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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : InnerProductSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace 𝕜 H
f : E →L[𝕜] F
x : E ⊗[𝕜] G
n : ℕ
e : Fin n → E
g : Fin n → G
hx : x = ∑ j, e j ⊗ₜ[𝕜] g j
c : E ⊗[𝕜] G →₀ 𝕜
hc_supp : ↑c.support ⊆ {t | ∃ m n, m ⊗ₜ[𝕜] n = t}
hc : (c.sum fun mi r ↦ r • mi) = x
m : ℕ
A : Fin m → Fin n → 𝕜
hA :
  ∀ (i j : Fin n),
    ‖f‖ ^ 2 • inner 𝕜 (e i) (e j) - (∑ i, Matrix.vecMulVec (A i) (star (A i))) i j = inner 𝕜 (f (e i)) (f (e j))
⊢ 0 ≤ RCLike.re (∑ x, ∑ x_1, ∑ i, inner 𝕜 (g x_1) (g x) * Matrix.vecMulVec (A i) (star (A i)) x_1 x)

Tactic:
Finset.sum_comm_cycle,

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : InnerProductSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace 𝕜 H
f : E →L[𝕜] F
x : E ⊗[𝕜] G
n : ℕ
e : Fin n → E
g : Fin n → G
hx : x = ∑ j, e j ⊗ₜ[𝕜] g j
c : E ⊗[𝕜] G →₀ 𝕜
hc_supp : ↑c.support ⊆ {t | ∃ m n, m ⊗ₜ[𝕜] n = t}
hc : (c.sum fun mi r ↦ r • mi) = x
m : ℕ
A : Fin m → Fin n → 𝕜
hA :
  ∀ (i j : Fin n),
    ‖f‖ ^ 2 • inner 𝕜 (e i) (e j) - (∑ i, Matrix.vecMulVec (A i) (star (A i))) i j = inner 𝕜 (f (e i)) (f (e j))
⊢ 0 ≤ RCLike.re (∑ z, ∑ x, ∑ y, inner 𝕜 (g y) (g x) * Matrix.vecMulVec (A z) (star (A z)) y x)
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
⊢ eLpNorm u (↑q) μ ≤ ↑(eLpNormLESNormFDerivOfLeConst F μ s p q) * eLpNorm (fderiv ℝ u) (↑p) μ

Tactic:
by_cases hq0 : q = 0

Goal state after:
case pos
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
⊢ eLpNorm u (↑q) μ ≤ ↑(eLpNormLESNormFDerivOfLeConst F μ s p q) * eLpNorm (fderiv ℝ u) (↑p) μ

case neg
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
⊢ eLpNorm u (↑q) μ ≤ ↑(eLpNormLESNormFDerivOfLeConst F μ s p q) * eLpNorm (fderiv ℝ u) (↑p) μ
Module: Mathlib.Analysis.InnerProductSpace.TensorProduct

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : InnerProductSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace 𝕜 H
f₁ : E →L[𝕜] F
f₂ : H →L[𝕜] E
⊢ lTensor G (f₁ ∘SL f₂) = lTensor G f₁ ∘SL lTensor G f₂

Tactic:
ext; simp [LinearMap.lTensor_comp]

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

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹⁷ : EMetricSpace X
inst✝¹⁶ : MeasurableSpace X
inst✝¹⁵ : BorelSpace X
inst✝¹⁴ : EMetricSpace Y
inst✝¹³ : MeasurableSpace Y
inst✝¹² : BorelSpace Y
V : Type u_3
P : Type u_4
inst✝¹¹ : NormedAddCommGroup V
inst✝¹⁰ : InnerProductSpace ℝ V
inst✝⁹ : MeasurableSpace V
inst✝⁸ : BorelSpace V
inst✝⁷ : FiniteDimensional ℝ V
inst✝⁶ : MetricSpace P
inst✝⁵ : MeasurableSpace P
inst✝⁴ : BorelSpace P
inst✝³ : NormedAddTorsor V P
α : Type u_5
inst✝² : AddGroup α
inst✝¹ : AddAction α X
inst✝ : IsIsometricVAdd α X
d : ℕ
⊢ VAddInvariantMeasure α X μHE[d]

Tactic:
by
  rw [euclideanHausdorffMeasure_def]
  infer_instance

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

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹⁶ : EMetricSpace X
inst✝¹⁵ : MeasurableSpace X
inst✝¹⁴ : BorelSpace X
inst✝¹³ : EMetricSpace Y
inst✝¹² : MeasurableSpace Y
inst✝¹¹ : BorelSpace Y
V : Type u_3
P : Type u_4
inst✝¹⁰ : NormedAddCommGroup V
inst✝⁹ : InnerProductSpace ℝ V
inst✝⁸ : MeasurableSpace V
inst✝⁷ : BorelSpace V
inst✝⁶ : FiniteDimensional ℝ V
inst✝⁵ : MetricSpace P
inst✝⁴ : MeasurableSpace P
inst✝³ : BorelSpace P
inst✝² : NormedAddTorsor V P
inst✝¹ : AddGroup X
inst✝ : IsIsometricVAdd X X
d : ℕ
⊢ μHE[d].IsAddLeftInvariant

Tactic:
by
  rw [euclideanHausdorffMeasure_def]
  infer_instance

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

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹⁶ : EMetricSpace X
inst✝¹⁵ : MeasurableSpace X
inst✝¹⁴ : BorelSpace X
inst✝¹³ : EMetricSpace Y
inst✝¹² : MeasurableSpace Y
inst✝¹¹ : BorelSpace Y
V : Type u_3
P : Type u_4
inst✝¹⁰ : NormedAddCommGroup V
inst✝⁹ : InnerProductSpace ℝ V
inst✝⁸ : MeasurableSpace V
inst✝⁷ : BorelSpace V
inst✝⁶ : FiniteDimensional ℝ V
inst✝⁵ : MetricSpace P
inst✝⁴ : MeasurableSpace P
inst✝³ : BorelSpace P
inst✝² : NormedAddTorsor V P
inst✝¹ : AddGroup X
inst✝ : IsIsometricVAdd Xᵃᵒᵖ X
d : ℕ
⊢ μHE[d].IsAddRightInvariant

Tactic:
by
  rw [euclideanHausdorffMeasure_def]
  infer_instance

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

Goal state before:
pp: V : Type u_3
P : Type u_4
inst✝⁸ : NormedAddCommGroup V
inst✝⁷ : InnerProductSpace ℝ V
inst✝⁶ : MeasurableSpace V
inst✝⁵ : BorelSpace V
inst✝⁴ : FiniteDimensional ℝ V
inst✝³ : MetricSpace P
inst✝² : MeasurableSpace P
inst✝¹ : BorelSpace P
inst✝ : NormedAddTorsor V P
s : AffineSubspace ℝ P
hs : Nonempty ↥s
t : Set P
ht : MeasurableSet t
p : ↥s
⊢ MeasurableEmbedding ⇑(IsometryEquiv.vaddConst ?m.162)

Tactic:
by simpa using (IsometryEquiv.vaddConst p).toHomeomorph.measurableEmbedding

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

Goal state before:
pp: 𝕜 : Type u_1
U : Type u_2
V : Type u_3
inst✝⁷ : RCLike 𝕜
inst✝⁶ : NormedAddCommGroup U
inst✝⁵ : InnerProductSpace 𝕜 U
inst✝⁴ : FiniteDimensional 𝕜 U
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace 𝕜 V
ι : Type u_5
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
f : U →ₗ[𝕜] V
bu : OrthonormalBasis ι 𝕜 U
bv : OrthonormalBasis ι 𝕜 ↥f.range
hrank : finrank 𝕜 U = finrank 𝕜 ↥f.range
h : Nonempty (OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 ↥f.range)
h1 : bu.toBasis.toMatrix ⇑(stdOrthonormalBasis 𝕜 U).toBasis * (stdOrthonormalBasis 𝕜 U).toBasis.toMatrix ⇑bu.toBasis = 1
h2 : (stdOrthonormalBasis 𝕜 U).toBasis.toMatrix ⇑bu.toBasis * bu.toBasis.toMatrix ⇑(stdOrthonormalBasis 𝕜 U).toBasis = 1
⊢ bu.toBasis.toMatrix ⇑(stdOrthonormalBasis 𝕜 U).toBasis * h.some.toBasis.toMatrix ⇑bv.toBasis *
        (h.some.toBasis.toMatrix ⇑bv.toBasis).conjTranspose *
      (bu.toBasis.toMatrix ⇑(stdOrthonormalBasis 𝕜 U).toBasis).conjTranspose =
    1

Tactic:
Matrix.mul_assoc (bu.toBasis.toMatrix (stdOrthonormalBasis 𝕜 U).toBasis)

Goal state after:
𝕜 : Type u_1
U : Type u_2
V : Type u_3
inst✝⁷ : RCLike 𝕜
inst✝⁶ : NormedAddCommGroup U
inst✝⁵ : InnerProductSpace 𝕜 U
inst✝⁴ : FiniteDimensional 𝕜 U
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace 𝕜 V
ι : Type u_5
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
f : U →ₗ[𝕜] V
bu : OrthonormalBasis ι 𝕜 U
bv : OrthonormalBasis ι 𝕜 ↥f.range
hrank : finrank 𝕜 U = finrank 𝕜 ↥f.range
h : Nonempty (OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 ↥f.range)
h1 : bu.toBasis.toMatrix ⇑(stdOrthonormalBasis 𝕜 U).toBasis * (stdOrthonormalBasis 𝕜 U).toBasis.toMatrix ⇑bu.toBasis = 1
h2 : (stdOrthonormalBasis 𝕜 U).toBasis.toMatrix ⇑bu.toBasis * bu.toBasis.toMatrix ⇑(stdOrthonormalBasis 𝕜 U).toBasis = 1
⊢ bu.toBasis.toMatrix ⇑(stdOrthonormalBasis 𝕜 U).toBasis *
        (h.some.toBasis.toMatrix ⇑bv.toBasis * (h.some.toBasis.toMatrix ⇑bv.toBasis).conjTranspose) *
      (bu.toBasis.toMatrix ⇑(stdOrthonormalBasis 𝕜 U).toBasis).conjTranspose =
    1
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: X : Type u_2
inst✝² : EMetricSpace X
μ : OuterMeasure X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
hm : μ.IsMetric
⊢ inst✝¹ ≤ μ.caratheodory

Tactic:
BorelSpace.measurable_eq (α := X)

Goal state after:
X : Type u_2
inst✝² : EMetricSpace X
μ : OuterMeasure X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
hm : μ.IsMetric
⊢ borel X ≤ μ.caratheodory
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: X : Type u_2
Y : Type u_3
inst✝¹ : EMetricSpace X
inst✝ : EMetricSpace Y
m : ℝ≥0∞ → ℝ≥0∞
f : X → Y
hf : Isometry f
H : Monotone m ∨ Surjective f
ε : ℝ≥0∞
hε : id ε > 0
⊢ (comap f) (boundedBy (extend fun s x ↦ m (ediam s))) = boundedBy (extend fun s x ↦ m (ediam s))

Tactic:
comap_boundedBy _ (H.imp _ id)

Goal state after:
X : Type u_2
Y : Type u_3
inst✝¹ : EMetricSpace X
inst✝ : EMetricSpace Y
m : ℝ≥0∞ → ℝ≥0∞
f : X → Y
hf : Isometry f
H : Monotone m ∨ Surjective f
ε : ℝ≥0∞
hε : id ε > 0
⊢ (boundedBy fun s ↦ extend (fun s x ↦ m (ediam s)) (f '' s)) = boundedBy (extend fun s x ↦ m (ediam s))

X : Type u_2
Y : Type u_3
inst✝¹ : EMetricSpace X
inst✝ : EMetricSpace Y
m : ℝ≥0∞ → ℝ≥0∞
f : X → Y
hf : Isometry f
H : Monotone m ∨ Surjective f
ε : ℝ≥0∞
hε : id ε > 0
⊢ Monotone m → Monotone fun s ↦ extend (fun s x ↦ m (ediam s)) ↑s
Module: Mathlib.Analysis.InnerProductSpace.NormDet

Goal state before:
pp: 𝕜 : Type u_1
U : Type u_2
V : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup U
inst✝⁴ : InnerProductSpace 𝕜 U
inst✝³ : FiniteDimensional 𝕜 U
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace 𝕜 V
inst✝ : FiniteDimensional 𝕜 V
f : U →ₗ[𝕜] V
⊢ ↑(f.normDet ^ 2) = ↑(∏ x ∈ Finset.range (finrank 𝕜 U), f.singularValues x ^ 2)

Tactic:
← Fin.prod_univ_eq_prod_range,

Goal state after:
𝕜 : Type u_1
U : Type u_2
V : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup U
inst✝⁴ : InnerProductSpace 𝕜 U
inst✝³ : FiniteDimensional 𝕜 U
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace 𝕜 V
inst✝ : FiniteDimensional 𝕜 V
f : U →ₗ[𝕜] V
⊢ ↑(f.normDet ^ 2) = ↑(∏ i, f.singularValues ↑i ^ 2)
Module: Mathlib.Analysis.InnerProductSpace.NormDet

Goal state before:
pp: 𝕜 : Type u_1
U : Type u_2
V : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup U
inst✝⁴ : InnerProductSpace 𝕜 U
inst✝³ : FiniteDimensional 𝕜 U
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace 𝕜 V
inst✝ : FiniteDimensional 𝕜 V
f : U →ₗ[𝕜] V
⊢ LinearMap.det (adjoint f ∘ₗ f) = ↑(∏ i, f.singularValues ↑i ^ 2)

Tactic:
sq_singularValues_fin

Goal state after:
𝕜 : Type u_1
U : Type u_2
V : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup U
inst✝⁴ : InnerProductSpace 𝕜 U
inst✝³ : FiniteDimensional 𝕜 U
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace 𝕜 V
inst✝ : FiniteDimensional 𝕜 V
f : U →ₗ[𝕜] V
⊢ LinearMap.det (adjoint f ∘ₗ f) = ↑(∏ x, ⋯.eigenvalues ⋯ x)
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: case a
X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
x : X
⊢ 1 ≤ μH[0] {x}

Tactic:
rw [hausdorffMeasure_apply]
    suffices
      (1 : ℝ≥0∞) ≤
        ⨅ (t : ℕ → Set X) (_ : {x} ⊆ ⋃ n, t n) (_ : ∀ n, ediam (t n) ≤ 1),
          ∑' n, ⨆ _ : (t n).Nonempty, ediam (t n) ^ (0 : ℝ) by
      apply le_trans this _
      convert! le_iSup₂ (α := ℝ≥0∞) (1 : ℝ≥0∞) zero_lt_one
      rfl
    simp only [ENNReal.rpow_zero, le_iInf_iff]
    intro t hst _
    rcases mem_iUnion.1 (hst (mem_singleton x)) with ⟨m, hm⟩
    have A : (t m).Nonempty := ⟨x, hm⟩
    calc
      (1 : ℝ≥0∞) = ⨆ h : (t m).Nonempty, 1 := by simp only [A, ciSup_pos]
      _ ≤ ∑' n, ⨆ h : (t n).Nonempty, 1 := ENNReal.le_tsum _

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

Goal state before:
pp: 𝕜 : Type u_1
H : Type u_2
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : InnerProductSpace 𝕜 H
inst✝³ : CompleteSpace H
inst✝² : Algebra ℝ (H →L[𝕜] H)
inst✝¹ : IsScalarTower ℝ 𝕜 (H →L[𝕜] H)
inst✝ : ContinuousFunctionalCalculus ℝ (H →L[𝕜] H) IsSelfAdjoint
f g : H →L[𝕜] H
h : (g - f).IsPositive
p : H →L[𝕜] H
hp₁ : IsSelfAdjoint p
hp₃ : p * p = g - f
⊢ f + p ^ 2 = g

Tactic:
rwa [add_comm, ← eq_sub_iff_add_eq]

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

Goal state before:
pp: 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 := fun n ↦ (i : ι) → Fin ⌈(↑(b i) - ↑(a i)) * ↑n⌉₊
t : (n : ℕ) → γ n → Set (ι → ℝ) := fun n f ↦ univ.pi fun i ↦ Icc (↑(a i) + ↑↑(f i) / ↑n) (↑(a i) + (↑↑(f i) + 1) / ↑n)
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 := fun i ↦ ⟨⌊(x i - ↑(a i)) * ↑n⌋₊, ⋯⟩
⊢ ∃ i, ∀ (i_1 : ι), x i_1 ∈ Icc (↑(a i_1) + ↑↑(i i_1) / ↑n) (↑(a i_1) + (↑↑(i i_1) + 1) / ↑n)

Tactic:
refine ⟨f, fun i => ⟨?_, ?_⟩⟩

Goal state after:
case refine_1
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 := fun n ↦ (i : ι) → Fin ⌈(↑(b i) - ↑(a i)) * ↑n⌉₊
t : (n : ℕ) → γ n → Set (ι → ℝ) := fun n f ↦ univ.pi fun i ↦ Icc (↑(a i) + ↑↑(f i) / ↑n) (↑(a i) + (↑↑(f i) + 1) / ↑n)
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 := fun i ↦ ⟨⌊(x i - ↑(a i)) * ↑n⌋₊, ⋯⟩
i : ι
⊢ ↑(a i) + ↑↑(f i) / ↑n ≤ x i

case refine_2
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 := fun n ↦ (i : ι) → Fin ⌈(↑(b i) - ↑(a i)) * ↑n⌉₊
t : (n : ℕ) → γ n → Set (ι → ℝ) := fun n f ↦ univ.pi fun i ↦ Icc (↑(a i) + ↑↑(f i) / ↑n) (↑(a i) + (↑↑(f i) + 1) / ↑n)
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 := fun i ↦ ⟨⌊(x i - ↑(a i)) * ↑n⌋₊, ⋯⟩
i : ι
⊢ x i ≤ ↑(a i) + (↑↑(f i) + 1) / ↑n
Module: Mathlib.Analysis.InnerProductSpace.TwoDim

Goal state before:
pp: case neg.mpr
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
hx : ¬x = 0
⊢ SameRay ℝ x y → 0 ≤ ⟪x, y⟫ ∧ (o.areaForm x) y = 0

Tactic:
· intro h
    obtain ⟨r, hr, rfl⟩ := h.exists_nonneg_left hx
    simp only [inner_smul_right, real_inner_self_eq_norm_sq, map_smulₛₗ, areaForm_apply_self,
      smul_eq_mul, mul_zero, and_true]
    positivity

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹⁶ : RCLike 𝕜
inst✝¹⁵ : AddCommGroup E
inst✝¹⁴ : Module 𝕜 E
inst✝¹³ : AddCommGroup F
inst✝¹² : Module 𝕜 F
inst✝¹¹ : Module ℝ E
inst✝¹⁰ : IsScalarTower ℝ 𝕜 E
inst✝⁹ : Module ℝ F
inst✝⁸ : IsScalarTower ℝ 𝕜 F
inst✝⁷ : TopologicalSpace E
inst✝⁶ : IsTopologicalAddGroup E
inst✝⁵ : ContinuousSMul 𝕜 E
inst✝⁴ : LocallyConvexSpace ℝ E
inst✝³ : TopologicalSpace F
inst✝² : IsTopologicalAddGroup F
inst✝¹ : ContinuousSMul 𝕜 F
inst✝ : LocallyConvexSpace ℝ F
s : Set E
hs : Convex ℝ s
e : E ≃ₗ[𝕜] F
he₁ : ∀ (f : StrongDual 𝕜 F), Continuous ⇑(e.dualMap ↑f)
he₂ : ∀ (f : StrongDual 𝕜 E), Continuous ⇑(e.symm.dualMap ↑f)
⊢ closure (⇑e '' s) ⊆ ⇑e '' closure s

Tactic:
simp only [← Set.image_subset_image_iff e.symm.injective]

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹⁶ : RCLike 𝕜
inst✝¹⁵ : AddCommGroup E
inst✝¹⁴ : Module 𝕜 E
inst✝¹³ : AddCommGroup F
inst✝¹² : Module 𝕜 F
inst✝¹¹ : Module ℝ E
inst✝¹⁰ : IsScalarTower ℝ 𝕜 E
inst✝⁹ : Module ℝ F
inst✝⁸ : IsScalarTower ℝ 𝕜 F
inst✝⁷ : TopologicalSpace E
inst✝⁶ : IsTopologicalAddGroup E
inst✝⁵ : ContinuousSMul 𝕜 E
inst✝⁴ : LocallyConvexSpace ℝ E
inst✝³ : TopologicalSpace F
inst✝² : IsTopologicalAddGroup F
inst✝¹ : ContinuousSMul 𝕜 F
inst✝ : LocallyConvexSpace ℝ F
s : Set E
hs : Convex ℝ s
e : E ≃ₗ[𝕜] F
he₁ : ∀ (f : StrongDual 𝕜 F), Continuous ⇑(e.dualMap ↑f)
he₂ : ∀ (f : StrongDual 𝕜 E), Continuous ⇑(e.symm.dualMap ↑f)
⊢ ⇑e.symm '' closure (⇑e '' s) ⊆ ⇑e.symm '' ⇑e '' closure s
Module: Mathlib.Analysis.MellinInversion

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
⊢ mellin f s = ∫ (u : ℝ), cexp (-s * ↑u) • f (rexp (-u))

Tactic:
mellin,

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
⊢ ∫ (t : ℝ) in Ioi 0, ↑t ^ (s - 1) • f t = ∫ (u : ℝ), cexp (-s * ↑u) • f (rexp (-u))
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 : a ≠ 0
t : ℝ
ht : t ∈ Ioi 0
⊢ ↑t ^ (s - 1) • f (t ^ a) = ↑t ^ (↑a - ↑1 + ↑a * (s / ↑a - 1)) • f (t ^ a)

Tactic:
ofReal_one,

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
ha : a ≠ 0
t : ℝ
ht : t ∈ Ioi 0
⊢ ↑t ^ (s - 1) • f (t ^ a) = ↑t ^ (↑a - 1 + ↑a * (s / ↑a - 1)) • f (t ^ a)
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: f : ℝ → ℂ
s a : ℂ
⊢ mellin (fun t ↦ f t / a) s = mellin f s / a

Tactic:
mellin,

Goal state after:
f : ℝ → ℂ
s a : ℂ
⊢ ∫ (t : ℝ) in Ioi 0, ↑t ^ (s - 1) • (f t / a) = (∫ (t : ℝ) in Ioi 0, ↑t ^ (s - 1) • f t) / a
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: case inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
ha : a ≠ 0
t : ℝ
ht : t ∈ Ioi 0
⊢ ↑t ^ (s - 1) • f (t ^ a) = ↑t ^ (↑a - ↑1 + ↑a * (s / ↑a - 1)) • f (t ^ a)

Tactic:
ofReal_one,

Goal state after:
case inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
ha : a ≠ 0
t : ℝ
ht : t ∈ Ioi 0
⊢ ↑t ^ (s - 1) • f (t ^ a) = ↑t ^ (↑a - 1 + ↑a * (s / ↑a - 1)) • f (t ^ a)
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: ⊢ logTaylor 0 = fun x ↦ 0

Tactic:
funext

Goal state after:
x✝ : ℂ
⊢ logTaylor 0 x✝ = 0
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

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

Tactic:
funext

Goal state after:
n : ℕ
x✝ : ℂ
⊢ logTaylor (n + 1) x✝ = (logTaylor n + fun z ↦ (-1) ^ (n + 1) * z ^ n / ↑n) x✝
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
⊢ mellin f (s / -↑1) = mellin f (-s)

Tactic:
ofReal_one,

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
⊢ mellin f (s / -1) = mellin f (-s)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Deriv

Goal state before:
pp: case pos
m : ℕ
hs : m ≠ 0
⊢ 1 ≤ m

Tactic:
← Nat.one_le_iff_ne_zero

Goal state after:
case pos
m : ℕ
hs : 1 ≤ m
⊢ 1 ≤ m
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: n : ℕ
z : ℂ
hz : ‖z‖ < 1
help : IntervalIntegrable (fun t ↦ t ^ n * (1 - ‖z‖)⁻¹) MeasureTheory.volume 0 1
f : ℂ → ℂ := fun z ↦ log (1 + z) - logTaylor (n + 1) z
f' : ℂ → ℂ := fun z ↦ (-z) ^ n * (1 + z)⁻¹
hderiv : ∀ t ∈ Set.Icc 0 1, HasDerivAt f (f' (0 + ↑t * z)) (0 + ↑t * z)
hcont : ContinuousOn (fun t ↦ f' (0 + ↑t * z)) (Set.Icc 0 1)
H : log (1 + z) - logTaylor (n + 1) z = z * ∫ (t : ℝ) in 0..1, (-(↑t * z)) ^ n * (1 + ↑t * z)⁻¹
⊢ ‖z‖ * (1 ^ n * (‖z‖ ^ n * ‖∫ (x : ℝ) in 0..1, ↑x ^ n * (1 + ↑x * z)⁻¹‖)) ≤ ‖z‖ ^ (n + 1) * (1 - ‖z‖)⁻¹ / (↑n + 1)

Tactic:
one_pow,

Goal state after:
n : ℕ
z : ℂ
hz : ‖z‖ < 1
help : IntervalIntegrable (fun t ↦ t ^ n * (1 - ‖z‖)⁻¹) MeasureTheory.volume 0 1
f : ℂ → ℂ := fun z ↦ log (1 + z) - logTaylor (n + 1) z
f' : ℂ → ℂ := fun z ↦ (-z) ^ n * (1 + z)⁻¹
hderiv : ∀ t ∈ Set.Icc 0 1, HasDerivAt f (f' (0 + ↑t * z)) (0 + ↑t * z)
hcont : ContinuousOn (fun t ↦ f' (0 + ↑t * z)) (Set.Icc 0 1)
H : log (1 + z) - logTaylor (n + 1) z = z * ∫ (t : ℝ) in 0..1, (-(↑t * z)) ^ n * (1 + ↑t * z)⁻¹
⊢ ‖z‖ * (1 * (‖z‖ ^ n * ‖∫ (x : ℝ) in 0..1, ↑x ^ n * (1 + ↑x * z)⁻¹‖)) ≤ ‖z‖ ^ (n + 1) * (1 - ‖z‖)⁻¹ / (↑n + 1)
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: case e'_2
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) =
    ∫ (x : ℝ) in 0..π / 2, ↑(sin x) * ↑(cos x) ^ (n - 1) * Complex.sin (2 * z * ↑x)

Tactic:
refine integral_congr fun x _ => ?_
    ring_nf

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)
⊢ Tendsto (fun x ↦ (1 + g x) ^ ↑x) atTop (𝓝 (cexp t))

Tactic:
by
  apply ((continuous_exp.tendsto _).comp (tendsto_mul_log_one_add_of_tendsto hg)).congr'
  have hg0 := tendsto_zero_of_isBoundedUnder_smul_of_tendsto_cobounded
    hg.norm.isBoundedUnder_le (RCLike.tendsto_ofReal_atTop_cobounded ℂ)
  filter_upwards [hg0.eventually_ne (show 0 ≠ -1 by simp)] with x hg1
  dsimp
  rw [cpow_def_of_ne_zero, mul_comm]
  intro hg0
  rw [← add_eq_zero_iff_neg_eq.mp hg0] at hg1
  norm_num at hg1

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

Goal state before:
pp: u : ℂ
hu : 0 < u.re
⊢ (↑1 ^ (u - 1 + 1) - 0 ^ (u - 1 + 1)) / (u - 1 + 1) = 1 / u

Tactic:
ofReal_one,

Goal state after:
u : ℂ
hu : 0 < u.re
⊢ (1 ^ (u - 1 + 1) - 0 ^ (u - 1 + 1)) / (u - 1 + 1) = 1 / u
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: f : ℝ → ℂ
hf : ContinuousOn f (Icc 0 (π / 2))
c_lt : ∀ y ∈ Icc 0 (π / 2), y ≠ 0 → cos y < cos 0
c_nonneg : ∀ x ∈ Icc 0 (π / 2), 0 ≤ cos x
c_zero_pos : 0 < cos 0
zero_mem : 0 ∈ closure (interior (Icc 0 (π / 2)))
⊢ Tendsto
    (fun n ↦ (∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n ∂volume)⁻¹ • ∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n • f x ∂volume)
    atTop (𝓝 (f 0))

Tactic:
exact
    tendsto_setIntegral_pow_smul_of_unique_maximum_of_isCompact_of_continuousOn isCompact_Icc
      continuousOn_cos c_lt c_nonneg c_zero_pos zero_mem hf

Goal state after:
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: s : ℂ
hs : 0 < s.re
aux1 : -1 < (s - 1).re
aux2 : s ≠ 0
aux3 : MeasurableSet (Ioc 0 1)
⊢ MellinConvergent ((Ioc 0 1).indicator fun x ↦ 1) s ∧ mellin ((Ioc 0 1).indicator fun x ↦ 1) s = 1 / s

Tactic:
mellin,

Goal state after:
s : ℂ
hs : 0 < s.re
aux1 : -1 < (s - 1).re
aux2 : s ≠ 0
aux3 : MeasurableSet (Ioc 0 1)
⊢ MellinConvergent ((Ioc 0 1).indicator fun x ↦ 1) s ∧
    ∫ (t : ℝ) in Ioi 0, ↑t ^ (s - 1) • (Ioc 0 1).indicator (fun x ↦ 1) t = 1 / s
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: s : ℂ
hs : 0 < s.re
aux1 : -1 < (s - 1).re
aux2 : s ≠ 0
aux3 : MeasurableSet (Ioc 0 1)
⊢ (↑1 ^ s - 0 ^ s) / s = 1 / s

Tactic:
ofReal_one,

Goal state after:
s : ℂ
hs : 0 < s.re
aux1 : -1 < (s - 1).re
aux2 : s ≠ 0
aux3 : MeasurableSet (Ioc 0 1)
⊢ (1 ^ s - 0 ^ s) / s = 1 / s
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: case e'_12
⊢ (fun s ↦ log (Γ (s / 2))) = (log ∘ Γ) ∘ ⇑(DistribSMul.toLinearMap ℝ ℝ (1 / 2)).toAffineMap

Tactic:
ext1 x
      simp only [LinearMap.coe_toAffineMap, Function.comp_apply, DistribSMul.toLinearMap_apply]
      rw [smul_eq_mul, mul_comm, mul_one_div]

Goal state after: