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

Goal state before:
pp: x y : ℝ≥0
hn : x ≠ y
n : ℕ
⊢ (x.agmSequences y n).1.agm (x.agmSequences y n).2 < (x.agmSequences y n).2

Tactic:
set p := (agmSequences x y n).1

Goal state after:
x y : ℝ≥0
hn : x ≠ y
n : ℕ
p : ℝ≥0 := (x.agmSequences y n).1
⊢ p.agm (x.agmSequences y n).2 < (x.agmSequences y n).2
Module: Mathlib.Analysis.SpecialFunctions.Complex.Arctan

Goal state before:
pp: case e'_6
z : ℂ
hz : ‖z‖ < 1
this :
  HasSum (fun x ↦ -I / 2 * (((-1) ^ (2 * x.1 + ↑x.2 + 1) + 1) * (z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)))
    (-I / 2 * log ((1 + z * I) / (1 - z * I)))
k : ℕ
⊢ (-1) ^ k * z ^ (2 * k + 1) / ↑(2 * k + 1) =
    -I / 2 * (((-1) ^ (2 * (k + 1)) + 1) * (z * I) ^ (2 * k + 1) / ↑(2 * k + 1))

Tactic:
Even.neg_one_pow (n := 2 * (k + 1)) (by simp),

Goal state after:
case e'_6
z : ℂ
hz : ‖z‖ < 1
this :
  HasSum (fun x ↦ -I / 2 * (((-1) ^ (2 * x.1 + ↑x.2 + 1) + 1) * (z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)))
    (-I / 2 * log ((1 + z * I) / (1 - z * I)))
k : ℕ
⊢ (-1) ^ k * z ^ (2 * k + 1) / ↑(2 * k + 1) = -I / 2 * ((1 + 1) * (z * I) ^ (2 * k + 1) / ↑(2 * k + 1))
Module: Mathlib.Analysis.SpecialFunctions.BinaryEntropy

Goal state before:
pp: p : ℝ
hp : p < 0
hp' : p < -1
⊢ log (-p) < log (1 - p)

Tactic:
exact log_lt_log (Left.neg_pos_iff.mpr hp) (by linarith)

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.CompareExp

Goal state before:
pp: l : Filter ℂ
hl : IsExpCmpFilter l
z : ℂ
hz : 1 ≤ z.re
h2 : 0 < √2
⊢ (fun x ↦ ‖Real.log ‖x‖‖) z ≤ (fun z ↦ Real.log √2 + Real.log (max z.re |z.im|)) z

Tactic:
have hz' : 1 ≤ ‖z‖ := hz.trans (re_le_norm z)

Goal state after:
l : Filter ℂ
hl : IsExpCmpFilter l
z : ℂ
hz : 1 ≤ z.re
h2 : 0 < √2
hz' : 1 ≤ ‖z‖
⊢ (fun x ↦ ‖Real.log ‖x‖‖) z ≤ (fun z ↦ Real.log √2 + Real.log (max z.re |z.im|)) z
Module: Mathlib.Analysis.SpecialFunctions.BinaryEntropy

Goal state before:
pp: ⊢ StrictMonoOn binEntropy (Icc 0 2⁻¹)

Tactic:
by
  rw [show Icc (0 : ℝ) 2⁻¹ = Icc 0 (1 - 1 / 2) by norm_num, ← qaryEntropy_two]
  exact qaryEntropy_strictMonoOn (by rfl)

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Order

Goal state before:
pp: A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ
hp : p ∈ Icc 0 1
⊢ Monotone fun a ↦ a ^ p

Tactic:
let q : ℝ≥0 := ⟨p, hp.1⟩

Goal state after:
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ
hp : p ∈ Icc 0 1
q : ℝ≥0 := ⟨p, ⋯⟩
⊢ Monotone fun a ↦ a ^ p
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Order

Goal state before:
pp: A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ
hp : p ∈ Icc 0 1
⊢ ConcaveOn ℝ (Ici 0) fun a ↦ a ^ p

Tactic:
let q : ℝ≥0 := ⟨p, hp.1⟩

Goal state after:
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ
hp : p ∈ Icc 0 1
q : ℝ≥0 := ⟨p, ⋯⟩
⊢ ConcaveOn ℝ (Ici 0) fun a ↦ a ^ p
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation

Goal state before:
pp: case ae_le
p x : ℝ
hp : p ∈ Ioo 0 1
hx : 0 ≤ x
⊢ ∀ᵐ (a : ℝ) ∂volume.restrict (Ioi 1), ‖p.rpowIntegrand₀₁ a x‖ ≤ a ^ (p - 2) * x

Tactic:
case ae_le =>
      refine ae_restrict_of_forall_mem measurableSet_Ioi fun t (ht : 1 < t) => ?_
      rw [Real.norm_of_nonneg (rpowIntegrand₀₁_nonneg hp.1 (by positivity) hx)]
      exact rpowIntegrand₀₁_le_rpow_sub_two_mul_self hp (by positivity) hx

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation

Goal state before:
pp: case refine_2
p : ℝ
hp : p ∈ Ioo 0 1
C : ℝ≥0 := NNReal.mk (∫ (t : ℝ) in Ioi 0, p.rpowIntegrand₀₁ t 1)⁻¹ ⋯
x : ℝ
hx : x ∈ Ici 0
⊢ x ^ p = ∫ (t : ℝ), p.rpowIntegrand₀₁ t x ∂C • volume.restrict (Ioi 0)

Tactic:
integral_smul_nnreal_measure,

Goal state after:
case refine_2
p : ℝ
hp : p ∈ Ioo 0 1
C : ℝ≥0 := NNReal.mk (∫ (t : ℝ) in Ioi 0, p.rpowIntegrand₀₁ t 1)⁻¹ ⋯
x : ℝ
hx : x ∈ Ici 0
⊢ x ^ p = C • ∫ (t : ℝ) in Ioi 0, p.rpowIntegrand₀₁ t x ∂volume
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation

Goal state before:
pp: case refine_2
p : ℝ
hp : p ∈ Ioo 1 2
C : ℝ≥0 := NNReal.mk (∫ (t : ℝ) in Ioi 0, (p - 1).rpowIntegrand₀₁ t 1)⁻¹ ⋯
μ : Measure ℝ := C • volume
x : ℝ
hx : x ∈ Ici 0
⊢ x ^ p = ∫ (t : ℝ), p.rpowIntegrand₁₂ t x ∂C • volume.restrict (Ioi 0)

Tactic:
integral_smul_nnreal_measure,

Goal state after:
case refine_2
p : ℝ
hp : p ∈ Ioo 1 2
C : ℝ≥0 := NNReal.mk (∫ (t : ℝ) in Ioi 0, (p - 1).rpowIntegrand₀₁ t 1)⁻¹ ⋯
μ : Measure ℝ := C • volume
x : ℝ
hx : x ∈ Ici 0
⊢ x ^ p = C • ∫ (x_1 : ℝ) in Ioi 0, p.rpowIntegrand₁₂ x_1 x ∂volume
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Digamma

Goal state before:
pp: s : ℂ
hs : ∀ (m : ℕ), s ≠ -↑m
⊢ (s + 1).digamma = s.digamma + s⁻¹

Tactic:
have hs0 : s ≠ 0 := by simpa using hs 0

Goal state after:
s : ℂ
hs : ∀ (m : ℕ), s ≠ -↑m
hs0 : s ≠ 0
⊢ (s + 1).digamma = s.digamma + s⁻¹
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation

Goal state before:
pp: A : Type u_1
inst✝⁹ : NonUnitalNormedRing A
inst✝⁸ : StarRing A
inst✝⁷ : NormedSpace ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : CompleteSpace A
p : ℝ≥0
hp : p ∈ Ioo 0 1
μ : Measure ℝ
hμ :
  ∀ x ∈ Ici 0,
    IntegrableOn (fun t ↦ (↑p).rpowIntegrand₀₁ t x) (Ioi 0) μ ∧ x ^ ↑p = ∫ (t : ℝ) in Ioi 0, (↑p).rpowIntegrand₀₁ t x ∂μ
a : A
ha : 0 ≤ a
a✝ : Nontrivial A
p_pos : 0 < ↑p
f : ℝ → ℝ → ℝ := fun t ↦ (↑p).rpowIntegrand₀₁ t
maxr : ℝ := sSup (quasispectrum ℝ a)
maxr_nonneg : 0 ≤ maxr
bound : ℝ → ℝ := fun t ↦ ‖f t maxr‖
hf : ContinuousOn (Function.uncurry f) (Ioi 0 ×ˢ quasispectrum ℝ a)
t : ℝ
ht : t ∈ Ioi 0
z : ℝ
hz : z ∈ quasispectrum ℝ a
⊢ ‖f t z‖ ≤ bound t

Tactic:
have hz' : 0 ≤ z := by grind

Goal state after:
A : Type u_1
inst✝⁹ : NonUnitalNormedRing A
inst✝⁸ : StarRing A
inst✝⁷ : NormedSpace ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : CompleteSpace A
p : ℝ≥0
hp : p ∈ Ioo 0 1
μ : Measure ℝ
hμ :
  ∀ x ∈ Ici 0,
    IntegrableOn (fun t ↦ (↑p).rpowIntegrand₀₁ t x) (Ioi 0) μ ∧ x ^ ↑p = ∫ (t : ℝ) in Ioi 0, (↑p).rpowIntegrand₀₁ t x ∂μ
a : A
ha : 0 ≤ a
a✝ : Nontrivial A
p_pos : 0 < ↑p
f : ℝ → ℝ → ℝ := fun t ↦ (↑p).rpowIntegrand₀₁ t
maxr : ℝ := sSup (quasispectrum ℝ a)
maxr_nonneg : 0 ≤ maxr
bound : ℝ → ℝ := fun t ↦ ‖f t maxr‖
hf : ContinuousOn (Function.uncurry f) (Ioi 0 ×ˢ quasispectrum ℝ a)
t : ℝ
ht : t ∈ Ioi 0
z : ℝ
hz : z ∈ quasispectrum ℝ a
hz' : 0 ≤ z
⊢ ‖f t z‖ ≤ bound t
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation

Goal state before:
pp: A : Type u_1
inst✝⁹ : NonUnitalNormedRing A
inst✝⁸ : StarRing A
inst✝⁷ : NormedSpace ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : CompleteSpace A
p : ℝ≥0
hp : p ∈ Ioo 1 2
μ : Measure ℝ
hμ :
  ∀ x ∈ Ici 0,
    IntegrableOn (fun t ↦ (↑p).rpowIntegrand₁₂ t x) (Ioi 0) μ ∧ x ^ ↑p = ∫ (t : ℝ) in Ioi 0, (↑p).rpowIntegrand₁₂ t x ∂μ
a : A
ha : 0 ≤ a
hpcoe : ↑p ∈ Ioo 1 2
f : ℝ → ℝ → ℝ := fun t ↦ (↑p).rpowIntegrand₁₂ t
maxr : ℝ := sSup (quasispectrum ℝ a)
maxr_nonneg : 0 ≤ maxr
bound : ℝ → ℝ := fun t ↦ ‖f t maxr‖
⊢ IntegrableOn (fun t ↦ cfcₙ ((↑p).rpowIntegrand₁₂ t) a) (Ioi 0) μ ∧
    a ^ p = ∫ (t : ℝ) in Ioi 0, cfcₙ ((↑p).rpowIntegrand₁₂ t) a ∂μ

Tactic:
have hf : ContinuousOn (Function.uncurry f) (Ioi (0 : ℝ) ×ˢ quasispectrum ℝ a) :=
    continuousOn_rpowIntegrand₁₂_uncurry hpcoe.1 (quasispectrum ℝ a) (by grind)

Goal state after:
A : Type u_1
inst✝⁹ : NonUnitalNormedRing A
inst✝⁸ : StarRing A
inst✝⁷ : NormedSpace ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : CompleteSpace A
p : ℝ≥0
hp : p ∈ Ioo 1 2
μ : Measure ℝ
hμ :
  ∀ x ∈ Ici 0,
    IntegrableOn (fun t ↦ (↑p).rpowIntegrand₁₂ t x) (Ioi 0) μ ∧ x ^ ↑p = ∫ (t : ℝ) in Ioi 0, (↑p).rpowIntegrand₁₂ t x ∂μ
a : A
ha : 0 ≤ a
hpcoe : ↑p ∈ Ioo 1 2
f : ℝ → ℝ → ℝ := fun t ↦ (↑p).rpowIntegrand₁₂ t
maxr : ℝ := sSup (quasispectrum ℝ a)
maxr_nonneg : 0 ≤ maxr
bound : ℝ → ℝ := fun t ↦ ‖f t maxr‖
hf : ContinuousOn (Function.uncurry f) (Ioi 0 ×ˢ quasispectrum ℝ a)
⊢ IntegrableOn (fun t ↦ cfcₙ ((↑p).rpowIntegrand₁₂ t) a) (Ioi 0) μ ∧
    a ^ p = ∫ (t : ℝ) in Ioi 0, cfcₙ ((↑p).rpowIntegrand₁₂ t) a ∂μ
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation

Goal state before:
pp: A : Type u_1
inst✝⁹ : NonUnitalNormedRing A
inst✝⁸ : StarRing A
inst✝⁷ : NormedSpace ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : CompleteSpace A
p : ℝ≥0
hp : p ∈ Ioo 1 2
μ : Measure ℝ
hμ :
  ∀ x ∈ Ici 0,
    IntegrableOn (fun t ↦ (↑p).rpowIntegrand₁₂ t x) (Ioi 0) μ ∧ x ^ ↑p = ∫ (t : ℝ) in Ioi 0, (↑p).rpowIntegrand₁₂ t x ∂μ
a : A
ha : 0 ≤ a
hpcoe : ↑p ∈ Ioo 1 2
f : ℝ → ℝ → ℝ := fun t ↦ (↑p).rpowIntegrand₁₂ t
maxr : ℝ := sSup (quasispectrum ℝ a)
maxr_nonneg : 0 ≤ maxr
bound : ℝ → ℝ := fun t ↦ ‖f t maxr‖
hf : ContinuousOn (Function.uncurry f) (Ioi 0 ×ˢ quasispectrum ℝ a)
t : ℝ
ht : t ∈ Ioi 0
z : ℝ
hz : z ∈ quasispectrum ℝ a
⊢ ‖f t z‖ ≤ bound t

Tactic:
have hz' : 0 ≤ z := by grind

Goal state after:
A : Type u_1
inst✝⁹ : NonUnitalNormedRing A
inst✝⁸ : StarRing A
inst✝⁷ : NormedSpace ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : CompleteSpace A
p : ℝ≥0
hp : p ∈ Ioo 1 2
μ : Measure ℝ
hμ :
  ∀ x ∈ Ici 0,
    IntegrableOn (fun t ↦ (↑p).rpowIntegrand₁₂ t x) (Ioi 0) μ ∧ x ^ ↑p = ∫ (t : ℝ) in Ioi 0, (↑p).rpowIntegrand₁₂ t x ∂μ
a : A
ha : 0 ≤ a
hpcoe : ↑p ∈ Ioo 1 2
f : ℝ → ℝ → ℝ := fun t ↦ (↑p).rpowIntegrand₁₂ t
maxr : ℝ := sSup (quasispectrum ℝ a)
maxr_nonneg : 0 ≤ maxr
bound : ℝ → ℝ := fun t ↦ ‖f t maxr‖
hf : ContinuousOn (Function.uncurry f) (Ioi 0 ×ˢ quasispectrum ℝ a)
t : ℝ
ht : t ∈ Ioi 0
z : ℝ
hz : z ∈ quasispectrum ℝ a
hz' : 0 ≤ z
⊢ ‖f t z‖ ≤ bound t
Module: Mathlib.Analysis.SpecialFunctions.Gaussian.PoissonSummation

Goal state before:
pp: a : ℂ
ha : 0 < a.re
b : ℂ
f : ℝ → ℂ := fun x ↦ cexp (-↑π * a * ↑x ^ 2 + 2 * ↑π * b * ↑x)
hFf : 𝓕 f = fun x ↦ 1 / a ^ (1 / 2) * cexp (-↑π / a * (↑x + I * b) ^ 2)
h1 : 0 < (↑π * a).re
h2 : 0 < (↑π / a).re
f_bd : f =O[cocompact ℝ] fun x ↦ |x| ^ (-2)
this : ∀ (x : ℝ), -↑π / a * (↑x + I * b) ^ 2 = -↑π / a * ↑x ^ 2 + -2 * ↑π * I * b / a * ↑x + ↑π * b ^ 2 / a
⊢ (-↑π / a).re < 0

Tactic:
rwa [neg_div, neg_re, neg_lt_zero]

Goal state after:
Module: Mathlib.NumberTheory.Harmonic.GammaDeriv

Goal state before:
pp: f : ℂ → ℂ := fun s ↦ ↑π ^ (-s / 2)
g : ℂ → ℂ := fun s ↦ Gamma (s / 2)
aux : ↑π ^ (1 / 2) = ↑√π
aux2 : ↑√π ≠ 0
hf : HasDerivAt f (-log ↑π / 2 / ↑√π) 1
hg : HasDerivAt g (-↑√π * (↑γ + 2 * log 2) / 2) 1
⊢ -(↑γ + ↑(Real.log (↑2 ^ 2) + Real.log π)) / ↑↑2 = -(↑γ + log (4 * ↑π)) / ↑↑2

Tactic:
← Real.log_mul (by positivity) (by positivity),

Goal state after:
f : ℂ → ℂ := fun s ↦ ↑π ^ (-s / 2)
g : ℂ → ℂ := fun s ↦ Gamma (s / 2)
aux : ↑π ^ (1 / 2) = ↑√π
aux2 : ↑√π ≠ 0
hf : HasDerivAt f (-log ↑π / 2 / ↑√π) 1
hg : HasDerivAt g (-↑√π * (↑γ + 2 * log 2) / 2) 1
⊢ -(↑γ + ↑(Real.log (↑2 ^ 2 * π))) / ↑↑2 = -(↑γ + log (4 * ↑π)) / ↑↑2
Module: Mathlib.Analysis.SpecialFunctions.Log.Base

Goal state before:
pp: b : ℝ
hb : 1 < b
x : ℝ
hx : x < 0
y : ℝ
hy : y < 0
hxy : x < y
⊢ |y| < |x|

Tactic:
rwa [abs_of_neg hy, abs_of_neg hx, neg_lt_neg_iff]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Log.Base

Goal state before:
pp: b : ℝ
b_pos : 0 < b
b_lt_one : b < 1
x : ℝ
hx : x < 0
y : ℝ
hy : y < 0
hxy : x < y
⊢ |y| < |x|

Tactic:
rwa [abs_of_neg hy, abs_of_neg hx, neg_lt_neg_iff]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.BoundedContinuousFunction

Goal state before:
pp: case e'_3
X : Type u_1
inst✝ : TopologicalSpace X
f : X →ᵇ ℝ≥0
x : X
⊢ f x = nndist (0 x) (f x)

Tactic:
simp only [coe_zero, Pi.zero_apply, NNReal.nndist_zero_eq_val]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.MulExpNegMulSqIntegral

Goal state before:
pp: case neg
ε : ℝ
E : Type u_2
inst✝⁶ : MeasurableSpace E
inst✝⁵ : PseudoEMetricSpace E
inst✝⁴ : BorelSpace E
inst✝³ : CompleteSpace E
inst✝² : SecondCountableTopology E
P P' : Measure E
inst✝¹ : IsFiniteMeasure P
inst✝ : IsFiniteMeasure P'
f : E →ᵇ ℝ
A : Subalgebra ℝ (E →ᵇ ℝ)
hA : (Subalgebra.map (toContinuousMapₐ ℝ) A).SeparatesPoints
heq : ∀ g ∈ A, ∫ (x : E), g x ∂P = ∫ (x : E), g x ∂P'
hε : 0 < ε
hPP' : ¬(P = 0 ∧ P' = 0)
const : ℝ := max (P.real Set.univ) (P'.real Set.univ)
pos_of_measure : 0 < const
KP : Set E
left✝¹ : KP ⊆ Set.univ
hKPco : IsCompact KP
hKPcl : IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] KP
hKP : P (Set.univ \ KP) < ENNReal.ofReal ε
KP' : Set E
left✝ : KP' ⊆ Set.univ
hKP'co : IsCompact KP'
hKP'cl : IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] KP'
hKP' : P' (Set.univ \ KP') < ENNReal.ofReal ε
⊢ |∫ (x : E), ε.mulExpNegMulSq (f x) ∂P - ∫ (x : E), ε.mulExpNegMulSq (f x) ∂P'| ≤ 6 * √ε

Tactic:
let K := KP ∪ KP'

Goal state after:
case neg
ε : ℝ
E : Type u_2
inst✝⁶ : MeasurableSpace E
inst✝⁵ : PseudoEMetricSpace E
inst✝⁴ : BorelSpace E
inst✝³ : CompleteSpace E
inst✝² : SecondCountableTopology E
P P' : Measure E
inst✝¹ : IsFiniteMeasure P
inst✝ : IsFiniteMeasure P'
f : E →ᵇ ℝ
A : Subalgebra ℝ (E →ᵇ ℝ)
hA : (Subalgebra.map (toContinuousMapₐ ℝ) A).SeparatesPoints
heq : ∀ g ∈ A, ∫ (x : E), g x ∂P = ∫ (x : E), g x ∂P'
hε : 0 < ε
hPP' : ¬(P = 0 ∧ P' = 0)
const : ℝ := max (P.real Set.univ) (P'.real Set.univ)
pos_of_measure : 0 < const
KP : Set E
left✝¹ : KP ⊆ Set.univ
hKPco : IsCompact KP
hKPcl : IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] KP
hKP : P (Set.univ \ KP) < ENNReal.ofReal ε
KP' : Set E
left✝ : KP' ⊆ Set.univ
hKP'co : IsCompact KP'
hKP'cl : IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] KP'
hKP' : P' (Set.univ \ KP') < ENNReal.ofReal ε
K : Set E := KP ∪ KP'
⊢ |∫ (x : E), ε.mulExpNegMulSq (f x) ∂P - ∫ (x : E), ε.mulExpNegMulSq (f x) ∂P'| ≤ 6 * √ε
Module: Mathlib.MeasureTheory.Constructions.HaarToSphere

Goal state before:
pp: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : MeasurableSpace E
μ : Measure E
inst✝ : BorelSpace E
s : Set ↑(sphere 0 1)
hs : MeasurableSet s
⊢ dim E •
      (comap (Subtype.val ∘ ⇑(homeomorphUnitSphereProd E).symm) μ)
        (Prod.fst ⁻¹' s ∩ univ ×ˢ Iio ⟨1, toSphere._proof_1⟩) =
    ↑(dim E) * μ (Ioo 0 1 • Subtype.val '' s)

Tactic:
((MeasurableEmbedding.subtype_coe (measurableSet_singleton _).compl).comp
      (Homeomorph.measurableEmbedding _)).comap_apply,

Goal state after:
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : MeasurableSpace E
μ : Measure E
inst✝ : BorelSpace E
s : Set ↑(sphere 0 1)
hs : MeasurableSet s
⊢ dim E •
      μ (Subtype.val ∘ ⇑(homeomorphUnitSphereProd E).symm '' (Prod.fst ⁻¹' s ∩ univ ×ˢ Iio ⟨1, toSphere._proof_1⟩)) =
    ↑(dim E) * μ (Ioo 0 1 • Subtype.val '' s)
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: case refine_2
L : PeriodPair
β : ℚ
h₂ : β.den = 1
α : ℤ
x✝ : (↑α).den = 1 ∧ β.den = 1
⊢ ↑↑α * L.ω₁ + ↑β * L.ω₂ ∈ L.lattice

Tactic:
lift β to ℤ using h₂

Goal state after:
case refine_2
L : PeriodPair
α β : ℤ
x✝ : (↑α).den = 1 ∧ (↑β).den = 1
⊢ ↑↑α * L.ω₁ + ↑↑β * L.ω₂ ∈ L.lattice
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
⊢ L.lattice = Submodule.span ℤ (Set.range ⇑L.basis)

Tactic:
have : Finset.univ (α := Fin 2) = {0, 1} := rfl

Goal state after:
L : PeriodPair
this : Finset.univ = {0, 1}
⊢ L.lattice = Submodule.span ℤ (Set.range ⇑L.basis)
Module: Mathlib.Analysis.SpecialFunctions.Pow.Integral

Goal state before:
pp: E : Type u_2
F : Type u_3
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : NormedAddCommGroup F
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hd : 1 ≤ Module.finrank ℝ E
f : E → F
C α r : ℝ
hα : α < ↑(Module.finrank ℝ E)
h_decay : ∀ᵐ (x : E) ∂μ.restrict (ball 0 r), ‖f x‖ ≤ C * ‖x‖ ^ (-α)
h_meas : AEStronglyMeasurable f μ
this : Nontrivial E
⊢ IntegrableOn (fun y ↦ y ^ (↑(Module.finrank ℝ E) - 1 - α)) (Ioo 0 r) volume

Tactic:
by_cases! hr : 0 < r

Goal state after:
case pos
E : Type u_2
F : Type u_3
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : NormedAddCommGroup F
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hd : 1 ≤ Module.finrank ℝ E
f : E → F
C α r : ℝ
hα : α < ↑(Module.finrank ℝ E)
h_decay : ∀ᵐ (x : E) ∂μ.restrict (ball 0 r), ‖f x‖ ≤ C * ‖x‖ ^ (-α)
h_meas : AEStronglyMeasurable f μ
this : Nontrivial E
hr : 0 < r
⊢ IntegrableOn (fun y ↦ y ^ (↑(Module.finrank ℝ E) - 1 - α)) (Ioo 0 r) volume

case neg
E : Type u_2
F : Type u_3
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : NormedAddCommGroup F
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hd : 1 ≤ Module.finrank ℝ E
f : E → F
C α r : ℝ
hα : α < ↑(Module.finrank ℝ E)
h_decay : ∀ᵐ (x : E) ∂μ.restrict (ball 0 r), ‖f x‖ ≤ C * ‖x‖ ^ (-α)
h_meas : AEStronglyMeasurable f μ
this : Nontrivial E
hr : r ≤ 0
⊢ IntegrableOn (fun y ↦ y ^ (↑(Module.finrank ℝ E) - 1 - α)) (Ioo 0 r) volume
Module: Mathlib.Analysis.SpecialFunctions.RegularizedHypergeometric

Goal state before:
pp: n : ℕ
a b : Multiset ℂ
hn : n ≠ 0
⊢ (Multiset.map (fun x ↦ x + ↑n) a).prod / ((Multiset.map (fun x ↦ x + ↑n) b).prod * (↑n + 1)) =
    ↑n ^ (↑a.card - ↑b.card - 1) * (Multiset.map (fun x ↦ x / ↑n + 1) a).prod /
      ((Multiset.map (fun x ↦ x / ↑n + 1) b).prod * (1 + (↑n)⁻¹))

Tactic:
rw [multiset_prod_eq_pow_mul_multiset_prod a hn, multiset_prod_eq_pow_mul_multiset_prod b hn]

Goal state after:
n : ℕ
a b : Multiset ℂ
hn : n ≠ 0
⊢ ↑n ^ a.card * (Multiset.map (fun x ↦ x / ↑n + 1) a).prod /
      (↑n ^ b.card * (Multiset.map (fun x ↦ x / ↑n + 1) b).prod * (↑n + 1)) =
    ↑n ^ (↑a.card - ↑b.card - 1) * (Multiset.map (fun x ↦ x / ↑n + 1) a).prod /
      ((Multiset.map (fun x ↦ x / ↑n + 1) b).prod * (1 + (↑n)⁻¹))
Module: Mathlib.Analysis.SpecialFunctions.RegularizedHypergeometric

Goal state before:
pp: a b : Multiset ℂ
h : a.card ≤ b.card
ha : ∀ j ∈ a, ∀ (k : ℕ), j ≠ -↑k
h₁ : Tendsto (fun n ↦ ↑n ^ (↑a.card - ↑b.card - 1)) atTop (𝓝 0)
this :
  Tendsto
    (fun x ↦
      ‖↑x ^ (↑a.card - ↑b.card - 1) *
          ((Multiset.map (fun x_1 ↦ x_1 / ↑x + 1) a).prod /
            ((Multiset.map (fun x_1 ↦ x_1 / ↑x + 1) b).prod * (1 + (↑x)⁻¹)))‖)
    atTop (𝓝 0)
h_ne : ∀ᶠ (n : ℕ) in atTop, regularizedHGFunCoeff a b n ≠ 0
n : ℕ
hn₁ : regularizedHGFunCoeff a b n ≠ 0
hn₂ : n ≠ 0
⊢ ‖↑n ^ (↑a.card - ↑b.card - 1) *
        ((Multiset.map (fun x ↦ x / ↑n + 1) a).prod / ((Multiset.map (fun x ↦ x / ↑n + 1) b).prod * (1 + (↑n)⁻¹)))‖ =
    ‖regularizedHGFunCoeff a b (n + 1)‖ / ‖regularizedHGFunCoeff a b n‖

Tactic:
← Complex.norm_div,

Goal state after:
a b : Multiset ℂ
h : a.card ≤ b.card
ha : ∀ j ∈ a, ∀ (k : ℕ), j ≠ -↑k
h₁ : Tendsto (fun n ↦ ↑n ^ (↑a.card - ↑b.card - 1)) atTop (𝓝 0)
this :
  Tendsto
    (fun x ↦
      ‖↑x ^ (↑a.card - ↑b.card - 1) *
          ((Multiset.map (fun x_1 ↦ x_1 / ↑x + 1) a).prod /
            ((Multiset.map (fun x_1 ↦ x_1 / ↑x + 1) b).prod * (1 + (↑x)⁻¹)))‖)
    atTop (𝓝 0)
h_ne : ∀ᶠ (n : ℕ) in atTop, regularizedHGFunCoeff a b n ≠ 0
n : ℕ
hn₁ : regularizedHGFunCoeff a b n ≠ 0
hn₂ : n ≠ 0
⊢ ‖↑n ^ (↑a.card - ↑b.card - 1) *
        ((Multiset.map (fun x ↦ x / ↑n + 1) a).prod / ((Multiset.map (fun x ↦ x / ↑n + 1) b).prod * (1 + (↑n)⁻¹)))‖ =
    ‖regularizedHGFunCoeff a b (n + 1) / regularizedHGFunCoeff a b n‖
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
l₀ : ↥L.lattice
z : ℂ
⊢ ∑' (b : ↥L.lattice), ((if ↑b = ↑l₀ then 0 else -2 / (z - ↑b) ^ 3) + if ↑b = ↑l₀ then -2 / (z - ↑l₀) ^ 3 else 0) =
    -∑' (l : ↥L.lattice), 2 / (z - ↑l) ^ 3

pp: case hf
L : PeriodPair
l₀ : ↥L.lattice
z : ℂ
⊢ Summable fun l ↦ if ↑l = ↑l₀ then 0 else -2 / (z - ↑l) ^ 3

pp: case hg
L : PeriodPair
l₀ : ↥L.lattice
z : ℂ
⊢ Summable fun i ↦ if ↑i = ↑l₀ then -2 / (z - ↑l₀) ^ 3 else 0

Tactic:
← tsum_neg

Goal state after:
L : PeriodPair
l₀ : ↥L.lattice
z : ℂ
⊢ ∑' (b : ↥L.lattice), ((if ↑b = ↑l₀ then 0 else -2 / (z - ↑b) ^ 3) + if ↑b = ↑l₀ then -2 / (z - ↑l₀) ^ 3 else 0) =
    ∑' (b : ↥L.lattice), -(2 / (z - ↑b) ^ 3)

case hf
L : PeriodPair
l₀ : ↥L.lattice
z : ℂ
⊢ Summable fun l ↦ if ↑l = ↑l₀ then 0 else -2 / (z - ↑l) ^ 3

case hg
L : PeriodPair
l₀ : ↥L.lattice
z : ℂ
⊢ Summable fun i ↦ if ↑i = ↑l₀ then -2 / (z - ↑l₀) ^ 3 else 0
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
l₀ : ℂ
hl : l₀ ∉ L.lattice
⊢ (fun z ↦ ∑' (l : ↥L.lattice), if ↑l = l₀ then 0 else -2 / (z - ↑l) ^ 3) = fun z ↦
    -∑' (l : ↥L.lattice), 2 / (z - ↑l) ^ 3

Tactic:
← tsum_neg

Goal state after:
L : PeriodPair
l₀ : ℂ
hl : l₀ ∉ L.lattice
⊢ (fun z ↦ ∑' (l : ↥L.lattice), if ↑l = l₀ then 0 else -2 / (z - ↑l) ^ 3) = fun z ↦
    ∑' (b : ↥L.lattice), -(2 / (z - ↑b) ^ 3)
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
z : ℂ
⊢ ℘'[L] (-z) = -℘'[L] z

Tactic:
simp only [derivWeierstrassP]

Goal state after:
L : PeriodPair
z : ℂ
⊢ -∑' (l : ↥L.lattice), 2 / (-z - ↑l) ^ 3 = - -∑' (l : ↥L.lattice), 2 / (z - ↑l) ^ 3
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
z : ℂ
l : ↥L.lattice
⊢ ℘'[L] (z + ↑l) = ℘'[L] z

Tactic:
simp only [derivWeierstrassP]

Goal state after:
L : PeriodPair
z : ℂ
l : ↥L.lattice
⊢ -∑' (l_1 : ↥L.lattice), 2 / (z + ↑l - ↑l_1) ^ 3 = -∑' (l : ↥L.lattice), 2 / (z - ↑l) ^ 3
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.ContDiff

Goal state before:
pp: E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
h : ContDiffOn ℝ 1 f (Icc a b)
hab : a ≤ b
⊢ ‖f b - f a‖ₑ ≤ ∫⁻ (x : ℝ) in Icc a b, ‖derivWithin f (Icc a b) x‖ₑ

Tactic:
by
  apply (enorm_sub_le_lintegral_deriv_of_contDiffOn_Icc h hab).trans_eq
  apply lintegral_congr_ae
  rw [← restrict_Ioo_eq_restrict_Icc]
  filter_upwards [self_mem_ae_restrict measurableSet_Ioo] with x hx
  rw [derivWithin_of_mem_nhds (Icc_mem_nhds hx.1 hx.2)]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Stirling

Goal state before:
pp: n : ℕ
this : log (rexp 1) - log √2 - log (stirlingSeq (n + 1)) ≤ 12⁻¹
⊢ 1 - 12⁻¹ - log 2 / 2 ≤ log (stirlingSeq (n + 1))

pp: n : ℕ
this : log (rexp 1 / √2) - log (stirlingSeq (n + 1)) ≤ 12⁻¹
⊢ √2 ≠ 0

Tactic:
log_exp,

Goal state after:
n : ℕ
this : 1 - log √2 - log (stirlingSeq (n + 1)) ≤ 12⁻¹
⊢ 1 - 12⁻¹ - log 2 / 2 ≤ log (stirlingSeq (n + 1))

n : ℕ
this : log (rexp 1 / √2) - log (stirlingSeq (n + 1)) ≤ 12⁻¹
⊢ √2 ≠ 0