### diff OD.agda @ 234:e06b76e5b682

ac from LEM in abstract ordinal
author Shinji KONO Tue, 13 Aug 2019 22:21:10 +0900 49736efc822b 846e0926bb89
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```--- a/OD.agda	Mon Aug 12 13:28:59 2019 +0900
+++ b/OD.agda	Tue Aug 13 22:21:10 2019 +0900
@@ -33,7 +33,7 @@
eq→ : ∀ { x : Ordinal  } → def a x → def b x
eq← : ∀ { x : Ordinal  } → def b x → def a x

-id : {n : Level} {A : Set n} → A → A
+id : {A : Set n} → A → A
id x = x

eq-refl :  {  x :  OD  } → x == x
@@ -193,13 +193,13 @@
lemma : ps ∋ minimul ps (λ eq →  ¬p (eqo∅ eq))
lemma = x∋minimul ps (λ eq →  ¬p (eqo∅ eq))

-∋-p : ( p : Set n ) → Dec p   -- assuming axiom of choice
-∋-p  p with p∨¬p p
-∋-p  p | case1 x = yes x
-∋-p  p | case2 x = no x
+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 ∋-p  A                         -- assuming axiom of choice
+double-neg-eilm  {A} notnot with decp  A                         -- assuming axiom of choice
... | yes p = p
... | no ¬p = ⊥-elim ( notnot ¬p )

@@ -477,6 +477,53 @@
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∋minimul 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 = IsOrdinals.TransFinite (Ordinals.isOrdinal O) {ψ} induction ox where
+                    ∋-p : (A x : OD ) → Dec ( A ∋ x )
+                    ∋-p A x with p∨¬p (Lift (suc n) ( A ∋ x ))
+                    ∋-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))
+                    ∀-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 with lemma-ord (od→ord X )
+                 have_to_find | t = dont-or  t ¬¬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 )
+                        }
+
+
_,_ = ZF._,_ OD→ZF
Union = ZF.Union OD→ZF
Power = ZF.Power OD→ZF```