Mercurial > hg > Members > kono > Proof > ZF-in-agda
changeset 1047:aebab71cc386
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Wed, 07 Dec 2022 11:06:36 +0900 |
| parents | e99e2bcb885c |
| children | a8d6ac036d88 |
| files | src/zorn.agda |
| diffstat | 1 files changed, 77 insertions(+), 33 deletions(-) [+] |
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--- a/src/zorn.agda Mon Dec 05 15:09:41 2022 +0900 +++ b/src/zorn.agda Wed Dec 07 11:06:36 2022 +0900 @@ -1094,50 +1094,94 @@ order : {a b : Ordinal} {w : Ordinal} → b o≤ x → supf1 a o< supf1 b → FClosure A f (supf1 a) w → w ≤ supf1 b - order {a} {b} {w} b≤x sa<sb fc with osuc-≡< b≤x - ... | case2 b<x = - subst ( λ k → w ≤ k ) (sym (sf1=sf0 ?)) ( ZChain.order zc (zc-b<x _ b<x) - (subst₂ (λ j k → j o< k ) (sf1=sf0 ?) (sf1=sf0 ?) sa<sb) (fcup fc ?) ) - ... | case1 eq with zc43 (supf1 a) b - ... | case1 sa<b = subst (λ k → w ≤ k ) (sym (sf1=sp1 ? )) ( MinSUP.x≤sup sup1 ?) where - z26 : odef pchainpx w - z26 = zc11 (chain-mono f mf ay supf1 supf1-mono ? (cfcs ? b≤x ? fc)) - ... | case2 b≤sa = ⊥-elim ( o≤> z27 sa<sb ) where - z28 : supf1 (supf0 a) ≡ supf1 a -- x o≤ supf1 a → - z28 with zc43 (supf0 a) x - ... | case1 sa<x = subst₂ (λ j k → j ≡ k) ? ? ( ZChain.supf-idem zc ? ? ) - ... | case2 x≤sa with osuc-≡< ( supf1-mono x≤sa ) -- = ? -- sp1 ≡ supf0 a --- sp1 o≤ supf0 a - ... | case1 eq = sym (trans z29 eq ) where - z30 : supf1 (supf0 a) ≡ supf1 (supf0 a) - z30 = ? - z29 : supf1 a ≡ supf1 x - z29 = ? - z32 : supf1 x ≡ supf1 (supf0 a) -- supf1 (supf0 a) ≡ supf1 a - z32 = eq - ... | case2 lt = ? where - z31 : supf1 x o< supf1 (supf0 a) - z31 = lt + order {a} {b} {w} b≤x sa<sb fc = z20 where + a<b : a o< b + a<b = supf-inject0 supf1-mono sa<sb + z20 : w ≤ supf1 b + z20 with trio< b px + ... | tri< b<px ¬b ¬c = ZChain.order zc (o<→≤ b<px) (subst (λ k → k o< supf0 b) (sf1=sf0 (o<→≤ (ordtrans a<b b<px))) sa<sb) + (fcup fc (o<→≤ (ordtrans a<b b<px))) + ... | tri≈ ¬a b=px ¬c = IsMinSUP.x≤sup (ZChain.is-minsup zc (o≤-refl0 b=px)) z26 where + sa<b : supf1 a o< b -- px + sa<b = ? + z26 : odef ( UnionCF A f ay supf0 b ) w + z26 = ? + z27 : odef ( UnionCF A f ay supf1 b ) w + z27 = cfcs a<b b≤x sa<b fc + ... | tri> ¬a ¬b px<b = MinSUP.x≤sup sup1 (zc11 ( chain-mono f mf ay supf1 supf1-mono b≤x (cfcs a<b b≤x sa<b fc))) where + sa<b : supf1 a o< b -- b ≡ x + sa<b = ? - z27 : supf1 b o≤ supf1 a - z27 = begin - supf1 b ≤⟨ ? ⟩ - supf1 (supf1 a) ≡⟨ ? ⟩ - supf1 a ∎ where open o≤-Reasoning O - + -- + -- sup=u : {b : Ordinal} (ab : odef A b) → b o≤ x → IsSUP A (UnionCF A f ay supf1 b) b ∧ (¬ HasPrev A (UnionCF A f ay supf1 b) f b ) → supf1 b ≡ b sup=u {b} ab b≤x is-sup with trio< b px ... | tri< a ¬b ¬c = ? -- ZChain.sup=u zc ab (o<→≤ a) is-sup ... | tri≈ ¬a b ¬c = ? -- ZChain.sup=u zc ab (o≤-refl0 b) is-sup - ... | tri> ¬a ¬b px<b = ? where - zc30 : x ≡ b - zc30 with osuc-≡< b≤x + ... | tri> ¬a ¬b px<b = trans zc36 x=b where + x=b : x ≡ b + x=b with osuc-≡< b≤x ... | case1 eq = sym (eq) ... | case2 b<x = ⊥-elim (¬p<x<op ⟪ px<b , subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) b<x ⟫ ) zc31 : sp1 o≤ b zc31 = MinSUP.minsup sup1 ab zc32 where zc32 : {w : Ordinal } → odef pchainpx w → (w ≡ b) ∨ (w << b) - zc32 = ? + zc32 {w} (case1 uw) = IsSUP.x≤sup (proj1 is-sup) ? + zc32 {w} (case2 fc) = IsSUP.x≤sup (proj1 is-sup) ⟪ A∋fc _ f mf (proj1 fc) , ch-is-sup (supf0 px) sa<x su=u fc1 ⟫ where + su=u : supf1 (supf0 px) ≡ supf0 px + su=u = trans (sf1=sf0 (zc-b<x _ (proj2 fc))) ( ZChain.supf-idem zc o≤-refl (zc-b<x _ (proj2 fc)) ) + fc1 : FClosure A f (supf1 (supf0 px)) w + fc1 = subst (λ k → FClosure A f k w ) (sym su=u) (proj1 fc) + sa<x : supf0 px o< b + sa<x = subst (λ k → supf0 px o< k ) x=b ( proj2 fc) + zc36 : sp1 ≡ x -- this cannot heppen because px o< supf0 px ( px o≤ spuf0 px o< supf1 x ≡ x → ⊥ ) + zc36 with osuc-≡< zc31 + ... | case1 eq = trans eq (sym x=b) + ... | case2 sp1<b with osuc-≡< ( zc-b<x _ (subst (λ k → sp1 o< k ) ? sp1<b)) + ... | case1 sp1=px = ⊥-elim ( <<-irr ? z42 ) where + z40 : odef pchainpx (f (supf1 sp1)) + z40 = case2 ⟪ fsuc _ (init ? ?) , ? ⟫ + z41 : f (supf1 sp1) ≤ sp1 + z41 = MinSUP.x≤sup sup1 z40 + z44 : sp1 ≤ supf1 sp1 + z44 = ? + z43 : px o≤ supf0 px + z43 with zc43 (supf0 px) px + ... | case2 le = le + ... | case1 spx<px = ⊥-elim ( <<-irr z45 (proj1 ( mf< (supf0 px) (ZChain.asupf zc)))) where + z46 : odef (UnionCF A f ay supf1 px) (f (supf1 px)) + z46 = ⟪ proj2 ( mf (supf1 px) ?) , ch-is-sup (supf0 px) spx<px z48 (fsuc _ (init ? z47 )) ⟫ where + z47 : supf1 (supf0 px) ≡ supf1 px + z47 = subst₂ (λ j k → j ≡ k ) (sym (sf1=sf0 ?)) (sym (sf1=sf0 ?)) ( ZChain.supf-idem zc o≤-refl (o<→≤ spx<px) ) + z48 : supf1 (supf0 px) ≡ supf0 px + z48 = subst (λ k → k ≡ supf0 px ) (sym (sf1=sf0 ?)) ( ZChain.supf-idem zc o≤-refl (o<→≤ spx<px) ) + z45 : f (supf0 px) ≤ supf0 px + z45 = subst (λ k → f k ≤ k ) ? ( IsMinSUP.x≤sup (is-minsup ? ) z46 ) + z42 : (supf1 sp1) << f (supf1 sp1) + z42 = proj1 ( mf< (supf1 sp1) ? ) + ... | case2 sp1<px = ⊥-elim ( o≤> z39 z44 ) where + z39 : supf1 px o≤ supf1 x -- supf0 px o≤ px + z39 = supf1-mono (o<→≤ px<x) + z43 : px o≤ supf0 px + z43 with zc43 (supf0 px) px + ... | case2 le = le + ... | case1 spx<px = ⊥-elim ( <<-irr z45 (proj1 ( mf< (supf0 px) (ZChain.asupf zc)))) where + z46 : odef (UnionCF A f ay supf1 px) (f (supf1 px)) + z46 = ⟪ proj2 ( mf (supf1 px) ?) , ch-is-sup (supf0 px) spx<px z48 (fsuc _ (init ? z47 )) ⟫ where + z47 : supf1 (supf0 px) ≡ supf1 px + z47 = subst₂ (λ j k → j ≡ k ) (sym (sf1=sf0 ?)) (sym (sf1=sf0 ?)) ( ZChain.supf-idem zc o≤-refl (o<→≤ spx<px) ) + z48 : supf1 (supf0 px) ≡ supf0 px + z48 = subst (λ k → k ≡ supf0 px ) (sym (sf1=sf0 ?)) ( ZChain.supf-idem zc o≤-refl (o<→≤ spx<px) ) + z45 : f (supf0 px) ≤ supf0 px + z45 = subst (λ k → f k ≤ k ) ? ( IsMinSUP.x≤sup (is-minsup ? ) z46 ) + z44 : supf1 x o< supf1 px + z44 = osucprev ( begin + osuc (supf1 x) ≡⟨ cong osuc (sf1=sp1 px<x) ⟩ + osuc sp1 ≤⟨ osucc sp1<px ⟩ + px ≤⟨ z43 ⟩ + supf0 px ≡⟨ sym (sf1=sf0 o≤-refl ) ⟩ + supf1 px ∎ ) where open o≤-Reasoning O ... | no lim with trio< x o∅ ... | tri< a ¬b ¬c = ⊥-elim ( ¬x<0 a )