rotation — Mathlib

English

As above, in a variant formulation ensuring the existence of n with t(n) < x ≤ t(n+1).

Русский

Как выше, в варианте формулировки существования n так, чтобы t(n) < x ≤ t(n+1).

LaTeX

$$$ \exists n \,( t(n) < x \land x \le t(n+1) ).$$$

Lean4

/-- The rotation in `E × E` with angle `θ`, as a continuous linear map. -/
noncomputable def _root_.ContinuousLinearMap.rotation (θ : ℝ) : E × E →L[ℝ] E × E
    where
  toFun := fun x ↦ (Real.cos θ • x.1 + Real.sin θ • x.2, -Real.sin θ • x.1 + Real.cos θ • x.2)
  map_add' x
    y := by
    simp only [Prod.fst_add, smul_add, Prod.snd_add, neg_smul, Prod.mk_add_mk]
    abel_nf
  map_smul' c x := by simp [smul_comm c]
  cont := by fun_prop

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