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

Goal state before:
pp: X : Type u_1
inst✝¹ : MetricSpace X
inst✝ : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
reformulation : ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / 2 ^ k * ϕ x' ↔ 2 ^ k * ϕ x ≤ ϕ x'
this : Nonempty X
F : ℕ → X → X
hF : ∀ (k : ℕ) (x' : X), d x' x ≤ 2 * ε ∧ 2 ^ k * ϕ x ≤ ϕ x' → d x' (F k x') ≤ ε / 2 ^ k ∧ 2 * ϕ x' < ϕ (F k x')
u : ℕ → X := fun n ↦ Nat.recOn n x F
hu :
  ∀ (n : ℕ), d (u n) x ≤ 2 * ε ∧ 2 ^ n * ϕ x ≤ ϕ (u n) → d (u n) (u (n + 1)) ≤ ε / 2 ^ n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
key : ∀ (n : ℕ), d (u n) (u (n + 1)) ≤ ε / 2 ^ n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
key₁ : ∀ (x : ℕ), d (u x) (u (x + 1)) ≤ ε / 2 ^ x
key₂ : ∀ (x : ℕ), 2 * ϕ (u x) < ϕ (u (x + 1))
⊢ False

Tactic:
have cauchy_u : CauchySeq u := by
    refine cauchySeq_of_le_geometric _ ε one_half_lt_one fun n => ?_
    simpa only [one_div, inv_pow] using! key₁ n

Goal state after:
X : Type u_1
inst✝¹ : MetricSpace X
inst✝ : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
reformulation : ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / 2 ^ k * ϕ x' ↔ 2 ^ k * ϕ x ≤ ϕ x'
this : Nonempty X
F : ℕ → X → X
hF : ∀ (k : ℕ) (x' : X), d x' x ≤ 2 * ε ∧ 2 ^ k * ϕ x ≤ ϕ x' → d x' (F k x') ≤ ε / 2 ^ k ∧ 2 * ϕ x' < ϕ (F k x')
u : ℕ → X := fun n ↦ Nat.recOn n x F
hu :
  ∀ (n : ℕ), d (u n) x ≤ 2 * ε ∧ 2 ^ n * ϕ x ≤ ϕ (u n) → d (u n) (u (n + 1)) ≤ ε / 2 ^ n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
key : ∀ (n : ℕ), d (u n) (u (n + 1)) ≤ ε / 2 ^ n ∧ 2 * ϕ (u n) < ϕ (u (n + 1))
key₁ : ∀ (x : ℕ), d (u x) (u (x + 1)) ≤ ε / 2 ^ x
key₂ : ∀ (x : ℕ), 2 * ϕ (u x) < ϕ (u (x + 1))
cauchy_u : CauchySeq u
⊢ False
Module: Mathlib.Analysis.InnerProductSpace.JointEigenspace

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
A B : E →ₗ[𝕜] E
hA : A.IsSymmetric
hB : B.IsSymmetric
i j : 𝕜 × 𝕜
hij : i ≠ j
v : E
x✝ : v ∈ eigenspace A i.2 ⊓ eigenspace B i.1
hv1 : v ∈ ↑(eigenspace A i.2)
hv2 : v ∈ ↑(eigenspace B i.1)
⊢ v ∈ (eigenspace A j.2 ⊓ eigenspace B j.1)ᗮ

Tactic:
by
    obtain (h₁ | h₂) : i.1 ≠ j.1 ∨ i.2 ≠ j.2 := by rwa [Ne.eq_def, Prod.ext_iff, not_and_or] at hij
    all_goals intro w ⟨hw1, hw2⟩
    · exact hB.orthogonalFamily_eigenspaces.pairwise h₁ hv2 w hw2
    · exact hA.orthogonalFamily_eigenspaces.pairwise h₂ hv1 w hw1

Goal state after:
Module: Mathlib.Dynamics.BirkhoffSum.Average

Goal state before:
pp: R : Type u_1
α : Type u_2
M : Type u_3
inst✝² : DivisionSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
f : α → α
g : α → M
x : α
⊢ birkhoffAverage R f g 0 x = 0

Tactic:
by simp [birkhoffAverage]

Goal state after:
Module: Mathlib.Dynamics.BirkhoffSum.Average

Goal state before:
pp: R : Type u_1
α : Type u_2
M : Type u_3
inst✝² : DivisionSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
f : α → α
g : α → M
x : α
⊢ birkhoffAverage R f g 1 x = g x

Tactic:
by simp [birkhoffAverage]

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.SingularValues

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : FiniteDimensional 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
inst✝ : FiniteDimensional 𝕜 F
T : E →ₗ[𝕜] F
i : ℕ
⊢ 0 ≤ T.singularValues i

Tactic:
by
  rw [singularValues, Finsupp.embDomain_apply, Finsupp.ofSupportFinite_coe]
  split_ifs <;> positivity

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.SingularValues

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : FiniteDimensional 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
inst✝ : FiniteDimensional 𝕜 F
T : E →ₗ[𝕜] F
i j : ℕ
⊢ i ≤ j → T.singularValues j ≤ T.singularValues i

Tactic:
hij

Goal state after:
𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : FiniteDimensional 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
inst✝ : FiniteDimensional 𝕜 F
T : E →ₗ[𝕜] F
i j : ℕ
hij : i ≤ j
⊢ T.singularValues j ≤ T.singularValues i
Module: Mathlib.Analysis.FunctionalSpaces.SobolevInequality

Goal state before:
pp: E : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F' : Type u_5
inst✝¹ : NormedAddCommGroup F'
inst✝ : InnerProductSpace ℝ F'
u : E → F'
hu : ContDiff ℝ 1 u
h2u : HasCompactSupport u
p p' : ℝ≥0
hp✝ : 1 ≤ p
hp'0 : ¬p' = 0
n : ℕ := finrank ℝ E
hn✝ : 0 < n
hp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹
n' : ℝ≥0 := (↑n).conjExponent
h2p : ↑p < ↑n
h0n : 2 ≤ n
hn : (↑n).HolderConjugate n'
h1n : 1 ≤ ↑n
h2n : 0 < ↑n - 1
hnp : 0 < ↑n - ↑p
hp : 1 < p
q : ℝ := (↑p).conjExponent
hq : (↑p).HolderConjugate q
h0p : p ≠ 0
h1p : ↑p ≠ 1
h3p : ↑p - 1 ≠ 0
h2q : 1 / ↑n' - 1 / q = 1 / ↑p'
γ : ℝ≥0 := NNReal.mk (↑p * (↑n - 1) / (↑n - ↑p)) ⋯
h0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)
h1γ : 1 < ↑γ
h2γ : γ * n' = p'
h3γ : (↑γ - 1) * q = ↑p'
h4γ : ↑γ ≠ 0
h3u : ¬∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ = 0
h4u : ∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ ≠ ∞
h5u : (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / q) ≠ 0
h6u : (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / q) ≠ ∞
h7u : Continuous u
v : E → ℝ := fun x ↦ ‖u x‖ ^ ↑γ
hv : ContDiff ℝ 1 v
h2v : HasCompactSupport v
C : ℝ≥0 := eLpNormLESNormFDerivOneConst μ ↑n'
this :
  (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / ↑n') ≤
    ↑C * ↑γ * (∫⁻ (x : E), ‖fderiv ℝ u x‖ₑ ^ ↑p ∂μ) ^ (1 / ↑p) * (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / q)
⊢ (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / ↑p') ≤ ↑C * ↑γ * (∫⁻ (x : E), ‖fderiv ℝ u x‖ₑ ^ ↑p ∂μ) ^ (1 / ↑p)

Tactic:
by
      rwa [← h2q, ENNReal.rpow_sub _ _ h3u h4u, ENNReal.div_le_iff h5u h6u]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: X : Type u_2
inst✝ : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
t : Set X
ht : t ∈ {s | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s}
s : Set X
S : ℕ → Set X := fun n ↦ {x | x ∈ s ∧ (↑n)⁻¹ ≤ infEDist x t}
Ssep : ∀ (n : ℕ), AreSeparated (S n) t
Ssep' : ∀ (n : ℕ), AreSeparated (S n) (s ∩ t)
S_sub : ∀ (n : ℕ), S n ⊆ s \ t
hSs : ∀ (n : ℕ), μ (s ∩ t) + μ (S n) ≤ μ s
iUnion_S : ⋃ n, S n = s \ t
htop : ¬μ (s \ t) = ∞
this : μ (⋃ n, S n) ≤ ⨆ n, μ (S n)
⊢ μ (s ∩ t) + μ (s \ t) = μ (s ∩ t) + μ (⋃ n, S n)

Tactic:
by rw [iUnion_S]

Goal state after:
Module: Mathlib.Geometry.Euclidean.Volume.Measure

Goal state before:
pp: 𝕜 : Type u_3
V : Type u_4
P : Type u_5
inst✝⁶ : NormedField 𝕜
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedSpace 𝕜 V
inst✝³ : MeasurableSpace P
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
inst✝ : BorelSpace P
d : ℕ
x : P
c : 𝕜
hc : c ≠ 0
s : Set P
⊢ μHE[d] (⇑(AffineMap.homothety x c) ⁻¹' s) = ‖c‖₊⁻¹ ^ d • μHE[d] s

Tactic:
simp_rw [euclideanHausdorffMeasure_def, Measure.smul_apply]
  rw [hausdorffMeasure_homothety_preimage (by simp) x hc, smul_comm, NNReal.rpow_natCast]

Goal state after:
Module: Mathlib.Topology.MetricSpace.HausdorffDimension

Goal state before:
pp: X : Type u_2
Y : Type u_3
inst✝² : EMetricSpace X
inst✝¹ : EMetricSpace Y
inst✝ : SecondCountableTopology X
r : ℝ≥0
f : X → Y
hr : 0 < r
hf : ∀ (x : X), ∃ C, ∃ s ∈ 𝓝 x, HolderOnWith C r f s
⊢ dimH (range f) ≤ dimH univ / ↑r

Tactic:
rw [← image_univ]

Goal state after:
X : Type u_2
Y : Type u_3
inst✝² : EMetricSpace X
inst✝¹ : EMetricSpace Y
inst✝ : SecondCountableTopology X
r : ℝ≥0
f : X → Y
hr : 0 < r
hf : ∀ (x : X), ∃ C, ∃ s ∈ 𝓝 x, HolderOnWith C r f s
⊢ dimH (f '' univ) ≤ dimH univ / ↑r
Module: Mathlib.Topology.MetricSpace.HausdorffDimension

Goal state before:
pp: X : Type u_2
Y : Type u_3
inst✝² : EMetricSpace X
inst✝¹ : EMetricSpace Y
inst✝ : SecondCountableTopology X
f : X → Y
hf : ∀ (x : X), ∃ C, ∃ s ∈ 𝓝 x, LipschitzOnWith C f s
⊢ dimH (range f) ≤ dimH univ

Tactic:
rw [← image_univ]

Goal state after:
X : Type u_2
Y : Type u_3
inst✝² : EMetricSpace X
inst✝¹ : EMetricSpace Y
inst✝ : SecondCountableTopology X
f : X → Y
hf : ∀ (x : X), ∃ C, ∃ s ∈ 𝓝 x, LipschitzOnWith C f s
⊢ dimH (f '' univ) ≤ dimH univ
Module: Mathlib.Analysis.InnerProductSpace.NormDet

Goal state before:
pp: 𝕜 : Type u_1
U : Type u_2
V : Type u_3
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup U
inst✝³ : InnerProductSpace 𝕜 U
inst✝² : FiniteDimensional 𝕜 U
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace 𝕜 V
f : U →ₗ[𝕜] V
c : 𝕜
hc : c = 0
a✝ : Nontrivial U
⊢ (c • f).normDet = ‖c‖ ^ finrank 𝕜 U * f.normDet

Tactic:
simp [hc, zero_pow finrank_pos.ne.symm]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
d₁ d₂ : ℝ
h : d₁ ≤ d₂
s : Set X
⊢ μH[d₂] s ≤ μH[d₁] s

Tactic:
rcases h.eq_or_lt with (rfl | h)

Goal state after:
case inl
X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
d₁ : ℝ
s : Set X
h : d₁ ≤ d₁
⊢ μH[d₁] s ≤ μH[d₁] s

case inr
X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
d₁ d₂ : ℝ
h✝ : d₁ ≤ d₂
s : Set X
h : d₁ < d₂
⊢ μH[d₂] s ≤ μH[d₁] s
Module: Mathlib.Analysis.InnerProductSpace.Reproducing

Goal state before:
pp: case refine_1
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
X : Type u_2
V : Type u_3
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace 𝕜 V
inst✝ : CompleteSpace V
K : Matrix X X (V →L[𝕜] V)
this : ∀ {h p1 p2 p3 : Prop}, (h → [p1, p2, p3].TFAE) → [h ∧ p1, h ∧ p2, h ∧ p3].TFAE
hHerm : K.IsHermitian
h : Subsingleton V
⊢ [∀ (x : X →₀ V →L[𝕜] V) (x_1 : V), 0 ≤ RCLike.re (x.sum fun i a ↦ x.sum fun i_1 a_1 ↦ ⟪(K i i_1) (a_1 x_1), a x_1⟫_𝕜),
      ∀ (f : X × V →₀ 𝕜),
        0 ≤ RCLike.re (f.sum fun xv z ↦ f.sum fun xv' w ↦ (starRingEnd 𝕜) z * w * ⟪(K xv'.1 xv.1) xv.2, xv'.2⟫_𝕜),
      ∀ (vv : X →₀ V), 0 ≤ RCLike.re (vv.sum fun x w ↦ vv.sum fun x' w' ↦ ⟪(K x' x) w, w'⟫_𝕜)].TFAE

Tactic:
have : ∀ v : V, v = 0 := fun v ↦ Subsingleton.elim v 0

Goal state after:
case refine_1
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
X : Type u_2
V : Type u_3
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace 𝕜 V
inst✝ : CompleteSpace V
K : Matrix X X (V →L[𝕜] V)
this✝ : ∀ {h p1 p2 p3 : Prop}, (h → [p1, p2, p3].TFAE) → [h ∧ p1, h ∧ p2, h ∧ p3].TFAE
hHerm : K.IsHermitian
h : Subsingleton V
this : ∀ (v : V), v = 0
⊢ [∀ (x : X →₀ V →L[𝕜] V) (x_1 : V), 0 ≤ RCLike.re (x.sum fun i a ↦ x.sum fun i_1 a_1 ↦ ⟪(K i i_1) (a_1 x_1), a x_1⟫_𝕜),
      ∀ (f : X × V →₀ 𝕜),
        0 ≤ RCLike.re (f.sum fun xv z ↦ f.sum fun xv' w ↦ (starRingEnd 𝕜) z * w * ⟪(K xv'.1 xv.1) xv.2, xv'.2⟫_𝕜),
      ∀ (vv : X →₀ V), 0 ≤ RCLike.re (vv.sum fun x w ↦ vv.sum fun x' w' ↦ ⟪(K x' x) w, w'⟫_𝕜)].TFAE
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: case inr
X : Type u_2
Y : Type u_3
inst✝⁵ : EMetricSpace X
inst✝⁴ : EMetricSpace Y
inst✝³ : MeasurableSpace X
inst✝² : BorelSpace X
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
C r : ℝ≥0
f : X → Y
s : Set X
h : HolderOnWith C r f s
hr : 0 < r
d : ℝ
hd : 0 ≤ d
hC0 : 0 < C
⊢ μH[d] (f '' s) ≤ ↑C ^ d * μH[↑r * d] s

Tactic:
have hCd0 : (C : ℝ≥0∞) ^ d ≠ 0 := by simp [hC0.ne']
    have hCd : (C : ℝ≥0∞) ^ d ≠ ∞ := by simp [hd]
    simp only [hausdorffMeasure_apply, ENNReal.mul_iSup, ENNReal.mul_iInf_of_ne hCd0 hCd,
      ← ENNReal.tsum_mul_left]
    refine iSup_le fun R => iSup_le fun hR => ?_
    have : Tendsto (fun d : ℝ≥0∞ => (C : ℝ≥0∞) * d ^ (r : ℝ)) (𝓝 0) (𝓝 0) :=
      ENNReal.tendsto_const_mul_rpow_nhds_zero_of_pos ENNReal.coe_ne_top hr
    rcases ENNReal.nhds_zero_basis_Iic.eventually_iff.1 (this.eventually (gt_mem_nhds hR)) with
      ⟨δ, δ0, H⟩
    refine le_iSup₂_of_le δ δ0 <| iInf₂_mono' fun t hst ↦
      ⟨fun n => f '' (t n ∩ s), ?_, iInf_mono' fun htδ ↦
        ⟨fun n => (h.ediam_image_inter_le (t n)).trans (H (htδ n)).le, ?_⟩⟩
    · grw [← image_iUnion, ← iUnion_inter, ← hst, inter_self]
    · refine ENNReal.tsum_le_tsum fun n => ?_
      simp only [iSup_le_iff, image_nonempty]
      intro hft
      simp only [Nonempty.mono ((t n).inter_subset_left) hft, ciSup_pos]
      rw [ENNReal.rpow_mul, ← ENNReal.mul_rpow_of_nonneg _ _ hd]
      gcongr
      exact h.ediam_image_inter_le _

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: case hab
ι : Type u_4
inst✝ : Fintype ι
a b : ι → ℚ
H : ∀ (i : ι), a i < b i
I : ∀ (i : ι), 0 ≤ ↑(b i) - ↑(a i)
γ : ℕ → Type u_4 := ⋯
t : (n : ℕ) → γ n → Set (ι → ℝ) := ⋯
A : Tendsto (fun n ↦ 1 / ↑n) atTop (𝓝 0)
B : ∀ᶠ (n : ℕ) in atTop, ∀ (i : γ n), ediam (t n i) ≤ 1 / ↑n
n : ℕ
hn : 1 ≤ n
npos : 0 < ↑n
x : ι → ℝ
hx : ∀ (i : ι), ↑(a i) < x i ∧ x i < ↑(b i)
f : γ n := ⋯
i : ι
⊢ (x i - ↑(a i)) * ↑n ≤ ↑⌊(x i - ↑(a i)) * ↑n⌋₊ + 1

Tactic:
exact (Nat.lt_floor_add_one _).le

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.WeakOperatorTopology

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : TopologicalSpace E
inst✝³ : Module 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
inst✝ : CompleteSpace F
⊢ Continuous[instTopologicalSpace, instTopologicalSpace] star

Tactic:
simp_rw [continuous_iff, star_apply, ContinuousLinearMap.star_eq_adjoint,
      ContinuousLinearMap.adjoint_inner_right, coe_toCLM]
    rw [forall_comm]
    conv in ⟪_, _⟫_𝕜 => rw [← inner_conj_symm, ← RCLike.star_def]
    fun_prop

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.ContinuousOfBounded

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
E : Type u_3
F : Type u_4
inst✝¹¹ : AddCommGroup E
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : IsTopologicalAddGroup E
inst✝⁸ : AddCommGroup F
inst✝⁷ : TopologicalSpace F
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : Module 𝕜 E
inst✝⁴ : ContinuousSMul 𝕜 E
inst✝³ : NormedField 𝕜'
inst✝² : Module 𝕜' F
σ : 𝕜 →+* 𝕜'
inst✝¹ : RingHomIsometric σ
inst✝ : IsTopologicalAddGroup F
f : E →ₛₗ[σ] F
h : ∃ V ∈ 𝓝 0, IsVonNBounded 𝕜' (⇑f '' V)
⊢ ∀ A ∈ 𝓝 0, ⇑f ⁻¹' A ∈ 𝓝 0

Tactic:
intro U hU

Goal state after:
𝕜 : Type u_1
𝕜' : Type u_2
E : Type u_3
F : Type u_4
inst✝¹¹ : AddCommGroup E
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : IsTopologicalAddGroup E
inst✝⁸ : AddCommGroup F
inst✝⁷ : TopologicalSpace F
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : Module 𝕜 E
inst✝⁴ : ContinuousSMul 𝕜 E
inst✝³ : NormedField 𝕜'
inst✝² : Module 𝕜' F
σ : 𝕜 →+* 𝕜'
inst✝¹ : RingHomIsometric σ
inst✝ : IsTopologicalAddGroup F
f : E →ₛₗ[σ] F
h : ∃ V ∈ 𝓝 0, IsVonNBounded 𝕜' (⇑f '' V)
U : Set F
hU : U ∈ 𝓝 0
⊢ ⇑f ⁻¹' U ∈ 𝓝 0
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: ι : Type u_1
X : Type u_2
Y : Type u_3
inst✝¹⁰ : EMetricSpace X
inst✝⁹ : EMetricSpace Y
inst✝⁸ : MeasurableSpace X
inst✝⁷ : BorelSpace X
inst✝⁶ : MeasurableSpace Y
inst✝⁵ : BorelSpace Y
E : Type u_4
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
U : Set E
hU : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
hU' : U.Nonempty
e : E ≃L[ℝ] Fin (finrank ℝ E) → ℝ := ContinuousLinearEquiv.ofFinrankEq ⋯
⊢ μH[↑(finrank ℝ E)] U ≠ 0

Tactic:
suffices 0 < μH[finrank ℝ E] (e '' U) from
      (ENNReal.mul_pos_iff.mp (lt_of_lt_of_le this <|
        e.lipschitz.hausdorffMeasure_image_le (by simp) _)).2.ne'

Goal state after:
ι : Type u_1
X : Type u_2
Y : Type u_3
inst✝¹⁰ : EMetricSpace X
inst✝⁹ : EMetricSpace Y
inst✝⁸ : MeasurableSpace X
inst✝⁷ : BorelSpace X
inst✝⁶ : MeasurableSpace Y
inst✝⁵ : BorelSpace Y
E : Type u_4
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
U : Set E
hU : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
hU' : U.Nonempty
e : E ≃L[ℝ] Fin (finrank ℝ E) → ℝ := ContinuousLinearEquiv.ofFinrankEq ⋯
⊢ 0 < μH[↑(finrank ℝ E)] (⇑e '' U)
Module: Mathlib.Analysis.InnerProductSpace.TwoDim

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x y : E
h : ⟪x, y⟫ = 0
i j : Fin 2
⊢ i ≠ j → (fun i j ↦ ⟪![x, y] i, ![x, y] j⟫ = 0) i j

Tactic:
hij

Goal state after:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x y : E
h : ⟪x, y⟫ = 0
i j : Fin 2
hij : i ≠ j
⊢ ⟪![x, y] i, ![x, y] j⟫ = 0
Module: Mathlib.Analysis.InnerProductSpace.TwoDim

Goal state before:
pp: case refine_2
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x : E
⊢ ‖x‖ ≤ ‖o.rightAngleRotationAux₁ x‖

Tactic:
let K : Submodule ℝ E := ℝ ∙ x

Goal state after:
case refine_2
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x : E
K : Submodule ℝ E := ℝ ∙ x
⊢ ‖x‖ ≤ ‖o.rightAngleRotationAux₁ x‖
Module: Mathlib.Analysis.InnerProductSpace.TwoDim

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x : E
hx : x ≠ 0
i j : Fin 2
⊢ i ≠ j → (fun i j ↦ ⟪![x, o.rightAngleRotation x] i, ![x, o.rightAngleRotation x] j⟫ = 0) i j

Tactic:
hij

Goal state after:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x : E
hx : x ≠ 0
i j : Fin 2
hij : i ≠ j
⊢ ⟪![x, o.rightAngleRotation x] i, ![x, o.rightAngleRotation x] j⟫ = 0
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
ha : 0 < a
this :
  IntegrableOn (fun x ↦ ↑(a * x) ^ (s - 1) • f (a * x)) (Ioi 0) volume ↔
    IntegrableOn (fun t ↦ ↑t ^ (s - 1) • f t) (Ioi 0) volume
⊢ MellinConvergent (fun t ↦ f (a * t)) s ↔ MellinConvergent f s

Tactic:
have h1 : EqOn (fun t : ℝ => (↑(a * t) : ℂ) ^ (s - 1) • f (a * t))
      ((a : ℂ) ^ (s - 1) • fun t : ℝ => (t : ℂ) ^ (s - 1) • f (a * t)) (Ioi 0) := fun t ht ↦ by
    simp only [ofReal_mul, mul_cpow_ofReal_nonneg ha.le (le_of_lt ht), mul_smul, Pi.smul_apply]

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
ha : 0 < a
this :
  IntegrableOn (fun x ↦ ↑(a * x) ^ (s - 1) • f (a * x)) (Ioi 0) volume ↔
    IntegrableOn (fun t ↦ ↑t ^ (s - 1) • f t) (Ioi 0) volume
h1 : EqOn (fun t ↦ ↑(a * t) ^ (s - 1) • f (a * t)) (↑a ^ (s - 1) • fun t ↦ ↑t ^ (s - 1) • f (a * t)) (Ioi 0)
⊢ MellinConvergent (fun t ↦ f (a * t)) s ↔ MellinConvergent f s
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Deriv

Goal state before:
pp: s✝ : ℂ
hs✝ : ∀ (m : ℕ), s✝ ≠ -↑m
n : ℕ
IH : ∀ (s : ℂ), -↑n < s.re → (∀ (m : ℕ), s ≠ -↑m) → DifferentiableAt ℂ Gamma s
s : ℂ
hsre : -↑(n + 1) < s.re
hs : ∀ (m : ℕ), s ≠ -↑m
hsne : s ≠ 0
m : ℕ
⊢ s + 1 ≠ -↑m

Tactic:
grind [hs (m + 1)]

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

Goal state before:
pp: n : ℕ
ih : ContinuousAt Gamma (-(↑n + 1))
h0 : -(↑n + 1) ≠ 0
⊢ ContinuousAt (fun s ↦ Gamma (s + 1)) (-(↑n + 1))

Tactic:
exact ((continuousAt_id.mul ih).continuousWithinAt.congr Gamma_add_one
      (Gamma_add_one (-(n + 1)) h0)).continuousAt (compl_singleton_mem_nhds h0)

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

Goal state before:
pp: case c0
z : ℂ
hz : ‖z‖ ≤ 1 / 2
hz3 : (1 - ‖z‖)⁻¹ ≤ 2
⊢ 0 ≤ (1 - ‖z‖)⁻¹

Tactic:
rw [inv_nonneg]

Goal state after:
case c0
z : ℂ
hz : ‖z‖ ≤ 1 / 2
hz3 : (1 - ‖z‖)⁻¹ ≤ 2
⊢ 0 ≤ 1 - ‖z‖
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)
⊢ (fun x ↦ ↑x * (log (1 + g x) - g x)) =O[atTop] fun x ↦ ↑x * g x ^ 2

Tactic:
by
      have hg0 := tendsto_zero_of_isBoundedUnder_smul_of_tendsto_cobounded hg.norm.isBoundedUnder_le
        (RCLike.tendsto_ofReal_atTop_cobounded ℂ)
      exact (isBigO_refl _ _).mul (log_sub_self_isBigO.comp_tendsto hg0)

Goal state after:
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: case hbc.inr
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 : t < 1
⊢ s.re - v - 1 ≤ (z - 1).re

Tactic:
rw [sub_re, one_re, sub_le_iff_le_add, sub_add_cancel]

Goal state after:
case hbc.inr
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 : t < 1
⊢ s.re - v ≤ z.re
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
⊢ f ↑n = f 1 + log ↑(n - 1)!

Tactic:
refine Nat.le_induction (by simp) (fun m hm IH => ?_) n (Nat.one_le_iff_ne_zero.2 hn)
  have A : 0 < (m : ℝ) := Nat.cast_pos.2 hm
  simp only [hf_feq A, Nat.cast_add, Nat.cast_one, Nat.add_succ_sub_one, add_zero]
  rw [IH, add_assoc, ← log_mul (Nat.cast_ne_zero.mpr (Nat.factorial_ne_zero _)) A.ne', ←
    Nat.cast_mul]
  conv_rhs => rw [← Nat.succ_pred_eq_of_pos hm, Nat.factorial_succ, mul_comm]
  congr
  exact (Nat.succ_pred_eq_of_pos hm).symm

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

Goal state before:
pp: case succ
f : ℝ → ℝ
x : ℝ
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hx : 0 < x
n : ℕ
hn : f (x + ↑n) = f x + ∑ m ∈ Finset.range n, log (x + ↑m)
this : x + ↑n.succ = x + ↑n + 1
⊢ f (x + ↑n + 1) = f x + ∑ m ∈ Finset.range (n + 1), log (x + ↑m)

Tactic:
hf_feq,

Goal state after:
case succ
f : ℝ → ℝ
x : ℝ
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hx : 0 < x
n : ℕ
hn : f (x + ↑n) = f x + ∑ m ∈ Finset.range n, log (x + ↑m)
this : x + ↑n.succ = x + ↑n + 1
⊢ f (x + ↑n) + log (x + ↑n) = f x + ∑ m ∈ Finset.range (n + 1), log (x + ↑m)

case succ
f : ℝ → ℝ
x : ℝ
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hx : 0 < x
n : ℕ
hn : f (x + ↑n) = f x + ∑ m ∈ Finset.range n, log (x + ↑m)
this : x + ↑n.succ = x + ↑n + 1
⊢ 0 < x + ↑n