mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: case e'_3
z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
der1 :
  ∀ x ∈ uIcc 0 (π / 2),
    HasDerivAt (fun y ↦ ↑(sin y) * ↑(cos y) ^ (n - 1)) (↑(cos x) ^ n - (↑n - 1) * ↑(sin x) ^ 2 * ↑(cos x) ^ (n - 2)) x
⊢ ∫ (x : ℝ) in 0..π / 2,
      ↑n / (2 * z) * (Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n) -
        (↑n - 1) / (2 * z) * (Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (n - 2)) =
    ∫ (x : ℝ) in 0..π / 2,
      -((↑(cos x) ^ n - (↑n - 1) * ↑(sin x) ^ 2 * ↑(cos x) ^ (n - 2)) * (-Complex.cos (2 * z * ↑x) / (2 * z)))

pp: case e'_3.hf
z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
der1 :
  ∀ x ∈ uIcc 0 (π / 2),
    HasDerivAt (fun y ↦ ↑(sin y) * ↑(cos y) ^ (n - 1)) (↑(cos x) ^ n - (↑n - 1) * ↑(sin x) ^ 2 * ↑(cos x) ^ (n - 2)) x
⊢ IntervalIntegrable (fun x ↦ ↑n / (2 * z) * (Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n)) volume 0 (π / 2)

pp: case e'_3.hg
z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
der1 :
  ∀ x ∈ uIcc 0 (π / 2),
    HasDerivAt (fun y ↦ ↑(sin y) * ↑(cos y) ^ (n - 1)) (↑(cos x) ^ n - (↑n - 1) * ↑(sin x) ^ 2 * ↑(cos x) ^ (n - 2)) x
⊢ IntervalIntegrable (fun x ↦ (↑n - 1) / (2 * z) * (Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (n - 2))) volume 0 (π / 2)

pp: case e'_3
z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
der1 :
  ∀ x ∈ uIcc 0 (π / 2),
    HasDerivAt (fun y ↦ ↑(sin y) * ↑(cos y) ^ (n - 1)) (↑(cos x) ^ n - (↑n - 1) * ↑(sin x) ^ 2 * ↑(cos x) ^ (n - 2)) x
⊢ n - 1 ≠ 0

Tactic:
rotate_left

Goal state after:
case e'_3.hf
z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
der1 :
  ∀ x ∈ uIcc 0 (π / 2),
    HasDerivAt (fun y ↦ ↑(sin y) * ↑(cos y) ^ (n - 1)) (↑(cos x) ^ n - (↑n - 1) * ↑(sin x) ^ 2 * ↑(cos x) ^ (n - 2)) x
⊢ IntervalIntegrable (fun x ↦ ↑n / (2 * z) * (Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n)) volume 0 (π / 2)

case e'_3.hg
z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
der1 :
  ∀ x ∈ uIcc 0 (π / 2),
    HasDerivAt (fun y ↦ ↑(sin y) * ↑(cos y) ^ (n - 1)) (↑(cos x) ^ n - (↑n - 1) * ↑(sin x) ^ 2 * ↑(cos x) ^ (n - 2)) x
⊢ IntervalIntegrable (fun x ↦ (↑n - 1) / (2 * z) * (Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (n - 2))) volume 0 (π / 2)

case e'_3
z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
der1 :
  ∀ x ∈ uIcc 0 (π / 2),
    HasDerivAt (fun y ↦ ↑(sin y) * ↑(cos y) ^ (n - 1)) (↑(cos x) ^ n - (↑n - 1) * ↑(sin x) ^ 2 * ↑(cos x) ^ (n - 2)) x
⊢ n - 1 ≠ 0

case e'_3
z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
der1 :
  ∀ x ∈ uIcc 0 (π / 2),
    HasDerivAt (fun y ↦ ↑(sin y) * ↑(cos y) ^ (n - 1)) (↑(cos x) ^ n - (↑n - 1) * ↑(sin x) ^ 2 * ↑(cos x) ^ (n - 2)) x
⊢ ∫ (x : ℝ) in 0..π / 2,
      ↑n / (2 * z) * (Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n) -
        (↑n - 1) / (2 * z) * (Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (n - 2)) =
    ∫ (x : ℝ) in 0..π / 2,
      -((↑(cos x) ^ n - (↑n - 1) * ↑(sin x) ^ 2 * ↑(cos x) ^ (n - 2)) * (-Complex.cos (2 * z * ↑x) / (2 * z)))
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: f : ℝ → ℝ
n : ℕ
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hn : n ≠ 0
m : ℕ
hm : 1 ≤ m
IH : f ↑m = f 1 + log ↑(m - 1)!
A : 0 < ↑m
| f 1 + log ↑m !

Tactic:
← Nat.succ_pred_eq_of_pos hm,

Goal state after:
f : ℝ → ℝ
n : ℕ
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hn : n ≠ 0
m : ℕ
hm : 1 ≤ m
IH : f ↑m = f 1 + log ↑(m - 1)!
A : 0 < ↑m
| f 1 + log ↑m.pred.succ !
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: x : ℝ
n : ℕ
⊢ (x + 1) * log ↑n + log ↑n ! - ∑ m ∈ Finset.range (n + 1), log (x + 1 + ↑m) =
    x * log ↑(n + 1) + (log ↑(n + 1) + log ↑n !) - (∑ k ∈ Finset.range (n + 1), log (x + ↑(k + 1)) + log x) + log x -
      (x + 1) * (log (↑n + 1) - log ↑n)

pp: case hx
x : ℝ
n : ℕ
⊢ ↑(n + 1) ≠ 0

pp: case hy
x : ℝ
n : ℕ
⊢ ↑n ! ≠ 0

Tactic:
rotate_left

Goal state after:
case hx
x : ℝ
n : ℕ
⊢ ↑(n + 1) ≠ 0

case hy
x : ℝ
n : ℕ
⊢ ↑n ! ≠ 0

x : ℝ
n : ℕ
⊢ (x + 1) * log ↑n + log ↑n ! - ∑ m ∈ Finset.range (n + 1), log (x + 1 + ↑m) =
    x * log ↑(n + 1) + (log ↑(n + 1) + log ↑n !) - (∑ k ∈ Finset.range (n + 1), log (x + ↑(k + 1)) + log x) + log x -
      (x + 1) * (log (↑n + 1) - log ↑n)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: case hf
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
F : ℝ → ℂ := fun x ↦ ↑x ^ u * (1 - ↑x) ^ v
hu' : 0 < (u + 1).re
hv' : 0 < (v + 1).re
hc : ContinuousOn F (Icc 0 1)
hder : ∀ x ∈ Ioo 0 1, HasDerivAt F (u * (↑x ^ (u - 1) * (1 - ↑x) ^ v) - v * (↑x ^ u * (1 - ↑x) ^ (v - 1))) x
h_int : IntervalIntegrable (fun x ↦ u * (↑x ^ (u - 1) * (1 - ↑x) ^ v) - v * (↑x ^ u * (1 - ↑x) ^ (v - 1))) volume 0 1
int_ev : ∫ (y : ℝ) in 0..1, u * (↑y ^ (u - 1) * (1 - ↑y) ^ v) - v * (↑y ^ u * (1 - ↑y) ^ (v - 1)) = 0
hF0 : F 0 = 0
hF1 : F 1 = 0
⊢ IntervalIntegrable (fun y ↦ u * (↑y ^ (u - 1) * (1 - ↑y) ^ v)) volume 0 1

Tactic:
apply IntervalIntegrable.const_mul

Goal state after:
case hf
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
F : ℝ → ℂ := ⋯
hu' : 0 < (u + 1).re
hv' : 0 < (v + 1).re
hc : ContinuousOn F (Icc 0 1)
hder : ∀ x ∈ Ioo 0 1, HasDerivAt F (u * (↑x ^ (u - 1) * (1 - ↑x) ^ v) - v * (↑x ^ u * (1 - ↑x) ^ (v - 1))) x
h_int : IntervalIntegrable (fun x ↦ u * (↑x ^ (u - 1) * (1 - ↑x) ^ v) - v * (↑x ^ u * (1 - ↑x) ^ (v - 1))) volume 0 1
int_ev : ∫ (y : ℝ) in 0..1, u * (↑y ^ (u - 1) * (1 - ↑y) ^ v) - v * (↑y ^ u * (1 - ↑y) ^ (v - 1)) = 0
hF0 : F 0 = 0
hF1 : F 1 = 0
⊢ IntervalIntegrable (fun x ↦ ↑x ^ (u - 1) * (1 - ↑x) ^ v) volume 0 1
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: case hg
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
F : ℝ → ℂ := fun x ↦ ↑x ^ u * (1 - ↑x) ^ v
hu' : 0 < (u + 1).re
hv' : 0 < (v + 1).re
hc : ContinuousOn F (Icc 0 1)
hder : ∀ x ∈ Ioo 0 1, HasDerivAt F (u * (↑x ^ (u - 1) * (1 - ↑x) ^ v) - v * (↑x ^ u * (1 - ↑x) ^ (v - 1))) x
h_int : IntervalIntegrable (fun x ↦ u * (↑x ^ (u - 1) * (1 - ↑x) ^ v) - v * (↑x ^ u * (1 - ↑x) ^ (v - 1))) volume 0 1
int_ev : ∫ (y : ℝ) in 0..1, u * (↑y ^ (u - 1) * (1 - ↑y) ^ v) - v * (↑y ^ u * (1 - ↑y) ^ (v - 1)) = 0
hF0 : F 0 = 0
hF1 : F 1 = 0
⊢ IntervalIntegrable (fun y ↦ v * (↑y ^ u * (1 - ↑y) ^ (v - 1))) volume 0 1

Tactic:
apply IntervalIntegrable.const_mul

Goal state after:
case hg
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
F : ℝ → ℂ := ⋯
hu' : 0 < (u + 1).re
hv' : 0 < (v + 1).re
hc : ContinuousOn F (Icc 0 1)
hder : ∀ x ∈ Ioo 0 1, HasDerivAt F (u * (↑x ^ (u - 1) * (1 - ↑x) ^ v) - v * (↑x ^ u * (1 - ↑x) ^ (v - 1))) x
h_int : IntervalIntegrable (fun x ↦ u * (↑x ^ (u - 1) * (1 - ↑x) ^ v) - v * (↑x ^ u * (1 - ↑x) ^ (v - 1))) volume 0 1
int_ev : ∫ (y : ℝ) in 0..1, u * (↑y ^ (u - 1) * (1 - ↑y) ^ v) - v * (↑y ^ u * (1 - ↑y) ^ (v - 1)) = 0
hF0 : F 0 = 0
hF1 : F 1 = 0
⊢ IntervalIntegrable (fun x ↦ ↑x ^ u * (1 - ↑x) ^ (v - 1)) volume 0 1
Module: Mathlib.Topology.Algebra.AsymptoticCone

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : Field k
inst✝⁴ : LinearOrder k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AddTorsor V P
inst✝ : TopologicalSpace V
u v : V
p : P
⊢ u +ᵥ asymptoticNhds k P v = asymptoticNhds k P v

Tactic:
← asymptoticNhds_vadd_pure v p

Goal state after:
k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : Field k
inst✝⁴ : LinearOrder k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AddTorsor V P
inst✝ : TopologicalSpace V
u v : V
p : P
⊢ u +ᵥ asymptoticNhds k V v +ᵥ pure p = asymptoticNhds k P v
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: m : ℕ
s : ℂ
IH : Tendsto (s + 1).GammaSeq atTop (𝓝 (s * Gamma s))
hs : ⌊1 - s.re⌋₊ = m + 1
hsne : s ≠ 0
this : Tendsto (fun n ↦ s.GammaSeq n * (↑n / (↑n + 1 + s))) atTop (𝓝 (Gamma s * 1))
⊢ Tendsto (fun n ↦ ↑n / (↑n + 1 + s)) atTop (𝓝 1)

Tactic:
simp_rw [add_assoc]

Goal state after:
m : ℕ
s : ℂ
IH : Tendsto (s + 1).GammaSeq atTop (𝓝 (s * Gamma s))
hs : ⌊1 - s.re⌋₊ = m + 1
hsne : s ≠ 0
this : Tendsto (fun n ↦ s.GammaSeq n * (↑n / (↑n + 1 + s))) atTop (𝓝 (Gamma s * 1))
⊢ Tendsto (fun n ↦ ↑n / (↑n + (1 + s))) atTop (𝓝 1)
Module: Mathlib.LinearAlgebra.AffineSpace.Ceva

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p : ι → P
hp : AffineIndependent k p
s : Set ι
hs : s.Nonempty
fs : ↑s → Finset ι
w : ↑s → ι → k
hw : ∀ (i : ↑s), ∑ j ∈ fs i, w i j = 1
p' : P
hp' : ∀ (i : ↑s), p' ∈ line[k, p ↑i, (Finset.affineCombination k (fs i) p) (w i)]
⊢ ∃ w' fs',
    ∑ j ∈ fs', w' j = 1 ∧
      (Finset.affineCombination k fs' p) w' = p' ∧
        ∀ (i : ↑s), ∃ r, ∀ (j : ι), r * (↑(fs i) \ {↑i}).indicator (w i) j = (↑fs' \ {↑i}).indicator w' j

Tactic:
by
  classical
  let fsx : s → Finset ι := fun i ↦ insert (i : ι) (fs i)
  have hfsx : ∀ i, (i : ι) ∈ fsx i := by simp [fsx]
  let wx : s → ι → k := fun i ↦ Set.indicator (fs i) (w i)
  have hwx : ∀ i, ∑ j ∈ fsx i, wx i j = 1 := by
    intro i
    simp_rw [← hw i, fsx, wx]
    by_cases hi : (i : ι) ∈ fs i <;> simpa [hi] using Finset.sum_congr rfl (by aesop)
  have hp'x : ∀ i : s, p' ∈ line[k, p i, (fsx i).affineCombination k p (wx i)] := by
    intro i
    convert! hp' i using 4
    simp_rw [fsx, wx]
    exact (Finset.affineCombination_indicator_subset _ _ (by simp)).symm
  obtain ⟨w', fs', h⟩ := hp.exists_affineCombination_eq_smul_eq_aux hs hfsx hwx hp'x
  refine ⟨w', fs', h.1, h.2.1, fun i ↦ ?_⟩
  obtain ⟨r, hr⟩ := h.2.2 i
  refine ⟨r, fun j ↦ ?_⟩
  convert! hr j using 2
  simp only [Set.indicator_apply, Set.mem_sdiff, SetLike.mem_coe, Set.mem_singleton_iff,
    Finset.coe_insert, Set.insert_sdiff_of_mem, fsx, wx]
  grind

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

Goal state before:
pp: R : Type u_1
inst✝ : Ring R
p : RingSeminorm R
hp : p 1 ≤ 1
s : ℕ → ℕ
hs_le : ∀ (n : ℕ), s n ≤ n
x : R
ψ : ℕ → ℕ
h_le : ∀ (m : ℕ), p (x ^ s (ψ m)) ^ (1 / ↑(ψ m)) ≤ p x ^ (↑(s (ψ m)) / ↑(ψ m))
⊢ IsBoundedUnder LE.le atTop fun n ↦ p (x ^ s (ψ n)) ^ (1 / ↑(ψ n))

Tactic:
apply isBoundedUnder_of

Goal state after:
R : Type u_1
inst✝ : Ring R
p : RingSeminorm R
hp : p 1 ≤ 1
s : ℕ → ℕ
hs_le : ∀ (n : ℕ), s n ≤ n
x : R
ψ : ℕ → ℕ
h_le : ∀ (m : ℕ), p (x ^ s (ψ m)) ^ (1 / ↑(ψ m)) ≤ p x ^ (↑(s (ψ m)) / ↑(ψ m))
⊢ ∃ b, ∀ (x_1 : ℕ), p (x ^ s (ψ x_1)) ^ (1 / ↑(ψ x_1)) ≤ b
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
x : R
hx : f x = 0
y : R
⊢ f (x * y) = 0

Tactic:
replace f_mul : f (x * y) ≤ 0 := by simpa [hx] using f_mul x y
  exact le_antisymm f_mul (f_nonneg _)

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

Goal state before:
pp: K : Type u_1
inst✝ : NormedField K
f g : K[X]
ε : ℝ
hε : 0 < ε
a : K
ha : eval a f = 0
hfm : f.Monic
hgm : g.Monic
hdeg : g.natDegree = f.natDegree
hcoeff : ∀ (i : ℕ), ‖g.coeff i - f.coeff i‖ < ε
hg : g.Splits
⊢ ∑ i ∈ Finset.range (g.natDegree + 1), ‖(g.coeff i - f.coeff i) * a ^ i‖ <
    (↑f.natDegree + 1) * ε * max ‖a‖ 1 ^ ↑f.natDegree

Tactic:
rw [hdeg]
    convert!
      Finset.sum_lt_sum_of_nonempty (g := fun i ↦ ε * (‖a‖ ⊔ 1) ^ ↑f.natDegree)
        (Finset.nonempty_range_add_one) ?_
    · simp [mul_assoc]
    · simp only [Finset.mem_range, norm_mul, norm_pow]
      intro i hi
      apply mul_lt_mul_of_lt_of_le_of_nonneg_of_pos
      · simpa [← map_sub] using hcoeff i
      · refine (pow_le_pow_left₀ (norm_nonneg a) (le_max_left ‖a‖ 1) i).trans ?_
        exact pow_le_pow_right₀ (le_max_right ‖a‖ 1) (Nat.le_of_lt_succ hi)
      all_goals positivity

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

Goal state before:
pp: case a
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_ne_zero : f ≠ 0
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
x : R
⊢ f x / f x ≤ 1

Tactic:
by_cases hx : f x = 0

Goal state after:
case pos
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_ne_zero : f ≠ 0
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
x : R
hx : f x = 0
⊢ f x / f x ≤ 1

case neg
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_ne_zero : f ≠ 0
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
x : R
hx : ¬f x = 0
⊢ f x / f x ≤ 1
Module: Mathlib.Topology.Algebra.Order.LiminfLimsup

Goal state before:
pp: ι : Type u_1
R : Type u_4
inst✝⁷ : ConditionallyCompleteLinearOrder R
inst✝⁶ : TopologicalSpace R
inst✝⁵ : OrderTopology R
F : Filter ι
inst✝⁴ : AddCommSemigroup R
inst✝³ : Sub R
inst✝² : ContinuousSub R
inst✝¹ : OrderedSub R
inst✝ : AddLeftMono R
f : ι → R
c : R
cobdd : IsCoboundedUnder (fun x1 x2 ↦ x1 ≥ x2) F f
bdd_below : IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) F f
⊢ limsup (fun i ↦ c - f i) F = c - liminf f F

Tactic:
by
  rcases F.eq_or_neBot with rfl | _
  · simp only [liminf, limsInf, limsup, limsSup, map_bot, eventually_bot, Set.ofPred_true]
    simp only [IsCoboundedUnder, IsCobounded, map_bot, eventually_bot, true_implies] at cobdd
    rcases cobdd with ⟨x, hx⟩
    refine (csInf_le ?_ (Set.mem_univ _)).antisymm
      (tsub_le_iff_tsub_le.1 (le_csSup ?_ (Set.mem_univ _)))
    · refine ⟨x - x, mem_lowerBounds.2 fun y ↦ ?_⟩
      simp only [Set.mem_univ, true_implies]
      exact tsub_le_iff_tsub_le.1 (hx (x - y))
    · refine ⟨x, mem_upperBounds.2 fun y ↦ ?_⟩
      simp only [Set.mem_univ, hx y, implies_true]
  · exact (Antitone.map_limsInf_of_continuousAt (F := F.map f) (f := fun (x : R) ↦ c - x)
    (fun _ _ h ↦ tsub_le_tsub_left h c) (continuous_sub_left c).continuousAt cobdd bdd_below).symm

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

Goal state before:
pp: case a
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
⊢ f x✝ * (f x / f x) ≤ f x✝

Tactic:
by_cases hx : f x = 0

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✝ * (f x / f x) ≤ f x✝

case neg
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✝ * (f x / f x) ≤ f x✝
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
x : R
hx : ∀ (y : R), f (x * y) = f x * f y
f_ne_zero : f ≠ 0
| f x

Tactic:
← mul_one (f x)

Goal state after:
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
f_ne_zero : f ≠ 0
| f x * 1
Module: Mathlib.Analysis.Normed.Unbundled.FiniteExtension

Goal state before:
pp: case h.h
K : Type u_1
L : Type u_2
inst✝⁴ : NormedField K
inst✝³ : Ring L
inst✝² : Algebra K L
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
B : Basis ι K L
i : ι
hBi : B i = 1
hna : IsNonarchimedean Norm.norm
c : ι × ι
left✝¹ : c ∈ univ
hc : (univ.sup' ⋯ fun i ↦ B.norm (B i.1 * B i.2)) = B.norm (B c.1 * B c.2)
x y : L
ixy : ι
left✝ : ixy ∈ univ
hixy_def : (univ.sup' ⋯ fun i ↦ ‖(B.repr (x * y)) i‖) = ‖(B.repr (x * y)) ixy‖
⊢ ‖(B.repr ((∑ i, (B.repr x) i • B i) * ∑ i, (B.repr y) i • B i)) ixy‖ ≤ B.norm (B c.1 * B c.2) * B.norm x * B.norm y

Tactic:
rw [sum_mul, map_finsetSum]

Goal state after:
case h.h
K : Type u_1
L : Type u_2
inst✝⁴ : NormedField K
inst✝³ : Ring L
inst✝² : Algebra K L
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
B : Basis ι K L
i : ι
hBi : B i = 1
hna : IsNonarchimedean Norm.norm
c : ι × ι
left✝¹ : c ∈ univ
hc : (univ.sup' ⋯ fun i ↦ B.norm (B i.1 * B i.2)) = B.norm (B c.1 * B c.2)
x y : L
ixy : ι
left✝ : ixy ∈ univ
hixy_def : (univ.sup' ⋯ fun i ↦ ‖(B.repr (x * y)) i‖) = ‖(B.repr (x * y)) ixy‖
⊢ ‖(∑ i, B.repr ((B.repr x) i • B i * ∑ i, (B.repr y) i • B i)) ixy‖ ≤ B.norm (B c.1 * B c.2) * B.norm x * B.norm y
Module: Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm

Goal state before:
pp: case hg
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
L : ℝ
hL : 0 ≤ L
ε : ℝ
hε : 0 < ε
m1 : ℕ
hm1 : 0 < m1
x : R
hx : μ x ≠ 0
h_exp : Tendsto (fun n ↦ ↑(n % m1) / ↑n) atTop (𝓝 0)
⊢ Tendsto (fun x_1 ↦ (μ x ^ ↑(x_1 % m1)) ^ (1 / ↑x_1)) atTop (𝓝 1)

Tactic:
← rpow_mul (apply_nonneg μ x),

Goal state after:
case hg
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
L : ℝ
hL : 0 ≤ L
ε : ℝ
hε : 0 < ε
m1 : ℕ
hm1 : 0 < m1
x : R
hx : μ x ≠ 0
h_exp : Tendsto (fun n ↦ ↑(n % m1) / ↑n) atTop (𝓝 0)
⊢ Tendsto (fun x_1 ↦ μ x ^ (↑(x_1 % m1) * (1 / ↑x_1))) atTop (𝓝 1)
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
x : L
n : ℕ
hn : 1 ≤ n
⊢ ⨆ σ, (algNormOfAlgEquiv σ) (x ^ n) = ⨆ i, (algNormOfAlgEquiv i) x ^ n

Tactic:
exact iSup_congr fun σ ↦ isPowMul_algNormOfAlgEquiv σ _ hn

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 : 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
hε2 : 0 < ε / 2
hL2 : ∀ ε_1 > 0, ∃ N, ∀ n ≥ N, dist ((L + ε / 2) ^ (-(↑(n % ↑m1) / ↑n)) * (μ x ^ (n % ↑m1)) ^ (1 / ↑n)) 1 < ε_1
δ : ℝ := ε / (2 * (L + ε / 2))
hδ_def : δ = ε / (2 * (L + ε / 2))
hδ : 0 < δ
N n : ℕ
hN : |(L + ε / 2) ^ (-(↑(n % ↑m1) / ↑n)) * (μ x ^ (n % ↑m1)) ^ (1 / ↑n) - 1| < δ
hn : n ≥ N
⊢ (L + ε / 2) ^ (-(↑(n % ↑m1) / ↑n)) * (μ x ^ (n % ↑m1)) ^ (1 / ↑n) - 1 ≤ δ

Tactic:
abs_lt

Goal state after:
case h
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
hε2 : 0 < ε / 2
hL2 : ∀ ε_1 > 0, ∃ N, ∀ n ≥ N, dist ((L + ε / 2) ^ (-(↑(n % ↑m1) / ↑n)) * (μ x ^ (n % ↑m1)) ^ (1 / ↑n)) 1 < ε_1
δ : ℝ := ε / (2 * (L + ε / 2))
hδ_def : δ = ε / (2 * (L + ε / 2))
hδ : 0 < δ
N n : ℕ
hN :
  -δ < (L + ε / 2) ^ (-(↑(n % ↑m1) / ↑n)) * (μ x ^ (n % ↑m1)) ^ (1 / ↑n) - 1 ∧
    (L + ε / 2) ^ (-(↑(n % ↑m1) / ↑n)) * (μ x ^ (n % ↑m1)) ^ (1 / ↑n) - 1 < δ
hn : n ≥ N
⊢ (L + ε / 2) ^ (-(↑(n % ↑m1) / ↑n)) * (μ x ^ (n % ↑m1)) ^ (1 / ↑n) - 1 ≤ δ
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 : 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
⊢ |smoothingSeminormSeq μ x n - smoothingFun μ x| < ε

Tactic:
abs_lt

Goal state after:
case h
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
⊢ -ε < smoothingSeminormSeq μ x n - smoothingFun μ x ∧ smoothingSeminormSeq μ x n - smoothingFun μ x < ε
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromConst

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
c : R
f : RingSeminorm R
hf1 : f 1 ≤ 1
hc : f c ≠ 0
hpm : IsPowMul ⇑f
⊢ seminormFromConst' c f 1 = 1

Tactic:
apply tendsto_nhds_unique_of_eventuallyEq (tendsto_seminormFromConst_seq_atTop hf1 hc hpm 1)
    tendsto_const_nhds

Goal state after:
R : Type u_1
inst✝ : CommRing R
c : R
f : RingSeminorm R
hf1 : f 1 ≤ 1
hc : f c ≠ 0
hpm : IsPowMul ⇑f
⊢ seminormFromConst_seq c f 1 =ᶠ[atTop] fun x ↦ 1
Module: Mathlib.Analysis.Normed.Field.Krasner

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝⁵ : NormedField L
inst✝⁴ : NontriviallyNormedField K
inst✝³ : CompleteSpace K
inst✝² : IsUltrametricDist K
inst✝¹ : NormedAlgebra K L
inst✝ : Algebra.IsAlgebraic K L
x y : L
xsep : IsSeparable K x
sp : (Polynomial.map (algebraMap K L) (minpoly K x)).Splits
yint : IsIntegral K y
kr : ∀ (x' : L), IsConjRoot K x x' → x ≠ x' → ‖x - y‖ < ‖x - x'‖
C : Type u_1 := AlgebraicClosure K
this✝ : NontriviallyNormedField C := spectralNorm.nontriviallyNormedField K C
this : NormedAlgebra K C := spectralNorm.normedAlgebra K C
⊢ x ∈ K⟮y⟯

Tactic:
let iL : L →ₐ[K] C := IsAlgClosed.lift

Goal state after:
K : Type u_1
L : Type u_2
inst✝⁵ : NormedField L
inst✝⁴ : NontriviallyNormedField K
inst✝³ : CompleteSpace K
inst✝² : IsUltrametricDist K
inst✝¹ : NormedAlgebra K L
inst✝ : Algebra.IsAlgebraic K L
x y : L
xsep : IsSeparable K x
sp : (Polynomial.map (algebraMap K L) (minpoly K x)).Splits
yint : IsIntegral K y
kr : ∀ (x' : L), IsConjRoot K x x' → x ≠ x' → ‖x - y‖ < ‖x - x'‖
C : Type u_1 := AlgebraicClosure K
this✝ : NontriviallyNormedField C := spectralNorm.nontriviallyNormedField K C
this : NormedAlgebra K C := spectralNorm.normedAlgebra K C
iL : L →ₐ[K] C := IsAlgClosed.lift
⊢ x ∈ K⟮y⟯
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
⊢ Tendsto (smoothingSeminormSeq μ x) atTop (𝓝 (smoothingFun μ x))

Tactic:
· exact tendsto_smoothingFun_of_ne_zero μ hμ1 hx

Goal state after:
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
hab✝ :
  ‖a - b‖ < ((↑f.natDegree + 1) * ((δ / max ‖a‖ 1) ^ f.natDegree / (↑f.natDegree + 1))) ^ (↑f.natDegree)⁻¹ * max ‖a‖ 1
hab : ‖a - b‖ < δ
bCp : L
left✝ : bCp ∈ (map (algebraMap K L) g).roots
hbCp : (algebraMap L f.SplittingField) bCp = b
⊢ b ∈ ⊥

Tactic:
rw [IntermediateField.mem_bot]

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
hab✝ :
  ‖a - b‖ < ((↑f.natDegree + 1) * ((δ / max ‖a‖ 1) ^ f.natDegree / (↑f.natDegree + 1))) ^ (↑f.natDegree)⁻¹ * max ‖a‖ 1
hab : ‖a - b‖ < δ
bCp : L
left✝ : bCp ∈ (map (algebraMap K L) g).roots
hbCp : (algebraMap L f.SplittingField) bCp = b
⊢ b ∈ Set.range ⇑(algebraMap L f.SplittingField)