Mercurial > hg > Members > kono > Proof > ZF-in-agda
changeset 211:6bb5d57c9561 release
Axiom of choice from exclude middle
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Thu, 01 Aug 2019 12:24:26 +0900 |
| parents | f5b3f30fcb16 (current diff) 2c7d45734e3b (diff) |
| children | fe8392f527eb |
| files | HOD.agda ordinal-definable.agda |
| diffstat | 8 files changed, 789 insertions(+), 1038 deletions(-) [+] |
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--- a/.hgtags Sat Jul 20 08:04:20 2019 +0900 +++ b/.hgtags Thu Aug 01 12:24:26 2019 +0900 @@ -7,3 +7,5 @@ a402881cc341fb6499f60bd0f55795dbef5efc70 current a402881cc341fb6499f60bd0f55795dbef5efc70 current b06f5d2f34b1a16ff39aae15680a1c0d640e6b93 current +b06f5d2f34b1a16ff39aae15680a1c0d640e6b93 current +ecb329ba38ac904913313f2dd03ae2329039ffa6 current
--- a/HOD.agda Sat Jul 20 08:04:20 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 ) -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 - -- double-neg-eilm {n} {A} notnot = ⊥-elim (notnot (λ A → {!!} )) - -- - -- 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 existence of minimul ) - -- ∀ 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 - -- - -- {-# TERMINATING #-} - 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 - 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 - ∎ - lemma2 : { lx : Nat } → lx < Suc lx - lemma2 {Zero} = s≤s z≤n - lemma2 {Suc lx} = s≤s (lemma2 {lx}) - -- lx Suc lx (1) z(a) <lx by case1 - -- ly(1) ly(2) (2) z(b) <lx by case1 - -- z(a) z(b) z(c) z(c) <lx by case2 ( ly==z==x) - -- - 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 = -- (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 = -- z(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 -- z(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 refl ¬c with d<→lv lz=ly -- z(c) - ... | eq = lemma6 {ly} {Φ lx} {oy} refl (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 Thu Aug 01 12:24:26 2019 +0900 @@ -0,0 +1,624 @@ +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)) + -- mimimul and x∋minimul is an 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 ) ) + -- minimulity (may proved by ε-induction ) + 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 y : Ordinal {n} } → def (Ord a) x → def (Ord x) y → def (Ord a) y +otrans x<a y<x = ordtrans y<x x<a + +def→o< : {n : Level } {X : OD {suc n}} → {x : Ordinal {suc n}} → def X x → x o< od→ord X +def→o< {n} {X} {x} lt = o<-subst {suc n} {_} {_} {x} {od→ord X} ( c<→o< ( def-subst {suc n} {X} {x} lt (sym oiso) (sym diso) )) diso diso + +∅3 : {n : Level} → { x : Ordinal {suc n}} → ( ∀(y : Ordinal {suc n}) → ¬ (y o< x ) ) → x ≡ o∅ {suc n} +∅3 {n} {x} = TransFinite {n} c2 c3 x where + c0 : Nat → Ordinal {suc n} → Set (suc n) + c0 lx x = (∀(y : Ordinal {suc n}) → ¬ (y o< x)) → x ≡ o∅ {suc n} + c2 : (lx : Nat) → ((x₁ : Ordinal) → x₁ o< ordinal lx (Φ lx) → c0 (lv x₁) (record { lv = lv x₁ ; ord = ord x₁ }))→ 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∅≡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∅ ) ) + +∅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()) + +ppp : { n : Level } → { p : Set (suc n) } { a : OD {suc n} } → record { def = λ x → p } ∋ a → p +ppp {n} {p} {a} d = d + +-- +-- Axiom of choice in intutionistic logic implies the exclude middle +-- https://plato.stanford.edu/entries/axiom-choice/#AxiChoLog +-- +p∨¬p : { n : Level } → ( p : Set (suc n) ) → p ∨ ( ¬ p ) -- assuming axiom of choice +p∨¬p {n} p with is-o∅ ( od→ord ( record { def = λ x → p } )) +p∨¬p {n} p | yes eq = case2 (¬p eq) where + ps = record { def = λ x → p } + lemma : ps == od∅ → p → ⊥ + lemma eq p0 = ¬x<0 {n} {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 {n} p | no ¬p = case1 (ppp {n} {p} {minimul ps (λ eq → ¬p (eqo∅ eq))} lemma) where + ps = record { def = λ x → p } + eqo∅ : ps == od∅ {suc n} → od→ord ps ≡ o∅ {suc n} + 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)) + +∋-p : { n : Level } → ( p : Set (suc n) ) → Dec p -- assuming axiom of choice +∋-p {n} p with p∨¬p p +∋-p {n} p | case1 x = yes x +∋-p {n} p | case2 x = no x + +double-neg-eilm : {n : Level } {A : Set (suc n)} → ¬ ¬ A → A -- we don't have this in intutionistic logic +double-neg-eilm {n} {A} notnot with ∋-p A -- assuming axiom of choice +... | yes p = p +... | no ¬p = ⊥-elim ( notnot ¬p ) + +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 } -- roughly x = A → Set + +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 )) ) ) + + +_⊆_ : {n : Level} ( A B : OD {suc n} ) → ∀{ x : OD {suc n} } → Set (suc n) +_⊆_ A B {x} = A ∋ x → B ∋ x + +infixr 220 _⊆_ + +subset-lemma : {n : Level} → {A x y : OD {suc n} } → ( x ∋ y → ZFSubset A x ∋ y ) ⇔ ( _⊆_ x A {y} ) +subset-lemma {n} {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 {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 + 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 _∩_ _∪_ + 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 = λ {A} {B} {w} → extensionality {A} {B} {w} + ; ε-induction = ε-induction + ; infinity∅ = infinity∅ + ; infinity = infinity + ; selection = λ {X} {ψ} {y} → selection {X} {ψ} {y} + ; replacement← = replacement← + ; replacement→ = replacement→ + ; choice-func = choice-func + ; choice = choice + } 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 ()) + + 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< + + -- + -- 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 ))) + + -- 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 {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) )) + + extensionality0 : {A B : OD {suc n}} → ((z : OD) → (A ∋ z) ⇔ (B ∋ z)) → A == B + eq→ (extensionality0 {A} {B} eq ) {x} d = def-iso {suc n} {A} {B} (sym diso) (proj1 (eq (ord→od x))) d + eq← (extensionality0 {A} {B} eq ) {x} d = def-iso {suc n} {B} {A} (sym diso) (proj2 (eq (ord→od x))) d + + extensionality : {A B w : OD {suc n} } → ((z : OD {suc n}) → (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∅ {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 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 : (z : OD) → ord→od record { lv = ly ; ord = oy } ∋ z → od→ord z o< record { lv = lx ; ord = ox } + lemma z lt with osuc-≡< y<x + lemma z lt | case1 refl = o<-subst (c<→o< lt) refl diso + lemma z 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) + + --- + --- With assuption of OD is ordered, p ∨ ( ¬ p ) <=> axiom of choice + --- + record choiced {n : Level} ( X : OD {suc n}) : Set (suc (suc n)) where + field + a-choice : OD {suc n} + is-in : X ∋ a-choice + choice-func' : (X : OD {suc n} ) → (p∨¬p : { n : Level } → ( p : Set (suc n) ) → p ∨ ( ¬ p )) → ¬ ( X == od∅ ) → choiced X + choice-func' X p∨¬p not = have_to_find + where + ψ : ( ox : Ordinal {suc n}) → Set (suc (suc n)) + ψ ox = (( x : Ordinal {suc n}) → x o< ox → ( ¬ def X x )) ∨ choiced X + lemma-ord : ( ox : Ordinal {suc n} ) → ψ ox + lemma-ord ox = TransFinite {n} {suc (suc n)} {ψ} caseΦ caseOSuc ox where + ∋-p' : (A x : OD {suc n} ) → Dec ( A ∋ x ) + ∋-p' A x with p∨¬p ( A ∋ x ) + ∋-p' A x | case1 t = yes t + ∋-p' A x | case2 t = no t + ∀-imply-or : {n : Level} {A : Ordinal {suc n} → Set (suc n) } {B : Set (suc (suc n)) } + → ((x : Ordinal {suc n}) → A x ∨ B) → ((x : Ordinal {suc n}) → A x) ∨ B + ∀-imply-or {n} {A} {B} ∀AB with p∨¬p ((x : Ordinal {suc n}) → A x) + ∀-imply-or {n} {A} {B} ∀AB | case1 t = case1 t + ∀-imply-or {n} {A} {B} ∀AB | case2 x = case2 (lemma x) where + lemma : ¬ ((x : Ordinal {suc n}) → 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 )) + caseΦ : (lx : Nat) → ( (x : Ordinal {suc n} ) → x o< ordinal lx (Φ lx) → ψ x) → ψ (ordinal lx (Φ lx) ) + caseΦ lx prev with ∋-p' X ( ord→od (ordinal lx (Φ lx) )) + caseΦ lx prev | yes p = case2 ( record { a-choice = ord→od (ordinal lx (Φ lx)) ; is-in = p } ) + caseΦ lx prev | no ¬p = lemma where + lemma1 : (x : Ordinal) → (((Suc (lv x) ≤ lx) ∨ (ord x d< Φ lx) → def X x → ⊥) ∨ choiced X) + lemma1 x with trio< x (ordinal lx (Φ lx)) + lemma1 x | tri< a ¬b ¬c with prev (osuc x) (lemma2 a) where + lemma2 : x o< (ordinal lx (Φ lx)) → osuc x o< ordinal lx (Φ lx) + lemma2 (case1 lt) = case1 lt + lemma1 x | tri< a ¬b ¬c | case2 found = case2 found + lemma1 x | tri< a ¬b ¬c | case1 not_found = case1 ( λ lt df → not_found x <-osuc df ) + lemma1 x | tri≈ ¬a refl ¬c = case1 ( λ lt → ⊥-elim (o<¬≡ refl lt )) + lemma1 x | tri> ¬a ¬b c = case1 ( λ lt → ⊥-elim (o<> lt c )) + lemma : ((x : Ordinal) → (Suc (lv x) ≤ lx) ∨ (ord x d< Φ lx) → def X x → ⊥) ∨ choiced X + lemma = ∀-imply-or lemma1 + caseOSuc : (lx : Nat) (x : OrdinalD lx) → ψ ( ordinal lx x ) → ψ ( ordinal lx (OSuc lx x) ) + caseOSuc lx x prev with ∋-p' X (ord→od record { lv = lx ; ord = x } ) + caseOSuc lx x prev | yes p = case2 (record { a-choice = ord→od record { lv = lx ; ord = x } ; is-in = p }) + caseOSuc lx x (case1 not_found) | no ¬p = case1 lemma where + lemma : (y : Ordinal) → (Suc (lv y) ≤ lx) ∨ (ord y d< OSuc lx x) → def X y → ⊥ + lemma y lt with trio< y (ordinal lx x ) + lemma y lt | tri< a ¬b ¬c = not_found y a + lemma y lt | tri≈ ¬a refl ¬c = subst (λ k → def X k → ⊥ ) diso ¬p + lemma y lt | tri> ¬a ¬b c with osuc-≡< lt + lemma y lt | tri> ¬a ¬b c | case1 refl = ⊥-elim ( ¬b refl ) + lemma y lt | tri> ¬a ¬b c | case2 lt1 = ⊥-elim (o<> c lt1 ) + caseOSuc lx x (case2 found) | no ¬p = case2 found + 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) → (Suc (lv x) ≤ lv (od→ord X)) ∨ (ord x d< ord (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 ) + } +
--- a/Todo Sat Jul 20 08:04:20 2019 +0900 +++ b/Todo Thu Aug 01 12:24:26 2019 +0900 @@ -1,3 +1,13 @@ +Tue Jul 23 11:02:50 JST 2019 + + define cardinals + prove CH in OD→ZF + define Ultra filter + define L M : ZF ZFSet = M is an OD + define L N : ZF ZFSet = N = G M (G is a generic fitler on M ) + prove ¬ CH on L N + prove no choice function on L N + Mon Jul 8 19:43:37 JST 2019 ordinal-definable.agda assumes all ZF Set are ordinals, that it too restrictive
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/filter.agda Thu Aug 01 12:24:26 2019 +0900 @@ -0,0 +1,80 @@ +open import Level +open import OD +open import zf +open import ordinal +module filter ( n : Level ) where + +open import Relation.Nullary +open import Relation.Binary +open import Data.Empty +open import Relation.Binary +open import Relation.Binary.Core +open import Relation.Binary.PropositionalEquality +open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) + +od = OD→ZF {n} + + +record Filter {n : Level} ( P max : OD {suc n} ) : Set (suc (suc n)) where + field + _⊇_ : OD {suc n} → OD {suc n} → Set (suc n) + G : OD {suc n} + G∋1 : G ∋ max + Gmax : { p : OD {suc n} } → P ∋ p → p ⊇ max + Gless : { p q : OD {suc n} } → G ∋ p → P ∋ q → p ⊇ q → G ∋ q + Gcompat : { p q : OD {suc n} } → G ∋ p → G ∋ q → ¬ ( + ( r : OD {suc n}) → (( p ⊇ r ) ∧ ( p ⊇ r ))) + +dense : {n : Level} → Set (suc (suc n)) +dense {n} = { D P p : OD {suc n} } → ({x : OD {suc n}} → P ∋ p → ¬ ( ( q : OD {suc n}) → D ∋ q → od→ord p o< od→ord q )) + +record NatFilter {n : Level} ( P : Nat → Set n) : Set (suc n) where + field + GN : Nat → Set n + GN∋1 : GN 0 + GNmax : { p : Nat } → P p → 0 ≤ p + GNless : { p q : Nat } → GN p → P q → q ≤ p → GN q + Gr : ( p q : Nat ) → GN p → GN q → Nat + GNcompat : { p q : Nat } → (gp : GN p) → (gq : GN q ) → ( Gr p q gp gq ≤ p ) ∨ ( Gr p q gp gq ≤ q ) + +record NatDense {n : Level} ( P : Nat → Set n) : Set (suc n) where + field + Gmid : { p : Nat } → P p → Nat + GDense : { D : Nat → Set n } → {x p : Nat } → (pp : P p ) → D (Gmid {p} pp) → Gmid pp ≤ p + +open OD.OD + +-- H(ω,2) = Power ( Power ω ) = Def ( Def ω)) + +Pred : {n : Level} ( Dom : OD {suc n} ) → OD {suc n} +Pred {n} dom = record { + def = λ x → def dom x → Set n + } + +Hω2 : {n : Level} → OD {suc n} +Hω2 {n} = record { def = λ x → {dom : Ordinal {suc n}} → x ≡ od→ord ( Pred ( ord→od dom )) } + +Hω2Filter : {n : Level} → Filter {n} Hω2 od∅ +Hω2Filter {n} = record { + _⊇_ = _⊇_ + ; G = {!!} + ; G∋1 = {!!} + ; Gmax = {!!} + ; Gless = {!!} + ; Gcompat = {!!} + } where + P = Hω2 + _⊇_ : OD {suc n} → OD {suc n} → Set (suc n) + _⊇_ = {!!} + G : OD {suc n} + G = {!!} + G∋1 : G ∋ od∅ + G∋1 = {!!} + Gmax : { p : OD {suc n} } → P ∋ p → p ⊇ od∅ + Gmax = {!!} + Gless : { p q : OD {suc n} } → G ∋ p → P ∋ q → p ⊇ q → G ∋ q + Gless = {!!} + Gcompat : { p q : OD {suc n} } → G ∋ p → G ∋ q → ¬ ( + ( r : OD {suc n}) → (( p ⊇ r ) ∧ ( p ⊇ r ))) + Gcompat = {!!} +
--- a/ordinal-definable.agda Sat Jul 20 08:04:20 2019 +0900 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,494 +0,0 @@ -{-# OPTIONS --allow-unsolved-metas #-} - -open import Level -module ordinal-definable 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 import Data.Unit - -open Ordinal - --- Ordinal in OD ( and ZFSet ) -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} = record { def = λ _ → Lift n ⊥ } -od∅ : {n : Level} → OD {n} -od∅ {n} = Ord o∅ - -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) } - -ord→od : {n : Level} → Ordinal {n} → OD {n} -ord→od a = Ord a - -o<→c< : {n : Level} {x y : Ordinal {n} } → x o< y → def (ord→od y) x -o<→c< {n} {x} {y} lt = lt - -postulate - -- OD can be iso to a subset of Ordinal ( by means of Godel Set ) - od→ord : {n : Level} → OD {n} → Ordinal {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 - -- 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 - -- 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 ) - -_∋_ : { 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 ) - -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→od ( 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} {_} {_} {sup-od ψ} {od→ord ( ψ x )} - ( o<→c< sup-o< ) refl (cong ( λ k → od→ord (ψ k) ) oiso) - -∅1 : {n : Level} → ( x : OD {n} ) → ¬ ( x c< od∅ {n} ) -∅1 {n} x (case1 ()) -∅1 {n} x (case2 ()) - -∅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 | () - -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 - -∅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<→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≡ : {n : Level} → { x y : Ordinal {suc n} } → ord→od x == ord→od y → x ≡ y -==→o≡ {n} {x} {y} eq with trio< {n} x y -==→o≡ {n} {x} {y} eq | tri< a ¬b ¬c = ⊥-elim ( o<→o> eq (o<-subst a (sym ord-iso) (sym ord-iso ))) -==→o≡ {n} {x} {y} eq | tri≈ ¬a b ¬c = b -==→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≡ {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≡ (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≡ 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} } → 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<∋→ : {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∅≡od∅ : {n : Level} → ord→od (o∅ {suc n}) ≡ od∅ {suc n} -o∅≡od∅ {n} with trio< {n} (o∅ {suc n}) (od→ord (od∅ {suc n} )) -o∅≡od∅ {n} | tri< a ¬b ¬c = ⊥-elim (lemma a) where - lemma : o∅ {suc n } o< (od→ord (od∅ {suc n} )) → ⊥ - lemma lt with def-subst (o<→c< lt) oiso refl - lemma lt | case1 () - lemma lt | case2 () -o∅≡od∅ {n} | tri≈ ¬a b ¬c = trans (cong (λ k → ord→od k ) b ) oiso -o∅≡od∅ {n} | tri> ¬a ¬b c = ⊥-elim (¬x<0 c) - -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 - -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) - -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 - -∅< : {n : Level} → { x y : OD {n} } → def x (od→ord y ) → ¬ ( x == od∅ {n} ) -∅< {n} {x} {y} d eq with eq→ eq d -∅< {n} {x} {y} d eq | case1 () -∅< {n} {x} {y} d eq | case2 () - -∅6 : {n : Level} → { x : OD {suc n} } → ¬ ( x ∋ x ) -- no Russel paradox -∅6 {n} {x} x∋x = c<> {n} {x} {x} 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-∋ : {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 - -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()) - -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 )) - -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 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 } - -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 )) ) ) - --- Constructible Set on α -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 ) - record { def = λ y → osuc y o< (od→ord (L {n} (record { lv = lx ; ord = Φ lx }) )) } - -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 = o<→c< (o<-subst (sup-lb (o<-subst (c<→o< P∋t) 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< lemma-t ) 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} - ( o<→c< {suc n} {od→ord (ZFSubset A (ord→od (od→ord t)) )} {sup-o (λ x → od→ord (ZFSubset A (ord→od x)))} - 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 = lemma <-osuc 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 (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 (proj2 lt)) 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 : (x : OD {suc n} ) → ¬ (x == od∅ )→ OD {suc n} - minimul 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 ) = ¬∅=→∅∈ not - 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 (proj1 (proj2 t )) ) - - 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≡ (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 -
--- a/ordinal.agda Sat Jul 20 08:04:20 2019 +0900 +++ b/ordinal.agda Thu Aug 01 12:24:26 2019 +0900 @@ -13,13 +13,11 @@ OSuc : (lv : Nat) → OrdinalD {n} lv → OrdinalD lv record Ordinal {n : Level} : Set n where + constructor ordinal field lv : Nat ord : OrdinalD {n} lv --- --- Φ (Suc lv) < ℵ lv < OSuc (Suc lv) (ℵ lv) < OSuc ... < OSuc (Suc lv) (Φ (Suc lv)) < OSuc ... < ℵ (Suc lv) --- data _d<_ {n : Level} : {lx ly : Nat} → OrdinalD {n} lx → OrdinalD {n} ly → Set n where Φ< : {lx : Nat} → {x : OrdinalD {n} lx} → Φ lx d< OSuc lx x s< : {lx : Nat} → {x y : OrdinalD {n} lx} → x d< y → OSuc lx x d< OSuc lx y @@ -114,10 +112,22 @@ ¬a≤a : {la : Nat} → Suc la ≤ la → ⊥ ¬a≤a (s≤s lt) = ¬a≤a lt +a<sa : {la : Nat} → la < Suc la +a<sa {Zero} = s≤s z≤n +a<sa {Suc la} = s≤s a<sa + =→¬< : {x : Nat } → ¬ ( x < x ) =→¬< {Zero} () =→¬< {Suc x} (s≤s lt) = =→¬< lt +<-∨ : { x y : Nat } → x < Suc y → ( (x ≡ y ) ∨ (x < y) ) +<-∨ {Zero} {Zero} (s≤s z≤n) = case1 refl +<-∨ {Zero} {Suc y} (s≤s lt) = case2 (s≤s z≤n) +<-∨ {Suc x} {Zero} (s≤s ()) +<-∨ {Suc x} {Suc y} (s≤s lt) with <-∨ {x} {y} lt +<-∨ {Suc x} {Suc y} (s≤s lt) | case1 eq = case1 (cong (λ k → Suc k ) eq) +<-∨ {Suc x} {Suc y} (s≤s lt) | case2 lt1 = case2 (s≤s lt1) + case12-⊥ : {n : Level} {x y : Ordinal {suc n}} → lv x < lv y → ord x d< ord y → ⊥ case12-⊥ {x} {y} lt1 lt2 with d<→lv lt2 ... | refl = nat-≡< refl lt1 @@ -126,7 +136,7 @@ case21-⊥ {x} {y} lt1 lt2 with d<→lv lt2 ... | refl = nat-≡< refl lt1 -o<¬≡ : {n : Level } { ox oy : Ordinal {n}} → ox ≡ oy → ox o< oy → ⊥ +o<¬≡ : {n : Level } { ox oy : Ordinal {suc n}} → ox ≡ oy → ox o< oy → ⊥ o<¬≡ {_} {ox} {ox} refl (case1 lt) = =→¬< lt o<¬≡ {_} {ox} {ox} refl (case2 (s< lt)) = trio<≡ refl lt @@ -224,6 +234,12 @@ lemma1 (case1 x) = ¬a x lemma1 (case2 x) = ≡→¬d< x +xo<ab : {n : Level} {oa ob : Ordinal {suc n}} → ( {ox : Ordinal {suc n}} → ox o< oa → ox o< ob ) → oa o< osuc ob +xo<ab {n} {oa} {ob} a→b with trio< oa ob +xo<ab {n} {oa} {ob} a→b | tri< a ¬b ¬c = ordtrans a <-osuc +xo<ab {n} {oa} {ob} a→b | tri≈ ¬a refl ¬c = <-osuc +xo<ab {n} {oa} {ob} a→b | tri> ¬a ¬b c = ⊥-elim ( o<¬≡ refl (a→b c ) ) + maxα : {n : Level} → Ordinal {suc n} → Ordinal → Ordinal maxα x y with trio< x y maxα x y | tri< a ¬b ¬c = y @@ -294,8 +310,6 @@ proj2 (osuc2 {n} x y) (case2 lt) with d<→lv lt ... | refl = case2 (s< lt) --- omax≡ (omax x x ) (osuc x) (omxx x) - OrdTrans : {n : Level} → Transitive {suc n} _o≤_ OrdTrans (case1 refl) (case1 refl) = case1 refl OrdTrans (case1 refl) (case2 lt2) = case2 lt2 @@ -313,15 +327,31 @@ } } -TransFinite : {n m : Level} → { ψ : Ordinal {n} → Set m } - → ( ∀ (lx : Nat ) → ψ ( record { lv = lx ; ord = Φ lx } ) ) +TransFinite : {n m : Level} → { ψ : Ordinal {suc n} → Set m } + → ( ∀ (lx : Nat ) → ( (x : Ordinal {suc n} ) → x o< ordinal lx (Φ lx) → ψ x ) → ψ ( record { lv = lx ; ord = Φ lx } ) ) → ( ∀ (lx : Nat ) → (x : OrdinalD lx ) → ψ ( record { lv = lx ; ord = x } ) → ψ ( record { lv = lx ; ord = OSuc lx x } ) ) → ∀ (x : Ordinal) → ψ x -TransFinite caseΦ caseOSuc record { lv = lv ; ord = (Φ (lv)) } = caseΦ lv -TransFinite caseΦ caseOSuc record { lv = lx ; ord = (OSuc lx ox) } = - caseOSuc lx ox (TransFinite caseΦ caseOSuc record { lv = lx ; ord = ox }) +TransFinite {n} {m} {ψ} caseΦ caseOSuc x = proj1 (TransFinite1 (lv x) (ord x) ) where + TransFinite1 : (lx : Nat) (ox : OrdinalD lx ) → ψ (ordinal lx ox) ∧ ( ( (x : Ordinal {suc n} ) → x o< ordinal lx (Φ lx) → ψ x ) ) + TransFinite1 Zero (Φ 0) = record { proj1 = caseΦ Zero lemma ; proj2 = lemma1 } where + lemma : (x : Ordinal) → x o< ordinal Zero (Φ Zero) → ψ x + lemma x (case1 ()) + lemma x (case2 ()) + lemma1 : (x : Ordinal) → x o< ordinal Zero (Φ Zero) → ψ x + lemma1 x (case1 ()) + lemma1 x (case2 ()) + TransFinite1 (Suc lx) (Φ (Suc lx)) = record { proj1 = caseΦ (Suc lx) (λ x → lemma (lv x) (ord x)) ; proj2 = (λ x → lemma (lv x) (ord x)) } where + lemma0 : (ly : Nat) (oy : OrdinalD ly ) → ordinal ly oy o< ordinal lx (Φ lx) → ψ (ordinal ly oy) + lemma0 ly oy lt = proj2 ( TransFinite1 lx (Φ lx) ) (ordinal ly oy) lt + lemma : (ly : Nat) (oy : OrdinalD ly ) → ordinal ly oy o< ordinal (Suc lx) (Φ (Suc lx)) → ψ (ordinal ly oy) + lemma lx1 ox1 (case1 lt) with <-∨ lt + lemma lx (Φ lx) (case1 lt) | case1 refl = proj1 ( TransFinite1 lx (Φ lx) ) + lemma lx (OSuc lx ox1) (case1 lt) | case1 refl = caseOSuc lx ox1 ( lemma lx ox1 (case1 a<sa)) + lemma lx (Φ lx) (case1 lt) | case2 (s≤s lt1) = lemma0 lx (Φ lx) (case1 (s≤s lt1)) + lemma lx1 (OSuc lx1 ox1) (case1 lt) | case2 lt1 = caseOSuc lx1 ox1 ( lemma lx1 ox1 (case1 (<-trans lt1 a<sa ))) + TransFinite1 lx (OSuc lx ox) = record { proj1 = caseOSuc lx ox (proj1 (TransFinite1 lx ox )) ; proj2 = proj2 (TransFinite1 lx ox )} --- we cannot prove this in intutonistic logic +-- we cannot prove this in intutionistic logic -- (¬ (∀ y → ¬ ( ψ y ))) → (ψ y → p ) → p -- postulate -- TransFiniteExists : {n m l : Level} → ( ψ : Ordinal {n} → Set m ) @@ -337,4 +367,3 @@ → ¬ p TransFiniteExists {n} {m} {l} ψ {p} P = contra-position ( λ p y ψy → P {y} ψy p ) -
--- a/zf.agda Sat Jul 20 08:04:20 2019 +0900 +++ b/zf.agda Thu Aug 01 12:24:26 2019 +0900 @@ -21,6 +21,8 @@ open import Relation.Nullary open import Relation.Binary +open import Data.Empty + contra-position : {n m : Level } {A : Set n} {B : Set m} → (A → B) → ¬ B → ¬ A contra-position {n} {m} {A} {B} f ¬b a = ¬b ( f a ) @@ -28,6 +30,24 @@ double-neg : {n : Level } {A : Set n} → A → ¬ ¬ A double-neg A notnot = notnot A +double-neg2 : {n : Level } {A : Set n} → ¬ ¬ ¬ A → ¬ A +double-neg2 notnot A = notnot ( double-neg A ) + +de-morgan : {n : Level } {A B : Set n} → A ∧ B → ¬ ( (¬ A ) ∨ (¬ B ) ) +de-morgan {n} {A} {B} and (case1 ¬A) = ⊥-elim ( ¬A ( _∧_.proj1 and )) +de-morgan {n} {A} {B} and (case2 ¬B) = ⊥-elim ( ¬B ( _∧_.proj2 and )) + +dont-or : {n m : Level} {A : Set n} { B : Set m } → A ∨ B → ¬ A → B +dont-or {A} {B} (case1 a) ¬A = ⊥-elim ( ¬A a ) +dont-or {A} {B} (case2 b) ¬A = b + +dont-orb : {n m : Level} {A : Set n} { B : Set m } → A ∨ B → ¬ B → A +dont-orb {A} {B} (case2 b) ¬B = ⊥-elim ( ¬B b ) +dont-orb {A} {B} (case1 a) ¬B = a + +-- mid-ex-neg : {n : Level } {A : Set n} → ( ¬ ¬ A ) ∨ (¬ A) +-- mid-ex-neg {n} {A} = {!!} + infixr 130 _∧_ infixr 140 _∨_ infixr 150 _⇔_ @@ -70,10 +90,15 @@ power→ : ∀( A t : ZFSet ) → Power A ∋ t → ∀ {x} → t ∋ x → ¬ ¬ ( A ∋ x ) -- _⊆_ t A {x} power← : ∀( A t : ZFSet ) → ( ∀ {x} → _⊆_ t A {x}) → Power A ∋ t -- extensionality : ∀ z ( z ∈ x ⇔ z ∈ y ) ⇒ ∀ w ( x ∈ w ⇔ y ∈ w ) - extensionality : { A B : ZFSet } → ( (z : ZFSet) → ( A ∋ z ) ⇔ (B ∋ z) ) → A ≈ B + extensionality : { A B w : ZFSet } → ( (z : ZFSet) → ( A ∋ z ) ⇔ (B ∋ z) ) → ( A ∈ w ⇔ B ∈ w ) + -- This form of regurality forces choice function -- regularity : ∀ x ( x ≠ ∅ → ∃ y ∈ x ( y ∩ x = ∅ ) ) - minimul : (x : ZFSet ) → ¬ (x ≈ ∅) → ZFSet - regularity : ∀( x : ZFSet ) → (not : ¬ (x ≈ ∅)) → ( minimul x not ∈ x ∧ ( minimul x not ∩ x ≈ ∅ ) ) + -- minimul : (x : ZFSet ) → ¬ (x ≈ ∅) → ZFSet + -- regularity : ∀( x : ZFSet ) → (not : ¬ (x ≈ ∅)) → ( minimul x not ∈ x ∧ ( minimul x not ∩ x ≈ ∅ ) ) + -- another form of regularity + ε-induction : { ψ : ZFSet → Set m} + → ( {x : ZFSet } → ({ y : ZFSet } → x ∋ y → ψ y ) → ψ x ) + → (x : ZFSet ) → ψ x -- infinity : ∃ A ( ∅ ∈ A ∧ ∀ x ∈ A ( x ∪ { x } ∈ A ) ) infinity∅ : ∅ ∈ infinite infinity : ∀( x : ZFSet ) → x ∈ infinite → ( x ∪ ｛ x ｝) ∈ infinite @@ -81,9 +106,9 @@ -- replacement : ∀ x ∀ y ∀ z ( ( ψ ( x , y ) ∧ ψ ( x , z ) ) → y = z ) → ∀ X ∃ A ∀ y ( y ∈ A ↔ ∃ x ∈ X ψ ( x , y ) ) replacement← : {ψ : ZFSet → ZFSet} → ∀ ( X x : ZFSet ) → x ∈ X → ψ x ∈ Replace X ψ replacement→ : {ψ : ZFSet → ZFSet} → ∀ ( X x : ZFSet ) → ( lt : x ∈ Replace X ψ ) → ¬ ( ∀ (y : ZFSet) → ¬ ( x ≈ ψ y ) ) - -- -- ∀ z [ ∀ x ( x ∈ z → ¬ ( x ≈ ∅ ) ) ∧ ∀ x ∀ y ( x , y ∈ z ∧ ¬ ( x ≈ y ) → x ∩ y ≈ ∅ ) → ∃ u ∀ x ( x ∈ z → ∃ t ( u ∩ x) ≈ ｛ t ｝) ] - -- axiom-of-choice : Set (suc n) - -- axiom-of-choice = ? + -- ∀ X [ ∅ ∉ X → (∃ f : X → ⋃ X ) → ∀ A ∈ X ( f ( A ) ∈ A ) ] + choice-func : (X : ZFSet ) → {x : ZFSet } → ¬ ( x ≈ ∅ ) → ( X ∋ x ) → ZFSet + choice : (X : ZFSet ) → {A : ZFSet } → ( X∋A : X ∋ A ) → (not : ¬ ( A ≈ ∅ )) → A ∋ choice-func X not X∋A record ZF {n m : Level } : Set (suc (n ⊔ m)) where infixr 210 _,_