translationNumber_of_eq_add_int — Mathlib

English

If there exists x with f(x) = x + m for some integer m, then τ(f) = m.

Русский

Если существует x such that f(x) = x + m, для некоторого m ∈ ℤ, тогда τ(f) = m.

LaTeX

$$$$ \\exists x \\in \\mathbb{R}, \\ f(x) = x + m \\quad \\Rightarrow \\quad \\tau f = m. $$$$

Lean4

/-- If `f x - x` is an integer number `m` for some point `x`, then `τ f = m`.
On the circle this means that a map with a fixed point has rotation number zero. -/
theorem translationNumber_of_eq_add_int {x : ℝ} {m : ℤ} (h : f x = x + m) : τ f = m :=
  le_antisymm (translationNumber_le_of_le_add_int f <| le_of_eq h)
    (le_translationNumber_of_add_int_le f <| le_of_eq h.symm)

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