open import Level
open import Ordinals
module OD {n : Level } (O : Ordinals {n} ) where
open import zf
open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ )
open import Relation.Binary.PropositionalEquality
open import Data.Nat.Properties
open import Data.Empty
open import Relation.Nullary
open import Relation.Binary
open import Relation.Binary.Core
open import logic
open import nat
open inOrdinal O
-- Ordinal Definable Set
record OD : Set (suc n ) where
field
def : (x : Ordinal ) → Set n
open OD
open _∧_
open _∨_
open Bool
record _==_ ( a b : OD ) : Set n where
field
eq→ : ∀ { x : Ordinal } → def a x → def b x
eq← : ∀ { x : Ordinal } → def b x → def a x
id : {A : Set n} → A → A
id x = x
==-refl : { x : OD } → x == x
==-refl {x} = record { eq→ = id ; eq← = id }
open _==_
==-trans : { x y z : OD } → x == y → y == z → x == z
==-trans x=y y=z = record { eq→ = λ {m} t → eq→ y=z (eq→ x=y t) ; eq← = λ {m} t → eq← x=y (eq← y=z t) }
==-sym : { x y : OD } → x == y → y == x
==-sym x=y = record { eq→ = λ {m} t → eq← x=y t ; eq← = λ {m} t → eq→ x=y t }
⇔→== : { x y : OD } → ( {z : Ordinal } → def x z ⇔ def y z) → x == y
eq→ ( ⇔→== {x} {y} eq ) {z} m = proj1 eq m
eq← ( ⇔→== {x} {y} eq ) {z} m = proj2 eq m
-- Ordinal in OD ( and ZFSet ) Transitive Set
Ord : ( a : Ordinal ) → OD
Ord a = record { def = λ y → y o< a }
od∅ : OD
od∅ = Ord o∅
-- next assumptions are our axiom
-- it defines a subset of OD, which is called HOD, usually defined as
-- HOD = { x | TC x ⊆ OD }
-- where TC x is a transitive clusure of x, i.e. Union of all elemnts of all subset of x
postulate
-- OD can be iso to a subset of Ordinal ( by means of Godel Set )
od→ord : OD → Ordinal
ord→od : Ordinal → OD
c<→o< : {x y : OD } → def y ( od→ord x ) → od→ord x o< od→ord y
oiso : {x : OD } → ord→od ( od→ord x ) ≡ x
diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x
==→o≡ : { x y : OD } → (x == y) → x ≡ y
-- next assumption causes ∀ x ∋ ∅ . It menas only an ordinal is allowed as OD
-- o<→c< : {n : Level} {x y : Ordinal } → x o< y → def (ord→od y) x
-- ord→od x ≡ Ord x results the same
-- supermum as Replacement Axiom ( this assumes Ordinal has some upper bound )
sup-o : ( Ordinal → Ordinal ) → Ordinal
sup-o< : { ψ : Ordinal → Ordinal } → ∀ {x : Ordinal } → ψ x o< sup-o ψ
-- contra-position of mimimulity of supermum required in Power Set Axiom
-- sup-x : {n : Level } → ( Ordinal → Ordinal ) → Ordinal
-- sup-lb : {n : Level } → { ψ : Ordinal → Ordinal } → {z : Ordinal } → z o< sup-o ψ → z o< osuc (ψ (sup-x ψ))
-- mimimul and x∋minimal is an Axiom of choice
minimal : (x : OD ) → ¬ (x == od∅ )→ OD
-- this should be ¬ (x == od∅ )→ ∃ ox → x ∋ Ord ox ( minimum of x )
x∋minimal : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → def x ( od→ord ( minimal x ne ) )
-- minimality (may proved by ε-induction )
minimal-1 : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → (y : OD ) → ¬ ( def (minimal x ne) (od→ord y)) ∧ (def x (od→ord y) )
o<→c<→OD=Ord : ( {x y : Ordinal } → x o< y → def (ord→od y) x ) → {x : OD } → x ≡ Ord (od→ord x)
o<→c<→OD=Ord o<→c< {x} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where
lemma1 : {y : Ordinal} → def x y → def (Ord (od→ord x)) y
lemma1 {y} lt = subst ( λ k → k o< od→ord x ) diso (c<→o< {ord→od y} {x} (subst (λ k → def x k ) (sym diso) lt))
lemma2 : {y : Ordinal} → def (Ord (od→ord x)) y → def x y
lemma2 {y} lt = subst (λ k → def k y ) oiso (o<→c< {y} {od→ord x} lt )
_∋_ : ( a x : OD ) → Set n
_∋_ a x = def a ( od→ord x )
_c<_ : ( x a : OD ) → Set n
x c< a = a ∋ x
cseq : {n : Level} → OD → OD
cseq x = record { def = λ y → def x (osuc y) } where
def-subst : {Z : OD } {X : Ordinal }{z : OD } {x : Ordinal }→ def Z X → Z ≡ z → X ≡ x → def z x
def-subst df refl refl = df
sup-od : ( OD → OD ) → OD
sup-od ψ = Ord ( sup-o ( λ x → od→ord (ψ (ord→od x ))) )
sup-c< : ( ψ : OD → OD ) → ∀ {x : OD } → def ( sup-od ψ ) (od→ord ( ψ x ))
sup-c< ψ {x} = def-subst {_} {_} {Ord ( sup-o ( λ x → od→ord (ψ (ord→od x ))) )} {od→ord ( ψ x )}
lemma refl (cong ( λ k → od→ord (ψ k) ) oiso) where
lemma : od→ord (ψ (ord→od (od→ord x))) o< sup-o (λ x → od→ord (ψ (ord→od x)))
lemma = subst₂ (λ j k → j o< k ) refl diso (o<-subst (sup-o< ) refl (sym diso) )
otrans : {n : Level} {a x y : Ordinal } → def (Ord a) x → def (Ord x) y → def (Ord a) y
otrans x ¬a ¬b c = no ¬b
_,_ : OD → OD → OD
x , y = record { def = λ t → (t ≡ od→ord x ) ∨ ( t ≡ od→ord y ) } -- Ord (omax (od→ord x) (od→ord y))
<_,_> : (x y : OD) → OD
< x , y > = (x , x ) , (x , y )
exg-pair : { x y : OD } → (x , y ) == ( y , x )
exg-pair {x} {y} = record { eq→ = left ; eq← = right } where
left : {z : Ordinal} → def (x , y) z → def (y , x) z
left (case1 t) = case2 t
left (case2 t) = case1 t
right : {z : Ordinal} → def (y , x) z → def (x , y) z
right (case1 t) = case2 t
right (case2 t) = case1 t
ord≡→≡ : { x y : OD } → od→ord x ≡ od→ord y → x ≡ y
ord≡→≡ eq = subst₂ (λ j k → j ≡ k ) oiso oiso ( cong ( λ k → ord→od k ) eq )
od≡→≡ : { x y : Ordinal } → ord→od x ≡ ord→od y → x ≡ y
od≡→≡ eq = subst₂ (λ j k → j ≡ k ) diso diso ( cong ( λ k → od→ord k ) eq )
eq-prod : { x x' y y' : OD } → x ≡ x' → y ≡ y' → < x , y > ≡ < x' , y' >
eq-prod refl refl = refl
prod-eq : { x x' y y' : OD } → < x , y > == < x' , y' > → (x ≡ x' ) ∧ ( y ≡ y' )
prod-eq {x} {x'} {y} {y'} eq = record { proj1 = lemmax ; proj2 = lemmay } where
lemma0 : {x y z : OD } → ( x , x ) == ( z , y ) → x ≡ y
lemma0 {x} {y} eq with trio< (od→ord x) (od→ord y)
lemma0 {x} {y} eq | tri< a ¬b ¬c with eq← eq {od→ord y} (case2 refl)
lemma0 {x} {y} eq | tri< a ¬b ¬c | case1 s = ⊥-elim ( o<¬≡ (sym s) a )
lemma0 {x} {y} eq | tri< a ¬b ¬c | case2 s = ⊥-elim ( o<¬≡ (sym s) a )
lemma0 {x} {y} eq | tri≈ ¬a b ¬c = ord≡→≡ b
lemma0 {x} {y} eq | tri> ¬a ¬b c with eq← eq {od→ord y} (case2 refl)
lemma0 {x} {y} eq | tri> ¬a ¬b c | case1 s = ⊥-elim ( o<¬≡ s c )
lemma0 {x} {y} eq | tri> ¬a ¬b c | case2 s = ⊥-elim ( o<¬≡ s c )
lemma2 : {x y z : OD } → ( x , x ) == ( z , y ) → z ≡ y
lemma2 {x} {y} {z} eq = trans (sym (lemma0 lemma3 )) ( lemma0 eq ) where
lemma3 : ( x , x ) == ( y , z )
lemma3 = ==-trans eq exg-pair
lemma1 : {x y : OD } → ( x , x ) == ( y , y ) → x ≡ y
lemma1 {x} {y} eq with eq← eq {od→ord y} (case2 refl)
lemma1 {x} {y} eq | case1 s = ord≡→≡ (sym s)
lemma1 {x} {y} eq | case2 s = ord≡→≡ (sym s)
lemma4 : {x y z : OD } → ( x , y ) == ( x , z ) → y ≡ z
lemma4 {x} {y} {z} eq with eq← eq {od→ord z} (case2 refl)
lemma4 {x} {y} {z} eq | case1 s with ord≡→≡ s -- x ≡ z
... | refl with lemma2 (==-sym eq )
... | refl = refl
lemma4 {x} {y} {z} eq | case2 s = ord≡→≡ (sym s) -- y ≡ z
lemmax : x ≡ x'
lemmax with eq→ eq {od→ord (x , x)} (case1 refl)
lemmax | case1 s = lemma1 (ord→== s ) -- (x,x)≡(x',x')
lemmax | case2 s with lemma2 (ord→== s ) -- (x,x)≡(x',y') with x'≡y'
... | refl = lemma1 (ord→== s )
lemmay : y ≡ y'
lemmay with lemmax
... | refl with lemma4 eq -- with (x,y)≡(x,y')
... | eq1 = lemma4 (ord→== (cong (λ k → od→ord k ) eq1 ))
data ord-pair : (p : Ordinal) → Set n where
pair : (x y : Ordinal ) → ord-pair ( od→ord ( < ord→od x , ord→od y > ) )
ZFProduct : OD
ZFProduct = record { def = λ x → ord-pair x }
-- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ )
-- eq-pair : { x x' y y' : Ordinal } → x ≡ x' → y ≡ y' → pair x y ≅ pair x' y'
-- eq-pair refl refl = HE.refl
pi1 : { p : Ordinal } → ord-pair p → Ordinal
pi1 ( pair x y) = x
π1 : { p : OD } → ZFProduct ∋ p → OD
π1 lt = ord→od (pi1 lt )
pi2 : { p : Ordinal } → ord-pair p → Ordinal
pi2 ( pair x y ) = y
π2 : { p : OD } → ZFProduct ∋ p → OD
π2 lt = ord→od (pi2 lt )
op-cons : { ox oy : Ordinal } → ZFProduct ∋ < ord→od ox , ord→od oy >
op-cons {ox} {oy} = pair ox oy
p-cons : ( x y : OD ) → ZFProduct ∋ < x , y >
p-cons x y = def-subst {_} {_} {ZFProduct} {od→ord (< x , y >)} (pair (od→ord x) ( od→ord y )) refl (
let open ≡-Reasoning in begin
od→ord < ord→od (od→ord x) , ord→od (od→ord y) >
≡⟨ cong₂ (λ j k → od→ord < j , k >) oiso oiso ⟩
od→ord < x , y >
∎ )
op-iso : { op : Ordinal } → (q : ord-pair op ) → od→ord < ord→od (pi1 q) , ord→od (pi2 q) > ≡ op
op-iso (pair ox oy) = refl
p-iso : { x : OD } → (p : ZFProduct ∋ x ) → < π1 p , π2 p > ≡ x
p-iso {x} p = ord≡→≡ (op-iso p)
p-pi1 : { x y : OD } → (p : ZFProduct ∋ < x , y > ) → π1 p ≡ x
p-pi1 {x} {y} p = proj1 ( prod-eq ( ord→== (op-iso p) ))
p-pi2 : { x y : OD } → (p : ZFProduct ∋ < x , y > ) → π2 p ≡ y
p-pi2 {x} {y} p = proj2 ( prod-eq ( ord→== (op-iso p)))
--
-- Axiom of choice in intutionistic logic implies the exclude middle
-- https://plato.stanford.edu/entries/axiom-choice/#AxiChoLog
--
ppp : { p : Set n } { a : OD } → record { def = λ x → p } ∋ a → p
ppp {p} {a} d = d
p∨¬p : ( p : Set n ) → p ∨ ( ¬ p ) -- assuming axiom of choice
p∨¬p p with is-o∅ ( od→ord ( record { def = λ x → p } ))
p∨¬p p | yes eq = case2 (¬p eq) where
ps = record { def = λ x → p }
lemma : ps == od∅ → p → ⊥
lemma eq p0 = ¬x<0 {od→ord ps} (eq→ eq p0 )
¬p : (od→ord ps ≡ o∅) → p → ⊥
¬p eq = lemma ( subst₂ (λ j k → j == k ) oiso o∅≡od∅ ( o≡→== eq ))
p∨¬p p | no ¬p = case1 (ppp {p} {minimal ps (λ eq → ¬p (eqo∅ eq))} lemma) where
ps = record { def = λ x → p }
eqo∅ : ps == od∅ → od→ord ps ≡ o∅
eqo∅ eq = subst (λ k → od→ord ps ≡ k) ord-od∅ ( cong (λ k → od→ord k ) (==→o≡ eq))
lemma : ps ∋ minimal ps (λ eq → ¬p (eqo∅ eq))
lemma = x∋minimal ps (λ eq → ¬p (eqo∅ eq))
decp : ( p : Set n ) → Dec p -- assuming axiom of choice
decp p with p∨¬p p
decp p | case1 x = yes x
decp p | case2 x = no x
double-neg-eilm : {A : Set n} → ¬ ¬ A → A -- we don't have this in intutionistic logic
double-neg-eilm {A} notnot with decp A -- assuming axiom of choice
... | yes p = p
... | no ¬p = ⊥-elim ( notnot ¬p )
OrdP : ( x : Ordinal ) ( y : OD ) → Dec ( Ord x ∋ y )
OrdP x y with trio< x (od→ord y)
OrdP x y | tri< a ¬b ¬c = no ¬c
OrdP x y | tri≈ ¬a refl ¬c = no ( o<¬≡ refl )
OrdP x y | tri> ¬a ¬b c = yes c
-- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ )
-- postulate f-extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality n (suc n)
in-codomain : (X : OD ) → ( ψ : OD → OD ) → OD
in-codomain X ψ = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( def X y ∧ ( x ≡ od→ord (ψ (ord→od y ))))) }
-- Power Set of X ( or constructible by λ y → def X (od→ord y )
ZFSubset : (A x : OD ) → OD
ZFSubset A x = record { def = λ y → def A y ∧ def x y } -- roughly x = A → Set
Def : (A : OD ) → OD
Def A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) )
-- _⊆_ : ( A B : OD ) → ∀{ x : OD } → Set n
-- _⊆_ A B {x} = A ∋ x → B ∋ x
record _⊆_ ( A B : OD ) : Set (suc n) where
field
incl : { x : OD } → A ∋ x → B ∋ x
open _⊆_
infixr 220 _⊆_
subset-lemma : {A x : OD } → ( {y : OD } → x ∋ y → ZFSubset A x ∋ y ) ⇔ ( x ⊆ A )
subset-lemma {A} {x} = record {
proj1 = λ lt → record { incl = λ x∋z → proj1 (lt x∋z) }
; proj2 = λ x⊆A lt → record { proj1 = incl x⊆A lt ; proj2 = lt }
}
open import Data.Unit
ε-induction : { ψ : OD → Set (suc n)}
→ ( {x : OD } → ({ y : OD } → x ∋ y → ψ y ) → ψ x )
→ (x : OD ) → ψ x
ε-induction {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (osuc (od→ord x)) <-osuc ) where
induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (ord→od oy)) → ψ (ord→od ox)
induction ox prev = ind ( λ {y} lt → subst (λ k → ψ k ) oiso (prev (od→ord y) (o<-subst (c<→o< lt) refl diso )))
ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (ord→od oy)
ε-induction-ord ox {oy} lt = TransFinite {λ oy → ψ (ord→od oy)} induction oy
-- minimal-2 : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → (y : OD ) → ¬ ( def (minimal x ne) (od→ord y)) ∧ (def x (od→ord y) )
-- minimal-2 x ne y = contra-position ( ε-induction (λ {z} ind → {!!} ) x ) ( λ p → {!!} )
OD→ZF : ZF
OD→ZF = record {
ZFSet = OD
; _∋_ = _∋_
; _≈_ = _==_
; ∅ = od∅
; _,_ = _,_
; Union = Union
; Power = Power
; Select = Select
; Replace = Replace
; infinite = infinite
; isZF = isZF
} where
ZFSet = OD
Select : (X : OD ) → ((x : OD ) → Set n ) → OD
Select X ψ = record { def = λ x → ( def X x ∧ ψ ( ord→od x )) }
Replace : OD → (OD → OD ) → OD
Replace X ψ = record { def = λ x → (x o< sup-o ( λ x → od→ord (ψ (ord→od x )))) ∧ def (in-codomain X ψ) x }
_∩_ : ( A B : ZFSet ) → ZFSet
A ∩ B = record { def = λ x → def A x ∧ def B x }
Union : OD → OD
Union U = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((def U u) ∧ (def (ord→od u) x))) }
_∈_ : ( A B : ZFSet ) → Set n
A ∈ B = B ∋ A
Power : OD → OD
Power A = Replace (Def (Ord (od→ord A))) ( λ x → A ∩ x )
{_} : ZFSet → ZFSet
{ x } = ( x , x )
data infinite-d : ( x : Ordinal ) → Set n where
iφ : infinite-d o∅
isuc : {x : Ordinal } → infinite-d x →
infinite-d (od→ord ( Union (ord→od x , (ord→od x , ord→od x ) ) ))
infinite : OD
infinite = record { def = λ x → infinite-d x }
infixr 200 _∈_
-- infixr 230 _∩_ _∪_
isZF : IsZF (OD ) _∋_ _==_ od∅ _,_ Union Power Select Replace infinite
isZF = record {
isEquivalence = record { refl = ==-refl ; sym = ==-sym; trans = ==-trans }
; pair→ = pair→
; pair← = pair←
; union→ = union→
; union← = union←
; empty = empty
; power→ = power→
; power← = power←
; extensionality = λ {A} {B} {w} → extensionality {A} {B} {w}
-- ; ε-induction = {!!}
; infinity∅ = infinity∅
; infinity = infinity
; selection = λ {X} {ψ} {y} → selection {X} {ψ} {y}
; replacement← = replacement←
; replacement→ = replacement→
; choice-func = choice-func
; choice = choice
} where
pair→ : ( x y t : ZFSet ) → (x , y) ∋ t → ( t == x ) ∨ ( t == y )
pair→ x y t (case1 t≡x ) = case1 (subst₂ (λ j k → j == k ) oiso oiso (o≡→== t≡x ))
pair→ x y t (case2 t≡y ) = case2 (subst₂ (λ j k → j == k ) oiso oiso (o≡→== t≡y ))
pair← : ( x y t : ZFSet ) → ( t == x ) ∨ ( t == y ) → (x , y) ∋ t
pair← x y t (case1 t==x) = case1 (cong (λ k → od→ord k ) (==→o≡ t==x))
pair← x y t (case2 t==y) = case2 (cong (λ k → od→ord k ) (==→o≡ t==y))
empty : (x : OD ) → ¬ (od∅ ∋ x)
empty x = ¬x<0
o<→c< : {x y : Ordinal } → x o< y → (Ord x) ⊆ (Ord y)
o<→c< lt = record { incl = λ z → ordtrans z lt }
⊆→o< : {x y : Ordinal } → (Ord x) ⊆ (Ord y) → 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 (incl lt) (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 = FExists _ lemma UX∋z where
lemma : {y : Ordinal} → def X y ∧ def (ord→od y) (od→ord z) → ¬ ((u : OD) → ¬ (X ∋ u) ∧ (u ∋ z))
lemma {y} xx not = not (ord→od y) record { proj1 = subst ( λ k → def X k ) (sym diso) (proj1 xx ) ; proj2 = proj2 xx }
ψiso : {ψ : OD → Set n} {x y : OD } → ψ x → x ≡ y → ψ y
ψiso {ψ} t refl = t
selection : {ψ : OD → Set n} {X y : OD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y)
selection {ψ} {X} {y} = record {
proj1 = λ cond → record { proj1 = proj1 cond ; proj2 = ψiso {ψ} (proj2 cond) (sym oiso) }
; proj2 = λ select → record { proj1 = proj1 select ; proj2 = ψiso {ψ} (proj2 select) oiso }
}
replacement← : {ψ : OD → OD} (X x : OD) → X ∋ x → Replace X ψ ∋ ψ x
replacement← {ψ} X x lt = record { proj1 = sup-c< ψ {x} ; proj2 = lemma } where
lemma : def (in-codomain X ψ) (od→ord (ψ x))
lemma not = ⊥-elim ( not ( od→ord x ) (record { proj1 = lt ; proj2 = cong (λ k → od→ord (ψ k)) (sym oiso)} ))
replacement→ : {ψ : OD → OD} (X x : OD) → (lt : Replace X ψ ∋ x) → ¬ ( (y : OD) → ¬ (x == ψ y))
replacement→ {ψ} X x lt = contra-position lemma (lemma2 (proj2 lt)) where
lemma2 : ¬ ((y : Ordinal) → ¬ def X y ∧ ((od→ord x) ≡ od→ord (ψ (ord→od y))))
→ ¬ ((y : Ordinal) → ¬ def X y ∧ (ord→od (od→ord x) == ψ (ord→od y)))
lemma2 not not2 = not ( λ y d → not2 y (record { proj1 = proj1 d ; proj2 = lemma3 (proj2 d)})) where
lemma3 : {y : Ordinal } → (od→ord x ≡ od→ord (ψ (ord→od y))) → (ord→od (od→ord x) == ψ (ord→od y))
lemma3 {y} eq = subst (λ k → ord→od (od→ord x) == k ) oiso (o≡→== eq )
lemma : ( (y : OD) → ¬ (x == ψ y)) → ( (y : Ordinal) → ¬ def X y ∧ (ord→od (od→ord x) == ψ (ord→od y)) )
lemma not y not2 = not (ord→od y) (subst (λ k → k == ψ (ord→od y)) oiso ( proj2 not2 ))
---
--- Power Set
---
--- First consider ordinals in OD
---
--- ZFSubset A x = record { def = λ y → def A y ∧ def x y } subset of A
--
--
∩-≡ : { a b : OD } → ({x : OD } → (a ∋ x → b ∋ x)) → a == ( b ∩ a )
∩-≡ {a} {b} inc = record {
eq→ = λ {x} x axiom of choice
---
record choiced ( X : OD) : Set (suc n) where
field
a-choice : OD
is-in : X ∋ a-choice
choice-func' : (X : OD ) → (p∨¬p : ( p : Set (suc n)) → p ∨ ( ¬ p )) → ¬ ( X == od∅ ) → choiced X
choice-func' X p∨¬p not = have_to_find where
ψ : ( ox : Ordinal ) → Set (suc n)
ψ ox = (( x : Ordinal ) → x o< ox → ( ¬ def X x )) ∨ choiced X
lemma-ord : ( ox : Ordinal ) → ψ ox
lemma-ord ox = TransFinite {ψ} induction ox where
∋-p : (A x : OD ) → Dec ( A ∋ x )
∋-p A x with p∨¬p (Lift (suc n) ( A ∋ x )) -- LEM
∋-p A x | case1 (lift t) = yes t
∋-p A x | case2 t = no (λ x → t (lift x ))
∀-imply-or : {A : Ordinal → Set n } {B : Set (suc n) }
→ ((x : Ordinal ) → A x ∨ B) → ((x : Ordinal ) → A x) ∨ B
∀-imply-or {A} {B} ∀AB with p∨¬p (Lift ( suc n ) ((x : Ordinal ) → A x)) -- LEM
∀-imply-or {A} {B} ∀AB | case1 (lift t) = case1 t
∀-imply-or {A} {B} ∀AB | case2 x = case2 (lemma (λ not → x (lift not ))) where
lemma : ¬ ((x : Ordinal ) → A x) → B
lemma not with p∨¬p B
lemma not | case1 b = b
lemma not | case2 ¬b = ⊥-elim (not (λ x → dont-orb (∀AB x) ¬b ))
induction : (x : Ordinal) → ((y : Ordinal) → y o< x → ψ y) → ψ x
induction x prev with ∋-p X ( ord→od x)
... | yes p = case2 ( record { a-choice = ord→od x ; is-in = p } )
... | no ¬p = lemma where
lemma1 : (y : Ordinal) → (y o< x → def X y → ⊥) ∨ choiced X
lemma1 y with ∋-p X (ord→od y)
lemma1 y | yes y