# HG changeset patch # User Shinji KONO # Date 1562582927 -32400 # Node ID e022c0716936afe6fa569a8f401d974a3ec295a5 # Parent 02d421f1cc06b67696f501d92719ee3a36a7f7b4# Parent 6e767ad3edc20dfae4ed99c2af5b2e9ce5cb6212 only ordinal-definable.agda is finished. it assmues all ZF Set are Ordinals. HOD is incomplete, but we leave this for a while. diff -r 02d421f1cc06 -r e022c0716936 HOD.agda --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/HOD.agda Mon Jul 08 19:48:47 2019 +0900 @@ -0,0 +1,511 @@ +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 import Data.Unit + +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 + ==→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 + 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 ) + 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) ) + -- we should prove this in agda, but simply put here + ===-≡ : {n : Level} { x y : OD {suc n}} → x == y → x ≡ 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→¬< : {x y : Nat } → (x < y ) → ¬ ( y < x ) +>→¬< (s≤s x→¬< x : {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()) + + +-- 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 )) ) ) + +OrdSubset : {n : Level} → (A x : Ordinal {suc n} ) → ZFSubset (Ord A) (Ord x) ≡ Ord ( minα A x ) +OrdSubset {n} A x = ===-≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where + lemma1 : {y : Ordinal} → def (ZFSubset (Ord A) (Ord x)) y → def (Ord (minα A x)) y + lemma1 {y} s with trio< A x + lemma1 {y} s | tri< a ¬b ¬c = proj1 s + lemma1 {y} s | tri≈ ¬a refl ¬c = proj1 s + lemma1 {y} s | tri> ¬a ¬b c = proj2 s + lemma2 : {y : Ordinal} → def (Ord (minα A x)) y → def (ZFSubset (Ord A) (Ord x)) y + lemma2 {y} lt with trio< A x + lemma2 {y} lt | tri< a ¬b ¬c = record { proj1 = lt ; proj2 = ordtrans lt a } + lemma2 {y} lt | tri≈ ¬a refl ¬c = record { proj1 = lt ; proj2 = lt } + lemma2 {y} lt | tri> ¬a ¬b c = record { proj1 = ordtrans lt c ; 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}) → α o< β → 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 + +omega : { n : Level } → Ordinal {n} +omega = record { lv = Suc Zero ; ord = Φ 1 } + +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 = Ord omega + ; 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 y → def U (osuc y) } + _∈_ : ( 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 ) + + infixr 200 _∈_ + -- infixr 230 _∩_ _∪_ + infixr 220 _⊆_ + 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 + ; union-u = λ X z UX∋z → union-u {X} {z} UX∋z + ; union→ = union→ + ; union← = union← + ; empty = empty + ; power→ = power→ + ; power← = power← + ; extensionality = extensionality + ; minimul = minimul + ; regularity = regularity + ; infinity∅ = infinity∅ + ; infinity = λ _ → 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 : (x : OD {suc n} ) → ¬ (od∅ ∋ x) + empty x (case1 ()) + empty x (case2 ()) + + union-d : (X : OD {suc n}) → OD {suc n} + union-d X = record { def = λ y → def X (osuc y) } + 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 : 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 = def-subst lemma1 (sym lemma0) diso where + lemma0 : X ≡ Ord (od→ord X) + lemma0 = sym {!!} + lemma : osuc (od→ord z) o< od→ord X + lemma = ordtrans c ( c<→o< ( proj1 xx ) ) + lemma1 : Ord ( od→ord X) ∋ ord→od (osuc (od→ord z) ) + lemma1 = o<-subst lemma (sym diso) refl + 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 UX∋z = record { proj1 = lemma ; proj2 = <-osuc } where + lemma : X ∋ union-u {X} {z} UX∋z + lemma = def-subst {suc n} {_} {_} {X} {od→ord (Ord (osuc ( od→ord z )))} UX∋z refl {!!} + + 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)) + {!!} } )) + 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 {!!}) 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 )) + + --- + --- 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 + -- + -- if Power A ∋ t, from a propertiy of minimum sup there is osuc ZFSubset A ∋ t + -- then ZFSubset A ≡ t or ZFSubset A ∋ t. 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But first, please read +. diff -r 02d421f1cc06 -r e022c0716936 Todo --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/Todo Mon Jul 08 19:48:47 2019 +0900 @@ -0,0 +1,6 @@ +Mon Jul 8 19:43:37 JST 2019 + + ordinal-definable.agda assumes all ZF Set are ordinals, that it too restrictive + + remove ord-Ord and prove with some assuption in HOD.agda + union, power set, replace, inifinite diff -r 02d421f1cc06 -r e022c0716936 ordinal-definable.agda --- a/ordinal-definable.agda Mon Jun 10 09:53:45 2019 +0900 +++ b/ordinal-definable.agda Mon Jul 08 19:48:47 2019 +0900 @@ -1,3 +1,5 @@ +{-# OPTIONS --allow-unsolved-metas #-} + open import Level module ordinal-definable where @@ -23,6 +25,15 @@ 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 @@ -42,16 +53,16 @@ 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) } -od∅ : {n : Level} → OD {n} -od∅ {n} = record { def = λ _ → Lift n ⊥ } +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} - 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 - o<→c< : {n : Level} {x y : Ordinal {n} } → x o< y → def (ord→od y) x 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 @@ -82,7 +93,8 @@ ( 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 (lift ()) +∅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 @@ -109,11 +121,6 @@ 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 -record Minimumo {n : Level } (x : Ordinal {n}) : Set (suc n) where - field - mino : Ordinal {n} - min : {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<¬≡ (od→ord x) (od→ord x) refl (c<→o< oyx ) +... | 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<¬≡ (od→ord x) (od→ord x) refl (c<→o< oyx ) +... | 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 @@ -202,7 +209,8 @@ 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 | lift () + 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) @@ -213,7 +221,7 @@ 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<¬≡ (od→ord x) (od→ord y) (orefl oeq ) (c<→o< lt) +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) @@ -229,7 +237,8 @@ ∅< : {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 | lift () +∅< {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 @@ -250,6 +259,9 @@ 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} @@ -264,7 +276,10 @@ -- 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→od ( osuc ( od→ord x )) +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 ) @@ -272,7 +287,7 @@ 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→od ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) ) +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} @@ -281,8 +296,8 @@ 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 }) )) } -OD→ZF : {n : Level} → ZF {suc (suc n)} {suc n} -OD→ZF {n} = record { +Ord→ZF : {n : Level} → ZF {suc (suc n)} {suc n} +Ord→ZF {n} = record { ZFSet = OD {suc n} ; _∋_ = _∋_ ; _≈_ = _==_ @@ -295,12 +310,14 @@ ; infinite = ord→od ( record { lv = Suc Zero ; ord = Φ 1 } ) ; isZF = isZF } where - Replace : OD {suc n} → (OD {suc n} → OD {suc n} ) → OD {suc n} - Replace X ψ = sup-od ψ 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 ) @@ -311,8 +328,6 @@ 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) ) -- _∪_ : ( A B : ZFSet ) → ZFSet -- A ∪ B = Select (A , B) ( λ x → (A ∋ x) ∨ ( B ∋ x ) ) infixr 200 _∈_ @@ -334,15 +349,15 @@ ; infinity∅ = infinity∅ ; infinity = λ _ → infinity ; selection = λ {ψ} {X} {y} → selection {ψ} {X} {y} - ; replacement = replacement + ; replacement← = replacement← + ; replacement→ = replacement→ } where - open _∧_ - open Minimumo 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 () + 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 @@ -398,8 +413,19 @@ 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) → Replace X ψ ∋ ψ x - replacement {ψ} X x = sup-c< ψ {x} + 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 )) ∅-iso : {x : OD} → ¬ (x == od∅) → ¬ ((ord→od (od→ord x)) == od∅) ∅-iso {x} neq = subst (λ k → ¬ k) (=-iso {n} ) neq minimul : (x : OD {suc n} ) → ¬ (x == od∅ )→ OD {suc n} @@ -407,10 +433,12 @@ 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← = λ () } where + 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 with proj1 t - ... | x∈∅ = x∈∅ + 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 diff -r 02d421f1cc06 -r e022c0716936 ordinal.agda --- a/ordinal.agda Mon Jun 10 09:53:45 2019 +0900 +++ b/ordinal.agda Mon Jul 08 19:48:47 2019 +0900 @@ -97,6 +97,11 @@ osuc-lveq : {n : Level} { x : Ordinal {n} } → lv x ≡ lv ( osuc x ) osuc-lveq {n} = refl +osucc : {n : Level} {ox oy : Ordinal {n}} → oy o< ox → osuc oy o< osuc ox +osucc {n} {ox} {oy} (case1 x) = case1 x +osucc {n} {ox} {oy} (case2 x) with d<→lv x +... | refl = case2 (s< x) + nat-<> : { x y : Nat } → x < y → y < x → ⊥ nat-<> (s≤s x x ¬a ¬b c = x +minαd : {n : Level} → { lx : Nat } → OrdinalD {n} lx → OrdinalD lx → OrdinalD lx +minαd x y with triOrdd x y +minαd x y | tri< a ¬b ¬c = x +minαd x y | tri≈ ¬a b ¬c = y +minαd x y | tri> ¬a ¬b c = x + _o≤_ : {n : Level} → Ordinal → Ordinal → Set (suc n) a o≤ b = (a ≡ b) ∨ ( a o< b ) @@ -205,11 +224,23 @@ lemma1 (case1 x) = ¬a x lemma1 (case2 x) = ≡→¬d< x -maxα : {n : Level} → Ordinal {n} → Ordinal → Ordinal -maxα x y with <-cmp (lv x) (lv y) -maxα x y | tri< a ¬b ¬c = x -maxα x y | tri> ¬a ¬b c = y -maxα x y | tri≈ ¬a refl ¬c = record { lv = lv x ; ord = maxαd (ord x) (ord y) } +maxα : {n : Level} → Ordinal {suc n} → Ordinal → Ordinal +maxα x y with trio< x y +maxα x y | tri< a ¬b ¬c = y +maxα x y | tri> ¬a ¬b c = x +maxα x y | tri≈ ¬a refl ¬c = x + +minα : {n : Level} → Ordinal {suc n} → Ordinal → Ordinal +minα {n} x y with trio< {n} x y +minα x y | tri< a ¬b ¬c = x +minα x y | tri> ¬a ¬b c = y +minα x y | tri≈ ¬a refl ¬c = x + +min1 : {n : Level} → {x y z : Ordinal {suc n} } → z o< x → z o< y → z o< minα x y +min1 {n} {x} {y} {z} z ¬a ¬b c = z