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comparison src/zorn.agda @ 943:df68bb333cd0

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sun, 30 Oct 2022 18:19:33 +0900
parents d396af76c559
children e1d727251e25
comparison
equal deleted inserted replaced
942:d396af76c559 943:df68bb333cd0
1422 sp1 : SUP A (ZChain.chain zc) 1422 sp1 : SUP A (ZChain.chain zc)
1423 sp1 = sp0 (cf nmx) (cf-is-≤-monotonic nmx) as0 zc 1423 sp1 = sp0 (cf nmx) (cf-is-≤-monotonic nmx) as0 zc
1424 c : Ordinal 1424 c : Ordinal
1425 c = & ( SUP.sup sp1 ) 1425 c = & ( SUP.sup sp1 )
1426 mc = MinSUP.sup msp1 1426 mc = MinSUP.sup msp1
1427 mc<A : mc o< & A
1428 mc<A = ∈∧P→o< ⟪ MinSUP.asm msp1 , lift true ⟫
1427 c=mc : c ≡ mc 1429 c=mc : c ≡ mc
1428 c=mc = &iso 1430 c=mc = &iso
1429 z20 : mc << cf nmx mc 1431 z20 : mc << cf nmx mc
1430 z20 = proj1 (cf-is-<-monotonic nmx mc (MinSUP.asm msp1) ) 1432 z20 = proj1 (cf-is-<-monotonic nmx mc (MinSUP.asm msp1) )
1431 asc : odef A (supf mc) 1433 asc : odef A (supf mc)
1438 msup : MinSUP A (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 supf d) 1440 msup : MinSUP A (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 supf d)
1439 msup = ZChain.minsup zc (o<→≤ d<A) 1441 msup = ZChain.minsup zc (o<→≤ d<A)
1440 sd=ms : supf d ≡ MinSUP.sup ( ZChain.minsup zc (o<→≤ d<A) ) 1442 sd=ms : supf d ≡ MinSUP.sup ( ZChain.minsup zc (o<→≤ d<A) )
1441 sd=ms = ZChain.supf-is-minsup zc (o<→≤ d<A) 1443 sd=ms = ZChain.supf-is-minsup zc (o<→≤ d<A)
1442 1444
1443 sc<<d : {mc : Ordinal } → {asc : odef A (supf mc)} → (spd : MinSUP A (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc )) 1445 sc<<d : {mc : Ordinal } → (asc : odef A (supf mc)) → (spd : MinSUP A (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc ))
1444 → supf mc << MinSUP.sup spd 1446 → supf mc << MinSUP.sup spd
1445 sc<<d {mc} {asc} spd = z25 where 1447 sc<<d {mc} asc spd = z25 where
1446 d1 : Ordinal 1448 d1 : Ordinal
1447 d1 = MinSUP.sup spd -- supf d1 ≡ d 1449 d1 = MinSUP.sup spd -- supf d1 ≡ d
1448 z24 : (supf mc ≡ d1) ∨ ( supf mc << d1 ) 1450 z24 : (supf mc ≡ d1) ∨ ( supf mc << d1 )
1449 z24 = MinSUP.x<sup spd (init asc refl) 1451 z24 = MinSUP.x<sup spd (init asc refl)
1450 -- 1452 --
1461 z29 : (* (cf nmx d1) ≡ * d1) ∨ (* (cf nmx d1) < * d1) 1463 z29 : (* (cf nmx d1) ≡ * d1) ∨ (* (cf nmx d1) < * d1)
1462 z29 with z32 1464 z29 with z32
1463 ... | case1 eq1 = case1 (cong (*) (trans (cong (cf nmx) (sym eq)) eq1) ) 1465 ... | case1 eq1 = case1 (cong (*) (trans (cong (cf nmx) (sym eq)) eq1) )
1464 ... | case2 lt = case2 (subst (λ k → * k < * d1 ) (cong (cf nmx) eq) lt) 1466 ... | case2 lt = case2 (subst (λ k → * k < * d1 ) (cong (cf nmx) eq) lt)
1465 1467
1468 smc<<d : supf mc << d
1469 smc<<d = sc<<d asc spd
1470
1471 sz<<c : {z : Ordinal } → z o< & A → supf z <= mc
1472 sz<<c z<A = MinSUP.x<sup msp1 (ZChain.csupf zc (z09 (ZChain.asupf zc) ))
1473
1474 sc=c : supf mc ≡ mc
1475 sc=c = ZChain.sup=u zc (MinSUP.asm msp1) (o<→≤ (∈∧P→o< ⟪ MinSUP.asm msp1 , lift true ⟫ )) ⟪ is-sup , not-hasprev ⟫ where
1476 is-sup : IsSup A (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 (ZChain.supf zc) mc) (MinSUP.asm msp1)
1477 is-sup = record { x<sup = λ zy → MinSUP.x<sup msp1 (chain-mono (cf nmx) (cf-is-≤-monotonic nmx) as0 supf (ZChain.supf-mono zc) (o<→≤ mc<A) zy )}
1478 not-hasprev : ¬ HasPrev A (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 (ZChain.supf zc) mc) mc (cf nmx)
1479 not-hasprev hp = <-irr z26 z30 where
1480 z30 : * mc < * (cf nmx mc)
1481 z30 = proj1 (cf-is-<-monotonic nmx mc (MinSUP.asm msp1))
1482 z26 : (* (cf nmx mc) ≡ * mc) ∨ (* (cf nmx mc) < * mc)
1483 z26 with MinSUP.x<sup msp1 ?
1484 ... | case1 eq = case1 (cong (*) eq)
1485 ... | case2 lt = case2 lt
1486
1466 is-sup : IsSup A (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 (ZChain.supf zc) d) (MinSUP.asm spd) 1487 is-sup : IsSup A (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 (ZChain.supf zc) d) (MinSUP.asm spd)
1467 is-sup = record { x<sup = z22 } where 1488 is-sup = record { x<sup = z22 } where
1468 z23 : {z : Ordinal } → odef (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc) z → (z ≡ MinSUP.sup spd) ∨ (z << MinSUP.sup spd) 1489 z23 : {z : Ordinal } → odef (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc) z → (z ≡ MinSUP.sup spd) ∨ (z << MinSUP.sup spd)
1469 z23 lt = MinSUP.x<sup spd lt 1490 z23 lt = MinSUP.x<sup spd lt
1470 z22 : {y : Ordinal} → odef (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 (ZChain.supf zc) d) y → 1491 z22 : {y : Ordinal} → odef (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 (ZChain.supf zc) d) y →
1473 z22 {a} ⟪ aa , ch-is-sup u u<x is-sup fc ⟫ = ? 1494 z22 {a} ⟪ aa , ch-is-sup u u<x is-sup fc ⟫ = ?
1474 -- u<x : ZChain.supf zc u o< ZChain.supf zc d 1495 -- u<x : ZChain.supf zc u o< ZChain.supf zc d
1475 -- supf u o< spuf c → order 1496 -- supf u o< spuf c → order
1476 1497
1477 not-hasprev : ¬ HasPrev A (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 (ZChain.supf zc) d) d (cf nmx) 1498 not-hasprev : ¬ HasPrev A (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 (ZChain.supf zc) d) d (cf nmx)
1478 not-hasprev hp = ⊥-elim (z29 {mc} {asc} spd hp) where 1499 not-hasprev hp = ⊥-elim (z29 {mc} {asc} spd hp ) where
1479 z29 : {mc : Ordinal } {asc : odef A (supf mc)} (spd : MinSUP A (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc )) 1500 z29 : {mc : Ordinal } {asc : odef A (supf mc)} (spd : MinSUP A (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc ))
1480 → HasPrev A (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 (ZChain.supf zc) (MinSUP.sup spd)) (MinSUP.sup spd) (cf nmx) 1501 → HasPrev A (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 (ZChain.supf zc) (MinSUP.sup spd)) (MinSUP.sup spd) (cf nmx)
1481 → ⊥ 1502 → ⊥
1482 z29 {mc} {asc} spd hp = <-irr z26 z30 where 1503 z29 {mc} {asc} spd hp = <-irr z26 z30 where
1483 y : Ordinal 1504 y : Ordinal
1484 y = HasPrev.y hp 1505 y = HasPrev.y hp
1485 d1 : Ordinal 1506 d1 : Ordinal
1486 d1 = MinSUP.sup spd -- supf d1 ≡ d 1507 d1 = MinSUP.sup spd -- supf d1 ≡ d
1508 z31 : odef (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc) (cf nmx (MinSUP.sup spd))
1509 z31 = ?
1487 z30 : * d1 < * (cf nmx d1) 1510 z30 : * d1 < * (cf nmx d1)
1488 z30 = proj1 (cf-is-<-monotonic nmx d1 (MinSUP.asm spd)) 1511 z30 = proj1 (cf-is-<-monotonic nmx d1 (MinSUP.asm spd))
1489 z31 : odef (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc) (cf nmx d1)
1490 z31 = ?
1491 z24 : y << d1 1512 z24 : y << d1
1492 z24 = subst (λ k → y << k) (sym (HasPrev.x=fy hp)) ( proj1 (cf-is-<-monotonic nmx y (proj1 (HasPrev.ay hp) ) )) 1513 z24 = subst (λ k → y << k) (sym (HasPrev.x=fy hp)) ( proj1 (cf-is-<-monotonic nmx y (proj1 (HasPrev.ay hp) ) ))
1493 z26 : (* (cf nmx d1) ≡ * d1) ∨ (* (cf nmx d1) < * d1) 1514 z26 : (* (cf nmx d1) ≡ * d1) ∨ (* (cf nmx d1) < * d1)
1494 z26 with MinSUP.x<sup spd z31 1515 z26 with MinSUP.x<sup spd z31
1495 ... | case1 eq = case1 (cong (*) eq) 1516 ... | case1 eq = case1 (cong (*) eq)
1496 ... | case2 lt = case2 lt 1517 ... | case2 lt = case2 lt
1518 -- case1 : FClosure of y
1519 -- case2 : u o< supf mc
1520 -- case3 : u ≡ supf mc z31
1521 -- case4 : supf mc o< u ⊥ ( why ? )
1497 z25 : odef (ZChain.chain zc) (cf nmx d) 1522 z25 : odef (ZChain.chain zc) (cf nmx d)
1498 z25 = ZChain.f-next zc (subst (λ k → odef (ZChain.chain zc) k) (sym (HasPrev.x=fy hp)) 1523 z25 = ZChain.f-next zc (subst (λ k → odef (ZChain.chain zc) k) (sym (HasPrev.x=fy hp))
1499 (ZChain.f-next zc 1524 (ZChain.f-next zc
1500 (chain-mono (cf nmx) (cf-is-≤-monotonic nmx) as0 supf (ZChain.supf-mono zc) (o<→≤ d<A) ( HasPrev.ay hp )))) 1525 (chain-mono (cf nmx) (cf-is-≤-monotonic nmx) as0 supf (ZChain.supf-mono zc) (o<→≤ d<A) ( HasPrev.ay hp ))))
1501 1526
1508 ... | case2 lt = lt 1533 ... | case2 lt = lt
1509 1534
1510 sms<sa : supf mc o< supf (& A) 1535 sms<sa : supf mc o< supf (& A)
1511 sms<sa with osuc-≡< ( ZChain.supf-mono zc (o<→≤ ( ∈∧P→o< ⟪ MinSUP.asm msp1 , lift true ⟫) )) 1536 sms<sa with osuc-≡< ( ZChain.supf-mono zc (o<→≤ ( ∈∧P→o< ⟪ MinSUP.asm msp1 , lift true ⟫) ))
1512 ... | case2 lt = lt 1537 ... | case2 lt = lt
1513 ... | case1 eq = ⊥-elim ( o<¬≡ eq ( ordtrans<-≤ (sc<sd (subst (λ k → supf mc << k ) (sym sd=d) (sc<<d {mc} {asc} spd)) ) 1538 ... | case1 eq = ⊥-elim ( o<¬≡ eq ( ordtrans<-≤ (sc<sd (subst (λ k → supf mc << k ) (sym sd=d) (sc<<d {mc} asc spd)) )
1514 ( ZChain.supf-mono zc (o<→≤ d<A )))) 1539 ( ZChain.supf-mono zc (o<→≤ d<A ))))
1515 1540
1516 ss<sa : supf c o< supf (& A) 1541 ss<sa : supf c o< supf (& A)
1517 ss<sa = subst (λ k → supf k o< supf (& A)) (sym c=mc) sms<sa 1542 ss<sa = subst (λ k → supf k o< supf (& A)) (sym c=mc) sms<sa
1518 1543