef X) (sym &iso) xz))) x
+ − 410 → Replaced1 X (λ z xz → & (fx (* z) (subst (odef X) (sym &iso) xz))) x
+ − 411 ri1 {x} record { z = z ; az = az ; x=ψz = x=ψz } = record { z = z ; az = az ; x=ψz = trans x=ψz (cong (&) (sym ( eq _ xz xz ))) } where
+ − 412 xz : X ∋ * z
+ − 413 xz = subst (λ k → odef X k ) (sym &iso) az
+ − 414
+ − 415 Replace'-iso1 : (X : HOD) → (ψ : (x : HOD) → X ∋ x → HOD) → {C : HOD} → (rc : RXCod X C ψ )
1285
+ − 416 → Replace' (* (& X)) (λ y xy → ψ y (subst (λ k → k ∋ y ) *iso xy) ) (cod-conv X ψ rc)
+ − 417 ≡ Replace' X ( λ y xy → ψ y xy ) rc
1294
+ − 418 Replace'-iso1 X ψ rc = Replace'-iso {* (& X)} {X} {λ y xy → ψ y (subst (λ k → k ∋ y ) *iso xy) } { λ y xy → ψ y xy }
+ − 419 (cod-conv X ψ rc) rc
+ − 420 *iso (λ x xx yx → fi00 x xx yx ) where
+ − 421 fi00 : (x : HOD ) → (xx : (* (& X)) ∋ x ) → (yx : X ∋ x) → ψ x (subst (λ k → k ∋ x) *iso xx) ≡ ψ x yx
+ − 422 fi00 x xx yx = cong (λ k → ψ x k ) ( HE.≅-to-≡ ( ∋-irr {X} {& x} (subst (λ k → k ∋ x) *iso xx) yx ) )
1218
+ − 423
1095
+ − 424 -- replacement←1 : {ψ : HOD → HOD} (X x : HOD) → X ∋ x → Replace1 X ψ ∋ ψ x
+ − 425 -- replacement←1 {ψ} X x lt = record { z = & x ; az = lt ; x=ψz = cong (λ k → & (ψ k)) (sym *iso) }
+ − 426 -- replacement→1 : {ψ : HOD → HOD} (X x : HOD) → (lt : Replace1 X ψ ∋ x) → ¬ ( (y : HOD) → ¬ (x =h= ψ y))
+ − 427 -- replacement→1 {ψ} X x lt eq = eq (* (Replaced.z lt)) (ord→== (Replaced.x=ψz lt))
+ − 428
431
+ − 429 _∈_ : ( A B : HOD ) → Set n
+ − 430 A ∈ B = B ∋ A
+ − 431
1095
+ − 432 Power : HOD → HOD
+ − 433 Power A = record { od = record { def = λ x → ( ( z : Ordinal) → odef (* x) z → odef A z ) } ; odmax = osuc (& A)
+ − 434 ; <odmax = p00 } where
+ − 435 p00 : {y : Ordinal} → ((z : Ordinal) → odef (* y) z → odef A z) → y o< osuc (& A)
+ − 436 p00 {y} y⊆A = p01 where
+ − 437 p01 : y o≤ & A
+ − 438 p01 = subst (λ k → k o≤ & A) &iso ( ⊆→o≤ (λ {x} yx → y⊆A x yx ))
431
+ − 439
1095
+ − 440 power→ : ( A t : HOD) → Power A ∋ t → {x : HOD} → t ∋ x → A ∋ x
+ − 441 power→ A t P∋t {x} t∋x = P∋t (& x) (subst (λ k → odef k (& x) ) (sym *iso) t∋x )
+ − 442
+ − 443 power← : (A t : HOD) → ({x : HOD} → (t ∋ x → A ∋ x)) → Power A ∋ t
+ − 444 power← A t t⊆A z xz = subst (λ k → odef A k ) &iso ( t⊆A (subst₂ (λ j k → odef j k) *iso (sym &iso) xz ))
+ − 445
1180
+ − 446 Intersection : (X : HOD ) → HOD -- ∩ X
1186
+ − 447 Intersection X = record { od = record { def = λ x → (x o≤ & X ) ∧ ( {y : Ordinal} → odef X y → odef (* y) x )} ; odmax = osuc (& X) ; <odmax = λ lt → proj1 lt }
1180
+ − 448
1300
+ − 449 empty : (x : HOD ) → ¬ (od∅ ∋ x)
+ − 450 empty x = ¬x<0
+ − 451
1180
+ − 452
431
+ − 453 -- {_} : ZFSet → ZFSet
+ − 454 -- { x } = ( x , x ) -- better to use (x , x) directly
+ − 455
1300
+ − 456 data Omega-d : ( x : Ordinal ) → Set n where
+ − 457 iφ : Omega-d o∅
+ − 458 isuc : {x : Ordinal } → Omega-d x →
+ − 459 Omega-d (& ( Union (* x , (* x , * x ) ) ))
431
+ − 460
+ − 461 -- ω can be diverged in our case, since we have no restriction on the corresponding ordinal of a pair.
1300
+ − 462 -- We simply assumes Omega-d y has a maximum.
1091
+ − 463 --
431
+ − 464 -- This means that many of OD may not be HODs because of the & mapping divergence.
1300
+ − 465 -- We should have some axioms to prevent this .
1091
+ − 466 --
1300
+ − 467
+ − 468 Omega-od : OD
+ − 469 Omega-od = record { def = λ x → Omega-d x }
+ − 470
+ − 471 o∅<x : {x : Ordinal} → o∅ o≤ x
+ − 472 o∅<x {x} with trio< o∅ x
+ − 473 ... | tri< a ¬b ¬c = o<→≤ a
+ − 474 ... | tri≈ ¬a b ¬c = o≤-refl0 b
+ − 475 ... | tri> ¬a ¬b c = ⊥-elim (¬x<0 c)
431
+ − 476
1300
+ − 477 ¬0=ux : {x : HOD} → ¬ o∅ ≡ & (Union ( x , ( x , x)))
+ − 478 ¬0=ux {x} eq = ⊥-elim ( o<¬≡ eq (ordtrans≤-< o∅<x (subst (λ k → k o< & (Union (x , (x , x)))) &iso (c<→o< lemma ) ))) where
+ − 479 lemma : Own (x , (x , x)) (& ( * (& x )))
+ − 480 lemma = record { owner = _ ; ao = case2 refl ; ox = subst₂ (λ j k → odef j k ) (sym *iso) (sym &iso) (case1 refl) }
1297
+ − 481
1300
+ − 482 ux-2cases : {x y : HOD } → Union ( x , ( x , x)) ∋ y → ( x ≡ y ) ∨ ( x ∋ y )
+ − 483 ux-2cases {x} {y} record { owner = owner ; ao = (case1 eq) ; ox = ox } = case2 (subst (λ k → odef k (& y)) (trans (cong (*) eq) *iso) ox)
+ − 484 ux-2cases {x} {y} record { owner = owner ; ao = (case2 eq) ; ox = ox } with subst (λ k → odef k (& y)) (trans (cong (*) eq) *iso) ox
+ − 485 ... | case1 eq = case1 (sym (&≡&→≡ eq))
+ − 486 ... | case2 eq = case1 (sym (&≡&→≡ eq))
+ − 487
+ − 488 ux-transitve : {x y : HOD} → x ∋ y → Union ( x , ( x , x)) ∋ y
+ − 489 ux-transitve {x} {y} ox = record { owner = _ ; ao = case1 refl ; ox = subst (λ k → odef k (& y)) (sym *iso) ox }
+ − 490
+ − 491 --
+ − 492 -- Possible Ordinal Limit
+ − 493 --
+ − 494
+ − 495 -- our Ordinals is greater than Union ( x , ( x , x)) transitive closure
+ − 496 --
1297
+ − 497 record ODAxiom-ho< : Set (suc n) where
+ − 498 field
+ − 499 omega : Ordinal
1300
+ − 500 ho< : {x : Ordinal } → Omega-d x → x o< omega
1297
+ − 501
+ − 502 postulate
+ − 503 odaxion-ho< : ODAxiom-ho<
+ − 504
+ − 505 open ODAxiom-ho< odaxion-ho<
+ − 506
1300
+ − 507 Omega : HOD
+ − 508 Omega = record { od = record { def = λ x → Omega-d x } ; odmax = omega ; <odmax = ho<}
431
+ − 509
1300
+ − 510 infinity∅ : Omega ∋ od∅
+ − 511 infinity∅ = subst (λ k → odef Omega k ) lemma iφ where
+ − 512 lemma : o∅ ≡ & od∅
+ − 513 lemma = let open ≡-Reasoning in begin
+ − 514 o∅
+ − 515 ≡⟨ sym &iso ⟩
+ − 516 & ( * o∅ )
+ − 517 ≡⟨ cong ( λ k → & k ) o∅≡od∅ ⟩
+ − 518 & od∅
+ − 519 ∎
+ − 520
+ − 521 infinity : (x : HOD) → Omega ∋ x → Omega ∋ Union (x , (x , x ))
+ − 522 infinity x lt = subst (λ k → odef Omega k ) lemma (isuc {& x} lt) where
+ − 523 lemma : & (Union (* (& x) , (* (& x) , * (& x))))
+ − 524 ≡ & (Union (x , (x , x)))
+ − 525 lemma = cong (λ k → & (Union ( k , ( k , k ) ))) *iso
431
+ − 526
1091
+ − 527 pair→ : ( x y t : HOD ) → (x , y) ∋ t → ( t =h= x ) ∨ ( t =h= y )
431
+ − 528 pair→ x y t (case1 t≡x ) = case1 (subst₂ (λ j k → j =h= k ) *iso *iso (o≡→== t≡x ))
+ − 529 pair→ x y t (case2 t≡y ) = case2 (subst₂ (λ j k → j =h= k ) *iso *iso (o≡→== t≡y ))
+ − 530
1091
+ − 531 pair← : ( x y t : HOD ) → ( t =h= x ) ∨ ( t =h= y ) → (x , y) ∋ t
431
+ − 532 pair← x y t (case1 t=h=x) = case1 (cong (λ k → & k ) (==→o≡ t=h=x))
+ − 533 pair← x y t (case2 t=h=y) = case2 (cong (λ k → & k ) (==→o≡ t=h=y))
+ − 534
1091
+ − 535 o<→c< : {x y : Ordinal } → x o< y → (Ord x) ⊆ (Ord y)
1096
+ − 536 o<→c< lt {z} ox = ordtrans ox lt
431
+ − 537
+ − 538 ⊆→o< : {x y : Ordinal } → (Ord x) ⊆ (Ord y) → x o< osuc y
1091
+ − 539 ⊆→o< {x} {y} lt with trio< x y
431
+ − 540 ⊆→o< {x} {y} lt | tri< a ¬b ¬c = ordtrans a <-osuc
+ − 541 ⊆→o< {x} {y} lt | tri≈ ¬a b ¬c = subst ( λ k → k o< osuc y) (sym b) <-osuc
1096
+ − 542 ⊆→o< {x} {y} lt | tri> ¬a ¬b c with lt (o<-subst c (sym &iso) refl )
431
+ − 543 ... | ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt &iso refl ))
+ − 544
+ − 545 ψiso : {ψ : HOD → Set n} {x y : HOD } → ψ x → x ≡ y → ψ y
+ − 546 ψiso {ψ} t refl = t
+ − 547 selection : {ψ : HOD → Set n} {X y : HOD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y)
+ − 548 selection {ψ} {X} {y} = ⟪
+ − 549 ( λ cond → ⟪ proj1 cond , ψiso {ψ} (proj2 cond) (sym *iso) ⟫ )
+ − 550 , ( λ select → ⟪ proj1 select , ψiso {ψ} (proj2 select) *iso ⟫ )
+ − 551 ⟫
+ − 552
1091
+ − 553 selection-in-domain : {ψ : HOD → Set n} {X y : HOD} → Select X ψ ∋ y → X ∋ y
431
+ − 554 selection-in-domain {ψ} {X} {y} lt = proj1 ((proj2 (selection {ψ} {X} )) lt)
+ − 555
+ − 556 ---
+ − 557 --- Power Set
+ − 558 ---
+ − 559 --- First consider ordinals in HOD
+ − 560 ---
+ − 561 --- A ∩ x = record { def = λ y → odef A y ∧ odef x y } subset of A
+ − 562 --
+ − 563 --
+ − 564 ∩-≡ : { a b : HOD } → ({x : HOD } → (a ∋ x → b ∋ x)) → a =h= ( b ∩ a )
+ − 565 ∩-≡ {a} {b} inc = record {
+ − 566 eq→ = λ {x} x<a → ⟪ (subst (λ k → odef b k ) &iso (inc (d→∋ a x<a))) , x<a ⟫ ;
+ − 567 eq← = λ {x} x<a∩b → proj2 x<a∩b }
+ − 568
+ − 569 extensionality0 : {A B : HOD } → ((z : HOD) → (A ∋ z) ⇔ (B ∋ z)) → A =h= B
1091
+ − 570 eq→ (extensionality0 {A} {B} eq ) {x} d = odef-iso {A} {B} (sym &iso) (proj1 (eq (* x))) d
+ − 571 eq← (extensionality0 {A} {B} eq ) {x} d = odef-iso {B} {A} (sym &iso) (proj2 (eq (* x))) d
431
+ − 572
+ − 573 extensionality : {A B w : HOD } → ((z : HOD ) → (A ∋ z) ⇔ (B ∋ z)) → (w ∋ A) ⇔ (w ∋ B)
+ − 574 proj1 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) ( ==→o≡ (extensionality0 {A} {B} eq) ) d
1091
+ − 575 proj2 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) (sym ( ==→o≡ (extensionality0 {A} {B} eq) )) d
431
+ − 576
1284
+ − 577 open import zf
+ − 578
1285
+ − 579 record ODAxiom-sup : Set (suc n) where
+ − 580 field
+ − 581 sup-o : (A : HOD) → ( ( x : Ordinal ) → def (od A) x → Ordinal ) → Ordinal -- required in Replace
+ − 582 sup-o≤ : (A : HOD) → { ψ : ( x : Ordinal ) → def (od A) x → Ordinal }
+ − 583 → ∀ {x : Ordinal } → (lt : def (od A) x ) → ψ x lt o≤ sup-o A ψ
+ − 584 sup-c≤ : (ψ : HOD → HOD) → {X x : HOD} → def (od X) (& x) → & (ψ x) o≤ (sup-o X (λ y X∋y → & (ψ (* y))))
+ − 585 sup-c≤ ψ {X} {x} lt = subst (λ k → & (ψ k) o< _ ) *iso (sup-o≤ X lt )
+ − 586
+ − 587 -- sup-o may contradict
+ − 588 -- If we have open monotonic function in Ordinal, there is no sup-o.
+ − 589 -- for example, if we may have countable sequence of Ordinal, which contains some ordinal larger than any given Ordinal.
+ − 590 -- This happens when we have a coutable model. In this case, we have to have codomain restriction in Replacement axiom.
+ − 591 -- that is, Replacement axiom does not create new ZF set.
+ − 592
+ − 593 open ODAxiom-sup
+ − 594
+ − 595 ZFReplace : ODAxiom-sup → HOD → (HOD → HOD) → HOD
+ − 596 ZFReplace os X ψ = record { od = record { def = λ x → Replaced X (λ z → & (ψ (* z))) x } ; odmax = rmax ; <odmax = rmax< } where
+ − 597 rmax : Ordinal
+ − 598 rmax = osuc ( sup-o os X (λ y X∋y → & (ψ (* y) )) )
+ − 599 rmax< : {y : Ordinal} → Replaced X (λ z → & (ψ (* z))) y → y o< rmax
+ − 600 rmax< {y} lt = subst (λ k → k o< rmax) r01 ( sup-o≤ os X (Replaced.az lt) ) where
+ − 601 r01 : & (ψ ( * (Replaced.z lt ) )) ≡ y
+ − 602 r01 = sym (Replaced.x=ψz lt )
+ − 603
+ − 604 zf-replacement← : (os : ODAxiom-sup) → {ψ : HOD → HOD} (X x : HOD) → X ∋ x → ZFReplace os X ψ ∋ ψ x
+ − 605 zf-replacement← os {ψ} X x lt = record { z = & x ; az = lt ; x=ψz = cong (λ k → & (ψ k)) (sym *iso) }
+ − 606 zf-replacement→ : (os : ODAxiom-sup ) → {ψ : HOD → HOD} (X x : HOD) → (lt : ZFReplace os X ψ ∋ x) → ¬ ( (y : HOD) → ¬ (x =h= ψ y))
+ − 607 zf-replacement→ os {ψ} X x lt eq = eq (* (Replaced.z lt)) (ord→== (Replaced.x=ψz lt))
+ − 608
1300
+ − 609 isZF : (os : ODAxiom-sup) → IsZF HOD _∋_ _=h=_ od∅ _,_ Union Power Select (ZFReplace os) Omega
1285
+ − 610 isZF os = record {
431
+ − 611 isEquivalence = record { refl = ==-refl ; sym = ==-sym; trans = ==-trans }
+ − 612 ; pair→ = pair→
+ − 613 ; pair← = pair←
+ − 614 ; union→ = union→
+ − 615 ; union← = union←
+ − 616 ; empty = empty
1091
+ − 617 ; power→ = power→
+ − 618 ; power← = power←
+ − 619 ; extensionality = λ {A} {B} {w} → extensionality {A} {B} {w}
431
+ − 620 ; ε-induction = ε-induction
+ − 621 ; infinity∅ = infinity∅
+ − 622 ; infinity = infinity
+ − 623 ; selection = λ {X} {ψ} {y} → selection {X} {ψ} {y}
1285
+ − 624 ; replacement← = zf-replacement← os
+ − 625 ; replacement→ = λ {ψ} → zf-replacement→ os {ψ}
1091
+ − 626 }
431
+ − 627
1285
+ − 628 HOD→ZF : ODAxiom-sup → ZF
+ − 629 HOD→ZF os = record {
1091
+ − 630 ZFSet = HOD
+ − 631 ; _∋_ = _∋_
+ − 632 ; _≈_ = _=h=_
431
+ − 633 ; ∅ = od∅
+ − 634 ; _,_ = _,_
+ − 635 ; Union = Union
+ − 636 ; Power = Power
+ − 637 ; Select = Select
1285
+ − 638 ; Replace = ZFReplace os
1300
+ − 639 ; infinite = Omega
1285
+ − 640 ; isZF = isZF os
1091
+ − 641 }
431
+ − 642
1091
+ − 643