Mercurial > hg > Members > kono > Proof > ZF-in-agda
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use supf of zchain for (nmx : ¬ Maximal A ) → ⊥
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Wed, 19 Oct 2022 08:41:42 +0900 |
parents | 507f6582c31d |
children | 88fae58f89f5 |
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{-# OPTIONS --allow-unsolved-metas #-} 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 hiding ( [_] ) open import Data.Nat.Properties open import Data.Empty open import Relation.Nullary open import Relation.Binary hiding (_⇔_) open import Relation.Binary.Core hiding (_⇔_) open import logic import OrdUtil open import nat open Ordinals.Ordinals O open Ordinals.IsOrdinals isOrdinal open Ordinals.IsNext isNext open OrdUtil 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 ==-refl : { x : OD } → x == x ==-refl {x} = record { eq→ = λ x → x ; eq← = λ x → x } 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. -- 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) & : HOD → Ordinal * : Ordinal → HOD c<→o< : {x y : HOD } → def (od y) ( & x ) → & x o< & y ⊆→o≤ : {y z : HOD } → ({x : Ordinal} → def (od y) x → def (od z) x ) → & y o< osuc (& z) *iso : {x : HOD } → * ( & x ) ≡ x &iso : {x : Ordinal } → & ( * 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 -- possible order restriction ho< : {x : HOD} → & x o< next (odmax x) postulate odAxiom : ODAxiom open ODAxiom odAxiom -- odmax minimality -- -- since we have ==→o≡ , so odmax have to be unique. We should have odmaxmin in HOD. -- We can calculate the minimum using sup but it is tedius. -- Only Select has non minimum odmax. -- We have the same problem on 'def' itself, but we leave it. odmaxmin : Set (suc n) odmaxmin = (y : HOD) (z : Ordinal) → ((x : Ordinal)→ def (od y) x → x o< z) → odmax y o< osuc z -- OD ⇔ Ordinal leads a contradiction, so we need bounded HOD ¬OD-order : ( & : OD → Ordinal ) → ( * : Ordinal → OD ) → ( { x y : OD } → def y ( & x ) → & x o< & y) → ⊥ ¬OD-order & * c<→o< = o≤> <-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 _∋_ : ( a x : HOD ) → Set n _∋_ a x = odef a ( & x ) -- _c<_ : ( x a : HOD ) → Set n -- x c< a = a ∋ x d→∋ : ( a : HOD ) { x : Ordinal} → odef a x → a ∋ (* x) d→∋ a lt = subst (λ k → odef a k ) (sym &iso) lt -- 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 : {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 -- If we have reverse of c<→o<, everything becomes Ordinal ∈→c<→HOD=Ord : ( o<→c< : {x y : Ordinal } → x o< y → odef (* y) x ) → {x : HOD } → x ≡ Ord (& x) ∈→c<→HOD=Ord o<→c< {x} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where lemma1 : {y : Ordinal} → odef x y → odef (Ord (& x)) y lemma1 {y} lt = subst ( λ k → k o< & x ) &iso (c<→o< {* y} {x} (d→∋ x lt)) lemma2 : {y : Ordinal} → odef (Ord (& x)) y → odef x y lemma2 {y} lt = subst (λ k → odef k y ) *iso (o<→c< {y} {& x} lt ) -- avoiding lv != Zero error orefl : { x : HOD } → { y : Ordinal } → & x ≡ y → & x ≡ y orefl refl = refl ==-iso : { x y : HOD } → od (* (& x)) == od (* (& y)) → od x == od y ==-iso {x} {y} eq = record { eq→ = λ {z} d → lemma ( eq→ eq (subst (λ k → odef k z ) (sym *iso) d )) ; eq← = λ {z} d → lemma ( eq← eq (subst (λ k → odef k z ) (sym *iso) d )) } where lemma : {x : HOD } {z : Ordinal } → odef (* (& x)) z → odef x z lemma {x} {z} d = subst (λ k → odef k z) (*iso) d =-iso : {x y : HOD } → (od x == od y) ≡ (od (* (& x)) == od y) =-iso {_} {y} = cong ( λ k → od k == od y ) (sym *iso) ord→== : { x y : HOD } → & x ≡ & y → od x == od y ord→== {x} {y} eq = ==-iso (lemma (& x) (& y) (orefl eq)) where lemma : ( ox oy : Ordinal ) → ox ≡ oy → od (* ox) == od (* oy) lemma ox ox refl = ==-refl o≡→== : { x y : Ordinal } → x ≡ y → od (* x) == od (* y) o≡→== {x} {.x} refl = ==-refl *≡*→≡ : { x y : Ordinal } → * x ≡ * y → x ≡ y *≡*→≡ eq = subst₂ (λ j k → j ≡ k ) &iso &iso ( cong (&) eq ) &≡&→≡ : { x y : HOD } → & x ≡ & y → x ≡ y &≡&→≡ eq = subst₂ (λ j k → j ≡ k ) *iso *iso ( cong (*) eq ) o∅≡od∅ : * (o∅ ) ≡ od∅ o∅≡od∅ = ==→o≡ lemma where lemma0 : {x : Ordinal} → odef (* o∅) x → odef od∅ x lemma0 {x} lt with c<→o< {* x} {* o∅} (subst (λ k → odef (* o∅) k ) (sym &iso) lt) ... | t = subst₂ (λ j k → j o< k ) &iso &iso t lemma1 : {x : Ordinal} → odef od∅ x → odef (* o∅) x lemma1 {x} lt = ⊥-elim (¬x<0 lt) lemma : od (* o∅) == od od∅ lemma = record { eq→ = lemma0 ; eq← = lemma1 } ord-od∅ : & (od∅ ) ≡ o∅ ord-od∅ = sym ( subst (λ k → k ≡ & (od∅ ) ) &iso (cong ( λ k → & k ) o∅≡od∅ ) ) ≡o∅→=od∅ : {x : HOD} → & x ≡ o∅ → od x == od od∅ ≡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)))) ; eq← = λ {y} lt → ⊥-elim ( ¬x<0 lt )} =od∅→≡o∅ : {x : HOD} → od x == od od∅ → & x ≡ o∅ =od∅→≡o∅ {x} eq = trans (cong (λ k → & k ) (==→o≡ {x} {od∅} eq)) ord-od∅ ≡od∅→=od∅ : {x : HOD} → x ≡ od∅ → od x == od od∅ ≡od∅→=od∅ {x} eq = ≡o∅→=od∅ (subst (λ k → & x ≡ k ) ord-od∅ ( cong & eq ) ) ∅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 (& y ) → ¬ ( od x == od od∅ ) ∅< {x} {y} d eq with eq→ (==-trans eq (==-sym ∅0) ) d ∅< {x} {y} d eq | lift () ∈∅< : { x : HOD } {y : Ordinal } → odef x y → o∅ o< (& x) ∈∅< {x} {y} d with trio< o∅ (& x) ... | tri< a ¬b ¬c = a ... | tri≈ ¬a b ¬c = ⊥-elim ( ∅< {x} {* y} (subst (λ k → odef x k ) (sym &iso) d ) ( ≡o∅→=od∅ (sym b) ) ) ... | tri> ¬a ¬b c = ⊥-elim ( ¬x<0 c ) ∅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 ≡ & x ) ∨ ( t ≡ & y ) } ; odmax = omax (& x) (& y) ; <odmax = lemma } where lemma : {t : Ordinal} → (t ≡ & x) ∨ (t ≡ & y) → t o< omax (& x) (& y) lemma {t} (case1 refl) = omax-x _ _ lemma {t} (case2 refl) = omax-y _ _ pair<y : {x y : HOD } → y ∋ x → & (x , x) o< osuc (& 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 -- another possible restriction. We require no minimality on odmax, so it may arbitrary larger. odmax<& : { x y : HOD } → x ∋ y → Set n odmax<& {x} {y} x∋y = odmax x o< & x in-codomain : (X : HOD ) → ( ψ : HOD → HOD ) → OD in-codomain X ψ = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( odef X y ∧ ( x ≡ & (ψ (* y ))))) } _∩_ : ( A B : HOD ) → HOD 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))} record _⊆_ ( A B : HOD ) : Set (suc n) where field incl : { x : HOD } → A ∋ x → B ∋ x open _⊆_ infixr 220 _⊆_ -- if we have & (x , x) ≡ osuc (& x), ⊆→o≤ → c<→o< ⊆→o≤→c<→o< : ({x : HOD} → & (x , x) ≡ osuc (& x) ) → ({y z : HOD } → ({x : Ordinal} → def (od y) x → def (od z) x ) → & y o< osuc (& z) ) → {x y : HOD } → def (od y) ( & x ) → & x o< & y ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x with trio< (& x) (& 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 ≡ & x) ∨ (z ≡ & x) → & x ≡ z lemma (case1 refl) = refl lemma (case2 refl) = refl y⊆x,x : {z : Ordinal} → 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 (& y) o< & (x , x) lemma1 = subst (λ k → osuc (& y) o< k ) (sym (peq {x})) (osucc c ) ε-induction : { ψ : HOD → Set (suc 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 ) *iso (ε-induction-ord (osuc (& x)) <-osuc ) where induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (* oy)) → ψ (* ox) induction ox prev = ind ( λ {y} lt → subst (λ k → ψ k ) *iso (prev (& y) (o<-subst (c<→o< lt) refl &iso ))) ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (* oy) ε-induction-ord ox {oy} lt = TransFinite {λ oy → ψ (* oy)} induction oy Select : (X : HOD ) → ((x : HOD ) → Set n ) → HOD Select X ψ = record { od = record { def = λ x → ( odef X x ∧ ψ ( * 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 → & (ψ (* y)))) ∧ def (in-codomain X ψ) x } ; odmax = rmax ; <odmax = rmax<} where rmax : Ordinal rmax = sup-o X (λ y X∋y → & (ψ (* y))) rmax< : {y : Ordinal} → (y o< rmax) ∧ def (in-codomain X ψ) y → y o< rmax rmax< lt = proj1 lt -- -- If we have LEM, Replace' is equivalent to Replace -- in-codomain' : (X : HOD ) → ((x : HOD) → X ∋ x → HOD) → OD in-codomain' X ψ = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( odef X y ∧ ((lt : odef X y) → x ≡ & (ψ (* y ) (d→∋ X lt) )))) } Replace' : (X : HOD) → ((x : HOD) → X ∋ x → HOD) → HOD Replace' X ψ = record { od = record { def = λ x → (x o< sup-o X (λ y X∋y → & (ψ (* y) (d→∋ X X∋y) ))) ∧ def (in-codomain' X ψ) x } ; odmax = rmax ; <odmax = rmax< } where rmax : Ordinal rmax = sup-o X (λ y X∋y → & (ψ (* y) (d→∋ X X∋y))) rmax< : {y : Ordinal} → (y o< rmax) ∧ def (in-codomain' X ψ) y → y o< rmax rmax< lt = proj1 lt Union : HOD → HOD Union U = record { od = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((odef U u) ∧ (odef (* u) x))) } ; odmax = osuc (& U) ; <odmax = umax< } where umax< : {y : Ordinal} → ¬ ((u : Ordinal) → ¬ def (od U) u ∧ def (od (* u)) y) → y o< osuc (& U) umax< {y} not = lemma (FExists _ lemma1 not ) where lemma0 : {x : Ordinal} → def (od (* x)) y → y o< x lemma0 {x} x<y = subst₂ (λ j k → j o< k ) &iso &iso (c<→o< (d→∋ (* x) x<y )) lemma2 : {x : Ordinal} → def (od U) x → x o< & U lemma2 {x} x<U = subst (λ k → k o< & U ) &iso (c<→o< (d→∋ U x<U)) lemma1 : {x : Ordinal} → def (od U) x ∧ def (od (* x)) y → ¬ (& U o< y) lemma1 {x} lt u<y = o<> u<y (ordtrans (lemma0 (proj2 lt)) (lemma2 (proj1 lt)) ) lemma : ¬ ((& U) o< y ) → y o< osuc (& U) lemma not with trio< y (& 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 : HOD ) → Set n A ∈ B = B ∋ A OPwr : (A : HOD ) → HOD OPwr A = Ord ( sup-o (Ord (osuc (& A))) ( λ x A∋x → & ( A ∩ (* x)) ) ) Power : HOD → HOD Power A = Replace (OPwr (Ord (& A))) ( λ x → A ∩ x ) -- {_} : ZFSet → ZFSet -- { x } = ( x , x ) -- better to use (x , x) directly union→ : (X z u : HOD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z union→ X z u xx not = ⊥-elim ( not (& u) ( ⟪ proj1 xx , subst ( λ k → odef k (& z)) (sym *iso) (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 (* y) (& z) → ¬ ((u : HOD) → ¬ (X ∋ u) ∧ (u ∋ z)) lemma {y} xx not = not (* y) ⟪ d→∋ X (proj1 xx) , proj2 xx ⟫ data infinite-d : ( x : Ordinal ) → Set n where iφ : infinite-d o∅ isuc : {x : Ordinal } → infinite-d x → infinite-d (& ( Union (* x , (* x , * 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 & mapping divergence. -- We should have some axioms to prevent this such as & 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 : HOD infinite = record { od = record { def = λ x → infinite-d x } ; odmax = next o∅ ; <odmax = lemma } where u : (y : Ordinal ) → HOD u y = Union (* y , (* y , * y)) -- next< : {x y z : Ordinal} → x o< next z → y o< next x → y o< next z lemma8 : {y : Ordinal} → & (* y , * y) o< next (odmax (* y , * y)) lemma8 = ho< --- (x,y) < next (omax x y) < next (osuc y) = next y lemmaa : {x y : HOD} → & x o< & y → & (x , y) o< next (& y) lemmaa {x} {y} x<y = subst (λ k → & (x , y) o< k ) (sym nexto≡) (subst (λ k → & (x , y) o< next k ) (sym (omax< _ _ x<y)) ho< ) lemma81 : {y : Ordinal} → & (* y , * y) o< next (& (* y)) lemma81 {y} = nexto=n (subst (λ k → & (* y , * y) o< k ) (cong (λ k → next k) (omxx _)) lemma8) lemma9 : {y : Ordinal} → & (* y , (* y , * y)) o< next (& (* y , * y)) lemma9 = lemmaa (c<→o< (case1 refl)) lemma71 : {y : Ordinal} → & (* y , (* y , * y)) o< next (& (* y)) lemma71 = next< lemma81 lemma9 lemma1 : {y : Ordinal} → & (u y) o< next (osuc (& (* y , (* y , * y)))) lemma1 = ho< --- main recursion lemma : {y : Ordinal} → infinite-d y → y o< next o∅ lemma {o∅} iφ = x<nx lemma (isuc {y} x) = next< (lemma x) (next< (subst (λ k → & (* y , (* y , * y)) o< next k) &iso lemma71 ) (nexto=n lemma1)) empty : (x : HOD ) → ¬ (od∅ ∋ x) empty x = ¬x<0 _=h=_ : (x y : HOD) → Set n x =h= y = od x == od y pair→ : ( x y t : HOD ) → (x , y) ∋ t → ( t =h= x ) ∨ ( t =h= y ) pair→ x y t (case1 t≡x ) = case1 (subst₂ (λ j k → j =h= k ) *iso *iso (o≡→== t≡x )) pair→ x y t (case2 t≡y ) = case2 (subst₂ (λ j k → j =h= k ) *iso *iso (o≡→== t≡y )) pair← : ( x y t : HOD ) → ( t =h= x ) ∨ ( t =h= y ) → (x , y) ∋ t pair← x y t (case1 t=h=x) = case1 (cong (λ k → & k ) (==→o≡ t=h=x)) pair← x y t (case2 t=h=y) = case2 (cong (λ k → & k ) (==→o≡ t=h=y)) 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 &iso) refl ) ... | ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt &iso refl )) 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} = ⟪ ( λ cond → ⟪ proj1 cond , ψiso {ψ} (proj2 cond) (sym *iso) ⟫ ) , ( λ select → ⟪ proj1 select , ψiso {ψ} (proj2 select) *iso ⟫ ) ⟫ selection-in-domain : {ψ : HOD → Set n} {X y : HOD} → Select X ψ ∋ y → X ∋ y selection-in-domain {ψ} {X} {y} lt = proj1 ((proj2 (selection {ψ} {X} )) lt) sup-c< : (ψ : HOD → HOD) → {X x : HOD} → X ∋ x → & (ψ x) o< (sup-o X (λ y X∋y → & (ψ (* y)))) sup-c< ψ {X} {x} lt = subst (λ k → & (ψ k) o< _ ) *iso (sup-o< X lt ) replacement← : {ψ : HOD → HOD} (X x : HOD) → X ∋ x → Replace X ψ ∋ ψ x replacement← {ψ} X x lt = ⟪ sup-c< ψ {X} {x} lt , lemma ⟫ where lemma : def (in-codomain X ψ) (& (ψ x)) lemma not = ⊥-elim ( not ( & x ) ⟪ lt , cong (λ k → & (ψ k)) (sym *iso)⟫ ) 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 ∧ ((& x) ≡ & (ψ (* y)))) → ¬ ((y : Ordinal) → ¬ odef X y ∧ (* (& x) =h= ψ (* y))) lemma2 not not2 = not ( λ y d → not2 y ⟪ proj1 d , lemma3 (proj2 d)⟫) where lemma3 : {y : Ordinal } → (& x ≡ & (ψ (* y))) → (* (& x) =h= ψ (* y)) lemma3 {y} eq = subst (λ k → * (& x) =h= k ) *iso (o≡→== eq ) lemma : ( (y : HOD) → ¬ (x =h= ψ y)) → ( (y : Ordinal) → ¬ odef X y ∧ (* (& x) =h= ψ (* y)) ) lemma not y not2 = not (* y) (subst (λ k → k =h= ψ (* y)) *iso ( proj2 not2 )) --- --- Power Set --- --- First consider ordinals in HOD --- --- 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 → ⟪ (subst (λ k → odef b k ) &iso (inc (d→∋ a x<a))) , x<a ⟫ ; 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 (Ord a) ∩ t =h= t -- OPwr (Ord a) is a sup of (Ord a) ∩ t, so OPwr (Ord a) ∋ t -- OPwr A = Ord ( sup-o ( λ x → & ( A ∩ (* x )) ) ) -- ord-power← : (a : Ordinal ) (t : HOD) → ({x : HOD} → (t ∋ x → (Ord a) ∋ x)) → OPwr (Ord a) ∋ t ord-power← a t t→A = subst (λ k → odef (OPwr (Ord a)) k ) (lemma1 lemma-eq) lemma where lemma-eq : ((Ord a) ∩ t) =h= t eq→ lemma-eq {z} w = proj2 w eq← lemma-eq {z} w = ⟪ subst (λ k → odef (Ord a) k ) &iso ( t→A (d→∋ t w)) , w ⟫ lemma1 : {a : Ordinal } { t : HOD } → (eq : ((Ord a) ∩ t) =h= t) → & ((Ord a) ∩ (* (& t))) ≡ & t lemma1 {a} {t} eq = subst (λ k → & ((Ord a) ∩ k) ≡ & t ) (sym *iso) (cong (λ k → & k ) (==→o≡ eq )) lemma2 : (& t) o< (osuc (& (Ord a))) lemma2 = ⊆→o≤ {t} {Ord a} (λ {x} x<t → subst (λ k → def (od (Ord a)) k) &iso (t→A (d→∋ t x<t))) lemma : & ((Ord a) ∩ (* (& t)) ) o< sup-o (Ord (osuc (& (Ord a)))) (λ x lt → & ((Ord a) ∩ (* x))) lemma = sup-o< _ lemma2 -- -- Every set in HOD is a subset of Ordinals, so make OPwr (Ord (& A)) first -- then replace of all elements of the Power set by A ∩ y -- -- Power A = Replace (OPwr (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 = & A lemma2 : ¬ ( (y : HOD) → ¬ (t =h= (A ∩ y))) lemma2 = replacement→ {λ x → A ∩ x} (OPwr (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 ∩ * y))) lemma4 not = lemma2 ( λ y not1 → not (& y) (subst (λ k → t =h= ( A ∩ k )) (sym *iso) not1 )) lemma5 : {y : Ordinal} → t =h= (A ∩ * y) → ¬ ¬ (odef A (& x)) lemma5 {y} eq not = (lemma3 (* y) eq) not power← : (A t : HOD) → ({x : HOD} → (t ∋ x → A ∋ x)) → Power A ∋ t power← A t t→A = ⟪ lemma1 , lemma2 ⟫ where a = & 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 ∩ * (& t)) ≡ t lemma4 = let open ≡-Reasoning in begin A ∩ * (& t) ≡⟨ cong (λ k → A ∩ k) *iso ⟩ A ∩ t ≡⟨ sym (==→o≡ ( ∩-≡ {t} {A} t→A )) ⟩ t ∎ sup1 : Ordinal sup1 = sup-o (Ord (osuc (& (Ord (& A))))) (λ x A∋x → & ((Ord (& A)) ∩ (* x))) lemma9 : def (od (Ord (Ordinals.osuc O (& (Ord (& A)))))) (& (Ord (& A))) lemma9 = <-osuc lemmab : & ((Ord (& A)) ∩ (* (& (Ord (& A) )))) o< sup1 lemmab = sup-o< (Ord (osuc (& (Ord (& A))))) lemma9 lemmad : Ord (osuc (& A)) ∋ t lemmad = ⊆→o≤ (λ {x} lt → subst (λ k → def (od A) k ) &iso (t→A (d→∋ t lt))) lemmac : ((Ord (& A)) ∩ (* (& (Ord (& A) )))) =h= Ord (& A) lemmac = record { eq→ = lemmaf ; eq← = lemmag } where lemmaf : {x : Ordinal} → def (od ((Ord (& A)) ∩ (* (& (Ord (& A)))))) x → def (od (Ord (& A))) x lemmaf {x} lt = proj1 lt lemmag : {x : Ordinal} → def (od (Ord (& A))) x → def (od ((Ord (& A)) ∩ (* (& (Ord (& A)))))) x lemmag {x} lt = ⟪ lt , subst (λ k → def (od k) x) (sym *iso) lt ⟫ lemmae : & ((Ord (& A)) ∩ (* (& (Ord (& A))))) ≡ & (Ord (& A)) lemmae = cong (λ k → & k ) ( ==→o≡ lemmac) lemma7 : def (od (OPwr (Ord (& A)))) (& 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≤ {* (& t)} {* (& (Ord (& t)))} (λ {x} lt → lemmah lt )) where lemmah : {x : Ordinal } → def (od (* (& t))) x → def (od (* (& (Ord (& t))))) x lemmah {x} lt = subst (λ k → def (od k) x ) (sym *iso) (subst (λ k → k o< (& t)) &iso (c<→o< (subst₂ (λ j k → def (od j) k) *iso (sym &iso) lt ))) lemma7 | case1 eq | case1 eq1 = subst (λ k → k o< sup1) (trans lemmae lemmai) lemmab where lemmai : & (Ord (& A)) ≡ & t lemmai = let open ≡-Reasoning in begin & (Ord (& A)) ≡⟨ sym (cong (λ k → & (Ord k)) eq) ⟩ & (Ord (& t)) ≡⟨ sym &iso ⟩ & (* (& (Ord (& t)))) ≡⟨ sym eq1 ⟩ & (* (& t)) ≡⟨ &iso ⟩ & t ∎ lemma7 | case1 eq | case2 lt = ordtrans lemmaj (subst (λ k → k o< sup1) lemmae lemmab ) where lemmak : & (* (& (Ord (& t)))) ≡ & (Ord (& A)) lemmak = let open ≡-Reasoning in begin & (* (& (Ord (& t)))) ≡⟨ &iso ⟩ & (Ord (& t)) ≡⟨ cong (λ k → & (Ord k)) eq ⟩ & (Ord (& A)) ∎ lemmaj : & t o< & (Ord (& A)) lemmaj = subst₂ (λ j k → j o< k ) &iso lemmak lt lemma1 : & t o< sup-o (OPwr (Ord (& A))) (λ x lt → & (A ∩ (* x))) lemma1 = subst (λ k → & k o< sup-o (OPwr (Ord (& A))) (λ x lt → & (A ∩ (* x)))) lemma4 (sup-o< (OPwr (Ord (& A))) lemma7 ) lemma2 : def (in-codomain (OPwr (Ord (& A))) (_∩_ A)) (& t) lemma2 not = ⊥-elim ( not (& t) ⟪ lemma3 , lemma6 ⟫ ) where lemma6 : & t ≡ & (A ∩ * (& t)) lemma6 = cong ( λ k → & k ) (==→o≡ (subst (λ k → t =h= (A ∩ k)) (sym *iso) ( ∩-≡ {t} {A} t→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 &iso) (proj1 (eq (* x))) d eq← (extensionality0 {A} {B} eq ) {x} d = odef-iso {B} {A} (sym &iso) (proj2 (eq (* 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∅ = subst (λ k → odef infinite k ) lemma iφ where lemma : o∅ ≡ & od∅ lemma = let open ≡-Reasoning in begin o∅ ≡⟨ sym &iso ⟩ & ( * o∅ ) ≡⟨ cong ( λ k → & k ) o∅≡od∅ ⟩ & od∅ ∎ infinity : (x : HOD) → infinite ∋ x → infinite ∋ Union (x , (x , x )) infinity x lt = subst (λ k → odef infinite k ) lemma (isuc {& x} lt) where lemma : & (Union (* (& x) , (* (& x) , * (& x)))) ≡ & (Union (x , (x , x))) lemma = cong (λ k → & (Union ( k , ( k , k ) ))) *iso 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} } HOD→ZF : ZF HOD→ZF = record { ZFSet = HOD ; _∋_ = _∋_ ; _≈_ = _=h=_ ; ∅ = od∅ ; _,_ = _,_ ; Union = Union ; Power = Power ; Select = Select ; Replace = Replace ; infinite = infinite ; isZF = isZF }