Members/kono/Proof/ZF-in-agda: OD.agda comparison

comparison OD.agda @ 276:6f10c47e4e7a

separate choice fix sup-o

author	Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date	Sat, 09 May 2020 09:02:52 +0900
parents	29a85a427ed2
children	d9d3654baee1

comparison

equal deleted inserted replaced

-:455792eaa611
+:6f10c47e4e7a
 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 )
+-- supermum as Replacement Axiom ( corresponding Ordinal of OD has maximum )
-sup-o  :  ( Ordinal  → Ordinal ) →  Ordinal
+sup-o  :  ( OD → Ordinal ) →  Ordinal
-sup-o< :  { ψ : Ordinal  →  Ordinal } → ∀ {x : Ordinal } → ψ x  o<  sup-o ψ
+sup-o< :  { ψ : OD →  Ordinal } → ∀ {x : OD } → 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
+data One : Set n where
--- this should be ¬ (x == od∅ )→ ∃ ox → x ∋ Ord ox  ( minimum of x )
+OneObj : One
-x∋minimal : (x : OD  ) → ( ne : ¬ (x == od∅ ) ) → def x ( od→ord ( minimal x ne ) )
--- minimality (may proved by ε-induction )
+Ords : OD
-minimal-1 : (x : OD  ) → ( ne : ¬ (x == od∅ ) ) → (y : OD ) → ¬ ( def (minimal x ne) (od→ord y)) ∧ (def x (od→ord  y) )
+Ords = record { def = λ x → One }
+maxod : {x : OD} → od→ord x o< od→ord Ords
+maxod {x} = c<→o< OneObj
 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))
 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-od ψ = Ord ( sup-o ( λ x → od→ord (ψ 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 )}
+sup-c<   ψ {x} = def-subst  {_} {_} {Ord ( sup-o  ( λ x → od→ord (ψ 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 : od→ord (ψ (ord→od (od→ord x))) o< sup-o (λ x → od→ord (ψ 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 y<x = ordtrans y<x x<a
 is-o∅ x | tri> ¬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))
---
--- 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 )))))  }
 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 )) ) )
+Def  A = Ord ( sup-o  ( λ x → od→ord ( ZFSubset A x) ) )
 -- _⊆_ :  ( A B : OD   ) → ∀{ x : OD  } →  Set n
 -- _⊆_ A B {x} = A ∋ x →  B ∋ x
 record _⊆_ ( A B : OD   ) : Set (suc n) where
 } 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 }
+Replace X ψ = record { def = λ x → (x o< sup-o  ( λ x → od→ord (ψ 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
 proj1 = def-subst  {_} {_} {(Ord a)} {z}
 ( t→A (def-subst  {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso  }
 lemma1 :  {a : Ordinal } { t : OD }
 → (eq : ZFSubset (Ord a) t == t)  → od→ord (ZFSubset (Ord a) (ord→od (od→ord t))) ≡ od→ord t
 lemma1  {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 :  od→ord (ZFSubset (Ord a) (ord→od (od→ord t)) ) o< sup-o  (λ x → od→ord (ZFSubset (Ord a) x))
 lemma = sup-o<
 --
 -- Every set in OD is a subset of Ordinals, so make Def (Ord (od→ord A)) first
 -- then replace of all elements of the Power set by A ∩ y
 ≡⟨ 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 : od→ord t o< sup-o  (λ x → od→ord (A ∩ x))
-lemma1 = subst (λ k → od→ord k 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 ∩ x)))
-lemma4 (sup-o<  {λ x → od→ord (A ∩ ord→od x)} {od→ord t} )
+lemma4 (sup-o<  {λ x → od→ord (A ∩ x)}  )
 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  )))
 lemma : {x y : OD} → od→ord x o< osuc a → x ∋ y →  Ord a ∋ y
 lemma lt y<x with osuc-≡< lt
 lemma lt y<x | case1 refl = c<→o< y<x
 lemma lt y<x | case2 x<a = ordtrans (c<→o< y<x) x<a
--- continuum-hyphotheis : (a : Ordinal) → Power (Ord a) ⊆ Ord (osuc a)
+continuum-hyphotheis : (a : Ordinal) → Set (suc n)
--- continuum-hyphotheis a = ?
+continuum-hyphotheis a = Power (Ord a) ⊆ Ord (osuc a)
---  assuming axiom of choice
-regularity :  (x : OD) (not : ¬ (x == od∅)) →
-(x ∋ minimal x not) ∧ (Select (minimal x not) (λ x₁ → (minimal x not ∋ x₁) ∧ (x ∋ x₁)) == od∅)
-proj1 (regularity x not ) = x∋minimal x not
-proj2 (regularity x not ) = record { eq→ = lemma1 ; eq← = λ {y} d → lemma2 {y} d } where
-lemma1 : {x₁ : Ordinal} → def (Select (minimal x not) (λ x₂ → (minimal x not ∋ x₂) ∧ (x ∋ x₂))) x₁ → def od∅ x₁
-lemma1 {x₁} s = ⊥-elim  ( minimal-1 x not (ord→od x₁) lemma3 ) where
-lemma3 : def (minimal x not) (od→ord (ord→od x₁)) ∧ def x (od→ord (ord→od x₁))
-lemma3 = record { proj1 = def-subst  {_} {_} {minimal x not} {_} (proj1 s) refl (sym diso)
-; proj2 = proj2 (proj2 s) }
-lemma2 : {x₁ : Ordinal} → def od∅ x₁ → def (Select (minimal x not) (λ x₂ → (minimal x not ∋ x₂) ∧ (x ∋ x₂))) x₁
-lemma2 {y} d = ⊥-elim (empty (ord→od y) (def-subst  {_} {_} {od∅} {od→ord (ord→od y)} d refl (sym diso) ))
 extensionality0 : {A B : OD } → ((z : OD) → (A ∋ z) ⇔ (B ∋ z)) → A == B
 eq→ (extensionality0 {A} {B} eq ) {x} d = def-iso  {A} {B} (sym diso) (proj1 (eq (ord→od x))) d
 eq← (extensionality0 {A} {B} eq ) {x} d = def-iso  {B} {A} (sym diso) (proj2 (eq (ord→od x))) d
 infinity x lt = def-subst  {_} {_} {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 minimal in our case )
--- ∀ X [ ∅ ∉ X → (∃ f : X → ⋃ X ) → ∀ A ∈ X ( f ( A ) ∈ A ) ]
-choice-func : (X : OD  ) → {x : OD } → ¬ ( x == od∅ ) → ( X ∋ x ) → OD
-choice-func X {x} not X∋x = minimal x not
-choice : (X : OD  ) → {A : OD } → ( X∋A : X ∋ A ) → (not : ¬ ( A == od∅ )) → A ∋ choice-func X not X∋A
-choice X {A} X∋A not = x∋minimal A not
----
---- With assuption of OD is ordered,  p ∨ ( ¬ p ) <=> 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<X = case2 ( record { a-choice = ord→od y ; is-in = y<X } )
-lemma1 y | no ¬y<X = case1 ( λ lt y<X → ¬y<X (subst (λ k → def X k ) (sym diso) y<X ) )
-lemma :  ((y : Ordinals.ord O) → (O Ordinals.o< y) x → def X y → ⊥) ∨ choiced X
-lemma = ∀-imply-or lemma1
-have_to_find : choiced X
-have_to_find = dont-or  (lemma-ord (od→ord X )) ¬¬X∋x where
-¬¬X∋x : ¬ ((x : Ordinal) → x o< (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 )
-}
 Union = ZF.Union OD→ZF
 Power = ZF.Power OD→ZF
 Select = ZF.Select OD→ZF
 Replace = ZF.Replace OD→ZF

Mercurial > hg > Members > kono > Proof > ZF-in-agda

comparison OD.agda @ 276:6f10c47e4e7a