open import Level open import Ordinals open import logic open import Relation.Nullary module LEMC {n : Level } (O : Ordinals {n} ) (p∨¬p : ( p : Set (suc n)) → p ∨ ( ¬ p )) 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.Binary open import Relation.Binary.Core open import nat import OD open inOrdinal O open OD O open OD.OD open OD._==_ open ODAxiom odAxiom open import zfc --- 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 open choiced 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 = λ A {X} not A∋X → a-choice (choice-func X not ); choice = λ A {X} A∋X not → is-in (choice-func X not) } where choice-func : (X : OD ) → ¬ ( X == od∅ ) → choiced X choice-func X 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