English
Let f: S → R be injective. If R has a NonUnitalCommSemiring structure, then S inherits a NonUnitalCommSemiring structure pulling back along f.
Русский
Пусть f: S → R инъективна. Если R имеет структуру NonUnitalCommSemiring, то S наследует её через перенос по f.
LaTeX
$$$\\exists\\, \\mathcal{S}: \\text{NonUnitalCommSemiring}(S),\\quad f(0)=0,\\quad f(1)=1,\\quad \\forall x,y, f(x+y)=f(x)+f(y),\\quad \\forall x,y, f(x\\cdot y)=f(x)\\cdot f(y),\\quad \\forall n, f(n\\cdot x)=n\\cdot f(x).$$$
Lean4
/-- Pullback a `NonUnitalCommSemiring` instance along an injective function. -/
-- See note [reducible non-instances]
protected abbrev nonUnitalCommSemiring [NonUnitalCommSemiring R] (f : S → R) (hf : Injective f) (zero : f 0 = 0)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
(nsmul : ∀ (n : ℕ) (x), f (n • x) = n • f x) : NonUnitalCommSemiring S
where
toNonUnitalSemiring := hf.nonUnitalSemiring f zero add mul nsmul
__ := hf.commSemigroup f mul
