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

comparison OD.agda @ 234:e06b76e5b682

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ac from LEM in abstract ordinal
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Tue, 13 Aug 2019 22:21:10 +0900
parents 49736efc822b
children 846e0926bb89
comparison
equal deleted inserted replaced
233:af60c40298a4 234:e06b76e5b682
31 record _==_ ( a b : OD ) : Set n where 31 record _==_ ( a b : OD ) : Set n where
32 field 32 field
33 eq→ : ∀ { x : Ordinal } → def a x → def b x 33 eq→ : ∀ { x : Ordinal } → def a x → def b x
34 eq← : ∀ { x : Ordinal } → def b x → def a x 34 eq← : ∀ { x : Ordinal } → def b x → def a x
35 35
36 id : {n : Level} {A : Set n} → A → A 36 id : {A : Set n} → A → A
37 id x = x 37 id x = x
38 38
39 eq-refl : { x : OD } → x == x 39 eq-refl : { x : OD } → x == x
40 eq-refl {x} = record { eq→ = id ; eq← = id } 40 eq-refl {x} = record { eq→ = id ; eq← = id }
41 41
191 eqo∅ : ps == od∅ → od→ord ps ≡ o∅ 191 eqo∅ : ps == od∅ → od→ord ps ≡ o∅
192 eqo∅ eq = subst (λ k → od→ord ps ≡ k) ord-od∅ ( cong (λ k → od→ord k ) (==→o≡ eq)) 192 eqo∅ eq = subst (λ k → od→ord ps ≡ k) ord-od∅ ( cong (λ k → od→ord k ) (==→o≡ eq))
193 lemma : ps ∋ minimul ps (λ eq → ¬p (eqo∅ eq)) 193 lemma : ps ∋ minimul ps (λ eq → ¬p (eqo∅ eq))
194 lemma = x∋minimul ps (λ eq → ¬p (eqo∅ eq)) 194 lemma = x∋minimul ps (λ eq → ¬p (eqo∅ eq))
195 195
196 ∋-p : ( p : Set n ) → Dec p -- assuming axiom of choice 196 decp : ( p : Set n ) → Dec p -- assuming axiom of choice
197 ∋-p p with p∨¬p p 197 decp p with p∨¬p p
198 ∋-p p | case1 x = yes x 198 decp p | case1 x = yes x
199 ∋-p p | case2 x = no x 199 decp p | case2 x = no x
200 200
201 double-neg-eilm : {A : Set n} → ¬ ¬ A → A -- we don't have this in intutionistic logic 201 double-neg-eilm : {A : Set n} → ¬ ¬ A → A -- we don't have this in intutionistic logic
202 double-neg-eilm {A} notnot with ∋-p A -- assuming axiom of choice 202 double-neg-eilm {A} notnot with decp A -- assuming axiom of choice
203 ... | yes p = p 203 ... | yes p = p
204 ... | no ¬p = ⊥-elim ( notnot ¬p ) 204 ... | no ¬p = ⊥-elim ( notnot ¬p )
205 205
206 OrdP : ( x : Ordinal ) ( y : OD ) → Dec ( Ord x ∋ y ) 206 OrdP : ( x : Ordinal ) ( y : OD ) → Dec ( Ord x ∋ y )
207 OrdP x y with trio< x (od→ord y) 207 OrdP x y with trio< x (od→ord y)
475 choice-func : (X : OD ) → {x : OD } → ¬ ( x == od∅ ) → ( X ∋ x ) → OD 475 choice-func : (X : OD ) → {x : OD } → ¬ ( x == od∅ ) → ( X ∋ x ) → OD
476 choice-func X {x} not X∋x = minimul x not 476 choice-func X {x} not X∋x = minimul x not
477 choice : (X : OD ) → {A : OD } → ( X∋A : X ∋ A ) → (not : ¬ ( A == od∅ )) → A ∋ choice-func X not X∋A 477 choice : (X : OD ) → {A : OD } → ( X∋A : X ∋ A ) → (not : ¬ ( A == od∅ )) → A ∋ choice-func X not X∋A
478 choice X {A} X∋A not = x∋minimul A not 478 choice X {A} X∋A not = x∋minimul A not
479 479
480 ---
481 --- With assuption of OD is ordered, p ∨ ( ¬ p ) <=> axiom of choice
482 ---
483 record choiced ( X : OD) : Set (suc n) where
484 field
485 a-choice : OD
486 is-in : X ∋ a-choice
487
488 choice-func' : (X : OD ) → (p∨¬p : ( p : Set (suc n)) → p ∨ ( ¬ p )) → ¬ ( X == od∅ ) → choiced X
489 choice-func' X p∨¬p not = have_to_find where
490 ψ : ( ox : Ordinal ) → Set (suc n)
491 ψ ox = (( x : Ordinal ) → x o< ox → ( ¬ def X x )) ∨ choiced X
492 lemma-ord : ( ox : Ordinal ) → ψ ox
493 lemma-ord ox = IsOrdinals.TransFinite (Ordinals.isOrdinal O) {ψ} induction ox where
494 ∋-p : (A x : OD ) → Dec ( A ∋ x )
495 ∋-p A x with p∨¬p (Lift (suc n) ( A ∋ x ))
496 ∋-p A x | case1 (lift t) = yes t
497 ∋-p A x | case2 t = no (λ x → t (lift x ))
498 ∀-imply-or : {A : Ordinal → Set n } {B : Set (suc n) }
499 → ((x : Ordinal ) → A x ∨ B) → ((x : Ordinal ) → A x) ∨ B
500 ∀-imply-or {A} {B} ∀AB with p∨¬p (Lift ( suc n ) ((x : Ordinal ) → A x))
501 ∀-imply-or {A} {B} ∀AB | case1 (lift t) = case1 t
502 ∀-imply-or {A} {B} ∀AB | case2 x = case2 (lemma (λ not → x (lift not ))) where
503 lemma : ¬ ((x : Ordinal ) → A x) → B
504 lemma not with p∨¬p B
505 lemma not | case1 b = b
506 lemma not | case2 ¬b = ⊥-elim (not (λ x → dont-orb (∀AB x) ¬b ))
507 induction : (x : Ordinal) → ((y : Ordinal) → y o< x → ψ y) → ψ x
508 induction x prev with ∋-p X ( ord→od x)
509 ... | yes p = case2 ( record { a-choice = ord→od x ; is-in = p } )
510 ... | no ¬p = lemma where
511 lemma1 : (y : Ordinal) → (y o< x → def X y → ⊥) ∨ choiced X
512 lemma1 y with ∋-p X (ord→od y)
513 lemma1 y | yes y<X = case2 ( record { a-choice = ord→od y ; is-in = y<X } )
514 lemma1 y | no ¬y<X = case1 ( λ lt y<X → ¬y<X (subst (λ k → def X k ) (sym diso) y<X ) )
515 lemma : ((y : Ordinals.ord O) → (O Ordinals.o< y) x → def X y → ⊥) ∨ choiced X
516 lemma = ∀-imply-or lemma1
517 have_to_find : choiced X
518 have_to_find with lemma-ord (od→ord X )
519 have_to_find | t = dont-or t ¬¬X∋x where
520 ¬¬X∋x : ¬ ((x : Ordinal) → x o< (od→ord X) → def X x → ⊥)
521 ¬¬X∋x nn = not record {
522 eq→ = λ {x} lt → ⊥-elim (nn x (def→o< lt) lt)
523 ; eq← = λ {x} lt → ⊥-elim ( ¬x<0 lt )
524 }
525
526
480 _,_ = ZF._,_ OD→ZF 527 _,_ = ZF._,_ OD→ZF
481 Union = ZF.Union OD→ZF 528 Union = ZF.Union OD→ZF
482 Power = ZF.Power OD→ZF 529 Power = ZF.Power OD→ZF
483 Select = ZF.Select OD→ZF 530 Select = ZF.Select OD→ZF
484 Replace = ZF.Replace OD→ZF 531 Replace = ZF.Replace OD→ZF