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)
(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
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⊢"ʜཧࣜͷ༗ݶྻϵ͔Βɺཧࣜ"͕ূ໌Ͱ͖Δ
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)) ਪنଇ ϞʔμεϙωϯεͷΈ (A → ¬A) → ¬A ¬A → (A → B)
/// `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]
/// `A, B ===> A ∧ B` func andIntro(_ a: A, _ b: B) -> (A, B) { (a, b) } /// `A ∧ B ===> A` func andElim1(_ ab: (A, B)) -> A { ab.0 } /// `A ∧ B ===> B` func andElim2(_ ab: (A, B)) -> B { ab.1 } A B A ∧ B A ∧ B A A ∧ B B
/// `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]
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]
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 ਪنଇ ϞʔμεϙωϯεͷΈ ¬¬A → A ⊥ → A (A → ¬A) → ¬A ¬A → (A → B)
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)) ਪنଇ 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˵ͷΈΛͬͯهड़Ͱ͖Δ B → (A → B)
/// `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)) } } }
/// `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
/// `((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 } }
/// `((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)) } } } } ݟ্͔͚࣮Ͱ͖͍ͯΔ͕ɺ ഉதΛ͍ͬͯΔͷͰɺ
ܕ͖ϥϜμܭࢉʢུ֓ʣ 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 ܕ͚نଇͷʮܕͷΈʯʹ͢Δͱʜ ࣗવԋ៷ͱ ಉ λ߲ͷܕ͚نଇ ม Еந ؔద༻
/// `¬¬(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) } }
/// `¬¬¬(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) } }
/// `¬¬((A → B) → A) → ¬¬A` (Peirce's Law in IL) func peirceLaw_IL() -> (Not B) -> A>>) -> Not> { { notNotF in Not> { notA in let notABA = Not 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) } } }
˲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
∀(X // 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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