mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.NumberTheory.Harmonic.GammaDeriv

Goal state before:
pp: n✝ : ℕ
f : ℝ → ℝ := log ∘ Gamma
hc : ConvexOn ℝ (Ioi 0) f
h_rec : ∀ (x : ℝ), 0 < x → f (x + 1) = f x + log x
hder : ∀ {x : ℝ}, 0 < x → DifferentiableAt ℝ f x
hder_rec : ∀ (x : ℝ), 0 < x → deriv f (x + 1) = deriv f x + 1 / x
hder_nat : ∀ (n : ℕ), deriv f (↑n + 1) = deriv f 1 + ↑(harmonic n)
n : ℕ
hn : 0 < n
⊢ log ↑n = (f (↑n + 1) - f ↑n) / (↑n + 1 - ↑n)

Tactic:
show n + 1 - n = (1 : ℝ) by ring,

Goal state after:
n✝ : ℕ
f : ℝ → ℝ := log ∘ Gamma
hc : ConvexOn ℝ (Ioi 0) f
h_rec : ∀ (x : ℝ), 0 < x → f (x + 1) = f x + log x
hder : ∀ {x : ℝ}, 0 < x → DifferentiableAt ℝ f x
hder_rec : ∀ (x : ℝ), 0 < x → deriv f (x + 1) = deriv f x + 1 / x
hder_nat : ∀ (n : ℕ), deriv f (↑n + 1) = deriv f 1 + ↑(harmonic n)
n : ℕ
hn : 0 < n
⊢ log ↑n = (f (↑n + 1) - f ↑n) / 1
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation

Goal state before:
pp: A : Type u_1
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace ℝ A
inst✝⁵ : SMulCommClass ℝ A A
inst✝⁴ : IsScalarTower ℝ A A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : NonnegSpectrumClass ℝ A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
p t : ℝ
hp : p ∈ Ioo 0 1
ht : 0 < t
a : A
ha : 0 ≤ a
⊢ quasispectrum ℝ a ⊆ Ici 0

Tactic:
grind

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

Goal state before:
pp: A : Type u_1
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace ℝ A
inst✝⁵ : SMulCommClass ℝ A A
inst✝⁴ : IsScalarTower ℝ A A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : NonnegSpectrumClass ℝ A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
p t : ℝ
hp : p ∈ Ioo 0 1
ht : 0 < t
a : A
ha : 0 ≤ a
hspec : quasispectrum ℝ a ⊆ Ici 0
h_mapsTo : MapsTo (fun x ↦ t⁻¹ • x) (Ici 0) (Ici 0)
⊢ t ^ (p - 1) • cfcₙ (fun x ↦ p.rpowIntegrand₀₁ 1 (t⁻¹ • x)) a = t ^ (p - 1) • cfcₙ (p.rpowIntegrand₀₁ 1) (t⁻¹ • a)

Tactic:
by
          congr! 1
          refine cfcₙ_comp_smul (R := ℝ) t⁻¹ (fun x => rpowIntegrand₀₁ p 1 x) a ?_
          exact continuousOn_rpowIntegrand₀₁_Ici hp zero_lt_one |>.mono <|
            (h_mapsTo.mono_left hspec).image_subset

Goal state after:
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)
⊢ IsCompact (quasispectrum ℝ a)

Tactic:
grind

Goal state after:
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‖
⊢ quasispectrum ℝ a ⊆ Ici 0

Tactic:
grind

Goal state after:
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
⊢ 0 ≤ z

Tactic:
grind

Goal state after:
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)
hbound : ∀ᵐ (t : ℝ) ∂μ.restrict (Ioi 0), ∀ z ∈ quasispectrum ℝ a, ‖f t z‖ ≤ bound t
hbound_finite_integral : HasFiniteIntegral bound (μ.restrict (Ioi 0))
hmapzero : ∀ᵐ (x : ℝ) ∂μ.restrict (Ioi 0), (↑p).rpowIntegrand₀₁ x 0 = 0
r : ℝ
x✝ : r ∈ quasispectrum ℝ a
⊢ r ∈ Ici 0

Tactic:
grind

Goal state after:
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‖
⊢ quasispectrum ℝ a ⊆ Ici 0

Tactic:
grind

Goal state after:
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
⊢ 0 ≤ z

Tactic:
grind

Goal state after:
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
hz' : 0 ≤ z
⊢ 1 < ↑p

Tactic:
grind

Goal state after:
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
hz' : 0 ≤ z
⊢ 0 ≤ t

Tactic:
grind

Goal state after:
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
hz' : 0 ≤ z
⊢ ↑p ∈ Ioo 1 2

Tactic:
grind

Goal state after:
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
hz' : 0 ≤ z
⊢ 0 < t

Tactic:
grind

Goal state after:
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)
hbound : ∀ᵐ (t : ℝ) ∂μ.restrict (Ioi 0), ∀ z ∈ quasispectrum ℝ a, ‖f t z‖ ≤ bound t
hbound_finite_integral : HasFiniteIntegral bound (μ.restrict (Ioi 0))
hmapzero : ∀ᵐ (x : ℝ) ∂μ.restrict (Ioi 0), (↑p).rpowIntegrand₁₂ x 0 = 0
r : ℝ
hr : r ∈ quasispectrum ℝ a
⊢ 0 ≤ r

Tactic:
grind

Goal state after:
Module: Mathlib.Data.Int.Log

Goal state before:
pp: case inr.inl
R : Type u_1
inst✝³ : Semifield R
inst✝² : LinearOrder R
inst✝¹ : IsStrictOrderedRing R
inst✝ : FloorSemiring R
b : ℕ
r₁ : R
h₀ : 0 < r₁
h₁ : 1 ≤ r₁
h : r₁ ≤ r₁
h₂ : r₁ ≤ 1
⊢ log b r₁ ≤ log b r₁

Tactic:
rfl

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

Goal state before:
pp: A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p t : ℝ
hp : p ∈ Ioo 0 1
ht : 0 < t
x : A
hx : x ∈ Ici 0
⊢ p ≠ 1

Tactic:
grind

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

Goal state before:
pp: a b c : ℝ
h₁ : b ≠ 0
h₂ : b ≠ 1
h₃ : b ≠ -1
⊢ logb a b * logb b c = logb a c

Tactic:
by
  unfold logb
  rw [mul_comm, div_mul_div_cancel₀ (log_ne_zero.mpr ⟨h₁, h₂, h₃⟩)]

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

Goal state before:
pp: A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p t : ℝ
hp : p ∈ Ioo 0 1
ht : 0 < t
x : A
hx : x ∈ Ici 0
hg : ContinuousOn (fun z ↦ (t + z)⁻¹) (spectrum ℝ x)
hf : ContinuousOn (fun z ↦ 1 + z) (spectrum ℝ x)
⊢ ∀ r ∈ spectrum ℝ x, t + r ≠ 0

Tactic:
grind

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

Goal state before:
pp: b x : ℝ
b_pos : 0 < b
b_ne_one : b ≠ 1
hx : x < 0
⊢ b ^ logb b x = -x

Tactic:
rw [rpow_logb_eq_abs b_pos b_ne_one (ne_of_lt hx)]

Goal state after:
b x : ℝ
b_pos : 0 < b
b_ne_one : b ≠ 1
hx : x < 0
⊢ |x| = -x
Module: Mathlib.MeasureTheory.Measure.RegularityCompacts

Goal state before:
pp: case refine_2
α : Type u_1
inst✝⁵ : MeasurableSpace α
μ : Measure α
inst✝⁴ : TopologicalSpace α
inst✝³ : SecondCountableTopology α
inst✝² : IsCompletelyPseudoMetrizableSpace α
inst✝¹ : BorelSpace α
P : Measure α
inst✝ : IsFiniteMeasure P
⊢ ∀ ⦃s U : Set α⦄, IsCompact s → IsOpen U → IsCompact (s \ U)

Tactic:
· exact fun s t hs_compact ht_open ↦ hs_compact.inter_right ht_open.isClosed_compl

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

Goal state before:
pp: n : ℕ
⊢ ConvexOn ℝ (Set.Ici (↑n - 1)) fun x ↦ Polynomial.eval x (descPochhammer ℝ n)

Tactic:
rcases n.eq_zero_or_pos with h_eq | _

Goal state after:
case inl
n : ℕ
h_eq : n = 0
⊢ ConvexOn ℝ (Set.Ici (↑n - 1)) fun x ↦ Polynomial.eval x (descPochhammer ℝ n)

case inr
n : ℕ
h✝ : n > 0
⊢ ConvexOn ℝ (Set.Ici (↑n - 1)) fun x ↦ Polynomial.eval x (descPochhammer ℝ n)
Module: Mathlib.MeasureTheory.Constructions.HaarToSphere

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : MeasurableSpace E
μ : Measure E
s : Set ↑(sphere 0 1)
r : ↑(Ioi 0)
⊢ μ ((fun x ↦ ↑((homeomorphUnitSphereProd E).symm x)) '' s ×ˢ Iio r) = μ ((fun x ↦ ↑x.2 • ↑x.1) '' s ×ˢ Iio r)

Tactic:
rfl

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.NthRootLemmas

Goal state before:
pp: n a guess : ℕ
hle : guess ≤ 0
⊢ guess = 0

Tactic:
grind

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.NthRootLemmas

Goal state before:
pp: n a fuel : ℕ
ih : ∀ {guess : ℕ}, guess ≤ fuel → go n a fuel guess ^ (n + 2) ≤ a
guess : ℕ
hle : guess ≤ fuel + 1
h : (a / guess ^ (n + 1) + (n + 1) * guess) / (n + 2) < guess
⊢ go n a fuel ((a / guess ^ (n + 1) + (n + 1) * guess) / (n + 2)) ^ (n + 2) ≤ a

Tactic:
grind

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.NthRootLemmas

Goal state before:
pp: n a fuel : ℕ
ih : ∀ {guess : ℕ}, guess ≤ fuel → go n a fuel guess ^ (n + 2) ≤ a
guess : ℕ
hle : guess ≤ fuel + 1
h : ¬(a / guess ^ (n + 1) + (n + 1) * guess) / (n + 2) < guess
this : guess * guess ^ (n + 1) ≤ a
⊢ guess ^ (n + 2) ≤ a

Tactic:
grind

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.NthRootLemmas

Goal state before:
pp: case zero
a : ℕ
⊢ nthRoot 0 a ^ 0 ≤ a ↔ 0 ≠ 0 ∨ a ≠ 0

Tactic:
grind

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.NthRootLemmas

Goal state before:
pp: case succ.zero
a : ℕ
⊢ (0 + 1).nthRoot a ^ (0 + 1) ≤ a ↔ 0 + 1 ≠ 0 ∨ a ≠ 0

Tactic:
grind

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.NthRootLemmas

Goal state before:
pp: case succ.succ
a n✝ : ℕ
⊢ (n✝ + 1 + 1).nthRoot a ^ (n✝ + 1 + 1) ≤ a ↔ n✝ + 1 + 1 ≠ 0 ∨ a ≠ 0

Tactic:
grind

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.HaarToSphere

Goal state before:
pp: E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
hx : ‖x‖ = 1
ε : ℝ
hε : 0 < ε
hε2 : ε ≤ 2
y : E
hy : dist y ((1 - ε / 4) • x) < ε / 4
habs : |1 - ε / 4| = 1 - ε / 4
hy₀ : y ≠ 0
hy₁ : ‖y‖ < 1
u : E := ‖y‖⁻¹ • y
hu₁ : ‖u‖ = 1
hyx : dist y x < ε / 2
⊢ u - y = (1 - ‖y‖) • u

Tactic:
simp [u, hy₀, sub_smul]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.NthRootLemmas

Goal state before:
pp: case inl
n a fuel : ℕ
ih : ∀ {guess : ℕ}, a < (guess + 1) ^ (n + 2) → a < (go n a fuel guess + 1) ^ (n + 2)
hlt : a < (0 + 1) ^ (n + 2)
h : (a / 0 ^ (n + 1) + (n + 1) * 0) / (n + 2) < 0
⊢ a < (go n a fuel ((a / 0 ^ (n + 1) + (n + 1) * 0) / (n + 2)) + 1) ^ (n + 2)

Tactic:
grind

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.NthRootLemmas

Goal state before:
pp: case inl
a b : ℕ
h₁ : a ^ 0 ≤ b
h₂ : b < (a + 1) ^ 0
⊢ nthRoot 0 b = a

Tactic:
grind

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.NthRootLemmas

Goal state before:
pp: case inr
n a b : ℕ
hn : n ≠ 0
h₁ : a ≤ n.nthRoot b
h₂ : n.nthRoot b < a + 1
⊢ n.nthRoot b = a

Tactic:
grind

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.HaarToSphere

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
μ : Measure E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
x : ↑(sphere 0 1)
this : Nontrivial E
ε : ℝ
hε : 0 < ε
hε₂ : ε ≤ 2
⊢ ↑(toSphereBallBound (dim E) ε) * μ (ball 0 1) ≤ ↑(dim E) * μ (Ioo 0 1 • (ball (↑x) ε ∩ sphere 0 1))

Tactic:
← ball_subset_sector_of_small_epsilon

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
μ : Measure E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
x : ↑(sphere 0 1)
this : Nontrivial E
ε : ℝ
hε : 0 < ε
hε₂ : ε ≤ 2
⊢ ↑(toSphereBallBound (dim E) ε) * μ (ball 0 1) ≤ ↑(dim E) * μ (ball ((1 - ε / 4) • ↑x) (ε / 4))

case hx
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
μ : Measure E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
x : ↑(sphere 0 1)
this : Nontrivial E
ε : ℝ
hε : 0 < ε
hε₂ : ε ≤ 2
⊢ ‖↑x‖ = 1

case hε
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
μ : Measure E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
x : ↑(sphere 0 1)
this : Nontrivial E
ε : ℝ
hε : 0 < ε
hε₂ : ε ≤ 2
⊢ 0 < ε

case hε2
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
μ : Measure E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
x : ↑(sphere 0 1)
this : Nontrivial E
ε : ℝ
hε : 0 < ε
hε₂ : ε ≤ 2
⊢ ε ≤ 2
Module: Mathlib.Analysis.SpecialFunctions.Pow.NthRootLemmas

Goal state before:
pp: case mpr
n a : ℕ
hn : n ≠ 0
⊢ n.nthRoot a ^ n = a → ∃ x, x ^ n = a

Tactic:
grind

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

Goal state before:
pp: a b : Multiset ℂ
n : ℕ
⊢ ((∃ a_1 ∈ a, ∃ k < n, ↑k = -a_1) ∨ n ! = 0 ∨ ∃ a ∈ b, Gamma (a + ↑n) = 0) ↔
    (∃ k, -↑k ∈ a ∧ k < n) ∨ ∃ j ∈ b, ∃ m, j + ↑n = -↑m

Tactic:
grind

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

Goal state before:
pp: a b : Multiset ℂ
n m : ℕ
hb : -↑n - ↑m ∈ b
⊢ regularizedHGFunCoeff a b n = 0

Tactic:
grind

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

Goal state before:
pp: a b : Multiset ℂ
n m : ℕ
ha : -↑m ∈ a
hm : m < n
⊢ regularizedHGFunCoeff a b n = 0

Tactic:
grind

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

Goal state before:
pp: case refine_2
a b : Multiset ℂ
n : ℕ
hb : ∀ k ∈ b, k ≠ -↑n
⊢ ↑((n + 1) * n !) = ↑n ! * (↑n + 1)

Tactic:
grind

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

Goal state before:
pp: case refine_3
a b : Multiset ℂ
n : ℕ
hb : ∀ k ∈ b, k ≠ -↑n
j : ℂ
hj : j ∈ b
⊢ Gamma (j + ↑n + 1) = Gamma (j + ↑n) * (j + ↑n)

Tactic:
grind

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

Goal state before:
pp: case neg
a b : Multiset ℂ
n : ℕ
h : regularizedHGFunCoeff a b n ≠ 0
j : ℂ
hj : j ∈ b ∧ j = -↑n
⊢ regularizedHGFunCoeff a b (n + 1) / regularizedHGFunCoeff a b n =
    (Multiset.map (fun x ↦ x + ↑n) a).prod / ((Multiset.map (fun x ↦ x + ↑n) b).prod * (↑n + 1))

Tactic:
have h₁ : (b.map (· + (n : ℂ))).prod = 0 := by
      grind [Multiset.prod_eq_zero, Multiset.mem_map]

Goal state after:
case neg
a b : Multiset ℂ
n : ℕ
h : regularizedHGFunCoeff a b n ≠ 0
j : ℂ
hj : j ∈ b ∧ j = -↑n
h₁ : (Multiset.map (fun x ↦ x + ↑n) b).prod = 0
⊢ regularizedHGFunCoeff a b (n + 1) / regularizedHGFunCoeff a b n =
    (Multiset.map (fun x ↦ x + ↑n) a).prod / ((Multiset.map (fun x ↦ x + ↑n) b).prod * (↑n + 1))
Module: Mathlib.Analysis.SpecialFunctions.RegularizedHypergeometric

Goal state before:
pp: case e_a
n : ℕ
a b : Multiset ℂ
hn : n ≠ 0
⊢ ↑n ^ ↑a.card = ↑n ^ (↑(b.card + 1) + (↑a.card - ↑b.card - 1))

Tactic:
grind

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

Goal state before:
pp: case h
a b : Multiset ℂ
n : ℕ
j : ℂ
ha : j ∈ a
hj : j = -↑n
⊢ ∀ (b_1 : ℕ), n + 1 ≤ b_1 → regularizedHGFunSeries a b b_1 = 0

Tactic:
grind

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

Goal state before:
pp: case h.inl
a b : Multiset ℂ
h : ∀ j ∈ a, ∀ (k : ℕ), j ≠ -↑k
n : ℕ
hn : (b.toFinset.sup fun x ↦ ⌈-x.re⌉₊) + 1 ≤ n
h' : ∃ j ∈ a, ∃ k < n, j = -↑k
⊢ False

Tactic:
grind

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

Goal state before:
pp: a b : Multiset ℂ
h : ∀ j ∈ a, ∀ (k : ℕ), j ≠ -↑k
n : ℕ
hn : (b.toFinset.sup fun x ↦ ⌈-x.re⌉₊) + 1 ≤ n
j : ℂ
hj : j ∈ b
m : ℕ
h' : j + ↑n = -↑m
this : ↑m < 0
⊢ False

Tactic:
grind

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

Goal state before:
pp: a b : Multiset ℂ
h : ∀ j ∈ a, ∀ (k : ℕ), j ≠ -↑k
n : ℕ
hn : (b.toFinset.sup fun x ↦ ⌈-x.re⌉₊) + 1 ≤ n
j : ℂ
hj : j ∈ b
m : ℕ
h' : j + ↑n = -↑m
this : -j.re < ↑n
⊢ j = -↑m - ↑n

Tactic:
grind

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

Goal state before:
pp: a b : Multiset ℂ
h : ∀ j ∈ a, ∀ (k : ℕ), j ≠ -↑k
n : ℕ
hn : (b.toFinset.sup fun x ↦ ⌈-x.re⌉₊) + 1 ≤ n
j : ℂ
hj : j ∈ b
m : ℕ
h' : j + ↑n = -↑m
⊢ j ∈ b.toFinset

Tactic:
grind

Goal state after:
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
this : Tendsto (fun x ↦ (1 / ↑x) ^ (b.card + 1 - a.card)) atTop (𝓝 (0 ^ (b.card + 1 - a.card)))
⊢ b.card + 1 - a.card ≠ 0

Tactic:
grind

Goal state after: