Mercurial > hg > Members > kono > Proof > ZF-in-agda
diff OD.agda @ 234:e06b76e5b682
ac from LEM in abstract ordinal
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Tue, 13 Aug 2019 22:21:10 +0900 |
parents | 49736efc822b |
children | 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