Mercurial > hg > Members > kono > Proof > ZF-in-agda
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author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Mon, 13 Jul 2020 22:40:37 +0900 |
parents | 06f10815d0b3 |
children | 08d94fec239c |
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open import Level open import Ordinals module OD {n : Level } (O : Ordinals {n} ) where open import zf open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) open import Relation.Binary.PropositionalEquality open import Data.Nat.Properties open import Data.Empty open import Relation.Nullary open import Relation.Binary open import Relation.Binary.Core open import logic open import nat open inOrdinal O -- Ordinal Definable Set record OD : Set (suc n ) where field def : (x : Ordinal ) → Set n open OD open _∧_ open _∨_ open Bool record _==_ ( a b : OD ) : Set n where field eq→ : ∀ { x : Ordinal } → def a x → def b x eq← : ∀ { x : Ordinal } → def b x → def a x id : {A : Set n} → A → A id x = x ==-refl : { x : OD } → x == x ==-refl {x} = record { eq→ = id ; eq← = id } open _==_ ==-trans : { x y z : OD } → x == y → y == z → x == z ==-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) } ==-sym : { x y : OD } → x == y → y == x ==-sym x=y = record { eq→ = λ {m} t → eq← x=y t ; eq← = λ {m} t → eq→ x=y t } ⇔→== : { x y : OD } → ( {z : Ordinal } → def x z ⇔ def y z) → x == y eq→ ( ⇔→== {x} {y} eq ) {z} m = proj1 eq m eq← ( ⇔→== {x} {y} eq ) {z} m = proj2 eq m -- next assumptions are our axiom -- -- 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. -- 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 -- Ordinals in OD , the maximum Ords : OD Ords = record { def = λ x → One } 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 -- possible order restriction hod-ord< : {x : HOD } → Set n hod-ord< {x} = od→ord x o< next (odmax x) -- OD ⇔ Ordinal leads a contradiction, so we need bounded HOD ¬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) → ⊥ ¬OD-order od→ord ord→od c<→o< = osuc-< <-osuc (c<→o< {Ords} OneObj ) -- Open supreme upper bound leads a contradition, so we use domain restriction on sup ¬open-sup : ( sup-o : (Ordinal → Ordinal ) → Ordinal) → ((ψ : Ordinal → Ordinal ) → (x : Ordinal) → ψ x o< sup-o ψ ) → ⊥ ¬open-sup sup-o sup-o< = o<> <-osuc (sup-o< next-ord (sup-o next-ord)) where next-ord : Ordinal → Ordinal next-ord x = osuc x -- Ordinal in OD ( and ZFSet ) Transitive Set 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∅ odef : HOD → Ordinal → Set n odef A x = def ( od A ) x -- If we have reverse of c<→o<, everything becomes Ordinal o<→c<→HOD=Ord : ( o<→c< : {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 ) _∋_ : ( a x : HOD ) → Set n _∋_ a x = odef a ( od→ord x ) _c<_ : ( x a : HOD ) → Set n x c< a = a ∋ x 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 ) 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 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 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 : HOD } → { y : Ordinal } → od→ord x ≡ y → od→ord x ≡ y orefl refl = refl ==-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 (odef-subst d (sym oiso) refl )) ; eq← = λ d → lemma ( eq← eq (odef-subst d (sym oiso) refl )) } where 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 : 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 : 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 → od (ord→od ox) == od (ord→od oy) lemma ox ox refl = ==-refl 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} → 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 : 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 od∅ eq→ ∅0 {w} (lift ()) eq← ∅0 {w} lt = lift (¬x<0 lt) ∅< : { 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 : HOD } → ¬ ( x ∋ x ) -- no Russel paradox ∅6 {x} x∋x = o<¬≡ refl ( c<→o< {x} {x} x∋x ) 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∅ is-o∅ x | tri< a ¬b ¬c = no ¬b is-o∅ x | tri≈ ¬a b ¬c = yes b is-o∅ x | tri> ¬a ¬b c = no ¬b -- the pair _,_ : 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 _ _ pair-xx<xy : {x y : HOD} → od→ord (x , x) o< osuc (od→ord (x , y) ) pair-xx<xy {x} {y} = ⊆→o≤ lemma where lemma : {z : Ordinal} → def (od (x , x)) z → def (od (x , y)) z lemma {z} (case1 refl) = case1 refl lemma {z} (case2 refl) = case1 refl -- another form of infinite -- pair-ord< : {x : Ordinal } → Set n pair-ord< : {x : HOD } → ( {y : HOD } → od→ord y o< next (odmax y) ) → od→ord ( x , x ) o< next (od→ord x) pair-ord< {x} ho< = subst (λ k → od→ord (x , x) o< k ) lemmab0 lemmab1 where lemmab0 : next (odmax (x , x)) ≡ next (od→ord x) lemmab0 = trans (cong (λ k → next k) (omxx _)) (sym nexto≡) lemmab1 : od→ord (x , x) o< next ( odmax (x , x)) lemmab1 = ho< pair<y : {x y : HOD } → y ∋ x → od→ord (x , x) o< osuc (od→ord y) pair<y {x} {y} y∋x = ⊆→o≤ lemma where lemma : {z : Ordinal} → def (od (x , x)) z → def (od y) z lemma (case1 refl) = y∋x lemma (case2 refl) = y∋x -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) -- postulate f-extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality n (suc n) 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 ))))) } 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)) record _⊆_ ( A B : HOD ) : Set (suc n) where field incl : { x : HOD } → A ∋ x → B ∋ x open _⊆_ infixr 220 _⊆_ 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 ))) -- if we have od→ord (x , x) ≡ osuc (od→ord x), ⊆→o≤ → c<→o< ⊆→o≤→c<→o< : ({x : HOD} → od→ord (x , x) ≡ osuc (od→ord x) ) → ({y z : HOD } → ({x : Ordinal} → def (od y) x → def (od z) x ) → od→ord y o< osuc (od→ord z) ) → {x y : HOD } → def (od y) ( od→ord x ) → od→ord x o< od→ord y ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x with trio< (od→ord x) (od→ord y) ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x | tri< a ¬b ¬c = a ⊆→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 ))) ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x | tri> ¬a ¬b c = ⊥-elim ( o<> (⊆→o≤ {x , x} {y} y⊆x,x ) lemma1 ) where lemma : {z : Ordinal} → (z ≡ od→ord x) ∨ (z ≡ od→ord x) → od→ord x ≡ z lemma (case1 refl) = refl lemma (case2 refl) = refl y⊆x,x : {z : Ordinals.ord O} → def (od (x , x)) z → def (od y) z y⊆x,x {z} lt = subst (λ k → def (od y) k ) (lemma lt) y∋x lemma1 : osuc (od→ord y) o< od→ord (x , x) lemma1 = subst (λ k → osuc (od→ord y) o< k ) (sym (peq {x})) (osucc c ) 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 } } 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 : 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 -- level trick (what'a shame) 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 HOD→ZF : ZF HOD→ZF = record { ZFSet = HOD ; _∋_ = _∋_ ; _≈_ = _=h=_ ; ∅ = od∅ ; _,_ = _,_ ; Union = Union ; Power = Power ; Select = Select ; Replace = Replace ; infinite = infinite ; isZF = isZF } where 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 { 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 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 ) -- better to use (x , x) directly data infinite-d : ( x : Ordinal ) → Set n where iφ : infinite-d o∅ isuc : {x : Ordinal } → infinite-d x → infinite-d (od→ord ( Union (ord→od x , (ord→od x , ord→od x ) ) )) -- ω can be diverged in our case, since we have no restriction on the corresponding ordinal of a pair. -- We simply assumes infinite-d y has a maximum. -- -- This means that many of OD may not be HODs because of the od→ord mapping divergence. -- We should have some axioms to prevent this such as od→ord x o< next (odmax x). -- 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 } infinite' : ({x : HOD} → od→ord x o< next (odmax x)) → HOD infinite' ho< = record { od = record { def = λ x → infinite-d x } ; odmax = next o∅ ; <odmax = lemma } where u : (y : Ordinal ) → HOD u y = Union (ord→od y , (ord→od y , ord→od y)) lemma : {y : Ordinal} → infinite-d y → y o< next o∅ lemma {o∅} iφ = proj1 next-limit lemma (isuc {y} x) = lemma2 where lemma0 : y o< next o∅ lemma0 = lemma x lemma8 : od→ord (ord→od y , ord→od y) o< next (odmax (ord→od y , ord→od y)) lemma8 = ho< --- (x,y) < next (omax x y) < next (osuc y) = next y lemmaa : {x y : HOD} → od→ord x o< od→ord y → od→ord (x , y) o< next (od→ord y) 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< ) lemma81 : od→ord (ord→od y , ord→od y) o< next (od→ord (ord→od y)) lemma81 = nexto=n (subst (λ k → od→ord (ord→od y , ord→od y) o< k ) (cong (λ k → next k) (omxx _)) lemma8) lemma9 : od→ord (ord→od y , (ord→od y , ord→od y)) o< next (od→ord (ord→od y , ord→od y)) lemma9 = lemmaa (c<→o< (case1 refl)) lemma71 : od→ord (ord→od y , (ord→od y , ord→od y)) o< next (od→ord (ord→od y)) lemma71 = next< lemma81 lemma9 lemma1 : od→ord (u y) o< next (osuc (od→ord (ord→od y , (ord→od y , ord→od y)))) lemma1 = ho< lemma2 : od→ord (u y) o< next o∅ lemma2 = next< lemma0 (next< (subst (λ k → od→ord (ord→od y , (ord→od y , ord→od y)) o< next k) diso lemma71 ) (nexto=n lemma1)) _=h=_ : (x y : HOD) → Set n x =h= y = od x == od y infixr 200 _∈_ -- infixr 230 _∩_ _∪_ isZF : IsZF (HOD ) _∋_ _=h=_ od∅ _,_ Union Power Select Replace infinite isZF = record { isEquivalence = record { refl = ==-refl ; sym = ==-sym; trans = ==-trans } ; pair→ = pair→ ; pair← = pair← ; union→ = union→ ; union← = union← ; empty = empty ; power→ = power→ ; power← = power← ; extensionality = λ {A} {B} {w} → extensionality {A} {B} {w} ; ε-induction = ε-induction ; infinity∅ = infinity∅ ; infinity = infinity ; selection = λ {X} {ψ} {y} → selection {X} {ψ} {y} ; replacement← = replacement← ; replacement→ = λ {ψ} → replacement→ {ψ} -- ; choice-func = choice-func -- ; choice = choice } where 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 =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 : HOD ) → ¬ (od∅ ∋ x) empty x = ¬x<0 o<→c< : {x y : Ordinal } → x o< y → (Ord x) ⊆ (Ord y) o<→c< lt = record { incl = λ z → ordtrans z lt } ⊆→o< : {x y : Ordinal } → (Ord x) ⊆ (Ord y) → x o< osuc y ⊆→o< {x} {y} lt with trio< x y ⊆→o< {x} {y} lt | tri< a ¬b ¬c = ordtrans a <-osuc ⊆→o< {x} {y} lt | tri≈ ¬a b ¬c = subst ( λ k → k o< osuc y) (sym b) <-osuc ⊆→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 : 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 → 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} → 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 : HOD → Set n} {x y : HOD } → ψ x → x ≡ y → ψ y ψiso {ψ} t refl = t 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 } } 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 : 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) → ¬ 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) =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 HOD --- --- ZFSubset A x = record { def = λ y → odef A y ∧ odef x y } subset of 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 = 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 =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 : 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 =h= t eq→ lemma-eq {z} w = proj2 w eq← lemma-eq {z} w = record { proj2 = w ; 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 )) 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 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 (OPwr (Ord (od→ord A))) ( λ y → A ∩ y ) -- we have oly double negation form because of the replacement axiom -- 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 : 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 =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 : 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 : HOD} → t ∋ x → Ord a ∋ x lemma0 {x} t∋x = c<→o< (t→A t∋x) 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} t→A )) ⟩ 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 =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 : 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 continuum-hyphotheis : (a : Ordinal) → Set (suc n) continuum-hyphotheis a = Power (Ord a) ⊆ Ord (osuc a) 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 : 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∅ = odef-subst {_} {_} {infinite} {od→ord (od∅ )} iφ refl lemma where lemma : o∅ ≡ od→ord od∅ lemma = let open ≡-Reasoning in begin o∅ ≡⟨ sym diso ⟩ od→ord ( ord→od o∅ ) ≡⟨ cong ( λ k → od→ord k ) o∅≡od∅ ⟩ od→ord od∅ ∎ 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 HOD→ZF Power = ZF.Power HOD→ZF Select = ZF.Select HOD→ZF Replace = ZF.Replace HOD→ZF isZF = ZF.isZF HOD→ZF