Swiftプログラミングと論理〜そして帰ってきた圏論〜 / Swift and Logic, and Category Theory

by Yasuhiro Inami

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Swiftͱ࿦ཧ ʙͦͯ͠ؼ͖ͬͯͨݍ࿦ʙ 2019/09/06 iOSDC 2019 Yasuhiro Inami / @inamiy

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࿦ཧֶ ࢥߟͷಓےΛʮਅཧ 5SVUI ʯ ͱ͍͏֓೦Λ༻͍ͯઆ໌͢Δֶ໰

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J04%$͸J04ΧϯϑΝϨϯε J04ΧϯϑΝϨϯε͸ͨͷ͠ʔ ΑͬͯɺJ04%$͸ͨͷ͠ʔ

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΋͠"ͳΒ͹# ΋͠#ͳΒ͹$ Αͬͯɺ΋͠"ͳΒ͹$

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2͜Ε͸ৗʹਖ਼͍͠ʁ

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൴ࢯ͸ΰϦϥ ΰϦϥ͸Πέϝϯ Αͬͯɺ൴ࢯ͸Πέϝϯ

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ʮਖ਼͍͠ʯͱ͸ͳΜͧ΍ʁ ԿΛਖ਼͍͠ͱߟ͑ɺࢥߟΛల։͍͚ͯ͠Δ͔ʁ

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࿦ཧͷʮਖ਼͠͞ʯΛ෼͚ͯߟ͑Δ ࿦ཧͷʮܗʯ͕ਖ਼͍͠ ࿦ཧͷʮ಺༰ʯ͕ਖ਼͍͠

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ߏจ࿦(Syntax) ه߸తͳ࿦ཧࣜͱਪ࿦نଇΛ࢖ͬͯɺ ʢػցతʹʣ৽͍͠࿦ཧࣜΛܭࢉʢূ໌ʣ͢Δ

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࿦ཧͷύλʔϯ "ͳΒ͹# "ͱ# "·ͨ͸# "Ͱͳ͍ A → B A ∧ B A ∨ B ¬A

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໋୊ͱ࿦ཧࣜ ࿦ཧه߸ ໋୊ม߲ ໋୊ఆ߲ ∧ , ∨ , ¬, → A, B, C, ⋯ ⊤ , ⊥

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࿦ཧࣜͷྫ A ∧ (B → C) (A ∨ B) ∧ (C → D) (¬A → ⊥ ) ∨ (¬B ∧ ⊤ )

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ਪ࿦نଇͷྫɿϞʔμεϙωϯε A → B A B جຊతͳਪ࿦نଇΛ ެཧܥͱͯ͠ఆΊΔ લఏ̍ɿ"ͳΒ͹# લఏ̎ɿ"Ͱ͋Δ ݁࿦ɹɿ#Ͱ͋Δ

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ূ໌ਤͷྫʢਪ࿦نଇͷద༻աఔͷ໦ߏ଄ʣ

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ҙຯ࿦(Semantics) ࿦ཧࣜʹʮਅʯ·ͨ͸ʮِʯͷ ਅཧ஋Λ෇༩ͯ͠ղऍ͢Δ

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࿦ཧࣜʢه߸ʣˠਅཧ஋ͷׂΓ౰ͯ A B ¬A A∧B A∨B A→B 1 1 0 1 1 1 1 0 0 0 1 0 0 1 1 0 1 1 0 0 1 0 0 1 ෇஋ ਅ ِ A → B A ∧ B A ∨ B ¬A ֤ม߲ͷਅِΛܾΊΔ ʹߦΛબͿ

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ਅཧ஋දͷྫ A B ¬A ¬A∨B A→B (¬A∨B) → (A→B) (A→B) → (¬A∨B) 1 1 0 1 1 1 1 1 0 0 0 0 1 1 0 1 1 1 1 1 1 0 0 1 1 1 1 1 ৗʹʹͳΔ ࿦ཧ͕ࣜଘࡏ

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(A → B) ↔ (¬A ∨ B) ͸ɺ໋୊"ɺ#ʹΑΒͣɺৗʹਅ ߃ਅ τʔτϩδʔ P ↔ Q := (P → Q) ∧ (Q → P)

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߃ਅʢτʔτϩδʔʣͷྫ ྗͱ͸ύϫʔ ຖ೔͕ΤϒϦσΠ A → A ಉҰ཯

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߃ਅʢτʔτϩδʔʣͷྫ ੈͷதʹ͸छྨͷஉ͔͠ډͳ͍ Զ͔ɺԶҎ֎͔ A ∨ ¬A ഉத཯

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߃ਅʢτʔτϩδʔʣͷྫ (A → B) → ((B → C) → (A → C)) ((((A → B) → (¬C → ¬D)) → C) → E) → ((E → A) → (D → A)) ¬(A ∧ ¬A) (A ∧ ¬A) → B ((A → B) → A) → A ¬¬A → A (A → B) ↔ (¬B → ¬A) ¬(A ∧ B) ↔ (¬A ∨ ¬B) ¬(A ∨ B) ↔ (¬A ∧ ¬B)

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(A → B) → ((B → C) → (A → C)) ((((A → B) → (¬C → ¬D)) → C) → E) → ((E → A) → (D → A)) ¬(A ∧ ¬A) (A ∧ ¬A) → B ((A → B) → A) → A ¬¬A → A (A → B) ↔ (¬B → ¬A) ¬(A ∧ B) ↔ (¬A ∨ ¬B) ¬(A ∨ B) ↔ (¬A ∧ ¬B) ߃ਅʢτʔτϩδʔʣͷྫ ໃ६཯ രൃ཯ ೋॏ൱ఆআڈ ύʔεͷ๏ଇ ରۮ཯ υɾϞϧΨϯͷ๏ଇ ਪҠ཯ ϝϨσΟεͷެཧ

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߃ਅʢτʔτϩδʔʣͷྫ ߃ਅͳ࿦ཧࣜ͸ແݶʹଘࡏ͢Δ  ˠ༗ݶͷنଇ͔Βಋ͖ग़ͤͳ͍͔ʁ A → ¬¬A A → ¬¬¬¬A A → ¬¬¬¬¬¬A ⋯

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߃ਅͳ࿦ཧ͔ࣜΒ ܗࣜతମܥΛߟ͑Δ

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ܗࣜతମܥʢه߸ʴจ๏ʴެཧʴਪ࿦نଇʣ ެཧʹલఏͱͯ͠ಋೖ͞ΕΔجຊతͳԾఆ ʢҰ෦ͷ߃ਅͳ࿦ཧࣜʣ ਪ࿦نଇʹ࿦ཧ͔ࣜΒଞͷ࿦ཧࣜΛಋ͘͜ͱ ެཧͱਪ࿦نଇʢܗࣜతମܥʣΛ࢖ͬͯ ͢΂ͯͷ߃ਅͳ࿦ཧࣜʢఆཧʣΛಋ͘

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ܗࣜతମܥͷछྨ w ήϯπΣϯྲྀ w ࣗવԋ៷ʜ೔ৗͷਪ࿦ʹ͍ۙɺެཧͳ͠ɺਪ࿦نଇ͕ଟ͍ w γʔΫΤϯτܭࢉʜެཧ͕̍ͭʢಉҰੑʣɺਪ࿦نଇ͕ଟ͍ w ώϧϕϧτྲྀ w ެཧ͕ଟ͍ɺਪ࿦نଇ͕̍ͭʢϞʔμεϙωϯεʣ w ܗࣜతମܥ͔Βಋग़͞ΕΔ࿦ཧࣜʢఆཧʣ͸ɺ߃ਅʢ݈શੑʣ

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ήϯπΣϯྲྀɾࣗવԋ៷ A B A ∧ B A ∧ B A A ∧ B B A A ∨ B B A ∨ B A ∨ B C C C A → B A B B A → B ⊥ ¬A A ¬A ⊥ ⊥ A ⋮ [A] ¬¬A A ⋮ [A] ⋮ [B] ⋮ [A] <">ʜ"ͷԾఆΛด͡Δ

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ήϯπΣϯྲྀɾࣗવԋ៷ʢγʔΫΤϯτܭࢉ෩ʣ Γ ⊢ A Γ ⊢ B Γ ⊢ A ∧ B Γ ⊢ A ∧ B Γ ⊢ A Γ ⊢ A ∧ B Γ ⊢ B Γ ⊢ A Γ ⊢ A ∨ B Γ ⊢ B Γ ⊢ A ∨ B Γ ⊢ A ∨ B Γ, A ⊢ C Γ, B ⊢ C Γ ⊢ C Γ ⊢ A → B Γ ⊢ A Γ ⊢ B Γ, A ⊢ B Γ ⊢ A → B Γ, A ⊢ ⊥ Γ ⊢ ¬A Γ ⊢ A Γ ⊢ ¬A Γ ⊢ ⊥ Γ ⊢ ⊥ Γ ⊢ A Γ ⊢ ¬¬A Γ ⊢ A A ∈ Γ Γ ⊢ A ϵ⊢"ʜ࿦ཧࣜͷ༗ݶྻϵ͔Βɺ࿦ཧࣜ"͕ূ໌Ͱ͖Δ

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ώϧϕϧτྲྀ A → B A B B → (A → B) (A → (B → C)) → ((A → B) → (A → C)) ¬¬A → A A → A ∨ B ⊥ → A B → A ∨ B (A → C) → ((B → C) → ((A ∨ B) → C)) A ∧ B → A A ∧ B → B A → (B → (A ∧ B)) ਪ࿦نଇ͸ ϞʔμεϙωϯεͷΈ (A → ¬A) → ¬A ¬A → (A → B)

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ͦΕͱͳ͘  ʮ4XJGUϓϩάϥϛϯάʯ ͬΆ͍

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࿦ཧ˱4XJGUϓϩάϥϛϯά A → B A ∧ B A ∨ B ¬A := A → ⊥ ରԠ 㱻 ؔ਺ܕ ௚ੵܕ ௚࿨ܕ ⊥ Either Never typealias Not = (A) -> Never (A, B) (A) -> B ϘτϜܕ 㲄Λ࢖ͬͯ ൱ఆΛఆٛͰ͖Δ

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ʲ࠶ܝʳήϯπΣϯྲྀɾࣗવԋ៷ A B A ∧ B A ∧ B A A ∧ B B A A ∨ B B A ∨ B A ∨ B C C C A → B A B B A → B ⊥ ¬A A ¬A ⊥ ⊥ A ⋮ [A] ¬¬A A ⋮ [A] ⋮ [B] ⋮ [A]

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/// `B ===> A → B` /// - Note: More formally, A ⊢ B ===> ⊢ A → B. func implyIntro(_ b: B) -> ((A) -> B) { { _ in b } } /// `A → B, A ===> B` (Modus ponens) func implyElim (_ f: (A) -> B, _ a: A) -> B { f(a) } A → B A B B A → B ⋮ [A]

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/// À, B ===> A ∧ B` func andIntro(_ a: A, _ b: B) -> (A, B) { (a, b) } /// À ∧ B ===> A` func andElim1 (_ ab: (A, B)) -> A { ab.0 } /// À ∧ B ===> B` func andElim2 (_ ab: (A, B)) -> B { ab.1 } A B A ∧ B A ∧ B A A ∧ B B

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/// `A ===> A ∨ B` func orIntro1(_ a: A) -> Either { .left(a) } /// `B ===> A ∨ B` func orIntro2 (_ b: B) -> Either { .right(b) } /// `A ∨ B, A → C, B → C ===> C` func orElim ( _ e: Either , _ ac: (A) -> C, _ bc: (B) -> C ) -> C { switch e { case let .left(a): return ac(a) case let .right(b): return bc(b) } } A A ∨ B B A ∨ B A ∨ B C C C ⋮ [A] ⋮ [B]

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struct Not { let f: (A) -> Never } /// `A ⊢ ⊥ ===> ⊢ ¬A` func notIntro (_ x: Never) -> Not { Not { _ in x } } /// `A, ¬A ===> ⊥` (NOTE: not using `fatalError`) func notElim (_ a: A, _ notA: Not ) -> Never { notA.f(a) } ⊥ ¬A A ¬A ⊥ ⋮ [A]

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/// `⊥ ===> A` func absurd(_ x: Never) -> A { // Do nothing, but it compiles. } ⊥ A

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ʲ࠶ܝʳώϧϕϧτྲྀ B → (A → B) (A → (B → C)) → ((A → B) → (A → C)) A → A ∨ B B → A ∨ B (A → C) → ((B → C) → ((A ∨ B) → C)) A ∧ B → A A ∧ B → B A → (B → (A ∧ B)) A → B A B ਪ࿦نଇ͸ ϞʔμεϙωϯεͷΈ ¬¬A → A ⊥ → A (A → ¬A) → ¬A ¬A → (A → B)

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ʲ࠶ܝʳώϧϕϧτྲྀ (A → C) → ((B → C) → ((A ∨ B) → C)) A → B A B A → A ∨ B B → A ∨ B A ∧ B → A A ∧ B → B A → (B → (A ∧ B)) ਪ࿦نଇ͸ ϞʔμεϙωϯεͷΈ ¬¬A → A ⊥ → A (A → ¬A) → ¬A ¬A → (A → B) (A → (B → C)) → ((A → B) → (A → C)) A ∨ B := ¬A → B := ¬B → A A ∧ B := ¬(A → ¬B) := ¬(B → ¬A) ¬A := A → ⊥ /05&ˮ ˭ ͸ɺˠͱ˵ͷΈΛ࢖ͬͯهड़Ͱ͖Δ B → (A → B)

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ίϯϏωʔλ࿦ཧ Kxy = x Sxyz = xz(yz) I = SKK K : X → Y → X S : (Z → Y → X) → (Z → Y) → Z → X I : X → X ஋ͷؔ਺Խ ҰൠԽ   3FBEFSϞφυͷQVSF Ϟʔμεϙωϯε ͷҰൠԽ  3FBEFSϞφυͷBQ

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/// `K = A → (B → A)` func k(_ a: A) -> (B) -> A { { _ in a } } /// `S = (A → (B → C)) → ((A → B) → (A → C))` func s (_ a: @escaping (C) -> (A) -> B) -> (_ b: @escaping (C) -> A) -> (C) -> B { { b in { c in a(c)(b(c)) } } }

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/// `I = SKK: A → A` func i(_ a: A) -> A { let k_: (A) -> (A) -> A = k let skk: (A) -> A = s(k)(k_) return skk(a) } i("hello") // "hello" i(123) // 123

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࿦ཧ ⇕ 4XJGUϓϩάϥϛϯά

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((A → B) → A) → A A ∨ ¬A ¬¬A → A ¬(A ∧ B) → (¬A ∨ ¬B) (A → B) → (¬A ∨ B) (¬B → ¬A) → (A → B) ഉத཯ ೋॏ൱ఆআڈ ରۮ཯ ൱ఆআڈWFS υɾϞϧΨϯͷ๏ଇ 㱸 ؚ࣮࣭ҙ ύʔεͷ๏ଇ /05& ʮٯʯ͸߃ਅ 4XJGUͰ࣮૷Ͱ͖ͳ͍߃ਅͳ࿦ཧࣜ

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/// `A ∨ ¬A` (Law of excluded middle) func excludedMiddle() -> Either > { fatalError("Can't impl in Swift”) }

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/// `A ∨ ¬A` (Law of excluded middle) func excludedMiddle() -> Either > { fatalError("Can't impl in Swift”) } ANY LANGUAGES!

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/// `((A → B) → A) → A` func peirceLaw() -> (((A) -> B) -> A) -> A { { (aba: ((A) -> B) -> A) in let ab: (A) -> B = { fatalError("Can't impl ") }() let a = aba(ab) return a } }

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/// `((A → B) → A) → A` func peirceLaw() -> (((A) -> B) -> A) -> A { { (aba: ((A) -> B) -> A) in // WARNING: This compiles, but using invalid `excludedMiddle()`. let either: Either > = excludedMiddle() switch either { case let .left(a): return a case let .right(notA): return aba { a in absurd(notA.f(a)) } } } } ݟ্͔͚͸࣮૷Ͱ͖͍ͯΔ͕ɺ ഉத཯Λ࢖͍ͬͯΔͷͰɺ

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͜Ε·Ͱͷ࿦ཧ͸ɾɾɾ ݹయ࿦ཧ ϒʔϧ࿦ཧ ࿦ཧࣜʹʮਅʯ·ͨ͸ʮِʯͷ  ਅཧ஋Λ෇༩ͯ͠ղऍ͢Δ

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ݹయ࿦ཧʹ͓͚Δʮˠʯ A → B Bool true false USVFPSGBMTF ௨Γ͔͠ͳ͍

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4XJGUϓϩάϥϛϯάʹ͓͚Δʮˠʯ Types Int Bool (A) -> B (Int)->Bool ৽͍͠ܕ͕ ߏ੒͞ΕΔ

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ݹయ࿦ཧ ϒʔϧ࿦ཧ ˣ  ߏ੒త࿦ཧ  ௚؍ओٛ࿦ཧ ഉத཯΍ೋॏ൱ఆআڈ͕ ੒Γཱͨͳ͘ͳΔ

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௚؍ओٛ࿦ཧ ⇕ 4XJGUϓϩάϥϛϯά

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௚؍ओ໋ٛ୊࿦ཧʹ͓͚Δ ࣗવԋ៷ ⇕ ୯७ܕ෇͖ϥϜμܭࢉ  ؔ਺ܕϓϩάϥϛϯάͷཧ࿦తج൫

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ܕͳ͠ϥϜμܭࢉʢུ֓ʣ t ::= x | λx . t | t t ʢม਺ʣ ʢλந৅ɺؔ਺ఆٛʣ ʢؔ਺ద༻ʣ λx . x λx . (λy . x) (λx . xx)(λx . xx) λ߲ͷྫ ϥϜμ߲

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ܕ෇͖ϥϜμܭࢉʢུ֓ʣ T ::= X | T → T ʢܕม਺ʣ ʢؔ਺ܕʣ t ::= x | λ(x : T) . t | t t ʢม਺ʣ ʢλந৅ʣ ʢؔ਺ద༻ʣ : T ϥϜμ߲ ܕɹ

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ܕ෇͖ϥϜμܭࢉʢུ֓ʣ A ∈ Γ Γ ⊢ A Γ, A ⊢ B Γ ⊢ A → B Γ ⊢ A → B Γ ⊢ A Γ ⊢ B (x : A) ∈ Γ Γ ⊢ x : A Γ, x : A ⊢ t : B Γ ⊢ λ(x : A) . t : A → B Γ ⊢ t12 : A → B Γ ⊢ t1 : A Γ ⊢ t12 t1 : B ܕ෇͚نଇͷʮܕͷΈʯʹ஫໨͢Δͱʜ ࣗવԋ៷ͱ ಉ౳ λ߲ͷܕ෇͚نଇ ม਺ Еந৅ ؔ਺ద༻

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௚؍ओ໋ٛ୊࿦ཧʹ͓͚Δ ࣗવԋ៷ ⇕ ୯७ܕ෇͖ϥϜμܭࢉ  ؔ਺ܕϓϩάϥϛϯάͷཧ࿦తج൫ ΧϦʔɾϋϫʔυ ಉܕରԠ

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ΧϦʔɾϋϫʔυಉܕରԠ ௚؍ओ໋ٛ୊࿦ཧʹ͓͚Δ  ࣗવԋ៷ ୯७ ܕ෇͖ϥϜμܭࢉ  ؔ਺ܕϓϩάϥϛϯά ໋୊ɾఆཧ ܕ ূ໌ਤ ϥϜμ߲ʢϓϩάϥϜʣ ؚҙಋೖ ϥϜμந৅ʢؔ਺ఆٛʣ Ϟʔμεϙωϯε ؔ਺ద༻ ެཧ άϩʔόϧม਺ Ծఆ ࣗ༝ม਺ ԾఆΛด͡Δ ଋറม਺ ଞͷ࿦ཧԋࢉ 㱸㱹ͳͲ ͷ ֦ுͱಉܕରԠ΋Մೳ

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௚؍ओٛ࿦ཧ  ࿦ཧࣜʹʮՄೳੈքʯΛߟྀͨ͠ ʮਅʯʮِʯͷਅཧ஋Λ෇༩ͯ͠ղऍ͢Δ

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௚؍ओٛ࿦ཧͱʮՄೳੈքʯʢ஌ࣝঢ়ଶʣ A 0 B 0 A 1 B 0 A 1 B 1

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֤ੈքͱਅཧ஋ͷҨ఻ؔ܎ A 0 B 0 A 1 B 0 A 1 B 0 ͸ະདྷ΁Ҩ఻͢Δ ʹ໭Βͳ͍ ৽͍͠஌ݟ

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֤ੈքͱਅཧ஋ͷҨ఻ؔ܎ A 0 ¬A 0 A 0 ¬A 1 A 0 ¬A 1 "ͷ৔߹ɺ ݱࡏ͔Βະདྷʹ͔͚ͯ" ʢ"ʹͳΔ͜ͱ͸ͳ͍ʣ Ҩ఻

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A 0 ¬A 0 A 0 ¬A 1 A 0 ¬A 1 "ͷ৔߹ɺ ݱࡏ͔Βະདྷʹ͔͚ͯ" ʢ"ʹͳΔ͜ͱ͸ͳ͍ʣ Ҩ఻ ֤ੈքͱਅཧ஋දͷҨ఻ؔ܎ A ∨ ¬A ഉத཯͸߃ਅͰ͸ͳ͍

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֤ੈքͱਅཧ஋ͷҨ఻ؔ܎ A → B 1 A 0 B 0 A → B 1 A 1 B 1 A → B 1 A 1 B 1 Ҩ఻ "͕ಘΒΕΔͱɺ  #΋ࣗಈతʹಘΒΕΔ " #ʹ ͳΔ͜ͱ͸ͳ͍

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֤ੈքͱਅཧ஋ͷҨ఻ؔ܎ "͕ಘΒΕΔͱɺ  #΋ࣗಈతʹಘΒΕΔ A 0 B 0 A 1 B 0 A 1 B 1 A 0 B 1

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ΫϦϓΩɾϞσϧʢࡶͳղઆʣ wΫϦϓΩɾϑϨʔϜʜˠˠͳͲͷੈքͷॱংू߹ wΫϦϓΩɾϞσϧʜϑϨʔϜͷ֤ੈքͷ্ʹҨ఻͢Δ  ɹɹɹɹɹɹɹɹɹਅཧ஋ͷׂΓ౰ͯΛߦͬͨ΋ͷ w௚؍ओٛ࿦ཧʹ͓͚Δ߃ਅ  ʜ೚ҙͷϑϨʔϜʴ೚ҙͷਅཧ஋ͷׂΓ౰ͯ  ɹʹ೚ҙͷϞσϧʹ͓͍ͯਅʹͳΔ࿦ཧࣜͷ͜ͱ "͕ಘΒΕΔͱɺ  #΋ࣗಈతʹಘΒΕΔ

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ݹయ࿦ཧ˩௚؍ओٛ࿦ཧ ࠷খ࿦ཧ ௚؍ओٛ࿦ཧ ݹయ࿦ཧ A ∨ ¬A ⊥ → A → , ∧ , ∨ , ¬, ⊥ Λ࢖ͬͨ࿦ཧࣜ A → A ∨ B A ∧ B → A ໃ६཯ ഉத཯ ⋮ ʲ஫ҙʳ௚؍ओٛ࿦ཧ͕ഉத཯Λ௚઀ؚΉΘ͚Ͱ͸ͳ͍ʢ࣍ϖʔδࢀরʣ

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άϦϕϯίͷఆཧɿݹయ࿦ཧˠ௚؍ओٛ࿦ཧ΁ͷຒΊࠐΈ ௚؍ओٛ࿦ཧ ݹయ࿦ཧ A ∨ ¬A ¬¬(A ∨ ¬A) ¬¬ ೋॏ൱ఆಋೖ ഉத཯ ೋॏ൱ఆഉத཯

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((A → B) → A) → A A ∨ ¬A ¬¬A → A ¬(A ∧ B) → (¬A ∨ ¬B) (A → B) → (¬A ∨ B) (¬B → ¬A) → (A → B) ഉத཯ ೋॏ൱ఆআڈ ରۮ཯ ൱ఆআڈWFS υɾϞϧΨϯͷ๏ଇ 㱸 ؚ࣮࣭ҙ ύʔεͷ๏ଇ ʲ࠶ܝʳ௚؍ओٛ࿦ཧͰ੒Γཱͨͳ͍࿦ཧࣜ /05& ʮٯʯ͸߃ਅ

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¬¬(((A → B) → A) → A) ¬¬(A ∨ ¬A) ¬¬(¬¬A → A) ¬¬(¬(A ∧ B) → (¬A ∨ ¬B)) ¬¬((A → B) → (¬A ∨ B)) ¬¬((¬B → ¬A) → (A → B)) ௚؍ओٛ࿦ཧͰ߃ਅͳ࿦ཧࣜ ͢΂ͯ ௚؍ओٛ࿦ཧ Ͱ੒Γཱͭ

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¬¬((A → B) → A) → ¬¬A ¬¬(A ∨ ¬A) ¬¬¬¬A → ¬¬A ¬¬¬(A ∧ B) → ¬¬(¬A ∨ ¬B) ¬¬(A → B) → ¬¬(¬A ∨ B) ¬¬(¬B → ¬A) → ¬¬(A → B) Tautologies in Intuitionistic Logic Using helper rule ¬¬(A → B) ¬¬A → ¬¬B

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/// `¬¬(A ∨ ¬A)` (Law of excluded middle in Intuitionistic Logic) func excludedMiddle_IL() -> Not>>> { Not>>> { notEither in let notA: Not = Not { a in notEither.f(.left(a)) } let either: Either > = .right(notA) return notEither.f(either) } }

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/// `¬¬¬¬A ===> ¬¬A` (Double negation elimination in IL) func doubleNegationElim_IL( _ notNotNotNotA: Not>>> ) -> Not> { Not> { notA in notNotNotNotA.f( Not>> { notNotA in notNotA.f(notA) } ) } }

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/// `¬¬¬(A ∧ B) ===> ¬¬(¬A ∨ ¬B)` (De-Morgan's Law in IL) func deMorgan4_IL(_ notNotNotAB: Not>>) -> Not, Not>>> { Not, Not>>> { (notEither: Not, Not>>) in let notNotAB = Not> { notAB in let notB = Not { b in let notA = Not { a in notAB.f((a, b)) } let either: Either, Not> = .left(notA) return notEither.f(either) } let either: Either, Not> = .right(notB) return notEither.f(either) } return notNotNotAB.f(notNotAB) } }

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/// `¬¬((A → B) → A) → ¬¬A` (Peirce's Law in IL) func peirceLaw_IL() -> (Not B) -> A>>) -> Not> { { notNotF in Not> { notA in let notABA = Not<((A) -> B) -> A> { aba in let ab: (A) -> B = { a in absurd(notElim(a, notA)) } let a = aba(ab) return notA.f(a) } return notNotF.f(notABA) } } }

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ύʔεͷ๏ଇˠDBMMDD ¬¬(A → B) ¬¬A → ¬¬B ((A → ¬¬B) → ¬¬A) → ¬¬A ͜͜Ͱ Λ࢖͏ͱ ¬¬(A → B) A → ¬¬B ¬¬(((A → B) → A) → A) ύʔεͷ๏ଇ

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/// Continuation. typealias Cont = (@escaping (A) -> R) -> R /// /// Call with Current Continuation. /// - Peirce: `((A → B) → A) → A` /// - CallCC: `((A → M) → M) → M ` /// /// - Note: `callCC` is like control operators e.g. `goto`, `return`, `throw`. /// func callCC ( _ f: @escaping (_ exit: @escaping (A) -> Cont) -> Cont ) -> Cont { { outer in f { a in { _ in outer(a) } }(outer) } }

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func doubleNegationElim_callCC( _ doubleNegation: @escaping ( _ neg: @escaping (A) -> Cont ) -> Cont ) -> Cont { callCC { exit -> Cont in flatMap(doubleNegation(exit), absurd) } }

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֊໋୊࿦ཧ  ͢΂ͯʮ˲ʯͱଘࡏʮ˳ʯ

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֊໋୊࿦ཧ A1 ∧ A2 ∧ ⋯ ≅ ∀X . X A1 ∨ A2 ∨ ⋯ ≅ ∃X . X (A1 → C) ∧ (A2 → C) ∧ ⋯ ≅ ∀X . (X → C) ʮ໋୊ʯͷશশྔԽ㱼ɺଘࡏྔԽ㱽  ֊ड़ޠ࿦ཧͷʮ߲ʯͷྔԽͱҟͳΔͷͰ஫ҙ 㱸ͱ㱼㱹ͱ㱽

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֊໋୊࿦ཧͷਪ࿦نଇ ∀X . A(X) A(T) A(Y) ∀X . A(X) ∃X . A(X) ∀X . (A(X) → C) C A(T) ∃X . A(X) ͸ɹɹɹʹࢸΔԋ៷ͷԾఆ͢΂ͯͱɹɹɹͷதͰ ࣗ༝ม਺ͱͯ͠࢖ΘΕ͍ͯͳ͍ܕม਺ Y A(Y) A(X) ͸ɹɹʹ ୅ೖՄೳͳܕ T A

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(A1 → C) ∧ (A2 → C) ∧ ⋯ ≅ ∀X . (X → C) let intToBool: (Int) -> Bool = { ... } let floatToBool: (Float) -> Bool = { ... } let strToBool: (String) -> Bool = { ... } // ... ͢΂ͯͷܕʹ͍ͭͯಉ͡Α͏ͳ࣮૷Λߦ͍͍ͨ // Swift ͷδΣωϦΫεʢଟ૬ؔ਺ʣΛ࢖͏ͱɺ // ্هͷॏෳίʔυΛ1ͭʹ·ͱΊͯந৅ԽͰ͖Δɻ func xToBool(_ x: X) -> Bool { ... }

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˲ͱ˳ͷؔ܎ (A1 ∨ A2 ) → C ≅ (A1 → C) ∧ (A2 → C) (∃X . X) → C ≅ ∀X . (X → C) (A1 ∨ A2 ∨ ⋯) → C ≅ (A1 → C) ∧ (A2 → C) ∧ ⋯ (A → C) → ((B → C) → ((A ∨ B) → C)) ࢀߟɿ

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∀X . (X → C) ≅ (∃X . X) → C func forallXToBool(_ x: X) -> Bool { anyToBool(x) } /// - Note: `∃X.X` is isomorphic to `Any`. func anyToBool(_ any: Any) -> Bool { forallXToBool(any) }

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∀(X <: P) . (X → C) ≅ (∃(X <: P) . X) → C protocol P { var value: String { get } } struct AnyP: P { let value: String } func forallXProtocolToBool(_ x: X) -> Bool { // Generic func protocolToBool(x) } func protocolToBool(_ p: P) -> Bool { // Protocol func (dynamic) forallXProtocolToBool(AnyP(value: p.value)) }

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∀(X <: P) . (X → C) ≅ (∃(X <: P) . X) → C // Reverse Opaque Result Type (static) func someProtocolToBool(_ p: some P) -> Bool { forallXProtocolToBool(p) } // ERROR: Isn’t supported yet in Swift 5.1

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֊໋୊࿦ཧ ⇕ δΣωϦΫε ଟ૬ੑ Ұ෦ɺϓϩτίϧϕʔεʹஔ͖׵͑΋Մೳ

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ϥϜμΩϡʔϒ

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ϥϜμΩϡʔϒ ֊ ଟ૬ ϥϜμ ֊௚؍ओ໋ٛ୊࿦ཧ ܕʹґଘͨ͠஋ ଟ૬ؔ਺ ͕࡞ΕΔ ୯७ܕ෇͖ϥϜμ ֊௚؍ओ໋ٛ୊࿦ཧ ஋ʹґଘͨ͠஋ ߴ֊ؔ਺ ͕࡞ΕΔ ґଘܕ ֊௚؍ओٛड़ޠ࿦ཧ ஋ʹґଘͨ͠ܕ͕࡞ΕΔ ߴ֊ଟ૬ϥϜμ ߴ֊ΧΠϯυʴଟ૬ؔ਺ ߴ֊ΧΠϯυܕ ܕʹґଘͨ͠ܕ ߴ֊ͷܕؔ਺ ͕࡞ΕΔ

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ϥϜμΩϡʔϒ 4XJGU͸ ΠϚίί ࠓճ ֶΜͩ͜ͱ

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ϥϜμΩϡʔϒ ֊ ଟ૬ ϥϜμ ֊௚؍ओ໋ٛ୊࿦ཧ ܕʹґଘͨ͠஋ ଟ૬ؔ਺ ͕࡞ΕΔ ୯७ܕ෇͖ϥϜμ ௚؍ओ໋ٛ୊࿦ཧ ஋ʹґଘͨ͠஋ ௨ৗͷߴ֊ؔ਺ ͕࡞ΕΔ ґଘܕ ֊௚؍ओٛड़ޠ࿦ཧ ஋ʹґଘͨ͠ܕ͕࡞ΕΔ ߴ֊ଟ૬ϥϜμ ଟ૬ͳߴ֊ܕߏஙࢠ͕࡞ΕΔ ߴ֊ΧΠϯυܕ ܕʹґଘͨ͠ܕ ߴ֊ͷܕߏஙࢠ ͕࡞ΕΔ Calculus of Constructions

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·ͱΊ w࿦ཧΛʮܗࣜʯͱʮҙຯʯͷͭͷଆ໘͔Βߟ͑Δ wܗࣜతମܥʜެཧʢҰ෦ͷ߃ਅͳ࿦ཧࣜʣͱਪ࿦نଇ wݹయ࿦ཧ͔Β௚؍ओٛ࿦ཧ΁ʢഉத཯ͷഇࢭͱΫϦϓΩҙຯ࿦ʣ w֊໋୊࿦ཧʢδΣωϦΫεͱϓϩτίϧʣ w௚؍ओٛ࿦ཧ˱4XJGUϓϩάϥϛϯάʢΧϦʔɾϋϫʔυରԠʣ w࿦ཧΛֶͿͱɺܕͷҙຯͱՁ஋͕෼͔Δ

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Sample Code  http://gist.github.com/inamiy/SwiftAndLogic

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One more thing…

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ʙؼ͖ͬͯͨݍ࿦ʙ

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ࡢ೥ͷJ04%$ͷൃද  IUUQTTQFBLFSEFDLDPNJOBNJZJPTEDKBQBO  IUUQTRJJUBDPNJOBNJZJUFNTFDEFBGCDE

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A B C ݍ

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A B C ݍ idA idB idC

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A B C ݍ f g ∘ f g

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߃౳ࣹͱ߹੒ A → B B → C A → C A → A

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࿦ཧԋࢉ˭ ͔ͭ A ∧ B A A ∧ B B A B A ∧ B

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A ∧ B A B A ∧ B A A ∧ B B

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(C → A) ∧ (C → B) C → (A ∧ B) A B C A ∧ B A B A ∧ B HomC (C, A) × HomC (C, B) ≅ HomC (C, A × B)

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࿦ཧԋࢉˮ ·ͨ͸ A A ∨ B B A ∨ B A ∨ B A → C B → C C

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A ∨ B A B A A ∨ B B A ∨ B

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A B C A ∨ B (A → C) ∧ (B → C) (A ∨ B) → C A ∨ B A → C B → C C HomC (A, C) × HomC (B, C) ≅ HomC (A + B, C)

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ถాͷิ୊ Hom[C,Set] (HomC (A, − ), F) ≅ F(A) ∀B . (A → B) → F(B) ≅ F(A) A ⟺ ∀B . (A → B) → B A ⟹ (A → ⊥ ) → ⊥ '߃౳ؔखͷͱ͖ #˵ʹಛघԽ  ˲আڈ ೋॏ൱ఆ ¬¬A

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άϦϕϯίͷఆཧɿݹయ࿦ཧˠ௚؍ओٛ࿦ཧ΁ͷຒΊࠐΈ ௚؍ओٛ࿦ཧ ݹయ࿦ཧ A ∨ ¬A ¬¬(A ∨ ¬A) ¬¬ ೋॏ൱ఆಋೖ ഉத཯ ೋॏ൱ఆഉத཯ $14ม׵ A ⟹ (A → ⊥ ) → ⊥

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F(C) D ݍ F C ݍ U(D) ࣹͷू߹ U ≅ rightAdjunct leftAdjunct

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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) unitC counitD

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A × X B ݍ − × X ( − )X A BX ݍ BX × X f f f × id eval ႈ ࢦ਺ର৅ ධՁࣹ

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B BX × X eval A × X − × X ( − )X A BX f f f × id ႈ ࢦ਺ର৅ ݍ ݍ ධՁࣹ X → B B B

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σΧϧτ ΧϧςγΞϯ ดݍ ऴର৅ ⊤ ௚ੵ"ʷ# "˭# ႈ ࢦ਺ର৅ #" "ˠ# Λ࣋ͭݍͷ͜ͱ ௚؍ओٛ ࿦ཧͷܗࣜతମܥ͸σΧϧτดݍʹΑͬͯ  ݍ࿦తʹҙຯ෇͚͢Δ͜ͱ͕Ͱ͖Δ σΧϧτดݍ ࿦ཧࣜͷূ໌ ܕ෇͖ϥϜμܭࢉ ର৅ ࿦ཧࣜ ܕ ࣹ ূ໌ ϥϜμ߲ ࣗ༝ม਺Λ࣋ͨͳ͍ ႈର৅ ˠΛ࢖ͬͨ࿦ཧࣜ ؔ਺ܕ ධՁࣹ Ϟʔμεϙωϯε ؔ਺ద༻ ΧϦʔɾϋϫʔυɾϥϯϕοΫରԠ

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"MHFCSBJD%BUB5ZQFJO4XJGU  IUUQTTQFBLFSEFDLDPNJOBNJZBMHFCSBJDEBUBUZQFJOTXJGU

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τϙεʹσΧϧτดݍʴ෦෼ର৅෼ྨࢠ {a ∈ A|p(a)} A τϙε Ω true 1 unit m p p.b.

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τϙεʹσΧϧτดݍʴ෦෼ର৅෼ྨࢠ {a ∈ A|p(a)} A τϙε Ω true 1 unit m ෦෼ର৅ 4VCPCKFDU Ϟϊࣹ ෦෼ର৅෼ྨࢠ 4VCPCKFDU$MBTTJpFS ྫɿτϙε͕ू߹ͷݍͷͱ͖  Њ\USVF GBMTF^ p p.b. Ҿ͖໭͠  1VMMCBDL QC ಛੑࣹ "্ͷड़ޠ

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ࢀߟจݙʢ࿦ཧֶɺϥϜμܭࢉɺܕγεςϜʣ w ʰܭࢉ࿦ཧֶʱߨٛࢿྉُࢁ޾ٛ w ΧϦʔʹϋϫʔυಉܕରԠ8JLJQFEJB w ΧϦʔɾϋϫʔυରԠͱ௚؍ओٛ࿦ཧͷ ҙຯ࿦ೖ໳໼ా෦ढ़հ w ௚؍ओٛ࿦ཧ΁ͷট଴রҪҰ੒ w ਺ֶಉ޷ձފా w $MBTTJDBM-PHJDJO)BTLFMM w "5ZQF4ZTUFN'SPN4DSBUDIr 3PCFSU8JENBOO w "4IPSU*OUSPEVDUJPOUP4ZTUFNT' BOE'Т1BCMP/PHVFJSB w ࿦ཧͱܭࢉͷ͘͠ΈԮ୩ণݾɺ੢࡚ਅ໵ w ৘ใՊֶʹ͓͚Δ࿦ཧখ໺׮᏷ w ϓϩάϥϜҙຯ࿦ԣ಺׮จ w ܕγεςϜೖ໳ 5B1- 1JFSDF

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ࢀߟจݙʢݍ࿦ɺτϙεʣ w ೥ݍ࿦ษڧձࢿྉதଜߊҰ w ᐻࢁਖ਼޾ͷΩϚΠϥࣂҭه w ఩ֶऀͷͨΊͷݍ࿦ೖ໳৿ాਅੜ w ݍ࿦ʹΑΔϓϩάϥϛϯάͱ࿦ཧ !TIJBUTVNBU !@@JOU w τϙεBMHE w 5IF%BSL4JEFPG'PSDJOH w 0MJWJB$BSBNFMMPTXFCTJUF w "O*OUSPEVDUJPOUP5PQPT5IFPSZ w ,SJQLFr+PZBMҙຯ࿦;JQIJM"MFTIMBT w (SPUIFOEJFDL5PQPT΁ͷೖ໳ !LSJQLFKPZBM w ϓϩάϥϛϯάݴޠͷҙຯ࿦ͱݍ࿦௕୩઒ ਅਓ w τϙεͱߴ֊࿦ཧ5BJDIJ6FNVSB w ݍ࿦"XPEFZ w ݍ࿦ʹΑΔ࿦ཧֶʕߴ֊࿦ཧͱτϙεਗ਼ ਫٛ෉ w ݍ࿦ͷา͖ํ೔ຊධ࿦ࣾ

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Thanks! Yasuhiro Inami @inamiy