toSemiring — Mathlib

English

If R is a semiring and S is a subsemiring of R, then S, with the inherited addition and multiplication, forms a semiring.

Русский

Если R — полупольный полусферийный полукольцо, и S — подпол semiring R, то S, наследующий сложение и умножение, образует полуринг.

LaTeX

$$$S \subseteq R$ and $S \text{ is a subsemiring of } R \Rightarrow S \text{ is a semiring with the induced operations.}$$$

Lean4

/-- A subsemiring of a `Semiring` is a `Semiring`. -/
instance (priority := 75) toSemiring {R} [Semiring R] [SetLike S R] [SubsemiringClass S R] : Semiring s :=
  fast_instance%Subtype.coe_injective.semiring Subtype.val rfl rfl (fun _ _ => rfl) (fun _ _ => rfl) (fun _ _ => rfl)
      (fun _ _ => rfl) fun _ => rfl

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