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

comparison OD.agda @ 206:684d70f1f26b

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Wed, 31 Jul 2019 17:48:08 +0900
parents 61ff37d51230
children 3e4eb4da1453
comparison
equal deleted inserted replaced
205:61ff37d51230 206:684d70f1f26b
562 record choiced {n : Level} ( X : OD {suc n}) : Set (suc (suc n)) where 562 record choiced {n : Level} ( X : OD {suc n}) : Set (suc (suc n)) where
563 field 563 field
564 a-choice : OD {suc n} 564 a-choice : OD {suc n}
565 is-in : X ∋ a-choice 565 is-in : X ∋ a-choice
566 choice-func' : (X : OD {suc n} ) → (∋-p : (A x : OD {suc n} ) → Dec ( A ∋ x ) ) → ¬ ( X == od∅ ) → choiced X 566 choice-func' : (X : OD {suc n} ) → (∋-p : (A x : OD {suc n} ) → Dec ( A ∋ x ) ) → ¬ ( X == od∅ ) → choiced X
567 choice-func' X ∋-p not = lemma0 567 choice-func' X ∋-p not = have_to_find
568 where 568 where
569 <-not : {X : OD {suc n}} → ( ox : Ordinal {suc n}) → Set (suc n) 569 <-not : {X : OD {suc n}} → ( ox : Ordinal {suc n}) → Set (suc n)
570 <-not {X} ox = ( y : Ordinal {suc n}) → y o< ox → ¬ (X ∋ (ord→od y)) 570 <-not {X} ox = ( y : Ordinal {suc n}) → y o< ox → ¬ (X ∋ (ord→od y))
571 lemma-ord : ( ox : Ordinal {suc n} ) → <-not {X} ox ∨ choiced X 571 lemma-ord : ( ox : Ordinal {suc n} ) → <-not {X} ox ∨ choiced X
572 lemma-ord ox = TransFinite {n} {suc (suc n)} {λ ox → <-not {X} ox ∨ choiced X } caseΦ caseOSuc ox where 572 lemma-ord ox = TransFinite {n} {suc (suc n)} {λ ox → <-not {X} ox ∨ choiced X } caseΦ caseOSuc ox where
573 caseΦ : (lx : Nat) → ((x : Ordinal) → x o< ordinal lx (Φ lx) → <-not {X} x ∨ choiced X) → 573 caseΦ : (lx : Nat) → ((x : Ordinal) → x o< ordinal lx (Φ lx) → <-not {X} x ∨ choiced X) →
574 <-not {X} (record { lv = lx ; ord = Φ lx }) ∨ choiced X 574 <-not {X} (record { lv = lx ; ord = Φ lx }) ∨ choiced X
575 caseΦ lx prev with ∋-p X ( ord→od (ordinal lx (Φ lx) )) 575 caseΦ lx prev with ∋-p X ( ord→od (ordinal lx (Φ lx) ))
576 caseΦ lx prev | yes p = case2 ( record { a-choice = ord→od (ordinal lx (Φ lx)) ; is-in = p } ) 576 caseΦ lx prev | yes p = case2 ( record { a-choice = ord→od (ordinal lx (Φ lx)) ; is-in = p } )
577 caseΦ lx prev | no ¬p = {!!} 577 caseΦ lx prev | no ¬p = lemma (ordinal lx (Φ lx)) <-osuc where
578 lemma : (x : Ordinal {suc n}) → x o< osuc (ordinal lx (Φ lx))
579 → ((y : Ordinal) → (Suc (lv y) ≤ lx) ∨ (ord y d< Φ lx) → def X (od→ord (ord→od y)) → ⊥) ∨ choiced X
580 lemma x lt with osuc-≡< lt
581 lemma x lt | case1 refl = case1 ?
582 lemma x lt | case2 lt1 with prev x lt1
583 lemma x lt | case2 lt1 | case1 lt2 = case1 {!!}
584 lemma x lt | case2 lt1 | case2 found = case2 found
578 caseOSuc : (lx : Nat) (x : OrdinalD lx) → (<-not {X} (record { lv = lx ; ord = x }) ∨ choiced X) → 585 caseOSuc : (lx : Nat) (x : OrdinalD lx) → (<-not {X} (record { lv = lx ; ord = x }) ∨ choiced X) →
579 <-not {X} (record { lv = lx ; ord = OSuc lx x }) ∨ choiced X 586 <-not {X} (record { lv = lx ; ord = OSuc lx x }) ∨ choiced X
580 caseOSuc lx x prev with ∋-p X (ord→od record { lv = lx ; ord = x } ) 587 caseOSuc lx x prev with ∋-p X (ord→od record { lv = lx ; ord = x } )
581 caseOSuc lx x prev | yes p = case2 (record { a-choice = ord→od record { lv = lx ; ord = x } ; is-in = p }) 588 caseOSuc lx x prev | yes p = case2 (record { a-choice = ord→od record { lv = lx ; ord = x } ; is-in = p })
582 caseOSuc lx x (case1 not_found) | no ¬p = case1 lemma where 589 caseOSuc lx x (case1 not_found) | no ¬p = case1 lemma where
586 lemma y lt | tri≈ ¬a refl ¬c = ¬p 593 lemma y lt | tri≈ ¬a refl ¬c = ¬p
587 lemma y lt | tri> ¬a ¬b c with osuc-≡< lt 594 lemma y lt | tri> ¬a ¬b c with osuc-≡< lt
588 lemma y lt | tri> ¬a ¬b c | case1 refl = ⊥-elim ( ¬b refl ) 595 lemma y lt | tri> ¬a ¬b c | case1 refl = ⊥-elim ( ¬b refl )
589 lemma y lt | tri> ¬a ¬b c | case2 lt1 = ⊥-elim (o<> c lt1 ) 596 lemma y lt | tri> ¬a ¬b c | case2 lt1 = ⊥-elim (o<> c lt1 )
590 caseOSuc lx x (case2 found) | no ¬p = case2 found 597 caseOSuc lx x (case2 found) | no ¬p = case2 found
591 lemma0 : choiced X 598 have_to_find : choiced X
592 lemma0 with lemma-ord (od→ord X ) 599 have_to_find with lemma-ord (od→ord X )
593 lemma0 | case1 not_found = ⊥-elim ( not ( record { 600 have_to_find | case1 not_found = ⊥-elim ( not ( record {
594 eq→ = λ {x} lt → ⊥-elim (not_found x (def→o< lt) (def-subst {suc n} {_} {_} {X} {_} lt refl (sym diso ))) ; 601 eq→ = λ {x} lt → ⊥-elim (not_found x (def→o< lt) (def-subst {suc n} {_} {_} {X} {_} lt refl (sym diso ))) ;
595 eq← = λ lt → ⊥-elim (¬x<0 lt) } ) ) 602 eq← = λ lt → ⊥-elim (¬x<0 lt) } ) )
596 lemma0 | case2 found = found 603 have_to_find | case2 found = found
597 604