Mercurial > hg > Members > kono > Proof > ZF-in-agda
diff src/filter.agda @ 516:286016848403
...
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
---|---|
date | Sat, 16 Apr 2022 12:20:08 +0900 |
parents | 1546541ed461 |
children | 9084a26445a7 |
line wrap: on
line diff
--- a/src/filter.agda Sat Apr 16 12:10:09 2022 +0900 +++ b/src/filter.agda Sat Apr 16 12:20:08 2022 +0900 @@ -228,20 +228,24 @@ F∋-p = incl U⊆F U∋-p f0 : filter F ∋ od∅ f0 = subst (λ k → odef (filter F) k ) (trans (cong (&) ∩-comm) (cong (&) [a-b]∩b=0 ) ) ( filter2 F F∋p F∋-p ( CAP {!!} {!!}) ) - --- open import zorn + +_⊆'_ : ( A B : HOD ) → Set n +_⊆'_ A B = (x : Ordinal ) → odef A x → odef B x + +import zorn +open zorn O _⊆'_ --- MaximumSubset : {L P : HOD} --- → o∅ o< & L → o∅ o< & P → P ⊆ L --- → IsPartialOrderSet P _⊆'_ --- → ( (B : HOD) → B ⊆ P → IsTotalOrderSet B _⊆'_ → SUP P B _⊆'_ ) --- → Maximal P (_⊆'_) --- MaximumSubset {L} {P} 0<L 0<P P⊆L PO SP = Zorn-lemma {P} {_⊆'_} 0<P PO SP +MaximumSubset : {L P : HOD} + → o∅ o< & L → o∅ o< & P → P ⊆ L + → IsPartialOrderSet P + → ( (B : HOD) → B ⊆ P → IsTotalOrderSet B → SUP P B ) + → Maximal P +MaximumSubset {L} {P} 0<L 0<P P⊆L PO SP = Zorn-lemma 0<P PO SP --- MaximumFilterExist : {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 LP) → o∅ o< & L → o∅ o< & (filter F) → (¬ (filter F ∋ od∅)) → MaximumFilter LP --- MaximumFilterExist {L} {P} LP NEG CAP F 0<L 0<F Fprop = {!!} where --- mf01 : Maximal O P (_⊆'_ O) --- mf01 = MaximumSubset O 0<L {!!} {!!} {!!} {!!} +MaximumFilterExist : {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 LP) → o∅ o< & L → o∅ o< & (filter F) → (¬ (filter F ∋ od∅)) → MaximumFilter LP +MaximumFilterExist {L} {P} LP NEG CAP F 0<L 0<F Fprop = record { mf = {!!} ; proper = {!!} ; is-maximum = {!!} } where + mf01 : Maximal P + mf01 = MaximumSubset 0<L {!!} {!!} {!!} {!!}