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EventuallyEq.{mul,add}_{right,left} #5

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20 changes: 9 additions & 11 deletions BirkhoffErgodicThm/FilterPR.lean
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import Mathlib.Tactic

/-- If two functions `f₁` and `f₂` are eventually equal with respect to a filter `f`,
then adding to right the same function `f₃` to both `f₁` and `f₂` results in two functions that
are also eventually equal with respect to the same filter `f`. -/
lemma Filter.EventuallyEq.add_right {f : Filter α} {f₁ f₂ f₃ : α → ℝ} (h : f₁ =ᶠ[f] f₂) :
f₁ + f₃ =ᶠ[f] f₂ + f₃ := h.mono fun x hx ↦ by simp [hx]

/-- If two functions `f₁` and `f₂` are eventually equal with respect to a filter `f`,
then adding to left the same function `f₃` to both `f₁` and `f₂` results in two functions that
are also eventually equal with respect to the same filter `f`. -/
lemma Filter.EventuallyEq.add_left {f : Filter α} {f₁ f₂ f₃ : α → ℝ} (h : f₁ =ᶠ[f] f₂) :
f₃ + f₁ =ᶠ[f] f₃ + f₂ := h.mono fun x hx ↦ by simp [hx]
namespace Filter.EventuallyEq

variable {f : Filter α} {f₁ f₂ f₃ : α → β} [Mul β]

@[to_additive]
lemma mul_right (h : f₁ =ᶠ[f] f₂) : f₁ * f₃ =ᶠ[f] f₂ * f₃ := mul h (by rfl)

@[to_additive]
lemma mul_left (h : f₁ =ᶠ[f] f₂) : f₃ * f₁ =ᶠ[f] f₃ * f₂ := mul (by rfl) h