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

Goal state before:
pp: F : Type u_3
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
E : Type u_4
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
u : E → F
hu : ContDiff ℝ 1 u
h2u : HasCompactSupport u
p : ℝ
hp✝ : (↑(finrank ℝ E)).HolderConjugate p
C : ℝ≥0 := lintegralPowLePowLIntegralFDerivConst μ p
ι : Type := Fin (finrank ℝ E)
hιcard : #ι = finrank ℝ E
this : finrank ℝ E = finrank ℝ (ι → ℝ)
e : E ≃L[ℝ] ι → ℝ := ContinuousLinearEquiv.ofFinrankEq this
hp : (↑#ι).HolderConjugate p
h0p : 0 ≤ p
c : ℝ≥0 := μ.addHaarScalarFactor (Measure.map (⇑e.symm) volume)
hc : 0 < c
h2c : μ = c • Measure.map (⇑e.symm) volume
h3c : ↑c ≠ 0
h0C : C = c * ‖↑e.symm‖₊ ^ p * (c ^ p)⁻¹
hC : C * c ^ p = c * ‖↑e.symm‖₊ ^ p
v : (ι → ℝ) → F := u ∘ ⇑e.symm
hv : ContDiff ℝ 1 v
h2v : HasCompactSupport v
⊢ ↑‖↑e.symm‖₊ ^ p * (∫⁻ (y : ι → ℝ), ‖fderiv ℝ u (e.symm y)‖ₑ) ^ p =
    ↑(‖↑e.symm‖₊ ^ p) * (∫⁻ (y : ι → ℝ), ‖fderiv ℝ u (e.symm y)‖ₑ) ^ p

Tactic:
← ENNReal.coe_rpow_of_nonneg _ h0p

Goal state after:
F : Type u_3
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
E : Type u_4
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
u : E → F
hu : ContDiff ℝ 1 u
h2u : HasCompactSupport u
p : ℝ
hp✝ : (↑(finrank ℝ E)).HolderConjugate p
C : ℝ≥0 := lintegralPowLePowLIntegralFDerivConst μ p
ι : Type := Fin (finrank ℝ E)
hιcard : #ι = finrank ℝ E
this : finrank ℝ E = finrank ℝ (ι → ℝ)
e : E ≃L[ℝ] ι → ℝ := ContinuousLinearEquiv.ofFinrankEq this
hp : (↑#ι).HolderConjugate p
h0p : 0 ≤ p
c : ℝ≥0 := μ.addHaarScalarFactor (Measure.map (⇑e.symm) volume)
hc : 0 < c
h2c : μ = c • Measure.map (⇑e.symm) volume
h3c : ↑c ≠ 0
h0C : C = c * ‖↑e.symm‖₊ ^ p * (c ^ p)⁻¹
hC : C * c ^ p = c * ‖↑e.symm‖₊ ^ p
v : (ι → ℝ) → F := u ∘ ⇑e.symm
hv : ContDiff ℝ 1 v
h2v : HasCompactSupport v
⊢ ↑(‖↑e.symm‖₊ ^ p) * (∫⁻ (y : ι → ℝ), ‖fderiv ℝ u (e.symm y)‖ₑ) ^ p =
    ↑(‖↑e.symm‖₊ ^ p) * (∫⁻ (y : ι → ℝ), ‖fderiv ℝ u (e.symm y)‖ₑ) ^ p
Module: Mathlib.Dynamics.BirkhoffSum.NormedSpace

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝² : NormedAddCommGroup E
𝕜 : Type u_3
inst✝¹ : RCLike 𝕜
inst✝ : NormedSpace 𝕜 E
f : α → α
g : α → E
n : ℕ
x y : α
⊢ dist (birkhoffAverage 𝕜 f g n x) (birkhoffAverage 𝕜 f g n y) = dist (birkhoffSum f g n x) (birkhoffSum f g n y) / ↑n

Tactic:
simp [birkhoffAverage, dist_smul₀, div_eq_inv_mul]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
T : E →ₗ.[𝕜] F
S : F →ₗ.[𝕜] E
inst✝ : CompleteSpace E
hT : Dense ↑T.domain
x y : ↥T.adjointDomain
z : E
zin : z ∈ ↑T.domain
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
by
      simp [InnerProductSpace.toDual_symm_apply, inner_add_left,
        adjointDomainMkCLMExtend_apply hT _ ⟨z, zin⟩, inner_add_left]

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
this : (adjoint T ∘ₗ T).ker = ⊥ ↔ ¬End.HasEigenvalue (adjoint T ∘ₗ T) 0
i : ℕ
h : i < finrank 𝕜 E
hz : T.singularValues i ≤ 0
⊢ 0 = ↑(⋯.eigenvalues ⋯ ⟨i, h⟩)

Tactic:
← sq_singularValues_of_lt,

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
this : (adjoint T ∘ₗ T).ker = ⊥ ↔ ¬End.HasEigenvalue (adjoint T ∘ₗ T) 0
i : ℕ
h : i < finrank 𝕜 E
hz : T.singularValues i ≤ 0
⊢ 0 = ↑(T.singularValues i ^ 2)
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
⊢ T.singularValues.support = Finset.range (finrank 𝕜 ↥T.range)

Tactic:
by
  obtain ⟨n, hn⟩ := T.isLowerSet_support_singularValues.eq_univ_or_Iio.resolve_left
    (fun h ↦ Set.infinite_univ.not_finite (h ▸ Finset.finite_toSet _))
  rw [← Finset.coe_Iio, Finset.coe_inj, Nat.Iio_eq_range] at hn
  simp [← card_support_singularValues, hn]

Goal state after:
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
hu : ContDiff ℝ 1 u
h2u : HasCompactSupport u
p p' : ℝ≥0
hp : 1 ≤ p
hn : 0 < finrank ℝ E
hp' : (↑p')⁻¹ = ↑p⁻¹ - (↑(finrank ℝ E))⁻¹
F' : Type := EuclideanSpace ℝ (Fin (finrank ℝ F))
e : F ≃L[ℝ] F' := toEuclidean
C₁ : ℝ≥0 := ‖↑e.symm‖₊
C : ℝ≥0 := eLpNormLESNormFDerivOfEqInnerConst μ ↑p
C₂ : ℝ≥0 := ‖↑e‖₊
v : E → F' := ⇑e ∘ u
hv : ContDiff ℝ 1 v
h2v : HasCompactSupport v
this : eLpNorm v (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ v) (↑p) μ
h4v : ∀ (x : E), ‖fderiv ℝ v x‖ ≤ ↑C₂ * ‖fderiv ℝ u x‖
⊢ eLpNorm (⇑e.symm ∘ v) (↑p') μ ≤ C₁ • eLpNorm v (↑p') μ

Tactic:
by
      apply eLpNorm_le_nnreal_smul_eLpNorm_of_ae_le_mul
      exact Eventually.of_forall (fun x ↦ (e.symm : F' →L[ℝ] F).le_opNNNorm _)

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 : Submodule ℝ V
p q : P
⊢ (s.measurableEquivProd p) q = (s.orthogonalProjectionOnto (q -ᵥ p), sᗮ.orthogonalProjectionOnto (q -ᵥ p))

Tactic:
by
  simp [measurableEquivProd]

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 : Submodule ℝ V
p : P
q : ↥s × ↥sᗮ
⊢ (s.measurableEquivProd p).symm q = (↑q.1 + ↑q.2) +ᵥ p

Tactic:
by
  simp [measurableEquivProd]

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

Goal state before:
pp: case inr.refine_1
ι : Type u_1
inst✝ : Fintype ι
x : ι → ℝ
r : ℝ
hr : 0 < r
h✝ : Nonempty ι
this : μH[↑(Fintype.card ι)] (Metric.ball x r) = ENNReal.ofReal ((2 * r) ^ Fintype.card ι)
⊢ ENNReal.ofReal ((2 * r) ^ Fintype.card ι) ≠ 0

Tactic:
simp [pow_pos (mul_pos (zero_lt_two' ℝ) hr)]

Goal state after:
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 ≃ᵢ Y
⊢ (map ⇑f) (mkMetric m) = mkMetric m

Tactic:
by
  rw [← isometryEquiv_comap_mkMetric _ f, map_comap_of_surjective f.surjective]

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
m : ℝ≥0∞ → ℝ≥0∞
s : Set X
⊢ (mkMetric m) s =
    ⨆ r,
      ⨆ (_ : 0 < r),
        ⨅ t,
          ⨅ (_ : s ⊆ iUnion t), ⨅ (_ : ∀ (n : ℕ), ediam (t n) ≤ r), ∑' (n : ℕ), ⨆ (_ : (t n).Nonempty), m (ediam (t n))

Tactic:
by
  classical
  -- We mostly unfold the definitions but we need to switch the order of `∑'` and `⨅`
  simp only [← OuterMeasure.coe_mkMetric, OuterMeasure.mkMetric, OuterMeasure.mkMetric',
    OuterMeasure.iSup_apply, OuterMeasure.mkMetric'.pre, OuterMeasure.boundedBy_apply, extend]
  refine
    surjective_id.iSup_congr id fun r =>
      iSup_congr_Prop Iff.rfl fun _ =>
        surjective_id.iInf_congr _ fun t => iInf_congr_Prop Iff.rfl fun ht => ?_
  dsimp
  by_cases htr : ∀ n, ediam (t n) ≤ r
  · rw [iInf_eq_if, if_pos htr]
    congr 1 with n : 1
    simp only [iInf_eq_if, htr n, if_true]
  · rw [iInf_eq_if, if_neg htr]
    push Not at htr; rcases htr with ⟨n, hn⟩
    refine ENNReal.tsum_eq_top_of_eq_top ⟨n, ?_⟩
    rw [iSup_eq_if, if_pos, iInf_eq_if, if_neg]
    · exact hn.not_ge
    rcases ediam_pos_iff.1 hn.pos with ⟨x, hx, -⟩
    exact ⟨x, hx⟩

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : RCLike 𝕜
X : Type u_2
V : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : InnerProductSpace 𝕜 V
H : Type u_4
inst✝⁴ : NormedAddCommGroup H
inst✝³ : InnerProductSpace 𝕜 H
inst✝² : RKHS 𝕜 H X V
inst✝¹ : CompleteSpace H
inst✝ : CompleteSpace V
C : ℝ
hC : ∀ (x : X), ‖kerFun H x‖ ≤ C
ι : Type u_5
l : Filter ι
F : ι → H
f : H
h : Tendsto F l (𝓝 f)
⊢ TendstoUniformly (fun n ↦ ⇑(F n)) (⇑f) l

Tactic:
rw [← tendstoUniformlyOn_univ]

Goal state after:
𝕜 : Type u_1
inst✝⁷ : RCLike 𝕜
X : Type u_2
V : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : InnerProductSpace 𝕜 V
H : Type u_4
inst✝⁴ : NormedAddCommGroup H
inst✝³ : InnerProductSpace 𝕜 H
inst✝² : RKHS 𝕜 H X V
inst✝¹ : CompleteSpace H
inst✝ : CompleteSpace V
C : ℝ
hC : ∀ (x : X), ‖kerFun H x‖ ≤ C
ι : Type u_5
l : Filter ι
F : ι → H
f : H
h : Tendsto F l (𝓝 f)
⊢ TendstoUniformlyOn (fun n ↦ ⇑(F n)) (⇑f) l Set.univ
Module: Mathlib.Analysis.InnerProductSpace.StarOrder

Goal state before:
pp: case mpr
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 p : H →L[𝕜] H
hp : p ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s)
⊢ p.IsPositive

Tactic:
induction hp using AddSubmonoid.closure_induction with
      | mem _ hf =>
        obtain ⟨f, rfl⟩ := hf
        simpa using! ContinuousLinearMap.IsPositive.adjoint_conj isPositive_one f
      | zero => exact isPositive_zero
      | add f g _ _ hf hg => exact hf.add hg

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁸ : RCLike 𝕜
X : Type u_2
V : Type u_3
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace 𝕜 V
H : Type u_4
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : InnerProductSpace 𝕜 H
inst✝³ : RKHS 𝕜 H X V
inst✝² : CompleteSpace H
inst✝¹ : CompleteSpace V
K : Matrix X X (V →L[𝕜] V)
inst✝ : Fact K.PosSemidef
f : OfKernel K
h : (ContinuousLinearMap.pi fun x ↦ adjoint (kerFun K x)) f = 0
i : X × V
a : 𝕜
f✝ : X × V →₀ 𝕜
a✝² : i ∉ f✝.support
a✝¹ : a ≠ 0
a✝ : ⟪↑f✝, f⟫_𝕜 = 0
this : ⟪i.2, (ContinuousLinearMap.pi fun x ↦ adjoint (kerFun K x)) f i.1⟫_𝕜 = ⟪i.2, 0 i.1⟫_𝕜
⊢ ?m.168

Tactic:
simpa [OfKernel.kerFun, adjoint_inner_right] using this

Goal state after:
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
K : Set E
hK : IsCompact K
e : E ≃L[ℝ] Fin (finrank ℝ E) → ℝ := ContinuousLinearEquiv.ofFinrankEq ⋯
⊢ μH[↑(Fintype.card (Fin (finrank ℝ E)))] (⇑e '' K) < ∞

Tactic:
rw [hausdorffMeasure_pi_real]

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
K : Set E
hK : IsCompact K
e : E ≃L[ℝ] Fin (finrank ℝ E) → ℝ := ContinuousLinearEquiv.ofFinrankEq ⋯
⊢ volume (⇑e '' K) < ∞
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 ⋯
⊢ 0 < μH[↑(Fintype.card (Fin (finrank ℝ E)))] (⇑e '' U)

Tactic:
rw [hausdorffMeasure_pi_real]

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 < volume (⇑e '' U)
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
⊢ μH[1] ((fun r ↦ r • v) '' s) = ‖v‖₊ • μH[1] s

Tactic:
obtain rfl | hv := eq_or_ne v 0
  · have := nullSingletonClass_hausdorff E one_pos
    obtain rfl | hs := s.eq_empty_or_nonempty
    · simp
    simp [hs]
  have hn : ‖v‖ ≠ 0 := norm_ne_zero_iff.mpr hv
  -- break lineMap into pieces
  suffices
      μH[1] ((‖v‖ • ·) '' LinearMap.toSpanSingleton ℝ E (‖v‖⁻¹ • v) '' s) = ‖v‖₊ • μH[1] s by
    simpa only [Set.image_image, smul_comm (norm _), inv_smul_smul₀ hn,
      LinearMap.toSpanSingleton_apply] using this
  have iso_smul : Isometry (LinearMap.toSpanSingleton ℝ E (‖v‖⁻¹ • v)) := by
    refine AddMonoidHomClass.isometry_of_norm _ fun x => (norm_smul _ _).trans ?_
    rw [norm_smul, norm_inv, norm_norm, inv_mul_cancel₀ hn, mul_one, LinearMap.id_apply]
  rw [Set.image_smul, Measure.hausdorffMeasure_smul₀ zero_le_one hn, nnnorm_norm,
      NNReal.rpow_one, iso_smul.hausdorffMeasure_image (Or.inl <| zero_le_one' ℝ)]

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
h : ⟪x, y⟫ = 0
⊢ |(o.areaForm x) y| = ‖x‖ * ‖y‖

Tactic:
rw [o.areaForm_to_volumeForm, o.abs_volumeForm_apply_of_pairwise_orthogonal]
  · simp [Fin.prod_univ_succ]
  intro i j hij
  fin_cases i <;> fin_cases j
  · simp_all
  · simpa using h
  · simpa [real_inner_comm] using h
  · simp_all

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)
⊢ (LinearIsometryEquiv.ofLinearIsometry o.rightAngleRotationAux₂ (-o.rightAngleRotationAux₁) ⋯ ⋯).symm =
    (LinearIsometryEquiv.ofLinearIsometry o.rightAngleRotationAux₂ (-o.rightAngleRotationAux₁) ⋯ ⋯).trans
      (LinearIsometryEquiv.neg ℝ)

Tactic:
exact LinearIsometryEquiv.toLinearIsometry_injective rfl

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
he : ∀ (f : StrongDual 𝕜 F), Continuous ⇑(e.dualMap ↑f)
he' : Continuous fun x ↦ (toWeakSpace 𝕜 F) (e ((toWeakSpace 𝕜 E).symm x))
h_convex : Convex ℝ (⇑e '' s)
⊢ ⇑e '' closure s ⊆ closure (⇑e '' s)

Tactic:
← Set.image_subset_image_iff (toWeakSpace 𝕜 F).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
he : ∀ (f : StrongDual 𝕜 F), Continuous ⇑(e.dualMap ↑f)
he' : Continuous fun x ↦ (toWeakSpace 𝕜 F) (e ((toWeakSpace 𝕜 E).symm x))
h_convex : Convex ℝ (⇑e '' s)
⊢ ⇑(toWeakSpace 𝕜 F) '' ⇑e '' closure s ⊆ ⇑(toWeakSpace 𝕜 F) '' closure (⇑e '' s)
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
⊢ (o.kahler x) y = (starRingEnd ℂ) ((o.kahler y) x)

Tactic:
simp only [kahler_apply_apply]
  rw [real_inner_comm, areaForm_swap]
  simp [Complex.conj_ofReal]

Goal state after:
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: 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

Tactic:
integral_div

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

Goal state before:
pp: s : ℂ
hs : ∀ (m : ℕ), s ≠ -↑m
⊢ DifferentiableAt ℂ Gamma s

Tactic:
suffices ∀ (n : ℕ) (s : ℂ) (hsre : -n < s.re) (hs : ∀ m : ℕ, s ≠ -m), DifferentiableAt ℂ _ s from
    this (⌊-s.re⌋₊ + 1) s (by grind [Nat.lt_floor_add_one (-s.re)]) hs

Goal state after:
s : ℂ
hs : ∀ (m : ℕ), s ≠ -↑m
⊢ ∀ (n : ℕ) (s : ℂ), -↑n < s.re → (∀ (m : ℕ), s ≠ -↑m) → DifferentiableAt ℂ Gamma s
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Deriv

Goal state before:
pp: x✝ : ℝ
h : x✝ ∈ Ioi 0
⊢ ∀ (m : ℕ), x✝ ≠ -↑m

Tactic:
bound [mem_Ioi.mp h]

Goal state after:
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: case hbc
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 := ⋯
F' : ℂ → ℝ → E := ⋯
v : ℝ
hv0 : 0 < v
hv1 : v < s.re - b
hv2 : v < a - s.re
bound : ℝ → ℝ := ⋯
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)‖ ≤ t ^ (s.re + v - 1) + t ^ (s.re - v - 1)

Tactic:
rw [norm_cpow_eq_rpow_re_of_pos ht]

Goal state after:
case hbc
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).re ≤ t ^ (s.re + v - 1) + t ^ (s.re - v - 1)
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))
h4 : ∀ᵐ (t : ℝ) ∂volume.restrict (Ioi 0), ∀ z ∈ Metric.ball s v, ‖F' z t‖ ≤ bound t
this : ∀ {j : ℝ}, b < j → j < a → IntegrableOn (fun t ↦ t ^ (j - 1) * (|log t| * ‖f t‖)) (Ioi 0) volume
⊢ IntegrableOn (fun t ↦ t ^ (s.re + v - 1) * (|log t| * ‖f t‖) + t ^ (s.re - v - 1) * (|log t| * ‖f t‖)) (Ioi 0) volume

Tactic:
refine Integrable.add (this ?_ ?_) (this ?_ ?_)
      all_goals linarith

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

Goal state before:
pp: case e_f
s t : ℂ
hs : 0 < s.re
ht : 0 < t.re
conv_int :
  ∫ (x : ℝ) in Ioi 0,
      ∫ (t_1 : ℝ) in 0..x, ↑(rexp (-t_1)) * ↑t_1 ^ (s - 1) * (↑(rexp (-(x - t_1))) * ↑(x - t_1) ^ (t - 1)) ∂volume =
    (∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (s - 1) ∂volume) * ∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (t - 1) ∂volume
hst : 0 < (s + t).re
x : ℝ
hx : x ∈ Ioi 0
y : ℝ
⊢ cexp (-↑x) = cexp (-↑y) * cexp (-(↑x - ↑y))

Tactic:
← Complex.exp_add

Goal state after:
case e_f
s t : ℂ
hs : 0 < s.re
ht : 0 < t.re
conv_int :
  ∫ (x : ℝ) in Ioi 0,
      ∫ (t_1 : ℝ) in 0..x, ↑(rexp (-t_1)) * ↑t_1 ^ (s - 1) * (↑(rexp (-(x - t_1))) * ↑(x - t_1) ^ (t - 1)) ∂volume =
    (∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (s - 1) ∂volume) * ∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (t - 1) ∂volume
hst : 0 < (s + t).re
x : ℝ
hx : x ∈ Ioi 0
y : ℝ
⊢ cexp (-↑x) = cexp (-↑y + -(↑x - ↑y))
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: s : ℝ
hs : s ≠ 0
⊢ Γ (s / 2 + 1 / 2) * Γ (s / 2 + 1) * 2 ^ (s - 1 + 1) / √π = s * (Γ (s / 2) * Γ (s / 2 + 1 / 2) * 2 ^ (s - 1) / √π)

Tactic:
Gamma_add_one (div_ne_zero hs two_ne_zero),

Goal state after:
s : ℝ
hs : s ≠ 0
⊢ Γ (s / 2 + 1 / 2) * (s / 2 * Γ (s / 2)) * 2 ^ (s - 1 + 1) / √π =
    s * (Γ (s / 2) * Γ (s / 2 + 1 / 2) * 2 ^ (s - 1) / √π)
Module: Mathlib.Topology.Algebra.AsymptoticCone

Goal state before:
pp: case a
k : Type u_1
V : Type u_2
P : Type u_3
inst✝¹⁰ : Field k
inst✝⁹ : LinearOrder k
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module k V
inst✝⁶ : AddTorsor V P
inst✝⁵ : TopologicalSpace V
inst✝⁴ : TopologicalSpace k
inst✝³ : OrderTopology k
inst✝² : IsStrictOrderedRing k
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul k V
v : V
p : P
⊢ ((𝓝 v).bind fun v ↦ map (fun x ↦ x +ᵥ p) (atTop • 𝓝 v)) ≤ map (fun x ↦ x +ᵥ p) (atTop • 𝓝 v)

Tactic:
← nhds_bind_nhds

Goal state after:
case a
k : Type u_1
V : Type u_2
P : Type u_3
inst✝¹⁰ : Field k
inst✝⁹ : LinearOrder k
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module k V
inst✝⁶ : AddTorsor V P
inst✝⁵ : TopologicalSpace V
inst✝⁴ : TopologicalSpace k
inst✝³ : OrderTopology k
inst✝² : IsStrictOrderedRing k
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul k V
v : V
p : P
⊢ ((𝓝 v).bind fun v ↦ map (fun x ↦ x +ᵥ p) (atTop • 𝓝 v)) ≤ map (fun x ↦ x +ᵥ p) (atTop • (𝓝 v).bind 𝓝)