Mercurial > hg > Members > kono > Proof > ZF-in-agda
view src/maximum-filter.agda @ 1284:45cd80181a29
remove import zf
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Sat, 20 May 2023 09:48:37 +0900 |
parents | 6216562a2bce |
children | 47d3cc596d68 |
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{-# OPTIONS --allow-unsolved-metas #-} open import Level open import Ordinals module maximum-filter {n : Level } (O : Ordinals {n}) where open import logic import OD open import Relation.Nullary open import Data.Empty open import Relation.Binary.Core open import Relation.Binary.PropositionalEquality import BAlgebra open BAlgebra O open inOrdinal O open OD O open OD.OD open ODAxiom odAxiom import OrdUtil import ODUtil open Ordinals.Ordinals O open Ordinals.IsOrdinals isOrdinal open Ordinals.IsNext isNext open OrdUtil O open ODUtil O import ODC open ODC O open _∧_ open _∨_ open Bool open import filter O open Filter open import Relation.Binary open import Relation.Binary.Structures PO : IsStrictPartialOrder _≡_ _⊂_ PO = record { isEquivalence = record { refl = refl ; sym = sym ; trans = trans } ; trans = λ {a} {b} {c} → trans-⊂ {a} {b} {c} ; irrefl = λ x=y x<y → o<¬≡ (cong (&) x=y) (proj1 x<y) ; <-resp-≈ = record { fst = ( λ {x} {y} {y1} y=y1 xy1 → subst (λ k → x ⊂ k) y=y1 xy1 ) ; snd = (λ {x} {x1} {y} x=x1 x1y → subst (λ k → k ⊂ x) x=x1 x1y ) } } import zorn open zorn O _⊂_ PO -- all filter contains F F⊆X : { L P : HOD } (LP : L ⊆ Power P) → Filter {L} {P} LP → HOD F⊆X {L} {P} LP F = record { od = record { def = λ x → IsFilter {L} {P} LP x ∧ ( filter F ⊆ * x) } ; odmax = osuc (& L) ; <odmax = λ {x} lt → fx00 lt } where fx00 : {x : Ordinal } → IsFilter LP x ∧ filter F ⊆ * x → x o< osuc (& L) fx00 {x} lt = subst (λ k → k o< osuc (& L)) &iso ( ⊆→o≤ (IsFilter.f⊆L (proj1 lt )) ) F→Maximum : {L P : HOD} (LP : L ⊆ Power 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 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 oF = & (filter F) mf11 : { p q : Ordinal } → odef L q → odef (* oF) p → (* p) ⊆ (* q) → odef (* oF) q mf11 {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 ) mf12 : { p q : Ordinal } → odef (* oF) p → odef (* oF) q → odef L (& ((* p) ∩ (* q))) → odef (* oF) (& ((* p) ∩ (* q))) mf12 {p} {q} Fp Fq Lpq = subst (λ k → odef k (& ((* p) ∩ (* q))) ) (sym *iso) ( 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) FX∋F : odef FX (& (filter F)) FX∋F = ⟪ record { f⊆L = subst (λ k → k ⊆ L) (sym *iso) (f⊆L F) ; filter1 = mf11 ; filter2 = mf12 ; proper = subst₂ (λ j k → ¬ (odef j k) ) (sym *iso) ord-od∅ Fprop } , subst (λ k → filter F ⊆ k ) (sym *iso) ( λ x → x ) ⟫ -- -- if filter B (which contains F) is transitive, Union B is also a filter which contains F -- and this is a SUP -- SUP⊆ : (B : HOD) → B ⊆ FX → IsTotalOrderSet B → SUP FX B SUP⊆ B B⊆FX OB with trio< (& B) o∅ ... | tri< a ¬b ¬c = ⊥-elim (¬x<0 a ) ... | tri≈ ¬a b ¬c = record { sup = filter F ; isSUP = record { ax = FX∋F ; x≤sup = λ {y} zy → ⊥-elim (o<¬≡ (sym b) (∈∅< zy)) } } ... | tri> ¬a ¬b 0<B = record { sup = Union B ; isSUP = record { ax = mf13 ; x≤sup = mf40 } } where mf20 : HOD mf20 = ODC.minimal O B (λ eq → (o<¬≡ (cong (&) (sym (==→o≡ eq))) (subst (λ k → k o< & B) (sym ord-od∅) 0<B ))) mf18 : odef B (& mf20 ) mf18 = ODC.x∋minimal O B (λ eq → (o<¬≡ (cong (&) (sym (==→o≡ eq))) (subst (λ k → k o< & B) (sym ord-od∅) 0<B ))) mf16 : Union B ⊆ L mf16 record { owner = b ; ao = Bb ; ox = bx } = IsFilter.f⊆L ( proj1 ( B⊆FX Bb )) bx mf17 : {p q : Ordinal} → odef L q → odef (* (& (Union B))) p → * p ⊆ * q → odef (* (& (Union B))) q mf17 {p} {q} Lq bp p⊆q with subst (λ k → odef k p ) *iso bp ... | record { owner = owner ; ao = ao ; ox = ox } = subst (λ k → odef k q) (sym *iso) record { owner = owner ; ao = ao ; ox = IsFilter.filter1 mf30 Lq ox p⊆q } where mf30 : IsFilter {L} {P} LP owner mf30 = proj1 ( B⊆FX ao ) mf31 : {p q : Ordinal} → odef (* (& (Union B))) p → odef (* (& (Union B))) q → odef L (& ((* p) ∩ (* q))) → odef (* (& (Union B))) (& ((* p) ∩ (* q))) mf31 {p} {q} bp bq Lpq with subst (λ k → odef k p ) *iso bp | subst (λ k → odef k q ) *iso bq ... | record { owner = bp ; ao = Bbp ; ox = bbp } | record { owner = bq ; ao = Bbq ; ox = bbq } with OB (subst (λ k → odef B k) (sym &iso) Bbp) (subst (λ k → odef B k) (sym &iso) Bbq) ... | tri< bp⊂bq ¬b ¬c = subst₂ (λ j k → odef j k ) (sym *iso) refl record { owner = bq ; ao = Bbq ; ox = mf36 } where mf36 : odef (* bq) (& ((* p) ∩ (* q))) mf36 = IsFilter.filter2 mf30 (proj2 bp⊂bq bbp) bbq Lpq where mf30 : IsFilter {L} {P} LP bq mf30 = proj1 ( B⊆FX Bbq ) ... | tri≈ ¬a bq=bp ¬c = subst₂ (λ j k → odef j k ) (sym *iso) refl record { owner = bp ; ao = Bbp ; ox = mf36 } where mf36 : odef (* bp) (& ((* p) ∩ (* q))) mf36 = IsFilter.filter2 mf30 bbp (subst (λ k → odef k q) (sym bq=bp) bbq) Lpq where mf30 : IsFilter {L} {P} LP bp mf30 = proj1 ( B⊆FX Bbp ) ... | tri> ¬a ¬b bq⊂bp = subst₂ (λ j k → odef j k ) (sym *iso) refl record { owner = bp ; ao = Bbp ; ox = mf36 } where mf36 : odef (* bp) (& ((* p) ∩ (* q))) mf36 = IsFilter.filter2 mf30 bbp (proj2 bq⊂bp bbq) Lpq where mf30 : IsFilter {L} {P} LP bp mf30 = proj1 ( B⊆FX Bbp ) mf32 : ¬ odef (Union B) o∅ mf32 record { owner = owner ; ao = bo ; ox = o0 } with proj1 ( B⊆FX bo ) ... | record { f⊆L = f⊆L ; filter1 = filter1 ; filter2 = filter2 ; proper = proper } = ⊥-elim ( proper o0 ) mf14 : IsFilter LP (& (Union B)) mf14 = record { f⊆L = subst (λ k → k ⊆ L) (sym *iso) mf16 ; filter1 = mf17 ; filter2 = mf31 ; proper = subst (λ k → ¬ odef k o∅) (sym *iso) mf32 } mf15 : filter F ⊆ Union B mf15 {x} Fx = record { owner = & mf20 ; ao = mf18 ; ox = subst (λ k → odef k x) (sym *iso) (mf22 Fx) } where mf22 : odef (filter F) x → odef mf20 x mf22 Fx = subst (λ k → odef k x) *iso ( proj2 (B⊆FX mf18) Fx ) mf13 : odef FX (& (Union B)) mf13 = ⟪ mf14 , subst (λ k → filter F ⊆ k ) (sym *iso) mf15 ⟫ mf42 : {z : Ordinal} → odef B z → * z ⊆ Union B mf42 {z} Bz {x} zx = record { owner = _ ; ao = Bz ; ox = zx } mf40 : {z : Ordinal} → odef B z → (z ≡ & (Union B)) ∨ ( * z ⊂ * (& (Union B)) ) mf40 {z} Bz with B⊆FX Bz ... | ⟪ filterz , F⊆z ⟫ with osuc-≡< ( ⊆→o≤ {* z} {Union B} (mf42 Bz) ) ... | case1 eq = case1 (trans (sym &iso) eq ) ... | case2 lt = case2 ⟪ subst₂ (λ j k → j o< & k ) refl (sym *iso) lt , subst (λ k → * z ⊆ k) (sym *iso) (mf42 Bz) ⟫ mx : Maximal FX mx = Zorn-lemma (∈∅< FX∋F) SUP⊆ imf : IsFilter {L} {P} LP (& (Maximal.maximal mx)) imf = proj1 (Maximal.as mx) mf00 : (* (& (Maximal.maximal mx))) ⊆ L mf00 = IsFilter.f⊆L imf mf01 : { p q : HOD } → L ∋ q → (* (& (Maximal.maximal mx))) ∋ p → p ⊆ q → (* (& (Maximal.maximal mx))) ∋ q mf01 {p} {q} Lq Fq p⊆q = IsFilter.filter1 imf Lq Fq (λ {x} lt → subst (λ k → odef k x) (sym *iso) ( p⊆q (subst (λ k → odef k x) *iso lt ) )) mf02 : { p q : HOD } → (* (& (Maximal.maximal mx))) ∋ p → (* (& (Maximal.maximal mx))) ∋ q → L ∋ (p ∩ q) → (* (& (Maximal.maximal mx))) ∋ (p ∩ q) mf02 {p} {q} Fp Fq Lpq = subst₂ (λ j k → odef (* (& (Maximal.maximal mx))) (& (j ∩ k ))) *iso *iso ( IsFilter.filter2 imf Fp Fq (subst₂ (λ j k → odef L (& (j ∩ k ))) (sym *iso) (sym *iso) Lpq )) mf : Filter {L} {P} LP 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 = subst (λ k → filter F ⊆ k ) (sym *iso) F⊆f F→ultra : {L P : HOD} (LP : L ⊆ Power 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∅)) → ultra-filter ( MaximumFilter.mf (F→Maximum {L} {P} LP CAP F 0<L 0<F proper )) F→ultra {L} {P} LP CAP F 0<L 0<F proper = max→ultra {L} {P} LP CAP F 0<F (F→Maximum {L} {P} LP CAP F 0<L 0<F proper )