Mercurial > hg > Members > kono > Proof > ZF-in-agda
comparison OD.agda @ 198:8589660ee388
another approach
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Mon, 29 Jul 2019 20:01:18 +0900 |
parents | b114cf5b9130 |
children | 68eecbb011ef |
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197:b114cf5b9130 | 198:8589660ee388 |
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561 record choiced {n : Level} ( X : OD {suc n}) : Set (suc (suc n)) where | 561 record choiced {n : Level} ( X : OD {suc n}) : Set (suc (suc n)) where |
562 field | 562 field |
563 a-choice : OD {suc n} | 563 a-choice : OD {suc n} |
564 is-in : X ∋ a-choice | 564 is-in : X ∋ a-choice |
565 choice-func' : (X : OD {suc n} ) → (∋-p : (A x : OD {suc n} ) → Dec ( A ∋ x ) ) → ¬ ( X == od∅ ) → choiced X | 565 choice-func' : (X : OD {suc n} ) → (∋-p : (A x : OD {suc n} ) → Dec ( A ∋ x ) ) → ¬ ( X == od∅ ) → choiced X |
566 choice-func' X ∋-p not = lemma-ord (od→ord X) lemma-init | 566 choice-func' X ∋-p not = proj2 ( lemma-ord (od→ord X) ) |
567 where | 567 where |
568 <-not : {X : OD {suc n}} → ( ox : Ordinal {suc n}) → Set (suc n) | 568 <-not : {X : OD {suc n}} → ( ox : Ordinal {suc n}) → Set (suc n) |
569 <-not {X} ox = ( y : Ordinal {suc n}) → ox o< osuc y → ¬ (X ∋ (ord→od y)) | 569 <-not {X} ox = ( y : Ordinal {suc n}) → ox o< osuc y → ¬ (X ∋ (ord→od y)) |
570 lemma-init : (y : Ordinal) → od→ord X o< osuc y → ¬ (X ∋ ord→od y) | 570 lemma-init : (y : Ordinal) → od→ord X o< osuc y → ¬ (X ∋ ord→od y) |
571 lemma-init y lt lt2 with osuc-≡< lt | 571 lemma-init y lt lt2 with osuc-≡< lt |
572 lemma-init y lt lt2 | case1 refl = o<¬≡ refl ( o<-subst (c<→o< {suc n} {_} {X} lt2) diso refl ) | 572 lemma-init y lt lt2 | case1 refl = o<¬≡ refl ( o<-subst (c<→o< {suc n} {_} {X} lt2) diso refl ) |
573 lemma-init y lt lt2 | case2 lt1 = o<> lt1 ( o<-subst (c<→o< lt2) diso refl ) | 573 lemma-init y lt lt2 | case2 lt1 = o<> lt1 ( o<-subst (c<→o< lt2) diso refl ) |
574 lemma-ord : ( ox : Ordinal {suc n} ) → <-not {X} ox → choiced X | 574 lemma-ord : ( ox : Ordinal {suc n} ) → <-not {X} ox ∧ choiced X |
575 lemma-ord ox = TransFinite {suc n} {suc (suc n)} {λ ox → <-not {X} ox → choiced X } caseΦ caseOSuc ox where | 575 lemma-ord ox = lemma1 (lv ox) (ord ox) where |
576 caseΦ : (lx : Nat) → <-not {X} (record { lv = lx ; ord = Φ lx }) → choiced X | 576 lemma1 : (lx : Nat) ( ox : OrdinalD lx ) → <-not {X} record {lv = lx ; ord = ox} ∧ choiced X |
577 caseΦ Zero n = ⊥-elim ( not {!!} ) | 577 lemma1 lx ox with ∋-p X (ord→od record { lv = lx ; ord = ox}) |
578 caseΦ (Suc lx) n = caseΦ lx ( λ y lt → {!!} ) | 578 ... | yes p = record { proj2 = record { a-choice = ord→od record { lv = lx ; ord = ox} ; is-in = p }; proj1 = {!!} } |
579 caseOSuc : (lx : Nat) (x : OrdinalD lx) → (<-not {X} (record { lv = lx ; ord = x }) → choiced X) → | 579 lemma1 Zero (Φ 0) | no ¬p = {!!} |
580 <-not {X} (record { lv = lx ; ord = OSuc lx x }) → choiced X | 580 lemma1 lx (OSuc lx ox) | no ¬p = record { proj2 = proj2 (lemma1 lx ox) ; proj1 = {!!} } |
581 caseOSuc lx x prev with ∋-p X (ord→od record { lv = lx ; ord = x } ) | 581 lemma1 (Suc lx) (Φ (Suc lx)) | no ¬p = record { proj2 = proj2 ( lemma1 lx (Φ lx) ) ; proj1 = {!!} } where |
582 caseOSuc lx x prev | yes p = λ _ → record { a-choice = ord→od record { lv = lx ; ord = x } ; is-in = p } | 582 -- not : ( y : Ordinal {suc n}) → (record { lv = Suc lx ; ord = Φ (Suc lx) }) o< osuc y → ¬ (X ∋ (ord→od y)) |
583 caseOSuc lx x prev | no ¬p = λ px → prev ( λ y lt → {!!} ) | 583 -- we also have lemma1 lx any |
584 | 584 lemmaΦ : ( y : Ordinal {suc n}) → (record { lv = lx ; ord = Φ lx }) o< osuc y → ¬ (X ∋ (ord→od y)) |
585 lemmaΦ y lt with trio< (record { lv = Suc lx ; ord = Φ (Suc lx) }) (osuc y ) | |
586 lemmaΦ y lt | tri< a ¬b ¬c = {!!} -- not y a | |
587 -- record { lv = lx ; ord = Φ lx } o< osuc y o< record { lv = Suc lx ; ord = Φ (Suc lx) } | |
588 lemmaΦ y lt | tri> ¬a ¬b c = {!!} | |
589 |