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

Proof state:
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
m : ℕ
hm : 1 ≤ m
⊢ 1 - ↑(mu (ψ m)) / ↑(ψ m) = ↑(nu (ψ m)) / ↑(ψ m)

Tactic:
have h0 : (ψ m : ℝ) ≠ 0 := cast_ne_zero.mpr (_root_.ne_of_gt (lt_of_le_of_lt (_root_.zero_le _)
  (hψ_mono (Nat.pos_of_ne_zero (one_le_iff_ne_zero.mp hm)))))

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
m : ℕ
hm : 1 ≤ m
h0 : ↑(ψ m) ≠ 0
⊢ 1 - ↑(mu (ψ m)) / ↑(ψ m) = ↑(nu (ψ m)) / ↑(ψ m)

Proof state:
case 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
ε : ℝ
hε : ε > 0
h_mul : smoothingFun μ x ^ a * smoothingFun μ y ^ b + ε ≤ smoothingFun μ x ⊔ smoothingFun μ y + ε
N : ℕ+
hN :
  μ (x ^ mu (ψ ↑N)) ^ (1 / ↑(ψ ↑N)) * μ (y ^ nu (ψ ↑N)) ^ (1 / ↑(ψ ↑N)) <
    smoothingFun μ x ^ a * smoothingFun μ y ^ b + ε
⊢ μ ((x + y) ^ ↑⟨ψ ↑N, ⋯⟩) ^ (1 / ↑↑⟨ψ ↑N, ⋯⟩) ≤ smoothingFun μ x ^ a * smoothingFun μ y ^ b + ε

Tactic:
apply le_trans _ hN.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
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 : ℝ
hε : ε > 0
h_mul : smoothingFun μ x ^ a * smoothingFun μ y ^ b + ε ≤ smoothingFun μ x ⊔ smoothingFun μ y + ε
N : ℕ+
hN :
  μ (x ^ mu (ψ ↑N)) ^ (1 / ↑(ψ ↑N)) * μ (y ^ nu (ψ ↑N)) ^ (1 / ↑(ψ ↑N)) <
    smoothingFun μ x ^ a * smoothingFun μ y ^ b + ε
⊢ μ ((x + y) ^ ↑⟨ψ ↑N, ⋯⟩) ^ (1 / ↑↑⟨ψ ↑N, ⋯⟩) ≤ μ (x ^ mu (ψ ↑N)) ^ (1 / ↑(ψ ↑N)) * μ (y ^ nu (ψ ↑N)) ^ (1 / ↑(ψ ↑N))

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)
⊢ smoothingFun μ (x + y) ≤ smoothingFun μ x ⊔ smoothingFun μ y

Tactic:
have b_in : b ∈ Set.Icc (0 : ℝ) 1 := unitInterval.mem_iff_one_sub_mem.mp a_in

Result:
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
⊢ smoothingFun μ (x + y) ≤ smoothingFun μ x ⊔ smoothingFun μ y

Proof state:
case neg
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 : ℝ
hε : ε > 0
h : ¬smoothingFun μ x ≤ smoothingFun μ y
⊢ smoothingFun μ x ^ a * smoothingFun μ x ^ b ≤ smoothingFun μ x

Tactic:
rw [hb, ← rpow_add_of_nonneg (smoothingFun_nonneg μ hμ1 _) a_in.1
  (sub_nonneg.mpr a_in.2), add_sub, add_sub_cancel_left, rpow_one]

Result:
no goals