Mercurial > hg > Members > kono > Proof > ZF-in-agda
changeset 1140:7515d1b0570b
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sat, 14 Jan 2023 07:47:44 +0900 |
| parents | 4517d0721b59 |
| children | e9a05e7c4e35 |
| files | src/filter.agda |
| diffstat | 1 files changed, 56 insertions(+), 11 deletions(-) [+] |
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--- a/src/filter.agda Fri Jan 13 13:16:49 2023 +0900 +++ b/src/filter.agda Sat Jan 14 07:47:44 2023 +0900 @@ -77,6 +77,20 @@ q∩q⊆q : {p q : HOD } → (q ∩ p) ⊆ q q∩q⊆q lt = proj1 lt +∩→≡1 : {p q : HOD } → p ⊆ q → (q ∩ p) ≡ p +∩→≡1 {p} {q} p⊆q = ==→o≡ record { eq→ = c00 ; eq← = c01 } where + c00 : {x : Ordinal} → odef (q ∩ p) x → odef p x + c00 {x} qpx = proj2 qpx + c01 : {x : Ordinal} → odef p x → odef (q ∩ p) x + c01 {x} px = ⟪ p⊆q px , px ⟫ + +∩→≡2 : {p q : HOD } → q ⊆ p → (q ∩ p) ≡ q +∩→≡2 {p} {q} q⊆p = ==→o≡ record { eq→ = c00 ; eq← = c01 } where + c00 : {x : Ordinal} → odef (q ∩ p) x → odef q x + c00 {x} qpx = proj1 qpx + c01 : {x : Ordinal} → odef q x → odef (q ∩ p) x + c01 {x} qx = ⟪ qx , q⊆p qx ⟫ + open HOD ----- @@ -174,7 +188,7 @@ mf : Filter {L} {P} LP F⊆mf : filter F ⊆ filter mf proper : ¬ (filter mf ∋ od∅) - is-maximum : ( f : Filter {L} {P} LP ) → ¬ (filter f ∋ od∅) → ¬ ( filter mf ⊂ filter f ) + is-maximum : ( f : Filter {L} {P} LP ) → ¬ (filter f ∋ od∅) → filter F ⊆ filter f → ¬ ( filter mf ⊂ filter f ) record Fp {L P : HOD} (LP : L ⊆ Power P) (F : Filter {L} {P} LP) (mx : MaximumFilter {L} {P} LP F ) (p x : Ordinal ) : Set n where field @@ -238,11 +252,12 @@ 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) ) ⟫ ) ) ) - mu40 = (MaximumFilter.is-maximum mx) + mu41 : filter F0 ⊆ F + mu41 {x} fx = ⟪ f⊆L F0 fx , record { y = ? ; mfy = ? ; y-p⊂x = ? } ⟫ 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 ? ⟪ lt , FisGreater ⟫ ) open _==_ @@ -251,8 +266,8 @@ → ({p q : HOD} → L ∋ p → L ∋ q → L ∋ (p ∩ q)) → (U : Filter {L} {P} LP) → ultra-filter U → MaximumFilter LP U ultra→max {L} {P} LP NG CAP U u = record { mf = U ; F⊆mf = λ x → x ; proper = ultra-filter.proper u ; is-maximum = is-maximum } where - is-maximum : (F : Filter {L} {P} LP) → (¬ (filter F ∋ od∅)) → (U⊂F : filter U ⊂ filter F ) → ⊥ - is-maximum F Prop ⟪ U<F , U⊆F ⟫ = Prop f0 where + is-maximum : (F : Filter {L} {P} LP) → (¬ (filter F ∋ od∅)) → filter U ⊆ filter F → (U⊂F : filter U ⊂ filter F ) → ⊥ + is-maximum F Prop F⊆U ⟪ U<F , U⊆F ⟫ = Prop f0 where GT : HOD GT = record { od = record { def = λ x → odef (filter F) x ∧ (¬ odef (filter U) x) } ; odmax = & L ; <odmax = um02 } where um02 : {y : Ordinal } → odef (filter F) y ∧ (¬ odef (filter U) y) → y o< & L @@ -309,6 +324,14 @@ filter2 : { p q : Ordinal } → odef (* filter) p → odef (* filter) q → odef L (& ((* p) ∩ (* q))) → odef (* filter) (& ((* p) ∩ (* q))) proper : ¬ (odef (* filter ) o∅) +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 = ? + ; proper = subst₂ (λ j k → ¬ odef j k ) (sym *iso) ord-od∅ proper + } + -- 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) @@ -319,7 +342,7 @@ 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 = ? - ; proper = subst (λ k → ¬ ( odef (filter mf ) k)) (sym ord-od∅) ( IsFilter.proper imf) ; is-maximum = {!!} } where + ; 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) @@ -341,7 +364,7 @@ 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 = ? } } where + ... | 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 ) @@ -358,9 +381,21 @@ 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 = ? - ... | tri≈ ¬a bq=bp ¬c = ? - ... | tri> ¬a ¬b bq⊂bp = ? + ... | 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 ) @@ -372,6 +407,13 @@ 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)) @@ -389,7 +431,10 @@ mf = record { filter = * (& (Maximal.maximal mx)) ; f⊆L = mf00 ; filter1 = mf01 ; filter2 = mf02 } - + 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 = ? 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∅))