mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: ⊢ EqOn (log ∘ doublingGamma) (fun s ↦ log (Γ (s / 2)) + log (Γ (s / 2 + 1 / 2)) + s * log 2 - log (2 * √π)) (Ioi 0)

Tactic:
intro s hs
  have h1 : √π ≠ 0 := sqrt_ne_zero'.mpr pi_pos
  have h2 : Gamma (s / 2) ≠ 0 := (Gamma_pos_of_pos <| div_pos hs two_pos).ne'
  have h3 : Gamma (s / 2 + 1 / 2) ≠ 0 :=
    (Gamma_pos_of_pos <| add_pos (div_pos hs two_pos) one_half_pos).ne'
  have h4 : (2 : ℝ) ^ (s - 1) ≠ 0 := (rpow_pos_of_pos two_pos _).ne'
  rw [Function.comp_apply, doublingGamma, log_div (mul_ne_zero (mul_ne_zero h2 h3) h4) h1,
    log_mul (mul_ne_zero h2 h3) h4, log_mul h2 h3, log_rpow two_pos, log_mul two_ne_zero h1]
  ring_nf

Goal state after:
Module: Mathlib.Topology.Algebra.AsymptoticCone

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝¹⁰ : Field k
inst✝⁹ : LinearOrder k
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module k V
inst✝⁶ : AddTorsor V P
inst✝⁵ : TopologicalSpace V
α : Type u_4
l : Filter α
inst✝⁴ : TopologicalSpace k
inst✝³ : OrderTopology k
inst✝² : IsStrictOrderedRing k
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul k V
v : V
p : P
⊢ (asymptoticNhds k P v).NeBot

Tactic:
asymptoticNhds_eq_smul_vadd v p

Goal state after:
k : Type u_1
V : Type u_2
P : Type u_3
inst✝¹⁰ : Field k
inst✝⁹ : LinearOrder k
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module k V
inst✝⁶ : AddTorsor V P
inst✝⁵ : TopologicalSpace V
α : Type u_4
l : Filter α
inst✝⁴ : TopologicalSpace k
inst✝³ : OrderTopology k
inst✝² : IsStrictOrderedRing k
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul k V
v : V
p : P
⊢ (atTop • 𝓝 v +ᵥ pure p).NeBot
Module: Mathlib.Topology.Algebra.AsymptoticCone

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝¹⁰ : Field k
inst✝⁹ : LinearOrder k
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module k V
inst✝⁶ : AddTorsor V P
inst✝⁵ : TopologicalSpace V
α : Type u_4
l : Filter α
inst✝⁴ : TopologicalSpace k
inst✝³ : OrderTopology k
inst✝² : IsStrictOrderedRing k
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul k V
v : V
p : P
⊢ (atTop • 𝓝 v +ᵥ pure p).NeBot

Tactic:
infer_instance

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: case e_a.e_a
s : ℂ
n : ℕ
hn : n ≠ 0
⊢ ↑n ^ s * ↑n = ↑n * ↑n ^ s

Tactic:
mul_comm

Goal state after:
case e_a.e_a
s : ℂ
n : ℕ
hn : n ≠ 0
⊢ ↑n * ↑n ^ s = ↑n * ↑n ^ s
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: case h
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := ⋯
f_ible : ∀ (n : ℕ), Integrable (f n) (volume.restrict (Ioi 0))
x : ℝ
hx : x ∈ Ioi 0
⊢ Tendsto (fun n ↦ ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)) atTop (𝓝 (↑(rexp (-x)) * ↑x ^ (s - 1)))

Tactic:
mul_comm

Goal state after:
case h
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n ↦ (Ioc 0 ↑n).indicator fun x ↦ ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
f_ible : ∀ (n : ℕ), Integrable (f n) (volume.restrict (Ioi 0))
x : ℝ
hx : x ∈ Ioi 0
⊢ Tendsto (fun n ↦ ↑x ^ (s - 1) * ↑((1 - x / ↑n) ^ n)) atTop (𝓝 (↑x ^ (s - 1) * ↑(rexp (-x))))
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: case pos
z : ℂ
pi_ne : ↑π ≠ 0
hs : ∃ k, ↑π * -z = ↑k * ↑π
⊢ Gamma z * Gamma (1 - z) = 0

Tactic:
mul_comm

Goal state after:
case pos
z : ℂ
pi_ne : ↑π ≠ 0
hs : ∃ k, -z * ↑π = ↑k * ↑π
⊢ Gamma z * Gamma (1 - z) = 0
Module: Mathlib.Analysis.Normed.Affine.Simplex

Goal state before:
pp: case inl.inl
R : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : Ring R
inst✝³ : SeminormedAddCommGroup V
inst✝² : PseudoMetricSpace P
inst✝¹ : Module R V
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex R P n
hs : s.Scalene
i₁ i₂ i₃ i₄ : Fin (n + 1)
h₁₂ : i₁ ≠ i₂
h₃₄ : i₃ ≠ i₄
h₁₂₃₄ : ¬i₁ = i₃ ∨ ¬i₂ = i₄
h₁₂₄₃ : ¬i₁ = i₄ ∨ ¬i₂ = i₃
h₁₂lt : i₁ < i₂
h₃₄lt : i₃ < i₄
⊢ ⟨(i₁, i₂), h₁₂lt⟩ ≠ ⟨(i₃, i₄), h₃₄lt⟩

Tactic:
cases h₁₂₃₄

Goal state after:
case inl.inl.inl
R : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : Ring R
inst✝³ : SeminormedAddCommGroup V
inst✝² : PseudoMetricSpace P
inst✝¹ : Module R V
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex R P n
hs : s.Scalene
i₁ i₂ i₃ i₄ : Fin (n + 1)
h₁₂ : i₁ ≠ i₂
h₃₄ : i₃ ≠ i₄
h₁₂₄₃ : ¬i₁ = i₄ ∨ ¬i₂ = i₃
h₁₂lt : i₁ < i₂
h₃₄lt : i₃ < i₄
h✝ : ¬i₁ = i₃
⊢ ⟨(i₁, i₂), h₁₂lt⟩ ≠ ⟨(i₃, i₄), h₃₄lt⟩

case inl.inl.inr
R : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : Ring R
inst✝³ : SeminormedAddCommGroup V
inst✝² : PseudoMetricSpace P
inst✝¹ : Module R V
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex R P n
hs : s.Scalene
i₁ i₂ i₃ i₄ : Fin (n + 1)
h₁₂ : i₁ ≠ i₂
h₃₄ : i₃ ≠ i₄
h₁₂₄₃ : ¬i₁ = i₄ ∨ ¬i₂ = i₃
h₁₂lt : i₁ < i₂
h₃₄lt : i₃ < i₄
h✝ : ¬i₂ = i₄
⊢ ⟨(i₁, i₂), h₁₂lt⟩ ≠ ⟨(i₃, i₄), h₃₄lt⟩
Module: Mathlib.Analysis.Normed.Affine.Simplex

Goal state before:
pp: case inr.inr
R : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : Ring R
inst✝³ : SeminormedAddCommGroup V
inst✝² : PseudoMetricSpace P
inst✝¹ : Module R V
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex R P n
hs : s.Scalene
i₁ i₂ i₃ i₄ : Fin (n + 1)
h₁₂ : i₁ ≠ i₂
h₃₄ : i₃ ≠ i₄
h₁₂₃₄ : ¬i₁ = i₃ ∨ ¬i₂ = i₄
h₁₂₄₃ : ¬i₁ = i₄ ∨ ¬i₂ = i₃
h₂₁lt : i₂ < i₁
h₄₃lt : i₄ < i₃
⊢ ⟨(i₂, i₁), h₂₁lt⟩ ≠ ⟨(i₄, i₃), h₄₃lt⟩

Tactic:
cases h₁₂₃₄

Goal state after:
case inr.inr.inl
R : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : Ring R
inst✝³ : SeminormedAddCommGroup V
inst✝² : PseudoMetricSpace P
inst✝¹ : Module R V
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex R P n
hs : s.Scalene
i₁ i₂ i₃ i₄ : Fin (n + 1)
h₁₂ : i₁ ≠ i₂
h₃₄ : i₃ ≠ i₄
h₁₂₄₃ : ¬i₁ = i₄ ∨ ¬i₂ = i₃
h₂₁lt : i₂ < i₁
h₄₃lt : i₄ < i₃
h✝ : ¬i₁ = i₃
⊢ ⟨(i₂, i₁), h₂₁lt⟩ ≠ ⟨(i₄, i₃), h₄₃lt⟩

case inr.inr.inr
R : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : Ring R
inst✝³ : SeminormedAddCommGroup V
inst✝² : PseudoMetricSpace P
inst✝¹ : Module R V
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex R P n
hs : s.Scalene
i₁ i₂ i₃ i₄ : Fin (n + 1)
h₁₂ : i₁ ≠ i₂
h₃₄ : i₃ ≠ i₄
h₁₂₄₃ : ¬i₁ = i₄ ∨ ¬i₂ = i₃
h₂₁lt : i₂ < i₁
h₄₃lt : i₄ < i₃
h✝ : ¬i₂ = i₄
⊢ ⟨(i₂, i₁), h₂₁lt⟩ ≠ ⟨(i₄, i₃), h₄₃lt⟩
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: s : ℝ
n : ℕ
⊢ ↑↑n ^ ↑s * ↑n ! / ∏ x ∈ Finset.range (n + 1), (↑s + ↑x) = ↑n ^ ↑s * ↑n ! / ∏ x ∈ Finset.range (n + 1), (↑s + ↑x)

Tactic:
Complex.ofReal_natCast

Goal state after:
s : ℝ
n : ℕ
⊢ ↑n ^ ↑s * ↑n ! / ∏ x ∈ Finset.range (n + 1), (↑s + ↑x) = ↑n ^ ↑s * ↑n ! / ∏ x ∈ Finset.range (n + 1), (↑s + ↑x)
Module: Mathlib.Analysis.Normed.Algebra.GelfandMazur

Goal state before:
pp: F : Type u_1
inst✝³ : NormedRing F
inst✝² : NormOneClass F
inst✝¹ : NormMulClass F
inst✝ : NormedAlgebra ℂ F
x : F
z : ℂ
hz : IsMinOn (fun x_1 ↦ ‖x - (algebraMap ℂ F) x_1‖) univ z
H : ∀ (z' : ℂ), ‖x - (algebraMap ℂ F) z'‖ ≠ 0
c : ℂ
M : ℝ := ‖x - (algebraMap ℂ F) z‖
hMdef : M = ‖x - (algebraMap ℂ F) z‖
hM₀ : 0 < M
y w : ℂ
hy : ‖x - (algebraMap ℂ F) y‖ = M
n : ℕ
hn : n > 0
p : ℂ[X]
hp : p.IsMonicOfDegree (n - 1)
hrel : p * (X - C (w - y)) = X ^ n - C (w - y) ^ n
⊢ ‖x - (algebraMap ℂ F) w‖ * ‖(aeval (x - (algebraMap ℂ F) y)) p‖ ≤ M ^ n + ‖w - y‖ ^ n

Tactic:
mul_comm

Goal state after:
F : Type u_1
inst✝³ : NormedRing F
inst✝² : NormOneClass F
inst✝¹ : NormMulClass F
inst✝ : NormedAlgebra ℂ F
x : F
z : ℂ
hz : IsMinOn (fun x_1 ↦ ‖x - (algebraMap ℂ F) x_1‖) univ z
H : ∀ (z' : ℂ), ‖x - (algebraMap ℂ F) z'‖ ≠ 0
c : ℂ
M : ℝ := ‖x - (algebraMap ℂ F) z‖
hMdef : M = ‖x - (algebraMap ℂ F) z‖
hM₀ : 0 < M
y w : ℂ
hy : ‖x - (algebraMap ℂ F) y‖ = M
n : ℕ
hn : n > 0
p : ℂ[X]
hp : p.IsMonicOfDegree (n - 1)
hrel : (X - C (w - y)) * p = X ^ n - C (w - y) ^ n
⊢ ‖x - (algebraMap ℂ F) w‖ * ‖(aeval (x - (algebraMap ℂ F) y)) p‖ ≤ M ^ n + ‖w - y‖ ^ n
Module: Mathlib.Analysis.Normed.Affine.Simplex

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : Ring R
inst✝³ : SeminormedAddCommGroup V
inst✝² : PseudoMetricSpace P
inst✝¹ : Module R V
inst✝ : NormedAddTorsor V P
t : Triangle R P
⊢ Simplex.Scalene t ↔
    dist (t.points 0) (t.points 1) ≠ dist (t.points 0) (t.points 2) ∧
      dist (t.points 0) (t.points 1) ≠ dist (t.points 1) (t.points 2) ∧
        dist (t.points 0) (t.points 2) ≠ dist (t.points 1) (t.points 2)

Tactic:
by
  refine ⟨fun h ↦
    ⟨h.dist_ne (by decide : (0 : Fin 3) ≠ 1) (by decide : (0 : Fin 3) ≠ 2) (by decide) (by decide),
     h.dist_ne (by decide : (0 : Fin 3) ≠ 1) (by decide : (1 : Fin 3) ≠ 2) (by decide) (by decide),
     h.dist_ne (by decide : (0 : Fin 3) ≠ 2) (by decide : (1 : Fin 3) ≠ 2) (by decide) (by decide)⟩,
    fun ⟨h₁, h₂, h₃⟩ ↦ ?_⟩
  intro ⟨⟨x₁, x₂⟩, hx⟩ ⟨⟨y₁, y₂⟩, hy⟩ hxy
  simp only at hx hy hxy
  simp only [Subtype.mk.injEq, Prod.mk.injEq]
  fin_cases x₁ <;> fin_cases x₂ <;> simp +decide only at hx <;>
    fin_cases y₁ <;> fin_cases y₂ <;> simp +decide only at hy <;>
    simp [h₁, h₂, h₃, h₁.symm, h₂.symm, h₃.symm] at hxy ⊢

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Series

Goal state before:
pp: z : ℂ
⊢ (fun n ↦ (-1) ^ n * (z ^ 2) ^ n / ↑(2 * n)!) = fun n ↦ (z ^ 2) ^ n * (-1) ^ n / ↑(2 * n)!

Tactic:
mul_comm

Goal state after:
Module: Mathlib.Analysis.Normed.Algebra.GelfandMazur

Goal state before:
pp: F : Type u_1
inst✝³ : NormedRing F
inst✝² : NormedAlgebra ℝ F
inst✝¹ : NormOneClass F
inst✝ : NormMulClass F
x : F
z : ℝ × ℝ
h : IsMinOn (fun x_1 ↦ ‖φ x x_1‖) Set.univ z
w✝ : ℝ × ℝ
M : ℝ := ‖φ x z‖
H : M ≠ 0
hM : M = ‖φ x z‖
hM₀ : 0 < M
w u : ℝ × ℝ
hw : ‖φ x w‖ = M
n : ℕ
hn : n > 0
q : ℝ × ℝ → ℝ[X] := fun y ↦ X ^ 2 - C y.1 * X + C y.2
hq : ∀ (y : ℝ × ℝ), (q y).IsMonicOfDegree 2
p : ℝ[X]
hp : p.IsMonicOfDegree (2 * (n - 1))
hrel : p * q u = q w ^ n - (q w - q u) ^ n
⊢ ‖φ x u‖ * ‖(aeval x) p‖ ≤ M ^ n + ‖φ x u - φ x w‖ ^ n

Tactic:
mul_comm

Goal state after:
F : Type u_1
inst✝³ : NormedRing F
inst✝² : NormedAlgebra ℝ F
inst✝¹ : NormOneClass F
inst✝ : NormMulClass F
x : F
z : ℝ × ℝ
h : IsMinOn (fun x_1 ↦ ‖φ x x_1‖) Set.univ z
w✝ : ℝ × ℝ
M : ℝ := ‖φ x z‖
H : M ≠ 0
hM : M = ‖φ x z‖
hM₀ : 0 < M
w u : ℝ × ℝ
hw : ‖φ x w‖ = M
n : ℕ
hn : n > 0
q : ℝ × ℝ → ℝ[X] := fun y ↦ X ^ 2 - C y.1 * X + C y.2
hq : ∀ (y : ℝ × ℝ), (q y).IsMonicOfDegree 2
p : ℝ[X]
hp : p.IsMonicOfDegree (2 * (n - 1))
hrel : q u * p = q w ^ n - (q w - q u) ^ n
⊢ ‖φ x u‖ * ‖(aeval x) p‖ ≤ M ^ n + ‖φ x u - φ x w‖ ^ n
Module: Mathlib.Topology.Algebra.Order.LiminfLimsup

Goal state before:
pp: ι : Type u_1
α : Type u_2
inst✝⁴ : AddCommGroup α
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : DenselyOrdered α
inst✝¹ : AddLeftMono α
f : Filter ι
inst✝ : f.NeBot
u v : ι → α
h₁ : IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) f u
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
h₃ : IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) f v
h₄ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≥ x2) f v
h : IsCoboundedUnder (fun x1 x2 ↦ x2 ≤ x1) f (u + v)
h' : IsBoundedUnder (fun x1 x2 ↦ x2 ≤ x1) f (u + v)
a : α
a_u : a > limsup u f
b : α
b_v : b > liminf v f
x✝¹ : α
c_ab : x✝¹ > a + b
x✝ : ι
ab_x : v x✝ < b ∧ u x✝ < a
⊢ (u + v) x✝ < x✝¹

Tactic:
exact (add_lt_add ab_x.2 ab_x.1).trans c_ab

Goal state after:
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
hna : IsNonarchimedean f
x y z : R
hf : f ((x + y) * z) / f z ≤ f (x * z) / f z ∨ f ((x + y) * z) / f z ≤ f (y * z) / f z
⊢ f ((x + y) * z) / f z ≤ seminormFromBounded' f x ∨ f ((x + y) * z) / f z ≤ seminormFromBounded' f y

Tactic:
by
    rcases hf with hfx | hfy
    · exact Or.inl <| le_ciSup_of_le (seminormFromBounded_bddAbove_range f_nonneg f_mul x) z hfx
    · exact Or.inr <| le_ciSup_of_le (seminormFromBounded_bddAbove_range f_nonneg f_mul y) z hfy

Goal state after:
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

Goal state before:
pp: case pos
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
x✝ : R
hx✝ : ∀ (y : R), f (x✝ * y) = f x✝ * f y
x : R
hx : f x = 0
⊢ f x✝ * (0 / 0) ≤ f x✝

Tactic:
div_zero,

Goal state after:
case pos
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
x✝ : R
hx✝ : ∀ (y : R), f (x✝ * y) = f x✝ * f y
x : R
hx : f x = 0
⊢ f x✝ * 0 ≤ f x✝
Module: Mathlib.Analysis.Normed.Unbundled.InvariantExtension

Goal state before:
pp: K : Type u_1
inst✝² : NormedField K
L : Type u_2
inst✝¹ : Field L
inst✝ : Algebra K L
h_fin : FiniteDimensional K L
hu : IsUltrametricDist K
⊢ ⨆ σ, (algNormOfAlgEquiv σ) 0 = 0

Tactic:
by simp only [map_zero, ciSup_const]

Goal state after:
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromConst

Goal state before:
pp: case f
R : Type u_1
inst✝ : CommRing R
c : R
f : RingSeminorm R
hf1 : f 1 ≤ 1
hc : f c ≠ 0
hpm : IsPowMul ⇑f
hna : IsNonarchimedean ⇑f
x y : R
n : ℕ
hmax : f ((x + y) * c ^ n) ≤ f (x * c ^ n) ∨ f ((x + y) * c ^ n) ≤ f (y * c ^ n)
⊢ f ((x + y) * c ^ n) ≤ f (x * c ^ n) → f ((x + y) * c ^ n) / f c ^ n ≤ f (x * c ^ n) / f c ^ n

Tactic:
intro

Goal state after:
case f
R : Type u_1
inst✝ : CommRing R
c : R
f : RingSeminorm R
hf1 : f 1 ≤ 1
hc : f c ≠ 0
hpm : IsPowMul ⇑f
hna : IsNonarchimedean ⇑f
x y : R
n : ℕ
hmax : f ((x + y) * c ^ n) ≤ f (x * c ^ n) ∨ f ((x + y) * c ^ n) ≤ f (y * c ^ n)
a✝ : f ((x + y) * c ^ n) ≤ f (x * c ^ n)
⊢ f ((x + y) * c ^ n) / f c ^ n ≤ f (x * c ^ n) / f c ^ n
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromConst

Goal state before:
pp: case g
R : Type u_1
inst✝ : CommRing R
c : R
f : RingSeminorm R
hf1 : f 1 ≤ 1
hc : f c ≠ 0
hpm : IsPowMul ⇑f
hna : IsNonarchimedean ⇑f
x y : R
n : ℕ
hmax : f ((x + y) * c ^ n) ≤ f (x * c ^ n) ∨ f ((x + y) * c ^ n) ≤ f (y * c ^ n)
⊢ f ((x + y) * c ^ n) ≤ f (y * c ^ n) → f ((x + y) * c ^ n) / f c ^ n ≤ f (y * c ^ n) / f c ^ n

Tactic:
intro

Goal state after:
case g
R : Type u_1
inst✝ : CommRing R
c : R
f : RingSeminorm R
hf1 : f 1 ≤ 1
hc : f c ≠ 0
hpm : IsPowMul ⇑f
hna : IsNonarchimedean ⇑f
x y : R
n : ℕ
hmax : f ((x + y) * c ^ n) ≤ f (x * c ^ n) ∨ f ((x + y) * c ^ n) ≤ f (y * c ^ n)
a✝ : f ((x + y) * c ^ n) ≤ f (y * c ^ n)
⊢ f ((x + y) * c ^ n) / f c ^ n ≤ f (y * c ^ n) / f c ^ n
Module: Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm

Goal state before:
pp: case neg
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
x : R
hx : μ x ≠ 0
L : ℝ := ⨅ n, μ (x ^ ↑n) ^ (1 / ↑↑n)
hL0 : 0 ≤ L
ε : ℝ
hε : ε > 0
m1 : ℕ+
hm1 : μ (x ^ ↑m1) ^ (1 / ↑↑m1) < (⨅ n, μ (x ^ ↑n) ^ (1 / ↑↑n)) + ε / 2
m2 : ℕ
hm2 : ∀ n ≥ m2, (L + ε / 2) ^ (-(↑(n % ↑m1) / ↑n)) * (μ x ^ (n % ↑m1)) ^ (1 / ↑n) - 1 ≤ ε / (2 * (L + ε / 2))
n : ℕ
hn : n ≥ max (↑m1) m2
hn0 : 0 < n
hL_le : L ≤ smoothingSeminormSeq μ x n
hxn : ¬μ (x ^ (n % ↑m1)) = 0
hxn' : 0 < μ (x ^ (n % ↑m1))
⊢ μ (x ^ (↑m1 * (n / ↑m1) + n % ↑m1)) ^ (1 / ↑n) < L + ε

Tactic:
have h : μ (x ^ ((m1 : ℕ) * (n / (m1 : ℕ)))) ^ (1 / (n : ℝ)) ≤
        (μ (x ^ (m1 : ℕ)) ^ (n / (m1 : ℕ))) ^ (1 / (n : ℝ)) := by
      gcongr
      rw [pow_mul]
      exact map_pow_le_pow μ (x ^ (m1 : ℕ))
        (pos_iff_ne_zero.mp (Nat.div_pos (le_trans (le_max_left (m1 : ℕ) m2) hn) (PNat.pos m1)))

Goal state after:
case neg
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
x : R
hx : μ x ≠ 0
L : ℝ := ⨅ n, μ (x ^ ↑n) ^ (1 / ↑↑n)
hL0 : 0 ≤ L
ε : ℝ
hε : ε > 0
m1 : ℕ+
hm1 : μ (x ^ ↑m1) ^ (1 / ↑↑m1) < (⨅ n, μ (x ^ ↑n) ^ (1 / ↑↑n)) + ε / 2
m2 : ℕ
hm2 : ∀ n ≥ m2, (L + ε / 2) ^ (-(↑(n % ↑m1) / ↑n)) * (μ x ^ (n % ↑m1)) ^ (1 / ↑n) - 1 ≤ ε / (2 * (L + ε / 2))
n : ℕ
hn : n ≥ max (↑m1) m2
hn0 : 0 < n
hL_le : L ≤ smoothingSeminormSeq μ x n
hxn : ¬μ (x ^ (n % ↑m1)) = 0
hxn' : 0 < μ (x ^ (n % ↑m1))
h : μ (x ^ (↑m1 * (n / ↑m1))) ^ (1 / ↑n) ≤ (μ (x ^ ↑m1) ^ (n / ↑m1)) ^ (1 / ↑n)
⊢ μ (x ^ (↑m1 * (n / ↑m1) + n % ↑m1)) ^ (1 / ↑n) < L + ε
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: case convert_4
R : Type u_1
inst✝¹ : SeminormedRing R
inst✝ : Nontrivial R
n : ℕ
⊢ Nonempty ℕ

Tactic:
infer_instance

Goal state after:
Module: Mathlib.Analysis.Normed.Field.Dense

Goal state before:
pp: case pos
K : Type u_1
L : Type u_2
inst✝⁶ : Field K
inst✝⁵ : NontriviallyNormedField L
inst✝⁴ : CompleteSpace L
inst✝³ : CharZero L
inst✝² : IsUltrametricDist L
inst✝¹ : Algebra K L
hi : DenseRange ⇑(algebraMap K L)
inst✝ : IsAlgClosed K
f : L[X]
fmon : f.Monic
firr : Irreducible f
fnatdeg0 : f.natDegree ≠ 0
F : Type u_2 := f.SplittingField
this✝ : NormedField F := spectralNorm.normedField L F
this : NormedAlgebra L F := spectralNorm.normedAlgebra L F
a : f.SplittingField := rootOfSplits ⋯ ⋯
fa0 : (aeval a) f = 0
S : Finset F := {x ∈ (f.rootSet F).toFinset | x ≠ a}
δ : ℝ := if hS : S.Nonempty then (Finset.image (fun x ↦ ‖a - x‖) S).min' ⋯ else 1
norm_sub_le : ∀ (a' : F), IsConjRoot L a a' → a ≠ a' → δ ≤ ‖a - a'‖
hS : S.Nonempty
⊢ (Finset.image (fun x ↦ ‖a - x‖) S).min' ⋯ > 0

Tactic:
simp only [gt_iff_lt, Finset.lt_min'_iff, Finset.mem_image, forall_exists_index, and_imp,
        forall_apply_eq_imp_iff₂]

Goal state after:
case pos
K : Type u_1
L : Type u_2
inst✝⁶ : Field K
inst✝⁵ : NontriviallyNormedField L
inst✝⁴ : CompleteSpace L
inst✝³ : CharZero L
inst✝² : IsUltrametricDist L
inst✝¹ : Algebra K L
hi : DenseRange ⇑(algebraMap K L)
inst✝ : IsAlgClosed K
f : L[X]
fmon : f.Monic
firr : Irreducible f
fnatdeg0 : f.natDegree ≠ 0
F : Type u_2 := f.SplittingField
this✝ : NormedField F := spectralNorm.normedField L F
this : NormedAlgebra L F := spectralNorm.normedAlgebra L F
a : f.SplittingField := rootOfSplits ⋯ ⋯
fa0 : (aeval a) f = 0
S : Finset F := {x ∈ (f.rootSet F).toFinset | x ≠ a}
δ : ℝ := if hS : S.Nonempty then (Finset.image (fun x ↦ ‖a - x‖) S).min' ⋯ else 1
norm_sub_le : ∀ (a' : F), IsConjRoot L a a' → a ≠ a' → δ ≤ ‖a - a'‖
hS : S.Nonempty
⊢ ∀ a_1 ∈ S, 0 < ‖a - a_1‖
Module: Mathlib.Analysis.Normed.Field.Dense

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝⁶ : Field K
inst✝⁵ : NontriviallyNormedField L
inst✝⁴ : CompleteSpace L
inst✝³ : CharZero L
inst✝² : IsUltrametricDist L
inst✝¹ : Algebra K L
hi : DenseRange ⇑(algebraMap K L)
inst✝ : IsAlgClosed K
f : L[X]
fmon : f.Monic
firr : Irreducible f
fnatdeg0 : f.natDegree ≠ 0
F : Type u_2 := f.SplittingField
this✝ : NormedField F := spectralNorm.normedField L F
this : NormedAlgebra L F := spectralNorm.normedAlgebra L F
a : f.SplittingField := rootOfSplits ⋯ ⋯
fa0 : (aeval a) f = 0
S : Finset F := {x ∈ (f.rootSet F).toFinset | x ≠ a}
δ : ℝ := if hS : S.Nonempty then (Finset.image (fun x ↦ ‖a - x‖) S).min' ⋯ else 1
norm_sub_le : ∀ (a' : F), IsConjRoot L a a' → a ≠ a' → δ ≤ ‖a - a'‖
δpos : δ > 0
hε : (δ / max ‖a‖ 1) ^ f.natDegree / (↑f.natDegree + 1) > 0
g : K[X]
gmon : g.Monic
gdeg : f.natDegree = g.natDegree
gcoeff : ∀ (n : ℕ), ‖(map (algebraMap K L) g).coeff n - f.coeff n‖ < (δ / max ‖a‖ 1) ^ f.natDegree / (↑f.natDegree + 1)
b : f.SplittingField
hb : b ∈ (map (algebraMap K L) g).aroots f.SplittingField
hab :
  ‖a - b‖ < ((↑f.natDegree + 1) * ((δ / max ‖a‖ 1) ^ ↑f.natDegree / (↑f.natDegree + 1))) ^ (↑f.natDegree)⁻¹ * max ‖a‖ 1
⊢ ‖a - b‖ < δ

Tactic:
← mul_comm_div,

Goal state after:
K : Type u_1
L : Type u_2
inst✝⁶ : Field K
inst✝⁵ : NontriviallyNormedField L
inst✝⁴ : CompleteSpace L
inst✝³ : CharZero L
inst✝² : IsUltrametricDist L
inst✝¹ : Algebra K L
hi : DenseRange ⇑(algebraMap K L)
inst✝ : IsAlgClosed K
f : L[X]
fmon : f.Monic
firr : Irreducible f
fnatdeg0 : f.natDegree ≠ 0
F : Type u_2 := f.SplittingField
this✝ : NormedField F := spectralNorm.normedField L F
this : NormedAlgebra L F := spectralNorm.normedAlgebra L F
a : f.SplittingField := rootOfSplits ⋯ ⋯
fa0 : (aeval a) f = 0
S : Finset F := {x ∈ (f.rootSet F).toFinset | x ≠ a}
δ : ℝ := if hS : S.Nonempty then (Finset.image (fun x ↦ ‖a - x‖) S).min' ⋯ else 1
norm_sub_le : ∀ (a' : F), IsConjRoot L a a' → a ≠ a' → δ ≤ ‖a - a'‖
δpos : δ > 0
hε : (δ / max ‖a‖ 1) ^ f.natDegree / (↑f.natDegree + 1) > 0
g : K[X]
gmon : g.Monic
gdeg : f.natDegree = g.natDegree
gcoeff : ∀ (n : ℕ), ‖(map (algebraMap K L) g).coeff n - f.coeff n‖ < (δ / max ‖a‖ 1) ^ f.natDegree / (↑f.natDegree + 1)
b : f.SplittingField
hb : b ∈ (map (algebraMap K L) g).aroots f.SplittingField
hab :
  ‖a - b‖ < ((↑f.natDegree + 1) / (↑f.natDegree + 1) * (δ / max ‖a‖ 1) ^ ↑f.natDegree) ^ (↑f.natDegree)⁻¹ * max ‖a‖ 1
⊢ ‖a - b‖ < δ
Module: Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
s : ℕ → ℕ
hs_le : ∀ (n : ℕ), s n ≤ n
x : R
ψ : ℕ → ℕ
hψ_lim : Tendsto ((fun n ↦ ↑(s n) / ↑n) ∘ ψ) atTop (𝓝 0)
c : ℝ
hc_bd : ∀ (x_1 : ℝ) (x_2 : ℕ), (∀ (b : ℕ), x_2 ≤ b → μ x ^ (↑(s (ψ b)) * (1 / ↑(ψ b))) ≤ x_1) → c ≤ x_1
hμx : ¬μ x < 1
hμ_lim : ∀ (U : Set ℝ), 1 ∈ U → IsOpen U → ∃ N, ∀ (n : ℕ), N ≤ n → μ x ^ (↑(s (ψ n)) * (1 / ↑(ψ n))) ∈ U
ε : ℝ
hε : 0 < ε
⊢ 1 ∈ Set.Ioo 0 (1 + ε)

Tactic:
simp only [Set.mem_Ioo, zero_lt_one, lt_add_iff_pos_right, hε, and_self]

Goal state after:
Module: Mathlib.Analysis.Normed.Group.SemiNormedGrp.Kernels

Goal state before:
pp: ⊢ HasCokernels SemiNormedGrp₁

Tactic:
infer_instance

Goal state after:
Module: Mathlib.Analysis.Normed.Group.SemiNormedGrp.Kernels

Goal state before:
pp: ⊢ HasCokernels SemiNormedGrp

Tactic:
infer_instance

Goal state after:
Module: Mathlib.Analysis.Normed.Group.SeparationQuotient

Goal state before:
pp: case h_above
M : Type u_1
inst✝¹ : SeminormedAddCommGroup M
inst✝ : NontrivialTopology M
⊢ ∀ (x : M), ‖normedMk x‖ ≤ 1 * ‖x‖

Tactic:
simpa only [normedMk_apply, one_mul] using! fun _ ↦ le_rfl

Goal state after:
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: K : Type u
inst✝⁴ : NontriviallyNormedField K
L : Type v
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra.IsAlgebraic K L
hu : IsUltrametricDist K
inst✝ : CompleteSpace K
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
x : L
E : Type v := id ↥K⟮x⟯
⊢ Field ↥K⟮x⟯

Tactic:
infer_instance

Goal state after:
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: K : Type u
inst✝⁴ : NontriviallyNormedField K
L : Type v
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra.IsAlgebraic K L
hu : IsUltrametricDist K
inst✝ : CompleteSpace K
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
x : L
E : Type v := id ↥K⟮x⟯
this : Field E :=
  id
    (have this := inferInstance;
    this)
⊢ Module K ↥K⟮x⟯

Tactic:
infer_instance

Goal state after:
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: R : Type u_1
K : Type u
inst✝⁴ : NontriviallyNormedField K
L : Type v
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra.IsAlgebraic K L
hu : IsUltrametricDist K
inst✝ : CompleteSpace K
this : NormedField L := normedField K L
⊢ SeminormedRing L

Tactic:
infer_instance

Goal state after:
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: R : Type u_1
K : Type u
inst✝⁴ : NontriviallyNormedField K
L : Type v
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra.IsAlgebraic K L
hu : IsUltrametricDist K
inst✝ : CompleteSpace K
this : NormedField L
⊢ NormedAddCommGroup L

Tactic:
infer_instance

Goal state after:
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: R : Type u_1
K : Type u
inst✝⁴ : NontriviallyNormedField K
L : Type v
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra.IsAlgebraic K L
hu : IsUltrametricDist K
inst✝ : CompleteSpace K
this : NormedField L
⊢ SeminormedAddCommGroup L

Tactic:
infer_instance

Goal state after: