Mercurial > hg > Members > kono > Proof > ZF-in-agda
changeset 1141:e9a05e7c4e35
Maximal Filter and Ultra Filter generation done
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
---|---|
date | Sat, 14 Jan 2023 12:36:07 +0900 (2023-01-14) |
parents | 7515d1b0570b |
children | 7b924ef65373 |
files | src/filter.agda |
diffstat | 1 files changed, 15 insertions(+), 7 deletions(-) [+] |
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--- a/src/filter.agda Sat Jan 14 07:47:44 2023 +0900 +++ b/src/filter.agda Sat Jan 14 12:36:07 2023 +0900 @@ -252,12 +252,14 @@ mu31 {x} zx with ODC.decp O (odef p x) ... | yes px = px ... | no npx = ⊥-elim ( ¬x<0 (subst (λ k → odef k x) *iso (z-p⊂x ⟪ zx , (λ px → npx (subst (λ k → odef k x) *iso px) ) ⟫ ) ) ) - mu41 : filter F0 ⊆ F - mu41 {x} fx = ⟪ f⊆L F0 fx , record { y = ? ; mfy = ? ; y-p⊂x = ? } ⟫ + F0⊆F : filter F0 ⊆ F + F0⊆F {x} fx = ⟪ f⊆L F0 fx , record { y = _ ; mfy = MaximumFilter.F⊆mf mx fx ; y-p⊂x = mu42 } ⟫ where + mu42 : (* x \ * (& p)) ⊆ * x + mu42 {z} ⟪ xz , ¬p ⟫ = xz F=mf : F ≡ filter mf F=mf with osuc-≡< ( ⊆→o≤ FisGreater ) ... | case1 eq = &≡&→≡ (sym eq) - ... | case2 lt = ⊥-elim ( MaximumFilter.is-maximum mx FisFilter FisProper ? ⟪ lt , FisGreater ⟫ ) + ... | case2 lt = ⊥-elim ( MaximumFilter.is-maximum mx FisFilter FisProper F0⊆F ⟪ lt , FisGreater ⟫ ) open _==_ @@ -327,8 +329,10 @@ Filter-is-Filter : { L P : HOD } (LP : L ⊆ Power P) → (F : Filter {L} {P} LP) → (proper : ¬ (filter F) ∋ od∅ ) → IsFilter {L} {P} LP (& (filter F)) Filter-is-Filter {L} {P} LP F proper = record { f⊆L = subst (λ k → k ⊆ L ) (sym *iso) (f⊆L F) - ; filter1 = ? - ; filter2 = ? + ; filter1 = λ {p} {q} Lq Fp p⊆q → subst₂ (λ j k → odef j k ) (sym *iso) &iso + ( filter1 F (subst (λ k → odef L k) (sym &iso) Lq) (subst₂ (λ j k → odef j k ) *iso (sym &iso) Fp) p⊆q ) + ; filter2 = λ {p} {q} Fp Fq Lpq → subst₂ (λ j k → odef j k ) (sym *iso) refl ( filter2 F (subst₂ (λ j k → odef j k ) *iso (sym &iso) Fp) + (subst₂ (λ j k → odef j k ) *iso (sym &iso) Fq) Lpq ) ; proper = subst₂ (λ j k → ¬ odef j k ) (sym *iso) ord-od∅ proper } @@ -341,7 +345,7 @@ F→Maximum : {L P : HOD} (LP : L ⊆ Power P) → ({p : HOD} → L ∋ p → L ∋ ( P \ p)) → ({p q : HOD} → L ∋ p → L ∋ q → L ∋ (p ∩ q)) → (F : Filter {L} {P} LP) → o∅ o< & L → {y : Ordinal } → odef (filter F) y → (¬ (filter F ∋ od∅)) → MaximumFilter {L} {P} LP F -F→Maximum {L} {P} LP NEG CAP F 0<L {y} 0<F Fprop = record { mf = mf ; F⊆mf = ? +F→Maximum {L} {P} LP NEG CAP F 0<L {y} 0<F Fprop = record { mf = mf ; F⊆mf = subst (λ k → filter F ⊆ k ) (sym *iso) mf52 ; proper = subst (λ k → ¬ ( odef (filter mf ) k)) (sym ord-od∅) ( IsFilter.proper imf) ; is-maximum = mf50 } where FX : HOD FX = F⊆X {L} {P} LP F @@ -431,10 +435,14 @@ mf = record { filter = * (& (Maximal.maximal mx)) ; f⊆L = mf00 ; filter1 = mf01 ; filter2 = mf02 } + mf52 : filter F ⊆ Maximal.maximal mx + mf52 = subst (λ k → filter F ⊆ k ) *iso (proj2 mf53) where + mf53 : FX ∋ Maximal.maximal mx + mf53 = Maximal.as mx mf50 : (f : Filter LP) → ¬ (filter f ∋ od∅) → filter F ⊆ filter f → ¬ (* (& (zorn.Maximal.maximal mx)) ⊂ filter f) mf50 f proper F⊆f = subst (λ k → ¬ ( k ⊂ filter f )) (sym *iso) (Maximal.¬maximal<x mx ⟪ Filter-is-Filter {L} {P} LP f proper , mf51 ⟫ ) where mf51 : filter F ⊆ * (& (filter f)) - mf51 = ? + mf51 = subst (λ k → filter F ⊆ k ) (sym *iso) F⊆f F→ultra : {L P : HOD} (LP : L ⊆ Power P) → (NEG : {p : HOD} → L ∋ p → L ∋ ( P \ p)) → (CAP : {p q : HOD} → L ∋ p → L ∋ q → L ∋ (p ∩ q)) → (F : Filter {L} {P} LP) → (0<L : o∅ o< & L) → {y : Ordinal} → (0<F : odef (filter F) y) → (proper : ¬ (filter F ∋ od∅))