Mercurial > hg > Members > kono > Proof > ZF-in-agda
diff Ordinals.agda @ 346:06f10815d0b3
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author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Mon, 13 Jul 2020 19:19:02 +0900 |
parents | b1ccdbb14c92 |
children | cfecd05a4061 |
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--- a/Ordinals.agda Mon Jul 13 14:46:03 2020 +0900 +++ b/Ordinals.agda Mon Jul 13 19:19:02 2020 +0900 @@ -244,6 +244,19 @@ nexto≡ {x} | tri> ¬a ¬b c = ⊥-elim ((proj2 (proj2 next-limit)) _ (ordtrans <-osuc (proj1 next-limit)) c (λ z eq → o<¬≡ (sym eq) ((proj1 (proj2 next-limit)) _ (osuc< (sym eq))))) + not-limit-p : ( x : Ordinal ) → Dec ( ¬ ((y : Ordinal) → ¬ (x ≡ osuc y) )) + not-limit-p x = TransFinite {λ x → Dec ( ¬ ((y : Ordinal) → ¬ (x ≡ osuc y)))} ind x where + ind : (x : Ordinal) → ((y : Ordinal) → y o< x → Dec (¬ ((y₁ : Ordinal) → ¬ y ≡ osuc y₁))) → Dec (¬ ((y : Ordinal) → ¬ x ≡ osuc y)) + ind x prev with trio< o∅ x + ind x prev | tri< a ¬b ¬c = ? + ind x prev | tri≈ ¬a refl ¬c = no (λ not → not lemma) where + lemma : (y : Ordinal) → o∅ ≡ osuc y → ⊥ + lemma y refl with trio< o∅ y + lemma y refl | tri< a ¬b ¬c = o<> a <-osuc + lemma y refl | tri≈ ¬a b ¬c = o<¬≡ (sym b) <-osuc + lemma y refl | tri> ¬a ¬b c = ¬x<0 c + ind x prev | tri> ¬a ¬b c = ⊥-elim ( ¬x<0 c ) + record OrdinalSubset (maxordinal : Ordinal) : Set (suc n) where field os→ : (x : Ordinal) → x o< maxordinal → Ordinal