Mercurial > hg > Members > kono > Proof > ZF-in-agda
diff src/ODC.agda @ 1120:e086a266c6b7
FIP fix
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Tue, 03 Jan 2023 09:28:23 +0900 |
parents | 55ab5de1ae02 |
children | 45cd80181a29 |
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--- a/src/ODC.agda Mon Jan 02 10:15:20 2023 +0900 +++ b/src/ODC.agda Tue Jan 03 09:28:23 2023 +0900 @@ -4,9 +4,9 @@ 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 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.Nat.Properties open import Data.Empty open import Relation.Nullary open import Relation.Binary @@ -35,9 +35,9 @@ open _∧_ -postulate +postulate -- mimimul and x∋minimal is an Axiom of choice - minimal : (x : HOD ) → ¬ (x =h= od∅ )→ HOD + minimal : (x : HOD ) → ¬ (x =h= od∅ )→ HOD -- this should be ¬ (x =h= od∅ )→ ∃ ox → x ∋ Ord ox ( minimum of x ) x∋minimal : (x : HOD ) → ( ne : ¬ (x =h= od∅ ) ) → odef x ( & ( minimal x ne ) ) -- minimality (proved by ε-induction with LEM) @@ -51,7 +51,7 @@ pred-od : ( p : Set n ) → HOD pred-od p = record { od = record { def = λ x → (x ≡ o∅) ∧ p } ; - odmax = osuc o∅; <odmax = λ x → subst (λ k → k o< osuc o∅) (sym (proj1 x)) <-osuc } + odmax = osuc o∅; <odmax = λ x → subst (λ k → k o< osuc o∅) (sym (proj1 x)) <-osuc } ppp : { p : Set n } { a : HOD } → pred-od p ∋ a → p ppp {p} {a} d = proj2 d @@ -59,32 +59,32 @@ p∨¬p : ( p : Set n ) → p ∨ ( ¬ p ) -- assuming axiom of choice p∨¬p p with is-o∅ ( & (pred-od p )) p∨¬p p | yes eq = case2 (¬p eq) where - ps = pred-od p - eqo∅ : ps =h= od∅ → & ps ≡ o∅ - eqo∅ eq = subst (λ k → & ps ≡ k) ord-od∅ ( cong (λ k → & k ) (==→o≡ eq)) + ps = pred-od p + eqo∅ : ps =h= od∅ → & ps ≡ o∅ + eqo∅ eq = subst (λ k → & ps ≡ k) ord-od∅ ( cong (λ k → & k ) (==→o≡ eq)) lemma : ps =h= od∅ → p → ⊥ lemma eq p0 = ¬x<0 {& ps} (eq→ eq record { proj1 = eqo∅ eq ; proj2 = p0 } ) ¬p : (& ps ≡ o∅) → p → ⊥ ¬p eq = lemma ( subst₂ (λ j k → j =h= k ) *iso o∅≡od∅ ( o≡→== eq )) p∨¬p p | no ¬p = case1 (ppp {p} {minimal ps (λ eq → ¬p (eqo∅ eq))} lemma) where - ps = pred-od p - eqo∅ : ps =h= od∅ → & ps ≡ o∅ - eqo∅ eq = subst (λ k → & ps ≡ k) ord-od∅ ( cong (λ k → & k ) (==→o≡ eq)) - lemma : ps ∋ minimal ps (λ eq → ¬p (eqo∅ eq)) + ps = pred-od p + eqo∅ : ps =h= od∅ → & ps ≡ o∅ + eqo∅ eq = subst (λ k → & ps ≡ k) ord-od∅ ( cong (λ k → & 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 : 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 -∋-p : (A x : HOD ) → Dec ( A ∋ x ) +∋-p : (A x : HOD ) → Dec ( A ∋ x ) ∋-p A x with p∨¬p ( A ∋ x ) -- LEM ∋-p A x | case1 t = yes t ∋-p A x | case2 t = no (λ x → t 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 +double-neg-elim : {A : Set n} → ¬ ¬ A → A -- we don't have this in intutionistic logic +double-neg-elim {A} notnot with decp A -- assuming axiom of choice ... | yes p = p ... | no ¬p = ⊥-elim ( notnot ¬p ) @@ -95,6 +95,18 @@ or-exclude {A} {B} (case2 b) | case1 a = case1 a or-exclude {A} {B} (case2 b) | case2 ¬a = case2 ⟪ ¬a , b ⟫ +-- record By-contradiction (A : Set n) (B : A → Set n) : Set (suc n) where +-- field +-- a : A +-- b : B a +-- +-- by-contradiction : {A : Set n} {B : A → Set n} → ¬ ( (a : A ) → ¬ B a ) → By-contradiction A B +-- by-contradiction {A} {B} not with p∨¬p A +-- ... | case2 ¬a = ⊥-elim (not (λ a → ⊥-elim (¬a a ))) +-- ... | case1 a with p∨¬p (B a) +-- ... | case1 b = record { a = a ; b = b } +-- ... | case2 ¬b = ⊥-elim ( not ( λ a b → ⊥-elim ( ¬b ? ))) +-- power→⊆ : ( A t : HOD) → Power A ∋ t → t ⊆ A power→⊆ A t PA∋t tx = subst (λ k → odef A k ) &iso ( power→ A t PA∋t (subst (λ k → odef t k) (sym &iso) tx ) ) @@ -112,10 +124,10 @@ open import zfc HOD→ZFC : ZFC -HOD→ZFC = record { - ZFSet = HOD - ; _∋_ = _∋_ - ; _≈_ = _=h=_ +HOD→ZFC = record { + ZFSet = HOD + ; _∋_ = _∋_ + ; _≈_ = _=h=_ ; ∅ = od∅ ; Select = Select ; isZFC = isZFC @@ -128,9 +140,9 @@ 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 ) ] + -- ∀ X [ ∅ ∉ X → (∃ f : X → ⋃ X ) → ∀ A ∈ X ( f ( A ) ∈ A ) ] choice-func : (X : HOD ) → {x : HOD } → ¬ ( x =h= od∅ ) → ( X ∋ x ) → HOD choice-func X {x} not X∋x = minimal x not - choice : (X : HOD ) → {A : HOD } → ( X∋A : X ∋ A ) → (not : ¬ ( A =h= od∅ )) → A ∋ choice-func X not X∋A + choice : (X : HOD ) → {A : HOD } → ( X∋A : X ∋ A ) → (not : ¬ ( A =h= od∅ )) → A ∋ choice-func X not X∋A choice X {A} X∋A not = x∋minimal A not