Mercurial > hg > Members > kono > Proof > ZF-in-agda
view OD.agda @ 246:3506f53c7d83
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author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Mon, 26 Aug 2019 02:50:16 +0900 |
parents | 846e0926bb89 |
children | d09437fcfc7c |
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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 eq-refl : { x : OD } → x == x eq-refl {x} = record { eq→ = id ; eq← = id } open _==_ eq-sym : { x y : OD } → x == y → y == x eq-sym eq = record { eq→ = eq← eq ; eq← = eq→ eq } eq-trans : { x y z : OD } → x == y → y == z → x == z eq-trans x=y y=z = record { eq→ = λ t → eq→ y=z ( eq→ x=y t) ; eq← = λ t → eq← x=y ( eq← y=z 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 -- Ordinal in OD ( and ZFSet ) Transitive Set Ord : ( a : Ordinal ) → OD Ord a = record { def = λ y → y o< a } od∅ : OD od∅ = Ord o∅ postulate -- OD can be iso to a subset of Ordinal ( by means of Godel Set ) 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 -- we should prove this in agda, but simply put here ==→o≡ : { x y : OD } → (x == y) → x ≡ y -- next assumption causes ∀ x ∋ ∅ . It menas only an ordinal becomes a set -- o<→c< : {n : Level} {x y : Ordinal } → x o< y → def (ord→od y) x -- ord→od x ≡ Ord x results the same -- supermum as Replacement Axiom sup-o : ( Ordinal → Ordinal ) → Ordinal sup-o< : { ψ : Ordinal → Ordinal } → ∀ {x : Ordinal } → ψ x o< sup-o ψ -- contra-position of mimimulity of supermum required in Power Set Axiom -- sup-x : {n : Level } → ( Ordinal → Ordinal ) → Ordinal -- sup-lb : {n : Level } → { ψ : Ordinal → Ordinal } → {z : Ordinal } → z o< sup-o ψ → z o< osuc (ψ (sup-x ψ)) -- mimimul and x∋minimul is an Axiom of choice minimul : (x : OD ) → ¬ (x == od∅ )→ OD -- this should be ¬ (x == od∅ )→ ∃ ox → x ∋ Ord ox ( minimum of x ) x∋minimul : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → def x ( od→ord ( minimul x ne ) ) -- minimulity (may proved by ε-induction ) minimul-1 : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → (y : OD ) → ¬ ( def (minimul x ne) (od→ord y)) ∧ (def x (od→ord y) ) _∋_ : ( a x : OD ) → Set n _∋_ a x = def a ( od→ord x ) _c<_ : ( x a : OD ) → Set n x c< a = a ∋ x _c≤_ : OD → OD → Set (suc n) a c≤ b = (a ≡ b) ∨ ( b ∋ a ) 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 sup-od : ( OD → OD ) → OD sup-od ψ = Ord ( sup-o ( λ x → od→ord (ψ (ord→od x ))) ) 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 (ord→od 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 (ord→od 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 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 -- avoiding lv != Zero error orefl : { x : OD } → { 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} 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 )) } 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 =-iso : {x y : OD } → (x == y) ≡ (ord→od (od→ord x) == y) =-iso {_} {y} = cong ( λ k → k == y ) (sym oiso) ord→== : { x y : OD } → od→ord x ≡ od→ord y → x == 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 ox refl = eq-refl o≡→== : { x y : Ordinal } → x ≡ y → ord→od x == ord→od y o≡→== {x} {.x} refl = eq-refl c≤-refl : {n : Level} → ( x : OD ) → x c≤ x c≤-refl x = case1 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 lemma1 {x} lt = ⊥-elim (¬x<0 lt) lemma : ord→od o∅ == 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∅ eq→ ∅0 {w} (lift ()) eq← ∅0 {w} lt = lift (¬x<0 lt) ∅< : { x y : OD } → def x (od→ord y ) → ¬ ( x == od∅ ) ∅< {x} {y} d eq with eq→ (eq-trans eq (eq-sym ∅0) ) d ∅< {x} {y} d eq | lift () ∅6 : { x : OD } → ¬ ( 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 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 ppp : { p : Set n } { a : OD } → record { def = λ x → p } ∋ a → p ppp {p} {a} d = d -- -- Axiom of choice in intutionistic logic implies the exclude middle -- https://plato.stanford.edu/entries/axiom-choice/#AxiChoLog -- p∨¬p : ( p : Set n ) → p ∨ ( ¬ p ) -- assuming axiom of choice p∨¬p p with is-o∅ ( od→ord ( record { def = λ x → p } )) p∨¬p p | yes eq = case2 (¬p eq) where ps = record { def = λ x → p } lemma : ps == od∅ → p → ⊥ lemma eq p0 = ¬x<0 {od→ord ps} (eq→ eq p0 ) ¬p : (od→ord ps ≡ o∅) → p → ⊥ ¬p eq = lemma ( subst₂ (λ j k → j == k ) oiso o∅≡od∅ ( o≡→== eq )) p∨¬p p | no ¬p = case1 (ppp {p} {minimul ps (λ eq → ¬p (eqo∅ eq))} lemma) where ps = record { def = λ x → p } eqo∅ : ps == od∅ → od→ord ps ≡ o∅ eqo∅ eq = subst (λ k → od→ord ps ≡ k) ord-od∅ ( cong (λ k → od→ord k ) (==→o≡ eq)) lemma : ps ∋ minimul ps (λ eq → ¬p (eqo∅ eq)) lemma = x∋minimul ps (λ eq → ¬p (eqo∅ eq)) decp : ( p : Set n ) → Dec p -- assuming axiom of choice decp p with p∨¬p p decp p | case1 x = yes x decp p | case2 x = no x double-neg-eilm : {A : Set n} → ¬ ¬ A → A -- we don't have this in intutionistic logic double-neg-eilm {A} notnot with decp A -- assuming axiom of choice ... | yes p = p ... | no ¬p = ⊥-elim ( notnot ¬p ) OrdP : ( x : Ordinal ) ( y : OD ) → Dec ( Ord x ∋ y ) OrdP x y with trio< x (od→ord y) OrdP x y | tri< a ¬b ¬c = no ¬c OrdP x y | tri≈ ¬a refl ¬c = no ( o<¬≡ refl ) OrdP x y | tri> ¬a ¬b c = yes c -- 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 ))))) } -- 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 Def : (A : OD ) → OD Def A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) ) _⊆_ : ( A B : OD ) → ∀{ x : OD } → Set n _⊆_ A B {x} = A ∋ x → B ∋ x infixr 220 _⊆_ subset-lemma : {A x y : OD } → ( x ∋ y → ZFSubset A x ∋ y ) ⇔ ( _⊆_ x A {y} ) subset-lemma {A} {x} {y} = record { proj1 = λ z lt → proj1 (z lt) ; proj2 = λ x⊆A lt → record { proj1 = x⊆A lt ; proj2 = lt } } -- Constructible Set on α -- Def X = record { def = λ y → ∀ (x : OD ) → y < X ∧ y < od→ord x } -- L (Φ 0) = Φ -- L (OSuc lv n) = { Def ( L n ) } -- L (Φ (Suc n)) = Union (L α) (α < Φ (Suc n) ) -- L : {n : Level} → (α : Ordinal ) → OD -- L record { lv = Zero ; ord = (Φ .0) } = od∅ -- L record { lv = lx ; ord = (OSuc lv ox) } = Def ( L ( record { lv = lx ; ord = ox } ) ) -- L record { lv = (Suc lx) ; ord = (Φ (Suc lx)) } = -- Union ( L α ) -- cseq ( Ord (od→ord (L (record { lv = lx ; ord = Φ lx })))) -- L0 : {n : Level} → (α : Ordinal ) → L (osuc α) ∋ L α -- L1 : {n : Level} → (α β : Ordinal ) → α o< β → ∀ (x : OD ) → L α ∋ x → L β ∋ x OD→ZF : ZF OD→ZF = record { ZFSet = OD ; _∋_ = _∋_ ; _≈_ = _==_ ; ∅ = od∅ ; _,_ = _,_ ; Union = Union ; Power = Power ; Select = Select ; Replace = Replace ; infinite = infinite ; isZF = isZF } where ZFSet = OD 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 (ord→od x )))) ∧ def (in-codomain X ψ) x } _,_ : OD → OD → OD x , y = Ord (omax (od→ord x) (od→ord y)) _∩_ : ( 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 : ZFSet ) → Set n A ∈ B = B ∋ A Power : OD → OD Power A = Replace (Def (Ord (od→ord A))) ( λ x → A ∩ x ) {_} : ZFSet → ZFSet { x } = ( x , x ) 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 ) ) )) infinite : OD infinite = record { def = λ x → infinite-d x } infixr 200 _∈_ -- infixr 230 _∩_ _∪_ isZF : IsZF (OD ) _∋_ _==_ od∅ _,_ Union Power Select Replace infinite isZF = record { isEquivalence = record { refl = eq-refl ; sym = eq-sym; trans = eq-trans } ; pair = pair ; union→ = union→ ; union← = union← ; empty = empty ; power→ = power→ ; power← = power← ; extensionality = λ {A} {B} {w} → extensionality {A} {B} {w} -- ; ε-induction = {!!} ; infinity∅ = infinity∅ ; infinity = infinity ; selection = λ {X} {ψ} {y} → selection {X} {ψ} {y} ; replacement← = replacement← ; replacement→ = replacement→ ; choice-func = choice-func ; choice = choice } where pair : (A B : OD ) → ((A , B) ∋ A) ∧ ((A , B) ∋ B) proj1 (pair A B ) = omax-x (od→ord A) (od→ord B) proj2 (pair A B ) = omax-y (od→ord A) (od→ord B) empty : (x : OD ) → ¬ (od∅ ∋ x) empty x = ¬x<0 o<→c< : {x y : Ordinal } {z : OD }→ x o< y → _⊆_ (Ord x) (Ord y) {z} o<→c< lt lt1 = ordtrans lt1 lt ⊆→o< : {x y : Ordinal } → (∀ (z : OD) → _⊆_ (Ord x) (Ord y) {z} ) → 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 lt (ord→od y) (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 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 ))) union← X z UX∋z = TransFiniteExists _ 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 } ψiso : {ψ : OD → Set n} {x y : OD } → 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} {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 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} 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 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 )) --- --- Power Set --- --- First consider ordinals in OD --- --- ZFSubset A x = record { def = λ y → def A y ∧ def x y } subset of A --- Power X = ord→od ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) ) Power X is a sup of all subset of A -- -- ∩-≡ : { a b : OD } → ({x : OD } → (a ∋ x → b ∋ x)) → a == ( 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 } ; eq← = λ {x} x<a∩b → proj2 x<a∩b } -- -- we have t ∋ x → A ∋ x means t is a subset of A, that is ZFSubset A t == t -- Power A is a sup of ZFSubset A t, so Power A ∋ t -- 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} lemma refl (lemma1 lemma-eq )where lemma-eq : ZFSubset (Ord a) t == 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 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) (ord→od x))) lemma = sup-o< -- -- Every set in OD is a subset of Ordinals -- -- Power A = Replace (Def (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 P∋t {x} t∋x = TransFiniteExists _ 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) 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)) 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 t→A = record { proj1 = lemma1 ; proj2 = lemma2 } where a = od→ord A lemma0 : {x : OD} → t ∋ x → Ord a ∋ x lemma0 {x} t∋x = c<→o< (t→A t∋x) lemma3 : Def (Ord a) ∋ t lemma3 = ord-power← a t lemma0 lt1 : od→ord (A ∩ ord→od (od→ord t)) o< sup-o (λ x → od→ord (A ∩ ord→od x)) lt1 = sup-o< {λ x → od→ord (A ∩ ord→od x)} {od→ord t} 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 ∎ lemma1 : od→ord t o< sup-o (λ x → od→ord (A ∩ ord→od x)) lemma1 = subst (λ k → od→ord k o< sup-o (λ x → od→ord (A ∩ ord→od x))) lemma4 (sup-o< {λ x → od→ord (A ∩ ord→od x)} {od→ord t}) lemma2 : def (in-codomain (Def (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 ))) -- assuming axiom of choice regularity : (x : OD) (not : ¬ (x == od∅)) → (x ∋ minimul x not) ∧ (Select (minimul x not) (λ x₁ → (minimul x not ∋ x₁) ∧ (x ∋ x₁)) == od∅) proj1 (regularity x not ) = x∋minimul x not proj2 (regularity x not ) = record { eq→ = lemma1 ; eq← = λ {y} d → lemma2 {y} d } where lemma1 : {x₁ : Ordinal} → def (Select (minimul x not) (λ x₂ → (minimul x not ∋ x₂) ∧ (x ∋ x₂))) x₁ → def od∅ x₁ lemma1 {x₁} s = ⊥-elim ( minimul-1 x not (ord→od x₁) lemma3 ) where lemma3 : def (minimul x not) (od→ord (ord→od x₁)) ∧ def x (od→ord (ord→od x₁)) lemma3 = record { proj1 = def-subst {_} {_} {minimul x not} {_} (proj1 s) refl (sym diso) ; proj2 = proj2 (proj2 s) } lemma2 : {x₁ : Ordinal} → def od∅ x₁ → def (Select (minimul x not) (λ x₂ → (minimul x not ∋ x₂) ∧ (x ∋ x₂))) x₁ lemma2 {y} d = ⊥-elim (empty (ord→od y) (def-subst {_} {_} {od∅} {od→ord (ord→od y)} d refl (sym diso) )) 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 extensionality : {A B w : OD } → ((z : OD ) → (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 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 : 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 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 -- Axiom of choice ( is equivalent to the existence of minimul in our case ) -- ∀ X [ ∅ ∉ X → (∃ f : X → ⋃ X ) → ∀ A ∈ X ( f ( A ) ∈ A ) ] choice-func : (X : OD ) → {x : OD } → ¬ ( x == od∅ ) → ( X ∋ x ) → OD choice-func X {x} not X∋x = minimul x not choice : (X : OD ) → {A : OD } → ( X∋A : X ∋ A ) → (not : ¬ ( A == od∅ )) → A ∋ choice-func X not X∋A choice X {A} X∋A not = x∋minimul A not --- --- With assuption of OD is ordered, p ∨ ( ¬ p ) <=> axiom of choice --- record choiced ( X : OD) : Set (suc n) where field a-choice : OD is-in : X ∋ a-choice choice-func' : (X : OD ) → (p∨¬p : ( p : Set (suc n)) → p ∨ ( ¬ p )) → ¬ ( X == od∅ ) → choiced X choice-func' X p∨¬p not = have_to_find where ψ : ( ox : Ordinal ) → Set (suc n) ψ ox = (( x : Ordinal ) → x o< ox → ( ¬ def X x )) ∨ choiced X lemma-ord : ( ox : Ordinal ) → ψ ox lemma-ord ox = TransFinite {ψ} induction ox where ∋-p : (A x : OD ) → Dec ( A ∋ x ) ∋-p A x with p∨¬p (Lift (suc n) ( A ∋ x )) ∋-p A x | case1 (lift t) = yes t ∋-p A x | case2 t = no (λ x → t (lift x )) ∀-imply-or : {A : Ordinal → Set n } {B : Set (suc n) } → ((x : Ordinal ) → A x ∨ B) → ((x : Ordinal ) → A x) ∨ B ∀-imply-or {A} {B} ∀AB with p∨¬p (Lift ( suc n ) ((x : Ordinal ) → A x)) ∀-imply-or {A} {B} ∀AB | case1 (lift t) = case1 t ∀-imply-or {A} {B} ∀AB | case2 x = case2 (lemma (λ not → x (lift not ))) where lemma : ¬ ((x : Ordinal ) → A x) → B lemma not with p∨¬p B lemma not | case1 b = b lemma not | case2 ¬b = ⊥-elim (not (λ x → dont-orb (∀AB x) ¬b )) induction : (x : Ordinal) → ((y : Ordinal) → y o< x → ψ y) → ψ x induction x prev with ∋-p X ( ord→od x) ... | yes p = case2 ( record { a-choice = ord→od x ; is-in = p } ) ... | no ¬p = lemma where lemma1 : (y : Ordinal) → (y o< x → def X y → ⊥) ∨ choiced X lemma1 y with ∋-p X (ord→od y) lemma1 y | yes y<X = case2 ( record { a-choice = ord→od y ; is-in = y<X } ) lemma1 y | no ¬y<X = case1 ( λ lt y<X → ¬y<X (subst (λ k → def X k ) (sym diso) y<X ) ) lemma : ((y : Ordinals.ord O) → (O Ordinals.o< y) x → def X y → ⊥) ∨ choiced X lemma = ∀-imply-or lemma1 have_to_find : choiced X have_to_find with lemma-ord (od→ord X ) have_to_find | t = dont-or t ¬¬X∋x where ¬¬X∋x : ¬ ((x : Ordinal) → x o< (od→ord X) → def X x → ⊥) ¬¬X∋x nn = not record { eq→ = λ {x} lt → ⊥-elim (nn x (def→o< lt) lt) ; eq← = λ {x} lt → ⊥-elim ( ¬x<0 lt ) } _,_ = ZF._,_ OD→ZF 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