Mercurial > hg > Members > kono > Proof > ZF-in-agda
diff HOD.agda @ 141:21b2654985c4
fix or
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Mon, 08 Jul 2019 00:20:30 +0900 |
| parents | 312e27aa3cb5 |
| children | c30bc9f5bd0d |
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--- a/HOD.agda Sun Jul 07 23:02:47 2019 +0900 +++ b/HOD.agda Mon Jul 08 00:20:30 2019 +0900 @@ -13,17 +13,17 @@ -- Ordinal Definable Set -record HOD {n : Level} : Set (suc n) where +record OD {n : Level} : Set (suc n) where field def : (x : Ordinal {n} ) → Set n -open HOD +open OD open import Data.Unit open Ordinal open _∧_ -record _==_ {n : Level} ( a b : HOD {n} ) : Set n where +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 @@ -31,37 +31,37 @@ id : {n : Level} {A : Set n} → A → A id x = x -eq-refl : {n : Level} { x : HOD {n} } → 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 : HOD {n} } → x == y → y == x +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 : HOD {n} } → x == y → y == z → x == z +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 : HOD {suc n} } → ( {z : Ordinal {suc n}} → def x z ⇔ def y z) → x == y +⇔→== : {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 HOD ( and ZFSet ) -Ord : { n : Level } → ( a : Ordinal {n} ) → HOD {n} +-- 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} → HOD {n} +od∅ : {n : Level} → OD {n} od∅ {n} = Ord o∅ postulate - -- HOD can be iso to a subset of Ordinal ( by means of Godel Set ) - od→ord : {n : Level} → HOD {n} → Ordinal {n} - ord→od : {n : Level} → Ordinal {n} → HOD {n} - c<→o< : {n : Level} {x y : HOD {n} } → def y ( od→ord x ) → od→ord x o< od→ord y - oiso : {n : Level} {x : HOD {n}} → ord→od ( od→ord x ) ≡ x + -- 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 ord-Ord :{n : Level} {x : Ordinal {n}} → x ≡ od→ord (Ord x) - ==→o≡ : {n : Level} → { x y : HOD {suc n} } → (x == y) → x ≡ y + ==→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 -- supermum as Replacement Axiom @@ -71,38 +71,38 @@ 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 ) - minimul : {n : Level } → (x : HOD {suc n} ) → ¬ (x == od∅ )→ HOD {suc n} + 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 : HOD {suc n} ) → ( ne : ¬ (x == od∅ ) ) → def x ( od→ord ( minimul x ne ) ) - minimul-1 : {n : Level } → (x : HOD {suc n} ) → ( ne : ¬ (x == od∅ ) ) → (y : HOD {suc n}) → ¬ ( def (minimul x ne) (od→ord y)) ∧ (def x (od→ord y) ) + 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) ) -- we should prove this in agda, but simply put here - ===-≡ : {n : Level} { x y : HOD {suc n}} → x == y → x ≡ y + ===-≡ : {n : Level} { x y : OD {suc n}} → x == y → x ≡ y Ord-ord : {n : Level } {ox : Ordinal {suc n}} → Ord ox ≡ ord→od ox Ord-ord {n} {px} = trans (sym oiso) (cong ( λ k → ord→od k ) (sym ord-Ord)) -_∋_ : { n : Level } → ( a x : HOD {n} ) → Set n +_∋_ : { n : Level } → ( a x : OD {n} ) → Set n _∋_ {n} a x = def a ( od→ord x ) -_c<_ : { n : Level } → ( x a : HOD {n} ) → Set n +_c<_ : { n : Level } → ( x a : OD {n} ) → Set n x c< a = a ∋ x -_c≤_ : {n : Level} → HOD {n} → HOD {n} → Set (suc n) +_c≤_ : {n : Level} → OD {n} → OD {n} → Set (suc n) a c≤ b = (a ≡ b) ∨ ( b ∋ a ) -cseq : {n : Level} → HOD {n} → HOD {n} +cseq : {n : Level} → OD {n} → OD {n} cseq x = record { def = λ y → def x (osuc y) } where -def-subst : {n : Level } {Z : HOD {n}} {X : Ordinal {n} }{z : HOD {n}} {x : Ordinal {n} }→ def Z X → Z ≡ z → X ≡ x → def z x +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 o<→c< : {n : Level} {x y : Ordinal {n} } → x o< y → Ord y ∋ Ord x o<→c< {n} {x} {y} lt = subst ( λ k → k o< y ) ord-Ord lt -sup-od : {n : Level } → ( HOD {n} → HOD {n}) → HOD {n} +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 : HOD {n} → HOD {n}) → ∀ {x : HOD {n}} → def ( sup-od ψ ) (od→ord ( ψ 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))) @@ -149,21 +149,21 @@ ord-iso = cong ( λ k → record { lv = lv k ; ord = ord k } ) diso -- avoiding lv != Zero error -orefl : {n : Level} → { x : HOD {n} } → { y : Ordinal {n} } → od→ord x ≡ y → od→ord x ≡ y +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 : HOD {n} } → ord→od (od→ord x) == ord→od (od→ord y) → x == y +==-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 : HOD {n} } {z : Ordinal {n}} → def (ord→od (od→ord x)) z → def x z + 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 : HOD {suc n} } → (x == y) ≡ (ord→od (od→ord x) == y) +=-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 : HOD {n} } → od→ord x ≡ od→ord y → x == y +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 @@ -174,10 +174,10 @@ >→¬< : {x y : Nat } → (x < y ) → ¬ ( y < x ) >→¬< (s≤s x<y) (s≤s y<x) = >→¬< x<y y<x -c≤-refl : {n : Level} → ( x : HOD {n} ) → x c≤ x +c≤-refl : {n : Level} → ( x : OD {n} ) → x c≤ x c≤-refl x = case1 refl -∋→o< : {n : Level} → { a x : HOD {suc n} } → a ∋ x → od→ord x o< od→ord a +∋→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 @@ -191,10 +191,10 @@ 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<→¬c> : {n : Level} → { x y : HOD {n} } → (od→ord x ) o< ( od→ord y) → ¬ (y c< x ) +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 : HOD {n} } → (od→ord x ) ≡ ( od→ord y) → ¬ x c< y +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} @@ -202,14 +202,14 @@ eq← ∅0 {w} (case1 ()) eq← ∅0 {w} (case2 ()) -∅< : {n : Level} → { x y : HOD {n} } → def x (od→ord y ) → ¬ ( x == od∅ {n} ) +∅< : {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 : HOD {suc n} } → ¬ ( x ∋ x ) -- no Russel paradox +∅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 : HOD {n}} {x y : Ordinal {n}} → x ≡ y → (def A y → def B y) → def A x → def B 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} ) @@ -221,15 +221,18 @@ -- 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} → HOD {suc n} → HOD {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 : HOD {suc n} ) → HOD {suc n} +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 : HOD {suc n}) → HOD {suc n} +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 )) ) ) OrdSubset : {n : Level} → (A x : Ordinal {suc n} ) → ZFSubset (Ord A) (Ord x) ≡ Ord ( minα A x ) @@ -250,7 +253,7 @@ -- 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}) → HOD {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 ) @@ -265,14 +268,14 @@ lemma0 {n} ox = trans (cong (λ k → od→ord k) (===-≡ (⇔→== lemma1) )) (sym ord-Ord) -- L0 : {n : Level} → (α : Ordinal {suc n}) → α o< β → L (osuc α) ∋ L α --- L1 : {n : Level} → (α β : Ordinal {suc n}) → α o< β → ∀ (x : HOD {suc n}) → L α ∋ x → L β ∋ x +-- L1 : {n : Level} → (α β : Ordinal {suc n}) → α o< β → ∀ (x : OD {suc n}) → L α ∋ x → L β ∋ x omega : { n : Level } → Ordinal {n} omega = record { lv = Suc Zero ; ord = Φ 1 } -HOD→ZF : {n : Level} → ZF {suc (suc n)} {suc n} -HOD→ZF {n} = record { - ZFSet = HOD {suc n} +OD→ZF : {n : Level} → ZF {suc (suc n)} {suc n} +OD→ZF {n} = record { + ZFSet = OD {suc n} ; _∋_ = _∋_ ; _≈_ = _==_ ; ∅ = od∅ @@ -284,23 +287,23 @@ ; infinite = Ord omega ; isZF = isZF } where - Select : (X : HOD {suc n} ) → ((x : HOD {suc n} ) → Set (suc n) ) → HOD {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 : HOD {suc n} → (HOD {suc n} → HOD {suc n} ) → HOD {suc n} - Replace X ψ = Select ( Ord (sup-o ( λ x → od→ord (ψ (ord→od x ))))) ( λ x → ¬ (∀ (y : Ordinal ) → ¬ ( def X y ∧ ( x == ψ (Ord y) )))) - _,_ : HOD {suc n} → HOD {suc n} → HOD {suc n} + 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)) - Union : HOD {suc n} → HOD {suc n} + Union : OD {suc n} → OD {suc n} Union U = cseq U -- power : ∀ X ∃ A ∀ t ( t ∈ A ↔ ( ∀ {x} → t ∋ x → X ∋ x ) - ZFSet = HOD {suc n} + 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) ) - Power : HOD {suc n} → HOD {suc n} + Power : OD {suc n} → OD {suc n} Power A = Replace (Def (Ord (od→ord A))) ( λ x → A ∩ x ) -- _∪_ : ( A B : ZFSet ) → ZFSet -- A ∪ B = Select (A , B) ( λ x → (A ∋ x) ∨ ( B ∋ x ) ) @@ -310,7 +313,7 @@ infixr 200 _∈_ -- infixr 230 _∩_ _∪_ infixr 220 _⊆_ - isZF : IsZF (HOD {suc n}) _∋_ _==_ od∅ _,_ Union Power Select Replace (Ord omega) + isZF : IsZF (OD {suc n}) _∋_ _==_ od∅ _,_ Union Power Select Replace (Ord omega) isZF = record { isEquivalence = record { refl = eq-refl ; sym = eq-sym; trans = eq-trans } ; pair = pair @@ -330,11 +333,11 @@ ; replacement→ = replacement→ } where - pair : (A B : HOD {suc n} ) → ((A , B) ∋ A) ∧ ((A , B) ∋ B) + 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 : HOD {suc n} ) → ¬ (od∅ ∋ x) + empty : (x : OD {suc n} ) → ¬ (od∅ ∋ x) empty x (case1 ()) empty x (case2 ()) @@ -346,18 +349,18 @@ -- 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 A ∋ x by transitivity -- - POrd : {a : Ordinal } {t : HOD} → Def (Ord a) ∋ t → Def (Ord a) ∋ Ord (od→ord t) + POrd : {a : Ordinal } {t : OD} → Def (Ord a) ∋ t → Def (Ord a) ∋ Ord (od→ord t) POrd {a} {t} P∋t = o<→c< P∋t - ord-power→ : (a : Ordinal ) ( t : HOD) → Def (Ord a) ∋ t → {x : HOD} → t ∋ x → Ord a ∋ x + ord-power→ : (a : Ordinal ) ( t : OD) → Def (Ord a) ∋ t → {x : OD} → t ∋ x → Ord a ∋ x ord-power→ a t P∋t {x} t∋x with osuc-≡< (sup-lb (POrd P∋t)) ... | case1 eq = proj1 (def-subst (Ltx t∋x) (sym (subst₂ (λ j k → j ≡ k ) oiso oiso ( cong (λ k → ord→od k) (sym eq) ))) refl ) where - Ltx : {n : Level} → {x t : HOD {suc n}} → t ∋ x → Ord (od→ord t) ∋ x + Ltx : {n : Level} → {x t : OD {suc n}} → t ∋ x → Ord (od→ord t) ∋ x Ltx {n} {x} {t} lt = c<→o< lt ... | case2 lt = {!!} where sp = sup-x (λ x → od→ord ( ZFSubset (Ord a) (ord→od x))) - minsup : HOD + minsup : OD minsup = ZFSubset (Ord a) ( ord→od ( sup-x (λ x → od→ord ( ZFSubset (Ord a) (ord→od x))))) - Ltx : {n : Level} → {x t : HOD {suc n}} → t ∋ x → Ord (od→ord t) ∋ x + Ltx : {n : Level} → {x t : OD {suc n}} → t ∋ x → Ord (od→ord t) ∋ x Ltx {n} {x} {t} lt = c<→o< lt -- lemma1 hold because minsup is Ord (minα a sp) lemma1 : od→ord (Ord (od→ord t)) o< od→ord minsup → minsup ∋ Ord (od→ord t) @@ -380,7 +383,7 @@ -- 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 : HOD) → ({x : HOD} → (t ∋ x → (Ord a) ∋ x)) → Def (Ord 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 @@ -388,48 +391,59 @@ 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 : HOD {suc n}} + 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 ) (===-≡ 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< -- Power A = Replace (Def (Ord (od→ord A))) ( λ x → A ∩ x ) - power→ : ( A t : HOD) → Power A ∋ t → {x : HOD} → t ∋ x → A ∋ x + power→ : ( A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → A ∋ x power→ = {!!} - power← : (A t : HOD) → ({x : HOD} → (t ∋ x → A ∋ x)) → Power A ∋ t + power← : (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t power← A t t→A = {!!} where a = od→ord A - ψ : HOD → HOD + ψ : OD → OD ψ y = Def (Ord a) ∩ y - union-u : {X z : HOD {suc n}} → (U>z : Union X ∋ z ) → HOD {suc n} + union-u : {X z : OD {suc n}} → (U>z : Union X ∋ z ) → OD {suc n} union-u {X} {z} U>z = Ord ( osuc ( od→ord z ) ) - union→ : (X z u : HOD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z + union→ : (X z u : OD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z union→ X z u xx with trio< ( od→ord u ) ( osuc ( od→ord z )) union→ X z u xx | tri< a ¬b ¬c with osuc-< a (c<→o< (proj2 xx)) union→ X z u xx | tri< a ¬b ¬c | () union→ X z u xx | tri≈ ¬a b ¬c = def-subst {suc n} {_} {_} {X} {osuc (od→ord z)} (proj1 xx) refl b union→ X z u xx | tri> ¬a ¬b c = {!!} - union← : (X z : HOD) (X∋z : Union X ∋ z) → (X ∋ union-u {X} {z} X∋z ) ∧ (union-u {X} {z} X∋z ∋ z ) + union← : (X z : OD) (X∋z : Union X ∋ z) → (X ∋ union-u {X} {z} X∋z ) ∧ (union-u {X} {z} X∋z ∋ z ) union← X z X∋z = record { proj1 = lemma ; proj2 = <-osuc } where lemma : X ∋ union-u {X} {z} X∋z lemma = def-subst {suc n} {_} {_} {X} {od→ord (Ord (osuc ( od→ord z )))} X∋z refl ord-Ord - -- ψiso : {ψ : HOD {suc n} → Set (suc n)} {x y : HOD {suc n}} → ψ x → x ≡ y → ψ y - -- ψiso {ψ} t refl = t - selection : {ψ : HOD → Set (suc n)} {X y : HOD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y) - selection {X} {ψ} {y} = {!!} - replacement← : {ψ : HOD → HOD} (X x : HOD) → X ∋ x → Replace X ψ ∋ ψ x - replacement← {ψ} X x lt = {!!} - replacement→ : {ψ : HOD → HOD} (X x : HOD) → (lt : Replace X ψ ∋ x) → ¬ ( (y : HOD) → ¬ (x == ψ y)) - replacement→ {ψ} X x lt = contra-position lemma {!!} where - lemma : ( (y : HOD) → ¬ (x == ψ y)) → ( (y : Ordinal) → ¬ def X y ∧ (ord→od (od→ord x) == ψ (Ord y)) ) - lemma not y not2 = not (ord→od y) (subst₂ ( λ k j → k == j ) oiso (cong (λ k → ψ k ) Ord-ord ) (proj2 not2 )) + 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 (subst (λ k → Ord (od→ord x) ≡ k) oiso (Ord-ord) )) } )) + 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 )) - ∅-iso : {x : HOD} → ¬ (x == od∅) → ¬ ((ord→od (od→ord x)) == od∅) + ∅-iso : {x : OD} → ¬ (x == od∅) → ¬ ((ord→od (od→ord x)) == od∅) ∅-iso {x} neq = subst (λ k → ¬ k) (=-iso {n} ) neq - regularity : (x : HOD) (not : ¬ (x == 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 ) = x∋minimul x not proj2 (regularity x not ) = record { eq→ = lemma1 ; eq← = λ {y} d → lemma2 {y} d } where @@ -440,12 +454,12 @@ 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 : HOD {suc n}} → ((z : HOD) → (A ∋ z) ⇔ (B ∋ z)) → A == B + 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 open import Relation.Binary.PropositionalEquality - uxxx-ord : {x : HOD {suc n}} → {y : Ordinal {suc n}} → def (Union (x , (x , x))) y ⇔ ( y o< osuc (od→ord x) ) + uxxx-ord : {x : OD {suc n}} → {y : Ordinal {suc n}} → def (Union (x , (x , x))) y ⇔ ( y o< osuc (od→ord x) ) uxxx-ord {x} {y} = subst (λ k → k ⇔ ( y o< osuc (od→ord x) )) (sym lemma) ( osuc2 y (od→ord x)) where lemma : {y : Ordinal {suc n}} → def (Union (x , (x , x))) y ≡ osuc y o< osuc (osuc (od→ord x)) lemma {y} = let open ≡-Reasoning in begin @@ -459,11 +473,11 @@ ≡⟨ cong (λ k → osuc y o< k ) (omxxx (od→ord x) ) ⟩ osuc y o< osuc (osuc (od→ord x)) ∎ - infinite : HOD {suc n} + infinite : OD {suc n} infinite = Ord omega infinity∅ : Ord omega ∋ od∅ {suc n} infinity∅ = o<-subst (case1 (s≤s z≤n) ) ord-Ord refl - infinity : (x : HOD) → infinite ∋ x → infinite ∋ Union (x , (x , x )) + infinity : (x : OD) → infinite ∋ x → infinite ∋ Union (x , (x , x )) infinity x lt = o<-subst ( lemma (od→ord x) lt ) eq refl where eq : osuc (od→ord x) ≡ od→ord (Union (x , (x , x))) eq = let open ≡-Reasoning in begin @@ -482,12 +496,12 @@ lemma record { lv = (Suc Zero) ; ord = (Φ .1) } (case2 ()) | refl -- ∀ X [ ∅ ∉ X → (∃ f : X → ⋃ X ) → ∀ A ∈ X ( f ( A ) ∈ A ) ] -- this form is no good since X is a transitive set -- ∀ z [ ∀ x ( x ∈ z → ¬ ( x ≈ ∅ ) ) ∧ ∀ x ∀ y ( x , y ∈ z ∧ ¬ ( x ≈ y ) → x ∩ y ≈ ∅ ) → ∃ u ∀ x ( x ∈ z → ∃ t ( u ∩ x) ≈ ｛ t ｝) ] - record Choice (z : HOD {suc n}) : Set (suc (suc n)) where + record Choice (z : OD {suc n}) : Set (suc (suc n)) where field - u : {x : HOD {suc n}} ( x∈z : x ∈ z ) → HOD {suc n} - t : {x : HOD {suc n}} ( x∈z : x ∈ z ) → (x : HOD {suc n} ) → HOD {suc n} - choice : { x : HOD {suc n} } → ( x∈z : x ∈ z ) → ( u x∈z ∩ x) == ｛ t x∈z x ｝ - -- choice : {x : HOD {suc n}} ( x ∈ z → ¬ ( x ≈ ∅ ) ) → - -- axiom-of-choice : { X : HOD } → ( ¬x∅ : ¬ ( X == od∅ ) ) → { A : HOD } → (A∈X : A ∈ X ) → choice ¬x∅ A∈X ∈ A + u : {x : OD {suc n}} ( x∈z : x ∈ z ) → OD {suc n} + t : {x : OD {suc n}} ( x∈z : x ∈ z ) → (x : OD {suc n} ) → OD {suc n} + choice : { x : OD {suc n} } → ( x∈z : x ∈ z ) → ( u x∈z ∩ x) == ｛ t x∈z x ｝ + -- choice : {x : OD {suc n}} ( x ∈ z → ¬ ( x ≈ ∅ ) ) → + -- axiom-of-choice : { X : OD } → ( ¬x∅ : ¬ ( X == od∅ ) ) → { A : OD } → (A∈X : A ∈ X ) → choice ¬x∅ A∈X ∈ A -- axiom-of-choice {X} nx {A} lt = ¬∅=→∅∈ {!!}