Mercurial > hg > Members > kono > Proof > ZF-in-agda
changeset 478:c6346d92f1a1
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author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Sun, 03 Apr 2022 07:59:55 +0900 |
parents | 24b4b854b310 |
children | fea0c2454b85 |
files | src/filter.agda src/zorn.agda |
diffstat | 2 files changed, 53 insertions(+), 11 deletions(-) [+] |
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--- a/src/filter.agda Sat Apr 02 08:37:17 2022 +0900 +++ b/src/filter.agda Sun Apr 03 07:59:55 2022 +0900 @@ -173,3 +173,15 @@ genf : Filter LP generic : (D : Dense LP ) → M ∋ Dense.dense D → ¬ ( (Dense.dense D ∩ Filter.filter genf ) ≡ od∅ ) +open import zorn + +record MaximumFilter {L P : HOD} (LP : L ⊆ Power P) : Set (suc n) where + field + mf : Filter LP + proper : ¬ (filter mf ∋ od∅) + is-maximum : ( f : Filter LP ) → filter f ⊆ filter mf + +max→ultra : {L P : HOD} (LP : L ⊆ Power P) → (mx : MaximumFilter LP ) → ultra-filter ( MaximumFilter.mf mx ) +max→ultra {L} {P} LP mx = record { proper = {!!} ; ultra = {!!} } + +
--- a/src/zorn.agda Sat Apr 02 08:37:17 2022 +0900 +++ b/src/zorn.agda Sun Apr 03 07:59:55 2022 +0900 @@ -1,3 +1,4 @@ +{-# OPTIONS --allow-unsolved-metas #-} open import Level open import Ordinals module zorn {n : Level } (O : Ordinals {n}) where @@ -60,13 +61,13 @@ field sup : HOD A∋maximal : A ∋ sup - x≤sup : {x : HOD} → B ∋ x → (x ≡ sup ) ∨ (x < sup ) + x≤sup : {x : HOD} → B ∋ x → (x ≡ sup ) ∨ (x < sup ) -- B is Total record Maximal ( A : HOD ) (_<_ : (x y : HOD) → Set n ) : Set (suc n) where field maximal : HOD A∋maximal : A ∋ maximal - ¬maximal<x : {x : HOD} → A ∋ x → ¬ maximal < x + ¬maximal<x : {x : HOD} → A ∋ x → ¬ maximal < x -- A is Partial, use negative open _==_ open _⊆_ @@ -82,11 +83,10 @@ Zorn-lemma : { A : HOD } → { _<_ : (x y : HOD) → Set n } → o∅ o< & A - → ( {a b c : HOD} → a < b → b < c → a < c ) → PartialOrderSet A _<_ - → ( ( B : HOD) → (B⊆A : B ⊆ A) → TotalOrderSet B _<_ → SUP A B _<_ ) + → ( ( B : HOD) → (B⊆A : B ⊆ A) → TotalOrderSet B _<_ → SUP A B _<_ ) -- SUP condition → Maximal A _<_ -Zorn-lemma {A} {_<_} 0<A TR PO supP = zorn00 where +Zorn-lemma {A} {_<_} 0<A PO supP = zorn00 where someA : HOD someA = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) isSomeA : A ∋ someA @@ -102,6 +102,7 @@ z01 : {a b : HOD} → A ∋ a → A ∋ b → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ z01 {a} {b} A∋a A∋b (case1 a=b) b<a = proj1 (proj2 (PO (me A∋b) (me A∋a)) (sym a=b)) b<a z01 {a} {b} A∋a A∋b (case2 a<b) b<a = proj1 (PO (me A∋b) (me A∋a)) b<a ⟪ a<b , (λ b=a → proj1 (proj2 (PO (me A∋b) (me A∋a)) b=a ) b<a ) ⟫ + -- ZChain is not compatible with the SUP condition ZChain→¬SUP : (z : ZChain A (& A) _<_ ) → ¬ (SUP A (ZChain.B z) _<_ ) ZChain→¬SUP z sp = ⊥-elim (z02 (ZChain.fb z (SUP.A∋maximal sp)) (ZChain.B∋fb z _ (SUP.A∋maximal sp)) (ZChain.¬x≤sup z _ (SUP.A∋maximal sp) z03 )) where z03 : & (SUP.sup sp) o< osuc (& A) @@ -137,29 +138,57 @@ ¬x<m : ¬ (* x < * m) ¬x<m x<m = ∅< {Gtx ax} {* m} ⟪ subst (λ k → odef A k) (sym &iso) am , subst (λ k → * x < k ) (cong (*) (sym &iso)) x<m ⟫ (≡o∅→=od∅ nogt) ... | no not = record { B = Bx -- we have larger element, let's create ZChain - ; B⊆A = B⊆A ; total = {!!} ; fb = {!!} ; B∋fb = {!!} ; ¬x≤sup = {!!} } where + ; B⊆A = B⊆A ; total = total ; fb = fb ; B∋fb = {!!} ; ¬x≤sup = {!!} } where B = ZChain.B zc1 Bx : HOD - Bx = record { od = record { def = λ y → (x ≡ y) ∨ odef B y } ; odmax = & A ; <odmax = {!!} } + Bx = record { od = record { def = λ y → (x ≡ y) ∨ odef B y } ; odmax = & A ; <odmax = {!!} } -- Union (B , x) B⊆A : Bx ⊆ A B⊆A = record { incl = λ {y} by → z07 y by } where z07 : (y : HOD) → Bx ∋ y → A ∋ y z07 y (case1 x=y) = subst (λ k → odef A k ) (trans &iso x=y) ax z07 y (case2 by) = incl (ZChain.B⊆A zc1 ) by m = ODC.minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq)) + p : odef A (& m) ∧ (* x < (* (& m))) + p = ODC.x∋minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq)) + fb : {y : HOD} → A ∋ y → HOD + fb {y} ay with trio< (& y) x + ... | tri< a ¬b ¬c = ZChain.fb zc1 ay + ... | tri≈ ¬a b ¬c = m + ... | tri> ¬a ¬b c = od∅ + total : TotalOrderSet Bx _<_ + total ex ey with is-elm ex | is-elm ey + ... | case1 eq | case1 eq1 = tri≈ {!!} {!!} {!!} + ... | case1 x | case2 x₁ = tri< {!!} {!!} {!!} + ... | case2 x | case1 x₁ = {!!} + ... | case2 x | case2 x₁ = ZChain.total zc1 (me x) (me x₁) ind nomx x prev | no ¬ox with trio< (& A) x --- limit ordinal case ... | tri< a ¬b ¬c = record { B = ZChain.B zc1 ; B⊆A = ZChain.B⊆A zc1 ; total = ZChain.total zc1 ; fb = ZChain.fb zc1 ; B∋fb = ZChain.B∋fb zc1 ; ¬x≤sup = {!!} } where zc1 : ZChain A (& A) _<_ zc1 = prev (& A) a ... | tri≈ ¬a b ¬c = record { B = B - ; B⊆A = {!!} ; total = {!!} ; fb = {!!} ; B∋fb = {!!} ; ¬x≤sup = {!!} } where - B : HOD - B = record { od = record { def = λ y → (y<x : y o< x ) → odef (ZChain.B (prev y y<x)) y } ; odmax = & A ; <odmax = {!!} } - ... | tri> ¬a ¬b c = {!!} + ; B⊆A = B⊆A ; total = {!!} ; fb = {!!} ; B∋fb = {!!} ; ¬x≤sup = {!!} } where + B : HOD -- Union (previous B) + B = record { od = record { def = λ y → (y o< x) ∧ ((y<x : y o< x ) → odef (ZChain.B (prev y y<x)) y) } ; odmax = & A ; <odmax = {!!} } + B⊆A : B ⊆ A + B⊆A = record { incl = λ {y} bx → incl (ZChain.B⊆A (prev (& y) (proj1 bx))) (proj2 bx (proj1 bx)) } + ... | tri> ¬a ¬b c with ODC.∋-p O A (* x) + ... | no ¬Ax = {!!} where + B : HOD -- Union (previous B) + B = record { od = record { def = λ y → (y o< x) ∧ ((y<x : y o< x ) → odef (ZChain.B (prev y y<x)) y) } ; odmax = & A ; <odmax = {!!} } + ... | yes ax with is-o∅ (& (Gtx ax)) + ... | yes nogt = ⊥-elim (no-maximum nomx x x-is-maximal ) where -- no larger element, so it is maximal + x-is-maximal : (m : Ordinal) → odef A m → odef A x ∧ (¬ (* x < * m)) + x-is-maximal m am = ⟪ subst (λ k → odef A k) &iso ax , ¬x<m ⟫ where + ¬x<m : ¬ (* x < * m) + ¬x<m x<m = ∅< {Gtx ax} {* m} ⟪ subst (λ k → odef A k) (sym &iso) am , subst (λ k → * x < k ) (cong (*) (sym &iso)) x<m ⟫ (≡o∅→=od∅ nogt) + ... | no not = {!!} where + B : HOD -- Union (x , previous B) + B = record { od = record { def = λ y → (y o< osuc x) ∧ ((y<x : y o< x ) → odef (ZChain.B (prev y y<x)) y) } ; odmax = & A ; <odmax = {!!} } zorn00 : Maximal A _<_ zorn00 with is-o∅ ( & HasMaximal ) ... | no not = record { maximal = ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ; A∋maximal = zorn01 ; ¬maximal<x = zorn02 } where + -- yes we have the maximal zorn03 : odef HasMaximal ( & ( ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ) ) zorn03 = ODC.x∋minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) zorn01 : A ∋ ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) @@ -167,6 +196,7 @@ zorn02 : {x : HOD} → A ∋ x → ¬ (ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) < x) zorn02 {x} ax m<x = proj2 (zorn03 (& x) ax) (subst₂ (λ j k → j < k) (sym *iso) (sym *iso) m<x ) ... | yes ¬Maximal = ⊥-elim ( ZChain→¬SUP (z (& A) (≡o∅→=od∅ ¬Maximal)) ( supP B (ZChain.B⊆A (z (& A) (≡o∅→=od∅ ¬Maximal))) (ZChain.total (z (& A) (≡o∅→=od∅ ¬Maximal))) )) where + -- if we have no maximal, make ZChain, which contradict SUP condition z : (x : Ordinal) → HasMaximal =h= od∅ → ZChain A x _<_ z x nomx = TransFinite (ind nomx) x B = ZChain.B (z (& A) (≡o∅→=od∅ ¬Maximal) )