Mercurial > hg > Members > kono > Proof > ZF-in-agda
changeset 182:9f3c0e0b2bc9
remove ordinal-definable
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sun, 21 Jul 2019 12:11:50 +0900 |
| parents | 7012158bf2d9 |
| children | de3d87b7494f |
| files | HOD.agda OD.agda ordinal-definable.agda |
| diffstat | 3 files changed, 525 insertions(+), 916 deletions(-) [+] |
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--- a/HOD.agda Sun Jul 21 12:09:50 2019 +0900 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,525 +0,0 @@ -open import Level -module HOD where - -open import zf -open import ordinal -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 - --- Ordinal Definable Set - -record OD {n : Level} : Set (suc n) where - field - def : (x : Ordinal {n} ) → Set n - -open OD - -open Ordinal -open _∧_ - -record _==_ {n : Level} ( a b : OD {n} ) : Set n where - field - eq→ : ∀ { x : Ordinal {n} } → def a x → def b x - eq← : ∀ { x : Ordinal {n} } → def b x → def a x - -id : {n : Level} {A : Set n} → A → A -id x = x - -eq-refl : {n : Level} { x : OD {n} } → x == x -eq-refl {n} {x} = record { eq→ = id ; eq← = id } - -open _==_ - -eq-sym : {n : Level} { x y : OD {n} } → x == y → y == x -eq-sym eq = record { eq→ = eq← eq ; eq← = eq→ eq } - -eq-trans : {n : Level} { x y z : OD {n} } → 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) } - -⇔→== : {n : Level} { x y : OD {suc n} } → ( {z : Ordinal {suc n}} → def x z ⇔ def y z) → x == y -eq→ ( ⇔→== {n} {x} {y} eq ) {z} m = proj1 eq m -eq← ( ⇔→== {n} {x} {y} eq ) {z} m = proj2 eq m - --- Ordinal in OD ( and ZFSet ) Transitive Set -Ord : { n : Level } → ( a : Ordinal {n} ) → OD {n} -Ord {n} a = record { def = λ y → y o< a } - -od∅ : {n : Level} → OD {n} -od∅ {n} = Ord o∅ - -postulate - -- OD can be iso to a subset of Ordinal ( by means of Godel Set ) - od→ord : {n : Level} → OD {n} → Ordinal {n} - ord→od : {n : Level} → Ordinal {n} → OD {n} - c<→o< : {n : Level} {x y : OD {n} } → def y ( od→ord x ) → od→ord x o< od→ord y - oiso : {n : Level} {x : OD {n}} → ord→od ( od→ord x ) ≡ x - diso : {n : Level} {x : Ordinal {n}} → od→ord ( ord→od x ) ≡ x - -- we should prove this in agda, but simply put here - ==→o≡ : {n : Level} → { x y : OD {suc n} } → (x == y) → x ≡ y - -- next assumption causes ∀ x ∋ ∅ . It menas only an ordinal becomes a set - -- o<→c< : {n : Level} {x y : Ordinal {n} } → x o< y → def (ord→od y) x - -- ord→od x ≡ Ord x results the same - -- supermum as Replacement Axiom - sup-o : {n : Level } → ( Ordinal {n} → Ordinal {n}) → Ordinal {n} - sup-o< : {n : Level } → { ψ : Ordinal {n} → Ordinal {n}} → ∀ {x : Ordinal {n}} → ψ x o< sup-o ψ - -- contra-position of mimimulity of supermum required in Power Set Axiom - -- sup-x : {n : Level } → ( Ordinal {n} → Ordinal {n}) → Ordinal {n} - -- sup-lb : {n : Level } → { ψ : Ordinal {n} → Ordinal {n}} → {z : Ordinal {n}} → z o< sup-o ψ → z o< osuc (ψ (sup-x ψ)) - -- sup-lb : {n : Level } → ( ψ : Ordinal {n} → Ordinal {n}) → ( ∀ {x : Ordinal {n}} → ψx o< z ) → z o< osuc ( sup-o ψ ) - -- mimimul and x∋minimul is a weaker form of Axiom of choice - minimul : {n : Level } → (x : OD {suc n} ) → ¬ (x == od∅ )→ OD {suc n} - -- this should be ¬ (x == od∅ )→ ∃ ox → x ∋ Ord ox ( minimum of x ) - x∋minimul : {n : Level } → (x : OD {suc n} ) → ( ne : ¬ (x == od∅ ) ) → def x ( od→ord ( minimul x ne ) ) - minimul-1 : {n : Level } → (x : OD {suc n} ) → ( ne : ¬ (x == od∅ ) ) → (y : OD {suc n}) → ¬ ( def (minimul x ne) (od→ord y)) ∧ (def x (od→ord y) ) - -_∋_ : { n : Level } → ( a x : OD {n} ) → Set n -_∋_ {n} a x = def a ( od→ord x ) - -_c<_ : { n : Level } → ( x a : OD {n} ) → Set n -x c< a = a ∋ x - -_c≤_ : {n : Level} → OD {n} → OD {n} → Set (suc n) -a c≤ b = (a ≡ b) ∨ ( b ∋ a ) - -cseq : {n : Level} → OD {n} → OD {n} -cseq x = record { def = λ y → def x (osuc y) } where - -def-subst : {n : Level } {Z : OD {n}} {X : Ordinal {n} }{z : OD {n}} {x : Ordinal {n} }→ def Z X → Z ≡ z → X ≡ x → def z x -def-subst df refl refl = df - -sup-od : {n : Level } → ( OD {n} → OD {n}) → OD {n} -sup-od ψ = Ord ( sup-o ( λ x → od→ord (ψ (ord→od x ))) ) - -sup-c< : {n : Level } → ( ψ : OD {n} → OD {n}) → ∀ {x : OD {n}} → def ( sup-od ψ ) (od→ord ( ψ x )) -sup-c< {n} ψ {x} = def-subst {n} {_} {_} {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 : Ordinal {n} } → def (Ord a) x → { y : Ordinal {n} } → y o< x → def (Ord a) y -otrans {n} {a} {x} x<a {y} y<x = ordtrans y<x x<a - -∅3 : {n : Level} → { x : Ordinal {n}} → ( ∀(y : Ordinal {n}) → ¬ (y o< x ) ) → x ≡ o∅ {n} -∅3 {n} {x} = TransFinite {n} c2 c3 x where - c0 : Nat → Ordinal {n} → Set n - c0 lx x = (∀(y : Ordinal {n}) → ¬ (y o< x)) → x ≡ o∅ {n} - c2 : (lx : Nat) → c0 lx (record { lv = lx ; ord = Φ lx } ) - c2 Zero not = refl - c2 (Suc lx) not with not ( record { lv = lx ; ord = Φ lx } ) - ... | t with t (case1 ≤-refl ) - c2 (Suc lx) not | t | () - c3 : (lx : Nat) (x₁ : OrdinalD lx) → c0 lx (record { lv = lx ; ord = x₁ }) → c0 lx (record { lv = lx ; ord = OSuc lx x₁ }) - c3 lx (Φ .lx) d not with not ( record { lv = lx ; ord = Φ lx } ) - ... | t with t (case2 Φ< ) - c3 lx (Φ .lx) d not | t | () - c3 lx (OSuc .lx x₁) d not with not ( record { lv = lx ; ord = OSuc lx x₁ } ) - ... | t with t (case2 (s< s<refl ) ) - c3 lx (OSuc .lx x₁) d not | t | () - -∅5 : {n : Level} → { x : Ordinal {n} } → ¬ ( x ≡ o∅ {n} ) → o∅ {n} o< x -∅5 {n} {record { lv = Zero ; ord = (Φ .0) }} not = ⊥-elim (not refl) -∅5 {n} {record { lv = Zero ; ord = (OSuc .0 ord) }} not = case2 Φ< -∅5 {n} {record { lv = (Suc lv) ; ord = ord }} not = case1 (s≤s z≤n) - -ord-iso : {n : Level} {y : Ordinal {n} } → record { lv = lv (od→ord (ord→od y)) ; ord = ord (od→ord (ord→od y)) } ≡ record { lv = lv y ; ord = ord y } -ord-iso = cong ( λ k → record { lv = lv k ; ord = ord k } ) diso - --- avoiding lv != Zero error -orefl : {n : Level} → { x : OD {n} } → { y : Ordinal {n} } → od→ord x ≡ y → od→ord x ≡ y -orefl refl = refl - -==-iso : {n : Level} → { x y : OD {n} } → ord→od (od→ord x) == ord→od (od→ord y) → x == y -==-iso {n} {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 {n} } {z : Ordinal {n}} → def (ord→od (od→ord x)) z → def x z - lemma {x} {z} d = def-subst d oiso refl - -=-iso : {n : Level } {x y : OD {suc n} } → (x == y) ≡ (ord→od (od→ord x) == y) -=-iso {_} {_} {y} = cong ( λ k → k == y ) (sym oiso) - -ord→== : {n : Level} → { x y : OD {n} } → od→ord x ≡ od→ord y → x == y -ord→== {n} {x} {y} eq = ==-iso (lemma (od→ord x) (od→ord y) (orefl eq)) where - lemma : ( ox oy : Ordinal {n} ) → ox ≡ oy → (ord→od ox) == (ord→od oy) - lemma ox ox refl = eq-refl - -o≡→== : {n : Level} → { x y : Ordinal {n} } → x ≡ y → ord→od x == ord→od y -o≡→== {n} {x} {.x} refl = eq-refl - ->→¬< : {x y : Nat } → (x < y ) → ¬ ( y < x ) ->→¬< (s≤s x<y) (s≤s y<x) = >→¬< x<y y<x - -c≤-refl : {n : Level} → ( x : OD {n} ) → x c≤ x -c≤-refl x = case1 refl - -∋→o< : {n : Level} → { a x : OD {suc n} } → a ∋ x → od→ord x o< od→ord a -∋→o< {n} {a} {x} lt = t where - t : (od→ord x) o< (od→ord a) - t = c<→o< {suc n} {x} {a} lt - -o∅≡od∅ : {n : Level} → ord→od (o∅ {suc n}) ≡ od∅ {suc n} -o∅≡od∅ {n} = ==→o≡ lemma where - lemma0 : {x : Ordinal} → def (ord→od o∅) x → def od∅ x - lemma0 {x} lt = o<-subst (c<→o< {suc n} {ord→od x} {ord→od o∅} (def-subst {suc n} {ord→od o∅} {x} lt refl (sym diso)) ) diso diso - lemma1 : {x : Ordinal} → def od∅ x → def (ord→od o∅) x - lemma1 (case1 ()) - lemma1 (case2 ()) - lemma : ord→od o∅ == od∅ - lemma = record { eq→ = lemma0 ; eq← = lemma1 } - -ord-od∅ : {n : Level} → od→ord (od∅ {suc n}) ≡ o∅ {suc n} -ord-od∅ {n} = sym ( subst (λ k → k ≡ od→ord (od∅ {suc n}) ) diso (cong ( λ k → od→ord k ) o∅≡od∅ ) ) - -o<→¬c> : {n : Level} → { x y : OD {n} } → (od→ord x ) o< ( od→ord y) → ¬ (y c< x ) -o<→¬c> {n} {x} {y} olt clt = o<> olt (c<→o< clt ) where - -o≡→¬c< : {n : Level} → { x y : OD {n} } → (od→ord x ) ≡ ( od→ord y) → ¬ x c< y -o≡→¬c< {n} {x} {y} oeq lt = o<¬≡ (orefl oeq ) (c<→o< lt) - -∅0 : {n : Level} → record { def = λ x → Lift n ⊥ } == od∅ {n} -eq→ ∅0 {w} (lift ()) -eq← ∅0 {w} (case1 ()) -eq← ∅0 {w} (case2 ()) - -∅< : {n : Level} → { x y : OD {n} } → def x (od→ord y ) → ¬ ( x == od∅ {n} ) -∅< {n} {x} {y} d eq with eq→ (eq-trans eq (eq-sym ∅0) ) d -∅< {n} {x} {y} d eq | lift () - -∅6 : {n : Level} → { x : OD {suc n} } → ¬ ( x ∋ x ) -- no Russel paradox -∅6 {n} {x} x∋x = o<¬≡ refl ( c<→o< {suc n} {x} {x} x∋x ) - -def-iso : {n : Level} {A B : OD {n}} {x y : Ordinal {n}} → x ≡ y → (def A y → def B y) → def A x → def B x -def-iso refl t = t - -is-o∅ : {n : Level} → ( x : Ordinal {suc n} ) → Dec ( x ≡ o∅ {suc n} ) -is-o∅ {n} record { lv = Zero ; ord = (Φ .0) } = yes refl -is-o∅ {n} record { lv = Zero ; ord = (OSuc .0 ord₁) } = no ( λ () ) -is-o∅ {n} record { lv = (Suc lv₁) ; ord = ord } = no (λ()) - -OrdP : {n : Level} → ( x : Ordinal {suc n} ) ( y : OD {suc n} ) → Dec ( Ord x ∋ y ) -OrdP {n} x y with trio< x (od→ord y) -OrdP {n} x y | tri< a ¬b ¬c = no ¬c -OrdP {n} x y | tri≈ ¬a refl ¬c = no ( o<¬≡ refl ) -OrdP {n} 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 (suc n) (suc (suc n)) - -in-codomain : {n : Level} → (X : OD {suc n} ) → ( ψ : OD {suc n} → OD {suc n} ) → OD {suc n} -in-codomain {n} X ψ = record { def = λ x → ¬ ( (y : Ordinal {suc n}) → ¬ ( 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 : {n : Level} → (A x : OD {suc n} ) → OD {suc n} -ZFSubset A x = record { def = λ y → def A y ∧ def x y } where - -Def : {n : Level} → (A : OD {suc n}) → OD {suc n} -Def {n} A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) ) -- Ord x does not help ord-power→ - --- 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 {suc n}) → OD {suc n} -L {n} record { lv = Zero ; ord = (Φ .0) } = od∅ -L {n} record { lv = lx ; ord = (OSuc lv ox) } = Def ( L {n} ( record { lv = lx ; ord = ox } ) ) -L {n} record { lv = (Suc lx) ; ord = (Φ (Suc lx)) } = -- Union ( L α ) - cseq ( Ord (od→ord (L {n} (record { lv = lx ; ord = Φ lx })))) - --- L0 : {n : Level} → (α : Ordinal {suc n}) → L (osuc α) ∋ L α --- L1 : {n : Level} → (α β : Ordinal {suc n}) → α o< β → ∀ (x : OD {suc n}) → L α ∋ x → L β ∋ x - - -OD→ZF : {n : Level} → ZF {suc (suc n)} {suc n} -OD→ZF {n} = record { - ZFSet = OD {suc n} - ; _∋_ = _∋_ - ; _≈_ = _==_ - ; ∅ = od∅ - ; _,_ = _,_ - ; Union = Union - ; Power = Power - ; Select = Select - ; Replace = Replace - ; infinite = infinite - ; isZF = isZF - } where - ZFSet = OD {suc n} - Select : (X : OD {suc n} ) → ((x : OD {suc n} ) → Set (suc n) ) → OD {suc n} - Select X ψ = record { def = λ x → ( def X x ∧ ψ ( ord→od x )) } - Replace : OD {suc n} → (OD {suc n} → OD {suc n} ) → OD {suc n} - 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 {suc n} → OD {suc n} → OD {suc n} - 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 {suc n} → OD {suc n} - Union U = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((def U u) ∧ (def (ord→od u) x))) } - _∈_ : ( A B : ZFSet ) → Set (suc n) - A ∈ B = B ∋ A - _⊆_ : ( A B : ZFSet ) → ∀{ x : ZFSet } → Set (suc n) - _⊆_ A B {x} = A ∋ x → B ∋ x - Power : OD {suc n} → OD {suc n} - Power A = Replace (Def (Ord (od→ord A))) ( λ x → A ∩ x ) - ｛_｝ : ZFSet → ZFSet - ｛ x ｝ = ( x , x ) - - data infinite-d : ( x : Ordinal {suc n} ) → Set (suc n) where - iφ : infinite-d o∅ - isuc : {x : Ordinal {suc n} } → infinite-d x → - infinite-d (od→ord ( Union (ord→od x , (ord→od x , ord→od x ) ) )) - - infinite : OD {suc n} - infinite = record { def = λ x → infinite-d x } - - infixr 200 _∈_ - -- infixr 230 _∩_ _∪_ - infixr 220 _⊆_ - isZF : IsZF (OD {suc n}) _∋_ _==_ 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 = extensionality - ; minimul = minimul - ; regularity = regularity - ; infinity∅ = infinity∅ - ; infinity = infinity - ; selection = λ {X} {ψ} {y} → selection {X} {ψ} {y} - ; replacement← = replacement← - ; replacement→ = replacement→ - } where - - pair : (A B : OD {suc n} ) → ((A , B) ∋ A) ∧ ((A , B) ∋ B) - proj1 (pair A B ) = omax-x {n} (od→ord A) (od→ord B) - proj2 (pair A B ) = omax-y {n} (od→ord A) (od→ord B) - - empty : {n : Level } (x : OD {suc n} ) → ¬ (od∅ ∋ x) - empty x (case1 ()) - empty x (case2 ()) - - ord-⊆ : ( t x : OD {suc n} ) → _⊆_ t (Ord (od→ord t )) {x} - ord-⊆ t x lt = c<→o< lt - o<→c< : {x y : Ordinal {suc n}} {z : OD {suc n}}→ x o< y → _⊆_ (Ord x) (Ord y) {z} - o<→c< lt lt1 = ordtrans lt1 lt - - ⊆→o< : {x y : Ordinal {suc n}} → (∀ (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 {suc n} → Set (suc n)} {x y : OD {suc n}} → ψ x → x ≡ y → ψ y - ψiso {ψ} t refl = t - selection : {ψ : OD → Set (suc 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 {suc n} } → ({x : OD {suc n} } → (a ∋ x → b ∋ x)) → a == ( b ∩ a ) - ∩-≡ {a} {b} inc = record { - eq→ = λ {x} x<a → record { proj2 = x<a ; - proj1 = def-subst {suc n} {_} {_} {b} {x} (inc (def-subst {suc n} {_} {_} {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 {suc n} {_} {_} {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 {suc n} {_} {_} {(Ord a)} {z} - ( t→A (def-subst {suc n} {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso } - lemma1 : {n : Level } {a : Ordinal {suc n}} { t : OD {suc n}} - → (eq : ZFSubset (Ord a) t == t) → od→ord (ZFSubset (Ord a) (ord→od (od→ord t))) ≡ od→ord t - lemma1 {n} {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< - - -- double-neg-eilm : {n : Level } {A : Set n} → ¬ ¬ A → A -- we don't have this in intutionistic logic - -- - -- 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< {suc n} {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< {suc n} {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 ))) - - 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 {suc n} {_} {_} {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 {suc n} {_} {_} {od∅} {od→ord (ord→od y)} d refl (sym diso) )) - - extensionality : {A B : OD {suc n}} → ((z : OD) → (A ∋ z) ⇔ (B ∋ z)) → A == B - eq→ (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {A} {B} (sym diso) (proj1 (eq (ord→od x))) d - eq← (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {B} {A} (sym diso) (proj2 (eq (ord→od x))) d - - infinity∅ : infinite ∋ od∅ {suc n} - infinity∅ = def-subst {suc n} {_} {_} {infinite} {od→ord (od∅ {suc n})} 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 {suc n} {_} {_} {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 {suc n} ) → {x : OD } → ¬ ( x == od∅ ) → ( X ∋ x ) → OD - choice-func X {x} not X∋x = minimul x not - choice : (X : OD {suc n} ) → {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 - - -- another form of regularity - -- - ε-induction : {n m : Level} { ψ : OD {suc n} → Set m} - → ( {x : OD {suc n} } → ({ y : OD {suc n} } → x ∋ y → ψ y ) → ψ x ) - → (x : OD {suc n} ) → ψ x - ε-induction {n} {m} {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (lv (osuc (od→ord x))) (ord (osuc (od→ord x))) <-osuc) where - ε-induction-ord : (lx : Nat) ( ox : OrdinalD {suc n} lx ) {ly : Nat} {oy : OrdinalD {suc n} ly } - → (ly < lx) ∨ (oy d< ox ) → ψ (ord→od (record { lv = ly ; ord = oy } ) ) - ε-induction-ord Zero (Φ 0) (case1 ()) - ε-induction-ord Zero (Φ 0) (case2 ()) - ε-induction-ord lx (OSuc lx ox) {ly} {oy} y<x = - ind {ord→od (record { lv = ly ; ord = oy })} ( λ {y} lt → subst (λ k → ψ k ) oiso (ε-induction-ord lx ox (lemma y lt ))) where - lemma : (y : OD) → ord→od record { lv = ly ; ord = oy } ∋ y → od→ord y o< record { lv = lx ; ord = ox } - lemma y lt with osuc-≡< y<x - lemma y lt | case1 refl = o<-subst (c<→o< lt) refl diso - lemma y lt | case2 lt1 = ordtrans (o<-subst (c<→o< lt) refl diso) lt1 - ε-induction-ord (Suc lx) (Φ (Suc lx)) {ly} {oy} (case1 y<sox ) = - ind {ord→od (record { lv = ly ; ord = oy })} ( λ {y} lt → lemma y lt ) where - -- - -- if lv of z if less than x Ok - -- else lv z = lv x. We can create OSuc of z which is larger than z and less than x in lemma - -- - -- lx Suc lx (1) lz(a) <lx by case1 - -- ly(1) ly(2) (2) lz(b) <lx by case1 - -- lz(a) lz(b) lz(c) lz(c) <lx by case2 ( ly==lz==lx) - -- - lemma0 : { lx ly : Nat } → ly < Suc lx → lx < ly → ⊥ - lemma0 {Suc lx} {Suc ly} (s≤s lt1) (s≤s lt2) = lemma0 lt1 lt2 - lemma1 : {n : Level } {ly : Nat} {oy : OrdinalD {suc n} ly} → lv (od→ord (ord→od (record { lv = ly ; ord = oy }))) ≡ ly - lemma1 {n} {ly} {oy} = let open ≡-Reasoning in begin - lv (od→ord (ord→od (record { lv = ly ; ord = oy }))) - ≡⟨ cong ( λ k → lv k ) diso ⟩ - lv (record { lv = ly ; ord = oy }) - ≡⟨⟩ - ly - ∎ - lemma : (z : OD) → ord→od record { lv = ly ; ord = oy } ∋ z → ψ z - lemma z lt with c<→o< {suc n} {z} {ord→od (record { lv = ly ; ord = oy })} lt - lemma z lt | case1 lz<ly with <-cmp lx ly - lemma z lt | case1 lz<ly | tri< a ¬b ¬c = ⊥-elim ( lemma0 y<sox a) -- can't happen - lemma z lt | case1 lz<ly | tri≈ ¬a refl ¬c = -- ly(1) - subst (λ k → ψ k ) oiso (ε-induction-ord lx (Φ lx) {_} {ord (od→ord z)} (case1 (subst (λ k → lv (od→ord z) < k ) lemma1 lz<ly ) )) - lemma z lt | case1 lz<ly | tri> ¬a ¬b c = -- lz(a) - subst (λ k → ψ k ) oiso (ε-induction-ord lx (Φ lx) {_} {ord (od→ord z)} (case1 (<-trans lz<ly (subst (λ k → k < lx ) (sym lemma1) c)))) - lemma z lt | case2 lz=ly with <-cmp lx ly - lemma z lt | case2 lz=ly | tri< a ¬b ¬c = ⊥-elim ( lemma0 y<sox a) -- can't happen - lemma z lt | case2 lz=ly | tri> ¬a ¬b c with d<→lv lz=ly -- lz(b) - ... | eq = subst (λ k → ψ k ) oiso - (ε-induction-ord lx (Φ lx) {_} {ord (od→ord z)} (case1 (subst (λ k → k < lx ) (trans (sym lemma1)(sym eq) ) c ))) - lemma z lt | case2 lz=ly | tri≈ ¬a lx=ly ¬c with d<→lv lz=ly -- lz(c) - ... | eq = lemma6 {ly} {Φ lx} {oy} lx=ly (sym (subst (λ k → lv (od→ord z) ≡ k) lemma1 eq)) where - lemma5 : (ox : OrdinalD lx) → (lv (od→ord z) < lx) ∨ (ord (od→ord z) d< ox) → ψ z - lemma5 ox lt = subst (λ k → ψ k ) oiso (ε-induction-ord lx ox lt ) - lemma6 : { ly : Nat } { ox : OrdinalD {suc n} lx } { oy : OrdinalD {suc n} ly } → - lx ≡ ly → ly ≡ lv (od→ord z) → ψ z - lemma6 {ly} {ox} {oy} refl refl = lemma5 (OSuc lx (ord (od→ord z)) ) (case2 s<refl) -
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/OD.agda Sun Jul 21 12:11:50 2019 +0900 @@ -0,0 +1,525 @@ +open import Level +module OD where + +open import zf +open import ordinal +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 + +-- Ordinal Definable Set + +record OD {n : Level} : Set (suc n) where + field + def : (x : Ordinal {n} ) → Set n + +open OD + +open Ordinal +open _∧_ + +record _==_ {n : Level} ( a b : OD {n} ) : Set n where + field + eq→ : ∀ { x : Ordinal {n} } → def a x → def b x + eq← : ∀ { x : Ordinal {n} } → def b x → def a x + +id : {n : Level} {A : Set n} → A → A +id x = x + +eq-refl : {n : Level} { x : OD {n} } → x == x +eq-refl {n} {x} = record { eq→ = id ; eq← = id } + +open _==_ + +eq-sym : {n : Level} { x y : OD {n} } → x == y → y == x +eq-sym eq = record { eq→ = eq← eq ; eq← = eq→ eq } + +eq-trans : {n : Level} { x y z : OD {n} } → 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) } + +⇔→== : {n : Level} { x y : OD {suc n} } → ( {z : Ordinal {suc n}} → def x z ⇔ def y z) → x == y +eq→ ( ⇔→== {n} {x} {y} eq ) {z} m = proj1 eq m +eq← ( ⇔→== {n} {x} {y} eq ) {z} m = proj2 eq m + +-- Ordinal in OD ( and ZFSet ) Transitive Set +Ord : { n : Level } → ( a : Ordinal {n} ) → OD {n} +Ord {n} a = record { def = λ y → y o< a } + +od∅ : {n : Level} → OD {n} +od∅ {n} = Ord o∅ + +postulate + -- OD can be iso to a subset of Ordinal ( by means of Godel Set ) + od→ord : {n : Level} → OD {n} → Ordinal {n} + ord→od : {n : Level} → Ordinal {n} → OD {n} + c<→o< : {n : Level} {x y : OD {n} } → def y ( od→ord x ) → od→ord x o< od→ord y + oiso : {n : Level} {x : OD {n}} → ord→od ( od→ord x ) ≡ x + diso : {n : Level} {x : Ordinal {n}} → od→ord ( ord→od x ) ≡ x + -- we should prove this in agda, but simply put here + ==→o≡ : {n : Level} → { x y : OD {suc n} } → (x == y) → x ≡ y + -- next assumption causes ∀ x ∋ ∅ . It menas only an ordinal becomes a set + -- o<→c< : {n : Level} {x y : Ordinal {n} } → x o< y → def (ord→od y) x + -- ord→od x ≡ Ord x results the same + -- supermum as Replacement Axiom + sup-o : {n : Level } → ( Ordinal {n} → Ordinal {n}) → Ordinal {n} + sup-o< : {n : Level } → { ψ : Ordinal {n} → Ordinal {n}} → ∀ {x : Ordinal {n}} → ψ x o< sup-o ψ + -- contra-position of mimimulity of supermum required in Power Set Axiom + -- sup-x : {n : Level } → ( Ordinal {n} → Ordinal {n}) → Ordinal {n} + -- sup-lb : {n : Level } → { ψ : Ordinal {n} → Ordinal {n}} → {z : Ordinal {n}} → z o< sup-o ψ → z o< osuc (ψ (sup-x ψ)) + -- sup-lb : {n : Level } → ( ψ : Ordinal {n} → Ordinal {n}) → ( ∀ {x : Ordinal {n}} → ψx o< z ) → z o< osuc ( sup-o ψ ) + -- mimimul and x∋minimul is a weaker form of Axiom of choice + minimul : {n : Level } → (x : OD {suc n} ) → ¬ (x == od∅ )→ OD {suc n} + -- this should be ¬ (x == od∅ )→ ∃ ox → x ∋ Ord ox ( minimum of x ) + x∋minimul : {n : Level } → (x : OD {suc n} ) → ( ne : ¬ (x == od∅ ) ) → def x ( od→ord ( minimul x ne ) ) + minimul-1 : {n : Level } → (x : OD {suc n} ) → ( ne : ¬ (x == od∅ ) ) → (y : OD {suc n}) → ¬ ( def (minimul x ne) (od→ord y)) ∧ (def x (od→ord y) ) + +_∋_ : { n : Level } → ( a x : OD {n} ) → Set n +_∋_ {n} a x = def a ( od→ord x ) + +_c<_ : { n : Level } → ( x a : OD {n} ) → Set n +x c< a = a ∋ x + +_c≤_ : {n : Level} → OD {n} → OD {n} → Set (suc n) +a c≤ b = (a ≡ b) ∨ ( b ∋ a ) + +cseq : {n : Level} → OD {n} → OD {n} +cseq x = record { def = λ y → def x (osuc y) } where + +def-subst : {n : Level } {Z : OD {n}} {X : Ordinal {n} }{z : OD {n}} {x : Ordinal {n} }→ def Z X → Z ≡ z → X ≡ x → def z x +def-subst df refl refl = df + +sup-od : {n : Level } → ( OD {n} → OD {n}) → OD {n} +sup-od ψ = Ord ( sup-o ( λ x → od→ord (ψ (ord→od x ))) ) + +sup-c< : {n : Level } → ( ψ : OD {n} → OD {n}) → ∀ {x : OD {n}} → def ( sup-od ψ ) (od→ord ( ψ x )) +sup-c< {n} ψ {x} = def-subst {n} {_} {_} {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 : Ordinal {n} } → def (Ord a) x → { y : Ordinal {n} } → y o< x → def (Ord a) y +otrans {n} {a} {x} x<a {y} y<x = ordtrans y<x x<a + +∅3 : {n : Level} → { x : Ordinal {n}} → ( ∀(y : Ordinal {n}) → ¬ (y o< x ) ) → x ≡ o∅ {n} +∅3 {n} {x} = TransFinite {n} c2 c3 x where + c0 : Nat → Ordinal {n} → Set n + c0 lx x = (∀(y : Ordinal {n}) → ¬ (y o< x)) → x ≡ o∅ {n} + c2 : (lx : Nat) → c0 lx (record { lv = lx ; ord = Φ lx } ) + c2 Zero not = refl + c2 (Suc lx) not with not ( record { lv = lx ; ord = Φ lx } ) + ... | t with t (case1 ≤-refl ) + c2 (Suc lx) not | t | () + c3 : (lx : Nat) (x₁ : OrdinalD lx) → c0 lx (record { lv = lx ; ord = x₁ }) → c0 lx (record { lv = lx ; ord = OSuc lx x₁ }) + c3 lx (Φ .lx) d not with not ( record { lv = lx ; ord = Φ lx } ) + ... | t with t (case2 Φ< ) + c3 lx (Φ .lx) d not | t | () + c3 lx (OSuc .lx x₁) d not with not ( record { lv = lx ; ord = OSuc lx x₁ } ) + ... | t with t (case2 (s< s<refl ) ) + c3 lx (OSuc .lx x₁) d not | t | () + +∅5 : {n : Level} → { x : Ordinal {n} } → ¬ ( x ≡ o∅ {n} ) → o∅ {n} o< x +∅5 {n} {record { lv = Zero ; ord = (Φ .0) }} not = ⊥-elim (not refl) +∅5 {n} {record { lv = Zero ; ord = (OSuc .0 ord) }} not = case2 Φ< +∅5 {n} {record { lv = (Suc lv) ; ord = ord }} not = case1 (s≤s z≤n) + +ord-iso : {n : Level} {y : Ordinal {n} } → record { lv = lv (od→ord (ord→od y)) ; ord = ord (od→ord (ord→od y)) } ≡ record { lv = lv y ; ord = ord y } +ord-iso = cong ( λ k → record { lv = lv k ; ord = ord k } ) diso + +-- avoiding lv != Zero error +orefl : {n : Level} → { x : OD {n} } → { y : Ordinal {n} } → od→ord x ≡ y → od→ord x ≡ y +orefl refl = refl + +==-iso : {n : Level} → { x y : OD {n} } → ord→od (od→ord x) == ord→od (od→ord y) → x == y +==-iso {n} {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 {n} } {z : Ordinal {n}} → def (ord→od (od→ord x)) z → def x z + lemma {x} {z} d = def-subst d oiso refl + +=-iso : {n : Level } {x y : OD {suc n} } → (x == y) ≡ (ord→od (od→ord x) == y) +=-iso {_} {_} {y} = cong ( λ k → k == y ) (sym oiso) + +ord→== : {n : Level} → { x y : OD {n} } → od→ord x ≡ od→ord y → x == y +ord→== {n} {x} {y} eq = ==-iso (lemma (od→ord x) (od→ord y) (orefl eq)) where + lemma : ( ox oy : Ordinal {n} ) → ox ≡ oy → (ord→od ox) == (ord→od oy) + lemma ox ox refl = eq-refl + +o≡→== : {n : Level} → { x y : Ordinal {n} } → x ≡ y → ord→od x == ord→od y +o≡→== {n} {x} {.x} refl = eq-refl + +>→¬< : {x y : Nat } → (x < y ) → ¬ ( y < x ) +>→¬< (s≤s x<y) (s≤s y<x) = >→¬< x<y y<x + +c≤-refl : {n : Level} → ( x : OD {n} ) → x c≤ x +c≤-refl x = case1 refl + +∋→o< : {n : Level} → { a x : OD {suc n} } → a ∋ x → od→ord x o< od→ord a +∋→o< {n} {a} {x} lt = t where + t : (od→ord x) o< (od→ord a) + t = c<→o< {suc n} {x} {a} lt + +o∅≡od∅ : {n : Level} → ord→od (o∅ {suc n}) ≡ od∅ {suc n} +o∅≡od∅ {n} = ==→o≡ lemma where + lemma0 : {x : Ordinal} → def (ord→od o∅) x → def od∅ x + lemma0 {x} lt = o<-subst (c<→o< {suc n} {ord→od x} {ord→od o∅} (def-subst {suc n} {ord→od o∅} {x} lt refl (sym diso)) ) diso diso + lemma1 : {x : Ordinal} → def od∅ x → def (ord→od o∅) x + lemma1 (case1 ()) + lemma1 (case2 ()) + lemma : ord→od o∅ == od∅ + lemma = record { eq→ = lemma0 ; eq← = lemma1 } + +ord-od∅ : {n : Level} → od→ord (od∅ {suc n}) ≡ o∅ {suc n} +ord-od∅ {n} = sym ( subst (λ k → k ≡ od→ord (od∅ {suc n}) ) diso (cong ( λ k → od→ord k ) o∅≡od∅ ) ) + +o<→¬c> : {n : Level} → { x y : OD {n} } → (od→ord x ) o< ( od→ord y) → ¬ (y c< x ) +o<→¬c> {n} {x} {y} olt clt = o<> olt (c<→o< clt ) where + +o≡→¬c< : {n : Level} → { x y : OD {n} } → (od→ord x ) ≡ ( od→ord y) → ¬ x c< y +o≡→¬c< {n} {x} {y} oeq lt = o<¬≡ (orefl oeq ) (c<→o< lt) + +∅0 : {n : Level} → record { def = λ x → Lift n ⊥ } == od∅ {n} +eq→ ∅0 {w} (lift ()) +eq← ∅0 {w} (case1 ()) +eq← ∅0 {w} (case2 ()) + +∅< : {n : Level} → { x y : OD {n} } → def x (od→ord y ) → ¬ ( x == od∅ {n} ) +∅< {n} {x} {y} d eq with eq→ (eq-trans eq (eq-sym ∅0) ) d +∅< {n} {x} {y} d eq | lift () + +∅6 : {n : Level} → { x : OD {suc n} } → ¬ ( x ∋ x ) -- no Russel paradox +∅6 {n} {x} x∋x = o<¬≡ refl ( c<→o< {suc n} {x} {x} x∋x ) + +def-iso : {n : Level} {A B : OD {n}} {x y : Ordinal {n}} → x ≡ y → (def A y → def B y) → def A x → def B x +def-iso refl t = t + +is-o∅ : {n : Level} → ( x : Ordinal {suc n} ) → Dec ( x ≡ o∅ {suc n} ) +is-o∅ {n} record { lv = Zero ; ord = (Φ .0) } = yes refl +is-o∅ {n} record { lv = Zero ; ord = (OSuc .0 ord₁) } = no ( λ () ) +is-o∅ {n} record { lv = (Suc lv₁) ; ord = ord } = no (λ()) + +OrdP : {n : Level} → ( x : Ordinal {suc n} ) ( y : OD {suc n} ) → Dec ( Ord x ∋ y ) +OrdP {n} x y with trio< x (od→ord y) +OrdP {n} x y | tri< a ¬b ¬c = no ¬c +OrdP {n} x y | tri≈ ¬a refl ¬c = no ( o<¬≡ refl ) +OrdP {n} 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 (suc n) (suc (suc n)) + +in-codomain : {n : Level} → (X : OD {suc n} ) → ( ψ : OD {suc n} → OD {suc n} ) → OD {suc n} +in-codomain {n} X ψ = record { def = λ x → ¬ ( (y : Ordinal {suc n}) → ¬ ( 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 : {n : Level} → (A x : OD {suc n} ) → OD {suc n} +ZFSubset A x = record { def = λ y → def A y ∧ def x y } where + +Def : {n : Level} → (A : OD {suc n}) → OD {suc n} +Def {n} A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) ) -- Ord x does not help ord-power→ + +-- 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 {suc n}) → OD {suc n} +L {n} record { lv = Zero ; ord = (Φ .0) } = od∅ +L {n} record { lv = lx ; ord = (OSuc lv ox) } = Def ( L {n} ( record { lv = lx ; ord = ox } ) ) +L {n} record { lv = (Suc lx) ; ord = (Φ (Suc lx)) } = -- Union ( L α ) + cseq ( Ord (od→ord (L {n} (record { lv = lx ; ord = Φ lx })))) + +-- L0 : {n : Level} → (α : Ordinal {suc n}) → L (osuc α) ∋ L α +-- L1 : {n : Level} → (α β : Ordinal {suc n}) → α o< β → ∀ (x : OD {suc n}) → L α ∋ x → L β ∋ x + + +OD→ZF : {n : Level} → ZF {suc (suc n)} {suc n} +OD→ZF {n} = record { + ZFSet = OD {suc n} + ; _∋_ = _∋_ + ; _≈_ = _==_ + ; ∅ = od∅ + ; _,_ = _,_ + ; Union = Union + ; Power = Power + ; Select = Select + ; Replace = Replace + ; infinite = infinite + ; isZF = isZF + } where + ZFSet = OD {suc n} + Select : (X : OD {suc n} ) → ((x : OD {suc n} ) → Set (suc n) ) → OD {suc n} + Select X ψ = record { def = λ x → ( def X x ∧ ψ ( ord→od x )) } + Replace : OD {suc n} → (OD {suc n} → OD {suc n} ) → OD {suc n} + 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 {suc n} → OD {suc n} → OD {suc n} + 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 {suc n} → OD {suc n} + Union U = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((def U u) ∧ (def (ord→od u) x))) } + _∈_ : ( A B : ZFSet ) → Set (suc n) + A ∈ B = B ∋ A + _⊆_ : ( A B : ZFSet ) → ∀{ x : ZFSet } → Set (suc n) + _⊆_ A B {x} = A ∋ x → B ∋ x + Power : OD {suc n} → OD {suc n} + Power A = Replace (Def (Ord (od→ord A))) ( λ x → A ∩ x ) + ｛_｝ : ZFSet → ZFSet + ｛ x ｝ = ( x , x ) + + data infinite-d : ( x : Ordinal {suc n} ) → Set (suc n) where + iφ : infinite-d o∅ + isuc : {x : Ordinal {suc n} } → infinite-d x → + infinite-d (od→ord ( Union (ord→od x , (ord→od x , ord→od x ) ) )) + + infinite : OD {suc n} + infinite = record { def = λ x → infinite-d x } + + infixr 200 _∈_ + -- infixr 230 _∩_ _∪_ + infixr 220 _⊆_ + isZF : IsZF (OD {suc n}) _∋_ _==_ 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 = extensionality + ; minimul = minimul + ; regularity = regularity + ; infinity∅ = infinity∅ + ; infinity = infinity + ; selection = λ {X} {ψ} {y} → selection {X} {ψ} {y} + ; replacement← = replacement← + ; replacement→ = replacement→ + } where + + pair : (A B : OD {suc n} ) → ((A , B) ∋ A) ∧ ((A , B) ∋ B) + proj1 (pair A B ) = omax-x {n} (od→ord A) (od→ord B) + proj2 (pair A B ) = omax-y {n} (od→ord A) (od→ord B) + + empty : {n : Level } (x : OD {suc n} ) → ¬ (od∅ ∋ x) + empty x (case1 ()) + empty x (case2 ()) + + ord-⊆ : ( t x : OD {suc n} ) → _⊆_ t (Ord (od→ord t )) {x} + ord-⊆ t x lt = c<→o< lt + o<→c< : {x y : Ordinal {suc n}} {z : OD {suc n}}→ x o< y → _⊆_ (Ord x) (Ord y) {z} + o<→c< lt lt1 = ordtrans lt1 lt + + ⊆→o< : {x y : Ordinal {suc n}} → (∀ (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 {suc n} → Set (suc n)} {x y : OD {suc n}} → ψ x → x ≡ y → ψ y + ψiso {ψ} t refl = t + selection : {ψ : OD → Set (suc 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 {suc n} } → ({x : OD {suc n} } → (a ∋ x → b ∋ x)) → a == ( b ∩ a ) + ∩-≡ {a} {b} inc = record { + eq→ = λ {x} x<a → record { proj2 = x<a ; + proj1 = def-subst {suc n} {_} {_} {b} {x} (inc (def-subst {suc n} {_} {_} {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 {suc n} {_} {_} {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 {suc n} {_} {_} {(Ord a)} {z} + ( t→A (def-subst {suc n} {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso } + lemma1 : {n : Level } {a : Ordinal {suc n}} { t : OD {suc n}} + → (eq : ZFSubset (Ord a) t == t) → od→ord (ZFSubset (Ord a) (ord→od (od→ord t))) ≡ od→ord t + lemma1 {n} {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< + + -- double-neg-eilm : {n : Level } {A : Set n} → ¬ ¬ A → A -- we don't have this in intutionistic logic + -- + -- 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< {suc n} {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< {suc n} {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 ))) + + 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 {suc n} {_} {_} {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 {suc n} {_} {_} {od∅} {od→ord (ord→od y)} d refl (sym diso) )) + + extensionality : {A B : OD {suc n}} → ((z : OD) → (A ∋ z) ⇔ (B ∋ z)) → A == B + eq→ (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {A} {B} (sym diso) (proj1 (eq (ord→od x))) d + eq← (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {B} {A} (sym diso) (proj2 (eq (ord→od x))) d + + infinity∅ : infinite ∋ od∅ {suc n} + infinity∅ = def-subst {suc n} {_} {_} {infinite} {od→ord (od∅ {suc n})} 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 {suc n} {_} {_} {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 {suc n} ) → {x : OD } → ¬ ( x == od∅ ) → ( X ∋ x ) → OD + choice-func X {x} not X∋x = minimul x not + choice : (X : OD {suc n} ) → {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 + + -- another form of regularity + -- + ε-induction : {n m : Level} { ψ : OD {suc n} → Set m} + → ( {x : OD {suc n} } → ({ y : OD {suc n} } → x ∋ y → ψ y ) → ψ x ) + → (x : OD {suc n} ) → ψ x + ε-induction {n} {m} {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (lv (osuc (od→ord x))) (ord (osuc (od→ord x))) <-osuc) where + ε-induction-ord : (lx : Nat) ( ox : OrdinalD {suc n} lx ) {ly : Nat} {oy : OrdinalD {suc n} ly } + → (ly < lx) ∨ (oy d< ox ) → ψ (ord→od (record { lv = ly ; ord = oy } ) ) + ε-induction-ord Zero (Φ 0) (case1 ()) + ε-induction-ord Zero (Φ 0) (case2 ()) + ε-induction-ord lx (OSuc lx ox) {ly} {oy} y<x = + ind {ord→od (record { lv = ly ; ord = oy })} ( λ {y} lt → subst (λ k → ψ k ) oiso (ε-induction-ord lx ox (lemma y lt ))) where + lemma : (y : OD) → ord→od record { lv = ly ; ord = oy } ∋ y → od→ord y o< record { lv = lx ; ord = ox } + lemma y lt with osuc-≡< y<x + lemma y lt | case1 refl = o<-subst (c<→o< lt) refl diso + lemma y lt | case2 lt1 = ordtrans (o<-subst (c<→o< lt) refl diso) lt1 + ε-induction-ord (Suc lx) (Φ (Suc lx)) {ly} {oy} (case1 y<sox ) = + ind {ord→od (record { lv = ly ; ord = oy })} ( λ {y} lt → lemma y lt ) where + -- + -- if lv of z if less than x Ok + -- else lv z = lv x. We can create OSuc of z which is larger than z and less than x in lemma + -- + -- lx Suc lx (1) lz(a) <lx by case1 + -- ly(1) ly(2) (2) lz(b) <lx by case1 + -- lz(a) lz(b) lz(c) lz(c) <lx by case2 ( ly==lz==lx) + -- + lemma0 : { lx ly : Nat } → ly < Suc lx → lx < ly → ⊥ + lemma0 {Suc lx} {Suc ly} (s≤s lt1) (s≤s lt2) = lemma0 lt1 lt2 + lemma1 : {n : Level } {ly : Nat} {oy : OrdinalD {suc n} ly} → lv (od→ord (ord→od (record { lv = ly ; ord = oy }))) ≡ ly + lemma1 {n} {ly} {oy} = let open ≡-Reasoning in begin + lv (od→ord (ord→od (record { lv = ly ; ord = oy }))) + ≡⟨ cong ( λ k → lv k ) diso ⟩ + lv (record { lv = ly ; ord = oy }) + ≡⟨⟩ + ly + ∎ + lemma : (z : OD) → ord→od record { lv = ly ; ord = oy } ∋ z → ψ z + lemma z lt with c<→o< {suc n} {z} {ord→od (record { lv = ly ; ord = oy })} lt + lemma z lt | case1 lz<ly with <-cmp lx ly + lemma z lt | case1 lz<ly | tri< a ¬b ¬c = ⊥-elim ( lemma0 y<sox a) -- can't happen + lemma z lt | case1 lz<ly | tri≈ ¬a refl ¬c = -- ly(1) + subst (λ k → ψ k ) oiso (ε-induction-ord lx (Φ lx) {_} {ord (od→ord z)} (case1 (subst (λ k → lv (od→ord z) < k ) lemma1 lz<ly ) )) + lemma z lt | case1 lz<ly | tri> ¬a ¬b c = -- lz(a) + subst (λ k → ψ k ) oiso (ε-induction-ord lx (Φ lx) {_} {ord (od→ord z)} (case1 (<-trans lz<ly (subst (λ k → k < lx ) (sym lemma1) c)))) + lemma z lt | case2 lz=ly with <-cmp lx ly + lemma z lt | case2 lz=ly | tri< a ¬b ¬c = ⊥-elim ( lemma0 y<sox a) -- can't happen + lemma z lt | case2 lz=ly | tri> ¬a ¬b c with d<→lv lz=ly -- lz(b) + ... | eq = subst (λ k → ψ k ) oiso + (ε-induction-ord lx (Φ lx) {_} {ord (od→ord z)} (case1 (subst (λ k → k < lx ) (trans (sym lemma1)(sym eq) ) c ))) + lemma z lt | case2 lz=ly | tri≈ ¬a lx=ly ¬c with d<→lv lz=ly -- lz(c) + ... | eq = lemma6 {ly} {Φ lx} {oy} lx=ly (sym (subst (λ k → lv (od→ord z) ≡ k) lemma1 eq)) where + lemma5 : (ox : OrdinalD lx) → (lv (od→ord z) < lx) ∨ (ord (od→ord z) d< ox) → ψ z + lemma5 ox lt = subst (λ k → ψ k ) oiso (ε-induction-ord lx ox lt ) + lemma6 : { ly : Nat } { ox : OrdinalD {suc n} lx } { oy : OrdinalD {suc n} ly } → + lx ≡ ly → ly ≡ lv (od→ord z) → ψ z + lemma6 {ly} {ox} {oy} refl refl = lemma5 (OSuc lx (ord (od→ord z)) ) (case2 s<refl) +
--- a/ordinal-definable.agda Sun Jul 21 12:09:50 2019 +0900 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,391 +0,0 @@ -{-# OPTIONS --allow-unsolved-metas #-} - -open import Level -module ordinal-definable where - -open import zf -open import ordinal -open import HOD - -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 - --- Ordinal Definable Set - -open OD -open import Data.Unit - -open Ordinal -open _==_ - - -postulate - od=ord : {n : Level } → { x : Ordinal {n}} → ord→od x ≡ Ord x - -- a property of supermum required in Power Set Axiom - sup-x : {n : Level } → ( Ordinal {n} → Ordinal {n}) → Ordinal {n} - sup-lb : {n : Level } → { ψ : Ordinal {n} → Ordinal {n}} → {z : Ordinal {n}} → z o< sup-o ψ → z o< osuc (ψ (sup-x ψ)) - -- sup-lb : {n : Level } → ( ψ : Ordinal {n} → Ordinal {n}) → ( ∀ {x : Ordinal {n}} → ψx o< z ) → z o< osuc ( sup-o ψ ) - -o<→c< : {n : Level} {x y : Ordinal {n} } → x o< y → def (ord→od y) x -o<→c< {n} {x} {y} lt = def-subst {n} {_} {_} {ord→od y} {x} lt (sym od=ord) refl - -ord=od : {n : Level } → { x : OD {n}} → x ≡ Ord (od→ord x) -ord=od {n} {x} = subst ( λ k → k ≡ Ord (od→ord x) ) oiso od=ord - -transitive : {n : Level } { z y x : OD {suc n} } → z ∋ y → y ∋ x → z ∋ x -transitive {n} {z} {y} {x} z∋y x∋y with ordtrans ( c<→o< {suc n} {x} {y} x∋y ) ( c<→o< {suc n} {y} {z} z∋y ) -... | t = lemma0 (lemma t) where - lemma : ( od→ord x ) o< ( od→ord z ) → def ( ord→od ( od→ord z )) ( od→ord x) - lemma xo<z = o<→c< xo<z - lemma0 : def ( ord→od ( od→ord z )) ( od→ord x) → def z (od→ord x) - lemma0 dz = def-subst {suc n} { ord→od ( od→ord z )} { od→ord x} dz (oiso) refl - -o<→o> : {n : Level} → { x y : OD {n} } → (x == y) → (od→ord x ) o< ( od→ord y) → ⊥ -o<→o> {n} {x} {y} record { eq→ = xy ; eq← = yx } (case1 lt) with - yx (def-subst {n} {ord→od (od→ord y)} {od→ord x} (o<→c< (case1 lt )) oiso refl ) -... | oyx with o<¬≡ refl (c<→o< {n} {x} oyx ) -... | () -o<→o> {n} {x} {y} record { eq→ = xy ; eq← = yx } (case2 lt) with - yx (def-subst {n} {ord→od (od→ord y)} {od→ord x} (o<→c< (case2 lt )) oiso refl ) -... | oyx with o<¬≡ refl (c<→o< {n} {x} oyx ) -... | () - -==→o≡o : {n : Level} → { x y : Ordinal {suc n} } → ord→od x == ord→od y → x ≡ y -==→o≡o {n} {x} {y} eq with trio< {n} x y -==→o≡o {n} {x} {y} eq | tri< a ¬b ¬c = ⊥-elim ( o<→o> eq (o<-subst a (sym ord-iso) (sym ord-iso ))) -==→o≡o {n} {x} {y} eq | tri≈ ¬a b ¬c = b -==→o≡o {n} {x} {y} eq | tri> ¬a ¬b c = ⊥-elim ( o<→o> (eq-sym eq) (o<-subst c (sym ord-iso) (sym ord-iso ))) - -≡-def : {n : Level} → { x : Ordinal {suc n} } → x ≡ od→ord (record { def = λ z → z o< x } ) -≡-def {n} {x} = ==→o≡o {n} (subst (λ k → ord→od x == k ) (sym oiso) lemma ) where - lemma : ord→od x == record { def = λ z → z o< x } - eq→ lemma {w} z = subst₂ (λ k j → k o< j ) diso refl (subst (λ k → (od→ord ( ord→od w)) o< k ) diso t ) where - t : (od→ord ( ord→od w)) o< (od→ord (ord→od x)) - t = c<→o< {suc n} {ord→od w} {ord→od x} (def-subst {suc n} {_} {_} {ord→od x} {_} z refl (sym diso)) - eq← lemma {w} z = def-subst {suc n} {_} {_} {ord→od x} {w} ( o<→c< {suc n} {_} {_} z ) refl refl - -od≡-def : {n : Level} → { x : Ordinal {suc n} } → ord→od x ≡ record { def = λ z → z o< x } -od≡-def {n} {x} = subst (λ k → ord→od x ≡ k ) oiso (cong ( λ k → ord→od k ) (≡-def {n} {x} )) - -==→o≡1 : {n : Level} → { x y : OD {suc n} } → x == y → od→ord x ≡ od→ord y -==→o≡1 eq = ==→o≡o (subst₂ (λ k j → k == j ) (sym oiso) (sym oiso) eq ) - -==-def-l : {n : Level } {x y : Ordinal {suc n} } { z : OD {suc n} }→ (ord→od x == ord→od y) → def z x → def z y -==-def-l {n} {x} {y} {z} eq z>x = subst ( λ k → def z k ) (==→o≡o eq) z>x - -==-def-r : {n : Level } {x y : OD {suc n} } { z : Ordinal {suc n} }→ (x == y) → def x z → def y z -==-def-r {n} {x} {y} {z} eq z>x = subst (λ k → def k z ) (subst₂ (λ j k → j ≡ k ) oiso oiso (cong (λ k → ord→od k) (==→o≡1 eq))) z>x - -o<∋→ : {n : Level} → { a x : OD {suc n} } → od→ord x o< od→ord a → a ∋ x -o<∋→ {n} {a} {x} lt = subst₂ (λ k j → def k j ) oiso refl t where - t : def (ord→od (od→ord a)) (od→ord x) - t = o<→c< {suc n} {od→ord x} {od→ord a} lt - -o<→¬== : {n : Level} → { x y : OD {n} } → (od→ord x ) o< ( od→ord y) → ¬ (x == y ) -o<→¬== {n} {x} {y} lt eq = o<→o> eq lt - -tri-c< : {n : Level} → Trichotomous _==_ (_c<_ {suc n}) -tri-c< {n} x y with trio< {n} (od→ord x) (od→ord y) -tri-c< {n} x y | tri< a ¬b ¬c = tri< (def-subst (o<→c< a) oiso refl) (o<→¬== a) ( o<→¬c> a ) -tri-c< {n} x y | tri≈ ¬a b ¬c = tri≈ (o≡→¬c< b) (ord→== b) (o≡→¬c< (sym b)) -tri-c< {n} x y | tri> ¬a ¬b c = tri> ( o<→¬c> c) (λ eq → o<→¬== c (eq-sym eq ) ) (def-subst (o<→c< c) oiso refl) - -c<> : {n : Level } { x y : OD {suc n}} → x c< y → y c< x → ⊥ -c<> {n} {x} {y} x<y y<x with tri-c< x y -c<> {n} {x} {y} x<y y<x | tri< a ¬b ¬c = ¬c y<x -c<> {n} {x} {y} x<y y<x | tri≈ ¬a b ¬c = o<→o> b ( c<→o< x<y ) -c<> {n} {x} {y} x<y y<x | tri> ¬a ¬b c = ¬a x<y - - -is-∋ : {n : Level} → ( x y : OD {suc n} ) → Dec ( x ∋ y ) -is-∋ {n} x y with tri-c< x y -is-∋ {n} x y | tri< a ¬b ¬c = no ¬c -is-∋ {n} x y | tri≈ ¬a b ¬c = no ¬c -is-∋ {n} x y | tri> ¬a ¬b c = yes c - - -open _∧_ - --- --- This menas OD is Ordinal here --- -¬∅=→∅∈ : {n : Level} → { x : OD {suc n} } → ¬ ( x == od∅ {suc n} ) → x ∋ od∅ {suc n} -¬∅=→∅∈ {n} {x} ne = def-subst (lemma (od→ord x) (subst (λ k → ¬ (k == od∅ {suc n} )) (sym oiso) ne )) oiso refl where - lemma : (ox : Ordinal {suc n}) → ¬ (ord→od ox == od∅ {suc n} ) → ord→od ox ∋ od∅ {suc n} - lemma ox ne with is-o∅ ox - lemma ox ne | yes refl with ne ( ord→== lemma1 ) where - lemma1 : od→ord (ord→od o∅) ≡ od→ord od∅ - lemma1 = cong ( λ k → od→ord k ) o∅≡od∅ - lemma o∅ ne | yes refl | () - lemma ox ne | no ¬p = subst ( λ k → def (ord→od ox) (od→ord k) ) o∅≡od∅ (o<→c< (subst (λ k → k o< ox ) (sym diso) (∅5 ¬p)) ) - --- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) --- postulate f-extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality (suc n) (suc (suc n)) - -csuc : {n : Level} → OD {suc n} → OD {suc n} -csuc x = Ord ( osuc ( od→ord x )) - -Ord→ZF : {n : Level} → ZF {suc (suc n)} {suc n} -Ord→ZF {n} = record { - ZFSet = OD {suc n} - ; _∋_ = _∋_ - ; _≈_ = _==_ - ; ∅ = od∅ - ; _,_ = _,_ - ; Union = Union - ; Power = Power - ; Select = Select - ; Replace = Replace - ; infinite = ord→od ( record { lv = Suc Zero ; ord = Φ 1 } ) - ; isZF = isZF - } where - Select : OD {suc n} → (OD {suc n} → Set (suc n) ) → OD {suc n} - Select X ψ = record { def = λ x → ( def X x ∧ ψ ( ord→od x )) } - _,_ : OD {suc n} → OD {suc n} → OD {suc n} - x , y = record { def = λ z → z o< (omax (od→ord x) (od→ord y)) } - _∩_ : ( A B : OD {suc n} ) → OD - A ∩ B = record { def = λ x → def A x ∧ def B x } - Replace : OD {suc n} → (OD {suc n} → OD {suc n} ) → OD {suc n} - 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 } - Union : OD {suc n} → OD {suc n} - Union U = record { def = λ y → osuc y o< (od→ord U) } - -- power : ∀ X ∃ A ∀ t ( t ∈ A ↔ ( ∀ {x} → t ∋ x → X ∋ x ) - Power : OD {suc n} → OD {suc n} - Power A = Def A - ZFSet = OD {suc n} - _∈_ : ( A B : ZFSet ) → Set (suc n) - A ∈ B = B ∋ A - _⊆_ : ( A B : ZFSet ) → ∀{ x : ZFSet } → Set (suc n) - _⊆_ A B {x} = A ∋ x → B ∋ x - -- _∪_ : ( A B : ZFSet ) → ZFSet - -- A ∪ B = Select (A , B) ( λ x → (A ∋ x) ∨ ( B ∋ x ) ) - infixr 200 _∈_ - -- infixr 230 _∩_ _∪_ - infixr 220 _⊆_ - isZF : IsZF (OD {suc n}) _∋_ _==_ od∅ _,_ Union Power Select Replace (ord→od ( record { lv = Suc Zero ; ord = Φ 1} )) - 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 = extensionality - ; minimul = minimul - ; regularity = regularity - ; infinity∅ = infinity∅ - ; infinity = infinity - ; selection = λ {ψ} {X} {y} → selection {ψ} {X} {y} - ; replacement← = replacement← - ; replacement→ = replacement→ - } where - - pair : (A B : OD {suc n} ) → ((A , B) ∋ A) ∧ ((A , B) ∋ B) - proj1 (pair A B ) = omax-x {n} (od→ord A) (od→ord B) - proj2 (pair A B ) = omax-y {n} (od→ord A) (od→ord B) - - empty : (x : OD {suc n} ) → ¬ (od∅ ∋ x) - empty x (case1 ()) - empty x (case2 ()) - - --- - --- 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 - -- - -- if Power A ∋ t, from a propertiy of minimum sup there is osuc ZFSubset A ∋ t - -- then ZFSubset A ≡ t or ZFSubset A ∋ t. In the former case ZFSubset A ∋ x implies A ∋ x - -- In case of later, ZFSubset A ∋ t and t ∋ x implies ZFSubset A ∋ x by transitivity - -- - power→ : (A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → ¬ ¬ (A ∋ x) - power→ A t P∋t {x} t∋x = double-neg (proj1 lemma-s) where - minsup : OD - minsup = ZFSubset A ( ord→od ( sup-x (λ x → od→ord ( ZFSubset A (ord→od x))))) - lemma-t : csuc minsup ∋ t - lemma-t = def-subst (o<→c< (o<-subst (sup-lb (o<-subst (c<→o< {!!}) refl diso )) refl refl ) ) {!!} {!!} - lemma-s : ZFSubset A ( ord→od ( sup-x (λ x → od→ord ( ZFSubset A (ord→od x))))) ∋ x - lemma-s with osuc-≡< ( o<-subst (c<→o< {!!} ) refl diso ) - lemma-s | case1 eq = def-subst ( ==-def-r (o≡→== eq) (subst (λ k → def k (od→ord x)) (sym oiso) t∋x ) ) oiso refl - lemma-s | case2 lt = transitive {n} {minsup} {t} {x} (def-subst (o<→c< lt) oiso refl ) t∋x - -- - -- 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 - -- - power← : (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t - power← A t t→A = def-subst {suc n} {_} {_} {Power A} {od→ord t} - lemma refl lemma1 where - lemma-eq : ZFSubset A t == t - eq→ lemma-eq {z} w = proj2 w - eq← lemma-eq {z} w = record { proj2 = w ; - proj1 = def-subst {suc n} {_} {_} {A} {z} ( t→A (def-subst {suc n} {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso } - lemma1 : od→ord (ZFSubset A (ord→od (od→ord t))) ≡ od→ord t - lemma1 = subst (λ k → od→ord (ZFSubset A k) ≡ od→ord t ) (sym oiso) (==→o≡1 (lemma-eq)) - lemma : od→ord (ZFSubset A (ord→od (od→ord t)) ) o< sup-o (λ x → od→ord (ZFSubset A (ord→od x))) - lemma = sup-o< - - union-lemma-u : {X z : OD {suc n}} → (U>z : Union X ∋ z ) → csuc z ∋ z - union-lemma-u {X} {z} U>z = def-subst (lemma <-osuc ) od=ord refl where - lemma : {oz ooz : Ordinal {suc n}} → oz o< ooz → def (ord→od ooz) oz - lemma {oz} {ooz} lt = def-subst {suc n} {ord→od ooz} (o<→c< lt) refl refl - union→ : (X z u : OD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z - union→ X y u xx with trio< ( od→ord u ) ( osuc ( od→ord y )) - union→ X y u xx | tri< a ¬b ¬c with osuc-< a (c<→o< (proj2 xx)) - union→ X y u xx | tri< a ¬b ¬c | () - union→ X y u xx | tri≈ ¬a b ¬c = lemma b (c<→o< (proj1 xx )) where - lemma : {oX ou ooy : Ordinal {suc n}} → ou ≡ ooy → ou o< oX → ooy o< oX - lemma refl lt = lt - union→ X y u xx | tri> ¬a ¬b c = ordtrans {suc n} {osuc ( od→ord y )} {od→ord u} {od→ord X} c ( c<→o< (proj1 xx )) - union← : (X z : OD) (X∋z : Union X ∋ z) → ¬ ( (u : OD ) → ¬ ((X ∋ u) ∧ (u ∋ z ))) -- (X ∋ csuc z) ∧ (csuc z ∋ z ) - union← X z X∋z not = not (csuc z) - record { proj1 = def-subst {suc n} {_} {_} {X} {od→ord (csuc z )} (o<→c< X∋z) oiso (trans (sym diso) {!!} ) ; proj2 = union-lemma-u X∋z } - - ψiso : {ψ : OD {suc n} → Set (suc n)} {x y : OD {suc n}} → ψ x → x ≡ y → ψ y - ψiso {ψ} t refl = t - selection : {ψ : OD → Set (suc 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 (def-subst (proj2 lt) {!!} refl )) where - lemma2 : ¬ ((y : Ordinal) → ¬ def X y ∧ ((od→ord x) ≡ od→ord (ψ (Ord y)))) - → ¬ ((y : Ordinal) → ¬ def X y ∧ (ord→od (od→ord x) == ψ (Ord 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 y))) → (ord→od (od→ord x) == ψ (Ord 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 y)) ) - lemma not y not2 = not (Ord y) (subst (λ k → k == ψ (Ord y)) oiso ( proj2 not2 )) - - minimul-o : (x : OD {suc n} ) → ¬ (x == od∅ )→ OD {suc n} - minimul-o x not = od∅ - 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 ) = def-subst (¬∅=→∅∈ not) {!!} refl - proj2 (regularity x not ) = record { eq→ = reg ; eq← = lemma } where - lemma : {ox : Ordinal} → def od∅ ox → def (Select (minimul x not) (λ y → (minimul x not ∋ y) ∧ (x ∋ y))) ox - lemma (case1 ()) - lemma (case2 ()) - reg : {y : Ordinal} → def (Select (minimul x not) (λ x₂ → (minimul x not ∋ x₂) ∧ (x ∋ x₂))) y → def od∅ y - reg {y} t = ⊥-elim ( ¬x<0 (def-subst (proj1 (proj2 t )) {!!} refl )) - - extensionality : {A B : OD {suc n}} → ((z : OD) → (A ∋ z) ⇔ (B ∋ z)) → A == B - eq→ (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {A} {B} (sym diso) (proj1 (eq (ord→od x))) d - eq← (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {B} {A} (sym diso) (proj2 (eq (ord→od x))) d - - xx-union : {x : OD {suc n}} → (x , x) ≡ record { def = λ z → z o< osuc (od→ord x) } - xx-union {x} = cong ( λ k → record { def = λ z → z o< k } ) (omxx (od→ord x)) - xxx-union : {x : OD {suc n}} → (x , (x , x)) ≡ record { def = λ z → z o< osuc (osuc (od→ord x))} - xxx-union {x} = cong ( λ k → record { def = λ z → z o< k } ) lemma where - lemma1 : {x : OD {suc n}} → od→ord x o< od→ord (x , x) - lemma1 {x} = c<→o< ( proj1 (pair x x ) ) - lemma2 : {x : OD {suc n}} → od→ord (x , x) ≡ osuc (od→ord x) - lemma2 = trans ( cong ( λ k → od→ord k ) xx-union ) (sym ≡-def) - lemma : {x : OD {suc n}} → omax (od→ord x) (od→ord (x , x)) ≡ osuc (osuc (od→ord x)) - lemma {x} = trans ( sym ( omax< (od→ord x) (od→ord (x , x)) lemma1 ) ) ( cong ( λ k → osuc k ) lemma2 ) - uxxx-union : {x : OD {suc n}} → Union (x , (x , x)) ≡ record { def = λ z → osuc z o< osuc (osuc (od→ord x)) } - uxxx-union {x} = cong ( λ k → record { def = λ z → osuc z o< k } ) lemma where - lemma : od→ord (x , (x , x)) ≡ osuc (osuc (od→ord x)) - lemma = trans ( cong ( λ k → od→ord k ) xxx-union ) (sym ≡-def ) - uxxx-2 : {x : OD {suc n}} → record { def = λ z → osuc z o< osuc (osuc (od→ord x)) } == record { def = λ z → z o< osuc (od→ord x) } - eq→ ( uxxx-2 {x} ) {m} lt = proj1 (osuc2 m (od→ord x)) lt - eq← ( uxxx-2 {x} ) {m} lt = proj2 (osuc2 m (od→ord x)) lt - uxxx-ord : {x : OD {suc n}} → od→ord (Union (x , (x , x))) ≡ osuc (od→ord x) - uxxx-ord {x} = trans (cong (λ k → od→ord k ) uxxx-union) (==→o≡o (subst₂ (λ j k → j == k ) (sym oiso) (sym od≡-def ) uxxx-2 )) - omega = record { lv = Suc Zero ; ord = Φ 1 } - infinite : OD {suc n} - infinite = ord→od ( omega ) - infinity∅ : ord→od ( omega ) ∋ od∅ {suc n} - infinity∅ = def-subst {suc n} {_} {o∅} {infinite} {od→ord od∅} - (o<→c< ( case1 (s≤s z≤n ))) refl (subst ( λ k → ( k ≡ od→ord od∅ )) diso (cong (λ k → od→ord k) o∅≡od∅ )) - infinite∋x : (x : OD) → infinite ∋ x → od→ord x o< omega - infinite∋x x lt = subst (λ k → od→ord x o< k ) diso t where - t : od→ord x o< od→ord (ord→od (omega)) - t = ∋→o< {n} {infinite} {x} lt - infinite∋uxxx : (x : OD) → osuc (od→ord x) o< omega → infinite ∋ Union (x , (x , x )) - infinite∋uxxx x lt = o<∋→ t where - t : od→ord (Union (x , (x , x))) o< od→ord (ord→od (omega)) - t = subst (λ k → od→ord (Union (x , (x , x))) o< k ) (sym diso ) ( subst ( λ k → k o< omega ) ( sym (uxxx-ord {x} ) ) lt ) - infinity : (x : OD) → infinite ∋ x → infinite ∋ Union (x , (x , x )) - infinity x lt = infinite∋uxxx x ( lemma (od→ord x) (infinite∋x x lt )) where - lemma : (ox : Ordinal {suc n} ) → ox o< omega → osuc ox o< omega - lemma record { lv = Zero ; ord = (Φ .0) } (case1 (s≤s x)) = case1 (s≤s z≤n) - lemma record { lv = Zero ; ord = (OSuc .0 ord₁) } (case1 (s≤s x)) = case1 (s≤s z≤n) - lemma record { lv = (Suc lv₁) ; ord = (Φ .(Suc lv₁)) } (case1 (s≤s ())) - lemma record { lv = (Suc lv₁) ; ord = (OSuc .(Suc lv₁) ord₁) } (case1 (s≤s ())) - lemma record { lv = 1 ; ord = (Φ 1) } (case2 c2) with d<→lv c2 - lemma record { lv = (Suc Zero) ; ord = (Φ .1) } (case2 ()) | refl - - -- 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 {suc n} ) → {x : OD } → ¬ ( x == od∅ ) → ( X ∋ x ) → OD - choice-func X {x} not X∋x = od∅ {suc n} - choice : (X : OD {suc n} ) → {A : OD } → ( X∋A : X ∋ A ) → (not : ¬ ( A == od∅ )) → A ∋ choice-func X not X∋A - choice X {A} X∋A not = ¬∅=→∅∈ not - - -- another form of regularity - -- - ε-induction : {n m : Level} { ψ : OD {suc n} → Set m} - → ( {x : OD {suc n} } → ({ y : OD {suc n} } → x ∋ y → ψ y ) → ψ x ) - → (x : OD {suc n} ) → ψ x - ε-induction {n} {m} {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (lv (osuc (od→ord x))) (ord (osuc (od→ord x))) <-osuc) where - ε-induction-ord : (lx : Nat) ( ox : OrdinalD {suc n} lx ) {ly : Nat} {oy : OrdinalD {suc n} ly } - → (ly < lx) ∨ (oy d< ox ) → ψ (ord→od (record { lv = ly ; ord = oy } ) ) - ε-induction-ord Zero (Φ 0) (case1 ()) - ε-induction-ord Zero (Φ 0) (case2 ()) - ε-induction-ord lx (OSuc lx ox) {ly} {oy} y<x = - ind {ord→od (record { lv = ly ; ord = oy })} ( λ {y} lt → subst (λ k → ψ k ) oiso (ε-induction-ord lx ox (lemma y lt ))) where - lemma : (y : OD) → ord→od record { lv = ly ; ord = oy } ∋ y → od→ord y o< record { lv = lx ; ord = ox } - lemma y lt with osuc-≡< y<x - lemma y lt | case1 refl = o<-subst (c<→o< lt) refl diso - lemma y lt | case2 lt1 = ordtrans (o<-subst (c<→o< lt) refl diso) lt1 - ε-induction-ord (Suc lx) (Φ (Suc lx)) {ly} {oy} (case1 y<sox ) = - ind {ord→od (record { lv = ly ; ord = oy })} ( λ {y} lt → lemma y lt ) where - -- - -- if lv of z if less than x Ok - -- else lv z = lv x. We can create OSuc of z which is larger than z and less than x in lemma - -- - -- lx Suc lx (1) lz(a) <lx by case1 - -- ly(1) ly(2) (2) lz(b) <lx by case1 - -- lz(a) lz(b) lz(c) lz(c) <lx by case2 ( ly==lz==lx) - -- - lemma0 : { lx ly : Nat } → ly < Suc lx → lx < ly → ⊥ - lemma0 {Suc lx} {Suc ly} (s≤s lt1) (s≤s lt2) = lemma0 lt1 lt2 - lemma1 : {n : Level } {ly : Nat} {oy : OrdinalD {suc n} ly} → lv (od→ord (ord→od (record { lv = ly ; ord = oy }))) ≡ ly - lemma1 {n} {ly} {oy} = let open ≡-Reasoning in begin - lv (od→ord (ord→od (record { lv = ly ; ord = oy }))) - ≡⟨ cong ( λ k → lv k ) diso ⟩ - lv (record { lv = ly ; ord = oy }) - ≡⟨⟩ - ly - ∎ - lemma : (z : OD) → ord→od record { lv = ly ; ord = oy } ∋ z → ψ z - lemma z lt with c<→o< {suc n} {z} {ord→od (record { lv = ly ; ord = oy })} lt - lemma z lt | case1 lz<ly with <-cmp lx ly - lemma z lt | case1 lz<ly | tri< a ¬b ¬c = ⊥-elim ( lemma0 y<sox a) -- can-t happen - lemma z lt | case1 lz<ly | tri≈ ¬a refl ¬c = -- ly(1) - subst (λ k → ψ k ) oiso (ε-induction-ord lx (Φ lx) {_} {ord (od→ord z)} (case1 (subst (λ k → lv (od→ord z) < k ) lemma1 lz<ly ) )) - lemma z lt | case1 lz<ly | tri> ¬a ¬b c = -- lz(a) - subst (λ k → ψ k ) oiso (ε-induction-ord lx (Φ lx) {_} {ord (od→ord z)} (case1 (<-trans lz<ly (subst (λ k → k < lx ) (sym lemma1) c)))) - lemma z lt | case2 lz=ly with <-cmp lx ly - lemma z lt | case2 lz=ly | tri< a ¬b ¬c = ⊥-elim ( lemma0 y<sox a) -- can-t happen - lemma z lt | case2 lz=ly | tri> ¬a ¬b c with d<→lv lz=ly -- lz(b) - ... | eq = subst (λ k → ψ k ) oiso - (ε-induction-ord lx (Φ lx) {_} {ord (od→ord z)} (case1 (subst (λ k → k < lx ) (trans (sym lemma1)(sym eq) ) c ))) - lemma z lt | case2 lz=ly | tri≈ ¬a lx=ly ¬c with d<→lv lz=ly -- lz(c) - ... | eq = lemma6 {ly} {Φ lx} {oy} lx=ly (sym (subst (λ k → lv (od→ord z) ≡ k) lemma1 eq)) where - lemma5 : (ox : OrdinalD lx) → (lv (od→ord z) < lx) ∨ (ord (od→ord z) d< ox) → ψ z - lemma5 ox lt = subst (λ k → ψ k ) oiso (ε-induction-ord lx ox lt ) - lemma6 : { ly : Nat } { ox : OrdinalD {suc n} lx } { oy : OrdinalD {suc n} ly } → - lx ≡ ly → ly ≡ lv (od→ord z) → ψ z - lemma6 {ly} {ox} {oy} refl refl = lemma5 (OSuc lx (ord (od→ord z)) ) (case2 s<refl) -