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

comparison src/OD.agda @ 1091:63c1167b2343

fix comments

author	Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date	Tue, 20 Dec 2022 11:20:52 +0900
parents	88fae58f89f5
children	08b6aa6870d9

comparison

equal deleted inserted replaced

-:2cf182f0f583
+:63c1167b2343
 {-# OPTIONS --allow-unsolved-metas #-}
 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 hiding ( [_] )
 open import Data.Nat.Properties
 open import Data.Empty
 open import Relation.Nullary
 open import Relation.Binary  hiding (_⇔_)
 open import Relation.Binary.Core hiding (_⇔_)
 open import logic
 import OrdUtil
 open import nat
 open Ordinals.Ordinals  O
 open Ordinals.IsOrdinals isOrdinal
 open Ordinals.IsNext isNext
 open OrdUtil O
 -- Ordinal Definable Set
 record OD : Set (suc n ) where
 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
 ==-refl :  {  x :  OD  } → x == x
 ==-refl  {x} = record { eq→ = λ x → x ; eq← = λ x → x }
 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
 -- next assumptions are our axiom
 --
 --  OD is an equation on Ordinals, so it contains Ordinals. If these Ordinals have one-to-one
 --  correspondence to the OD then the OD looks like a ZF Set.
 --
 --  If all ZF Set have supreme upper bound, the solutions of OD have to be bounded, i.e.
 --  bbounded ODs are ZF Set. Unbounded ODs are classes.
 --
 --  In classical Set Theory, HOD is used, as a subset of OD,
 --     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.
 --  This is not possible because we don't have V yet. So we assumes HODs are bounded OD.
 --
 --  We also assumes HODs are isomorphic to Ordinals, which is ususally proved by Goedel number tricks.
 odmax : Ordinal
 <odmax : {y : Ordinal} → def od y → y o< odmax
 open HOD
 record ODAxiom : Set (suc n) where
 field
 -- HOD is isomorphic to Ordinal (by means of Goedel number)
 & : HOD  → Ordinal
 * : Ordinal  → HOD
 c<→o<  :  {x y : HOD  }   → def (od y) ( & x ) → & x o< & y
 ⊆→o≤   :  {y z : HOD  }   → ({x : Ordinal} → def (od y) x → def (od z) x ) → & y o< osuc (& z)
 *iso   :  {x : HOD }      → * ( & x ) ≡ x
 &iso   :  {x : Ordinal }  → & ( * x ) ≡ x
 ==→o≡  :  {x y : HOD  }   → (od x == od y) → x ≡ y
 sup-o  :  (A : HOD) → (     ( x : Ordinal ) → def (od A) x →  Ordinal ) →  Ordinal
 sup-o≤ :  (A : HOD) → { ψ : ( x : Ordinal ) → def (od A) x →  Ordinal } → ∀ {x : Ordinal } → (lt : def (od A) x ) → ψ x lt o≤  sup-o A ψ
 -- possible order restriction
 ho< : {x : HOD} → & x o< next (odmax x)
 postulate  odAxiom : ODAxiom
 -- odmax minimality
 --
 -- since we have ==→o≡ , so odmax have to be unique. We should have odmaxmin in HOD.
 -- We can calculate the minimum using sup but it is tedius.
 -- Only Select has non minimum odmax.
 -- We have the same problem on 'def' itself, but we leave it.
 odmaxmin : Set (suc n)
 odmaxmin = (y : HOD) (z : Ordinal) → ((x : Ordinal)→ def (od y) x → x o< z) → odmax y o< osuc z
 -- OD ⇔ Ordinal leads a contradiction, so we need bounded HOD
 ¬OD-order : ( & : OD  → Ordinal ) → ( * : Ordinal  → OD ) → ( { x y : OD  }  → def y ( & x ) → & x o< & y) → ⊥
 ¬OD-order & * c<→o< = o≤> <-osuc (c<→o< {Ords} OneObj )
--- Open supreme upper bound leads a contradition, so we use domain restriction on sup
-¬open-sup : ( sup-o : (Ordinal →  Ordinal ) → Ordinal) → ((ψ : Ordinal →  Ordinal ) → (x : Ordinal) → ψ x  o<  sup-o ψ ) → ⊥
-¬open-sup sup-o sup-o< = o<> <-osuc (sup-o< next-ord (sup-o next-ord)) where
-next-ord : Ordinal → Ordinal
-next-ord x = osuc x
 -- Ordinal in OD ( and ZFSet ) Transitive Set
 Ord : ( a : Ordinal  ) → HOD
 Ord  a = record { od = record { def = λ y → y o< a } ; odmax = a ; <odmax = lemma } where
 lemma :  {x : Ordinal} → x o< a → x o< a
 lemma {x} lt = lt
 od∅ : HOD
 od∅  = Ord o∅
 odef : HOD → Ordinal → Set n
 odef A x = def ( od A ) x
 _∋_ : ( a x : HOD  ) → Set n
 _∋_  a x  = odef a ( & x )
 -- _c<_ : ( x a : HOD  ) → Set n
 -- x c< a = a ∋ x
 d→∋ : ( a : HOD  ) { x : Ordinal} → odef a x → a ∋ (* x)
 d→∋ a lt = subst (λ k → odef a k ) (sym &iso) lt
 -- odef-subst :  {Z : HOD } {X : Ordinal  }{z : HOD } {x : Ordinal  }→ odef Z X → Z ≡ z  →  X ≡ x  →  odef z x
 orefl : { x : HOD  } → { y : Ordinal  } → & x ≡ y → & x ≡ y
 orefl refl = refl
 ==-iso : { x y : HOD  } → od (* (& x)) == od (* (& y))  →  od x == od y
 ==-iso  {x} {y} eq = record {
 eq→ = λ {z} d →  lemma ( eq→  eq (subst (λ k → odef k z ) (sym *iso) d )) ;
 eq← = λ {z} d →  lemma ( eq←  eq (subst (λ k → odef k z ) (sym *iso) d )) }
 where
 lemma : {x : HOD  } {z : Ordinal } → odef (* (& x)) z → odef x z
 lemma {x} {z} d = subst (λ k → odef k z) (*iso) d
 =-iso :  {x y : HOD  } → (od x == od y) ≡ (od (* (& x)) == od y)
 =-iso  {_} {y} = cong ( λ k → od k == od y ) (sym *iso)
 ord→== : { x y : HOD  } → & x ≡  & y →  od x == od y
 *≡*→≡ eq = subst₂ (λ j k → j ≡ k ) &iso &iso ( cong (&) eq )
 &≡&→≡ : { x y : HOD  } → & x ≡  & y →  x ≡ y
 &≡&→≡ eq = subst₂ (λ j k → j ≡ k ) *iso *iso ( cong (*) eq )
 o∅≡od∅ : * (o∅ ) ≡ od∅
 o∅≡od∅  = ==→o≡ lemma where
 lemma0 :  {x : Ordinal} → odef (* o∅) x → odef od∅ x
 lemma0 {x} lt with c<→o< {* x} {* o∅} (subst (λ k → odef (* o∅) k ) (sym &iso) lt)
 ... | t = subst₂ (λ j k → j o< k ) &iso &iso t
 lemma1 :  {x : Ordinal} → odef od∅ x → odef (* o∅) x
 lemma1 {x} lt = ⊥-elim (¬x<0 lt)
 lemma : od (* o∅) == od od∅
 lemma = record { eq→ = lemma0 ; eq← = lemma1 }
 ord-od∅ : & (od∅ ) ≡ o∅
 ord-od∅  = sym ( subst (λ k → k ≡  & (od∅ ) ) &iso (cong ( λ k → & k ) o∅≡od∅ ) )
 ≡o∅→=od∅  : {x : HOD} → & x ≡ o∅ → od x == od od∅
 ≡o∅→=od∅ {x} eq = record { eq→ = λ {y} lt → ⊥-elim ( ¬x<0 {y} (subst₂ (λ j k → j o< k ) &iso eq ( c<→o< {* y} {x} (d→∋ x lt))))
 ; eq← = λ {y} lt → ⊥-elim ( ¬x<0 lt )}
 =od∅→≡o∅  : {x : HOD} → od x == od od∅ → & x ≡ o∅
 =od∅→≡o∅ {x} eq = trans (cong (λ k → & k ) (==→o≡ {x} {od∅} eq)) ord-od∅
 ≡od∅→=od∅  : {x : HOD} → x ≡ od∅ → od x == od od∅
 ≡od∅→=od∅ {x} eq = ≡o∅→=od∅ (subst (λ k → & x  ≡ k ) ord-od∅ ( cong & eq ) )
 ∅0 : record { def = λ x →  Lift n ⊥ } == od od∅
 eq→ ∅0 {w} (lift ())
 eq← ∅0 {w} lt = lift (¬x<0 lt)
 ∅< : { x y : HOD  } → odef x (& y ) → ¬ (  od x  == od od∅  )
 ∅<  {x} {y} d eq with eq→ (==-trans eq (==-sym ∅0) ) d
 is-o∅ x with trio< x o∅
 is-o∅ x | tri< a ¬b ¬c = no ¬b
 is-o∅ x | tri≈ ¬a b ¬c = yes b
 is-o∅ x | tri> ¬a ¬b c = no ¬b
+odef< : {b : Ordinal } { A : HOD } → odef A b → b o< & A
+odef< {b} {A} ab = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) ab))
+odef∧< : {A : HOD } {y : Ordinal} {n : Level } → {P : Set n} → odef A y ∧ P → y o< & A
+odef∧< {A } {y} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p )))
 -- the pair
 _,_ : HOD  → HOD  → HOD
 x , y = record { od = record { def = λ t → (t ≡ & x ) ∨ ( t ≡ & y ) } ; odmax = omax (& x)  (& y) ; <odmax = lemma }  where
 lemma : {t : Ordinal} → (t ≡ & x) ∨ (t ≡ & y) → t o< omax (& x) (& y)
 lemma {t} (case1 refl) = omax-x  _ _
 lemma {t} (case2 refl) = omax-y  _ _
 -- another possible restriction. We require no minimality on odmax, so it may arbitrary larger.
 odmax<&  : { x y : HOD } → x ∋ y →  Set n
 odmax<& {x} {y} x∋y = odmax x o< & x
 in-codomain : (X : HOD  ) → ( ψ : HOD  → HOD  ) → OD
 in-codomain  X ψ = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( odef X y ∧  ( x ≡ & (ψ (* y )))))  }
 _∩_ : ( A B : HOD ) → HOD
 A ∩ B = record { od = record { def = λ x → odef A x ∧ odef B x }
 ; odmax = omin (odmax A) (odmax B) ; <odmax = λ y → min1 (<odmax A (proj1 y)) (<odmax B (proj2 y))}
 record _⊆_ ( A B : HOD   ) : Set (suc n) where
 field
 incl : { x : HOD } → A ∋ x →  B ∋ x
 open _⊆_
 infixr  220 _⊆_
 -- if we have & (x , x) ≡ osuc (& x),  ⊆→o≤ → c<→o<
 ⊆→o≤→c<→o< : ({x : HOD} → & (x , x) ≡ osuc (& x) )
 →  ({y z : HOD  }   → ({x : Ordinal} → def (od y) x → def (od z) x ) → & y o< osuc (& z) )
 →   {x y : HOD  }   → def (od y) ( & x ) → & x o< & y
 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x with trio< (& x) (& y)
 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x | tri< a ¬b ¬c = a
 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x | tri≈ ¬a b ¬c = ⊥-elim ( o<¬≡ (peq {x}) (pair<y (subst (λ k → k ∋ x) (sym ( ==→o≡ {x} {y} (ord→== b))) y∋x )))
 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x | tri> ¬a ¬b c =
 ⊥-elim ( o<> (⊆→o≤ {x , x} {y} y⊆x,x ) lemma1 ) where
 lemma : {z : Ordinal} → (z ≡ & x) ∨ (z ≡ & x) → & x ≡ z
 lemma (case1 refl) = refl
 lemma (case2 refl) = refl
 y⊆x,x : {z : Ordinal} → def (od (x , x)) z → def (od y) z
 y⊆x,x {z} lt = subst (λ k → def (od y) k ) (lemma lt) y∋x
 lemma1 : osuc (& y) o< & (x , x)
 lemma1 = subst (λ k → osuc (& y) o< k ) (sym (peq {x})) (osucc c )
 ε-induction : { ψ : HOD  → Set (suc n)}
 → ( {x : HOD } → ({ y : HOD } →  x ∋ y → ψ y ) → ψ x )
 → (x : HOD ) → ψ x
 ε-induction {ψ} ind x = subst (λ k → ψ k ) *iso (ε-induction-ord (osuc (& x)) <-osuc )  where
 induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (* oy)) → ψ (* ox)
 induction ox prev = ind  ( λ {y} lt → subst (λ k → ψ k ) *iso (prev (& y) (o<-subst (c<→o< lt) refl &iso )))
 ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (* oy)
 ε-induction-ord ox {oy} lt = TransFinite {λ oy → ψ (* oy)} induction oy
-Select : (X : HOD  ) → ((x : HOD  ) → Set n ) → HOD
+record OSUP (A x : Ordinal ) ( ψ : ( x : Ordinal ) → odef (* A) x →  Ordinal ) : Set n where
+field
+sup-o'  :   Ordinal
+sup-o≤' :  {z : Ordinal } → z o< x → (lt : odef (* A) z ) → ψ z lt o≤  sup-o'
+-- o<-sup :  ( A x : Ordinal) { ψ : ( x : Ordinal ) → odef (* A) x →  Ordinal } → OSUP A x ψ
+-- o<-sup A x {ψ} = ? where
+--   m00 : (x : Ordinal ) → OSUP A x ψ
+--   m00 x = Ordinals.inOrdinal.TransFinite0 O ind x where
+--      ind : (x : Ordinal) → ((z : Ordinal) → z o< x → OSUP A z ψ ) → OSUP A x ψ
+--      ind x prev  =  ? where
+--        if has prev , od01 is empty use prev (cannot be odef (* A) x)
+--                              not empty, take larger
+--           limit ordinal, take address of Union
+--
+--          od01 : HOD
+--          od01 = record { od = record { def = λ z → odef A z ∧ ( z ≡ & x ) } ; odmax = & A ; <odmax = odef∧<  }
+--          m01 : OSUP A x ψ
+--          m01 with is-o∅ (& od01 )
+--          ... | case1 t=0 = record { sup-o' = prevjjk
+--          ... | case2 t>0 = ?
+-- Open supreme upper bound leads a contradition, so we use domain restriction on sup
+¬open-sup : ( sup-o : (Ordinal →  Ordinal ) → Ordinal) → ((ψ : Ordinal →  Ordinal ) → (x : Ordinal) → ψ x  o<  sup-o ψ ) → ⊥
+¬open-sup sup-o sup-o< = o<> <-osuc (sup-o< next-ord (sup-o next-ord)) where
+next-ord : Ordinal → Ordinal
+next-ord x = osuc x
+Select : (X : HOD  ) → ((x : HOD  ) → Set n ) → HOD
 Select X ψ = record { od = record { def = λ x →  ( odef X x ∧ ψ ( * x )) } ; odmax = odmax X ; <odmax = λ y → <odmax X (proj1 y) }
 Replace : HOD  → (HOD  → HOD) → HOD
 Replace X ψ = record { od = record { def = λ x → (x o≤ sup-o X (λ y X∋y → & (ψ (* y)))) ∧ def (in-codomain X ψ) x }
 ; odmax = rmax ; <odmax = rmax<} where
 rmax : Ordinal
 rmax = osuc ( sup-o X (λ y X∋y → & (ψ (* y))))
 rmax< :  {y : Ordinal} → (y o< rmax) ∧ def (in-codomain X ψ) y → y o< rmax
 rmax< lt = proj1 lt
 --
 -- If we have LEM, Replace' is equivalent to Replace
 --
 in-codomain' : (X : HOD  ) → ((x : HOD) → X ∋ x → HOD) → OD
 in-codomain'  X ψ = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( odef X y ∧  ((lt : odef X y) →  x ≡ & (ψ (* y ) (d→∋ X lt) ))))  }
 Replace' : (X : HOD)  → ((x : HOD) → X ∋ x → HOD) → HOD
 Replace' X ψ = record { od = record { def = λ x → (x o< sup-o X (λ y X∋y → & (ψ (* y) (d→∋ X X∋y) ))) ∧ def (in-codomain' X ψ) x }
 ; odmax = rmax ; <odmax = rmax< } where
 rmax : Ordinal
 rmax = sup-o X (λ y X∋y → & (ψ (* y) (d→∋ X X∋y)))
 rmax< :  {y : Ordinal} → (y o< rmax) ∧ def (in-codomain' X ψ) y → y o< rmax
 rmax< lt = proj1 lt
 Union : HOD  → HOD
 Union U = record { od = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((odef U u) ∧ (odef (* u) x)))  }
 ; odmax = osuc (& U) ; <odmax = umax< } where
 umax< :  {y : Ordinal} → ¬ ((u : Ordinal) → ¬ def (od U) u ∧ def (od (* u)) y) → y o< osuc (& U)
 umax< {y} not = lemma (FExists _ lemma1 not ) where
 lemma0 : {x : Ordinal} → def (od (* x)) y → y o< x
 lemma0 {x} x<y = subst₂ (λ j k → j o< k ) &iso  &iso (c<→o< (d→∋ (* x) x<y ))
 lemma2 : {x : Ordinal} → def (od U) x → x o< & U
 lemma2 {x} x<U = subst (λ k → k o< & U ) &iso (c<→o< (d→∋ U x<U))
 lemma1 : {x : Ordinal} → def (od U) x ∧ def (od (* x)) y → ¬ (& U o< y)
 lemma1 {x} lt u<y = o<> u<y (ordtrans (lemma0 (proj2 lt)) (lemma2 (proj1 lt)) )
 lemma : ¬ ((& U) o< y ) → y o< osuc (& U)
 lemma not with trio< y (& U)
 lemma not | tri< a ¬b ¬c = ordtrans a <-osuc
 lemma not | tri≈ ¬a refl ¬c = <-osuc
 lemma not | tri> ¬a ¬b c = ⊥-elim (not c)
 _∈_ : ( A B : HOD  ) → Set n
 A ∈ B = B ∋ A
 OPwr :  (A :  HOD ) → HOD
 OPwr  A = Ord ( osuc ( sup-o (Ord (osuc (& A))) ( λ x A∋x → & ( A ∩ (* x)) ) ) )
 Power : HOD  → HOD
 Power A = Replace (OPwr (Ord (& A))) ( λ x → A ∩ x )
 -- ｛_｝ : ZFSet → ZFSet
 -- ｛ x ｝ = ( x ,  x )     -- better to use (x , x) directly
 union→ :  (X z u : HOD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z
 isuc : {x : Ordinal  } →   infinite-d  x  →
 infinite-d  (& ( Union (* x , (* x , * x ) ) ))
 -- ω can be diverged in our case, since we have no restriction on the corresponding ordinal of a pair.
 -- We simply assumes infinite-d y has a maximum.
 --
 -- This means that many of OD may not be HODs because of the & mapping divergence.
 -- We should have some axioms to prevent this such as & x o< next (odmax x).
 --
 -- postulate
 --     ωmax : Ordinal
 --     <ωmax : {y : Ordinal} → infinite-d y → y o< ωmax
 --
 -- infinite : HOD
 -- infinite = record { od = record { def = λ x → infinite-d x } ; odmax = ωmax ; <odmax = <ωmax }
 infinite : HOD
 infinite = record { od = record { def = λ x → infinite-d x } ; odmax = next o∅ ; <odmax = lemma }  where
 u : (y : Ordinal ) → HOD
 u y = Union (* y , (* y , * y))
 --   next< : {x y z : Ordinal} → x o< next z  → y o< next x → y o< next z
 lemma8 : {y : Ordinal} → & (* y , * y) o< next (odmax (* y , * y))
 lemma8 = ho<
 ---           (x,y) < next (omax x y) < next (osuc y) = next y
 lemmaa : {x y : HOD} → & x o< & y → & (x , y) o< next (& y)
 lemmaa {x} {y} x<y = subst (λ k → & (x , y) o< k ) (sym nexto≡) (subst (λ k → & (x , y) o< next k ) (sym (omax< _ _ x<y)) ho< )
 lemma81 : {y : Ordinal} → & (* y , * y) o< next (& (* y))
 lemma81 {y} = nexto=n (subst (λ k →  & (* y , * y) o< k ) (cong (λ k → next k) (omxx _)) lemma8)
 lemma9 : {y : Ordinal} → & (* y , (* y , * y)) o< next (& (* y , * y))
 lemma : {y : Ordinal} → infinite-d y → y o< next o∅
 lemma {o∅} iφ = x<nx
 lemma (isuc {y} x) = next< (lemma x) (next< (subst (λ k → & (* y , (* y , * y)) o< next k) &iso lemma71 ) (nexto=n lemma1))
 empty : (x : HOD  ) → ¬  (od∅ ∋ x)
 empty x = ¬x<0
 _=h=_ : (x y : HOD) → Set n
 x =h= y  = od x == od y
 pair→ : ( x y t : HOD  ) →  (x , y)  ∋ t  → ( t =h= x ) ∨ ( t =h= y )
 pair→ x y t (case1 t≡x ) = case1 (subst₂ (λ j k → j =h= k ) *iso *iso (o≡→== t≡x ))
 pair→ x y t (case2 t≡y ) = case2 (subst₂ (λ j k → j =h= k ) *iso *iso (o≡→== t≡y ))
 pair← : ( x y t : HOD  ) → ( t =h= x ) ∨ ( t =h= y ) →  (x , y)  ∋ t
 pair← x y t (case1 t=h=x) = case1 (cong (λ k → & k ) (==→o≡ t=h=x))
 pair← x y t (case2 t=h=y) = case2 (cong (λ k → & k ) (==→o≡ t=h=y))
 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 &iso) refl )
 ... | ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt &iso refl ))
 selection {ψ} {X} {y} = ⟪
 ( λ cond → ⟪ proj1 cond , ψiso {ψ} (proj2 cond) (sym *iso)  ⟫ )
 ,  ( λ select → ⟪ proj1 select  , ψiso {ψ} (proj2 select) *iso  ⟫ )
 ⟫
 selection-in-domain : {ψ : HOD → Set n} {X y : HOD} → Select X ψ ∋ y → X ∋ y
 selection-in-domain {ψ} {X} {y} lt = proj1 ((proj2 (selection {ψ} {X} )) lt)
 sup-c≤ :  (ψ : HOD → HOD) → {X x : HOD} → X ∋ x  → & (ψ x) o≤ (sup-o X (λ y X∋y → & (ψ (* y))))
 sup-c≤ ψ {X} {x} lt = subst (λ k → & (ψ k) o< _ ) *iso (sup-o≤ X lt )
 replacement← : {ψ : HOD → HOD} (X x : HOD) →  X ∋ x → Replace X ψ ∋ ψ x
 replacement← {ψ} X x lt = ⟪ sup-c≤ ψ {X} {x} lt , lemma ⟫ where
 lemma : def (in-codomain X ψ) (& (ψ x))
 lemma not = ⊥-elim ( not ( & x ) ⟪ lt , cong (λ k → & (ψ k)) (sym *iso)⟫ )
 replacement→ : {ψ : HOD → HOD} (X x : HOD) → (lt : Replace X ψ ∋ x) → ¬ ( (y : HOD) → ¬ (x =h= ψ y))
 replacement→ {ψ} X x lt = contra-position lemma (lemma2 (proj2 lt)) where
 lemma2 :  ¬ ((y : Ordinal) → ¬ odef X y ∧ ((& x) ≡ & (ψ (* y))))
 → ¬ ((y : Ordinal) → ¬ odef X y ∧ (* (& x) =h= ψ (* y)))
 lemma2 not not2  = not ( λ y d →  not2 y ⟪ proj1 d , lemma3 (proj2 d)⟫) where
 lemma3 : {y : Ordinal } → (& x ≡ & (ψ (*  y))) → (* (& x) =h= ψ (* y))
 lemma3 {y} eq = subst (λ k  → * (& x) =h= k ) *iso (o≡→== eq )
 lemma :  ( (y : HOD) → ¬ (x =h= ψ y)) → ( (y : Ordinal) → ¬ odef X y ∧ (* (& x) =h= ψ (* y)) )
 lemma not y not2 = not (* y) (subst (λ k → k =h= ψ (* y)) *iso  ( proj2 not2 ))
 ---
 --
 ∩-≡ :  { a b : HOD  } → ({x : HOD  } → (a ∋ x → b ∋ x)) → a =h= ( b ∩ a )
 ∩-≡ {a} {b} inc = record {
 eq→ = λ {x} x<a → ⟪ (subst (λ k → odef b k ) &iso (inc (d→∋ a x<a))) , x<a  ⟫ ;
 eq← = λ {x} x<a∩b → proj2 x<a∩b }
 --
 -- Transitive Set case
 -- we have t ∋ x → Ord a ∋ x means t is a subset of Ord a, that is (Ord a) ∩ t =h= t
 -- OPwr (Ord a) is a sup of (Ord a) ∩ t, so OPwr (Ord a) ∋ t
 -- OPwr  A = Ord ( sup-o ( λ x → & ( A ∩ (* x )) ) )
 --
 ord-power← : (a : Ordinal ) (t : HOD) → ({x : HOD} → (t ∋ x → (Ord a) ∋ x)) → OPwr (Ord a) ∋ t
 ord-power← a t t→A  = subst (λ k → odef (OPwr (Ord a)) k ) (lemma1 lemma-eq) lemma where
 lemma-eq :  ((Ord a) ∩ t) =h= t
 eq→ lemma-eq {z} w = proj2 w
 eq← lemma-eq {z} w = ⟪ subst (λ k → odef (Ord a) k ) &iso ( t→A (d→∋ t w)) , w ⟫
 lemma1 :  {a : Ordinal } { t : HOD }
 → (eq : ((Ord a) ∩ t) =h= t)  → & ((Ord a) ∩ (* (& t))) ≡ & t
 lemma1  {a} {t} eq = subst (λ k → & ((Ord a) ∩ k) ≡ & t ) (sym *iso) (cong (λ k → & k ) (==→o≡ eq ))
 lemma2 : (& t) o< (osuc (& (Ord a)))
 lemma2 = ⊆→o≤  {t} {Ord a} (λ {x} x<t → subst (λ k → def (od (Ord a)) k) &iso (t→A (d→∋ t x<t)))
 lemma :  & ((Ord a) ∩ (* (& t)) ) o≤ sup-o (Ord (osuc (& (Ord a)))) (λ x lt → & ((Ord a) ∩ (* x)))
 lemma = sup-o≤ _ lemma2
 --
 -- Every set in HOD is a subset of Ordinals, so make OPwr (Ord (& A)) first
 -- then replace of all elements of the Power set by A ∩ y
 --
 -- Power A = Replace (OPwr (Ord (& A))) ( λ y → A ∩ y )
 -- we have oly double negation form because of the replacement axiom
 --
 power→ :  ( A t : HOD) → Power A ∋ t → {x : HOD} → t ∋ x → ¬ ¬ (A ∋ x)
 lemma4 not = lemma2 ( λ y not1 → not (& y) (subst (λ k → t =h= ( A ∩ k )) (sym *iso) not1 ))
 lemma5 : {y : Ordinal} → t =h= (A ∩ * y) →  ¬ ¬  (odef A (& x))
 lemma5 {y} eq not = (lemma3 (* y) eq) not
 power← :  (A t : HOD) → ({x : HOD} → (t ∋ x → A ∋ x)) → Power A ∋ t
 power← A t t→A = ⟪ lemma1 , lemma2 ⟫ where
 a = & A
 lemma0 : {x : HOD} → t ∋ x → Ord a ∋ x
 lemma0 {x} t∋x = c<→o< (t→A t∋x)
 lemma3 : OPwr (Ord a) ∋ t
 lemma3 = ord-power← a t lemma0
 t
 ∎
 sup1 : Ordinal
 sup1 =  sup-o (Ord (osuc (& (Ord (& A))))) (λ x A∋x → & ((Ord (& A)) ∩ (* x)))
 lemma9 : def (od (Ord (Ordinals.osuc O (& (Ord (& A)))))) (& (Ord (& A)))
 lemma9 = <-osuc
 lemmab : & ((Ord (& A)) ∩ (* (& (Ord (& A) )))) o≤ sup1
 lemmab = sup-o≤ (Ord (osuc (& (Ord (& A))))) lemma9
 lemmad : Ord (osuc (& A)) ∋ t
 lemmad = ⊆→o≤ (λ {x} lt → subst (λ k → def (od A) k ) &iso (t→A (d→∋ t lt)))
 lemmac : ((Ord (& A)) ∩  (* (& (Ord (& A) )))) =h= Ord (& A)
 lemmac = record { eq→ = lemmaf ; eq← = lemmag } where
 lemmaf : {x : Ordinal} → def (od ((Ord (& A)) ∩  (* (& (Ord (& A)))))) x → def (od (Ord (& A))) x
 lemmaf {x} lt = proj1 lt
 lemmag :  {x : Ordinal} → def (od (Ord (& A))) x → def (od ((Ord (& A)) ∩ (* (& (Ord (& A)))))) x
 lemmag {x} lt = ⟪ lt , subst (λ k → def (od k) x) (sym *iso) lt ⟫
 lemmae : & ((Ord (& A)) ∩ (* (& (Ord (& A))))) ≡ & (Ord (& A))
 lemmae = cong (λ k → & k ) ( ==→o≡ lemmac)
 lemma7 : def (od (OPwr (Ord (& A)))) (& t)
 lemma7 with osuc-≡< lemmad
 lemma7 | case2 lt = ordtrans (c<→o< lt) (subst (λ k → k o≤ sup1) lemmae lemmab )
 lemma7 | case1 eq with osuc-≡< (⊆→o≤ {* (& t)} {* (& (Ord (& t)))} (λ {x} lt → lemmah lt )) where
 lemmah : {x : Ordinal } → def (od (* (& t))) x → def (od (* (& (Ord (& t))))) x
 lemmah {x} lt = subst (λ k → def (od k) x ) (sym *iso) (subst (λ k → k o< (& t))
 &iso
 (c<→o< (subst₂ (λ j k → def (od j)  k) *iso (sym &iso) lt )))
 lemma7 | case1 eq | case1 eq1 = subst (λ k → k o≤ sup1) (trans lemmae lemmai) lemmab where
 lemmai : & (Ord (& A)) ≡ & t
 lemmai = let open ≡-Reasoning in begin
 & (Ord (& A))
 ≡⟨ sym (cong (λ k → & (Ord k)) eq) ⟩
 & (Ord (& t))
 ≡⟨ sym &iso ⟩
 & (* (& (Ord (& t))))
 ≡⟨ sym eq1 ⟩
 & (* (& t))
 ≡⟨ &iso ⟩
 & t
 ∎
 lemma7 | case1 eq | case2 lt = ordtrans lemmaj (subst (λ k → k o≤ sup1) lemmae lemmab ) where
 lemmak : & (* (& (Ord (& t)))) ≡ & (Ord (& A))
 lemmak = let open ≡-Reasoning in begin
 & (* (& (Ord (& t))))
 & (Ord (& t))
 ≡⟨ cong (λ k → & (Ord k)) eq ⟩
 & (Ord (& A))
 ∎
 lemmaj : & t o< & (Ord (& A))
 lemmaj = subst₂ (λ j k → j o< k ) &iso lemmak lt
 lemma1 : & t o≤ sup-o (OPwr (Ord (& A))) (λ x lt → & (A ∩ (* x)))
 lemma1 = subst (λ k → & k o≤ sup-o (OPwr (Ord (& A)))  (λ x lt → & (A ∩ (* x))))
 lemma4 (sup-o≤ (OPwr (Ord (& A))) lemma7 )
 lemma2 :  def (in-codomain (OPwr (Ord (& A))) (_∩_ A)) (& t)
 lemma2 not = ⊥-elim ( not (& t) ⟪ lemma3 , lemma6 ⟫ ) where
 lemma6 : & t ≡ & (A ∩ * (& t))
 lemma6 = cong ( λ k → & k ) (==→o≡ (subst (λ k → t =h= (A ∩ k)) (sym *iso) ( ∩-≡ {t} {A} t→A  )))
 extensionality0 : {A B : HOD } → ((z : HOD) → (A ∋ z) ⇔ (B ∋ z)) → A =h= B
 eq→ (extensionality0 {A} {B} eq ) {x} d = odef-iso  {A} {B} (sym &iso) (proj1 (eq (* x))) d
 eq← (extensionality0 {A} {B} eq ) {x} d = odef-iso  {B} {A} (sym &iso) (proj2 (eq (* x))) d
 extensionality : {A B w : HOD  } → ((z : HOD ) → (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∅
 infinity∅ = subst (λ k → odef infinite k ) lemma iφ where
 lemma : o∅ ≡ & od∅
 lemma =  let open ≡-Reasoning in begin
 o∅
 ≡⟨ sym &iso ⟩
 & ( * o∅ )
 ∎
 infinity : (x : HOD) → infinite ∋ x → infinite ∋ Union (x , (x , x ))
 infinity x lt = subst (λ k → odef infinite k ) lemma (isuc {& x} lt) where
 lemma :  & (Union (* (& x) , (* (& x) , * (& x))))
 ≡ & (Union (x , (x , x)))
 lemma = cong (λ k → & (Union ( k , ( k , k ) ))) *iso
 isZF : IsZF (HOD )  _∋_  _=h=_ 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 = ε-induction
 ;   infinity∅ = infinity∅
 ;   infinity = infinity
 ;   selection = λ {X} {ψ} {y} → selection {X} {ψ} {y}
 ;   replacement← = replacement←
 ;   replacement→ = λ {ψ} → replacement→ {ψ}
 }
 HOD→ZF : ZF
 HOD→ZF   = record {
 ZFSet = HOD
 ; _∋_ = _∋_
 ; _≈_ = _=h=_
 ; ∅  = od∅
 ; _,_ = _,_
 ; Union = Union
 ; Power = Power
 ; Select = Select
 ; Replace = Replace
 ; infinite = infinite
 ; isZF = isZF
 }

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

comparison src/OD.agda @ 1091:63c1167b2343