Mercurial > hg > Members > kono > Proof > ZF-in-agda

comparison LEMC.agda @ 396:8c092c042093

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Mon, 27 Jul 2020 15:11:54 +0900
parents 19687f3304c9
children 44a484f17f26
comparison
equal deleted inserted replaced
395:77c6123f49ee 396:8c092c042093
43 ... | no ¬p = ⊥-elim ( notnot ¬p ) 43 ... | no ¬p = ⊥-elim ( notnot ¬p )
44 44
45 power→⊆ : ( A t : HOD) → Power A ∋ t → t ⊆ A 45 power→⊆ : ( A t : HOD) → Power A ∋ t → t ⊆ A
46 power→⊆ A t PA∋t = record { incl = λ {x} t∋x → double-neg-eilm (λ not → t1 t∋x (λ x → not x) ) } where 46 power→⊆ A t PA∋t = record { incl = λ {x} t∋x → double-neg-eilm (λ not → t1 t∋x (λ x → not x) ) } where
47 t1 : {x : HOD } → t ∋ x → ¬ ¬ (A ∋ x) 47 t1 : {x : HOD } → t ∋ x → ¬ ¬ (A ∋ x)
48 t1 = zf.IsZF.power→ isZF A t PA∋t 48 t1 = power→ A t PA∋t
49 49
50 --- With assuption of HOD is ordered, p ∨ ( ¬ p ) <=> axiom of choice 50 --- With assuption of HOD is ordered, p ∨ ( ¬ p ) <=> axiom of choice
51 --- 51 ---
52 record choiced ( X : Ordinal ) : Set n where 52 record choiced ( X : Ordinal ) : Set n where
53 field 53 field
108 ... | yes p = case2 ( record { a-choice = x ; is-in = ∋oo p } ) 108 ... | yes p = case2 ( record { a-choice = x ; is-in = ∋oo p } )
109 ... | no ¬p = lemma where 109 ... | no ¬p = lemma where
110 lemma1 : (y : Ordinal) → (y o< x → odef X y → ⊥) ∨ choiced (od→ord X) 110 lemma1 : (y : Ordinal) → (y o< x → odef X y → ⊥) ∨ choiced (od→ord X)
111 lemma1 y with ∋-p X (ord→od y) 111 lemma1 y with ∋-p X (ord→od y)
112 lemma1 y | yes y<X = case2 ( record { a-choice = y ; is-in = ∋oo y<X } ) 112 lemma1 y | yes y<X = case2 ( record { a-choice = y ; is-in = ∋oo y<X } )
113 lemma1 y | no ¬y<X = case1 ( λ lt y<X → ¬y<X (subst (λ k → odef X k ) (sym diso) y<X ) ) 113 lemma1 y | no ¬y<X = case1 ( λ lt y<X → ¬y<X (d→∋ X y<X) )
114 lemma : ((y : Ordinals.ord O) → (O Ordinals.o< y) x → odef X y → ⊥) ∨ choiced (od→ord X) 114 lemma : ((y : Ordinals.ord O) → (O Ordinals.o< y) x → odef X y → ⊥) ∨ choiced (od→ord X)
115 lemma = ∀-imply-or lemma1 115 lemma = ∀-imply-or lemma1
116 have_to_find : choiced (od→ord X) 116 have_to_find : choiced (od→ord X)
117 have_to_find = dont-or (lemma-ord (od→ord X )) ¬¬X∋x where 117 have_to_find = dont-or (lemma-ord (od→ord X )) ¬¬X∋x where
118 ¬¬X∋x : ¬ ((x : Ordinal) → x o< (od→ord X) → odef X x → ⊥) 118 ¬¬X∋x : ¬ ((x : Ordinal) → x o< (od→ord X) → odef X x → ⊥)
146 p : HOD 146 p : HOD
147 p = record { od = record { def = λ y1 → odef x y1 ∧ odef u y1 } ; odmax = omin (odmax x) (odmax u) ; <odmax = lemma } where 147 p = record { od = record { def = λ y1 → odef x y1 ∧ odef u y1 } ; odmax = omin (odmax x) (odmax u) ; <odmax = lemma } where
148 lemma : {y : Ordinal} → OD.def (od x) y ∧ OD.def (od u) y → y o< omin (odmax x) (odmax u) 148 lemma : {y : Ordinal} → OD.def (od x) y ∧ OD.def (od u) y → y o< omin (odmax x) (odmax u)
149 lemma {y} lt = min1 (<odmax x (proj1 lt)) (<odmax u (proj2 lt)) 149 lemma {y} lt = min1 (<odmax x (proj1 lt)) (<odmax u (proj2 lt))
150 np : ¬ (p =h= od∅) 150 np : ¬ (p =h= od∅)
151 np p∅ = NP (λ y p∋y → ∅< {p} {_} (subst (λ k → odef p k) (sym diso) p∋y) p∅ ) 151 np p∅ = NP (λ y p∋y → ∅< {p} {_} (d→∋ p p∋y) p∅ )
152 y1choice : choiced (od→ord p) 152 y1choice : choiced (od→ord p)
153 y1choice = choice-func p np 153 y1choice = choice-func p np
154 y1 : HOD 154 y1 : HOD
155 y1 = ord→od (a-choice y1choice) 155 y1 = ord→od (a-choice y1choice)
156 y1prop : (x ∋ y1) ∧ (u ∋ y1) 156 y1prop : (x ∋ y1) ∧ (u ∋ y1)