mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b : ℝ
f : ℝ → E
s : ℂ
hfc : LocallyIntegrableOn f (Ioi 0) volume
hf_top : f =O[atTop] fun x ↦ x ^ (-a)
hs_top : s.re < a
hf_bot : f =O[𝓝[>] 0] fun x ↦ x ^ (-b)
hs_bot : b < s.re
F : ℂ → ℝ → E := fun z t ↦ ↑t ^ (z - 1) • f t
F' : ℂ → ℝ → E := fun z t ↦ (↑t ^ (z - 1) * ↑(log t)) • f t
⊢ ∃ v, 0 < v ∧ v < s.re - b ∧ v < a - s.re

Tactic:
obtain ⟨w, hw1, hw2⟩ := exists_between (sub_pos.mpr hs_top)
    obtain ⟨w', hw1', hw2'⟩ := exists_between (sub_pos.mpr hs_bot)
    exact
      ⟨min w w', lt_min hw1 hw1', (min_le_right _ _).trans_lt hw2', (min_le_left _ _).trans_lt hw2⟩

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: g : ℝ → ℂ
t : ℂ
hg : Tendsto (fun x ↦ ↑x * g x) atTop (𝓝 t)
x : ℝ
hx0 : x ≠ 0
⊢ ↑x * g x ^ 2 = (↑x * g x) ^ 2 * (↑x)⁻¹

Tactic:
eq_mul_inv_iff_mul_eq₀ (mod_cast hx0)

Goal state after:
g : ℝ → ℂ
t : ℂ
hg : Tendsto (fun x ↦ ↑x * g x) atTop (𝓝 t)
x : ℝ
hx0 : x ≠ 0
⊢ ↑x * g x ^ 2 * ↑x = (↑x * g x) ^ 2
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: case hbc.inl
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b : ℝ
f : ℝ → E
s : ℂ
hfc : LocallyIntegrableOn f (Ioi 0) volume
hf_top : f =O[atTop] fun x ↦ x ^ (-a)
hs_top : s.re < a
hf_bot : f =O[𝓝[>] 0] fun x ↦ x ^ (-b)
hs_bot : b < s.re
F : ℂ → ℝ → E := fun z t ↦ ↑t ^ (z - 1) • f t
F' : ℂ → ℝ → E := fun z t ↦ (↑t ^ (z - 1) * ↑(log t)) • f t
v : ℝ
hv0 : 0 < v
hv1 : v < s.re - b
hv2 : v < a - s.re
bound : ℝ → ℝ := fun t ↦ (t ^ (s.re + v - 1) + t ^ (s.re - v - 1)) * |log t| * ‖f t‖
h1 : ∀ᶠ (z : ℂ) in 𝓝 s, AEStronglyMeasurable (F z) (volume.restrict (Ioi 0))
h2 : IntegrableOn (F s) (Ioi 0) volume
h3 : AEStronglyMeasurable (F' s) (volume.restrict (Ioi 0))
t : ℝ
ht : t ∈ Ioi 0
z : ℂ
hz : z ∈ Metric.ball s v
h : 1 ≤ t
⊢ z.re - 1 ≤ s.re + v - 1

Tactic:
sub_le_sub_iff_right

Goal state after:
case hbc.inl
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b : ℝ
f : ℝ → E
s : ℂ
hfc : LocallyIntegrableOn f (Ioi 0) volume
hf_top : f =O[atTop] fun x ↦ x ^ (-a)
hs_top : s.re < a
hf_bot : f =O[𝓝[>] 0] fun x ↦ x ^ (-b)
hs_bot : b < s.re
F : ℂ → ℝ → E := fun z t ↦ ↑t ^ (z - 1) • f t
F' : ℂ → ℝ → E := fun z t ↦ (↑t ^ (z - 1) * ↑(log t)) • f t
v : ℝ
hv0 : 0 < v
hv1 : v < s.re - b
hv2 : v < a - s.re
bound : ℝ → ℝ := fun t ↦ (t ^ (s.re + v - 1) + t ^ (s.re - v - 1)) * |log t| * ‖f t‖
h1 : ∀ᶠ (z : ℂ) in 𝓝 s, AEStronglyMeasurable (F z) (volume.restrict (Ioi 0))
h2 : IntegrableOn (F s) (Ioi 0) volume
h3 : AEStronglyMeasurable (F' s) (volume.restrict (Ioi 0))
t : ℝ
ht : t ∈ Ioi 0
z : ℂ
hz : z ∈ Metric.ball s v
h : 1 ≤ t
⊢ z.re ≤ s.re + v
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: 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)
x : ℝ
hx : x ∈ Ioo 0 1
U : HasDerivAt (fun y ↦ y ^ u) (u * ↑x ^ (u - 1)) ↑x
A : HasDerivAt (fun x ↦ x ^ v) (v * (1 - ↑x) ^ (v - 1) * 1) (1 - ↑x)
⊢ HasDerivAt (fun x ↦ x) 1 ↑x

Tactic:
apply hasDerivAt_id

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

Goal state before:
pp: case inr
z : ℂ
n : ℕ
hz : z ≠ 0
⊢ Complex.sin (↑π * z) =
    ((↑π * z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) *
        ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) /
      ↑(∫ (x : ℝ) in 0..π / 2, cos x ^ (2 * n))

Tactic:
induction n with
  | zero =>
    simp_rw [mul_zero, pow_zero, mul_one, Finset.prod_range_zero, mul_one,
      integral_one, sub_zero]
    rw [integral_cos_mul_complex (mul_ne_zero two_ne_zero hz), Complex.ofReal_zero,
      mul_zero, Complex.sin_zero, zero_div, sub_zero,
      (by push_cast; ring : 2 * z * ↑(π / 2) = π * z)]
    simp [field]
  | succ n hn =>
    rw [hn, Finset.prod_range_succ]
    set A := ∏ j ∈ Finset.range n, ((1 : ℂ) - z ^ 2 / ((j : ℂ) + 1) ^ 2)
    set B := ∫ x in (0 : ℝ)..π / 2, Complex.cos (2 * z * x) * (cos x : ℂ) ^ (2 * n)
    set C := ∫ x in (0 : ℝ)..π / 2, cos x ^ (2 * n)
    have aux' : 2 * n.succ = 2 * n + 2 := by rw [Nat.succ_eq_add_one, mul_add, mul_one]
    have : (∫ x in (0 : ℝ)..π / 2, cos x ^ (2 * n.succ)) = (2 * (n : ℝ) + 1) / (2 * n + 2) * C := by
      rw [integral_cos_pow_eq]
      dsimp only [C]
      rw [integral_cos_pow_eq, aux', integral_sin_pow, sin_zero, sin_pi, pow_succ',
        zero_mul, zero_mul, zero_mul, sub_zero, zero_div,
        zero_add, ← mul_assoc, ← mul_assoc, mul_comm (1 / 2 : ℝ) _, Nat.cast_mul, Nat.cast_ofNat]
    rw [this]
    change
      π * z * A * B / C =
        (π * z * (A * ((1 : ℂ) - z ^ 2 / ((n : ℂ) + 1) ^ 2)) *
            ∫ x in (0 : ℝ)..π / 2, Complex.cos (2 * z * x) * (cos x : ℂ) ^ (2 * n.succ)) /
          ((2 * n + 1) / (2 * n + 2) * C : ℝ)
    have :
      (π * z * (A * ((1 : ℂ) - z ^ 2 / ((n : ℂ) + 1) ^ 2)) *
          ∫ x in (0 : ℝ)..π / 2, Complex.cos (2 * z * x) * (cos x : ℂ) ^ (2 * n.succ)) =
        π * z * A *
          (((1 : ℂ) - z ^ 2 / (n.succ : ℂ) ^ 2) *
            ∫ x in (0 : ℝ)..π / 2, Complex.cos (2 * z * x) * (cos x : ℂ) ^ (2 * n.succ)) := by
      grind
    rw [this]
    suffices
      (((1 : ℂ) - z ^ 2 / (n.succ : ℂ) ^ 2) *
          ∫ x in (0 : ℝ)..π / 2, Complex.cos (2 * z * x) * (cos x : ℂ) ^ (2 * n.succ)) =
        (2 * n + 1) / (2 * n + 2) * B by
      rw [this, Complex.ofReal_mul, Complex.ofReal_div]
      have : (C : ℂ) ≠ 0 := Complex.ofReal_ne_zero.mpr (integral_cos_pow_pos _).ne'
      have : 2 * (n : ℂ) + 1 ≠ 0 := by
        convert! (Nat.cast_add_one_ne_zero (2 * n) : (↑(2 * n) + 1 : ℂ) ≠ 0)
        simp
      have : (n : ℂ) + 1 ≠ 0 := Nat.cast_add_one_ne_zero n
      simp [field]
    convert! integral_cos_mul_cos_pow_even n hz
    rw [Nat.cast_succ]

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.AsymptoticCone

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : NormedSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
inst✝ : FiniteDimensional ℝ V
s : Set P
h : asymptoticCone ℝ s ⊆ {0}
⊢ Bornology.IsBounded s

Tactic:
isBounded_def,

Goal state after:
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : NormedSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
inst✝ : FiniteDimensional ℝ V
s : Set P
h : asymptoticCone ℝ s ⊆ {0}
⊢ sᶜ ∈ cobounded P
Module: Mathlib.Topology.Algebra.AsymptoticCone

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝¹² : Field k
inst✝¹¹ : LinearOrder k
inst✝¹⁰ : AddCommGroup V
inst✝⁹ : Module k V
inst✝⁸ : AddTorsor V P
inst✝⁷ : TopologicalSpace V
inst✝⁶ : TopologicalSpace k
inst✝⁵ : OrderTopology k
inst✝⁴ : IsStrictOrderedRing k
inst✝³ : IsTopologicalAddGroup V
inst✝² : ContinuousSMul k V
inst✝¹ : TopologicalSpace P
inst✝ : IsTopologicalAddTorsor P
v : V
s : Set P
p : P
t₁ : Set k
ht₁ : t₁ ∈ atTop
t₂ : Set V
hs : t₁ • t₂ ⊆ (fun x ↦ x +ᵥ p) ⁻¹' s
t₃ : Set V
ht₃ : t₃ ∈ 𝓝 v
ht₂ : ∀ x ∈ t₃, t₂ ∈ 𝓝 x
c : k
x✝ : c ∈ t₁ ∩ Set.Ioi 0
u : V
hu : u ∈ t₃
hc₁ : c ∈ t₁
hc₂ : c ∈ Set.Ioi 0
⊢ (fun x ↦ x +ᵥ p) ⁻¹' s ∈ 𝓝 ((fun x ↦ x +ᵥ p) (c • u) -ᵥ p)

Tactic:
vadd_vsub

Goal state after:
k : Type u_1
V : Type u_2
P : Type u_3
inst✝¹² : Field k
inst✝¹¹ : LinearOrder k
inst✝¹⁰ : AddCommGroup V
inst✝⁹ : Module k V
inst✝⁸ : AddTorsor V P
inst✝⁷ : TopologicalSpace V
inst✝⁶ : TopologicalSpace k
inst✝⁵ : OrderTopology k
inst✝⁴ : IsStrictOrderedRing k
inst✝³ : IsTopologicalAddGroup V
inst✝² : ContinuousSMul k V
inst✝¹ : TopologicalSpace P
inst✝ : IsTopologicalAddTorsor P
v : V
s : Set P
p : P
t₁ : Set k
ht₁ : t₁ ∈ atTop
t₂ : Set V
hs : t₁ • t₂ ⊆ (fun x ↦ x +ᵥ p) ⁻¹' s
t₃ : Set V
ht₃ : t₃ ∈ 𝓝 v
ht₂ : ∀ x ∈ t₃, t₂ ∈ 𝓝 x
c : k
x✝ : c ∈ t₁ ∩ Set.Ioi 0
u : V
hu : u ∈ t₃
hc₁ : c ∈ t₁
hc₂ : c ∈ Set.Ioi 0
⊢ (fun x ↦ x +ᵥ p) ⁻¹' s ∈ 𝓝 (c • u)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: 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)
n : ℕ
⊢ Integrable (f n) (volume.restrict (Ioi 0))

Tactic:
rw [integrable_indicator_iff (measurableSet_Ioc : MeasurableSet (Ioc (_ : ℝ) _)), IntegrableOn,
      Measure.restrict_restrict_of_subset Ioc_subset_Ioi_self, ← IntegrableOn, ←
      intervalIntegrable_iff_integrableOn_Ioc_of_le (by positivity : (0 : ℝ) ≤ n)]

Goal state after:
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)
n : ℕ
⊢ IntervalIntegrable (fun x ↦ ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)) volume 0 ↑n
Module: Mathlib.Analysis.Normed.Affine.MazurUlam

Goal state before:
pp: E : Type u_1
PE : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : MetricSpace PE
inst✝ : NormedAddTorsor E PE
x y : PE
z : PE := midpoint ℝ x y
s : Set (PE ≃ᵢ PE) := {e | e x = x ∧ e y = y}
this : Nonempty ↑s
e : PE ≃ᵢ PE
hx : e x = x
right✝ : e y = y
⊢ dist (e z) x + dist x z = dist (e x) (e z) + dist x z

Tactic:
hx,

Goal state after:
E : Type u_1
PE : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : MetricSpace PE
inst✝ : NormedAddTorsor E PE
x y : PE
z : PE := midpoint ℝ x y
s : Set (PE ≃ᵢ PE) := {e | e x = x ∧ e y = y}
this : Nonempty ↑s
e : PE ≃ᵢ PE
hx : e x = x
right✝ : e y = y
⊢ dist (e z) x + dist x z = dist x (e z) + dist x z
Module: Mathlib.Topology.Algebra.AsymptoticCone

Goal state before:
pp: k : Type u_1
V : Type u_2
inst✝⁹ : Field k
inst✝⁸ : LinearOrder k
inst✝⁷ : IsStrictOrderedRing k
inst✝⁶ : TopologicalSpace k
inst✝⁵ : OrderTopology k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : TopologicalSpace V
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul k V
s : Set V
c : k
v p : V
hs : Convex k s
hc : 0 ≤ c
hp : s ∈ 𝓝 p
hv : ∃ᶠ (a : k × V) in atTop ×ˢ 𝓝 v, a.1 • a.2 +ᵥ c • v +ᵥ p ∈ s
this : Tendsto (fun u ↦ -(c • u) +ᵥ c • v +ᵥ p) (𝓝 v) (𝓝 p)
t : k
u : V
ht : 0 < t
hu : -(c • u) +ᵥ c • v +ᵥ p ∈ s
h : t • u +ᵥ c • v +ᵥ p ∈ s
⊢ SameRay k (c • u) ((t • u +ᵥ c • v +ᵥ p) -ᵥ (c • v +ᵥ p))

Tactic:
vadd_vsub

Goal state after:
k : Type u_1
V : Type u_2
inst✝⁹ : Field k
inst✝⁸ : LinearOrder k
inst✝⁷ : IsStrictOrderedRing k
inst✝⁶ : TopologicalSpace k
inst✝⁵ : OrderTopology k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : TopologicalSpace V
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul k V
s : Set V
c : k
v p : V
hs : Convex k s
hc : 0 ≤ c
hp : s ∈ 𝓝 p
hv : ∃ᶠ (a : k × V) in atTop ×ˢ 𝓝 v, a.1 • a.2 +ᵥ c • v +ᵥ p ∈ s
this : Tendsto (fun u ↦ -(c • u) +ᵥ c • v +ᵥ p) (𝓝 v) (𝓝 p)
t : k
u : V
ht : 0 < t
hu : -(c • u) +ᵥ c • v +ᵥ p ∈ s
h : t • u +ᵥ c • v +ᵥ p ∈ s
⊢ SameRay k (c • u) (t • u)
Module: Mathlib.Topology.Algebra.AsymptoticCone

Goal state before:
pp: k : Type u_1
V : Type u_2
inst✝⁹ : Field k
inst✝⁸ : LinearOrder k
inst✝⁷ : IsStrictOrderedRing k
inst✝⁶ : TopologicalSpace k
inst✝⁵ : OrderTopology k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : TopologicalSpace V
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul k V
s : Set V
c : k
v p : V
hs : Convex k s
hc : 0 ≤ c
hp : s ∈ 𝓝 p
hv : ∃ᶠ (a : k × V) in atTop ×ˢ 𝓝 v, a.1 • a.2 +ᵥ c • v +ᵥ p ∈ s
this : Tendsto (fun u ↦ -(c • u) +ᵥ c • v +ᵥ p) (𝓝 v) (𝓝 p)
t : k
u : V
ht : 0 < t
hu : -(c • u) +ᵥ c • v +ᵥ p ∈ s
h : t • u +ᵥ c • v +ᵥ p ∈ s
⊢ SameRay k (c • u) (t • u)

Tactic:
exact (SameRay.sameRay_nonneg_smul_left _ hc).pos_smul_right ht

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

Goal state before:
pp: case pos
z : ℂ
pi_ne : ↑π ≠ 0
k : ℤ
hk : -z = ↑k ∨ ↑π = 0
⊢ Gamma z * Gamma (1 - z) = 0

Tactic:
eq_false (ofReal_ne_zero.mpr pi_pos.ne'),

Goal state after:
case pos
z : ℂ
pi_ne : ↑π ≠ 0
k : ℤ
hk : -z = ↑k ∨ False
⊢ Gamma z * Gamma (1 - z) = 0
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Series

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

Tactic:
mul_div_assoc,

Goal state after:
z : ℂ
⊢ (fun n ↦ (-1) ^ n * (z ^ 2) ^ n * (z / ↑(2 * n + 1)!)) = fun n ↦ (z ^ 2) ^ n * z * (-1) ^ n * (I / ↑(2 * n + 1)! / I)
Module: Mathlib.Analysis.Normed.Algebra.QuaternionExponential

Goal state before:
pp: case inr.calc_2.e_a
q : ℍ
hq : q.re = 0
n : ℕ
hq0 : q ≠ 0
hq2 : q ^ 2 = -↑(normSq q)
hqn : ‖q‖ ≠ 0
k : ℝ := ↑(2 * n + 1)!
⊢ k⁻¹ * ((-1) ^ n * ‖q‖ ^ (2 * n)) = (-1) ^ n * (‖q‖ ^ (2 * n) * ‖q‖) / k / ‖q‖

Tactic:
mul_div_assoc,

Goal state after:
case inr.calc_2.e_a
q : ℍ
hq : q.re = 0
n : ℕ
hq0 : q ≠ 0
hq2 : q ^ 2 = -↑(normSq q)
hqn : ‖q‖ ≠ 0
k : ℝ := ↑(2 * n + 1)!
⊢ k⁻¹ * ((-1) ^ n * ‖q‖ ^ (2 * n)) = (-1) ^ n * (‖q‖ ^ (2 * n) * (‖q‖ / k / ‖q‖))
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Series

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

Tactic:
mul_div_assoc,

Goal state after:
z : ℂ
⊢ (fun n ↦ (-1) ^ n * (z ^ 2) ^ n * (z / ↑(2 * n + 1)!)) = fun n ↦ (-1) ^ n * ((z ^ 2) ^ n * (z / ↑(2 * n + 1)!))
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 : 0 < f x
⊢ 0 ≤ c

Tactic:
specialize f_mul x 1

Goal state after:
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
x : R
f_mul : f (x * 1) ≤ c * f x * f 1
hx : 0 < f x
⊢ 0 ≤ c
Module: Mathlib.Analysis.Normed.Unbundled.RingSeminorm

Goal state before:
pp: R : Type u_1
K : Type u_2
inst✝ : Field K
f : RingSeminorm K
hnt : f ≠ 0
x : K
hx : f.toFun x = 0
c : K
hc : f c ≠ 0
hn0 : ¬x = 0
⊢ f.toFun x * f.toFun (c * x⁻¹) ≤ 0

Tactic:
hx,

Goal state after:
R : Type u_1
K : Type u_2
inst✝ : Field K
f : RingSeminorm K
hnt : f ≠ 0
x : K
hx : f.toFun x = 0
c : K
hc : f c ≠ 0
hn0 : ¬x = 0
⊢ 0 * f.toFun (c * x⁻¹) ≤ 0
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
x : R
f_mul : 1 ≤ c * f 1
hx : 0 < f x
f_nonneg : 0 ≤ f 1
h1 : f 1 = 0
⊢ 1 ≤ 0

Tactic:
simpa [h1] using f_mul

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
h1 : f 1 = 0
⊢ f x ≤ f 1 * seminormFromBounded' f x

Tactic:
specialize f_mul x 1

Goal state after:
case pos
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
x : R
f_mul : f (x * 1) ≤ c * f x * f 1
h1 : f 1 = 0
⊢ f x ≤ f 1 * seminormFromBounded' f x
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_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

Tactic:
hx,

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
⊢ 0 / 0 ≤ 1
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
⊢ ⨆ y, f (x * y) / f y = f x

Tactic:
hx,

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
⊢ ⨆ y, f x * f y / f y = f x
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✝ * (f x / f x) ≤ f x✝

Tactic:
hx,

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 / 0) ≤ f x✝
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

Goal state before:
pp: case neg
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
f_nonneg : 0 ≤ f
x : R
hx : ∀ (y : R), f (x * y) ≤ f x * f y
h_one : f 1 ≤ 1
y : R
hy0 : ¬f y = 0
⊢ f (x * y) ≤ f x * f y / f y * f y

Tactic:
mul_div_assoc,

Goal state after:
case neg
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
f_nonneg : 0 ≤ f
x : R
hx : ∀ (y : R), f (x * y) ≤ f x * f y
h_one : f 1 ≤ 1
y : R
hy0 : ¬f y = 0
⊢ f (x * y) ≤ f x * (f y / f y) * f y
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 : L ≃ₐ[K] L
x y : L
⊢ (Classical.choose ⋯) (σ x + σ y) ≤ max ((Classical.choose ⋯) (σ x)) ((Classical.choose ⋯) (σ y))

Tactic:
exact (Classical.choose_spec (exists_nonarchimedean_pow_mul_seminorm_of_finiteDimensional
    h_fin hu.isNonarchimedean_norm)).2.2 _ _

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

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : NormedCommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
f₁ : AlgebraNorm R S
hf₁_pm : IsPowMul ⇑f₁
f₂ : AlgebraNorm R S
hf₂_pm : IsPowMul ⇑f₂
h_eq : ∀ (y : S), ∃ C₁ C₂, ∃ (_ : 0 < C₁) (_ : 0 < C₂), ∀ (x : ↥R[y]), f₁ ↑x ≤ C₁ * f₂ ↑x ∧ f₂ ↑x ≤ C₂ * f₁ ↑x
x : S
g₁ : AlgebraNorm R ↥R[x] := AlgebraNorm.restriction R[x] f₁
g₂ : AlgebraNorm R ↥R[x] := AlgebraNorm.restriction R[x] f₂
hg₁_pm : IsPowMul ⇑g₁
hg₂_pm : IsPowMul ⇑g₂
⊢ f₁ x = f₂ x

Tactic:
let y : R[(x : S)] := ⟨x, self_mem_adjoin_singleton R x⟩

Goal state after:
R : Type u_1
S : Type u_2
inst✝² : NormedCommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
f₁ : AlgebraNorm R S
hf₁_pm : IsPowMul ⇑f₁
f₂ : AlgebraNorm R S
hf₂_pm : IsPowMul ⇑f₂
h_eq : ∀ (y : S), ∃ C₁ C₂, ∃ (_ : 0 < C₁) (_ : 0 < C₂), ∀ (x : ↥R[y]), f₁ ↑x ≤ C₁ * f₂ ↑x ∧ f₂ ↑x ≤ C₂ * f₁ ↑x
x : S
g₁ : AlgebraNorm R ↥R[x] := AlgebraNorm.restriction R[x] f₁
g₂ : AlgebraNorm R ↥R[x] := AlgebraNorm.restriction R[x] f₂
hg₁_pm : IsPowMul ⇑g₁
hg₂_pm : IsPowMul ⇑g₂
y : ↥R[x] := ⟨x, ⋯⟩
⊢ f₁ x = f₂ x
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromConst

Goal state before:
pp: case inr
R : Type u_1
inst✝ : CommRing R
c : R
f : RingSeminorm R
hf1 : f 1 ≤ 1
hc : f c ≠ 0
hpm : IsPowMul ⇑f
x : R
m n : ℕ
hmn : m ≤ n
hc_pos : 0 < f c
hlt : m < n
h1 : 1 ≤ n - m
⊢ f (x * c ^ m) * f c ^ (n - m) / f c ^ n ≤ f (x * c ^ m) / f c ^ m

Tactic:
mul_div_assoc,

Goal state after:
case inr
R : Type u_1
inst✝ : CommRing R
c : R
f : RingSeminorm R
hf1 : f 1 ≤ 1
hc : f c ≠ 0
hpm : IsPowMul ⇑f
x : R
m n : ℕ
hmn : m ≤ n
hc_pos : 0 < f c
hlt : m < n
h1 : 1 ≤ n - m
⊢ f (x * c ^ m) * (f c ^ (n - m) / f c ^ n) ≤ f (x * c ^ m) / f c ^ m
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
n : ℕ
⊢ f c * f c ^ n / f c ^ n = f c

Tactic:
mul_div_assoc,

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
n : ℕ
⊢ f c * (f c ^ n / f c ^ n) = f c
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) / ↑n) < (L + ε / 2) ^ (↑↑m1 * ↑(n / ↑m1) / ↑n)

Tactic:
mul_div_assoc,

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) / ↑n) < (L + ε / 2) ^ (↑↑m1 * (↑(n / ↑m1) / ↑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 : ∀ x_1 ∈ Set.range (spectralValueTerms p), x_1 < f x
n : ℕ
hn : n < p.natDegree
hexp : (‖p.coeff n‖ ^ (1 / (↑p.natDegree - ↑n))) ^ (p.natDegree - n) = ‖p.coeff n‖
h_rg : ‖p.coeff n‖ ^ (1 / (↑p.natDegree - ↑n)) ∈ Set.range (spectralValueTerms p)
⊢ ‖p.coeff n‖ ^ (1 / (↑p.natDegree - ↑n)) < f x

Tactic:
exact h_ge (‖p.coeff n‖₊ ^ (1 / (p.natDegree - n : ℝ))) h_rg

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

Goal state before:
pp: case pos
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 a)
ha : a = 0
⊢ limsup (fun n ↦ μ (x ^ s (ψ n)) ^ (1 / ↑(ψ n))) atTop ≤ smoothingFun μ x ^ a

Tactic:
rw [ha] at hψ_lim
    calc limsup (fun n : ℕ => μ (x ^ s (ψ n)) ^ (1 / (ψ n : ℝ))) atTop ≤
          limsup (fun n : ℕ => μ x ^ ((s (ψ n) : ℝ) * (1 / (ψ n : ℝ)))) atTop := by
          apply csInf_le_csInf _ (μ_nonempty μ hs_le ψ)
          · intro b hb
            simp only [eventually_map, eventually_atTop, Set.mem_ofPred_eq] at hb ⊢
            obtain ⟨m, hm⟩ := hb
            use m
            intro k hkm
            apply le_trans _ (hm k hkm)
            rw [rpow_mul (apply_nonneg μ x), rpow_natCast]
            gcongr
            exact map_pow_le_pow' hμ1 x _
          · use 0
            simp only [mem_lowerBounds, eventually_map, eventually_atTop,
              Set.mem_ofPred_eq, forall_exists_index]
            exact fun _ m hm ↦ le_trans (by positivity) (hm m (le_refl _))
      _ ≤ 1 := (μ_limsup_le_one μ hs_le hψ_lim)
      _ = smoothingFun μ x ^ a := by rw [ha, rpow_zero]

Goal state after: