Mercurial > hg > Members > kono > Proof > ZF-in-agda
diff OD.agda @ 334:ba3ebb9a16c6 release
HOD
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sun, 05 Jul 2020 16:59:13 +0900 |
| parents | 0faa7120e4b5 |
| children | daafa2213dd2 |
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--- a/OD.agda Sun Jun 07 20:35:14 2020 +0900 +++ b/OD.agda Sun Jul 05 16:59:13 2020 +0900 @@ -53,28 +53,27 @@ eq← ( ⇔→== {x} {y} eq ) {z} m = proj2 eq m -- next assumptions are our axiom --- it defines a subset of OD, which is called HOD, usually defined as +-- +-- OD is an equation on Ordinals, so it contains Ordinals. If these Ordinals have one-to-one +-- correspondence to the OD then the OD looks like a ZF Set. +-- +-- If all ZF Set have supreme upper bound, the solutions of OD have to be bounded, i.e. +-- bbounded ODs are ZF Set. Unbounded ODs are classes. +-- +-- In classical Set Theory, HOD is used, as a subset of OD, -- HOD = { x | TC x ⊆ OD } --- where TC x is a transitive clusure of x, i.e. Union of all elemnts of all subset of x - -record ODAxiom : Set (suc n) where - -- OD can be iso to a subset of Ordinal ( by means of Godel Set ) - field - od→ord : OD → Ordinal - ord→od : Ordinal → OD - c<→o< : {x y : OD } → def y ( od→ord x ) → od→ord x o< od→ord y - oiso : {x : OD } → ord→od ( od→ord x ) ≡ x - diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x - ==→o≡ : { x y : OD } → (x == y) → x ≡ y - -- supermum as Replacement Axiom ( corresponding Ordinal of OD has maximum ) - sup-o : ( OD → Ordinal ) → Ordinal - sup-o< : { ψ : OD → Ordinal } → ∀ {x : OD } → ψ x o< sup-o ψ - -- contra-position of mimimulity of supermum required in Power Set Axiom which we don't use - -- sup-x : {n : Level } → ( OD → Ordinal ) → Ordinal - -- sup-lb : {n : Level } → { ψ : OD → Ordinal } → {z : Ordinal } → z o< sup-o ψ → z o< osuc (ψ (sup-x ψ)) - -postulate odAxiom : ODAxiom -open ODAxiom odAxiom +-- where TC x is a transitive clusure of x, i.e. Union of all elemnts of all subset of x. +-- This is not possible because we don't have V yet. So we assumes HODs are bounded OD. +-- +-- We also assumes HODs are isomorphic to Ordinals, which is ususally proved by Goedel number tricks. +-- There two contraints on the HOD order, one is ∋, the other one is ⊂. +-- ODs have an ovbious maximum, but Ordinals are not, but HOD has no maximum because there is an aribtrary +-- bound on each HOD. +-- +-- In classical Set Theory, sup is defined by Uion, since we are working on constructive logic, +-- we need explict assumption on sup. +-- +-- ==→o≡ is necessary to prove axiom of extensionality. data One : Set n where OneObj : One @@ -83,100 +82,125 @@ Ords : OD Ords = record { def = λ x → One } -maxod : {x : OD} → od→ord x o< od→ord Ords -maxod {x} = c<→o< OneObj +record HOD : Set (suc n) where + field + od : OD + odmax : Ordinal + <odmax : {y : Ordinal} → def od y → y o< odmax + +open HOD + +record ODAxiom : Set (suc n) where + field + -- HOD is isomorphic to Ordinal (by means of Goedel number) + od→ord : HOD → Ordinal + ord→od : Ordinal → HOD + c<→o< : {x y : HOD } → def (od y) ( od→ord x ) → od→ord x o< od→ord y + ⊆→o≤ : {y z : HOD } → ({x : Ordinal} → def (od y) x → def (od z) x ) → od→ord y o< osuc (od→ord z) + oiso : {x : HOD } → ord→od ( od→ord x ) ≡ x + diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x + ==→o≡ : { x y : HOD } → (od x == od y) → x ≡ y + sup-o : (A : HOD) → (( x : Ordinal ) → def (od A) x → Ordinal ) → Ordinal + 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 + +postulate odAxiom : ODAxiom +open ODAxiom odAxiom + +-- maxod : {x : OD} → od→ord x o< od→ord Ords +-- maxod {x} = c<→o< OneObj + +-- we have not this contradiction +-- bad-bad : ⊥ +-- bad-bad = osuc-< <-osuc (c<→o< { record { od = record { def = λ x → One }; <odmax = {!!} } } OneObj) -- Ordinal in OD ( and ZFSet ) Transitive Set -Ord : ( a : Ordinal ) → OD -Ord a = record { def = λ y → y o< a } +Ord : ( a : Ordinal ) → HOD +Ord a = record { od = record { def = λ y → y o< a } ; odmax = a ; <odmax = lemma } where + lemma : {x : Ordinal} → x o< a → x o< a + lemma {x} lt = lt + +od∅ : HOD +od∅ = Ord o∅ -od∅ : OD -od∅ = Ord o∅ +odef : HOD → Ordinal → Set n +odef A x = def ( od A ) x + +o<→c<→HOD=Ord : ( {x y : Ordinal } → x o< y → odef (ord→od y) x ) → {x : HOD } → x ≡ Ord (od→ord x) +o<→c<→HOD=Ord o<→c< {x} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where + lemma1 : {y : Ordinal} → odef x y → odef (Ord (od→ord x)) y + 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)) + lemma2 : {y : Ordinal} → odef (Ord (od→ord x)) y → odef x y + lemma2 {y} lt = subst (λ k → odef k y ) oiso (o<→c< {y} {od→ord x} lt ) -o<→c<→OD=Ord : ( {x y : Ordinal } → x o< y → def (ord→od y) x ) → {x : OD } → x ≡ Ord (od→ord x) -o<→c<→OD=Ord o<→c< {x} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where - lemma1 : {y : Ordinal} → def x y → def (Ord (od→ord x)) y - lemma1 {y} lt = subst ( λ k → k o< od→ord x ) diso (c<→o< {ord→od y} {x} (subst (λ k → def x k ) (sym diso) lt)) - lemma2 : {y : Ordinal} → def (Ord (od→ord x)) y → def x y - lemma2 {y} lt = subst (λ k → def k y ) oiso (o<→c< {y} {od→ord x} lt ) +_∋_ : ( a x : HOD ) → Set n +_∋_ a x = odef a ( od→ord x ) -_∋_ : ( a x : OD ) → Set n -_∋_ a x = def a ( od→ord x ) - -_c<_ : ( x a : OD ) → Set n +_c<_ : ( x a : HOD ) → Set n x c< a = a ∋ x -cseq : {n : Level} → OD → OD -cseq x = record { def = λ y → def x (osuc y) } where - -def-subst : {Z : OD } {X : Ordinal }{z : OD } {x : Ordinal }→ def Z X → Z ≡ z → X ≡ x → def z x -def-subst df refl refl = df +cseq : {n : Level} → HOD → HOD +cseq x = record { od = record { def = λ y → odef x (osuc y) } ; odmax = osuc (odmax x) ; <odmax = lemma } where + lemma : {y : Ordinal} → def (od x) (osuc y) → y o< osuc (odmax x) + lemma {y} lt = ordtrans <-osuc (ordtrans (<odmax x lt) <-osuc ) -sup-od : ( OD → OD ) → OD -sup-od ψ = Ord ( sup-o ( λ x → od→ord (ψ x)) ) +odef-subst : {Z : HOD } {X : Ordinal }{z : HOD } {x : Ordinal }→ odef Z X → Z ≡ z → X ≡ x → odef z x +odef-subst df refl refl = df -sup-c< : ( ψ : OD → OD ) → ∀ {x : OD } → def ( sup-od ψ ) (od→ord ( ψ x )) -sup-c< ψ {x} = def-subst {_} {_} {Ord ( sup-o ( λ x → od→ord (ψ x)) )} {od→ord ( ψ x )} - lemma refl (cong ( λ k → od→ord (ψ k) ) oiso) where - lemma : od→ord (ψ (ord→od (od→ord x))) o< sup-o (λ x → od→ord (ψ x)) - lemma = subst₂ (λ j k → j o< k ) refl diso (o<-subst (sup-o< ) refl (sym diso) ) - -otrans : {n : Level} {a x y : Ordinal } → def (Ord a) x → def (Ord x) y → def (Ord a) y +otrans : {n : Level} {a x y : Ordinal } → odef (Ord a) x → odef (Ord x) y → odef (Ord a) y otrans x<a y<x = ordtrans y<x x<a -def→o< : {X : OD } → {x : Ordinal } → def X x → x o< od→ord X -def→o< {X} {x} lt = o<-subst {_} {_} {x} {od→ord X} ( c<→o< ( def-subst {X} {x} lt (sym oiso) (sym diso) )) diso diso - +odef→o< : {X : HOD } → {x : Ordinal } → odef X x → x o< od→ord X +odef→o< {X} {x} lt = o<-subst {_} {_} {x} {od→ord X} ( c<→o< ( odef-subst {X} {x} lt (sym oiso) (sym diso) )) diso diso -- avoiding lv != Zero error -orefl : { x : OD } → { y : Ordinal } → od→ord x ≡ y → od→ord x ≡ y +orefl : { x : HOD } → { y : Ordinal } → od→ord x ≡ y → od→ord x ≡ y orefl refl = refl -==-iso : { x y : OD } → ord→od (od→ord x) == ord→od (od→ord y) → x == y +==-iso : { x y : HOD } → od (ord→od (od→ord x)) == od (ord→od (od→ord y)) → od x == od y ==-iso {x} {y} eq = record { - eq→ = λ d → lemma ( eq→ eq (def-subst d (sym oiso) refl )) ; - eq← = λ d → lemma ( eq← eq (def-subst d (sym oiso) refl )) } + eq→ = λ d → lemma ( eq→ eq (odef-subst d (sym oiso) refl )) ; + eq← = λ d → lemma ( eq← eq (odef-subst d (sym oiso) refl )) } where - lemma : {x : OD } {z : Ordinal } → def (ord→od (od→ord x)) z → def x z - lemma {x} {z} d = def-subst d oiso refl + lemma : {x : HOD } {z : Ordinal } → odef (ord→od (od→ord x)) z → odef x z + lemma {x} {z} d = odef-subst d oiso refl -=-iso : {x y : OD } → (x == y) ≡ (ord→od (od→ord x) == y) -=-iso {_} {y} = cong ( λ k → k == y ) (sym oiso) +=-iso : {x y : HOD } → (od x == od y) ≡ (od (ord→od (od→ord x)) == od y) +=-iso {_} {y} = cong ( λ k → od k == od y ) (sym oiso) -ord→== : { x y : OD } → od→ord x ≡ od→ord y → x == y +ord→== : { x y : HOD } → od→ord x ≡ od→ord y → od x == od y ord→== {x} {y} eq = ==-iso (lemma (od→ord x) (od→ord y) (orefl eq)) where - lemma : ( ox oy : Ordinal ) → ox ≡ oy → (ord→od ox) == (ord→od oy) + lemma : ( ox oy : Ordinal ) → ox ≡ oy → od (ord→od ox) == od (ord→od oy) lemma ox ox refl = ==-refl -o≡→== : { x y : Ordinal } → x ≡ y → ord→od x == ord→od y +o≡→== : { x y : Ordinal } → x ≡ y → od (ord→od x) == od (ord→od y) o≡→== {x} {.x} refl = ==-refl o∅≡od∅ : ord→od (o∅ ) ≡ od∅ o∅≡od∅ = ==→o≡ lemma where - lemma0 : {x : Ordinal} → def (ord→od o∅) x → def od∅ x - lemma0 {x} lt = o<-subst (c<→o< {ord→od x} {ord→od o∅} (def-subst {ord→od o∅} {x} lt refl (sym diso)) ) diso diso - lemma1 : {x : Ordinal} → def od∅ x → def (ord→od o∅) x + lemma0 : {x : Ordinal} → odef (ord→od o∅) x → odef od∅ x + 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 + lemma1 : {x : Ordinal} → odef od∅ x → odef (ord→od o∅) x lemma1 {x} lt = ⊥-elim (¬x<0 lt) - lemma : ord→od o∅ == od∅ + lemma : od (ord→od o∅) == od od∅ lemma = record { eq→ = lemma0 ; eq← = lemma1 } ord-od∅ : od→ord (od∅ ) ≡ o∅ ord-od∅ = sym ( subst (λ k → k ≡ od→ord (od∅ ) ) diso (cong ( λ k → od→ord k ) o∅≡od∅ ) ) -∅0 : record { def = λ x → Lift n ⊥ } == od∅ +∅0 : record { def = λ x → Lift n ⊥ } == od od∅ eq→ ∅0 {w} (lift ()) eq← ∅0 {w} lt = lift (¬x<0 lt) -∅< : { x y : OD } → def x (od→ord y ) → ¬ ( x == od∅ ) +∅< : { x y : HOD } → odef x (od→ord y ) → ¬ ( od x == od od∅ ) ∅< {x} {y} d eq with eq→ (==-trans eq (==-sym ∅0) ) d ∅< {x} {y} d eq | lift () -∅6 : { x : OD } → ¬ ( x ∋ x ) -- no Russel paradox +∅6 : { x : HOD } → ¬ ( x ∋ x ) -- no Russel paradox ∅6 {x} x∋x = o<¬≡ refl ( c<→o< {x} {x} x∋x ) -def-iso : {A B : OD } {x y : Ordinal } → x ≡ y → (def A y → def B y) → def A x → def B x -def-iso refl t = t +odef-iso : {A B : HOD } {x y : Ordinal } → x ≡ y → (odef A y → odef B y) → odef A x → odef B x +odef-iso refl t = t is-o∅ : ( x : Ordinal ) → Dec ( x ≡ o∅ ) is-o∅ x with trio< x o∅ @@ -184,59 +208,72 @@ is-o∅ x | tri≈ ¬a b ¬c = yes b is-o∅ x | tri> ¬a ¬b c = no ¬b -_,_ : OD → OD → OD -x , y = record { def = λ t → (t ≡ od→ord x ) ∨ ( t ≡ od→ord y ) } -- Ord (omax (od→ord x) (od→ord y)) +_,_ : HOD → HOD → HOD +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 + lemma : {t : Ordinal} → (t ≡ od→ord x) ∨ (t ≡ od→ord y) → t o< omax (od→ord x) (od→ord y) + lemma {t} (case1 refl) = omax-x _ _ + lemma {t} (case2 refl) = omax-y _ _ + -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) -- postulate f-extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality n (suc n) -in-codomain : (X : OD ) → ( ψ : OD → OD ) → OD -in-codomain X ψ = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( def X y ∧ ( x ≡ od→ord (ψ (ord→od y ))))) } +in-codomain : (X : HOD ) → ( ψ : HOD → HOD ) → OD +in-codomain X ψ = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( odef X y ∧ ( x ≡ od→ord (ψ (ord→od y ))))) } --- Power Set of X ( or constructible by λ y → def X (od→ord y ) - -ZFSubset : (A x : OD ) → OD -ZFSubset A x = record { def = λ y → def A y ∧ def x y } -- roughly x = A → Set +-- Power Set of X ( or constructible by λ y → odef X (od→ord y ) -Def : (A : OD ) → OD -Def A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A x) ) ) +ZFSubset : (A x : HOD ) → HOD +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 + lemma : {y : Ordinal} → def (od A) y ∧ def (od x) y → y o< omin (odmax A) (odmax x) + lemma {y} and = min1 (<odmax A (proj1 and)) (<odmax x (proj2 and)) --- _⊆_ : ( A B : OD ) → ∀{ x : OD } → Set n --- _⊆_ A B {x} = A ∋ x → B ∋ x - -record _⊆_ ( A B : OD ) : Set (suc n) where +record _⊆_ ( A B : HOD ) : Set (suc n) where field - incl : { x : OD } → A ∋ x → B ∋ x + incl : { x : HOD } → A ∋ x → B ∋ x open _⊆_ - infixr 220 _⊆_ -subset-lemma : {A x : OD } → ( {y : OD } → x ∋ y → ZFSubset A x ∋ y ) ⇔ ( x ⊆ A ) +subset-lemma : {A x : HOD } → ( {y : HOD } → x ∋ y → ZFSubset A x ∋ y ) ⇔ ( x ⊆ A ) subset-lemma {A} {x} = record { proj1 = λ lt → record { incl = λ x∋z → proj1 (lt x∋z) } ; proj2 = λ x⊆A lt → record { proj1 = incl x⊆A lt ; proj2 = lt } } +od⊆→o≤ : {x y : HOD } → x ⊆ y → od→ord x o< osuc (od→ord y) +od⊆→o≤ {x} {y} lt = ⊆→o≤ {x} {y} (λ {z} x>z → subst (λ k → def (od y) k ) diso (incl lt (subst (λ k → def (od x) k ) (sym diso) x>z ))) + +power< : {A x : HOD } → x ⊆ A → Ord (osuc (od→ord A)) ∋ x +power< {A} {x} x⊆A = ⊆→o≤ (λ {y} x∋y → subst (λ k → def (od A) k) diso (lemma y x∋y ) ) where + lemma : (y : Ordinal) → def (od x) y → def (od A) (od→ord (ord→od y)) + lemma y x∋y = incl x⊆A (subst (λ k → def (od x) k ) (sym diso) x∋y ) + open import Data.Unit -ε-induction : { ψ : OD → Set (suc n)} - → ( {x : OD } → ({ y : OD } → x ∋ y → ψ y ) → ψ x ) - → (x : OD ) → ψ x +ε-induction : { ψ : HOD → Set n} + → ( {x : HOD } → ({ y : HOD } → x ∋ y → ψ y ) → ψ x ) + → (x : HOD ) → ψ x ε-induction {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (osuc (od→ord x)) <-osuc ) where induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (ord→od oy)) → ψ (ord→od ox) induction ox prev = ind ( λ {y} lt → subst (λ k → ψ k ) oiso (prev (od→ord y) (o<-subst (c<→o< lt) refl diso ))) ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (ord→od oy) ε-induction-ord ox {oy} lt = TransFinite {λ oy → ψ (ord→od oy)} induction oy --- minimal-2 : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → (y : OD ) → ¬ ( def (minimal x ne) (od→ord y)) ∧ (def x (od→ord y) ) --- minimal-2 x ne y = contra-position ( ε-induction (λ {z} ind → {!!} ) x ) ( λ p → {!!} ) +ε-induction1 : { ψ : HOD → Set (suc n)} + → ( {x : HOD } → ({ y : HOD } → x ∋ y → ψ y ) → ψ x ) + → (x : HOD ) → ψ x +ε-induction1 {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (osuc (od→ord x)) <-osuc ) where + induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (ord→od oy)) → ψ (ord→od ox) + induction ox prev = ind ( λ {y} lt → subst (λ k → ψ k ) oiso (prev (od→ord y) (o<-subst (c<→o< lt) refl diso ))) + ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (ord→od oy) + ε-induction-ord ox {oy} lt = TransFinite1 {λ oy → ψ (ord→od oy)} induction oy -OD→ZF : ZF -OD→ZF = record { - ZFSet = OD +HOD→ZF : ZF +HOD→ZF = record { + ZFSet = HOD ; _∋_ = _∋_ - ; _≈_ = _==_ + ; _≈_ = _=h=_ ; ∅ = od∅ ; _,_ = _,_ ; Union = Union @@ -246,19 +283,43 @@ ; infinite = infinite ; isZF = isZF } where - ZFSet = OD -- is less than Ords because of maxod - Select : (X : OD ) → ((x : OD ) → Set n ) → OD - Select X ψ = record { def = λ x → ( def X x ∧ ψ ( ord→od x )) } - Replace : OD → (OD → OD ) → OD - Replace X ψ = record { def = λ x → (x o< sup-o ( λ x → od→ord (ψ x))) ∧ def (in-codomain X ψ) x } + ZFSet = HOD -- is less than Ords because of maxod + Select : (X : HOD ) → ((x : HOD ) → Set n ) → HOD + 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) } + Replace : HOD → (HOD → HOD) → HOD + 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 } + ; odmax = rmax ; <odmax = rmax<} where + rmax : Ordinal + rmax = sup-o X (λ y X∋y → od→ord (ψ (ord→od y))) + rmax< : {y : Ordinal} → (y o< rmax) ∧ def (in-codomain X ψ) y → y o< rmax + rmax< lt = proj1 lt _∩_ : ( A B : ZFSet ) → ZFSet - A ∩ B = record { def = λ x → def A x ∧ def B x } - Union : OD → OD - Union U = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((def U u) ∧ (def (ord→od u) x))) } + A ∩ B = record { od = record { def = λ x → odef A x ∧ odef B x } + ; odmax = omin (odmax A) (odmax B) ; <odmax = λ y → min1 (<odmax A (proj1 y)) (<odmax B (proj2 y))} + Union : HOD → HOD + Union U = record { od = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((odef U u) ∧ (odef (ord→od u) x))) } + ; odmax = osuc (od→ord U) ; <odmax = umax< } where + umax< : {y : Ordinal} → ¬ ((u : Ordinal) → ¬ def (od U) u ∧ def (od (ord→od u)) y) → y o< osuc (od→ord U) + umax< {y} not = lemma (FExists _ lemma1 not ) where + lemma0 : {x : Ordinal} → def (od (ord→od x)) y → y o< x + 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)) + lemma2 : {x : Ordinal} → def (od U) x → x o< od→ord U + 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)) + lemma1 : {x : Ordinal} → def (od U) x ∧ def (od (ord→od x)) y → ¬ (od→ord U o< y) + lemma1 {x} lt u<y = o<> u<y (ordtrans (lemma0 (proj2 lt)) (lemma2 (proj1 lt)) ) + lemma : ¬ ((od→ord U) o< y ) → y o< osuc (od→ord U) + lemma not with trio< y (od→ord U) + lemma not | tri< a ¬b ¬c = ordtrans a <-osuc + lemma not | tri≈ ¬a refl ¬c = <-osuc + lemma not | tri> ¬a ¬b c = ⊥-elim (not c) _∈_ : ( A B : ZFSet ) → Set n A ∈ B = B ∋ A - Power : OD → OD - Power A = Replace (Def (Ord (od→ord A))) ( λ x → A ∩ x ) + + OPwr : (A : HOD ) → HOD + OPwr A = Ord ( sup-o (Ord (osuc (od→ord A))) ( λ x A∋x → od→ord ( ZFSubset A (ord→od x)) ) ) + + Power : HOD → HOD + Power A = Replace (OPwr (Ord (od→ord A))) ( λ x → A ∩ x ) -- ｛_｝ : ZFSet → ZFSet -- ｛ x ｝ = ( x , x ) -- it works but we don't use @@ -267,12 +328,25 @@ isuc : {x : Ordinal } → infinite-d x → infinite-d (od→ord ( Union (ord→od x , (ord→od x , ord→od x ) ) )) - infinite : OD - infinite = record { def = λ x → infinite-d x } + -- ω can be diverged in our case, since we have no restriction on the corresponding ordinal of a pair. + -- We simply assumes nfinite-d y has a maximum. + -- + -- This means that many of OD cannot be HODs because of the od→ord mapping divergence. + -- We should have some axioms to prevent this, but it may complicate thins. + -- + postulate + ωmax : Ordinal + <ωmax : {y : Ordinal} → infinite-d y → y o< ωmax + + infinite : HOD + infinite = record { od = record { def = λ x → infinite-d x } ; odmax = ωmax ; <odmax = <ωmax } + + _=h=_ : (x y : HOD) → Set n + x =h= y = od x == od y infixr 200 _∈_ -- infixr 230 _∩_ _∪_ - isZF : IsZF (OD ) _∋_ _==_ od∅ _,_ Union Power Select Replace infinite + isZF : IsZF (HOD ) _∋_ _=h=_ od∅ _,_ Union Power Select Replace infinite isZF = record { isEquivalence = record { refl = ==-refl ; sym = ==-sym; trans = ==-trans } ; pair→ = pair→ @@ -288,20 +362,20 @@ ; infinity = infinity ; selection = λ {X} {ψ} {y} → selection {X} {ψ} {y} ; replacement← = replacement← - ; replacement→ = replacement→ + ; replacement→ = λ {ψ} → replacement→ {ψ} -- ; choice-func = choice-func -- ; choice = choice } where - pair→ : ( x y t : ZFSet ) → (x , y) ∋ t → ( t == x ) ∨ ( t == y ) - pair→ x y t (case1 t≡x ) = case1 (subst₂ (λ j k → j == k ) oiso oiso (o≡→== t≡x )) - pair→ x y t (case2 t≡y ) = case2 (subst₂ (λ j k → j == k ) oiso oiso (o≡→== t≡y )) + pair→ : ( x y t : ZFSet ) → (x , y) ∋ t → ( t =h= x ) ∨ ( t =h= y ) + pair→ x y t (case1 t≡x ) = case1 (subst₂ (λ j k → j =h= k ) oiso oiso (o≡→== t≡x )) + pair→ x y t (case2 t≡y ) = case2 (subst₂ (λ j k → j =h= k ) oiso oiso (o≡→== t≡y )) - pair← : ( x y t : ZFSet ) → ( t == x ) ∨ ( t == y ) → (x , y) ∋ t - pair← x y t (case1 t==x) = case1 (cong (λ k → od→ord k ) (==→o≡ t==x)) - pair← x y t (case2 t==y) = case2 (cong (λ k → od→ord k ) (==→o≡ t==y)) + pair← : ( x y t : ZFSet ) → ( t =h= x ) ∨ ( t =h= y ) → (x , y) ∋ t + pair← x y t (case1 t=h=x) = case1 (cong (λ k → od→ord k ) (==→o≡ t=h=x)) + pair← x y t (case2 t=h=y) = case2 (cong (λ k → od→ord k ) (==→o≡ t=h=y)) - empty : (x : OD ) → ¬ (od∅ ∋ x) + empty : (x : HOD ) → ¬ (od∅ ∋ x) empty x = ¬x<0 o<→c< : {x y : Ordinal } → x o< y → (Ord x) ⊆ (Ord y) @@ -314,114 +388,167 @@ ⊆→o< {x} {y} lt | tri> ¬a ¬b c with (incl lt) (o<-subst c (sym diso) refl ) ... | ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt diso refl )) - union→ : (X z u : OD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z + union→ : (X z u : HOD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z union→ X z u xx not = ⊥-elim ( not (od→ord u) ( record { proj1 = proj1 xx - ; proj2 = subst ( λ k → def k (od→ord z)) (sym oiso) (proj2 xx) } )) - union← : (X z : OD) (X∋z : Union X ∋ z) → ¬ ( (u : OD ) → ¬ ((X ∋ u) ∧ (u ∋ z ))) + ; proj2 = subst ( λ k → odef k (od→ord z)) (sym oiso) (proj2 xx) } )) + union← : (X z : HOD) (X∋z : Union X ∋ z) → ¬ ( (u : HOD ) → ¬ ((X ∋ u) ∧ (u ∋ z ))) union← X z UX∋z = FExists _ lemma UX∋z where - lemma : {y : Ordinal} → def X y ∧ def (ord→od y) (od→ord z) → ¬ ((u : OD) → ¬ (X ∋ u) ∧ (u ∋ z)) - lemma {y} xx not = not (ord→od y) record { proj1 = subst ( λ k → def X k ) (sym diso) (proj1 xx ) ; proj2 = proj2 xx } + lemma : {y : Ordinal} → odef X y ∧ odef (ord→od y) (od→ord z) → ¬ ((u : HOD) → ¬ (X ∋ u) ∧ (u ∋ z)) + lemma {y} xx not = not (ord→od y) record { proj1 = subst ( λ k → odef X k ) (sym diso) (proj1 xx ) ; proj2 = proj2 xx } - ψiso : {ψ : OD → Set n} {x y : OD } → ψ x → x ≡ y → ψ y + ψiso : {ψ : HOD → Set n} {x y : HOD } → ψ x → x ≡ y → ψ y ψiso {ψ} t refl = t - selection : {ψ : OD → Set n} {X y : OD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y) + selection : {ψ : HOD → Set n} {X y : HOD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y) selection {ψ} {X} {y} = record { proj1 = λ cond → record { proj1 = proj1 cond ; proj2 = ψiso {ψ} (proj2 cond) (sym oiso) } ; proj2 = λ select → record { proj1 = proj1 select ; proj2 = ψiso {ψ} (proj2 select) oiso } } - replacement← : {ψ : OD → OD} (X x : OD) → X ∋ x → Replace X ψ ∋ ψ x - replacement← {ψ} X x lt = record { proj1 = sup-c< ψ {x} ; proj2 = lemma } where + 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)))) + sup-c< ψ {X} {x} lt = subst (λ k → od→ord (ψ k) o< _ ) oiso (sup-o< X lt ) + replacement← : {ψ : HOD → HOD} (X x : HOD) → X ∋ x → Replace X ψ ∋ ψ x + replacement← {ψ} X x lt = record { proj1 = sup-c< ψ {X} {x} lt ; proj2 = lemma } where lemma : def (in-codomain X ψ) (od→ord (ψ x)) lemma not = ⊥-elim ( not ( od→ord x ) (record { proj1 = lt ; proj2 = cong (λ k → od→ord (ψ k)) (sym oiso)} )) - replacement→ : {ψ : OD → OD} (X x : OD) → (lt : Replace X ψ ∋ x) → ¬ ( (y : OD) → ¬ (x == ψ y)) + replacement→ : {ψ : HOD → HOD} (X x : HOD) → (lt : Replace X ψ ∋ x) → ¬ ( (y : HOD) → ¬ (x =h= ψ y)) replacement→ {ψ} X x lt = contra-position lemma (lemma2 (proj2 lt)) where - lemma2 : ¬ ((y : Ordinal) → ¬ def X y ∧ ((od→ord x) ≡ od→ord (ψ (ord→od y)))) - → ¬ ((y : Ordinal) → ¬ def X y ∧ (ord→od (od→ord x) == ψ (ord→od y))) + lemma2 : ¬ ((y : Ordinal) → ¬ odef X y ∧ ((od→ord x) ≡ od→ord (ψ (ord→od y)))) + → ¬ ((y : Ordinal) → ¬ odef X y ∧ (ord→od (od→ord x) =h= ψ (ord→od y))) lemma2 not not2 = not ( λ y d → not2 y (record { proj1 = proj1 d ; proj2 = lemma3 (proj2 d)})) where - lemma3 : {y : Ordinal } → (od→ord x ≡ od→ord (ψ (ord→od y))) → (ord→od (od→ord x) == ψ (ord→od y)) - lemma3 {y} eq = subst (λ k → ord→od (od→ord x) == k ) oiso (o≡→== eq ) - lemma : ( (y : OD) → ¬ (x == ψ y)) → ( (y : Ordinal) → ¬ def X y ∧ (ord→od (od→ord x) == ψ (ord→od y)) ) - lemma not y not2 = not (ord→od y) (subst (λ k → k == ψ (ord→od y)) oiso ( proj2 not2 )) + lemma3 : {y : Ordinal } → (od→ord x ≡ od→ord (ψ (ord→od y))) → (ord→od (od→ord x) =h= ψ (ord→od y)) + lemma3 {y} eq = subst (λ k → ord→od (od→ord x) =h= k ) oiso (o≡→== eq ) + lemma : ( (y : HOD) → ¬ (x =h= ψ y)) → ( (y : Ordinal) → ¬ odef X y ∧ (ord→od (od→ord x) =h= ψ (ord→od y)) ) + lemma not y not2 = not (ord→od y) (subst (λ k → k =h= ψ (ord→od y)) oiso ( proj2 not2 )) --- --- Power Set --- - --- First consider ordinals in OD + --- First consider ordinals in HOD --- - --- ZFSubset A x = record { def = λ y → def A y ∧ def x y } subset of A + --- ZFSubset A x = record { def = λ y → odef A y ∧ odef x y } subset of A -- -- - ∩-≡ : { a b : OD } → ({x : OD } → (a ∋ x → b ∋ x)) → a == ( b ∩ a ) + ∩-≡ : { a b : HOD } → ({x : HOD } → (a ∋ x → b ∋ x)) → a =h= ( b ∩ a ) ∩-≡ {a} {b} inc = record { eq→ = λ {x} x<a → record { proj2 = x<a ; - proj1 = def-subst {_} {_} {b} {x} (inc (def-subst {_} {_} {a} {_} x<a refl (sym diso) )) refl diso } ; + proj1 = odef-subst {_} {_} {b} {x} (inc (odef-subst {_} {_} {a} {_} x<a refl (sym diso) )) refl diso } ; eq← = λ {x} x<a∩b → proj2 x<a∩b } -- -- Transitive Set case - -- we have t ∋ x → Ord a ∋ x means t is a subset of Ord a, that is ZFSubset (Ord a) t == t - -- Def (Ord a) is a sup of ZFSubset (Ord a) t, so Def (Ord a) ∋ t - -- Def A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) ) + -- we have t ∋ x → Ord a ∋ x means t is a subset of Ord a, that is ZFSubset (Ord a) t =h= t + -- OPwr (Ord a) is a sup of ZFSubset (Ord a) t, so OPwr (Ord a) ∋ t + -- OPwr A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) ) -- - ord-power← : (a : Ordinal ) (t : OD) → ({x : OD} → (t ∋ x → (Ord a) ∋ x)) → Def (Ord a) ∋ t - ord-power← a t t→A = def-subst {_} {_} {Def (Ord a)} {od→ord t} + ord-power← : (a : Ordinal ) (t : HOD) → ({x : HOD} → (t ∋ x → (Ord a) ∋ x)) → OPwr (Ord a) ∋ t + ord-power← a t t→A = odef-subst {_} {_} {OPwr (Ord a)} {od→ord t} lemma refl (lemma1 lemma-eq )where - lemma-eq : ZFSubset (Ord a) t == t + lemma-eq : ZFSubset (Ord a) t =h= t eq→ lemma-eq {z} w = proj2 w eq← lemma-eq {z} w = record { proj2 = w ; - proj1 = def-subst {_} {_} {(Ord a)} {z} - ( t→A (def-subst {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso } - lemma1 : {a : Ordinal } { t : OD } - → (eq : ZFSubset (Ord a) t == t) → od→ord (ZFSubset (Ord a) (ord→od (od→ord t))) ≡ od→ord t + proj1 = odef-subst {_} {_} {(Ord a)} {z} + ( t→A (odef-subst {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso } + lemma1 : {a : Ordinal } { t : HOD } + → (eq : ZFSubset (Ord a) t =h= t) → od→ord (ZFSubset (Ord a) (ord→od (od→ord t))) ≡ od→ord t 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 )) - lemma : od→ord (ZFSubset (Ord a) (ord→od (od→ord t)) ) o< sup-o (λ x → od→ord (ZFSubset (Ord a) x)) - lemma = sup-o< + lemma2 : (od→ord t) o< (osuc (od→ord (Ord a))) + 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))) + 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))) + lemma = sup-o< _ lemma2 -- - -- Every set in OD is a subset of Ordinals, so make Def (Ord (od→ord A)) first + -- Every set in HOD is a subset of Ordinals, so make OPwr (Ord (od→ord A)) first -- then replace of all elements of the Power set by A ∩ y -- - -- Power A = Replace (Def (Ord (od→ord A))) ( λ y → A ∩ y ) + -- Power A = Replace (OPwr (Ord (od→ord A))) ( λ y → A ∩ y ) -- we have oly double negation form because of the replacement axiom -- - power→ : ( A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → ¬ ¬ (A ∋ x) + power→ : ( A t : HOD) → Power A ∋ t → {x : HOD} → t ∋ x → ¬ ¬ (A ∋ x) power→ A t P∋t {x} t∋x = FExists _ lemma5 lemma4 where a = od→ord A - lemma2 : ¬ ( (y : OD) → ¬ (t == (A ∩ y))) - lemma2 = replacement→ (Def (Ord (od→ord A))) t P∋t - lemma3 : (y : OD) → t == ( A ∩ y ) → ¬ ¬ (A ∋ x) + lemma2 : ¬ ( (y : HOD) → ¬ (t =h= (A ∩ y))) + lemma2 = replacement→ {λ x → A ∩ x} (OPwr (Ord (od→ord A))) t P∋t + lemma3 : (y : HOD) → t =h= ( A ∩ y ) → ¬ ¬ (A ∋ x) lemma3 y eq not = not (proj1 (eq→ eq t∋x)) - lemma4 : ¬ ((y : Ordinal) → ¬ (t == (A ∩ ord→od y))) - lemma4 not = lemma2 ( λ y not1 → not (od→ord y) (subst (λ k → t == ( A ∩ k )) (sym oiso) not1 )) - lemma5 : {y : Ordinal} → t == (A ∩ ord→od y) → ¬ ¬ (def A (od→ord x)) + lemma4 : ¬ ((y : Ordinal) → ¬ (t =h= (A ∩ ord→od y))) + lemma4 not = lemma2 ( λ y not1 → not (od→ord y) (subst (λ k → t =h= ( A ∩ k )) (sym oiso) not1 )) + lemma5 : {y : Ordinal} → t =h= (A ∩ ord→od y) → ¬ ¬ (odef A (od→ord x)) lemma5 {y} eq not = (lemma3 (ord→od y) eq) not - power← : (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t + power← : (A t : HOD) → ({x : HOD} → (t ∋ x → A ∋ x)) → Power A ∋ t power← A t t→A = record { proj1 = lemma1 ; proj2 = lemma2 } where a = od→ord A - lemma0 : {x : OD} → t ∋ x → Ord a ∋ x + lemma0 : {x : HOD} → t ∋ x → Ord a ∋ x lemma0 {x} t∋x = c<→o< (t→A t∋x) - lemma3 : Def (Ord a) ∋ t + lemma3 : OPwr (Ord a) ∋ t lemma3 = ord-power← a t lemma0 lemma4 : (A ∩ ord→od (od→ord t)) ≡ t lemma4 = let open ≡-Reasoning in begin A ∩ ord→od (od→ord t) ≡⟨ cong (λ k → A ∩ k) oiso ⟩ A ∩ t - ≡⟨ sym (==→o≡ ( ∩-≡ t→A )) ⟩ + ≡⟨ sym (==→o≡ ( ∩-≡ {t} {A} t→A )) ⟩ t ∎ - lemma1 : od→ord t o< sup-o (λ x → od→ord (A ∩ x)) - lemma1 = subst (λ k → od→ord k o< sup-o (λ x → od→ord (A ∩ x))) - lemma4 (sup-o< {λ x → od→ord (A ∩ x)} ) - lemma2 : def (in-codomain (Def (Ord (od→ord A))) (_∩_ A)) (od→ord t) + sup1 : Ordinal + 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))) + lemma9 : def (od (Ord (Ordinals.osuc O (od→ord (Ord (od→ord A)))))) (od→ord (Ord (od→ord A))) + lemma9 = <-osuc + lemmab : od→ord (ZFSubset (Ord (od→ord A)) (ord→od (od→ord (Ord (od→ord A) )))) o< sup1 + lemmab = sup-o< (Ord (osuc (od→ord (Ord (od→ord A))))) lemma9 + lemmad : Ord (osuc (od→ord A)) ∋ t + 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))) + lemmac : ZFSubset (Ord (od→ord A)) (ord→od (od→ord (Ord (od→ord A) ))) =h= Ord (od→ord A) + lemmac = record { eq→ = lemmaf ; eq← = lemmag } where + 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 + lemmaf {x} lt = proj1 lt + 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 + lemmag {x} lt = record { proj1 = lt ; proj2 = subst (λ k → def (od k) x) (sym oiso) lt } + lemmae : od→ord (ZFSubset (Ord (od→ord A)) (ord→od (od→ord (Ord (od→ord A))))) ≡ od→ord (Ord (od→ord A)) + lemmae = cong (λ k → od→ord k ) ( ==→o≡ lemmac) + lemma7 : def (od (OPwr (Ord (od→ord A)))) (od→ord t) + lemma7 with osuc-≡< lemmad + lemma7 | case2 lt = ordtrans (c<→o< lt) (subst (λ k → k o< sup1) lemmae lemmab ) + 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 + lemmah : {x : Ordinal } → def (od (ord→od (od→ord t))) x → def (od (ord→od (od→ord (Ord (od→ord t))))) x + lemmah {x} lt = subst (λ k → def (od k) x ) (sym oiso) (subst (λ k → k o< (od→ord t)) + diso + (c<→o< (subst₂ (λ j k → def (od j) k) oiso (sym diso) lt ))) + lemma7 | case1 eq | case1 eq1 = subst (λ k → k o< sup1) (trans lemmae lemmai) lemmab where + lemmai : od→ord (Ord (od→ord A)) ≡ od→ord t + lemmai = let open ≡-Reasoning in begin + od→ord (Ord (od→ord A)) + ≡⟨ sym (cong (λ k → od→ord (Ord k)) eq) ⟩ + od→ord (Ord (od→ord t)) + ≡⟨ sym diso ⟩ + od→ord (ord→od (od→ord (Ord (od→ord t)))) + ≡⟨ sym eq1 ⟩ + od→ord (ord→od (od→ord t)) + ≡⟨ diso ⟩ + od→ord t + ∎ + lemma7 | case1 eq | case2 lt = ordtrans lemmaj (subst (λ k → k o< sup1) lemmae lemmab ) where + lemmak : od→ord (ord→od (od→ord (Ord (od→ord t)))) ≡ od→ord (Ord (od→ord A)) + lemmak = let open ≡-Reasoning in begin + od→ord (ord→od (od→ord (Ord (od→ord t)))) + ≡⟨ diso ⟩ + od→ord (Ord (od→ord t)) + ≡⟨ cong (λ k → od→ord (Ord k)) eq ⟩ + od→ord (Ord (od→ord A)) + ∎ + lemmaj : od→ord t o< od→ord (Ord (od→ord A)) + lemmaj = subst₂ (λ j k → j o< k ) diso lemmak lt + lemma1 : od→ord t o< sup-o (OPwr (Ord (od→ord A))) (λ x lt → od→ord (A ∩ (ord→od x))) + lemma1 = subst (λ k → od→ord k o< sup-o (OPwr (Ord (od→ord A))) (λ x lt → od→ord (A ∩ (ord→od x)))) + lemma4 (sup-o< (OPwr (Ord (od→ord A))) lemma7 ) + lemma2 : def (in-codomain (OPwr (Ord (od→ord A))) (_∩_ A)) (od→ord t) lemma2 not = ⊥-elim ( not (od→ord t) (record { proj1 = lemma3 ; proj2 = lemma6 }) ) where lemma6 : od→ord t ≡ od→ord (A ∩ ord→od (od→ord t)) - lemma6 = cong ( λ k → od→ord k ) (==→o≡ (subst (λ k → t == (A ∩ k)) (sym oiso) ( ∩-≡ t→A ))) + lemma6 = cong ( λ k → od→ord k ) (==→o≡ (subst (λ k → t =h= (A ∩ k)) (sym oiso) ( ∩-≡ {t} {A} t→A ))) + ord⊆power : (a : Ordinal) → (Ord (osuc a)) ⊆ (Power (Ord a)) ord⊆power a = record { incl = λ {x} lt → power← (Ord a) x (lemma lt) } where - lemma : {x y : OD} → od→ord x o< osuc a → x ∋ y → Ord a ∋ y + lemma : {x y : HOD} → od→ord x o< osuc a → x ∋ y → Ord a ∋ y lemma lt y<x with osuc-≡< lt lemma lt y<x | case1 refl = c<→o< y<x lemma lt y<x | case2 x<a = ordtrans (c<→o< y<x) x<a @@ -429,16 +556,16 @@ continuum-hyphotheis : (a : Ordinal) → Set (suc n) continuum-hyphotheis a = Power (Ord a) ⊆ Ord (osuc a) - extensionality0 : {A B : OD } → ((z : OD) → (A ∋ z) ⇔ (B ∋ z)) → A == B - eq→ (extensionality0 {A} {B} eq ) {x} d = def-iso {A} {B} (sym diso) (proj1 (eq (ord→od x))) d - eq← (extensionality0 {A} {B} eq ) {x} d = def-iso {B} {A} (sym diso) (proj2 (eq (ord→od x))) d + extensionality0 : {A B : HOD } → ((z : HOD) → (A ∋ z) ⇔ (B ∋ z)) → A =h= B + eq→ (extensionality0 {A} {B} eq ) {x} d = odef-iso {A} {B} (sym diso) (proj1 (eq (ord→od x))) d + eq← (extensionality0 {A} {B} eq ) {x} d = odef-iso {B} {A} (sym diso) (proj2 (eq (ord→od x))) d - extensionality : {A B w : OD } → ((z : OD ) → (A ∋ z) ⇔ (B ∋ z)) → (w ∋ A) ⇔ (w ∋ B) + extensionality : {A B w : HOD } → ((z : HOD ) → (A ∋ z) ⇔ (B ∋ z)) → (w ∋ A) ⇔ (w ∋ B) proj1 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) ( ==→o≡ (extensionality0 {A} {B} eq) ) d proj2 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) (sym ( ==→o≡ (extensionality0 {A} {B} eq) )) d infinity∅ : infinite ∋ od∅ - infinity∅ = def-subst {_} {_} {infinite} {od→ord (od∅ )} iφ refl lemma where + infinity∅ = odef-subst {_} {_} {infinite} {od→ord (od∅ )} iφ refl lemma where lemma : o∅ ≡ od→ord od∅ lemma = let open ≡-Reasoning in begin o∅ @@ -447,15 +574,15 @@ ≡⟨ cong ( λ k → od→ord k ) o∅≡od∅ ⟩ od→ord od∅ ∎ - infinity : (x : OD) → infinite ∋ x → infinite ∋ Union (x , (x , x )) - infinity x lt = def-subst {_} {_} {infinite} {od→ord (Union (x , (x , x )))} ( isuc {od→ord x} lt ) refl lemma where + infinity : (x : HOD) → infinite ∋ x → infinite ∋ Union (x , (x , x )) + infinity x lt = odef-subst {_} {_} {infinite} {od→ord (Union (x , (x , x )))} ( isuc {od→ord x} lt ) refl lemma where lemma : od→ord (Union (ord→od (od→ord x) , (ord→od (od→ord x) , ord→od (od→ord x)))) ≡ od→ord (Union (x , (x , x))) lemma = cong (λ k → od→ord (Union ( k , ( k , k ) ))) oiso -Union = ZF.Union OD→ZF -Power = ZF.Power OD→ZF -Select = ZF.Select OD→ZF -Replace = ZF.Replace OD→ZF -isZF = ZF.isZF OD→ZF +Union = ZF.Union HOD→ZF +Power = ZF.Power HOD→ZF +Select = ZF.Select HOD→ZF +Replace = ZF.Replace HOD→ZF +isZF = ZF.isZF HOD→ZF