Mercurial > hg > Members > kono > Proof > ZF-in-agda
diff LEMC.agda @ 396:8c092c042093
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author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Mon, 27 Jul 2020 15:11:54 +0900 |
parents | 19687f3304c9 |
children | 44a484f17f26 |
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--- a/LEMC.agda Mon Jul 27 09:29:41 2020 +0900 +++ b/LEMC.agda Mon Jul 27 15:11:54 2020 +0900 @@ -45,7 +45,7 @@ power→⊆ : ( A t : HOD) → Power A ∋ t → t ⊆ A power→⊆ A t PA∋t = record { incl = λ {x} t∋x → double-neg-eilm (λ not → t1 t∋x (λ x → not x) ) } where t1 : {x : HOD } → t ∋ x → ¬ ¬ (A ∋ x) - t1 = zf.IsZF.power→ isZF A t PA∋t + t1 = power→ A t PA∋t --- With assuption of HOD is ordered, p ∨ ( ¬ p ) <=> axiom of choice --- @@ -110,7 +110,7 @@ lemma1 : (y : Ordinal) → (y o< x → odef X y → ⊥) ∨ choiced (od→ord X) lemma1 y with ∋-p X (ord→od y) lemma1 y | yes y<X = case2 ( record { a-choice = y ; is-in = ∋oo y<X } ) - lemma1 y | no ¬y<X = case1 ( λ lt y<X → ¬y<X (subst (λ k → odef X k ) (sym diso) y<X ) ) + lemma1 y | no ¬y<X = case1 ( λ lt y<X → ¬y<X (d→∋ X y<X) ) lemma : ((y : Ordinals.ord O) → (O Ordinals.o< y) x → odef X y → ⊥) ∨ choiced (od→ord X) lemma = ∀-imply-or lemma1 have_to_find : choiced (od→ord X) @@ -148,7 +148,7 @@ lemma : {y : Ordinal} → OD.def (od x) y ∧ OD.def (od u) y → y o< omin (odmax x) (odmax u) lemma {y} lt = min1 (<odmax x (proj1 lt)) (<odmax u (proj2 lt)) np : ¬ (p =h= od∅) - np p∅ = NP (λ y p∋y → ∅< {p} {_} (subst (λ k → odef p k) (sym diso) p∋y) p∅ ) + np p∅ = NP (λ y p∋y → ∅< {p} {_} (d→∋ p p∋y) p∅ ) y1choice : choiced (od→ord p) y1choice = choice-func p np y1 : HOD