-- Mathlib/Analysis/Normed/Ring/SmoothingSeminorm.lean
theorem isNonarchimedean_smoothingFun (hμ1 : μ 1 ≤ 1) (hna : IsNonarchimedean μ) :
    IsNonarchimedean (smoothingFun μ)

Proof state:
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
hna : IsNonarchimedean ⇑μ
x y : R
hn : ∀ (n : ℕ), ∃ m < n + 1, μ ((x + y) ^ n) ^ (1 / ↑n) ≤ (μ (x ^ m) * μ (y ^ (n - m))) ^ (1 / ↑n)
mu : ℕ → ℕ := fun n => _root_.mu μ hn n
nu : ℕ → ℕ := fun n => n - mu n
hnu : nu = fun n => n - mu n
hmu_le : ∀ (n : ℕ), mu n ≤ n
hmu_bdd : ∀ (n : ℕ), ↑(mu n) / ↑n ∈ Set.Icc 0 1
hs : Bornology.IsBounded (Set.Icc 0 1)
a : ℝ
a_in : a ∈ Set.Icc 0 1
ψ : ℕ → ℕ
hψ_mono : StrictMono ψ
hψ_lim : Tendsto ((fun n => ↑(mu n) / ↑n) ∘ ψ) atTop (𝓝 a)
b : ℝ := 1 - a
hb : b = 1 - a
hb_lim : Tendsto ((fun n => ↑(nu n) / ↑n) ∘ ψ) atTop (𝓝 b)
b_in : b ∈ Set.Icc 0 1
hnu_le : ∀ (n : ℕ), nu n ≤ n
hx : limsup (fun n => μ (x ^ mu (ψ n)) ^ (1 / ↑(ψ n))) atTop ≤ smoothingFun μ x ^ a
hy : limsup (fun n => μ (y ^ nu (ψ n)) ^ (1 / ↑(ψ n))) atTop ≤ smoothingFun μ y ^ b
⊢ limsup (fun n => μ (x ^ mu (ψ n)) ^ (1 / ↑(ψ n)) * μ (y ^ nu (ψ n)) ^ (1 / ↑(ψ n))) atTop ≤
    smoothingFun μ x ^ a * smoothingFun μ y ^ b

Tactic:
have hxy' :
  limsup (fun n : ℕ => μ (x ^ mu (ψ n)) ^ (1 / (ψ n : ℝ)) * μ (y ^ nu (ψ n)) ^ (1 / (ψ n : ℝ)))
    atTop ≤ limsup (fun n : ℕ => μ (x ^ mu (ψ n)) ^ (1 / (ψ n : ℝ))) atTop *
      limsup (fun n : ℕ => μ (y ^ nu (ψ n)) ^ (1 / (ψ n : ℝ))) atTop :=
  limsup_mul_le (Eventually.of_forall (fun n => rpow_nonneg (apply_nonneg _ _) _))
    (μ_bddAbove μ hμ1 hmu_le x ψ).isBoundedUnder_of_range
    (Eventually.of_forall (fun n => rpow_nonneg (apply_nonneg _ _) _))
    (μ_bddAbove μ hμ1 hnu_le y ψ).isBoundedUnder_of_range

Result:
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
hna : IsNonarchimedean ⇑μ
x y : R
hn : ∀ (n : ℕ), ∃ m < n + 1, μ ((x + y) ^ n) ^ (1 / ↑n) ≤ (μ (x ^ m) * μ (y ^ (n - m))) ^ (1 / ↑n)
mu : ℕ → ℕ := fun n => _root_.mu μ hn n
nu : ℕ → ℕ := fun n => n - mu n
hnu : nu = fun n => n - mu n
hmu_le : ∀ (n : ℕ), mu n ≤ n
hmu_bdd : ∀ (n : ℕ), ↑(mu n) / ↑n ∈ Set.Icc 0 1
hs : Bornology.IsBounded (Set.Icc 0 1)
a : ℝ
a_in : a ∈ Set.Icc 0 1
ψ : ℕ → ℕ
hψ_mono : StrictMono ψ
hψ_lim : Tendsto ((fun n => ↑(mu n) / ↑n) ∘ ψ) atTop (𝓝 a)
b : ℝ := 1 - a
hb : b = 1 - a
hb_lim : Tendsto ((fun n => ↑(nu n) / ↑n) ∘ ψ) atTop (𝓝 b)
b_in : b ∈ Set.Icc 0 1
hnu_le : ∀ (n : ℕ), nu n ≤ n
hx : limsup (fun n => μ (x ^ mu (ψ n)) ^ (1 / ↑(ψ n))) atTop ≤ smoothingFun μ x ^ a
hy : limsup (fun n => μ (y ^ nu (ψ n)) ^ (1 / ↑(ψ n))) atTop ≤ smoothingFun μ y ^ b
hxy' :
  limsup (fun n => μ (x ^ mu (ψ n)) ^ (1 / ↑(ψ n)) * μ (y ^ nu (ψ n)) ^ (1 / ↑(ψ n))) atTop ≤
    limsup (fun n => μ (x ^ mu (ψ n)) ^ (1 / ↑(ψ n))) atTop * limsup (fun n => μ (y ^ nu (ψ n)) ^ (1 / ↑(ψ n))) atTop
⊢ limsup (fun n => μ (x ^ mu (ψ n)) ^ (1 / ↑(ψ n)) * μ (y ^ nu (ψ n)) ^ (1 / ↑(ψ n))) atTop ≤
    smoothingFun μ x ^ a * smoothingFun μ y ^ b

Proof state:
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
hna : IsNonarchimedean ⇑μ
x y : R
hn : ∀ (n : ℕ), ∃ m < n + 1, μ ((x + y) ^ n) ^ (1 / ↑n) ≤ (μ (x ^ m) * μ (y ^ (n - m))) ^ (1 / ↑n)
mu : ℕ → ℕ := fun n => _root_.mu μ hn n
nu : ℕ → ℕ := fun n => n - mu n
hnu : nu = fun n => n - mu n
hmu_le : ∀ (n : ℕ), mu n ≤ n
hmu_bdd : ∀ (n : ℕ), ↑(mu n) / ↑n ∈ Set.Icc 0 1
hs : Bornology.IsBounded (Set.Icc 0 1)
a : ℝ
a_in : a ∈ Set.Icc 0 1
ψ : ℕ → ℕ
hψ_mono : StrictMono ψ
hψ_lim : Tendsto ((fun n => ↑(mu n) / ↑n) ∘ ψ) atTop (𝓝 a)
b : ℝ := 1 - a
hb : b = 1 - a
⊢ ∃ a, ∀ (b : ℕ), a ≤ b → 1 - ↑(mu (ψ b)) / ↑(ψ b) = ↑(nu (ψ b)) / ↑(ψ b)

Tactic:
use 1

Result:
case h
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
hna : IsNonarchimedean ⇑μ
x y : R
hn : ∀ (n : ℕ), ∃ m < n + 1, μ ((x + y) ^ n) ^ (1 / ↑n) ≤ (μ (x ^ m) * μ (y ^ (n - m))) ^ (1 / ↑n)
mu : ℕ → ℕ := fun n => _root_.mu μ hn n
nu : ℕ → ℕ := fun n => n - mu n
hnu : nu = fun n => n - mu n
hmu_le : ∀ (n : ℕ), mu n ≤ n
hmu_bdd : ∀ (n : ℕ), ↑(mu n) / ↑n ∈ Set.Icc 0 1
hs : Bornology.IsBounded (Set.Icc 0 1)
a : ℝ
a_in : a ∈ Set.Icc 0 1
ψ : ℕ → ℕ
hψ_mono : StrictMono ψ
hψ_lim : Tendsto ((fun n => ↑(mu n) / ↑n) ∘ ψ) atTop (𝓝 a)
b : ℝ := 1 - a
hb : b = 1 - a
⊢ ∀ (b : ℕ), 1 ≤ b → 1 - ↑(mu (ψ b)) / ↑(ψ b) = ↑(nu (ψ b)) / ↑(ψ b)

Proof state:
case intro.intro.intro.intro
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
hna : IsNonarchimedean ⇑μ
x y : R
hn : ∀ (n : ℕ), ∃ m < n + 1, μ ((x + y) ^ n) ^ (1 / ↑n) ≤ (μ (x ^ m) * μ (y ^ (n - m))) ^ (1 / ↑n)
mu : ℕ → ℕ := fun n => _root_.mu μ hn n
nu : ℕ → ℕ := fun n => n - mu n
hnu : nu = fun n => n - mu n
hmu_le : ∀ (n : ℕ), mu n ≤ n
hmu_bdd : ∀ (n : ℕ), ↑(mu n) / ↑n ∈ Set.Icc 0 1
hs : Bornology.IsBounded (Set.Icc 0 1)
a : ℝ
a_in : a ∈ Set.Icc 0 1
ψ : ℕ → ℕ
hψ_mono : StrictMono ψ
hψ_lim : Tendsto ((fun n => ↑(mu n) / ↑n) ∘ ψ) atTop (𝓝 a)
b : ℝ := 1 - a
hb : b = 1 - a
hb_lim : Tendsto ((fun n => ↑(nu n) / ↑n) ∘ ψ) atTop (𝓝 b)
b_in : b ∈ Set.Icc 0 1
hnu_le : ∀ (n : ℕ), nu n ≤ n
hx : limsup (fun n => μ (x ^ mu (ψ n)) ^ (1 / ↑(ψ n))) atTop ≤ smoothingFun μ x ^ a
hy : limsup (fun n => μ (y ^ nu (ψ n)) ^ (1 / ↑(ψ n))) atTop ≤ smoothingFun μ y ^ b
hxy :
  limsup (fun n => μ (x ^ mu (ψ n)) ^ (1 / ↑(ψ n)) * μ (y ^ nu (ψ n)) ^ (1 / ↑(ψ n))) atTop ≤
    smoothingFun μ x ^ a * smoothingFun μ y ^ b
⊢ smoothingFun μ (x + y) ≤ smoothingFun μ x ⊔ smoothingFun μ y

Tactic:
apply le_of_forall_sub_le

Result:
case intro.intro.intro.intro.h
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
hna : IsNonarchimedean ⇑μ
x y : R
hn : ∀ (n : ℕ), ∃ m < n + 1, μ ((x + y) ^ n) ^ (1 / ↑n) ≤ (μ (x ^ m) * μ (y ^ (n - m))) ^ (1 / ↑n)
mu : ℕ → ℕ := fun n => _root_.mu μ hn n
nu : ℕ → ℕ := fun n => n - mu n
hnu : nu = fun n => n - mu n
hmu_le : ∀ (n : ℕ), mu n ≤ n
hmu_bdd : ∀ (n : ℕ), ↑(mu n) / ↑n ∈ Set.Icc 0 1
hs : Bornology.IsBounded (Set.Icc 0 1)
a : ℝ
a_in : a ∈ Set.Icc 0 1
ψ : ℕ → ℕ
hψ_mono : StrictMono ψ
hψ_lim : Tendsto ((fun n => ↑(mu n) / ↑n) ∘ ψ) atTop (𝓝 a)
b : ℝ := 1 - a
hb : b = 1 - a
hb_lim : Tendsto ((fun n => ↑(nu n) / ↑n) ∘ ψ) atTop (𝓝 b)
b_in : b ∈ Set.Icc 0 1
hnu_le : ∀ (n : ℕ), nu n ≤ n
hx : limsup (fun n => μ (x ^ mu (ψ n)) ^ (1 / ↑(ψ n))) atTop ≤ smoothingFun μ x ^ a
hy : limsup (fun n => μ (y ^ nu (ψ n)) ^ (1 / ↑(ψ n))) atTop ≤ smoothingFun μ y ^ b
hxy :
  limsup (fun n => μ (x ^ mu (ψ n)) ^ (1 / ↑(ψ n)) * μ (y ^ nu (ψ n)) ^ (1 / ↑(ψ n))) atTop ≤
    smoothingFun μ x ^ a * smoothingFun μ y ^ b
⊢ ∀ ε > 0, smoothingFun μ (x + y) - ε ≤ smoothingFun μ x ⊔ smoothingFun μ y

Proof state:
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
hna : IsNonarchimedean ⇑μ
x y : R
hn : ∀ (n : ℕ), ∃ m < n + 1, μ ((x + y) ^ n) ^ (1 / ↑n) ≤ (μ (x ^ m) * μ (y ^ (n - m))) ^ (1 / ↑n)
mu : ℕ → ℕ := fun n => _root_.mu μ hn n
nu : ℕ → ℕ := fun n => n - mu n
hnu : nu = fun n => n - mu n
hmu_le : ∀ (n : ℕ), mu n ≤ n
hmu_bdd : ∀ (n : ℕ), ↑(mu n) / ↑n ∈ Set.Icc 0 1
hs : Bornology.IsBounded (Set.Icc 0 1)
a : ℝ
a_in : a ∈ Set.Icc 0 1
ψ : ℕ → ℕ
hψ_mono : StrictMono ψ
hψ_lim : Tendsto ((fun n => ↑(mu n) / ↑n) ∘ ψ) atTop (𝓝 a)
b : ℝ := 1 - a
hb : b = 1 - a
⊢ (fun k => 1 - ((fun n => ↑(mu n) / ↑n) ∘ ψ) k) =ᶠ[atTop] (fun n => ↑(nu n) / ↑n) ∘ ψ

Tactic:
simp only [EventuallyEq, Function.comp_apply, eventually_atTop, ge_iff_le]

Result:
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
hna : IsNonarchimedean ⇑μ
x y : R
hn : ∀ (n : ℕ), ∃ m < n + 1, μ ((x + y) ^ n) ^ (1 / ↑n) ≤ (μ (x ^ m) * μ (y ^ (n - m))) ^ (1 / ↑n)
mu : ℕ → ℕ := fun n => _root_.mu μ hn n
nu : ℕ → ℕ := fun n => n - mu n
hnu : nu = fun n => n - mu n
hmu_le : ∀ (n : ℕ), mu n ≤ n
hmu_bdd : ∀ (n : ℕ), ↑(mu n) / ↑n ∈ Set.Icc 0 1
hs : Bornology.IsBounded (Set.Icc 0 1)
a : ℝ
a_in : a ∈ Set.Icc 0 1
ψ : ℕ → ℕ
hψ_mono : StrictMono ψ
hψ_lim : Tendsto ((fun n => ↑(mu n) / ↑n) ∘ ψ) atTop (𝓝 a)
b : ℝ := 1 - a
hb : b = 1 - a
⊢ ∃ a, ∀ (b : ℕ), a ≤ b → 1 - ↑(mu (ψ b)) / ↑(ψ b) = ↑(nu (ψ b)) / ↑(ψ b)