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!ϓϩάϥϚͷͨΊͷ" Ϟφυ ݍ࿦ ϓϩάϥϚͷͨΊͷݍ࿦ษڧձDBUQH Ҵݟହ޺!JOBNJZ

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Ҵݟହ޺!JOBNJZ "CFNB57
 J04ΤϯδχΞ 4XJGU)BTLFMM&MN ݍ࿦ྺ೥ J04%$ొஃ

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ϒϩάهࣄ w ݍ࿦ͷΦεεϝษڧ๏ʢϓϩάϥϚ޲͚ʣ2JJUB w J04%$+BQBOʮݍ࿦ͱ4XJGU΁ͷԠ༻ʯൃදεϥΠυϝϞ 2JJUB w J04%$+BQBOͰൃදͨ͠ʮݍ࿦ͱ4XJGU΁ͷԠ༻ʯͷิ଍ 2JJUB

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!ϓϩάϥϚͷͨΊͷ" Ϟφυೖ໳

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ϓϩάϥϚͷͨΊͷϞφυೖ໳ ശͰߟ͑Δ'VODUPSɺ"QQMJDBUJWFͦͯ͠.POBE2JJUB

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!ϓϩάϥϚͷͨΊͷ" Ϟφυ ݍ࿦

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ͳͥϓϩάϥϚ͸ Ϟφυʹऒ͔ΕΔͷ͔ʁ

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ؔ਺ܕϓϩάϥϛϯά αʔόʔαΠυ։ൃ

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ΫϥΠΞϯτ։ൃ ΦϒδΣΫτࢤ޲ʴؔ਺ܕ

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ΫϥΠΞϯτ։ൃݴޠͷॆ࣮ʢྫɿ4XJGUʣ w0QUJPOBMDIBJOJOH •view?.superview?.frame ==
 view.flatMap { $0.superview }.map { $0.frame } w1SPNJTF'VUVSF •sendRequest(req).then { nextRequest($0) } wؔ਺ܕϦΞΫςΟϒϓϩάϥϛϯά wObservable.flatMap(merge, concat, first, latest, race, …)

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// RxSwiftΛ࢖ͬͨɺΠϯΫϦϝϯλϧαʔνͷྫ searchBar.rx.text .debounce(0.1, scheduler: backgroundScheduler) .flatMapLatest { API.fetchData(query: $0) } .map { $0.results.map(SectionModel.init) } .observeOn(mainScheduler) .bind(to: tableView.rx.items( cellIdentifier: "Cell", configureCell: { index, model, cell in cell.textLabel?.text = model } ) .disposed(by: disposeBag)

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// RxSwiftΛ࢖ͬͨɺΠϯΫϦϝϯλϧαʔνͷྫ searchBar.rx.text .debounce(0.1, scheduler: backgroundScheduler) .flatMapLatest { API.fetchData(query: $0) } .map { $0.results.map(SectionModel.init) } .observeOn(mainScheduler) .bind(to: tableView.rx.items( cellIdentifier: "Cell", configureCell: { index, model, cell in cell.textLabel?.text = model } ) .disposed(by: disposeBag)

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Ϟφυ flatMap / then / >>=

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ͦͷଞͷϞφυ׆༻ࣄྫ w)BTLFMMEPߏจ4DBMBGPS಺แදه %4- w&JUIFSʢΤϥʔॲཧʣ w454UBUF ঢ়ଶૢ࡞ w*0ʢ෭࡞༻ʣ w$PODVSSFOUʢฒྻॲཧʣ w1BSTFDʢߏจղੳʣ

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Ϟφυߏ଄Խఆཧͷͭͷجຊ੍ޚߏ଄Λ΋ͭ ॱ࣍ ˺"QQMJDBUJWF 
 ϓϩάϥϜʹه͞Εͨॱʹɺஞ࣍ॲཧΛߦͳ͍ͬͯ͘ ൓෮ ˺'VODUPS 
 Ұఆͷ৚͕݅ຬͨ͞Ε͍ͯΔؒॲཧΛ܁Γฦ͢ ෼ذ ˺.POBE 
 ͋Δ৚͕݅੒ཱ͢ΔͳΒॲཧ"Λɺͦ͏Ͱͳ͚Ε͹ॲཧ#Λߦ͏

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A monad is just a monoid in the category of endofunctors, 
 what's the problem? ෆ׬શʹ͓ͯ͠Αͦਖ਼͘͠ͳ͍ϓϩάϥϛϯάݴޠখ࢙ ݩωλ͸ɺ.BD-BOFʮݍ࿦ͷجૅʯ

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Ϟφυ͸୯ͳΔࣗݾؔख ͷݍʹ͓͚ΔϞϊΠυର ৅ͩΑɻԿ͔໰୊Ͱ΋?

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Ϟφυ͸୯ͳΔࣗݾؔख ͷݍʹ͓͚ΔϞϊΠυର ৅ͩΑɻԿ͔໰୊Ͱ΋?

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ࣗݾؔखͷݍ ϞϊΠυର৅

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ؔख Functor

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ݍ ݍ

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ݍ ݍ

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ݍ ݍ

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ܕͷݍ String [String] Int [Int] Bool [Bool] Array ܕͷݍ f g map(f ) map(g) g ∘ f map(g ∘ f ) map(g) ∘ map(f ) =

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A B C ݍ F f g g ∘ f ݍ F(f) F(g) F(A) F(B) F(C) F(g ∘ f) F(g) ∘ F(f) =

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ؔखͷ߹੒ Functor Composition

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A B C ݍ f g g ∘ f ݍ F(f) F(g) F(A) F(B) F(C) F(g ∘ f) F(g) ∘ F(f) = F

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ݍ ݍ F A B C f g g ∘ f F( f ) F(g) F(A) F(B) F(C) F(g ∘ f ) F(g) ∘ F( f ) =

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খ͍͞ݍ খ͍͞ݍ (খ͍͞ݍͷݍ) F A B C f g g ∘ f F( f ) F(g) F(A) F(B) F(C) F(g ∘ f ) F(g) ∘ F( f ) = Cat

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খ͍͞ݍ খ͍͞ݍ (খ͍͞ݍͷݍ) F A B C f g g ∘ f F( f ) F(g) F(A) F(B) F(C) F(g ∘ f ) F(g) ∘ F( f ) = Cat খ͍͞ݍ G G(F( f )) G(F(g)) G(F(A)) G(F(B)) G(F(C)) G(F(g ∘ f )) G(F(g)) ∘ G(F( f )) =

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খ͍͞ݍ খ͍͞ݍ (খ͍͞ݍͷݍ) F A B C f g g ∘ f F( f ) F(g) F(A) F(B) F(C) F(g ∘ f ) F(g) ∘ F( f ) = Cat খ͍͞ݍ G G(F( f )) G(F(g)) G(F(A)) G(F(B)) G(F(C)) G(F(g ∘ f )) G(F(g)) ∘ G(F( f )) = G ∘ F

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A B C ݍ G ∘ F f g g ∘ f ݍ G(F(f)) G(F(g)) G(F(A)) G(F(B)) G(F(g ∘ f)) G(F(g)) ∘ G(F(f)) = G(F(C))

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(খ͍͞ݍͷݍ) F C D Cat

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(খ͍͞ݍͷݍ) F G C D E Cat G ∘ F ؔखͷ߹੒

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(খ͍͞ݍͷݍ) F G C D E Cat G ∘ F I I I ߃౳ؔख ؔखͷ߹੒

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(খ͍͞ݍͷݍ) Cat F G C D E G ∘ F E′ H H ∘ G (H ∘ G) ∘ F H ∘ (G ∘ F) ྆ऀ͸౳͍͠ (݁߹཯Λຬͨ͢)

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ؔखݍ Functor Category

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(খ͍͞ݍͷݍ) Cat F C D

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(খ͍͞ݍͷݍ) Cat C D F H G

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(খ͍͞ݍͷݍ) Cat C D F H G α β

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A B F f H F(A) F(B) F(f) α C D G(A) G(B) G(f) H(A) H(B) H(f) β G αA αB βA βB

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A B F f H F(A) F(B) F(f) α C D G(A) G(B) G(f) H(A) H(B) H(f) β G αA αB βA βB β ∘ α βA ∘ αA βB ∘ αB ࣗવม׵ͷ ਨ௚߹੒
 (݁߹཯Λຬͨ͢)

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(খ͍͞ݍͷݍ) Cat C D F H G α β ؔखݍ DC

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(ؔखݍ) α β F G H DC

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α β F G H β ∘ α id id id ߃౳ࣗવม׵ ࣗવม׵ͷ(ਨ௚)߹੒ (݁߹཯Λຬͨ͢) (ؔखݍ) DC

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ࣗݾؔखͷݍ Category of Endofunctors

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(খ͍͞ݍͷݍ) Cat C D F H G

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(খ͍͞ݍͷݍ) Cat C C F H G

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(খ͍͞ݍͷݍ) Cat C F G H

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(খ͍͞ݍͷݍ) Cat C G ∘ F F G H I

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(খ͍͞ݍͷݍ) Cat C G ∘ F F G H I ࣗݾؔखͷݍ CC α β

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α β F G H (ؔखݍ) DC

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α β F G H (ࣗݾؔखͷݍ) CC I G ∘ F ߃౳ؔख (୯Ґݩ) ؔखͷ߹੒ (ೋ߲ԋࢉ)

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ϞϊΠυ Monoid

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ϞϊΠυ ೋ߲ԋࢉ ୯Ґݩͷଘࡏ

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(Int, +, 0) a + 0 = a 0 + a = a (a + b) + c = a + (b + c) ϞϊΠυͷྫ

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(Int, *, 1) a * 1 = a 1 * a = a (a * b) * c = a * (b * c) ϞϊΠυͷྫ

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(String, +, ””) a + ”” = a ”” + a = a (a + b) + c = a + (b + c) ϞϊΠυͷྫ

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(Array, +, []) a + [] = a [] + a = a (a + b) + c = a + (b + c) ϞϊΠυͷྫ

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(Bool, &&, true) a && true = a true && a = a (a && b) && c = a && (b && c) ϞϊΠυͷྫ

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(Bool, ||, false) a || false = a false || a = a (a || b) || c = a || (b || c) ϞϊΠυͷྫ

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(Int, max, minBound) max(a, minBound) = a max(minBound, a) = a max(max(a, b), c)
 = max(a, max(b, c)) ϞϊΠυͷྫ

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(Int, min, maxBound) min(a, maxBound) = a min(maxBound, a) = a min(min(a, b), c)
 = min(a, min(b, c)) ϞϊΠυͷྫ

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ϞϊΠυ w ू߹ɺೋ߲ԋࢉ υοτ ɺ୯Ґݩͷͭ૊
 ͰɺҎԼͷ৚݅Λຬͨ͢ɿ w ݁߹཯೚ҙͷʹ͍ͭͯ w ୯Ґݩ೚ҙͷʹ͍ͭͯ (a ⋅ b) ⋅ c = a ⋅ (b ⋅ c) e ⋅ a = a ⋅ e = e a, b, c ∈ M a ∈ M M ⋅ e ∈ M (M, ⋅ ,e) (M, ⋅ ,e)

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// Swift ʹ͓͚ΔϞϊΠυ protocol Monoid { static func <> (lhs: Self, rhs: Self) -> Self static var empty: Self } -- Haskell ʹ͓͚ΔϞϊΠυ class Monoid a where (<>) :: a -> a -> a mempty :: a

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ݍ࿦ʹ͓͚ΔϞϊΠυ Monoid in Category Theory

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Int 1 … 3 (+): (Int,Int) -> Int ΛࣹͰදݱ͍ͨ͠l 0 2 ୯Ґݩ

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Int 0 1 2 … 3 ୯Ґݩ ϖΞΛ࡞ͬͯɾɾɾ ೋ߲ԋࢉʁ ͜ͷॻ͖ํ͸ݍ࿦Ͱ͸ͳ͍ ΛࣹͰදݱ͍ͨ͠l (+): (Int,Int) -> Int

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σΧϧτੵ ੵू߹ (0,0) (1,0) (0,2) … (1,2) Int 0 1 2 … 3 (Int,Int) ͱ΋ॻ͘ Int × Int Int 0 1 2 … 3 Int 0 1 2 … 3 ϖΞ λϓϧ (+)

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(0,0) (1,0) (0,2) … (1,2) Int 0 1 2 … 3 (+) ೋ߲ԋࢉΛIntͷ ʮ֎ʯͰఆٛ͢Δ ಺෦Ͱࣹ(ߏ଄)Λ ߟ͑ͳ͍ (Int,Int) Int 0 1 2 … 3 Int 0 1 2 … 3 ϖΞ λϓϧ σΧϧτੵ ੵू߹ σΧϧτੵ Int × Int

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(+) : (Int, Int) -> Int ΧϦʔԽ͢Δͱɺ (+) : Int -> (Int -> Int) ϖΞΛ࡞ͬͯɺIntʹม׵ ϖΞΛ࡞Βͣɺ IntΛʮࣗݾؔ਺ʯʹม׵ ͱ͜ΖͰɾɾɾ

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୯Ұର৅ݍ Int (0+) (1+) (2+) Int (0+) Int -> Int 0 1 2 … 3 (1+) (2+) … (3+) (+)

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୯Ұର৅ݍ Int (0+) (1+) (2+) Int (0+) Int -> Int 0 1 2 … 3 (1+) (2+) … (3+) (+) ߃౳ม׵ ࣹͳͷͰɺ ߹੒Մೳ

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0 + 1 + 2 (0+) ∘ (1+) ∘ (2+) (+)ͷ෦෼ద༻ ࣗݾؔ਺Խ Int (0+) Int -> Int 0 1 2 … 3 (1+) (2+) … (3+) (+)

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Int (0+) Int -> Int 0 1 2 … 3 (1+) (2+) … (3+) { f in f(0) } \f -> f 0 0 + 1 + 2 (0+) ∘ (1+) ∘ (2+) ୯ҐݩΛద༻ ஋ʹ໭͢ PS

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Int (0+) Int -> Int 0 1 2 … 3 (1+) (2+) … (3+) 0 + 1 + 2 (0+) ∘ (1+) ∘ (2+) ͷ෦෼ద༻ ࣗݾؔ਺Խ ୯ҐݩΛద༻ ஋ʹ໭͢ ୯Ґݩ ೋ߲ԋࢉ ୯Ґݩ ೋ߲ԋࢉ ࣗݾؔ਺͸ ϞϊΠυʂ

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0 + 1 + 2 ͷ෦෼ద༻ ؔ਺Խ ୯ҐݩΛద༻ ஋ʹ໭͢ (0+) ∘ (1+) ∘ (2+) ୯Ґݩ ೋ߲ԋࢉ έΠϦʔදݱ ೚ҙͷϞϊΠυ͸ɺͦͷࣗݾ४ಉܕ ˺ࣗݾؔ਺ ͷϞϊΠυͷ෦෼ϞϊΠυ Int (0+) Int -> Int 0 1 2 … 3 (1+) (2+) … (3+)

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ϞϊΠυର৅ Monoid Object

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σΧϧτੵ ੵू߹ (0,0) (1,0) (0,2) … (1,2) Int 0 1 2 … 3 Int 0 1 2 … 3 Int 0 1 2 … 3 ೋ߲ԋࢉΛIntͷ ʮ֎ʯͰఆٛ͢Δ ୯Ґݩ (Int,Int) ϖΞ ಺෦Ͱࣹ(ߏ଄)Λ ߟ͑ͳ͍ λϓϧ (+)

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(Int,Int) Int 0 1 2 … 3 ೋ߲ԋࢉΛIntͷ ʮ֎ʯͰఆٛ͢Δ ୯Ґݩ (+)

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Int 0 1 2 … 3 ೋ߲ԋࢉΛIntͷ ʮ֎ʯͰఆٛ͢Δ ୯Ґݩ () ୯ҐݩΛܾΊΔࣹΛ ʮ֎ʯͰఆٛ͢Δ(ҰൠԽݩ) Ұݩू߹ (Int,Int) 0 (+) Unit

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ϞϊΠυର৅ (M, μ, η) μ η 1 ୯Ґࣹ ೋ߲ԋࢉ M × M M

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Int 0 1 2 … 3 Unit () (Int,Int) ⟨ Int, Int ⟩ ϖΞ 0 (+) = Int ⊗ Int ⊗ = (, ) (Int × IntͷҰൠԽ)

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(Int × IntͷҰൠԽ) Int 0 1 2 … 3 () () ϖΞ(ܕͰ͸ͳ͍) λϓϧܕ ʮϖΞΛ࡞Δʯ͜ͱͱʮϖΞΛܕͱ Έͳ͢(λϓϧ)ʯ͜ͱΛ෼཭͢Δ ςϯιϧੵ (ޙड़) (Int,Int) Int ⊗ Int = ⟨ Int, Int ⟩ ϖΞ ⊗ = (, )

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ϞϊΠυର৅ (M, μ, η) μ η I M ⊗ M M ୯Ґࣹ ೋ߲ԋࢉ

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(σΧϧτੵͷҰൠԽ) ⊗ ୯Ґର৅ (Ұݩू߹ͷҰൠԽ) μ η I M ⊗ M M ⊗ ςϯιϧੵ

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ϞϊΠμϧݍ Monoidal Category

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ςϯιϧੵ ⊗ : C × C → C M I M ⊗ M ⊗ ୯Ґର৅ (ϞϊΠμϧݍ) C (C, ⊗ ,I, α, λ, ρ)

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M I M ⊗ M M ⊗ I M ⊗ M′ (M ⊗ M′) ⊗ M′′ طଘͷର৅ͷϖΞ͔Β ৽ͨͳର৅Λ࡞Γग़ͤΔ ςϯιϧੵ ⊗ : C × C → C (ϞϊΠμϧݍ) C (C, ⊗ ,I, α, λ, ρ) ⊗

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ςϯιϧੵ ⟨M ⊗ M′, M′′⟩ ੵͷݍ I M M ⊗ M ⟨M, M⟩ C × C M ⊗ M′ ⟨M, M′⟩ ⋯ ⊗ (M ⊗ M′) ⊗ M′′ ݍͷϖΞ ⊗ ⋯ (ϞϊΠμϧݍ) C (C, ⊗ ,I, α, λ, ρ) ૒ؔख

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ϞϊΠμϧݍ wςϯιϧੵ w୯Ґର৅ w݁߹ࢠ wࠨ୯Ґࢠ wӈ୯Ґࢠ ⊗ : C × C → C I ∈ Ob(C) αA,B,C : (A ⊗ B) ⊗ C ≅ ⟶ A ⊗ (B ⊗ C) λA : I ⊗ A ≅ ⟶ A ρA : A ⊗ I ≅ ⟶ A ౳߸ͷͱ͖ɺ ετϦΫτϞϊΠμϧݍ (C, ⊗ ,I, α, λ, ρ)

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ϞϊΠμϧݍ (C, ⊗ ,I, α, λ, ρ) ݁߹ࢠ ݁߹ॱংͷೖΕସ͑Մೳ ୯Ґࢠ ୯Ґݩ Λ௥ՃɾআڈՄೳ I

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μ η I M ⊗ M M ϞϊΠυʹʮϞϊΠμϧݍʯʹ͓͚ΔʮϞϊΠυର৅ʯ (ϞϊΠμϧݍ) (C, ⊗ ,I, α, λ, ρ) C

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࠶ߟɿࣗݾؔखͷݍ Revisiting Category of Endofunctors

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α β F G H I G ∘ F ߃౳ؔख (୯Ґݩ) ؔखͷ߹੒ (ೋ߲ԋࢉ) (ࣗݾؔखͷݍ) CC

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ؔखͷ߹੒ (ೋ߲ԋࢉ) (ࣗݾؔखͷݍ) CC ߃౳ؔख (୯Ґݩ) F I F ∘ F ∘

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M ⊗ M ςϯιϧੵ ⊗ : C × C → C ୯Ґର৅ M I M ⊗ M (ϞϊΠμϧݍ) C (C, ⊗ ,I, α, λ, ρ) ⊗

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ؔखͷ߹੒ˠςϯιϧੵ ߃౳ؔखˠ୯Ґର৅ ؔखͷ߹੒ʹ͍ͭͯ ݁߹཯ɾ୯Ґ཯͕੒Γཱͭ

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ࣗݾؔखͷݍ ʹετϦΫτϞϊΠμϧݍ

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Ϟφυ Monad

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F(C) D ݍ F C ݍ U(D) ࣹͷू߹ U ≅

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F(C) D ݍ F U C U(D) ݍ F(U(D)) U(F(C)) f f F(f) U(f) counitD ηC ϵD unitC = =

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F(C) D ݍ F U C ݍ F(U(D)) U(F(C)) ηC ϵD

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F(C) D ݍ F U C ݍ F(U(D)) U(F(C)) ηC ϵD U(F(U(F(C)))) F(U(F(C)))

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F(C) D ݍ F U C ݍ F(U(D)) U(F(C)) ηC U(F(U(F(C)))) F(U(F(C))) ϵD ϵF(C)

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F(C) D ݍ F U C ݍ F(U(D)) U(F(C)) ηC U(F(U(F(C)))) F(U(F(C))) ϵD ϵF(C) UϵF(C)

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C ݍ T(C) ηC T2(C) T = UF UϵF(C)

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C ݍ ηC T T ∘ T I T(C) T2(C) UϵF(C)

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C ݍ ηC T T ∘ T μ η I T(C) T2(C) UϵF = ࣗݾؔखͷݍ UϵF(C)

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μ η I M ⊗ M M ϞϊΠυʹʮϞϊΠμϧݍʯʹ͓͚ΔʮϞϊΠυର৅ʯ (ϞϊΠμϧݍ) (C, ⊗ ,I, α, λ, ρ) C

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T T ∘ T μ η Ϟφυʹʮࣗݾؔखͷݍ(ετϦΫτϞϊΠμϧݍ)ʯ ʹ͓͚ΔʮϞϊΠυର৅ʯ (ࣗݾؔखͷݍ) CC (CC, ∘ ,I) I

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Ϟφυ wࣗݾؔख wࣗવม׵ wࣗવม׵ T : C → C μ : T ⊗ T → T η : I → T μ ∘ Tμ = μ ∘ μT μ ∘ Tη = μ ∘ ηT = 1T Ϟφυଇ ݁߹཯ ୯Ґ཯ (T, μ, η)

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Ϟφυ (T, μ, η) μ ∘ Tμ = μ ∘ μT μ ∘ Tη = μ ∘ ηT = 1T ݁߹཯ ୯Ґ཯

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ʹࣗݾؔखͷݍʹ͓͚ΔϞϊΠυର৅ (T, μ, η) Ϟφυ

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)BTLFMMʹ͓͚ΔϞφυ class Functor f where fmap :: (a -> b) -> f a -> f b class Functor f => Applicative f where pure :: a -> f a (<*>) :: f (a -> b) -> f a -> f b class Applicative m => Monad m where (>>=) :: m a -> (a -> m b) -> m b return :: a -> m a return = pure (ࣗݾ)ؔख ΞϓϦΧςΟϒؔख Ϟφυ

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join :: (Monad m) => m (m a) -> m a join x = x >>= id )BTLFMMʹ͓͚ΔϞφυ class Applicative m => Monad m where (>>=) :: m a -> (a -> m b) -> m b return :: a -> m a return = pure Ϟφυ

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join :: (Monad m) => m (m a) -> m a join x = x >>= id )BTLFMMʹ͓͚ΔϞφυ class Applicative m => Monad m where (>>=) :: m a -> (a -> m b) -> m b return :: a -> m a return = pure ࣗવม׵ η : I → T

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join :: (Monad m) => m (m a) -> m a join x = x >>= id )BTLFMMʹ͓͚ΔϞφυ class Applicative m => Monad m where (>>=) :: m a -> (a -> m b) -> m b return :: a -> m a return = pure μ : T ⊗ T → T ࣗવม׵

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join :: (Monad m) => m (m a) -> m a join x = x >>= id )BTLFMMʹ͓͚ΔϞφυ class Applicative m => Monad m where (>>=) :: m a -> (a -> m b) -> m b return :: a -> m a return = pure flatMap

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ϓϩάϥϛϯάͷੈքͰ͸ɺ joinΑΓ΋ flatMap(>>=)ͷํ͕ॏཁ

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)BTLFMMʹ͓͚ΔflatMap == (>>=) (>>=) :: m a -> (a -> m b) -> m b ͜ͷܕͷҙຯ͸Կʁ

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ΫϥΠεϦ߹੒ͱ ΫϥΠεϦɾτϦϓϧ Kleisli Composition & Kleisli Triple

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C B A C ࣹͷ߹੒͕Մೳ

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T(B) C B A T(C) C Ϟφυ͕ίυϝΠϯ(ऴҬ)ͷ ࣹ͸߹੒Ͱ͖ͳ͍

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C B A C B A T(B) T(C) ⋯ C CT ΫϥΠεϦݍ ⋯ ⋯ f g gT fT ΫϥΠεϦࣹ (A -> T(B)ͷܗͷࣹ)

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C B A C B A T(B) T(C) ⋯ C CT ⋯ ⋯ f g gT fT ྆ऀ ͷࣹͷू߹ Λ ରԠ͍ͤͨ͞

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f g C B A C B A T(B) T(C) ⋯ ⋯ C CT ⋯ gT fT gT ∘ fT ରԠͰ͖Δͱɺ ݍͷࣹͷ߹੒A -> C͕ ݍ ͷΫϥΠεϦࣹͷ߹੒ A -> T(C)ʹରԠ͢Δ CT C

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F G C B A C B A T(B) T(C) ⋯ C CT ⋯ ⋯ f g gT fT ͱͷ ਵ൐ؔखΛߟ͑Δ CT C

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C B A C B A T(B) T(C) ⋯ C CT ⋯ ⋯ f g gT fT F G ຒΊࠐΈؔख ର৅ͱࣹΛ ͦͷ··ࣸ͢ fT ≠ F(f) ஫ҙɿ

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f g C B A F G C B A T(B) T(C) ⋯ ⋯ C CT T(A) ⋯ gT fT G(fT ) G(gT ) ֦ுؔख ͷର৅Λ ͷର৅ʹࣸ͢ A CT C T(A) G(A) = T(A) G(fT ) ≠ T(f) ஫ҙɿ

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f g C B A F G C B A T(B) T(C) ⋯ C CT T(A) F gT fT G(fT ) G(gT ) T2(C) T2(B) T(f ) T(g) μB ηB ηC μA ηA ͸ϞφυͳͷͰ ͜ΕΒͷࣹ͕ଘࡏ͢Δ T μC

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g C B A F G C B ⋯ C CT T(A) F gT fT G(fT ) G(gT ) T2(B) gT ∘ fT T(f ) μB ηB μA ηA f A T(B) T(C) T2(C) T(g) μC μC ∘ T(g) ∘ f ηC ͱͷΫϥΠεϦ߹੒ f g

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ΫϥΠεϦ߹੒ C A gT ∘ fT f g B A T(B) T(C) μC ∘ T(g) ∘ f g < = < f = def < = < /PUFΫϥΠεϦ߹੒͸ ݁߹཯ɾ୯Ґ཯Λຬͨ͢

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wΫϥΠεϦ߹੒ ࢀߟ ؔ਺߹੒ ࠨ୯Ґ཯ return <=< f ≡ f ӈ୯Ґ཯ f <=< return ≡ f ݁߹཯ (h <=< g) <=< f ≡ h <=< (g <=< f) Ϟφυଇ ΫϥΠεϦ߹੒ (<=<) :: (b -> m c) -> (a -> m b) -> (a -> m c) (.) :: (b -> c) -> (a -> b) -> (a -> c) ࠨ୯Ґ཯ id . f ≡ f ӈ୯Ґ཯ f . id ≡ f ݁߹཯ (h . g) . f ≡ h . (g . f)

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)BTLFMMʹ͓͚ΔΫϥΠεϦ߹੒ (<=<) :: Monad m => (a -> m b) -> m a -> m b g <=< f = join . fmap g . f μC ∘ T(g) ∘ f g < = < f = def

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)BTLFMMʹ͓͚ΔΫϥΠεϦ߹੒ ࣮ࡍͷίʔυ -- | Left-to-right composition of Kleisli arrows. (>=>) :: Monad m => (a -> m b) -> (b -> m c) -> (a -> m c) f >=> g = \x -> f x >>= g -- | Right-to-left composition of Kleisli arrows. -- > (.) :: (b -> c) -> (a -> b) -> a -> c -- > (<=<) :: Monad m => (b -> m c) -> (a -> m b) -> a -> m c (<=<) :: Monad m => (b -> m c) -> (a -> m b) -> (a -> m c) g <=< f = \x -> g =<< f x —- (<=<) = flip (>=>)

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-- | Left-to-right composition of Kleisli arrows. (>=>) :: Monad m => (a -> m b) -> (b -> m c) -> (a -> m c) f >=> g = \x -> f x >>= g -- | Right-to-left composition of Kleisli arrows. -- > (.) :: (b -> c) -> (a -> b) -> a -> c -- > (<=<) :: Monad m => (b -> m c) -> (a -> m b) -> a -> m c (<=<) :: Monad m => (b -> m c) -> (a -> m b) -> (a -> m c) g <=< f = \x -> g =<< f x —- (<=<) = flip (>=>) (=<<) :: Monad m => (a -> m b) -> m a -> m b f =<< x = x >>= f —- (=<<) = flip (>>=) )BTLFMMʹ͓͚ΔΫϥΠεϦ߹੒ ࣮ࡍͷίʔυ flip(flatMap)

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(=<<) :: (a -> m b) -> m a -> m b ͜ͷܕͷҙຯ͸Կʁ )BTLFMMʹ͓͚Δflip(flatMap) == (=<<)

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(=<<) :: (a -> m b) -> m a -> m b ͜ͷܕͷҙຯ͸Կʁ ʮa -> m bʯͷࣹ͔Β ʮm a -> m bʯͷࣹʹࣸؔ͢਺ )BTLFMMʹ͓͚Δflip(flatMap) == (=<<)

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A T(B) C T(A) ηA CT B A g C B T(C) ⋯ G(gT ) T2(C) T(g) ηB ηC μA μC C gT fT T2(B) T(f ) μB F G G(fT ) f

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A T(B) C T(A) ηA CT B A g C B T(C) ⋯ G(gT ) T2(C) T(g) ηB ηC μA μC C gT fT T2(B) T(f ) μB F G = < < ֦ுԋࢉ

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A T(B) C T(A) ηA CT B A g C B T(C) ⋯ G(gT ) T2(C) T(g) ηB ηC μA μC C gT fT T2(B) T(f ) μB = < < ֦ுԋࢉ F G ֦ுؔख

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F G f A T(B) C T(A) ηA CT B A g C B T(C) ⋯ G(gT ) T2(C) T(g) ηB ηC μA μC C gT fT G(fT ) T2(B) T(f ) μB G(fT ) = μB ∘ T(f) ӈล͸Λ Ҿ਺ʹऔΔؔ਺ f

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Ϟφυɾ֦ுԋࢉ = < < A T(B) T(A) B A fT T2(B) T(f ) μB μB ∘ T(f) f = < < = def f

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Ϟφυɾ֦ுԋࢉ f A T(B) T(A) B A fT T2(B) T(f ) μB f = < < Λ࢖Θͣ Λ࢖ͬͯϞφυΛදݱ͢Δ μ = < < = < < ֦ுԋࢉ = < <

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ΫϥΠεϦɾτϦϓϧ wࣗݾؔख wࣗવม׵ w֦ுԋࢉ (T, η, = < < ) T : C → C η : I → T = < <: (A → T(B)) → T(A) → T(B) (g = < < ) ∘ (f = < < ) = (((g = < < ) ∘ f) = < < ) (η = < < ) = 1T ݁߹཯ ୯Ґ཯ (f = < < ) ∘ η = f Ϟφυଇ ,MFJTMJ4UBSͱ΋ݴ͏ f* def = (f = < < )

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ΫϥΠεϦɾτϦϓϧ ʹϞφυͷjoinΛ࢖Θͣ flip(flatMap)Λ࢖ͬͨผͷܗ (T, η, = < < )

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)BTLFMMʹ͓͚Δ֦ுԋࢉ (=<<) :: Monad m => (a -> m b) -> m a -> m b (=<<) f = join . fmap f μB ∘ T(f) f = < < = def = < <

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ࢀߟɿjoin, <=<, =<< ͷؔ܎ -- from `join` join :: (Monad m) => m (m a) -> m a join x = id =<< x join x = (id <=< id) x -- from `<=<` (<=<) :: Monad m => (b -> m c) -> (a -> m b) -> (a -> m c) g <=< f = \a -> g =<< f a g <=< f = join . fmap g . f -- from `=<<` (=<<) :: Monad m => (a -> m b) -> m a -> m b f =<< m = (f <=< id) m f =<< m = (join . fmap f) m

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wΫϥΠεϦ߹੒ wϞφυflatMap ࠨ୯Ґ཯ return >=> g ≡ g ӈ୯Ґ཯ f >=> return ≡ f ݁߹཯ (f >=> g) >=> h ≡ f >=> (g >=> h) ࠨ୯Ґ཯ return a >>= f ≡ f a ӈ୯Ґ཯ m>>= return ≡ m ݁߹཯ (m >>= f) >>= g ≡ m >>= (\x -> f x >>= g) Ϟφυଇ(flatMap / >>=) (>=>) :: (a -> m b) -> (b -> m c) -> (a -> m c) (>>=) :: m a -> (a -> m b) -> m b (<=<)ͷٯ޲͖ ࠨ͔Βӈ΁ͷΫϥΠεϦ߹੒ f >=> g = \a -> f a >>= g

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·ͱΊ

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·ͱΊ wϞφυ wؔखݍɺࣗݾؔखͷݍ wϞϊΠυɺϞϊΠυର৅ɺϞϊΠμϧݍ wࣗݾؔखͷݍʹ͓͚ΔϞϊΠυର৅ wΫϥΠεϦɾτϦϓϧ wjoinͷ୅ΘΓʹflip(flatMap) /=<<Λ࢖͏

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(T, μ, η) (T, η, = < < )

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No content

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͞ΒʹֶͿͨΊʹ w ετϦϯάਤ 4USJOH%JBHSBN .POBET.BEF%J⒏DVMU
 4USJOHEJBHSBNT BEKVODUJPOTBOENPOBET:PV5VCF

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͞ΒʹֶͿͨΊʹ wϞφυม׵ࢠ .POBE5SBOTGPSNFS wࣗ༝Ϟφυ 'SFF.POBE w'SFFS0QFSBUJPOBMϞφυ ࣗ༝ؔखࣗ༝Ϟφυ wࢦඪ෇͖Ϟφυ *OEFYFE.POBE w&YUFOTJCMF'SFFS&GGFDUT 'SFFS0QFO6OJPO w(SBEFEϞφυ ϞφυͷҰൠԽ

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ࢀߟจݙ wϞφυ΢ΥʔΫεϧʔ)BTLFMM wϞφυʹ͍ͭͯϞϊΠυɾϞϊΠυݍɾϞϊΠυର৅ɾϞφυٙ೦͸୳ڀͷಈػͰ͋Γɺ୳ڀͷ།Ұͷ໨త͸৴ ೦ͷ֬ఆͰ͋Δɻ wϓϩάϥϛϯάʹ͓͚ΔϞφυͷॳظͷྺ࢙ʹ͍ͭͯ࠶ؼͷ൓෮ wʮϞφυ͸୯ͳΔࣗݾؔखͷݍʹ͓͚ΔϞϊΠυର৅ͩΑɻԿ͔໰୊Ͱ΋ʁʯ2JJUB w)BTLFMMͷ.POBEΛݍ࿦ͷ.POBE,MFJTMJ5SJQMFͱରൺ͢ΔΊ΋Ί΋ wϞφυͷఆٛͱ͔ᐻࢁਖ਼޾ͷΩϚΠϥࣂҭه wϞϊΠυ͔ΒϞφυΛ࡞Δᐻࢁਖ਼޾ͷΩϚΠϥࣂҭه w.POBET.BEF%JGpDVMU w/PUJPOTPG$PNQVUBUJPOBT.POPJET

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Thanks! Yasuhiro Inami @inamiy