mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm

Goal state before:
pp: 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) < (L + ε / 2) ^ ↑↑m1
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)
hL0' : 0 < L + ε / 2
hm10 : ↑↑m1 ≠ 0
h_lt : 0 < ↑(n / ↑m1) / ↑n
⊢ (μ (x ^ ↑m1) ^ (n / ↑m1)) ^ (1 / ↑n) <
    (L + ε / 2) ^ (↑(↑m1 * (n / ↑m1) + n % ↑m1) / ↑n) * (L + ε / 2) ^ (-(↑(n % ↑m1) / ↑n))

Tactic:
← rpow_natCast,

Goal state after:
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) < (L + ε / 2) ^ ↑↑m1
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)
hL0' : 0 < L + ε / 2
hm10 : ↑↑m1 ≠ 0
h_lt : 0 < ↑(n / ↑m1) / ↑n
⊢ (μ (x ^ ↑m1) ^ ↑(n / ↑m1)) ^ (1 / ↑n) <
    (L + ε / 2) ^ (↑(↑m1 * (n / ↑m1) + n % ↑m1) / ↑n) * (L + ε / 2) ^ (-(↑(n % ↑m1) / ↑n))
Module: Mathlib.Analysis.Normed.Field.ProperSpace

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : WeaklyLocallyCompactSpace 𝕜
r : ℝ
rpos : 0 < r
hr : IsCompact (closedBall 0 r)
c : 𝕜
hc : 1 < ‖c‖
⊢ ProperSpace 𝕜

Tactic:
have hC n : IsCompact (closedBall (0 : 𝕜) (‖c‖ ^ n * r)) := by
    have : c ^ n ≠ 0 := pow_ne_zero _ <| fun h ↦ by simp [h, zero_le_one.not_gt] at hc
    convert! hr.smul (c ^ n)
    ext
    simp only [mem_closedBall, dist_zero_right, Set.mem_smul_set_iff_inv_smul_mem₀ this,
      smul_eq_mul, norm_mul, norm_inv, norm_pow,
      inv_mul_le_iff₀ (by simpa only [norm_pow] using norm_pos_iff.mpr this)]

Goal state after:
𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : WeaklyLocallyCompactSpace 𝕜
r : ℝ
rpos : 0 < r
hr : IsCompact (closedBall 0 r)
c : 𝕜
hc : 1 < ‖c‖
hC : ∀ (n : ℕ), IsCompact (closedBall 0 (‖c‖ ^ n * r))
⊢ ProperSpace 𝕜
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: case pos
K : Type u_2
inst✝² : NormedField K
L : Type u_3
inst✝¹ : Field L
inst✝ : Algebra K L
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
hf_na : IsNonarchimedean ⇑f
p : K[X]
hp : p.Monic
x : L
hx : (aeval x) p = 0
hx0 : f x = 0
⊢ 0 ≤ spectralValue p

Tactic:
exact spectralValue_nonneg p

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

Goal state before:
pp: K : Type u_2
inst✝² : NormedField K
L : Type u_3
inst✝¹ : Field L
inst✝ : Algebra K L
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
hf_na : IsNonarchimedean ⇑f
p : K[X]
hp : p.Monic
x : L
hx : (aeval x) p = 0
hx0 : ¬f x = 0
h_ge : ¬f x ≤ spectralValue p
n : ℕ
hn : n < p.natDegree
⊢ (‖p.coeff n‖ ^ (1 / (↑p.natDegree - ↑n))) ^ (p.natDegree - n) = ‖p.coeff n‖

Tactic:
← rpow_natCast,

Goal state after:
K : Type u_2
inst✝² : NormedField K
L : Type u_3
inst✝¹ : Field L
inst✝ : Algebra K L
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
hf_na : IsNonarchimedean ⇑f
p : K[X]
hp : p.Monic
x : L
hx : (aeval x) p = 0
hx0 : ¬f x = 0
h_ge : ¬f x ≤ spectralValue p
n : ℕ
hn : n < p.natDegree
⊢ (‖p.coeff n‖ ^ (1 / (↑p.natDegree - ↑n))) ^ ↑(p.natDegree - n) = ‖p.coeff n‖
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: K : Type u_2
inst✝² : NormedField K
L : Type u_3
inst✝¹ : Field L
inst✝ : Algebra K L
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
hf_na : IsNonarchimedean ⇑f
p : K[X]
hp : p.Monic
x : L
hx : (aeval x) p = 0
hx0 : ¬f x = 0
h_ge : ¬f x ≤ spectralValue p
n : ℕ
hn : n < p.natDegree
⊢ (‖p.coeff n‖ ^ (1 / (↑p.natDegree - ↑n))) ^ (p.natDegree - n) = ‖p.coeff n‖

Tactic:
rw [← rpow_natCast, ← rpow_mul (norm_nonneg _), mul_comm, rpow_mul (norm_nonneg _),
          rpow_natCast, ← cast_sub (le_of_lt hn), one_div,
          pow_rpow_inv_natCast (norm_nonneg _) (_root_.ne_of_gt (tsub_pos_of_lt hn))]

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

Goal state before:
pp: K : Type u_2
inst✝² : NormedField K
L : Type u_3
inst✝¹ : Field L
inst✝ : Algebra K L
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
hf_na : IsNonarchimedean ⇑f
p : K[X]
hp : p.Monic
x : L
hx : (aeval x) p = 0
hx0 : ¬f x = 0
h_ge : ¬f x ≤ spectralValue p
n : ℕ
hn : n < p.natDegree
hexp : (‖p.coeff n‖ ^ (1 / (↑p.natDegree - ↑n))) ^ (p.natDegree - n) = ‖p.coeff n‖
h_base : ‖p.coeff n‖ ^ (1 / (↑p.natDegree - ↑n)) < f x
⊢ (‖p.coeff n‖ ^ (1 / (↑p.natDegree - ↑n))) ^ (p.natDegree - n) < f x ^ (p.natDegree - n)

Tactic:
← rpow_natCast,

Goal state after:
K : Type u_2
inst✝² : NormedField K
L : Type u_3
inst✝¹ : Field L
inst✝ : Algebra K L
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
hf_na : IsNonarchimedean ⇑f
p : K[X]
hp : p.Monic
x : L
hx : (aeval x) p = 0
hx0 : ¬f x = 0
h_ge : ¬f x ≤ spectralValue p
n : ℕ
hn : n < p.natDegree
hexp : (‖p.coeff n‖ ^ (1 / (↑p.natDegree - ↑n))) ^ (p.natDegree - n) = ‖p.coeff n‖
h_base : ‖p.coeff n‖ ^ (1 / (↑p.natDegree - ↑n)) < f x
⊢ (‖p.coeff n‖ ^ (1 / (↑p.natDegree - ↑n))) ^ ↑(p.natDegree - n) < f x ^ (p.natDegree - n)
Module: Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm

Goal state before:
pp: case h
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
s : ℕ → ℕ
hs : ∀ (n : ℕ), s n ≤ n
x : R
ψ : ℕ → ℕ
hψ : ∀ (n : ℕ), 0 ≤ 1 / ↑(ψ n)
hx : μ x ≤ 1
n : ℕ
⊢ (μ x ^ s (ψ n)) ^ (1 / ↑(ψ n)) ≤ 1

Tactic:
← rpow_natCast,

Goal state after:
case h
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
s : ℕ → ℕ
hs : ∀ (n : ℕ), s n ≤ n
x : R
ψ : ℕ → ℕ
hψ : ∀ (n : ℕ), 0 ≤ 1 / ↑(ψ n)
hx : μ x ≤ 1
n : ℕ
⊢ (μ x ^ ↑(s (ψ n))) ^ (1 / ↑(ψ n)) ≤ 1
Module: Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm

Goal state before:
pp: case h
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
s : ℕ → ℕ
hs : ∀ (n : ℕ), s n ≤ n
x : R
ψ : ℕ → ℕ
hψ : ∀ (n : ℕ), 0 ≤ 1 / ↑(ψ n)
hx : 1 < μ x
n : ℕ
⊢ (μ x ^ s (ψ n)) ^ (1 / ↑(ψ n)) ≤ μ x

Tactic:
← rpow_natCast,

Goal state after:
case h
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
s : ℕ → ℕ
hs : ∀ (n : ℕ), s n ≤ n
x : R
ψ : ℕ → ℕ
hψ : ∀ (n : ℕ), 0 ≤ 1 / ↑(ψ n)
hx : 1 < μ x
n : ℕ
⊢ (μ x ^ ↑(s (ψ n))) ^ (1 / ↑(ψ n)) ≤ μ x
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: case pos
K : Type u_2
inst✝³ : NormedField K
L : Type u_3
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : DecidableEq L
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
hf_na : IsNonarchimedean ⇑f
hf1 : f 1 = 1
p : K[X]
s : Multiset L
hp : (mapAlg K L) p = (Multiset.map (fun a ↦ X - C a) s).prod
h_le : 0 ≤ ⨆ x, if x ∈ s then f x else 0
m : ℕ
hm : m < p.natDegree
⊢ spectralValueTerms p m ≤ ⨆ x, if x ∈ s then f x else 0

Tactic:
rw [spectralValueTerms_of_lt_natDegree _ hm]

Goal state after:
case pos
K : Type u_2
inst✝³ : NormedField K
L : Type u_3
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : DecidableEq L
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
hf_na : IsNonarchimedean ⇑f
hf1 : f 1 = 1
p : K[X]
s : Multiset L
hp : (mapAlg K L) p = (Multiset.map (fun a ↦ X - C a) s).prod
h_le : 0 ≤ ⨆ x, if x ∈ s then f x else 0
m : ℕ
hm : m < p.natDegree
⊢ ‖p.coeff m‖ ^ (1 / (↑p.natDegree - ↑m)) ≤ ⨆ x, if x ∈ s then f x else 0
Module: Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm

Goal state before:
pp: case e'_5
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
⊢ 1 = μ x ^ 0

Tactic:
rpow_zero

Goal state after:
case e'_5
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
⊢ 1 = 1
Module: Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm

Goal state before:
pp: case e'_5
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
⊢ 1 = μ x ^ 0

Tactic:
· rw [rpow_zero]

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
s : ℕ → ℕ
hs_le : ∀ (n : ℕ), s n ≤ n
x : R
a : ℝ
a_in : a ∈ Set.Icc 0 1
ψ : ℕ → ℕ
hψ_mono : StrictMono ψ
hψ_lim : Tendsto ((fun n ↦ ↑(s n) / ↑n) ∘ ψ) atTop (𝓝 0)
ha : a = 0
⊢ 1 = smoothingFun μ x ^ 0

Tactic:
rpow_zero

Goal state after:
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
s : ℕ → ℕ
hs_le : ∀ (n : ℕ), s n ≤ n
x : R
a : ℝ
a_in : a ∈ Set.Icc 0 1
ψ : ℕ → ℕ
hψ_mono : StrictMono ψ
hψ_lim : Tendsto ((fun n ↦ ↑(s n) / ↑n) ∘ ψ) atTop (𝓝 0)
ha : a = 0
⊢ 1 = 1
Module: Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm

Goal state before:
pp: 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) ≤ max (smoothingFun μ x) (smoothingFun μ y)

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

Goal state after:
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) ≤ max (smoothingFun μ x) (smoothingFun μ y)
Module: Mathlib.Analysis.Normed.Group.ControlledClosure

Goal state before:
pp: G : Type u_1
inst✝³ : NormedAddCommGroup G
inst✝² : CompleteSpace G
H : Type u_2
inst✝¹ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : Type u_3
inst✝ : SeminormedAddCommGroup K
j : NormedAddGroupHom K H
hj : ∀ (x : K), ‖j x‖ = ‖x‖
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : ∀ (k : K), ∃ g, f g = j k ∧ ‖g‖ ≤ C * ‖k‖
k : K
h_in : j k ∈ j.range
⊢ ∃ g, f g = j k ∧ ‖g‖ ≤ C * ‖j k‖

Tactic:
hj

Goal state after:
G : Type u_1
inst✝³ : NormedAddCommGroup G
inst✝² : CompleteSpace G
H : Type u_2
inst✝¹ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : Type u_3
inst✝ : SeminormedAddCommGroup K
j : NormedAddGroupHom K H
hj : ∀ (x : K), ‖j x‖ = ‖x‖
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : ∀ (k : K), ∃ g, f g = j k ∧ ‖g‖ ≤ C * ‖k‖
k : K
h_in : j k ∈ j.range
⊢ ∃ g, f g = j k ∧ ‖g‖ ≤ C * ‖k‖
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: R : Type u_1
K : Type u_2
inst✝³ : NormedField K
L : Type u_3
inst✝² : Field L
inst✝¹ : Algebra K L
E : IntermediateField K L
y : L
h_fin : FiniteDimensional K L
hn : Normal K L
inst✝ : IsUltrametricDist K
⊢ ∀ (r s : L), spectralNorm K L (r + s) ≤ spectralNorm K L r + spectralNorm K L s

Tactic:
spectralNorm_eq_invariantExtension

Goal state after:
R : Type u_1
K : Type u_2
inst✝³ : NormedField K
L : Type u_3
inst✝² : Field L
inst✝¹ : Algebra K L
E : IntermediateField K L
y : L
h_fin : FiniteDimensional K L
hn : Normal K L
inst✝ : IsUltrametricDist K
⊢ ∀ (r s : L), (invariantExtension K L) (r + s) ≤ (invariantExtension K L) r + (invariantExtension K L) s
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: R : Type u_1
K : Type u_2
inst✝³ : NormedField K
L : Type u_3
inst✝² : Field L
inst✝¹ : Algebra K L
E : IntermediateField K L
y : L
h_fin : FiniteDimensional K L
hn : Normal K L
inst✝ : IsUltrametricDist K
⊢ ∀ (r : L), spectralNorm K L (-r) = spectralNorm K L r

Tactic:
spectralNorm_eq_invariantExtension

Goal state after:
R : Type u_1
K : Type u_2
inst✝³ : NormedField K
L : Type u_3
inst✝² : Field L
inst✝¹ : Algebra K L
E : IntermediateField K L
y : L
h_fin : FiniteDimensional K L
hn : Normal K L
inst✝ : IsUltrametricDist K
⊢ ∀ (r : L), (invariantExtension K L) (-r) = (invariantExtension K L) r
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: K : Type u_2
inst✝³ : NormedField K
L : Type u_3
inst✝² : Field L
inst✝¹ : Algebra K L
h_fin : FiniteDimensional K L
hn : Normal K L
inst✝ : IsUltrametricDist K
⊢ IsNonarchimedean (spectralNorm K L)

Tactic:
spectralNorm_eq_invariantExtension

Goal state after:
K : Type u_2
inst✝³ : NormedField K
L : Type u_3
inst✝² : Field L
inst✝¹ : Algebra K L
h_fin : FiniteDimensional K L
hn : Normal K L
inst✝ : IsUltrametricDist K
⊢ IsNonarchimedean ⇑(invariantExtension K L)
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: K : Type u_2
inst✝³ : NormedField K
L : Type u_3
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsUltrametricDist K
x y : L
hx : IsAlgebraic K x
hy : IsAlgebraic K y
E : IntermediateField K L := K⟮x, y⟯
h_finiteDimensional_E : FiniteDimensional K ↥E
gx : ↥K⟮x, y⟯ := AdjoinPair.gen₁ K x y
gy : ↥K⟮x, y⟯ := AdjoinPair.gen₂ K x y
⊢ spectralNorm K L (x * y) ≤ spectralNorm K L x * spectralNorm K L y

Tactic:
have hxy : x * y = (algebraMap K⟮x, y⟯ L) (gx * gy) := rfl

Goal state after:
K : Type u_2
inst✝³ : NormedField K
L : Type u_3
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsUltrametricDist K
x y : L
hx : IsAlgebraic K x
hy : IsAlgebraic K y
E : IntermediateField K L := K⟮x, y⟯
h_finiteDimensional_E : FiniteDimensional K ↥E
gx : ↥K⟮x, y⟯ := AdjoinPair.gen₁ K x y
gy : ↥K⟮x, y⟯ := AdjoinPair.gen₂ K x y
hxy : x * y = (algebraMap (↥K⟮x, y⟯) L) (gx * gy)
⊢ spectralNorm K L (x * y) ≤ spectralNorm K L x * spectralNorm K L y
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)
this : Module K E :=
  id
    (have this := inferInstance;
    this)
id1 : ↥K⟮x⟯ →ₗ[K] E := LinearMap.id
id2 : E →ₗ[K] ↥K⟮x⟯ := LinearMap.id
⊢ spectralNorm K L ↑(id2 0) = 0

Tactic:
by simp [map_zero, spectralNorm_zero, ZeroMemClass.coe_zero]

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
x : R
m : ℕ
hm : 1 ≤ m
hlim : Tendsto (fun n ↦ smoothingSeminormSeq μ x (m * n)) atTop (𝓝 (smoothingFun μ x))
n : ℕ
hm' : ↑m ≠ 0
⊢ (μ ((x ^ m) ^ n) ^ (1 / ↑(m * n))) ^ m = μ ((x ^ m) ^ n) ^ (1 / ↑n)

Tactic:
← rpow_natCast,

Goal state after:
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
x : R
m : ℕ
hm : 1 ≤ m
hlim : Tendsto (fun n ↦ smoothingSeminormSeq μ x (m * n)) atTop (𝓝 (smoothingFun μ x))
n : ℕ
hm' : ↑m ≠ 0
⊢ (μ ((x ^ m) ^ n) ^ (1 / ↑(m * n))) ^ ↑m = μ ((x ^ m) ^ n) ^ (1 / ↑n)
Module: Mathlib.Analysis.Normed.Group.Tannery

Goal state before:
pp: α : Type u_1
β : Type u_2
G : Type u_3
𝓕 : Filter α
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace G
f : α → β → G
g : β → G
bound : β → ℝ
h_sum : Summable bound
hab : ∀ (k : β), Tendsto (fun x ↦ f x k) 𝓕 (𝓝 (g k))
h_bound : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k
h✝¹ : Nonempty β
h✝ : 𝓕.NeBot
h_g_le : ∀ (k : β), ‖g k‖ ≤ bound k
h_sumg : Summable fun x ↦ ‖g x‖
n : α
h : ∀ (k : β), ‖f n k‖ ≤ bound k
⊢ ∀ (i : β), ‖‖f n i‖‖ ≤ bound i

Tactic:
simpa only [norm_norm] using h

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

Goal state before:
pp: case h
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
x : R
hx : ∀ (n : ℕ), 1 ≤ n → μ (x ^ n) = μ x ^ n
n : ℕ
hn : 1 ≤ n
hn0 : ↑n ≠ 0
⊢ (μ x ^ n) ^ (1 / ↑n) = μ x

Tactic:
← rpow_natCast,

Goal state after:
case h
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
x : R
hx : ∀ (n : ℕ), 1 ≤ n → μ (x ^ n) = μ x ^ n
n : ℕ
hn : 1 ≤ n
hn0 : ↑n ≠ 0
⊢ (μ x ^ ↑n) ^ (1 / ↑n) = μ x
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 : ∀ (y : R), μ (x * y) = μ x * μ y
n : ℕ
hn : 1 ≤ n
hx0 : ¬μ x = 0
h1 : μ 1 = 1
hn0 : ↑n ≠ 0
⊢ (μ x ^ n * μ 1) ^ (1 / ↑n) = μ x

Tactic:
← rpow_natCast,

Goal state after:
case neg
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
x : R
hx : ∀ (y : R), μ (x * y) = μ x * μ y
n : ℕ
hn : 1 ≤ n
hx0 : ¬μ x = 0
h1 : μ 1 = 1
hn0 : ↑n ≠ 0
⊢ (μ x ^ ↑n * μ 1) ^ (1 / ↑n) = μ x
Module: Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm

Goal state before:
pp: case h
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
x : R
hx : ∀ (y : R), μ (x * y) = μ x * μ y
y : R
hlim : Tendsto (fun n ↦ μ x * smoothingSeminormSeq μ y n) atTop (𝓝 (smoothingFun μ x * smoothingFun μ y))
n : ℕ
hn1 : 1 ≤ n
hn0 : ↑n ≠ 0
⊢ (μ x ^ n * μ (y ^ n)) ^ (1 / ↑n) = μ x * μ (y ^ n) ^ (1 / ↑n)

Tactic:
mul_rpow (pow_nonneg (apply_nonneg μ _) _)
    (apply_nonneg μ _),

Goal state after:
case h
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
x : R
hx : ∀ (y : R), μ (x * y) = μ x * μ y
y : R
hlim : Tendsto (fun n ↦ μ x * smoothingSeminormSeq μ y n) atTop (𝓝 (smoothingFun μ x * smoothingFun μ y))
n : ℕ
hn1 : 1 ≤ n
hn0 : ↑n ≠ 0
⊢ (μ x ^ n) ^ (1 / ↑n) * μ (y ^ n) ^ (1 / ↑n) = μ x * μ (y ^ n) ^ (1 / ↑n)
Module: Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm

Goal state before:
pp: case h
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
x : R
hx : ∀ (y : R), μ (x * y) = μ x * μ y
y : R
hlim : Tendsto (fun n ↦ μ x * smoothingSeminormSeq μ y n) atTop (𝓝 (smoothingFun μ x * smoothingFun μ y))
n : ℕ
hn1 : 1 ≤ n
hn0 : ↑n ≠ 0
⊢ (μ x ^ n) ^ (1 / ↑n) * μ (y ^ n) ^ (1 / ↑n) = μ x * μ (y ^ n) ^ (1 / ↑n)

Tactic:
← rpow_natCast,

Goal state after:
case h
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
x : R
hx : ∀ (y : R), μ (x * y) = μ x * μ y
y : R
hlim : Tendsto (fun n ↦ μ x * smoothingSeminormSeq μ y n) atTop (𝓝 (smoothingFun μ x * smoothingFun μ y))
n : ℕ
hn1 : 1 ≤ n
hn0 : ↑n ≠ 0
⊢ (μ x ^ ↑n) ^ (1 / ↑n) * μ (y ^ n) ^ (1 / ↑n) = μ x * μ (y ^ n) ^ (1 / ↑n)
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
x y : L
⊢ spectralNorm K L (x - y) = spectralNorm K L (y - x)

Tactic:
rw [← neg_sub, spectralNorm_neg (Algebra.IsAlgebraic.isAlgebraic _)]

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

Goal state before:
pp: K : Type u
inst✝⁹ : NontriviallyNormedField K
L : Type v
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
hu : IsUltrametricDist K
inst✝⁶ : CompleteSpace K
x : L
E : Type u_2
inst✝⁵ : Field E
inst✝⁴ : Algebra K E
inst✝³ : Algebra L E
inst✝² : IsScalarTower K L E
inst✝¹ : IsSplittingField L E ((mapAlg K L) (minpoly K x))
inst✝ : Algebra.IsAlgebraic K E
h_deg : (minpoly K x).natDegree = ((mapAlg K E) (minpoly K x)).roots.card
r : ℝ
hr : (spectralMulAlgNorm K E) ((algebraMap L E) x) = r
s : ℝ
hs : s ∈ Multiset.map (⇑(spectralMulAlgNorm K E)) ((mapAlg K E) (minpoly K x)).roots
a : E
ha : a ∈ ((mapAlg K E) (minpoly K x)).roots
has : (spectralMulAlgNorm K E) a = s
⊢ r = s

Tactic:
← hr,

Goal state after:
K : Type u
inst✝⁹ : NontriviallyNormedField K
L : Type v
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
hu : IsUltrametricDist K
inst✝⁶ : CompleteSpace K
x : L
E : Type u_2
inst✝⁵ : Field E
inst✝⁴ : Algebra K E
inst✝³ : Algebra L E
inst✝² : IsScalarTower K L E
inst✝¹ : IsSplittingField L E ((mapAlg K L) (minpoly K x))
inst✝ : Algebra.IsAlgebraic K E
h_deg : (minpoly K x).natDegree = ((mapAlg K E) (minpoly K x)).roots.card
r : ℝ
hr : (spectralMulAlgNorm K E) ((algebraMap L E) x) = r
s : ℝ
hs : s ∈ Multiset.map (⇑(spectralMulAlgNorm K E)) ((mapAlg K E) (minpoly K x)).roots
a : E
ha : a ∈ ((mapAlg K E) (minpoly K x)).roots
has : (spectralMulAlgNorm K E) a = s
⊢ (spectralMulAlgNorm K E) ((algebraMap L E) x) = s
Module: Mathlib.Analysis.SpecialFunctions.Log.Summable

Goal state before:
pp: case neg
ι : Type u_1
f : ι → ℂ
hf : Summable fun i ↦ log (f i)
hfn : ∀ (n : ι), f n ≠ 0
⊢ Multipliable f

Tactic:
· exact ⟨_, hasProd_of_hasSum_log hfn hf.hasSum⟩

Goal state after:
Module: Mathlib.Analysis.Normed.Module.MStructure

Goal state before:
pp: X : Type u_1
inst✝³ : NormedAddCommGroup X
M : Type u_2
inst✝² : Ring M
inst✝¹ : Module M X
inst✝ : FaithfulSMul M X
P : { P // IsLprojection X P }
⊢ ↑(P ⊓ Pᶜ) = ↑⊥

Tactic:
rw [coe_inf, coe_compl, coe_bot, ← coe_compl, mul_compl_self]

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Fredholm.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : AddCommGroup F
inst✝³ : Module 𝕜 E
inst✝² : Module 𝕜 F
inst✝¹ : TopologicalSpace E
inst✝ : TopologicalSpace F
u : E →L[𝕜] F
pkg : u.FredholmPackage
⊢ (↑pkg.quasiInverse).IsQuasiInverse ↑u

Tactic:
nth_rw 2 [pkg.eq_equiv]
  have hdom : IsQuasiInverse pkg.decDom.X₁.subtype pkg.decDom.proj :=
    have := pkg.decDom.finite_X₀
    isQuasiInverse_subtype_projectionOnto _
  have hcodom : IsQuasiInverse pkg.decCodom.X₁.subtype pkg.decCodom.proj :=
    have := pkg.decCodom.finite_X₀
    isQuasiInverse_subtype_projectionOnto _
  -- For some reason `exact` and `refine` are slow here!
  apply hdom.comp (pkg.equiv.isQuasiInverse.comp hcodom.symm)

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Fredholm.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : AddCommGroup E
inst✝⁶ : AddCommGroup F
inst✝⁵ : Module 𝕜 E
inst✝⁴ : Module 𝕜 F
inst✝³ : TopologicalSpace E
inst✝² : TopologicalSpace F
inst✝¹ : T2Space E
inst✝ : T2Space F
u : E →L[𝕜] F
v : F →L[𝕜] E
hvu : (↑v).IsQuasiInverse ↑u
⊢ ∃ E₁ F₁,
    IsClosed[inst✝³] ↑E₁ ∧
      IsClosed[inst✝²] ↑F₁ ∧ E₁.CoFG ∧ F₁.CoFG ∧ ∃ (h : MapsTo ⇑u ↑E₁ ↑F₁), (u.restrict h).IsInvertible

Tactic:
obtain ⟨hvu, huv⟩ := hvu
  rw [IsRightQuasiInverse, Setoid.comm, equiv_iff_eqLocus_coFG] at huv
  rw [IsLeftQuasiInverse, Setoid.comm, equiv_iff_eqLocus_coFG] at hvu
  set E₁ := (ContinuousLinearMap.id 𝕜 E).eqLocus (v ∘L u)
  set F₁ := (ContinuousLinearMap.id 𝕜 F).eqLocus (u ∘L v)
  have u_mapsto : MapsTo u E₁ F₁ := fun x hx ↦ congr(u $hx)
  have v_mapsto : MapsTo v F₁ E₁ := fun x hx ↦ congr(v $hx)
  refine ⟨E₁, F₁, isClosed_eqLocus _ _, isClosed_eqLocus _ _, hvu, huv, u_mapsto, ?_⟩
  refine .of_inverse (g := v.restrict v_mapsto) ?_ ?_
  · ext ⟨x, hx : x = u (v x)⟩
    simp [coe_restrict_apply u_mapsto, coe_restrict_apply v_mapsto, ← hx]
  · ext ⟨x, hx : x = v (u x)⟩
    simp [coe_restrict_apply u_mapsto, coe_restrict_apply v_mapsto, ← hx]

Goal state after:
Module: Mathlib.Analysis.Polynomial.Fourier

Goal state before:
pp: case monomial.inl
p : ℂ[X]
n : ℤ
m : ℕ
a : ℂ
hn : n < 0
⊢ fourierCoeff (⇑(toAddCircle ((monomial m) a))) n = if 0 ≤ n then ((monomial m) a).coeff n.natAbs else 0

Tactic:
| monomial m a =>

Goal state after: