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

Goal state before:
pp: V : Type u_2
P : Type u_3
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b c : P
t₁ t₂ : ℝ
h_inner : ⟪b -ᵥ a, c -ᵥ a⟫ = 0
hvec : (AffineMap.lineMap a b) t₁ -ᵥ (AffineMap.lineMap a c) t₂ = t₁ • (b -ᵥ a) - t₂ • (c -ᵥ a)
⊢ ‖t₁ • (b -ᵥ a)‖ ^ 2 - 2 * ⟪t₁ • (b -ᵥ a), t₂ • (c -ᵥ a)⟫ + ‖t₂ • (c -ᵥ a)‖ ^ 2 =
    t₁ ^ 2 * dist a b ^ 2 + t₂ ^ 2 * dist a c ^ 2

Tactic:
norm_smul,

Goal state after:
V : Type u_2
P : Type u_3
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b c : P
t₁ t₂ : ℝ
h_inner : ⟪b -ᵥ a, c -ᵥ a⟫ = 0
hvec : (AffineMap.lineMap a b) t₁ -ᵥ (AffineMap.lineMap a c) t₂ = t₁ • (b -ᵥ a) - t₂ • (c -ᵥ a)
⊢ (‖t₁‖ * ‖b -ᵥ a‖) ^ 2 - 2 * ⟪t₁ • (b -ᵥ a), t₂ • (c -ᵥ a)⟫ + ‖t₂ • (c -ᵥ a)‖ ^ 2 =
    t₁ ^ 2 * dist a b ^ 2 + t₂ ^ 2 * dist a c ^ 2
Module: Mathlib.Analysis.InnerProductSpace.Affine

Goal state before:
pp: V : Type u_2
P : Type u_3
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b c : P
t₁ t₂ : ℝ
h_inner : ⟪b -ᵥ a, c -ᵥ a⟫ = 0
hvec : (AffineMap.lineMap a b) t₁ -ᵥ (AffineMap.lineMap a c) t₂ = t₁ • (b -ᵥ a) - t₂ • (c -ᵥ a)
⊢ (‖t₁‖ * ‖b -ᵥ a‖) ^ 2 - 2 * ⟪t₁ • (b -ᵥ a), t₂ • (c -ᵥ a)⟫ + ‖t₂ • (c -ᵥ a)‖ ^ 2 =
    t₁ ^ 2 * dist a b ^ 2 + t₂ ^ 2 * dist a c ^ 2

Tactic:
norm_smul,

Goal state after:
V : Type u_2
P : Type u_3
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
a b c : P
t₁ t₂ : ℝ
h_inner : ⟪b -ᵥ a, c -ᵥ a⟫ = 0
hvec : (AffineMap.lineMap a b) t₁ -ᵥ (AffineMap.lineMap a c) t₂ = t₁ • (b -ᵥ a) - t₂ • (c -ᵥ a)
⊢ (‖t₁‖ * ‖b -ᵥ a‖) ^ 2 - 2 * ⟪t₁ • (b -ᵥ a), t₂ • (c -ᵥ a)⟫ + (‖t₂‖ * ‖c -ᵥ a‖) ^ 2 =
    t₁ ^ 2 * dist a b ^ 2 + t₂ ^ 2 * dist a c ^ 2
Module: Mathlib.Analysis.InnerProductSpace.Coalgebra

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : FiniteDimensional 𝕜 E
inst✝ : Coalgebra 𝕜 E
x : E
⊢ 1 * x = x

Tactic:
by
    dsimp [OfNat.ofNat]
    rw [← rTensor_tmul, ← comp_apply, ← adjoint_rTensor, ← adjoint_comp, rTensor_counit_comp_comul,
      ← toLinearMap_symm_lid, ← toLinearEquiv_lidIsometry, ← toLinearEquiv_symm,
      adjoint_toLinearMap_eq_symm]
    exact one_smul _ _

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : InnerProductSpace 𝕜 F
T : E →ₗ[𝕜] E
f : E ≃ₗᵢ[𝕜] F
⊢ (↑f.toLinearEquiv ∘ₗ T ∘ₗ ↑f.symm.toLinearEquiv).IsPositive ↔ T.IsPositive

Tactic:
by
  simp_rw [IsPositive, isSymmetric_linearIsometryEquiv_conj_iff, and_congr_right_iff,
    LinearIsometryEquiv.toLinearEquiv_symm, coe_comp, LinearEquiv.coe_coe,
    LinearIsometryEquiv.coe_toLinearEquiv, LinearIsometryEquiv.coe_symm_toLinearEquiv,
    Function.comp_apply, LinearIsometryEquiv.inner_map_eq_flip]
  exact fun _ => ⟨fun h x => by simpa using h (f x), fun h x => h _⟩

Goal state after:
Module: Mathlib.Analysis.FunctionalSpaces.SobolevInequality

Goal state before:
pp: ι : Type u_1
A : ι → Type u_2
inst✝³ : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝² : DecidableEq ι
inst✝¹ : Fintype ι
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
p : ℝ
hp : (↑#ι).HolderConjugate p
f : ((a : ι) → A a) → ℝ≥0∞
hf : Measurable f
this : Nontrivial ι
h0 : 1 < ↑#ι
h1 : 0 < ↑#ι - 1
h2 : 0 ≤ 1 / (↑#ι - 1)
h3 : (↑#ι - 1) * (1 / (↑#ι - 1)) ≤ 1
⊢ ∫⁻ (x : (i : ι) → A i), ∏ i, (∫⁻ (xᵢ : A i), f (update x i xᵢ) ∂μ i) ^ (1 / (↑#ι - 1)) ∂Measure.pi μ ≤
    (∫⁻ (x : (i : ι) → A i), f x ∂Measure.pi μ) ^ p

Tactic:
have h4 : p = 1 + 1 / (↑#ι - 1) := by simp [field]; rw [mul_comm, hp.sub_one_mul_conj]

Goal state after:
ι : Type u_1
A : ι → Type u_2
inst✝³ : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝² : DecidableEq ι
inst✝¹ : Fintype ι
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
p : ℝ
hp : (↑#ι).HolderConjugate p
f : ((a : ι) → A a) → ℝ≥0∞
hf : Measurable f
this : Nontrivial ι
h0 : 1 < ↑#ι
h1 : 0 < ↑#ι - 1
h2 : 0 ≤ 1 / (↑#ι - 1)
h3 : (↑#ι - 1) * (1 / (↑#ι - 1)) ≤ 1
h4 : p = 1 + 1 / (↑#ι - 1)
⊢ ∫⁻ (x : (i : ι) → A i), ∏ i, (∫⁻ (xᵢ : A i), f (update x i xᵢ) ∂μ i) ^ (1 / (↑#ι - 1)) ∂Measure.pi μ ≤
    (∫⁻ (x : (i : ι) → A i), f x ∂Measure.pi μ) ^ p
Module: Mathlib.Analysis.InnerProductSpace.ExteriorPower

Goal state before:
pp: n : ℕ
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
x : ↥(⋀[ℝ]^n E)
ι : Type u_2
inst✝¹ : Fintype ι
inst✝ : LinearOrder ι
b : OrthonormalBasis ι ℝ E
⊢ (innerProductForm x) x = ∑ s, ((Module.Basis.exteriorPower n b.toBasis).repr x) s ^ 2

Tactic:
simp_rw [innerProductForm_eq_sum b, pow_two]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : RCLike 𝕜
H : Type u_4
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace 𝕜 H
f : H →L[𝕜] H
c : ℝ≥0
hc : 0 < c
h : ∀ (x : H), ‖x‖ ^ 2 * ↑c ≤ ‖⟪f x, x⟫‖
x : H
⊢ ‖x‖ ≤ ‖f x‖ / ↑c

Tactic:
le_div_iff₀ (by exact_mod_cast hc)

Goal state after:
𝕜 : Type u_1
inst✝² : RCLike 𝕜
H : Type u_4
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace 𝕜 H
f : H →L[𝕜] H
c : ℝ≥0
hc : 0 < c
h : ∀ (x : H), ‖x‖ ^ 2 * ↑c ≤ ‖⟪f x, x⟫‖
x : H
⊢ ‖x‖ * ↑c ≤ ‖f x‖
Module: Mathlib.Analysis.InnerProductSpace.Positive

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
T : E →L[𝕜] E
hT : T.IsPositive
a : Fin (Module.finrank 𝕜 E) → E := fun i ↦ ↑√(⋯.eigenvalues ⋯ i) • (⋯.eigenvectorBasis ⋯) i
x✝ : E
⊢ T x✝ = ↑T (∑ x, ((⋯.eigenvectorBasis ⋯).repr x✝).ofLp x • (⋯.eigenvectorBasis ⋯) x)

Tactic:
OrthonormalBasis.sum_repr,

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
T : E →L[𝕜] E
hT : T.IsPositive
a : Fin (Module.finrank 𝕜 E) → E := fun i ↦ ↑√(⋯.eigenvalues ⋯ i) • (⋯.eigenvectorBasis ⋯) i
x✝ : E
⊢ T x✝ = ↑T x✝
Module: Mathlib.Analysis.FunctionalSpaces.SobolevInequality

Goal state before:
pp: case neg
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
⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ

Tactic:
have h2p : (p : ℝ) < n := by
    have : 0 < p⁻¹ - (n : ℝ)⁻¹ :=
      NNReal.coe_lt_coe.mpr (pos_iff_ne_zero.mpr (inv_ne_zero hp'0)) |>.trans_eq hp'
    rwa [NNReal.coe_inv, sub_pos,
      inv_lt_inv₀ _ (zero_lt_one.trans_le (NNReal.coe_le_coe.mpr hp))] at this
    exact_mod_cast hn

Goal state after:
case neg
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
⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ
Module: Mathlib.Analysis.FunctionalSpaces.SobolevInequality

Goal state before:
pp: F : Type u_3
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
E : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
μ : Measure E
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : FiniteDimensional ℝ F
u : E → F
s : Set E
hu : ContDiff ℝ 1 u
h2u : support u ⊆ s
p q : ℝ≥0
hp : 1 ≤ p
h2p : p < ↑(finrank ℝ E)
hpq : ↑p⁻¹ - (↑(finrank ℝ E))⁻¹ ≤ (↑q)⁻¹
hs : Bornology.IsBounded s
hq0 : ¬q = 0
p' : ℝ≥0 := (p⁻¹ - (↑(finrank ℝ E))⁻¹)⁻¹
⊢ ↑(p⁻¹ - (↑(finrank ℝ E))⁻¹) = ↑p⁻¹ - (↑(finrank ℝ E))⁻¹

Tactic:
NNReal.coe_sub

Goal state after:
F : Type u_3
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
E : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
μ : Measure E
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : FiniteDimensional ℝ F
u : E → F
s : Set E
hu : ContDiff ℝ 1 u
h2u : support u ⊆ s
p q : ℝ≥0
hp : 1 ≤ p
h2p : p < ↑(finrank ℝ E)
hpq : ↑p⁻¹ - (↑(finrank ℝ E))⁻¹ ≤ (↑q)⁻¹
hs : Bornology.IsBounded s
hq0 : ¬q = 0
p' : ℝ≥0 := (p⁻¹ - (↑(finrank ℝ E))⁻¹)⁻¹
⊢ ↑p⁻¹ - ↑(↑(finrank ℝ E))⁻¹ = ↑p⁻¹ - (↑(finrank ℝ E))⁻¹

F : Type u_3
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
E : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
μ : Measure E
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : FiniteDimensional ℝ F
u : E → F
s : Set E
hu : ContDiff ℝ 1 u
h2u : support u ⊆ s
p q : ℝ≥0
hp : 1 ≤ p
h2p : p < ↑(finrank ℝ E)
hpq : ↑p⁻¹ - (↑(finrank ℝ E))⁻¹ ≤ (↑q)⁻¹
hs : Bornology.IsBounded s
hq0 : ¬q = 0
p' : ℝ≥0 := (p⁻¹ - (↑(finrank ℝ E))⁻¹)⁻¹
⊢ (↑(finrank ℝ E))⁻¹ ≤ p⁻¹
Module: Mathlib.Topology.MetricSpace.HausdorffDimension

Goal state before:
pp: case refine_2
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
x : E
s : Set E
h : s ∈ 𝓝 x
e : E ≃L[ℝ] Fin (finrank ℝ E) → ℝ
this : ⇑e '' s ∈ 𝓝 (e x)
⊢ ↑(finrank ℝ E) ≤ dimH (⇑e '' s)

Tactic:
rcases Metric.nhds_basis_ball.mem_iff.1 this with ⟨r, hr0, hr⟩

Goal state after:
case refine_2
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
x : E
s : Set E
h : s ∈ 𝓝 x
e : E ≃L[ℝ] Fin (finrank ℝ E) → ℝ
this : ⇑e '' s ∈ 𝓝 (e x)
r : ℝ
hr0 : 0 < r
hr : Metric.ball (e x) r ⊆ ⇑e '' s
⊢ ↑(finrank ℝ E) ≤ dimH (⇑e '' s)
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
m₁ m₂ : ℝ≥0∞ → ℝ≥0∞
hle : m₁ ≤ᶠ[𝓝 0] m₂
⊢ mkMetric m₁ ≤ mkMetric m₂

Tactic:
convert! @mkMetric_mono_smul X _ _ _ _ m₂ _ ENNReal.one_ne_top one_ne_zero _

Goal state after:
case e'_4
X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
m₁ m₂ : ℝ≥0∞ → ℝ≥0∞
hle : m₁ ≤ᶠ[𝓝 0] m₂
⊢ mkMetric m₂ = 1 • mkMetric m₂

case convert_2
X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
m₁ m₂ : ℝ≥0∞ → ℝ≥0∞
hle : m₁ ≤ᶠ[𝓝 0] m₂
⊢ m₁ ≤ᶠ[𝓝[≥] 0] 1 • m₂
Module: Mathlib.Analysis.InnerProductSpace.OfNorm

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x y : E
hI : ¬I = 0
hI' : I * I = -1
v : E
⊢ ‖I • v‖ = ‖v‖

Tactic:
norm_smul,

Goal state after:
𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x y : E
hI : ¬I = 0
hI' : I * I = -1
v : E
⊢ ‖I‖ * ‖v‖ = ‖v‖
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: E : Type u_5
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
v : E
s : Set ℝ
hv : v ≠ 0
hn : ‖v‖ ≠ 0
x : ℝ
⊢ ‖LinearMap.id x‖ * ‖‖v‖⁻¹ • v‖ = ‖x‖

Tactic:
norm_smul,

Goal state after:
E : Type u_5
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
v : E
s : Set ℝ
hv : v ≠ 0
hn : ‖v‖ ≠ 0
x : ℝ
⊢ ‖LinearMap.id x‖ * (‖‖v‖⁻¹‖ * ‖v‖) = ‖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 y : E
⊢ x = 0 ∨ y = 0 → (o.kahler x) y = 0

Tactic:
rintro (rfl | rfl) <;> simp

Goal state after:
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
⊢ x = 0 ∨ y = 0 → ↑⟪x, y⟫ + ↑((o.areaForm x) y) * Complex.I = 0

Tactic:
rintro (rfl | rfl) <;> simp

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

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
hF : Fact (finrank ℝ F = 2)
φ : E ≃ₗᵢ[ℝ] F
x y : F
⊢ (((map (Fin 2) φ.toLinearEquiv) o).kahler x) y = (o.kahler (φ.symm x)) (φ.symm y)

Tactic:
simp [kahler_apply_apply, areaForm_map]

Goal state after:
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
h1 : EqOn (fun t ↦ ↑(a * t) ^ (s - 1) • f (a * t)) (↑a ^ (s - 1) • fun t ↦ ↑t ^ (s - 1) • f (a * t)) (Ioi 0)
h2 : ↑a ^ (s - 1) ≠ 0
⊢ Integrable (fun t ↦ ↑t ^ (s - 1) • f (a * t)) (volume.restrict (Ioi 0)) ↔
    Integrable (↑a ^ (s - 1) • fun t ↦ ↑t ^ (s - 1) • f (a * t)) (volume.restrict (Ioi 0))

Tactic:
integrable_smul_iff h2

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)
h2 : ↑a ^ (s - 1) ≠ 0
⊢ Integrable (fun t ↦ ↑t ^ (s - 1) • f (a * t)) (volume.restrict (Ioi 0)) ↔
    Integrable (fun t ↦ ↑t ^ (s - 1) • f (a * t)) (volume.restrict (Ioi 0))
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
T : Set ℝ
hT : T ⊆ Ioi 0
hT' : MeasurableSet T
hfc : AEStronglyMeasurable f (volume.restrict (Ioi 0))
s : ℂ
this : AEStronglyMeasurable (fun t ↦ ↑t ^ (s - 1) • f t) (volume.restrict T)
t : ℝ
ht : t ∈ T
⊢ ‖↑t ^ (s - 1) • f t‖ = t ^ (s.re - 1) * ‖f t‖

Tactic:
norm_smul,

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
T : Set ℝ
hT : T ⊆ Ioi 0
hT' : MeasurableSet T
hfc : AEStronglyMeasurable f (volume.restrict (Ioi 0))
s : ℂ
this : AEStronglyMeasurable (fun t ↦ ↑t ^ (s - 1) • f t) (volume.restrict T)
t : ℝ
ht : t ∈ T
⊢ ‖↑t ^ (s - 1)‖ * ‖f t‖ = t ^ (s.re - 1) * ‖f t‖
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: case e'_2.e'_4
z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
der1 : ∀ x ∈ uIcc 0 (π / 2), HasDerivAt (fun y ↦ ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x
x : ℝ
⊢ Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n = ↑(cos x) ^ n * Complex.cos (2 * z * ↑x)

Tactic:
rw [mul_comm]

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

Goal state before:
pp: n : ℕ
z : ℂ
hz : ‖z‖ < 1
⊢ ‖log (1 + -z) - logTaylor (n + 1) (-z)‖ ≤ ‖z‖ ^ (n + 1) * (1 - ‖z‖)⁻¹ / (↑n + 1)

Tactic:
convert! norm_log_sub_logTaylor_le n <| (norm_neg z).symm ▸ hz using 4 <;> rw [norm_neg]

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

Goal state before:
pp: z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
⊢ ∫ (x : ℝ) in 0..π / 2, Complex.sin (2 * z * ↑x) * ↑(sin x) * ↑(cos x) ^ (n - 1) =
    (↑n / (2 * z) * ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n) -
      (↑n - 1) / (2 * z) * ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (n - 2)

Tactic:
have der1 :
    ∀ x : ℝ,
      x ∈ uIcc 0 (π / 2) →
        HasDerivAt (fun y : ℝ => sin y * (cos y : ℂ) ^ (n - 1))
          ((cos x : ℂ) ^ n - (n - 1) * (sin x : ℂ) ^ 2 * (cos x : ℂ) ^ (n - 2)) x := by
    intro x _
    have c := HasDerivAt.comp (x : ℂ) (hasDerivAt_pow (n - 1) _) (Complex.hasDerivAt_cos x)
    convert! ((Complex.hasDerivAt_sin x).fun_mul c).comp_ofReal using 1
    · simp only [Complex.ofReal_sin, Complex.ofReal_cos, Function.comp]
    · simp only [Complex.ofReal_cos, Complex.ofReal_sin]
      rw [mul_neg, mul_neg, ← sub_eq_add_neg, Function.comp_apply]
      congr 1
      · rw [← pow_succ', Nat.sub_add_cancel (by lia : 1 ≤ n)]
      · have : ((n - 1 : ℕ) : ℂ) = (n : ℂ) - 1 := by
          rw [Nat.cast_sub (one_le_two.trans hn), Nat.cast_one]
        rw [Nat.sub_sub, this]
        ring

Goal state after:
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, Complex.sin (2 * z * ↑x) * ↑(sin x) * ↑(cos x) ^ (n - 1) =
    (↑n / (2 * z) * ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n) -
      (↑n - 1) / (2 * z) * ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (n - 2)
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 : 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
⊢ ‖(↑t ^ (z - 1) * ↑(log t)) • f t‖ ≤ (t ^ (s.re + v - 1) + t ^ (s.re - v - 1)) * |log t| * ‖f t‖

Tactic:
norm_smul,

Goal state after:
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
⊢ ‖↑t ^ (z - 1) * ↑(log t)‖ * ‖f t‖ ≤ (t ^ (s.re + v - 1) + t ^ (s.re - v - 1)) * |log t| * ‖f t‖
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
x : ℝ
| ↑(-x) ^ (u - 1) * (1 - ↑(-x)) ^ (v - 1)

Tactic:
rw [mul_comm]

Goal state after:
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
x : ℝ
| (1 - ↑(-x)) ^ (v - 1) * ↑(-x) ^ (u - 1)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: f : ℝ → ℝ
x : ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hx : 0 < x
hx' : x ≤ 1
n : ℕ
⊢ f (↑(n + 1) + x) ≤ f 1 + x * log (↑n + 1) - x * log ↑n + (x * log ↑n + log ↑n !)

Tactic:
refine (f_add_nat_le hf_conv (@hf_feq) (Nat.add_one_ne_zero n) hx hx').trans (le_of_eq ?_)

Goal state after:
f : ℝ → ℝ
x : ℝ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hx : 0 < x
hx' : x ≤ 1
n : ℕ
⊢ f ↑(n + 1) + x * log ↑(n + 1) = f 1 + x * log (↑n + 1) - x * log ↑n + (x * log ↑n + log ↑n !)
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: case e'_3.e_a.e_f
z : ℂ
this✝ : 𝓝 (Complex.sin (↑π * z)) = 𝓝 (Complex.sin (↑π * z) * 1)
A :
  Tendsto
    (fun n ↦
      (↑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))))
    atTop (𝓝 (Complex.sin (↑π * z) * 1))
this : ContinuousOn (fun x ↦ Complex.cos (2 * z * ↑x)) (Icc 0 (π / 2))
n : ℕ
x : ℝ
⊢ Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n = ↑(cos x) ^ n * Complex.cos (2 * z * ↑x)

Tactic:
rw [mul_comm]

Goal state after:
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 := ⋯
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

Tactic:
convert! betaIntegral_convergent hu hv'

Goal state after:
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
x✝ : ℝ
⊢ v = v + 1 - 1
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
p : P
t : V → Set ℝ
ht : ∀ i ∈ Metric.sphere 0 1, t i ∈ atTop
u : V → Set V
hu : ∀ i ∈ Metric.sphere 0 1, u i ∈ 𝓝 i
smul_subset_s : ∀ i ∈ Metric.sphere 0 1, t i • u i ⊆ (fun x ↦ x +ᵥ p) ⁻¹' s
cover : Finset V
h₁ : ∀ x ∈ cover, x ∈ Metric.sphere 0 1
h₂ : Metric.sphere 0 1 ⊆ ⋃ x ∈ cover, u x
x : P
hx : ‖x -ᵥ p‖ ∈ Set.Ioi 0 ∩ ⋂ x ∈ cover, t x
x' : V := ‖x -ᵥ p‖⁻¹ • (x -ᵥ p)
⊢ ‖x'‖ = 1

Tactic:
norm_smul,

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
p : P
t : V → Set ℝ
ht : ∀ i ∈ Metric.sphere 0 1, t i ∈ atTop
u : V → Set V
hu : ∀ i ∈ Metric.sphere 0 1, u i ∈ 𝓝 i
smul_subset_s : ∀ i ∈ Metric.sphere 0 1, t i • u i ⊆ (fun x ↦ x +ᵥ p) ⁻¹' s
cover : Finset V
h₁ : ∀ x ∈ cover, x ∈ Metric.sphere 0 1
h₂ : Metric.sphere 0 1 ⊆ ⋃ x ∈ cover, u x
x : P
hx : ‖x -ᵥ p‖ ∈ Set.Ioi 0 ∩ ⋂ x ∈ cover, t x
x' : V := ‖x -ᵥ p‖⁻¹ • (x -ᵥ p)
⊢ ‖‖x -ᵥ p‖⁻¹‖ * ‖x -ᵥ p‖ = 1
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: s : ℂ
n : ℕ
hn : n ≠ 0
| (↑n ^ (s + 1) * ↑n ! / ∏ j ∈ Finset.range (n + 1), (s + 1 + ↑j)) / s

Tactic:
Finset.prod_range_succ,

Goal state after:
s : ℂ
n : ℕ
hn : n ≠ 0
| ↑n ^ (s + 1) * ↑n ! / ((∏ x ∈ Finset.range n, (s + 1 + ↑x)) * (s + 1 + ↑n)) / s
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: case succ.inr
m : ℕ
IH : ∀ (s : ℂ), ⌊1 - s.re⌋₊ = m → Tendsto s.GammaSeq atTop (𝓝 (Gamma s))
s : ℂ
hs : ↑(m + 1) ≤ 1 - s.re ∧ 1 - s.re < ↑(m + 1) + 1
hsne : s ≠ 0
⊢ ⌊1 - (s + 1).re⌋₊ = m

Tactic:
have : s.re ≤ -m := by grind

Goal state after:
case succ.inr
m : ℕ
IH : ∀ (s : ℂ), ⌊1 - s.re⌋₊ = m → Tendsto s.GammaSeq atTop (𝓝 (Gamma s))
s : ℂ
hs : ↑(m + 1) ≤ 1 - s.re ∧ 1 - s.re < ↑(m + 1) + 1
hsne : s ≠ 0
this : s.re ≤ -↑m
⊢ ⌊1 - (s + 1).re⌋₊ = m