comparison ODC.agda @ 331:12071f79f3cf

HOD done
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sun, 05 Jul 2020 16:56:21 +0900
parents 0faa7120e4b5
children 7f919d6b045b
comparison
equal deleted inserted replaced
330:0faa7120e4b5 331:12071f79f3cf
21 open OD._==_ 21 open OD._==_
22 open ODAxiom odAxiom 22 open ODAxiom odAxiom
23 23
24 open HOD 24 open HOD
25 25
26 open _∧_
27
26 _=h=_ : (x y : HOD) → Set n 28 _=h=_ : (x y : HOD) → Set n
27 x =h= y = od x == od y 29 x =h= y = od x == od y
28 30
29 postulate 31 postulate
30 -- mimimul and x∋minimal is an Axiom of choice 32 -- mimimul and x∋minimal is an Axiom of choice
38 -- 40 --
39 -- Axiom of choice in intutionistic logic implies the exclude middle 41 -- Axiom of choice in intutionistic logic implies the exclude middle
40 -- https://plato.stanford.edu/entries/axiom-choice/#AxiChoLog 42 -- https://plato.stanford.edu/entries/axiom-choice/#AxiChoLog
41 -- 43 --
42 44
43 ppp : { p : Set n } { a : HOD } → record { od = record { def = λ x → p } ; odmax = {!!} ; <odmax = {!!} } ∋ a → p 45 pred-od : ( p : Set n ) → HOD
44 ppp {p} {a} d = d 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 }
45 48
46 -- p∨¬p : ( p : Set n ) → p ∨ ( ¬ p ) -- assuming axiom of choice 49 ppp : { p : Set n } { a : HOD } → pred-od p ∋ a → p
47 -- p∨¬p p with is-o∅ ( od→ord ( record { odef = λ x → p } )) 50 ppp {p} {a} d = proj2 d
48 -- p∨¬p p | yes eq = case2 (¬p eq) where
49 -- ps = record { odef = λ x → p }
50 -- lemma : ps =h= od∅ → p → ⊥
51 -- lemma eq p0 = ¬x<0 {od→ord ps} (eq→ eq p0 )
52 -- ¬p : (od→ord ps ≡ o∅) → p → ⊥
53 -- ¬p eq = lemma ( subst₂ (λ j k → j =h= k ) oiso o∅≡od∅ ( o≡→== eq ))
54 -- p∨¬p p | no ¬p = case1 (ppp {p} {minimal ps (λ eq → ¬p (eqo∅ eq))} lemma) where
55 -- ps = record { odef = λ x → 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 ∋ minimal ps (λ eq → ¬p (eqo∅ eq))
59 -- lemma = x∋minimal ps (λ eq → ¬p (eqo∅ eq))
60 51
61 postulate 52 p∨¬p : ( p : Set n ) → p ∨ ( ¬ p ) -- assuming axiom of choice
62 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))
63 68
64 decp : ( p : Set n ) → Dec p -- assuming axiom of choice 69 decp : ( p : Set n ) → Dec p -- assuming axiom of choice
65 decp p with p∨¬p p 70 decp p with p∨¬p p
66 decp p | case1 x = yes x 71 decp p | case1 x = yes x
67 decp p | case2 x = no x 72 decp p | case2 x = no x