Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate ODC.agda @ 358:811152bf2f47
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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| date | Tue, 14 Jul 2020 12:39:21 +0900 |
| parents | 12071f79f3cf |
| children | 7f919d6b045b |
| rev | line source |
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| 16 | 1 open import Level |
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2 open import Ordinals |
| 276 | 3 module ODC {n : Level } (O : Ordinals {n} ) where |
| 3 | 4 |
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5 open import zf |
| 23 | 6 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) |
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7 open import Relation.Binary.PropositionalEquality |
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8 open import Data.Nat.Properties |
| 6 | 9 open import Data.Empty |
| 10 open import Relation.Nullary | |
| 11 open import Relation.Binary | |
| 12 open import Relation.Binary.Core | |
| 13 | |
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14 open import logic |
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15 open import nat |
| 276 | 16 import OD |
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17 |
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18 open inOrdinal O |
| 276 | 19 open OD O |
| 20 open OD.OD | |
| 21 open OD._==_ | |
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22 open ODAxiom odAxiom |
| 258 | 23 |
| 329 | 24 open HOD |
| 25 | |
| 331 | 26 open _∧_ |
| 27 | |
| 329 | 28 _=h=_ : (x y : HOD) → Set n |
| 29 x =h= y = od x == od y | |
| 30 | |
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31 postulate |
| 258 | 32 -- mimimul and x∋minimal is an Axiom of choice |
| 329 | 33 minimal : (x : HOD ) → ¬ (x =h= od∅ )→ HOD |
| 34 -- this should be ¬ (x =h= od∅ )→ ∃ ox → x ∋ Ord ox ( minimum of x ) | |
| 35 x∋minimal : (x : HOD ) → ( ne : ¬ (x =h= od∅ ) ) → odef x ( od→ord ( minimal x ne ) ) | |
| 330 | 36 -- minimality (may proved by ε-induction with LEM) |
| 329 | 37 minimal-1 : (x : HOD ) → ( ne : ¬ (x =h= od∅ ) ) → (y : HOD ) → ¬ ( odef (minimal x ne) (od→ord y)) ∧ (odef x (od→ord y) ) |
| 258 | 38 |
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39 |
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40 -- |
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41 -- Axiom of choice in intutionistic logic implies the exclude middle |
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42 -- https://plato.stanford.edu/entries/axiom-choice/#AxiChoLog |
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43 -- |
| 257 | 44 |
| 331 | 45 pred-od : ( p : Set n ) → HOD |
| 46 pred-od p = record { od = record { def = λ x → (x ≡ o∅) ∧ p } ; | |
| 47 odmax = osuc o∅; <odmax = λ x → subst (λ k → k o< osuc o∅) (sym (proj1 x)) <-osuc } | |
| 48 | |
| 49 ppp : { p : Set n } { a : HOD } → pred-od p ∋ a → p | |
| 50 ppp {p} {a} d = proj2 d | |
| 257 | 51 |
| 331 | 52 p∨¬p : ( p : Set n ) → p ∨ ( ¬ p ) -- assuming axiom of choice |
| 53 p∨¬p p with is-o∅ ( od→ord (pred-od p )) | |
| 54 p∨¬p p | yes eq = case2 (¬p eq) where | |
| 55 ps = pred-od p | |
| 56 eqo∅ : ps =h= od∅ → od→ord ps ≡ o∅ | |
| 57 eqo∅ eq = subst (λ k → od→ord ps ≡ k) ord-od∅ ( cong (λ k → od→ord k ) (==→o≡ eq)) | |
| 58 lemma : ps =h= od∅ → p → ⊥ | |
| 59 lemma eq p0 = ¬x<0 {od→ord ps} (eq→ eq record { proj1 = eqo∅ eq ; proj2 = p0 } ) | |
| 60 ¬p : (od→ord ps ≡ o∅) → p → ⊥ | |
| 61 ¬p eq = lemma ( subst₂ (λ j k → j =h= k ) oiso o∅≡od∅ ( o≡→== eq )) | |
| 62 p∨¬p p | no ¬p = case1 (ppp {p} {minimal ps (λ eq → ¬p (eqo∅ eq))} lemma) where | |
| 63 ps = pred-od p | |
| 64 eqo∅ : ps =h= od∅ → od→ord ps ≡ o∅ | |
| 65 eqo∅ eq = subst (λ k → od→ord ps ≡ k) ord-od∅ ( cong (λ k → od→ord k ) (==→o≡ eq)) | |
| 66 lemma : ps ∋ minimal ps (λ eq → ¬p (eqo∅ eq)) | |
| 67 lemma = x∋minimal ps (λ eq → ¬p (eqo∅ eq)) | |
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68 |
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69 decp : ( p : Set n ) → Dec p -- assuming axiom of choice |
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70 decp p with p∨¬p p |
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71 decp p | case1 x = yes x |
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72 decp p | case2 x = no x |
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73 |
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74 double-neg-eilm : {A : Set n} → ¬ ¬ A → A -- we don't have this in intutionistic logic |
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75 double-neg-eilm {A} notnot with decp A -- assuming axiom of choice |
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76 ... | yes p = p |
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77 ... | no ¬p = ⊥-elim ( notnot ¬p ) |
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78 |
| 329 | 79 OrdP : ( x : Ordinal ) ( y : HOD ) → Dec ( Ord x ∋ y ) |
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80 OrdP x y with trio< x (od→ord y) |
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81 OrdP x y | tri< a ¬b ¬c = no ¬c |
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82 OrdP x y | tri≈ ¬a refl ¬c = no ( o<¬≡ refl ) |
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83 OrdP x y | tri> ¬a ¬b c = yes c |
| 119 | 84 |
| 276 | 85 open import zfc |
| 190 | 86 |
| 329 | 87 HOD→ZFC : ZFC |
| 88 HOD→ZFC = record { | |
| 89 ZFSet = HOD | |
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90 ; _∋_ = _∋_ |
| 329 | 91 ; _≈_ = _=h=_ |
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92 ; ∅ = od∅ |
| 28 | 93 ; Select = Select |
| 276 | 94 ; isZFC = isZFC |
| 28 | 95 } where |
| 276 | 96 -- infixr 200 _∈_ |
| 96 | 97 -- infixr 230 _∩_ _∪_ |
| 329 | 98 isZFC : IsZFC (HOD ) _∋_ _=h=_ od∅ Select |
| 276 | 99 isZFC = record { |
| 100 choice-func = choice-func ; | |
| 101 choice = choice | |
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102 } where |
| 258 | 103 -- Axiom of choice ( is equivalent to the existence of minimal in our case ) |
| 162 | 104 -- ∀ X [ ∅ ∉ X → (∃ f : X → ⋃ X ) → ∀ A ∈ X ( f ( A ) ∈ A ) ] |
| 329 | 105 choice-func : (X : HOD ) → {x : HOD } → ¬ ( x =h= od∅ ) → ( X ∋ x ) → HOD |
| 258 | 106 choice-func X {x} not X∋x = minimal x not |
| 329 | 107 choice : (X : HOD ) → {A : HOD } → ( X∋A : X ∋ A ) → (not : ¬ ( A =h= od∅ )) → A ∋ choice-func X not X∋A |
| 258 | 108 choice X {A} X∋A not = x∋minimal A not |
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109 |