min — Mathlib

English

If f is monotone on s, and x,y ∈ s, then f(x ⊔ y) = f x ⊔ f y when both x,y are in s.

Русский

Если f монотонна на s и x,y ∈ s, то при условиях f(x ⊔ y) = f x ⊔ f y.

LaTeX

$$$ [Preorder α] [LinearOrder α] [SemilatticeSup β] {f : α → β} (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) : f(x \sqcup y) = f(x) \sqcup f(y)$$$

Lean4

/-- Pointwise minimum of two monotone functions is a monotone function. -/
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
    (hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
  hf.inf hg

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