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changeset 1461:fa52d72f4bb3

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Mon, 01 Jan 2024 18:21:36 +0900
parents d1b6fb58aad0
children 76df157f6a3f
files ZF.agda-lib src/OD.agda src/OrdUtil.agda src/Ordinals.agda src/Tychonoff.agda src/ZProduct.agda src/bijection.agda src/logic.agda src/nat.agda src/ordinal.agda src/zf.agda
diffstat 11 files changed, 264 insertions(+), 263 deletions(-) [+]
line wrap: on
line diff
--- a/ZF.agda-lib	Sat Aug 26 10:36:09 2023 +0900
+++ b/ZF.agda-lib	Mon Jan 01 18:21:36 2024 +0900
@@ -1,3 +1,6 @@
 name: ZF
 depend: standard-library
 include: src
+flags:
+  --warning=noUnsupportedIndexedMatch
+
--- a/src/OD.agda	Sat Aug 26 10:36:09 2023 +0900
+++ b/src/OD.agda	Mon Jan 01 18:21:36 2024 +0900
@@ -1,7 +1,8 @@
-{-# OPTIONS --allow-unsolved-metas #-}
+{-# OPTIONS --cubical-compatible --safe #-}
 open import Level
 open import Ordinals
-module OD {n : Level } (O : Ordinals {n} ) where
+import HODBase
+module OD {n : Level } (O : Ordinals {n} ) (HODAxiom : HODBase.ODAxiom O) where
 
 open import Data.Nat renaming ( zero to Zero ; suc to Suc ;  ℕ to Nat ; _⊔_ to _n⊔_ )
 open import  Relation.Binary.PropositionalEquality hiding ( [_] )
@@ -23,87 +24,25 @@
 
 -- Ordinal Definable Set
 
-record OD : Set (suc n ) where
-  field
-    def : (x : Ordinal  ) → Set n
-
-open OD
+open HODBase.HOD 
+open HODBase.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
-
-==-refl :  {  x :  OD  } → x == x
-==-refl  {x} = record { eq→ = λ x → x ; eq← = λ x → x }
-
-open  _==_
+open  HODBase._==_
 
-==-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
+open HODBase.ODAxiom HODAxiom  
 
---
---  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.
---  There two contraints on the HOD order, one is ∋, the other one is ⊂.
---  ODs have an ovbious maximum, but Ordinals are not, but HOD has no maximum because there is an aribtrary
---  bound on each HOD.
---
---  In classical Set Theory, sup is defined by Uion, since we are working on constructive logic,
---  we need explict assumption on sup for unrestricted Replacement.
---
---  ==→o≡ is necessary to prove axiom of extensionality.
-
--- Ordinals in OD , the maximum
-Ords : OD
-Ords = record { def = λ x → Lift n ⊤ }
-
-record HOD : Set (suc n) where
-  field
-    od : OD
-    odmax : Ordinal
-    <odmax : {y : Ordinal} → def od y → y o< odmax
-
-open HOD
-
-open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ )
-
-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
-  ∋-irr : {X : HOD} {x : Ordinal } → (a b : def (od X) x) → a ≅ b
-
-postulate  odAxiom : ODAxiom
-open ODAxiom odAxiom
+HOD =  HODBase.HOD O 
+OD  =  HODBase.OD O 
+Ords  =  HODBase.Ords O 
+_==_  =  HODBase._==_ O 
+==-refl = HODBase.==-refl  O
+==-trans = HODBase.==-trans O
+==-sym = HODBase.==-sym O
+⇔→== = HODBase.⇔→== O
 
 -- possible order restriction (required in the axiom of Omega )
 
@@ -152,12 +91,12 @@
 otrans x<a y<x = ordtrans y<x x<a
 
 -- If we have reverse of c<→o<, everything becomes Ordinal
-∈→c<→HOD=Ord : ( o<→c< : {x y : Ordinal  } → x o< y → odef (* y) x ) → {x : HOD } →  x ≡ Ord (& x)
-∈→c<→HOD=Ord o<→c< {x} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where
+∈→c<→HOD=Ord : ( o<→c< : {x y : Ordinal  } → x o< y → odef (* y) x ) → {x : HOD } →  od x == od (Ord (& x))
+∈→c<→HOD=Ord o<→c< {x} = record { eq→ = lemma1 ; eq← = lemma2 }  where
    lemma1 : {y : Ordinal} → odef x y → odef (Ord (& x)) y
    lemma1 {y} lt = subst ( λ k → k o< & x ) &iso (c<→o< {* y} {x} (d→∋ x lt))
    lemma2 : {y : Ordinal} → odef (Ord (& x)) y → odef x y
-   lemma2 {y} lt = subst (λ k → odef k y ) *iso (o<→c< {y} {& x} lt )
+   lemma2 {y} lt = eq→  *iso (o<→c< {y} {& x} lt )
 
 -- avoiding lv != Zero error
 orefl : { x : HOD  } → { y : Ordinal  } → & x ≡ y → & x ≡ y
@@ -165,14 +104,11 @@
 
 ==-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
+      eq→ = λ {z} d →  eq→  *iso ( eq→ eq (eq←  *iso d) )  ;
+      eq← = λ {z} d →  eq→  *iso ( eq← eq (eq←  *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)
+-- =-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
 ord→==  {x} {y} eq = ==-iso (lemma (& x) (& y) (orefl eq)) where
@@ -185,11 +121,11 @@
 *≡*→≡ : { x y : Ordinal  } → * x ≡ * y →  x ≡ 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 )
+--- &≡&→≡ : { 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
+o∅==od∅ : od ( * (o∅ )) == od od∅
+o∅==od∅  = 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
@@ -199,14 +135,14 @@
      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∅ ) )
+ord-od∅  = trans (==→o≡ (==-sym o∅==od∅)) &iso  
 
 ≡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∅→≡o∅ {x} eq = trans (==→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 ) )
@@ -219,7 +155,7 @@
 ∅<  {x} {y} d eq with eq→ (==-trans eq (==-sym ∅0) ) d
 ∅<  {x} {y} d eq | lift ()
 
-¬x∋y→x≡od∅  : { x : HOD  } → ({y : Ordinal } → ¬ odef x y ) → x ≡ od∅ 
+¬x∋y→x≡od∅  : { x : HOD  } → ({y : Ordinal } → ¬ odef x y ) → (& x) ≡ & od∅ 
 ¬x∋y→x≡od∅ {x} nxy = ==→o≡ record { eq→ = λ {y} lt → ⊥-elim (nxy lt) ; eq← = λ {y} lt → ⊥-elim (¬x<0 lt)  }
 
 0<P→ne  : { x : HOD  } → o∅ o< & x → ¬ (  od x  == od od∅  )
@@ -284,7 +220,7 @@
    →   {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<¬≡ (peq {x}) (pair<y (eq← (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
@@ -298,20 +234,25 @@
 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
+ε-induction {ψ} ind x = ε-induction-hod _ {& x} <-osuc x <-osuc where
+     induction2 : (x₁ : Ordinal) →
+            ((y : Ordinal) → y o< x₁ → (y₁ : HOD) → & y₁ o< osuc y → ψ y₁) →
+            (y : HOD) → & y o< osuc x₁ → ψ y
+     induction2 x prev y y≤x = ind (λ {y₁} lt → prev (& y₁) (lemma1 y₁ lt)  y₁ <-osuc  ) where
+         lemma1 : (y₁ : HOD) → y ∋ y₁ →  & y₁ o< x
+         lemma1 y₁ lt with trio< (& y₁) x
+         ... | tri< a ¬b ¬c = a
+         ... | tri> ¬a ¬b c = ⊥-elim (o≤> (ordtrans (c<→o< lt)  y≤x)  c )
+         ... | tri≈ ¬a b ¬c with osuc-≡< y≤x
+         ... | case1 y=x = subst (λ k → & y₁ o< k ) y=x (c<→o< lt)
+         ... | case2 y<x = ⊥-elim ( o<¬≡ b ( (ordtrans (c<→o< lt) y<x)  )) 
+     ε-induction-hod : (ox : Ordinal) { oy : Ordinal } → oy o< ox → (y : HOD) → & y o< osuc oy  → ψ y
+     ε-induction-hod ox {oy} lt = TransFinite {λ oy → (y : HOD) → & y o< osuc oy →  ψ y} induction2 oy 
 
-ε-induction0 : { ψ : HOD  → Set n}
-   → ( {x : HOD } → ({ y : HOD } →  x ∋ y → ψ y ) → ψ x )
-   → (x : HOD ) → ψ x
-ε-induction0 {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 = inOrdinal.TransFinite0 O {λ oy → ψ (* oy)} induction oy
+-- we cannot prove this...
+-- ε-induction0 : { ψ : HOD  → Set n}
+--    → ( {x : HOD } → ({ y : HOD } →  x ∋ y → ψ y ) → ψ x )
+--    → (x : HOD ) → ψ x
 
 -- 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 ) → ⊥
@@ -339,7 +280,7 @@
             umax1 = odef< (Own.ao uy)
          
 union→ :  (X z u : HOD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z
-union→ X z u xx =  record { owner = & u ; ao = proj1 xx ; ox = subst (λ k → odef k (& z)) (sym *iso) (proj2 xx)   }
+union→ X z u xx =  record { owner = & u ; ao = proj1 xx ; ox = eq← *iso (proj2 xx) } 
 union← :  (X z : HOD) (X∋z : Union X ∋ z) →  ¬  ( (u : HOD ) → ¬ ((X ∋  u) ∧ (u ∋ z )))
 union← X z UX∋z not = ⊥-elim ( not (* (Own.owner UX∋z)) ⟪ subst (λ k → odef X k) (sym &iso) ( Own.ao UX∋z) , Own.ox UX∋z ⟫  )
 
@@ -366,7 +307,7 @@
             r01 = sym (Replaced.x=ψz lt )
 
 replacement← : {ψ : HOD → HOD} (X x : HOD) →  X ∋ x → {C : HOD} → (rc : RCod C ψ) → Replace X ψ rc ∋ ψ x
-replacement← {ψ} X x lt {C} rc = record { z = & x ; az = lt  ; x=ψz = cong (λ k → & (ψ k)) (sym *iso) }
+replacement← {ψ} X x lt {C} rc = record { z = & x ; az = lt  ; x=ψz = cong (λ k → & (ψ k)) ? }
 replacement→ : {ψ : HOD → HOD} (X x : HOD) → {C : HOD} → (rc : RCod C ψ ) → (lt : Replace X ψ rc ∋ x) 
    →  ¬ ( (y : HOD) → ¬ (x =h= ψ y))
 replacement→ {ψ} X x {C} rc lt eq = eq (* (Replaced.z lt)) (ord→== (Replaced.x=ψz lt)) 
@@ -378,6 +319,7 @@
 record RXCod (X COD : HOD) (ψ : (x : HOD) → X ∋ x → HOD)  : Set (suc n) where
  field
      ≤COD : ∀ {x : HOD } → (lt : X ∋ x) → ψ x lt ⊆ COD 
+     ψ-eq : ∀ {x : HOD } → (lt lt1 : X ∋ x) → ψ x lt ≡ ψ x lt1
 
 record Replaced1 (A : HOD) (ψ : (x : Ordinal ) → odef A x → Ordinal ) (x : Ordinal ) : Set n where
    field
@@ -393,14 +335,15 @@
             r01 = sym (Replaced1.x=ψz lt )
 
 cod-conv : (X : HOD) → (ψ : (x : HOD) → X ∋ x → HOD) → {C : HOD} → (rc : RXCod X C ψ   )
-      → RXCod (* (& X)) C (λ y xy → ψ y (subst (λ k → k ∋ y) *iso xy))
-cod-conv X ψ {C} rc = record { ≤COD = λ {x} lt → RXCod.≤COD rc (subst (λ k → odef k (& x)) *iso lt) }
+      → RXCod (* (& X)) C (λ y xy → ψ y (eq→ *iso xy)) 
+cod-conv X ψ {C} rc = record { ≤COD = λ {x} lt → RXCod.≤COD rc (eq→ *iso lt ) 
+        ; ψ-eq = λ {x} lt lt1 → RXCod.ψ-eq rc (eq→ *iso lt) (eq→ *iso lt1) } 
 
 Replace'-iso : {X Y : HOD} → {fx : (x : HOD) → X ∋ x → HOD} {fy : (x : HOD) → Y ∋ x → HOD}
     → {CX : HOD} → (rcx : RXCod X CX fx  ) → {CY : HOD} → (rcy : RXCod Y CY fy   )
       → X ≡ Y →  ( (x :  HOD) → (xx : X ∋ x ) → (yy : Y ∋ x ) → fx _ xx ≡ fy _ yy )
-      → Replace' X fx rcx ≡ Replace' Y fy rcy
-Replace'-iso {X} {X} {fx} {fy} _ _ refl eq  = ==→o≡ record { eq→ = ri0 ; eq← = ri1 } where
+      → od (Replace' X fx rcx ) == od (Replace' Y fy rcy)
+Replace'-iso {X} {X} {fx} {fy} _ _ refl eq  = record { eq→ = ri0 ; eq← = ri1 } where
      ri0 : {x : Ordinal} → Replaced1 X (λ z xz → & (fx (* z) (subst (odef X) (sym &iso) xz))) x 
                          → Replaced1 X (λ z xz → & (fy (* z) (subst (odef X) (sym &iso) xz))) x
      ri0 {x} record { z = z ; az = az ; x=ψz = x=ψz } = record { z = z ; az = az ; x=ψz = trans x=ψz (cong (&) ( eq _ xz xz ))  } where
@@ -413,18 +356,19 @@
          xz = subst (λ k → odef X k ) (sym &iso) az
 
 Replace'-iso1 : (X : HOD) → (ψ : (x : HOD) → X ∋ x → HOD) → {C : HOD} → (rc : RXCod X C ψ   )
-      → Replace' (* (& X)) (λ y xy → ψ y (subst (λ k → k ∋ y ) *iso xy) ) (cod-conv X ψ rc)
-         ≡ Replace' X ( λ y xy → ψ y xy ) rc 
-Replace'-iso1 X ψ rc = Replace'-iso {* (& X)} {X} {λ y xy → ψ y (subst (λ k → k ∋ y ) *iso xy) } { λ y xy → ψ y xy } 
-    (cod-conv X ψ rc) rc 
-    *iso (λ x xx yx → fi00 x xx yx ) where
-      fi00 : (x : HOD ) → (xx : (* (& X)) ∋ x ) → (yx : X ∋ x) →  ψ x (subst (λ k → k ∋ x) *iso xx) ≡ ψ x yx
-      fi00 x xx yx = cong (λ k → ψ x k ) ( HE.≅-to-≡ ( ∋-irr {X} {& x} (subst (λ k → k ∋ x) *iso xx) yx ) )
-
--- replacement←1 : {ψ : HOD → HOD} (X x : HOD) →  X ∋ x → Replace1 X ψ ∋ ψ x
--- replacement←1 {ψ} X x lt = record { z = & x ; az = lt  ; x=ψz = cong (λ k → & (ψ k)) (sym *iso) }
--- replacement→1 : {ψ : HOD → HOD} (X x : HOD) → (lt : Replace1 X ψ ∋ x) → ¬ ( (y : HOD) → ¬ (x =h= ψ y))
--- replacement→1 {ψ} X x lt eq = eq (* (Replaced.z lt)) (ord→== (Replaced.x=ψz lt)) 
+      → od (Replace' (* (& X)) (λ y xy → ψ y (eq→ *iso xy) ) (cod-conv X ψ rc))
+         == od ( Replace' X ( λ y xy → ψ y xy ) rc )
+Replace'-iso1 X ψ rc = record { eq→ = ri0 ; eq← = ri1 } where
+      ri0 : {x : Ordinal} → Replaced1 (* (& X))
+            (λ z xz → & (ψ (* z) (eq→ *iso (subst (odef (* (& X))) (sym &iso) xz)))) x →
+            Replaced1 X (λ z xz → & (ψ (* z) (subst (odef X) (sym &iso) xz))) x
+      ri0 {x} record { z = z ; az = az ; x=ψz = x=ψz } = record { z = z ; az = eq→  *iso az 
+          ; x=ψz = trans x=ψz (cong (&) (RXCod.ψ-eq rc _ _ ))  } 
+      ri1 : {x : Ordinal} → 
+            Replaced1 X (λ z xz → & (ψ (* z) (subst (odef X) (sym &iso) xz))) x →
+              Replaced1 (* (& X)) (λ z xz → & (ψ (* z) (eq→ *iso (subst (odef (* (& X))) (sym &iso) xz)))) x 
+      ri1 {x} record { z = z ; az = az ; x=ψz = x=ψz } = record { z = z ; az = eq←  *iso az 
+          ; x=ψz = trans x=ψz (cong (&) (RXCod.ψ-eq rc _ _ ))  } 
 
 _∈_ : ( A B : HOD  ) → Set n
 A ∈ B = B ∋ A
@@ -438,13 +382,13 @@
          p01 = subst (λ k → k o≤ & A) &iso ( ⊆→o≤ (λ {x} yx → y⊆A x yx ))
 
 power→ :  ( A t : HOD) → Power A ∋ t → {x : HOD} → t ∋ x → A ∋ x
-power→ A t P∋t {x} t∋x = P∋t (& x) (subst (λ k → odef k (& x) ) (sym *iso) t∋x )
+power→ A t P∋t {x} t∋x = P∋t (& x) (eq← *iso t∋x )
 
 power← :  (A t : HOD) → ({x : HOD} → (t ∋ x → A ∋ x)) → Power A ∋ t
-power← A t t⊆A z xz = subst (λ k → odef A k ) &iso ( t⊆A  (subst₂ (λ j k → odef j k) *iso (sym &iso) xz ))
+power← A t t⊆A z xz = subst (λ k → odef A k ) &iso ( t⊆A  (subst (λ  k → odef t k) (sym &iso) (eq→ *iso xz )))
 
 Power∋∅ : {S : HOD} → odef (Power S) o∅
-Power∋∅ z xz = ⊥-elim (¬x<0 (subst (λ k → odef k z) o∅≡od∅ xz)  )
+Power∋∅ z xz = ⊥-elim (¬x<0 ( eq→ o∅==od∅ xz)  )
 
 Intersection : (X : HOD ) → HOD   -- ∩ X
 Intersection X = record { od = record { def = λ x → (x o≤ & X ) ∧ ( {y : Ordinal} → odef X y → odef (* y) x )} ; odmax = osuc (& X) ; <odmax = λ lt → proj1 lt } 
@@ -480,16 +424,17 @@
 ¬0=ux : {x : HOD} → ¬ o∅ ≡ & (Union ( x , ( x ,  x)))
 ¬0=ux {x} eq = ⊥-elim ( o<¬≡ eq (ordtrans≤-< o∅<x (subst (λ k → k o< & (Union (x , (x , x)))) &iso (c<→o< lemma ) ))) where
     lemma : Own (x , (x , x)) (& ( * (& x )))
-    lemma = record { owner = _ ; ao = case2 refl ; ox = subst₂ (λ j k → odef j k ) (sym *iso) (sym &iso) (case1 refl) }
+    lemma = record { owner = _ ; ao = case2 refl ; ox = eq← *iso (subst (λ k → odef (x , x)  k) (sym &iso) (case1 refl)) }
 
-ux-2cases : {x y : HOD } → Union ( x , ( x ,  x)) ∋ y → ( x ≡ y ) ∨ ( x ∋ y )
-ux-2cases {x} {y} record { owner = owner ; ao = (case1 eq) ; ox = ox } = case2 (subst (λ k → odef k (& y)) (trans (cong (*) eq) *iso) ox)
-ux-2cases {x} {y} record { owner = owner ; ao = (case2 eq) ; ox = ox } with subst (λ k → odef k (& y))  (trans (cong (*) eq) *iso) ox
-... | case1 eq = case1 (sym (&≡&→≡ eq))
-... | case2 eq = case1 (sym (&≡&→≡ eq))
+ux-2cases : {x y : HOD } → Union ( x , ( x ,  x)) ∋ y → ( & x ≡ & y ) ∨ ( x ∋ y )
+ux-2cases {x} {y} record { owner = owner ; ao = (case1 eq) ; ox = ox } 
+    = case2 (eq→ *iso (subst (λ k → odef k (& y)) (cong (*) eq)  ox ))
+ux-2cases {x} {y} record { owner = owner ; ao = (case2 eq) ; ox = ox } with eq→ *iso (subst (λ k → odef k (& y))  (cong (*) eq) ox)
+... | case1 y=x = case1 (sym y=x)
+... | case2 y=x = case1 (sym y=x)
 
 ux-transitve  : {x y : HOD} → x ∋ y →  Union ( x , ( x ,  x)) ∋ y 
-ux-transitve {x} {y} ox  = record { owner = _ ; ao = case1 refl ; ox = subst (λ k → odef k (& y)) (sym *iso) ox }
+ux-transitve {x} {y} ox  = record { owner = _ ; ao = case1 refl ; ox = eq← *iso ox }
 
 --
 -- Possible Ordinal Limit
@@ -502,38 +447,40 @@
     omega : Ordinal  
     ho< : {x : Ordinal } → Omega-d x →  x o< omega
 
-postulate
-    odaxion-ho< : ODAxiom-ho< 
+-- postulate
+--    odaxion-ho< : ODAxiom-ho< 
 
-open ODAxiom-ho< odaxion-ho<
+-- open ODAxiom-ho< odaxion-ho<
 
-Omega : HOD
-Omega = record { od = record { def = λ x → Omega-d x } ; odmax = omega ; <odmax = ho<}  
+Omega : ODAxiom-ho< → HOD
+Omega ho< = record { od = record { def = λ x → Omega-d x } ; odmax = ODAxiom-ho<.omega ho< ; <odmax = λ lt → ODAxiom-ho<.ho< ho< lt }  
 
-infinity∅ : Omega  ∋ od∅
-infinity∅ = subst (λ k → odef Omega k ) lemma iφ where
+infinity∅ : (ho< : ODAxiom-ho<) →  Omega ho<  ∋ od∅
+infinity∅ ho< = subst (λ k → odef (Omega ho<) k ) lemma iφ where
     lemma : o∅ ≡ & od∅
-    lemma =  let open ≡-Reasoning in begin
-        o∅
-        ≡⟨ sym &iso ⟩
-        & ( * o∅ )
-        ≡⟨ cong ( λ k → & k ) o∅≡od∅ ⟩
-        & od∅
-        ∎
+    lemma =  sym ord-od∅ 
 
-infinity : (x : HOD) → Omega ∋ x → Omega ∋ Union (x , (x , x ))
-infinity x lt = subst (λ k → odef Omega k ) lemma (isuc {& x} lt) where
-    lemma :  & (Union (* (& x) , (* (& x) , * (& x))))
-        ≡ & (Union (x , (x , x)))
-    lemma = cong (λ k → & (Union ( k , ( k , k ) ))) *iso
+infinity : (ho< : ODAxiom-ho<) → (x : HOD) → Omega ho< ∋ x → Omega ho< ∋ Union (x , (x , x ))
+infinity ho< x lt = subst (λ k → odef (Omega ho<) k ) lemma (isuc {& x} lt) where
+    lemma :  & (Union (* (& x) , (* (& x) , * (& x)))) ≡ & (Union (x , (x , x)))
+    lemma = ==→o≡ record { eq→ = lemma2 ; eq← = lemma3 } where
+      lemma2 :  {y : Ordinal} → Own (* (& x) , (* (& x) , * (& x))) y → Own (x , (x , x)) y
+      lemma2 {y} record { owner = owner ; ao = case1 ao ; ox = ox } = record { owner = owner ; ao = case1 lemma4 ; ox = ox }  where
+          lemma4 : owner ≡ & x
+          lemma4 = trans ao ( ==→o≡ *iso )
+      lemma2 {y} record { owner = owner ; ao = case2 ao ; ox = ox } = record { owner = owner ; ao = case2 ? ; ox = ox }  where
+          lemma4 : owner ≡ & (x , x )
+          lemma4 = trans ao ( ==→o≡ record { eq→ = ? ; eq← = ? } )
+      lemma3 :  {y : Ordinal}  → Own (x , (x , x)) y → Own (* (& x) , (* (& x) , * (& x))) y
+      lemma3 {y} record { owner = owner ; ao = ao ; ox = ox } = record { owner = owner ; ao = ? ; ox = ox } 
 
 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 (case1 t≡x ) = case1 (subst₂ (λ j k → j =h= k ) ? ? (o≡→== t≡x ))
+pair→ x y t (case2 t≡y ) = case2 (subst₂ (λ j k → j =h= k ) ? ? (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))
+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 {z} ox = ordtrans ox lt
@@ -549,8 +496,8 @@
 ψiso {ψ} t refl = t
 selection : {ψ : HOD → Set n} {X y : HOD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y)
 selection {ψ} {X} {y} = ⟪
-     ( λ cond → ⟪ proj1 cond , ψiso {ψ} (proj2 cond) (sym *iso)  ⟫ )
-  ,  ( λ select → ⟪ proj1 select  , ψiso {ψ} (proj2 select) *iso  ⟫ )
+     ( λ cond → ⟪ proj1 cond , ψiso {ψ} (proj2 cond) (sym ?)  ⟫ )
+  ,  ( λ select → ⟪ proj1 select  , ψiso {ψ} (proj2 select) ?  ⟫ )
   ⟫
 
 selection-in-domain : {ψ : HOD → Set n} {X y : HOD} → Select X ψ ∋ y → X ∋ y
@@ -574,8 +521,8 @@
 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
+proj1 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) ? d -- ( ==→o≡ (extensionality0 {A} {B} eq) ) d
+proj2 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) ? d -- (sym ( ==→o≡ (extensionality0 {A} {B} eq) )) d
 
 open import zf
 
@@ -585,7 +532,7 @@
   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
  sup-c≤ :  (ψ : HOD → HOD) → {X x : HOD} → def (od 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 )
+ sup-c≤ ψ {X} {x} lt = subst (λ k → & (ψ k) o< _ ) ? (sup-o≤ X lt )
 
 -- sup-o may contradict
 --    If we have open monotonic function in Ordinal, there is no sup-o. 
@@ -605,12 +552,12 @@
             r01 = sym (Replaced.x=ψz lt )
 
 zf-replacement← : (os : ODAxiom-sup) → {ψ : HOD → HOD} (X x : HOD) →  X ∋ x → ZFReplace os X ψ ∋ ψ x
-zf-replacement← os {ψ} X x lt = record { z = & x ; az = lt  ; x=ψz = cong (λ k → & (ψ k)) (sym *iso) }
+zf-replacement← os {ψ} X x lt = record { z = & x ; az = lt  ; x=ψz = cong (λ k → & (ψ k)) (sym ?) }
 zf-replacement→ : (os : ODAxiom-sup ) → {ψ : HOD → HOD} (X x : HOD) → (lt : ZFReplace os X ψ ∋ x) → ¬ ( (y : HOD) → ¬ (x =h= ψ y))
 zf-replacement→ os {ψ} X x lt eq = eq (* (Replaced.z lt)) (ord→== (Replaced.x=ψz lt)) 
 
-isZF : (os : ODAxiom-sup) → IsZF HOD _∋_  _=h=_ od∅ _,_ Union Power Select (ZFReplace os) Omega
-isZF os = record {
+isZF : (os : ODAxiom-sup)  (ho< : ODAxiom-ho< ) → IsZF HOD _∋_  _=h=_ od∅ _,_ Union Power Select (ZFReplace os) (Omega ho<)
+isZF os ho< = record {
         isEquivalence  = record { refl = ==-refl ; sym = ==-sym; trans = ==-trans }
     ;   pair→  = pair→
     ;   pair←  = pair←
@@ -621,15 +568,15 @@
     ;   power← = power←
     ;   extensionality = λ {A} {B} {w} → extensionality {A} {B} {w}
     ;   ε-induction = ε-induction
-    ;   infinity∅ = infinity∅
-    ;   infinity = infinity
+    ;   infinity∅ = infinity∅ ho<
+    ;   infinity = infinity ho<
     ;   selection = λ {X} {ψ} {y} → selection {X} {ψ} {y}
     ;   replacement← = zf-replacement← os
     ;   replacement→ = λ {ψ} → zf-replacement→ os {ψ}
     }
 
-HOD→ZF : ODAxiom-sup → ZF
-HOD→ZF os  = record {
+HOD→ZF : ODAxiom-sup →  ODAxiom-ho< → ZF
+HOD→ZF os ho< = record {
     ZFSet = HOD
     ; _∋_ = _∋_
     ; _≈_ = _=h=_
@@ -639,8 +586,8 @@
     ; Power = Power
     ; Select = Select
     ; Replace = ZFReplace os
-    ; infinite = Omega
-    ; isZF = isZF os
+    ; infinite = Omega ho<
+    ; isZF = isZF os ho<
  }
 
 
--- a/src/OrdUtil.agda	Sat Aug 26 10:36:09 2023 +0900
+++ b/src/OrdUtil.agda	Mon Jan 01 18:21:36 2024 +0900
@@ -1,3 +1,5 @@
+{-# OPTIONS --cubical-compatible --safe #-}
+
 open import Level
 open import Ordinals
 module OrdUtil {n : Level} (O : Ordinals {n} ) where
@@ -152,7 +154,7 @@
 omax  x y with trio< x y
 omax  x y | tri< a ¬b ¬c = osuc y
 omax  x y | tri> ¬a ¬b c = osuc x
-omax  x y | tri≈ ¬a refl ¬c  = osuc x
+omax  x y | tri≈ ¬a b ¬c  = osuc x
 
 omax< :  ( x y : Ordinal  ) → x o< y → osuc y ≡ omax x y
 omax<  x y lt with trio< x y
@@ -190,7 +192,7 @@
 omxx  x with  trio< x x
 omxx  x | tri< a ¬b ¬c = ⊥-elim (¬b refl )
 omxx  x | tri> ¬a ¬b c = ⊥-elim (¬b refl )
-omxx  x | tri≈ ¬a refl ¬c = refl
+omxx  x | tri≈ ¬a b ¬c =  refl
 
 omxxx :  ( x : Ordinal  ) → omax x (omax x x ) ≡ osuc (osuc x)
 omxxx  x = trans ( cong ( λ k → omax x k ) (omxx x )) (sym ( omax< x (osuc x) <-osuc ))
--- a/src/Ordinals.agda	Sat Aug 26 10:36:09 2023 +0900
+++ b/src/Ordinals.agda	Mon Jan 01 18:21:36 2024 +0900
@@ -1,3 +1,5 @@
+{-# OPTIONS --cubical-compatible --safe #-}
+
 open import Level
 module Ordinals where
 
--- a/src/Tychonoff.agda	Sat Aug 26 10:36:09 2023 +0900
+++ b/src/Tychonoff.agda	Mon Jan 01 18:21:36 2024 +0900
@@ -376,7 +376,7 @@
 
 import Axiom.Extensionality.Propositional
 postulate f-extensionality : { n m : Level}  → Axiom.Extensionality.Propositional.Extensionality n m
-open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ )
+-- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ )
 
 --
 -- We have UFLP both in P and Q. Given an ultra filter F on P x Q. It has limits on P and Q because a projection of ultra filter
--- a/src/ZProduct.agda	Sat Aug 26 10:36:09 2023 +0900
+++ b/src/ZProduct.agda	Mon Jan 01 18:21:36 2024 +0900
@@ -340,7 +340,7 @@
 --  Set of All function from A to B
 --
 
-open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ )
+-- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ )
 
 Funcs : (A B : HOD) → HOD
 Funcs A B = record { od = record { def = λ x → FuncHOD A B x } ; odmax = osuc (& (ZFP A B))
@@ -361,6 +361,7 @@
 record Injection (A B : Ordinal ) : Set n where
    field
        i→  : (x : Ordinal ) → odef (* A)  x → Ordinal
+       irr : (x : Ordinal ) → ( lt1 lt2 : odef (* A)  x ) → i→ x lt1 ≡ i→ x lt2 
        iB  : (x : Ordinal ) → ( lt : odef (* A)  x ) → odef (* B) ( i→ x lt )
        inject : (x y : Ordinal ) → ( ltx : odef (* A)  x ) ( lty : odef (* A)  y ) → i→ x ltx ≡ i→ y lty → x ≡ y
 
--- a/src/bijection.agda	Sat Aug 26 10:36:09 2023 +0900
+++ b/src/bijection.agda	Mon Jan 01 18:21:36 2024 +0900
@@ -47,6 +47,27 @@
 bi-inject→ : {n m : Level} {R : Set n} {S : Set m} → (rs : Bijection R S) → {x y : R} → fun→ rs x ≡ fun→ rs y → x ≡ y
 bi-inject→ rs {x} {y} eq = subst₂ (λ j k → j ≡ k ) (fiso←  rs _) (fiso← rs _) (cong (fun← rs) eq)
 
+bi-∨  : {n m n1 m1 : Level } {A : Set n} {B : Set m} {C : Set n1} {D : Set m1}  → (ab : Bijection A B) → (cd : Bijection C D ) 
+       → Bijection (A ∨ C) (B ∨ D)           
+bi-∨  {_} {_} {_} {_} {A} {B} {C} {D} ab cd = record {
+         fun→  = fun→1
+       ; fun←  = fun←1
+       ; fiso→ = fiso→1
+       ; fiso← = fiso←1
+       } where
+    fun→1 : (A ∨ C) → (B ∨ D)
+    fun→1 (case1 a) = case1 (fun→ ab a)
+    fun→1 (case2 c) = case2 (fun→ cd c)
+    fun←1 : (B ∨ D) → (A ∨ C)
+    fun←1 (case1 a) = case1 (fun← ab a)
+    fun←1 (case2 c) = case2 (fun← cd c)
+    fiso→1 : (x : B ∨ D) → fun→1 (fun←1 x) ≡ x
+    fiso→1 (case1 a) = cong case1 (fiso→ ab a)
+    fiso→1 (case2 c) = cong case2 (fiso→ cd c)
+    fiso←1 : (x : A ∨ C) → fun←1 (fun→1 x) ≡ x 
+    fiso←1 (case1 a) = cong case1 (fiso← ab a)
+    fiso←1 (case2 c) = cong case2 (fiso← cd c)
+
 open import Relation.Binary.Structures
 
 bijIsEquivalence : {n : Level } → IsEquivalence  (Bijection {n} {n})
@@ -511,6 +532,42 @@
       a : A
       fa=c : f a ≡ c
 
+record IsImage0 (A B : Set ) (f : (x : A ) → B) (x : B ) : Set  where
+   field
+      y : A
+      x=fy : x ≡ f y 
+
+IsImage : (a b : Set) (iab : InjectiveF a b ) (x : b ) → Set 
+IsImage a b iab x = IsImage0 a b (InjectiveF.f iab) x
+
+Bernstein : (A B : Set) 
+     → (fi : InjectiveF A  B ) → (gi : InjectiveF  B A )
+     → (is-A : (b : B ) → Dec (Is A B (InjectiveF.f fi) b)  )
+     → (is-B : (a : A ) → Dec (Is B A (InjectiveF.f gi) a)  )
+     → Bijection A B
+Bernstein A B fi gi isa isb = ?  where
+    open InjectiveF
+    gfi : InjectiveF A A
+    gfi = record { f = λ x → f gi (f fi x) ; inject = λ {x} {y} eq → inject fi (inject gi eq) }
+    data gfImage :  (x : A) → Set where
+       a-g : {x : A} → (¬ib : ¬ ( IsImage B A gi x )) → gfImage  x
+       next-gf : {x : A} → (ix : IsImage A A gfi x) → (gfiy : gfImage (IsImage0.y ix) ) → gfImage  x
+    data ¬gfImage :  (x : A) → Set where
+       ngf : {x : A} → (¬gfiy : ¬ gfImage x) → ¬gfImage  x
+    gf02 : {x : A} → IsImage B A gi x ∨ (¬ IsImage B A gi x) ∨ ((ix : IsImage A A gfi x) →  ¬  gfImage (IsImage0.y ix)  ) → ¬ gfImage x
+    gf02 {x} c gf = ?
+    gfi∨ : (x : A) → gfImage x ∨ ¬gfImage x
+    gfi∨ x with isb x
+    ... | no ¬ib = case1 ( a-g (λ ib → ¬ib (record { a = IsImage0.y ib ; fa=c = sym (IsImage0.x=fy ib) })))
+    ... | yes ib with isa (f fi x)
+    ... | no ¬ia = case2 ( ngf ? )
+    ... | yes ia = case1 ( next-gf record { y = f gi (Is.a ib) ; x=fy = br00 }  (a-g br01 ) ) where
+         br00 :  x ≡ f gi (f fi (f gi (Is.a ib)))
+         br00 = ?
+         br01 :  ¬ IsImage B A gi (f gi (Is.a ib))
+         br01 record { y = y ; x=fy = x=fy } = ?
+
+
 Countable-Bernstein : (A B C : Set) → Bijection A ℕ → Bijection C ℕ
      → (fi : InjectiveF A  B ) → (gi : InjectiveF  B C )
      → (is-A : (c : C ) → Dec (Is A C (λ x → (InjectiveF.f gi (InjectiveF.f fi x))) c )) 
--- a/src/logic.agda	Sat Aug 26 10:36:09 2023 +0900
+++ b/src/logic.agda	Mon Jan 01 18:21:36 2024 +0900
@@ -1,3 +1,5 @@
+{-# OPTIONS --cubical-compatible --safe #-}
+
 module logic where
 
 open import Level
@@ -5,14 +7,11 @@
 open import Relation.Binary hiding (_⇔_ )
 open import Data.Empty
 
+
 data Bool : Set where
     true : Bool
     false : Bool
 
-data Two : Set where
-   i0 : Two
-   i1 : Two
-
 record  _∧_  {n m : Level} (A  : Set n) ( B : Set m ) : Set (n ⊔ m) where
    constructor ⟪_,_⟫
    field
@@ -26,13 +25,6 @@
 _⇔_ : {n m : Level } → ( A : Set n ) ( B : Set m )  → Set (n ⊔ m)
 _⇔_ A B =  ( A → B ) ∧ ( B → A )
 
-∧-exch : {n m : Level} {A  : Set n} { B : Set m } → A ∧ B → B ∧ A
-∧-exch p = ⟪ _∧_.proj2 p , _∧_.proj1 p ⟫
-
-∨-exch : {n m : Level} {A  : Set n} { B : Set m } → A ∨ B → B ∨ A
-∨-exch (case1 x) = case2 x
-∨-exch (case2 x) = case1 x
-
 contra-position : {n m : Level } {A : Set n} {B : Set m} → (A → B) → ¬ B → ¬ A
 contra-position {n} {m} {A} {B}  f ¬b a = ¬b ( f a )
 
@@ -46,10 +38,6 @@
 de-morgan {n} {A} {B} and (case1 ¬A) = ⊥-elim ( ¬A ( _∧_.proj1 and ))
 de-morgan {n} {A} {B} and (case2 ¬B) = ⊥-elim ( ¬B ( _∧_.proj2 and ))
 
-de-morgan∨ : {n  : Level } {A B : Set n} →  A ∨ B  → ¬ ( (¬ A ) ∧ (¬ B ) )
-de-morgan∨ {n} {A} {B} (case1 a) and = ⊥-elim (  _∧_.proj1 and a )
-de-morgan∨ {n} {A} {B} (case2 b) and = ⊥-elim (  _∧_.proj2 and b )
-
 dont-or :　{n m : Level} {A  : Set n} { B : Set m } →  A ∨ B → ¬ A → B
 dont-or {A} {B} (case1 a) ¬A = ⊥-elim ( ¬A a )
 dont-or {A} {B} (case2 b) ¬A = b
@@ -79,11 +67,10 @@
 false <=> false = true
 _ <=> _ = false
 
-open import Relation.Binary.PropositionalEquality
+infixr  130 _\/_
+infixr  140 _/\_
 
-not-not-bool : { b : Bool } → not (not b) ≡ b
-not-not-bool {true} = refl
-not-not-bool {false} = refl
+open import Relation.Binary.PropositionalEquality
 
 record Bijection {n m : Level} (R : Set n) (S : Set m) : Set (n Level.⊔ m)  where
    field
@@ -96,13 +83,14 @@
 injection R S f = (x y : R) → f x ≡ f y → x ≡ y
 
 
+not-not-bool : { b : Bool } → not (not b) ≡ b
+not-not-bool {true} = refl
+not-not-bool {false} = refl
+
 ¬t=f : (t : Bool ) → ¬ ( not t ≡ t) 
 ¬t=f true ()
 ¬t=f false ()
 
-infixr  130 _\/_
-infixr  140 _/\_
-
 ≡-Bool-func : {A B : Bool } → ( A ≡ true → B ≡ true ) → ( B ≡ true → A ≡ true ) → A ≡ B
 ≡-Bool-func {true} {true} a→b b→a = refl
 ≡-Bool-func {false} {true} a→b b→a with b→a refl
@@ -129,89 +117,57 @@
 ¬-bool refl ()
 
 lemma-∧-0 : {a b : Bool} → a /\ b ≡ true → a ≡ false → ⊥
-lemma-∧-0 {true} {true} refl ()
 lemma-∧-0 {true} {false} ()
 lemma-∧-0 {false} {true} ()
 lemma-∧-0 {false} {false} ()
+lemma-∧-0 {true} {true} eq1 ()
 
 lemma-∧-1 : {a b : Bool} → a /\ b ≡ true → b ≡ false → ⊥
-lemma-∧-1 {true} {true} refl ()
 lemma-∧-1 {true} {false} ()
 lemma-∧-1 {false} {true} ()
 lemma-∧-1 {false} {false} ()
+lemma-∧-1 {true} {true} eq1 ()
 
 bool-and-tt : {a b : Bool} → a ≡ true → b ≡ true → ( a /\ b ) ≡ true
 bool-and-tt refl refl = refl
 
 bool-∧→tt-0 : {a b : Bool} → ( a /\ b ) ≡ true → a ≡ true 
-bool-∧→tt-0 {true} {true} refl = refl
+bool-∧→tt-0 {true} {true} eq =  refl
 bool-∧→tt-0 {false} {_} ()
 
 bool-∧→tt-1 : {a b : Bool} → ( a /\ b ) ≡ true → b ≡ true 
-bool-∧→tt-1 {true} {true} refl = refl
+bool-∧→tt-1 {true} {true} eq = refl
 bool-∧→tt-1 {true} {false} ()
 bool-∧→tt-1 {false} {false} ()
 
 bool-or-1 : {a b : Bool} → a ≡ false → ( a \/ b ) ≡ b 
-bool-or-1 {false} {true} refl = refl
-bool-or-1 {false} {false} refl = refl
+bool-or-1 {false} {true} eq = refl
+bool-or-1 {false} {false} eq = refl
 bool-or-2 : {a b : Bool} → b ≡ false → (a \/ b ) ≡ a 
-bool-or-2 {true} {false} refl = refl
-bool-or-2 {false} {false} refl = refl
+bool-or-2 {true} {false} eq = refl
+bool-or-2 {false} {false} eq = refl
 
 bool-or-3 : {a : Bool} → ( a \/ true ) ≡ true 
 bool-or-3 {true} = refl
 bool-or-3 {false} = refl
 
 bool-or-31 : {a b : Bool} → b ≡ true  → ( a \/ b ) ≡ true 
-bool-or-31 {true} {true} refl = refl
-bool-or-31 {false} {true} refl = refl
+bool-or-31 {true} {true} eq = refl
+bool-or-31 {false} {true} eq = refl
 
 bool-or-4 : {a : Bool} → ( true \/ a ) ≡ true 
 bool-or-4 {true} = refl
 bool-or-4 {false} = refl
 
 bool-or-41 : {a b : Bool} → a ≡ true  → ( a \/ b ) ≡ true 
-bool-or-41 {true} {b} refl = refl
+bool-or-41 {true} {b} eq = refl
 
 bool-and-1 : {a b : Bool} →  a ≡ false → (a /\ b ) ≡ false
-bool-and-1 {false} {b} refl = refl
+bool-and-1 {false} {b} eq = refl
 bool-and-2 : {a b : Bool} →  b ≡ false → (a /\ b ) ≡ false
-bool-and-2 {true} {false} refl = refl
-bool-and-2 {false} {false} refl = refl
+bool-and-2 {true} {false} eq = refl
+bool-and-2 {false} {false} eq = refl
+bool-and-2 {true} {true} ()
+bool-and-2 {false} {true} ()
 
 
-open import Data.Nat
-open import Data.Nat.Properties
-
-_≥b_ : ( x y : ℕ) → Bool
-x ≥b y with <-cmp x y
-... | tri< a ¬b ¬c = false
-... | tri≈ ¬a b ¬c = true
-... | tri> ¬a ¬b c = true
-
-_>b_ : ( x y : ℕ) → Bool
-x >b y with <-cmp x y
-... | tri< a ¬b ¬c = false
-... | tri≈ ¬a b ¬c = false
-... | tri> ¬a ¬b c = true
-
-_≤b_ : ( x y : ℕ) → Bool
-x ≤b y  = y ≥b x
-
-_<b_ : ( x y : ℕ) → Bool
-x <b y  = y >b x
-
-open import Relation.Binary.PropositionalEquality
-
-¬i0≡i1 : ¬ ( i0 ≡ i1 )
-¬i0≡i1 ()
-
-¬i0→i1 : {x : Two} → ¬ (x ≡ i0 ) → x ≡ i1 
-¬i0→i1 {i0} ne = ⊥-elim ( ne refl )
-¬i0→i1 {i1} ne = refl
-
-¬i1→i0 : {x : Two} → ¬ (x ≡ i1 ) → x ≡ i0 
-¬i1→i0 {i0} ne = refl
-¬i1→i0 {i1} ne = ⊥-elim ( ne refl )
-
--- a/src/nat.agda	Sat Aug 26 10:36:09 2023 +0900
+++ b/src/nat.agda	Mon Jan 01 18:21:36 2024 +0900
@@ -1,4 +1,5 @@
-{-# OPTIONS --allow-unsolved-metas #-}
+{-# OPTIONS --cubical-compatible --safe #-}
+
 module nat where
 
 open import Data.Nat 
@@ -104,15 +105,15 @@
 div2-eq : (x : ℕ ) → div2-rev ( div2 x ) ≡ x
 div2-eq zero = refl
 div2-eq (suc zero) = refl
-div2-eq (suc (suc x)) with div2 x | inspect div2 x 
-... | ⟪ x1 , true ⟫ | record { eq = eq1 } = begin -- eq1 : div2 x ≡ ⟪ x1 , true ⟫
+div2-eq (suc (suc x)) with div2 x in eq1 
+... | ⟪ x1 , true ⟫ = begin -- eq1 : div2 x ≡ ⟪ x1 , true ⟫
      div2-rev ⟪ suc x1 , true ⟫ ≡⟨⟩
      suc (suc (x1 + suc x1)) ≡⟨ cong (λ k → suc (suc k )) (+-comm x1  _ ) ⟩
      suc (suc (suc (x1 + x1))) ≡⟨⟩    
      suc (suc (div2-rev ⟪ x1 , true ⟫)) ≡⟨ cong (λ k → suc (suc (div2-rev k ))) (sym eq1) ⟩ 
      suc (suc (div2-rev (div2 x)))      ≡⟨ cong (λ k → suc (suc k)) (div2-eq x) ⟩ 
      suc (suc x) ∎  where open ≡-Reasoning
-... | ⟪ x1 , false ⟫ | record { eq = eq1 } = begin
+... | ⟪ x1 , false ⟫ = begin
      div2-rev ⟪ suc x1 , false ⟫ ≡⟨⟩
      suc (x1 + suc x1) ≡⟨ cong (λ k → (suc k )) (+-comm x1  _ ) ⟩
      suc (suc (x1 + x1)) ≡⟨⟩    
@@ -138,6 +139,40 @@
 
 _-_ = minus
 
+sn-m=sn-m : {m n : ℕ } →  m ≤ n → suc n - m ≡ suc ( n - m )
+sn-m=sn-m {0} {n} z≤n = refl
+sn-m=sn-m {suc m} {suc n} (s≤s m<n) = sn-m=sn-m m<n
+
+si-sn=i-n : {i n : ℕ } → n < i  → suc (i - suc n) ≡ (i - n)
+si-sn=i-n {i} {n} n<i = begin
+   suc (i - suc n) ≡⟨ sym (sn-m=sn-m n<i )  ⟩
+   suc i - suc n ≡⟨⟩
+   i - n
+   ∎  where
+      open ≡-Reasoning
+
+refl-≤s : {x : ℕ } → x ≤ suc x
+refl-≤s {zero} = z≤n
+refl-≤s {suc x} = s≤s (refl-≤s {x})
+
+a≤sa = refl-≤s
+
+n-m<n : (n m : ℕ ) →  n - m ≤ n
+n-m<n zero zero = z≤n
+n-m<n (suc n) zero = s≤s (n-m<n n zero)
+n-m<n zero (suc m) = z≤n
+n-m<n (suc n) (suc m) = ≤-trans (n-m<n n m ) refl-≤s
+
+n-n-m=m : {m n : ℕ } → m ≤ n  → m ≡ (n - (n - m))
+n-n-m=m {0} {zero} z≤n = refl
+n-n-m=m {0} {suc n} z≤n = n-n-m=m {0} {n} z≤n
+n-n-m=m {suc m} {suc n} (s≤s m≤n) = sym ( begin
+   suc n - ( n - m )    ≡⟨ sn-m=sn-m (n-m<n  n m) ⟩
+   suc (n - ( n - m ))  ≡⟨ cong (λ k → suc k ) (sym (n-n-m=m m≤n)) ⟩
+   suc m
+   ∎  ) where
+      open ≡-Reasoning
+
 m+= : {i j  m : ℕ } → m + i ≡ m + j → i ≡ j
 m+= {i} {j} {zero} refl = refl
 m+= {i} {j} {suc m} eq = m+= {i} {j} {m} ( cong (λ k → pred k ) eq )
@@ -238,12 +273,6 @@
 ... | tri≈ ¬a refl ¬c = case2 ≤-refl
 ... | tri> ¬a ¬b c = case2 (<to≤ c)
 
-refl-≤s : {x : ℕ } → x ≤ suc x
-refl-≤s {zero} = z≤n
-refl-≤s {suc x} = s≤s (refl-≤s {x})
-
-a≤sa = refl-≤s
-
 refl-≤ : {x : ℕ } → x ≤ x
 refl-≤ {zero} = z≤n
 refl-≤ {suc x} = s≤s (refl-≤ {x})
@@ -287,6 +316,9 @@
 x≤y→x<sy {.zero} {y} z≤n = ≤-trans a<sa (s≤s z≤n)
 x≤y→x<sy {.(suc _)} {.(suc _)} (s≤s le) = s≤s ( x≤y→x<sy le) 
 
+sx≤y→x<y : {x y : ℕ } → suc x ≤ y → x < y 
+sx≤y→x<y {zero} {suc y} (s≤s le) = s≤s z≤n
+sx≤y→x<y {suc x} {suc y} (s≤s le) = s≤s ( sx≤y→x<y {x} {y} le )
 
 open import Data.Product
 
@@ -693,15 +725,15 @@
           m ∎  where open ≤-Reasoning  
 
 0<factor : { m k : ℕ } → k > 0 → m > 0 →  (d :  Dividable k m ) → Dividable.factor d > 0
-0<factor {m} {k} k>0 m>0 d with Dividable.factor d | inspect Dividable.factor d 
-... | zero | record { eq = eq1 } = ⊥-elim ( nat-≡< ff1 m>0 ) where
+0<factor {m} {k} k>0 m>0 d with Dividable.factor d in eq1 
+... | zero = ⊥-elim ( nat-≡< ff1 m>0 ) where
     ff1 : 0 ≡ m 
     ff1 = begin
           0 ≡⟨⟩
           0 * k + 0 ≡⟨ cong  (λ j → j * k + 0) (sym eq1) ⟩
           Dividable.factor d * k + 0 ≡⟨ Dividable.is-factor d  ⟩
           m ∎  where open ≡-Reasoning  
-... | suc t | _ = s≤s z≤n 
+... | suc t = s≤s z≤n 
 
 div→k≤m : { m k : ℕ } → k > 1 → m > 0 →  Dividable k m → m ≥ k
 div→k≤m {m} {k} k>1 m>0 d with <-cmp m k
--- a/src/ordinal.agda	Sat Aug 26 10:36:09 2023 +0900
+++ b/src/ordinal.agda	Mon Jan 01 18:21:36 2024 +0900
@@ -50,11 +50,11 @@
 o∅ : {n : Level} → Ordinal {n}
 o∅  = record { lv = Zero ; ord = Φ Zero }
 
-open import Relation.Binary.HeterogeneousEquality using (_≅_;refl)
+-- open import Relation.Binary.HeterogeneousEquality using (_≅_;refl)
 
-ordinal-cong : {n : Level} {x y : Ordinal {n}}  →
-      lv x  ≡ lv y → ord x ≅ ord y →  x ≡ y
-ordinal-cong refl refl = refl
+-- ordinal-cong : {n : Level} {x y : Ordinal {n}}  →
+--       lv x  ≡ lv y → ord x ≅ ord y →  x ≡ y
+-- ordinal-cong refl refl = refl
 
 ≡→¬d< : {n : Level} →  {lv : ℕ} → {x  : OrdinalD {n}  lv }  → x d< x → ⊥
 ≡→¬d<  {n} {lx} {OSuc lx y} (s< t) = ≡→¬d< t
--- a/src/zf.agda	Sat Aug 26 10:36:09 2023 +0900
+++ b/src/zf.agda	Mon Jan 01 18:21:36 2024 +0900
@@ -1,3 +1,4 @@
+{-# OPTIONS --cubical-compatible --safe #-}
 module zf where
 
 open import Level