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

Goal state before:
pp: n : ℕ
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : FiniteDimensional ℝ E
x : ↥(⋀[ℝ]^n E)
h : ∀ i ∈ Finset.univ, ((Module.Basis.exteriorPower n (stdOrthonormalBasis ℝ E).toBasis).repr x) i ^ 2 = 0
⊢ x = 0

Tactic:
apply Module.Basis.ext_elem ((stdOrthonormalBasis ℝ E).toBasis.exteriorPower n)

Goal state after:
n : ℕ
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : FiniteDimensional ℝ E
x : ↥(⋀[ℝ]^n E)
h : ∀ i ∈ Finset.univ, ((Module.Basis.exteriorPower n (stdOrthonormalBasis ℝ E).toBasis).repr x) i ^ 2 = 0
⊢ ∀ (i : ↑(Set.powersetCard (Fin (Module.finrank ℝ E)) n)),
    ((Module.Basis.exteriorPower n (stdOrthonormalBasis ℝ E).toBasis).repr x) i =
      ((Module.Basis.exteriorPower n (stdOrthonormalBasis ℝ E).toBasis).repr 0) i
Module: Mathlib.Analysis.FunctionalSpaces.SobolevInequality

Goal state before:
pp: case hfg
ι : Type u_1
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : Fintype ι
p : ℝ
hp : (↑#ι).HolderConjugate p
u : (ι → ℝ) → F
hu : ContDiff ℝ 1 u
h2u : HasCompactSupport u
this : 1 ≤ ↑#ι - 1
x : ι → ℝ
i : ι
a✝ : i ∈ Finset.univ
y : ℝ
⊢ ‖deriv (u ∘ update x i) y‖ₑ ≤ ‖fderiv ℝ u (update x i y)‖ₑ

Tactic:
calc ‖deriv (u ∘ update x i) y‖ₑ = ‖fderiv ℝ u (update x i y) (deriv (update x i) y)‖ₑ := by
        rw [fderiv_comp_deriv _ (hu.differentiable one_ne_zero).differentiableAt
          (hasDerivAt_update x i y).differentiableAt]
    _ ≤ ‖fderiv ℝ u (update x i y)‖ₑ * ‖deriv (update x i) y‖ₑ := ContinuousLinearMap.le_opENorm _ _
    _ ≤ ‖fderiv ℝ u (update x i y)‖ₑ := by simp [deriv_update, Pi.enorm_single]

Goal state after:
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
U : Submodule 𝕜 E
T : E →L[𝕜] E
hT : T.IsPositive
inst✝ : U.HasOrthogonalProjection
⊢ (∀ (x y : E), ⟪↑T (↑U.starProjection x), ↑U.starProjection y⟫ = ⟪x, ↑U.starProjection (↑T (↑U.starProjection y))⟫) ∧
    ∀ (x : E), 0 ≤ ⟪↑T (↑U.starProjection x), ↑U.starProjection x⟫

Tactic:
hT.isSymmetric _,

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
U : Submodule 𝕜 E
T : E →L[𝕜] E
hT : T.IsPositive
inst✝ : U.HasOrthogonalProjection
⊢ (∀ (x y : E), ⟪↑U.starProjection x, ↑T (↑U.starProjection y)⟫ = ⟪x, ↑U.starProjection (↑T (↑U.starProjection y))⟫) ∧
    ∀ (x : E), 0 ≤ ⟪↑U.starProjection x, ↑T (↑U.starProjection 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
T : E →L[𝕜] E
hT : T.IsPositive
U : Submodule 𝕜 E
inst✝ : U.HasOrthogonalProjection
⊢ (∀ (x y : ↥U), ⟪↑T ↑x, ↑y⟫ = ⟪↑x, ↑T ↑y⟫) ∧ ∀ (x : ↥U), 0 ≤ ⟪↑T ↑x, ↑x⟫

Tactic:
hT.isSymmetric _,

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
T : E →L[𝕜] E
hT : T.IsPositive
U : Submodule 𝕜 E
inst✝ : U.HasOrthogonalProjection
⊢ (∀ (x y : ↥U), True) ∧ ∀ (x : ↥U), 0 ≤ ⟪↑x, ↑T ↑x⟫
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
hn : 0 < finrank ℝ E
hp' : (↑p')⁻¹ = ↑p⁻¹ - (↑(finrank ℝ E))⁻¹
⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ

Tactic:
by_cases hp'0 : p' = 0

Goal state after:
case pos
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
hn : 0 < finrank ℝ E
hp' : (↑p')⁻¹ = ↑p⁻¹ - (↑(finrank ℝ E))⁻¹
hp'0 : p' = 0
⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ

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
hn : 0 < finrank ℝ E
hp' : (↑p')⁻¹ = ↑p⁻¹ - (↑(finrank ℝ E))⁻¹
hp'0 : ¬p' = 0
⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ
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)
⊢ 0 < ↑n

Tactic:
exact hn.coe.pos

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
inst✝ : CompleteSpace E
y : F
w : E
hw : ∀ (x : ↥T.domain), ⟪w, ↑x⟫ = ⟪y, ↑T x⟫
⊢ y ∈ T†.domain

Tactic:
T.mem_adjoint_domain_iff

Goal state after:
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
inst✝ : CompleteSpace E
y : F
w : E
hw : ∀ (x : ↥T.domain), ⟪w, ↑x⟫ = ⟪y, ↑T x⟫
⊢ Continuous ⇑((innerₛₗ 𝕜) y ∘ₗ T.toFun)
Module: Mathlib.Topology.MetricSpace.HausdorffDimension

Goal state before:
pp: X : Type u_2
Y : Type u_3
inst✝¹ : EMetricSpace X
inst✝ : EMetricSpace Y
K : ℝ≥0
f : X → Y
hf : AntilipschitzWith K f
s : Set Y
this✝³ : MeasurableSpace X := borel X
this✝² : BorelSpace X
this✝¹ : MeasurableSpace Y := borel Y
this✝ : BorelSpace Y
d : ℝ≥0
hd : μH[↑d] (f ⁻¹' s) = ∞
this : μH[↑d] s ≠ ∞
⊢ ↑K ^ ↑d * μH[↑d] s ≠ ∞

Tactic:
exact ENNReal.mul_ne_top (by simp) this

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✝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✝ : Subsingleton U
f : U →ₗ[𝕜] V
h : f.ker = ⊥
hrank : finrank 𝕜 U = 0
⊢ f.normDet = 1

Tactic:
let bu : OrthonormalBasis (Fin 0) 𝕜 U := (stdOrthonormalBasis 𝕜 U).reindex (by rw [hrank])

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✝ : Subsingleton U
f : U →ₗ[𝕜] V
h : f.ker = ⊥
hrank : finrank 𝕜 U = 0
bu : OrthonormalBasis (Fin 0) 𝕜 U := (stdOrthonormalBasis 𝕜 U).reindex (⋯.mpr (Equiv.refl (Fin 0)))
⊢ f.normDet = 1
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 : E
h₁ : ‖x - x‖ = 0
⊢ ‖x + x‖ = 2 • ‖x‖

Tactic:
convert norm_nsmul 𝕜 2 x

Goal state after:
𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
h₁ : ‖x - x‖ = 0
⊢ x + x = 2 • x
Module: Mathlib.Analysis.InnerProductSpace.NormDet

Goal state before:
pp: case neg
U : Type u_5
V : Type u_6
inst✝⁸ : NormedAddCommGroup U
inst✝⁷ : InnerProductSpace ℝ U
inst✝⁶ : FiniteDimensional ℝ U
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : MeasurableSpace U
inst✝² : BorelSpace U
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
f : U →ₗ[ℝ] V
s : Set U
h : ¬f.ker = ⊥
⊢ μH[↑(finrank ℝ U)] (⇑f.rangeRestrict '' s) = 0

Tactic:
have h : (finrank ℝ f.range : ℝ) < finrank ℝ U := by
      exact_mod_cast (f.normDet_eq_zero_tfae.out 1 3).mp h

Goal state after:
case neg
U : Type u_5
V : Type u_6
inst✝⁸ : NormedAddCommGroup U
inst✝⁷ : InnerProductSpace ℝ U
inst✝⁶ : FiniteDimensional ℝ U
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : MeasurableSpace U
inst✝² : BorelSpace U
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
f : U →ₗ[ℝ] V
s : Set U
h✝ : ¬f.ker = ⊥
h : ↑(finrank ℝ ↥f.range) < ↑(finrank ℝ U)
⊢ μH[↑(finrank ℝ U)] (⇑f.rangeRestrict '' s) = 0
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
this :
  K.IsHermitian ∧
    ∀ (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)
⊢ ∀ (x : H₀ K),
    0 ≤
      RCLike.re
        (Finsupp.sum x fun x_1 z ↦
          match x_1 with
          | (y, u) =>
            Finsupp.sum x fun x w ↦
              match x with
              | (x, v) => star z * w * ⟪(K x y) u, v⟫_𝕜)

Tactic:
exact this.2

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
Hle : volume ≤ μH[↑(Fintype.card ι)]
⊢ ∏ x, volume ((fun i ↦ Ioo ↑(a i) ↑(b i)) x) ≤ μH[↑(Fintype.card ι)] (univ.pi fun i ↦ Ioo ↑(a i) ↑(b i))

Tactic:
← volume_pi_pi fun i => Ioo (a i : ℝ) (b i)

Goal state after:
ι : Type u_4
inst✝ : Fintype ι
a b : ι → ℚ
H : ∀ (i : ι), a i < b i
Hle : volume ≤ μH[↑(Fintype.card ι)]
⊢ volume (univ.pi fun i ↦ Ioo ↑(a i) ↑(b i)) ≤ μH[↑(Fintype.card ι)] (univ.pi fun i ↦ Ioo ↑(a i) ↑(b i))
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:
by
    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.WeakOperatorTopology

Goal state before:
pp: 𝕜₁✝ : Type u_1
𝕜₂✝ : Type u_2
inst✝³¹ : NormedField 𝕜₁✝
inst✝³⁰ : NormedField 𝕜₂✝
σ : 𝕜₁✝ →+* 𝕜₂✝
E✝ : Type u_3
F✝ : Type u_4
inst✝²⁹ : AddCommGroup E✝
inst✝²⁸ : TopologicalSpace E✝
inst✝²⁷ : Module 𝕜₁✝ E✝
inst✝²⁶ : AddCommGroup F✝
inst✝²⁵ : TopologicalSpace F✝
inst✝²⁴ : Module 𝕜₂✝ F✝
𝕜₁ : Type u_5
𝕜₂ : Type u_6
𝕜₃ : Type u_7
𝕜₄ : Type u_8
E : Type u_9
F : Type u_10
G : Type u_11
H : Type u_12
inst✝²³ : NormedField 𝕜₁
inst✝²² : NormedField 𝕜₂
inst✝²¹ : NormedField 𝕜₃
inst✝²⁰ : NormedField 𝕜₄
σ₁₂ : 𝕜₁ →+* 𝕜₂
σ₁₃ : 𝕜₁ →+* 𝕜₃
σ₁₄ : 𝕜₁ →+* 𝕜₄
σ₂₃ : 𝕜₂ →+* 𝕜₃
σ₂₄ : 𝕜₂ →+* 𝕜₄
σ₃₄ : 𝕜₃ →+* 𝕜₄
inst✝¹⁹ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
inst✝¹⁸ : RingHomCompTriple σ₁₃ σ₃₄ σ₁₄
inst✝¹⁷ : RingHomCompTriple σ₁₂ σ₂₄ σ₁₄
inst✝¹⁶ : RingHomCompTriple σ₂₃ σ₃₄ σ₂₄
inst✝¹⁵ : AddCommGroup E
inst✝¹⁴ : TopologicalSpace E
inst✝¹³ : Module 𝕜₁ E
inst✝¹² : AddCommGroup F
inst✝¹¹ : TopologicalSpace F
inst✝¹⁰ : Module 𝕜₂ F
inst✝⁹ : AddCommGroup G
inst✝⁸ : TopologicalSpace G
inst✝⁷ : Module 𝕜₃ G
inst✝⁶ : AddCommGroup H
inst✝⁵ : TopologicalSpace H
inst✝⁴ : Module 𝕜₄ H
inst✝³ : IsTopologicalAddGroup F
inst✝² : ContinuousConstSMul 𝕜₂ F
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜₃ G
x✝ : F →WOT[𝕜₂] F
⊢ Continuous[instTopologicalSpace, instTopologicalSpace] fun x ↦ x✝ * x

Tactic:
simp_rw [mul_eq_comp]; fun_prop

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

Goal state before:
pp: 𝕜₁✝ : Type u_1
𝕜₂✝ : Type u_2
inst✝³¹ : NormedField 𝕜₁✝
inst✝³⁰ : NormedField 𝕜₂✝
σ : 𝕜₁✝ →+* 𝕜₂✝
E✝ : Type u_3
F✝ : Type u_4
inst✝²⁹ : AddCommGroup E✝
inst✝²⁸ : TopologicalSpace E✝
inst✝²⁷ : Module 𝕜₁✝ E✝
inst✝²⁶ : AddCommGroup F✝
inst✝²⁵ : TopologicalSpace F✝
inst✝²⁴ : Module 𝕜₂✝ F✝
𝕜₁ : Type u_5
𝕜₂ : Type u_6
𝕜₃ : Type u_7
𝕜₄ : Type u_8
E : Type u_9
F : Type u_10
G : Type u_11
H : Type u_12
inst✝²³ : NormedField 𝕜₁
inst✝²² : NormedField 𝕜₂
inst✝²¹ : NormedField 𝕜₃
inst✝²⁰ : NormedField 𝕜₄
σ₁₂ : 𝕜₁ →+* 𝕜₂
σ₁₃ : 𝕜₁ →+* 𝕜₃
σ₁₄ : 𝕜₁ →+* 𝕜₄
σ₂₃ : 𝕜₂ →+* 𝕜₃
σ₂₄ : 𝕜₂ →+* 𝕜₄
σ₃₄ : 𝕜₃ →+* 𝕜₄
inst✝¹⁹ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
inst✝¹⁸ : RingHomCompTriple σ₁₃ σ₃₄ σ₁₄
inst✝¹⁷ : RingHomCompTriple σ₁₂ σ₂₄ σ₁₄
inst✝¹⁶ : RingHomCompTriple σ₂₃ σ₃₄ σ₂₄
inst✝¹⁵ : AddCommGroup E
inst✝¹⁴ : TopologicalSpace E
inst✝¹³ : Module 𝕜₁ E
inst✝¹² : AddCommGroup F
inst✝¹¹ : TopologicalSpace F
inst✝¹⁰ : Module 𝕜₂ F
inst✝⁹ : AddCommGroup G
inst✝⁸ : TopologicalSpace G
inst✝⁷ : Module 𝕜₃ G
inst✝⁶ : AddCommGroup H
inst✝⁵ : TopologicalSpace H
inst✝⁴ : Module 𝕜₄ H
inst✝³ : IsTopologicalAddGroup F
inst✝² : ContinuousConstSMul 𝕜₂ F
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜₃ G
x✝ : F →WOT[𝕜₂] F
⊢ Continuous[instTopologicalSpace, instTopologicalSpace] fun x ↦ x * x✝

Tactic:
simp_rw [mul_eq_comp]; fun_prop

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
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)
⊢ Integrable (fun a ↦ ‖↑a ^ (s - 1) • f a‖) (volume.restrict T) ↔ IntegrableOn (fun t ↦ t ^ (s.re - 1) * ‖f t‖) T volume

Tactic:
← IntegrableOn

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)
⊢ IntegrableOn (fun a ↦ ‖↑a ^ (s - 1) • f a‖) T volume ↔ IntegrableOn (fun t ↦ t ^ (s.re - 1) * ‖f t‖) T volume
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: case refine_2
f : ℝ → ℝ
hfc : AEStronglyMeasurable f (volume.restrict (Ioi 0))
a s : ℝ
hf : f =O[atTop] fun x ↦ x ^ (-a)
hs : s < a
d e : ℝ
he : ∀ (b : ℝ), e ≤ b → ‖f b‖ ≤ d * ‖b ^ (-a)‖
he' : 0 < max e 1
⊢ HasFiniteIntegral (fun t ↦ t ^ (s - 1) * f t) (volume.restrict (Ioi (max e 1)))

Tactic:
have : ∀ᵐ t : ℝ ∂volume.restrict (Ioi <| max e 1),
        ‖t ^ (s - 1) * f t‖ ≤ t ^ (s - 1 + -a) * d := by
      refine (ae_restrict_mem measurableSet_Ioi).mono fun t ht => ?_
      have ht' : 0 < t := he'.trans ht
      rw [norm_mul, rpow_add ht', ← norm_of_nonneg (rpow_nonneg ht'.le (-a)), mul_assoc,
        mul_comm _ d, norm_of_nonneg (rpow_nonneg ht'.le _)]
      gcongr
      exact he t ((le_max_left e 1).trans_lt ht).le

Goal state after:
case refine_2
f : ℝ → ℝ
hfc : AEStronglyMeasurable f (volume.restrict (Ioi 0))
a s : ℝ
hf : f =O[atTop] fun x ↦ x ^ (-a)
hs : s < a
d e : ℝ
he : ∀ (b : ℝ), e ≤ b → ‖f b‖ ≤ d * ‖b ^ (-a)‖
he' : 0 < max e 1
this : ∀ᵐ (t : ℝ) ∂volume.restrict (Ioi (max e 1)), ‖t ^ (s - 1) * f t‖ ≤ t ^ (s - 1 + -a) * d
⊢ HasFiniteIntegral (fun t ↦ t ^ (s - 1) * f t) (volume.restrict (Ioi (max e 1)))
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Deriv

Goal state before:
pp: n : ℕ
⊢ ¬ContinuousAt Gamma (-↑n)

Tactic:
induction n
  case zero =>
    rw [Nat.cast_zero, neg_zero]
    exact not_continuousAt_Gamma_zero
  case succ n ih =>
    contrapose ih
    rw [Nat.cast_add, Nat.cast_one] at ih
    suffices ContinuousAt (fun s ↦ Gamma (s - 1 + 1)) (-n) by simpa using this
    suffices ContinuousAt (fun s ↦ Gamma (s + 1)) (-n - 1) from
      this.comp' (f := fun s ↦ s - 1) (continuous_sub_right 1).continuousAt
    rw [← neg_add']
    have h0 : -(n + 1) ≠ (0 : ℂ) := neg_ne_zero.mpr n.cast_add_one_ne_zero
    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.MellinTransform

Goal state before:
pp: case refine_2.h
b : ℝ
f : ℝ → ℝ
hfc : AEStronglyMeasurable f (volume.restrict (Ioi 0))
hf : f =O[𝓝[>] 0] fun x ↦ x ^ (-b)
s : ℝ
hs : b < s
d : ℝ
left✝ : d > 0
ε : ℝ
hε : 0 < ε
t : ℝ
hε' : ‖f t‖ ≤ d * ‖t ^ (-b)‖
ht : t ∈ Ioo 0 ε
⊢ d * ‖t ^ (-b)‖ * ‖t ^ (s - 1)‖ = d * t ^ (s - b - 1)

Tactic:
norm_of_nonneg (rpow_nonneg ht.1.le _),

Goal state after:
case refine_2.h
b : ℝ
f : ℝ → ℝ
hfc : AEStronglyMeasurable f (volume.restrict (Ioi 0))
hf : f =O[𝓝[>] 0] fun x ↦ x ^ (-b)
s : ℝ
hs : b < s
d : ℝ
left✝ : d > 0
ε : ℝ
hε : 0 < ε
t : ℝ
hε' : ‖f t‖ ≤ d * ‖t ^ (-b)‖
ht : t ∈ Ioo 0 ε
⊢ d * t ^ (-b) * t ^ (s - 1) = d * t ^ (s - b - 1)
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: case refine_2.h
b : ℝ
f : ℝ → ℝ
hfc : AEStronglyMeasurable f (volume.restrict (Ioi 0))
hf : f =O[𝓝[>] 0] fun x ↦ x ^ (-b)
s : ℝ
hs : b < s
d : ℝ
left✝ : d > 0
ε : ℝ
hε : 0 < ε
t : ℝ
hε' : ‖f t‖ ≤ d * ‖t ^ (-b)‖
ht : t ∈ Ioo 0 ε
⊢ ‖f t‖ * ‖t ^ (s - 1)‖ ≤ d * t ^ (s - b - 1)

Tactic:
· calc _ ≤ d * ‖t ^ (-b)‖ * ‖t ^ (s - 1)‖ := by gcongr
          _ = d * t ^ (s - b - 1) := ?_
        simp_rw [norm_of_nonneg (rpow_nonneg ht.1.le _), mul_assoc]
        rw [← rpow_add ht.1]
        congr 2
        abel

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

Goal state before:
pp: case e'_9.e_a
z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
x : ℝ
a✝ : x ∈ uIcc 0 (π / 2)
c : HasDerivAt ((fun x ↦ x ^ (n - 1)) ∘ Complex.cos) (↑(n - 1) * Complex.cos ↑x ^ (n - 1 - 1) * -Complex.sin ↑x) ↑x
this : ↑(n - 1) = ↑n - 1
⊢ (↑n - 1) * Complex.sin ↑x ^ 2 * Complex.cos ↑x ^ (n - 2) =
    Complex.sin ↑x * (↑(n - 1) * Complex.cos ↑x ^ (n - 1 - 1) * Complex.sin ↑x)

Tactic:
rw [Nat.sub_sub, this]

Goal state after:
case e'_9.e_a
z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
x : ℝ
a✝ : x ∈ uIcc 0 (π / 2)
c : HasDerivAt ((fun x ↦ x ^ (n - 1)) ∘ Complex.cos) (↑(n - 1) * Complex.cos ↑x ^ (n - 1 - 1) * -Complex.sin ↑x) ↑x
this : ↑(n - 1) = ↑n - 1
⊢ (↑n - 1) * Complex.sin ↑x ^ 2 * Complex.cos ↑x ^ (n - 2) =
    Complex.sin ↑x * ((↑n - 1) * Complex.cos ↑x ^ (n - (1 + 1)) * Complex.sin ↑x)
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: case hx
g : ℝ → ℂ
t : ℂ
hg : Tendsto (fun x ↦ ↑x * g x) atTop (𝓝 t)
hg0✝ : Tendsto g atTop (𝓝 0)
x : ℝ
hg1 : -1 ≠ -1
hg0 : 1 + g x = 0
⊢ False

Tactic:
norm_num at hg1

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

Goal state before:
pp: 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
⊢ u * u.betaIntegral (v + 1) = v * (u + 1).betaIntegral v

Tactic:
have h_int := ((betaIntegral_convergent hu hv').const_mul u).sub
    ((betaIntegral_convergent hu' hv).const_mul v)

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 + 1 - 1)) - v * (↑x ^ (u + 1 - 1) * (1 - ↑x) ^ (v - 1)))
    volume 0 1
⊢ u * u.betaIntegral (v + 1) = v * (u + 1).betaIntegral v
Module: Mathlib.Analysis.Meromorphic.LogDeriv

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
𝕜' : Type u_2
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
U : Set 𝕜
ι : Type u_3
s : Finset ι
F : ι → 𝕜 → 𝕜'
h : ∀ i ∈ s, MeromorphicOn (F i) U
h' : ∀ i ∈ s, ∀ x ∈ U, meromorphicOrderAt (F i) x ≠ ⊤
hA : ∀ᶠ (y : 𝕜) in codiscreteWithin U, ∀ i ∈ s, AnalyticAt 𝕜 (F i) y
hN : ∀ᶠ (y : 𝕜) in codiscreteWithin U, ∀ i ∈ s, F i y ≠ 0
⊢ logDeriv (∏ i ∈ s, F i) =ᶠ[codiscreteWithin U] ∑ i ∈ s, logDeriv (F i)

Tactic:
filter_upwards [hA, hN] with y h₁y h₂y

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
𝕜' : Type u_2
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
U : Set 𝕜
ι : Type u_3
s : Finset ι
F : ι → 𝕜 → 𝕜'
h : ∀ i ∈ s, MeromorphicOn (F i) U
h' : ∀ i ∈ s, ∀ x ∈ U, meromorphicOrderAt (F i) x ≠ ⊤
hA : ∀ᶠ (y : 𝕜) in codiscreteWithin U, ∀ i ∈ s, AnalyticAt 𝕜 (F i) y
hN : ∀ᶠ (y : 𝕜) in codiscreteWithin U, ∀ i ∈ s, F i y ≠ 0
y : 𝕜
h₁y : ∀ i ∈ s, AnalyticAt 𝕜 (F i) y
h₂y : ∀ i ∈ s, F i y ≠ 0
⊢ logDeriv (∏ i ∈ s, F i) y = (∑ i ∈ s, logDeriv (F i)) y
Module: Mathlib.Analysis.Meromorphic.LogDeriv

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
𝕜' : Type u_2
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
U : Set 𝕜
ι : Type u_3
F : ι → 𝕜 → 𝕜'
d : ι → ℤ
hd : (support d).Finite
h : ∀ (i : ι), MeromorphicOn (F i) U
h' : ∀ (i : ι), ∀ x ∈ U, meromorphicOrderAt (F i) x ≠ ⊤
hA : ∀ᶠ (y : 𝕜) in codiscreteWithin U, ∀ i ∈ hd.toFinset, AnalyticAt 𝕜 (F i) y
hN : ∀ᶠ (y : 𝕜) in codiscreteWithin U, ∀ i ∈ hd.toFinset, F i y ≠ 0
⊢ logDeriv (∏ᶠ (i : ι), F i ^ d i) =ᶠ[codiscreteWithin U] fun z ↦ ∑ᶠ (i : ι), d i • logDeriv (F i) z

Tactic:
filter_upwards [hA, hN] with y h₁y h₂y

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
𝕜' : Type u_2
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
U : Set 𝕜
ι : Type u_3
F : ι → 𝕜 → 𝕜'
d : ι → ℤ
hd : (support d).Finite
h : ∀ (i : ι), MeromorphicOn (F i) U
h' : ∀ (i : ι), ∀ x ∈ U, meromorphicOrderAt (F i) x ≠ ⊤
hA : ∀ᶠ (y : 𝕜) in codiscreteWithin U, ∀ i ∈ hd.toFinset, AnalyticAt 𝕜 (F i) y
hN : ∀ᶠ (y : 𝕜) in codiscreteWithin U, ∀ i ∈ hd.toFinset, F i y ≠ 0
y : 𝕜
h₁y : ∀ i ∈ hd.toFinset, AnalyticAt 𝕜 (F i) y
h₂y : ∀ i ∈ hd.toFinset, F i y ≠ 0
⊢ logDeriv (∏ᶠ (i : ι), F i ^ d i) y = ∑ᶠ (i : ι), d i • logDeriv (F i) y