→ 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)
∧ 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
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)
formally, A ⊢ B ===> ⊢ A → B. func implyIntro<A, B>(_ b: B) -> ((A) -> B) { { _ in b } } /// `A → B, A ===> B` (Modus ponens) func implyElim<A, B>(_ f: (A) -> B, _ a: A) -> B { f(a) } A → B A B B A → B ⋮ [A]
a: A, _ b: B) -> (A, B) { (a, b) } /// `A ∧ B ===> A` func andElim1<A, B>(_ ab: (A, B)) -> A { ab.0 } /// `A ∧ B ===> B` func andElim2<A, B>(_ ab: (A, B)) -> B { ab.1 } A B A ∧ B A ∧ B A A ∧ B B
A) -> Either<A, B> { .left(a) } /// `B ===> A ∨ B` func orIntro2<A, B>(_ b: B) -> Either<A, B> { .right(b) } /// `A ∨ B, A → C, B → C ===> C` func orElim<A, B, C>( _ e: Either<A, B>, _ 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]
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)
∨ 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)
B>(_ a: A) -> (B) -> A { { _ in a } } /// `S = (A → (B → C)) → ((A → B) → (A → C))` func s<A, B, C>(_ a: @escaping (C) -> (A) -> B) -> (_ b: @escaping (C) -> A) -> (C) -> B { { b in { c in a(c)(b(c)) } } }
B>() -> (((A) -> B) -> A) -> A { { (aba: ((A) -> B) -> A) in // WARNING: This compiles, but using invalid `excludedMiddle()`. let either: Either<A, Not<A>> = excludedMiddle() switch either { case let .left(a): return a case let .right(notA): return aba { a in absurd(notA.f(a)) } } } } ݟ্͔͚࣮Ͱ͖͍ͯΔ͕ɺ ഉதΛ͍ͬͯΔͷͰɺ
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༻
in IL) func deMorgan4_IL<A, B>(_ notNotNotAB: Not<Not<Not<(A, B)>>>) -> Not<Not<Either<Not<A>, Not<B>>>> { Not<Not<Either<Not<A>, Not<B>>>> { (notEither: Not<Either<Not<A>, Not<B>>>) in let notNotAB = Not<Not<(A, B)>> { notAB in let notB = Not<B> { b in let notA = Not<A> { a in notAB.f((a, b)) } let either: Either<Not<A>, Not<B>> = .left(notA) return notEither.f(either) } let either: Either<Not<A>, Not<B>> = .right(notB) return notEither.f(either) } return notNotNotAB.f(notNotAB) } }
in IL) func peirceLaw_IL<A, B>() -> (Not<Not<((A) -> B) -> A>>) -> Not<Not<A>> { { notNotF in Not<Not<A>> { 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) } } }
-> R /// /// Call with Current Continuation. /// - Peirce: `((A → B) → A) → A` /// - CallCC: `((A → M<B>) → M<A>) → M<A>` /// /// - Note: `callCC` is like control operators e.g. `goto`, `return`, `throw`. /// func callCC<A, B, R>( _ f: @escaping (_ exit: @escaping (A) -> Cont<R, B>) -> Cont<R, A> ) -> Cont<R, A> { { outer in f { a in { _ in outer(a) } }(outer) } }
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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