--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/ODC.agda	Sun Jun 07 20:31:30 2020 +0900
@@ -0,0 +1,96 @@
+open import Level
+open import Ordinals
+module ODC {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
+import OD
+
+open inOrdinal O
+open OD O
+open OD.OD
+open OD._==_
+open ODAxiom odAxiom
+
+postulate      
+  -- 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) )
+
+
+--
+-- 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 zfc
+
+OD→ZFC : ZFC
+OD→ZFC   = record { 
+    ZFSet = OD 
+    ; _∋_ = _∋_ 
+    ; _≈_ = _==_ 
+    ; ∅  = od∅
+    ; Select = Select
+    ; isZFC = isZFC
+ } where
+    -- infixr  200 _∈_
+    -- infixr  230 _∩_ _∪_
+    isZFC : IsZFC (OD )  _∋_  _==_ od∅ Select 
+    isZFC = record {
+       choice-func = choice-func ;
+       choice = choice
+     } where
+         -- 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
+