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

comparison src/zorn.agda @ 744:d92ad9e365b6

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Thu, 21 Jul 2022 09:03:28 +0900
parents 71556e611842
children dc208a885e0c
comparison
equal deleted inserted replaced
743:71556e611842 744:d92ad9e365b6
494 m03 = ⟪ proj1 ua , record { u = UChain.u (proj2 ua) ; u<x = m02 ; uchain = UChain.uchain (proj2 ua) } ⟫ 494 m03 = ⟪ proj1 ua , record { u = UChain.u (proj2 ua) ; u<x = m02 ; uchain = UChain.uchain (proj2 ua) } ⟫
495 m04 : odef (UnionCF A f mf ay (ZChain.supf zc) px) b 495 m04 : odef (UnionCF A f mf ay (ZChain.supf zc) px) b
496 m04 = ZChain1.is-max (prev px px<x) m03 b<px ab 496 m04 = ZChain1.is-max (prev px px<x) m03 b<px ab
497 (case2 record {x<sup = λ {z} lt → IsSup.x<sup is-sup (chain-mono2 x ( o<→≤ (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc )) o≤-refl lt) } ) a<b 497 (case2 record {x<sup = λ {z} lt → IsSup.x<sup is-sup (chain-mono2 x ( o<→≤ (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc )) o≤-refl lt) } ) a<b
498 ... | tri≈ ¬a b=px ¬c = ? -- b = px case 498 ... | tri≈ ¬a b=px ¬c = ? -- b = px case
499 ... | no lim = record { is-max = is-max ; chain-mono2 = chain-mono2 x ; fcy<sup = ? } where 499 ... | no lim = record { is-max = is-max ; chain-mono2 = chain-mono2 x ; fcy<sup = fcy<sup ; sup=u = sup=u ; order = order } where
500 fcy<sup : {u w : Ordinal} → u o< x → FClosure A f y w → w << ZChain.supf zc u 500 fcy<sup : {u w : Ordinal} → u o< x → FClosure A f y w → w << ZChain.supf zc u
501 fcy<sup {u} {w} u<x fc = ? 501 fcy<sup {u} {w} u<x fc = ZChain1.fcy<sup (prev (osuc u) (ob<x lim u<x)) <-osuc fc
502 sup=u : {b : Ordinal} {ab : odef A b} → b o< x →
503 IsSup A (UnionCF A f mf ay (ZChain.supf zc) (osuc b)) ab →
504 ZChain.supf zc b ≡ b
505 sup=u {b} {ab} b<x is-sup = ZChain1.sup=u (prev (osuc b) (ob<x lim b<x)) <-osuc is-sup
506 order : {b sup1 z1 : Ordinal} → b o< x → sup1 o< b →
507 FClosure A f (ZChain.supf zc sup1) z1 → z1 << ZChain.supf zc b
508 order {b} b<x s<b fc = ZChain1.order (prev (osuc b) (ob<x lim b<x)) <-osuc s<b fc
502 is-max : {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay (ZChain.supf zc) x) a → 509 is-max : {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay (ZChain.supf zc) x) a →
503 b o< x → (ab : odef A b) → 510 b o< x → (ab : odef A b) →
504 HasPrev A (UnionCF A f mf ay (ZChain.supf zc) x) ab f ∨ IsSup A (UnionCF A f mf ay (ZChain.supf zc) x) ab → 511 HasPrev A (UnionCF A f mf ay (ZChain.supf zc) x) ab f ∨ IsSup A (UnionCF A f mf ay (ZChain.supf zc) x) ab →
505 * a < * b → odef (UnionCF A f mf ay (ZChain.supf zc) x) b 512 * a < * b → odef (UnionCF A f mf ay (ZChain.supf zc) x) b
506 is-max {a} {b} ua b<x ab (case1 has-prev) a<b = is-max-hp x {a} {b} ua b<x ab has-prev a<b 513 is-max {a} {b} ua b<x ab (case1 has-prev) a<b = is-max-hp x {a} {b} ua b<x ab has-prev a<b
507 is-max {a} {b} ua b<x ab (case2 is-sup) a<b with IsSup.x<sup is-sup (init-uchain A f mf ay ) 514 is-max {a} {b} ua b<x ab (case2 is-sup) a<b with IsSup.x<sup is-sup (init-uchain A f mf ay )
508 ... | case1 b=y = ⊥-elim ( <-irr ( ZChain.initial zc (chain<ZA (chain-mono2 (osuc x) (o<→≤ <-osuc ) o≤-refl ua )) ) 515 ... | case1 b=y = ⊥-elim ( <-irr ( ZChain.initial zc (chain<ZA (chain-mono2 (osuc x) (o<→≤ <-osuc ) o≤-refl ua )) )
509 (subst (λ k → * a < * k ) (sym b=y) a<b ) ) 516 (subst (λ k → * a < * k ) (sym b=y) a<b ) )
510 ... | case2 y<b = chain-mono2 x (o<→≤ ob<x) o≤-refl m04 where 517 ... | case2 y<b = chain-mono2 x (o<→≤ (ob<x lim b<x) ) o≤-refl m04 where
511 y<s : y << ZChain.supf zc b 518 y<s : y << ZChain.supf zc b
512 y<s = y<s 519 y<s = y<s
513 ob<x : osuc b o< x
514 ob<x with trio< (osuc b) x
515 ... | tri< a ¬b ¬c = a
516 ... | tri≈ ¬a ob=x ¬c = ⊥-elim ( lim record { op = b ; oprev=x = ob=x } )
517 ... | tri> ¬a ¬b c = ⊥-elim ( ¬p<x<op ⟪ b<x , c ⟫ )
518 m07 : {z : Ordinal} → FClosure A f y z → z << ZChain.supf zc b 520 m07 : {z : Ordinal} → FClosure A f y z → z << ZChain.supf zc b
519 m07 {z} fc = ZChain1.fcy<sup (prev (osuc b) ob<x) <-osuc fc 521 m07 {z} fc = ZChain1.fcy<sup (prev (osuc b) (ob<x lim b<x)) <-osuc fc
520 m08 : {sup1 z1 : Ordinal} → sup1 o< b → FClosure A f (ZChain.supf zc sup1) z1 → z1 << ZChain.supf zc b 522 m08 : {sup1 z1 : Ordinal} → sup1 o< b → FClosure A f (ZChain.supf zc sup1) z1 → z1 << ZChain.supf zc b
521 m08 {sup1} {z1} s<b fc = ZChain1.order (prev (osuc b) ob<x) <-osuc s<b fc 523 m08 {sup1} {z1} s<b fc = ZChain1.order (prev (osuc b) (ob<x lim b<x) ) <-osuc s<b fc
522 m05 : b ≡ ZChain.supf zc b 524 m05 : b ≡ ZChain.supf zc b
523 m05 = sym (ZChain1.sup=u (prev (osuc b) ob<x) {_} {ab} <-osuc 525 m05 = sym (ZChain1.sup=u (prev (osuc b) (ob<x lim b<x)) {_} {ab} <-osuc
524 record { x<sup = λ lt → IsSup.x<sup is-sup (chain-mono2 x (o<→≤ ob<x) o≤-refl lt )} ) -- ZChain on x 526 record { x<sup = λ lt → IsSup.x<sup is-sup (chain-mono2 x (o<→≤ (ob<x lim b<x)) o≤-refl lt )} ) -- ZChain on x
525 m06 : ChainP A f mf ay (ZChain.supf zc) b b 527 m06 : ChainP A f mf ay (ZChain.supf zc) b b
526 m06 = record { fcy<sup = m07 ; csupz = subst (λ k → FClosure A f k b ) m05 (init ab) ; order = m08 ; y<s = y<s } 528 m06 = record { fcy<sup = m07 ; csupz = subst (λ k → FClosure A f k b ) m05 (init ab) ; order = m08 ; y<s = y<s
529 ; supfu=u = sym m05 }
527 m04 : odef (UnionCF A f mf ay (ZChain.supf zc) (osuc b)) b 530 m04 : odef (UnionCF A f mf ay (ZChain.supf zc) (osuc b)) b
528 m04 = ⟪ ab , record { u = b ; u<x = case1 <-osuc ; uchain = ch-is-sup m06 (subst (λ k → FClosure A f k b) m05 (init ab)) } ⟫ 531 m04 = ⟪ ab , record { u = b ; u<x = case1 <-osuc ; uchain = ch-is-sup m06 (subst (λ k → FClosure A f k b) m05 (init ab)) } ⟫
529 532
530 --- 533 ---
531 --- the maximum chain has fix point of any ≤-monotonic function 534 --- the maximum chain has fix point of any ≤-monotonic function