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

Goal state before:
pp: h_diff : ∀ {s : ℝ}, 0 < s → DifferentiableAt ℝ Gamma s
h_diff' : ∀ {s : ℝ}, 0 < s → DifferentiableAt ℝ (fun s ↦ Gamma (2 * s)) s
⊢ √π *
      (deriv (fun s ↦ Gamma (2 * s)) (1 / 2) * 2 ^ (1 - 2 * (1 / 2)) +
          Gamma (2 * (1 / 2)) * deriv (fun s ↦ 2 ^ (1 - 2 * s)) (1 / 2) +
        γ) =
    √π * (deriv (fun s ↦ Gamma (2 * s)) (1 / 2) + deriv (fun s ↦ 2 ^ (1 - 2 * s)) (1 / 2) + γ)

pp: case hc
h_diff : ∀ {s : ℝ}, 0 < s → DifferentiableAt ℝ Gamma s
h_diff' : ∀ {s : ℝ}, 0 < s → DifferentiableAt ℝ (fun s ↦ Gamma (2 * s)) s
⊢ DifferentiableAt ℝ (fun s ↦ Gamma (2 * s)) (1 / 2)

pp: case hd
h_diff : ∀ {s : ℝ}, 0 < s → DifferentiableAt ℝ Gamma s
h_diff' : ∀ {s : ℝ}, 0 < s → DifferentiableAt ℝ (fun s ↦ Gamma (2 * s)) s
⊢ DifferentiableAt ℝ (fun s ↦ 2 ^ (1 - 2 * s)) (1 / 2)

Tactic:
· simp

Goal state after:
case hc
h_diff : ∀ {s : ℝ}, 0 < s → DifferentiableAt ℝ Gamma s
h_diff' : ∀ {s : ℝ}, 0 < s → DifferentiableAt ℝ (fun s ↦ Gamma (2 * s)) s
⊢ DifferentiableAt ℝ (fun s ↦ Gamma (2 * s)) (1 / 2)

case hd
h_diff : ∀ {s : ℝ}, 0 < s → DifferentiableAt ℝ Gamma s
h_diff' : ∀ {s : ℝ}, 0 < s → DifferentiableAt ℝ (fun s ↦ Gamma (2 * s)) s
⊢ DifferentiableAt ℝ (fun s ↦ 2 ^ (1 - 2 * s)) (1 / 2)
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:
by 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:
by 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:
by 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:
by 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:
by grind

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:
by grind

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

Goal state before:
pp: ⊢ AntitoneOn (fun x ↦ log x / x) (Ici (rexp 1))

Tactic:
intro x hex y hey hxy

Goal state after:
x : ℝ
hex : x ∈ Ici (rexp 1)
y : ℝ
hey : y ∈ Ici (rexp 1)
hxy : x ≤ y
⊢ (fun x ↦ log x / x) y ≤ (fun x ↦ log x / x) x
Module: Mathlib.Analysis.SpecialFunctions.Log.Base

Goal state before:
pp: case a
b : ℕ
r : ℝ
hr✝ : 0 ≤ r
hr : 0 < r
hb : 1 < b
hb1' : 1 < ↑b
⊢ Int.log b r ≤ ⌊logb (↑b) r⌋

Tactic:
· rw [Int.le_floor, le_logb_iff_rpow_le hb1' hr, rpow_intCast]
      exact Int.zpow_log_le_self hb hr

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

Goal state before:
pp: case succ.succ.inl
b : ℕ
⊢ ⌊logb ↑(b + 1 + 1) ↑0⌋₊ = Nat.log (b + 1 + 1) 0

pp: case succ.succ.inr
n b : ℕ
hn : n ≠ 0
⊢ ⌊logb ↑(b + 1 + 1) ↑n⌋₊ = Nat.log (b + 1 + 1) n

Tactic:
· simp

Goal state after:
case succ.succ.inr
n b : ℕ
hn : n ≠ 0
⊢ ⌊logb ↑(b + 1 + 1) ↑n⌋₊ = Nat.log (b + 1 + 1) n
Module: Mathlib.Analysis.SpecialFunctions.Log.Base

Goal state before:
pp: case succ.succ.inl
b : ℕ
⊢ ⌈logb ↑(b + 1 + 1) ↑0⌉₊ = Nat.clog (b + 1 + 1) 0

pp: case succ.succ.inr
n b : ℕ
hn : n ≠ 0
⊢ ⌈logb ↑(b + 1 + 1) ↑n⌉₊ = Nat.clog (b + 1 + 1) n

Tactic:
· simp

Goal state after:
case succ.succ.inr
n b : ℕ
hn : n ≠ 0
⊢ ⌈logb ↑(b + 1 + 1) ↑n⌉₊ = Nat.clog (b + 1 + 1) n
Module: Mathlib.Analysis.SpecialFunctions.Log.Base

Goal state before:
pp: P : ℝ → Prop
x₀ r : ℝ
hr : 1 < r
hx₀ : 0 < x₀
base : ∀ x ∈ Ico x₀ (r * x₀), P x
step : ∀ n ≥ 1, (∀ z ∈ Ico x₀ (r ^ n * x₀), P z) → ∀ z ∈ Ico (r ^ n * x₀) (r ^ (n + 1) * x₀), P z
this : ∀ (n : ℕ), ∀ x ∈ Ico x₀ (r ^ (n + 1) * x₀), P x
x : ℝ
hx : x ≥ x₀
hx' : 0 < x / x₀
⊢ x₀ ≤ x ∧ logb r (x / x₀) < ↑(⌊logb r (x / x₀)⌋₊ + 1)

Tactic:
Nat.cast_add,

Goal state after:
P : ℝ → Prop
x₀ r : ℝ
hr : 1 < r
hx₀ : 0 < x₀
base : ∀ x ∈ Ico x₀ (r * x₀), P x
step : ∀ n ≥ 1, (∀ z ∈ Ico x₀ (r ^ n * x₀), P z) → ∀ z ∈ Ico (r ^ n * x₀) (r ^ (n + 1) * x₀), P z
this : ∀ (n : ℕ), ∀ x ∈ Ico x₀ (r ^ (n + 1) * x₀), P x
x : ℝ
hx : x ≥ x₀
hx' : 0 < x / x₀
⊢ x₀ ≤ x ∧ logb r (x / x₀) < ↑⌊logb r (x / x₀)⌋₊ + ↑1
Module: Mathlib.MeasureTheory.Constructions.HaarToSphere

Goal state before:
pp: n : ℕ
x : ↑(Ioi 0)
hr₀ : 0 ≤ ↑x
⊢ ENNReal.ofReal (∫ (x : ℝ) in 0..↑x, x ^ n) = ENNReal.ofReal (↑x ^ (n + 1) / (↑n + 1))

pp: n : ℕ
x : ↑(Ioi 0)
hr₀ : 0 ≤ ↑x
⊢ 0 ≤ᵐ[volume.restrict (Ioc 0 ↑x)] fun x ↦ x ^ n

Tactic:
· simp

Goal state after:
n : ℕ
x : ↑(Ioi 0)
hr₀ : 0 ≤ ↑x
⊢ 0 ≤ᵐ[volume.restrict (Ioc 0 ↑x)] fun x ↦ x ^ n
Module: Mathlib.Analysis.SpecialFunctions.Pow.NthRootLemmas

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

Tactic:
by grind

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: case pos
L : PeriodPair
l₀ : ℂ
s : Finset ↥L.lattice
i : ↥L.lattice
hi : i ∈ s
h✝ : ↑i = l₀
⊢ DifferentiableOn ℂ (fun z ↦ 0) (↑L.lattice \ {l₀})ᶜ

pp: case neg
L : PeriodPair
l₀ : ℂ
s : Finset ↥L.lattice
i : ↥L.lattice
hi : i ∈ s
h✝ : ¬↑i = l₀
⊢ DifferentiableOn ℂ (fun z ↦ 1 / (z - ↑i) ^ 2 - 1 / ↑i ^ 2) (↑L.lattice \ {l₀})ᶜ

Tactic:
· simp

Goal state after:
case neg
L : PeriodPair
l₀ : ℂ
s : Finset ↥L.lattice
i : ↥L.lattice
hi : i ∈ s
h✝ : ¬↑i = l₀
⊢ DifferentiableOn ℂ (fun z ↦ 1 / (z - ↑i) ^ 2 - 1 / ↑i ^ 2) (↑L.lattice \ {l₀})ᶜ
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
l₀ : ↥L.lattice
z : ℂ
⊢ ℘[L - ↑l₀] z + (1 / (z - ↑l₀) ^ 2 - 1 / ↑l₀ ^ 2) =
    ℘[L - ↑l₀] z + ∑' (i : ↥L.lattice), if ↑i = ↑l₀ then 1 / (z - ↑l₀) ^ 2 - 1 / ↑l₀ ^ 2 else 0

pp: L : PeriodPair
l₀ : ↥L.lattice
z : ℂ
⊢ (℘[L - ↑l₀] z + ∑' (i : ↥L.lattice), if ↑i = ↑l₀ then 1 / (z - ↑l₀) ^ 2 - 1 / ↑l₀ ^ 2 else 0) = ℘[L] z

Tactic:
· simp

Goal state after:
L : PeriodPair
l₀ : ↥L.lattice
z : ℂ
⊢ (℘[L - ↑l₀] z + ∑' (i : ↥L.lattice), if ↑i = ↑l₀ then 1 / (z - ↑l₀) ^ 2 - 1 / ↑l₀ ^ 2 else 0) = ℘[L] z
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:
by 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:
by grind

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: case pos
L : PeriodPair
l₀ : ℂ
r : ℝ
hr : r > 0
s : ℂ
hs : ‖s‖ < r
l : ↥L.lattice
h : 2 * r ≤ ‖↑l‖
h✝ : ↑l = l₀
⊢ ‖0‖ ≤ 16 * ‖l‖ ^ (-3)

pp: case neg
L : PeriodPair
l₀ : ℂ
r : ℝ
hr : r > 0
s : ℂ
hs : ‖s‖ < r
l : ↥L.lattice
h : 2 * r ≤ ‖↑l‖
h✝ : ¬↑l = l₀
⊢ ‖-2 / (s - ↑l) ^ 3‖ ≤ 16 * ‖l‖ ^ (-3)

Tactic:
· simp

Goal state after:
case neg
L : PeriodPair
l₀ : ℂ
r : ℝ
hr : r > 0
s : ℂ
hs : ‖s‖ < r
l : ↥L.lattice
h : 2 * r ≤ ‖↑l‖
h✝ : ¬↑l = l₀
⊢ ‖-2 / (s - ↑l) ^ 3‖ ≤ 16 * ‖l‖ ^ (-3)
Module: Mathlib.MeasureTheory.Constructions.HaarToSphere

Goal state before:
pp: 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

Tactic:
· simp

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: case pos
L : PeriodPair
l₀ : ℂ
s : Finset ↥L.lattice
i : ↥L.lattice
hi : i ∈ s
h✝ : ↑i = l₀
⊢ DifferentiableOn ℂ (fun z ↦ 0) (↑L.lattice \ {l₀})ᶜ

pp: case neg
L : PeriodPair
l₀ : ℂ
s : Finset ↥L.lattice
i : ↥L.lattice
hi : i ∈ s
h✝ : ¬↑i = l₀
⊢ DifferentiableOn ℂ (fun z ↦ -2 / (z - ↑i) ^ 3) (↑L.lattice \ {l₀})ᶜ

Tactic:
· simp

Goal state after:
case neg
L : PeriodPair
l₀ : ℂ
s : Finset ↥L.lattice
i : ↥L.lattice
hi : i ∈ s
h✝ : ¬↑i = l₀
⊢ DifferentiableOn ℂ (fun z ↦ -2 / (z - ↑i) ^ 3) (↑L.lattice \ {l₀})ᶜ
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:
by 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
⊢ ↑m < 0

Tactic:
have h : j = -m - n := by grind
      simpa [h] using this

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:
by grind

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: case pos
L : PeriodPair
l₀ : ℂ
s : Finset ↥L.lattice
l : ℂ
hl : l ∈ (↑L.lattice \ {l₀})ᶜ
x : ↥L.lattice
hl₁ : ↑x = l₀
⊢ 0 = deriv (fun x2 ↦ 0) l

pp: case neg
L : PeriodPair
l₀ : ℂ
s : Finset ↥L.lattice
l : ℂ
hl : l ∈ (↑L.lattice \ {l₀})ᶜ
x : ↥L.lattice
hl₁ : ¬↑x = l₀
⊢ -2 / (l - ↑x) ^ 3 = deriv (fun x2 ↦ 1 / (x2 - ↑x) ^ 2 - 1 / ↑x ^ 2) l

Tactic:
· simp

Goal state after:
case neg
L : PeriodPair
l₀ : ℂ
s : Finset ↥L.lattice
l : ℂ
hl : l ∈ (↑L.lattice \ {l₀})ᶜ
x : ↥L.lattice
hl₁ : ¬↑x = l₀
⊢ -2 / (l - ↑x) ^ 3 = deriv (fun x2 ↦ 1 / (x2 - ↑x) ^ 2 - 1 / ↑x ^ 2) l
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:
by 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)
n : ℕ
⊢ a.card ≤ b.card + 1

Tactic:
by grind

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

Goal state before:
pp: E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : MeasurableSpace E
F : Type u_2
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : Nontrivial E
μ : Measure E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : BorelSpace E
inst✝ : μ.IsAddHaarMeasure
f : ℝ → F
this :
  Integrable (fun x ↦ f ‖↑x‖) (Measure.comap Subtype.val μ) ↔
    Integrable (fun x ↦ f ↑x.2) (μ.toSphere.prod (Measure.volumeIoiPow (dim E - 1)))
⊢ (fun x ↦ (ENNReal.ofReal (↑x ^ (dim E - 1))).toReal • f ↑x) =ᵐ[Measure.comap Subtype.val volume] fun x ↦
    ↑x ^ (dim E - 1) • f ↑x

Tactic:
refine .of_forall ?_
    rintro ⟨x, hx : 0 < x⟩
    simp (disch := positivity) [ENNReal.toReal_ofReal]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: case pos
L : PeriodPair
l₀ : ℂ
s : Finset ↥L.lattice
l : ℂ
hl : l ∈ (↑L.lattice \ {l₀})ᶜ
x : ↥L.lattice
hxs : x ∈ s
hl₁ : ↑x = l₀
⊢ DifferentiableAt ℂ (fun x2 ↦ 0) l

pp: case neg
L : PeriodPair
l₀ : ℂ
s : Finset ↥L.lattice
l : ℂ
hl : l ∈ (↑L.lattice \ {l₀})ᶜ
x : ↥L.lattice
hxs : x ∈ s
hl₁ : ¬↑x = l₀
⊢ DifferentiableAt ℂ (fun x2 ↦ 1 / (x2 - ↑x) ^ 2 - 1 / ↑x ^ 2) l

Tactic:
· simp

Goal state after:
case neg
L : PeriodPair
l₀ : ℂ
s : Finset ↥L.lattice
l : ℂ
hl : l ∈ (↑L.lattice \ {l₀})ᶜ
x : ↥L.lattice
hxs : x ∈ s
hl₁ : ¬↑x = l₀
⊢ DifferentiableAt ℂ (fun x2 ↦ 1 / (x2 - ↑x) ^ 2 - 1 / ↑x ^ 2) l
Module: Mathlib.MeasureTheory.Constructions.HaarToSphere

Goal state before:
pp: case hflt
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : MeasurableSpace E
F : Type u_2
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : Nontrivial E
μ : Measure E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : BorelSpace E
inst✝ : μ.IsAddHaarMeasure
f : ℝ → F
this :
  Integrable (fun x ↦ f ‖↑x‖) (Measure.comap Subtype.val μ) ↔
    Integrable (fun x ↦ f ↑x.2) (μ.toSphere.prod (Measure.volumeIoiPow (dim E - 1)))
⊢ ∀ᵐ (x : ↑(Ioi 0)) ∂Measure.comap Subtype.val volume, ENNReal.ofReal (↑x ^ (dim E - 1)) < ∞

pp: E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : MeasurableSpace E
F : Type u_2
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : Nontrivial E
μ : Measure E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : BorelSpace E
inst✝ : μ.IsAddHaarMeasure
f : ℝ → F
this :
  Integrable (fun x ↦ f ‖↑x‖) (Measure.comap Subtype.val μ) ↔
    Integrable (fun x ↦ f ↑x.2) (μ.toSphere.prod (Measure.volumeIoiPow (dim E - 1)))
⊢ MeasurableSet (Ioi 0)

pp: E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : MeasurableSpace E
F : Type u_2
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : Nontrivial E
μ : Measure E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : BorelSpace E
inst✝ : μ.IsAddHaarMeasure
f : ℝ → F
this :
  Integrable (fun x ↦ f ‖↑x‖) (Measure.comap Subtype.val μ) ↔
    Integrable (fun x ↦ f ↑x.2) (μ.toSphere.prod (Measure.volumeIoiPow (dim E - 1)))
⊢ μ.toSphere ≠ 0

Tactic:
· simp

Goal state after:
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : MeasurableSpace E
F : Type u_2
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : Nontrivial E
μ : Measure E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : BorelSpace E
inst✝ : μ.IsAddHaarMeasure
f : ℝ → F
this :
  Integrable (fun x ↦ f ‖↑x‖) (Measure.comap Subtype.val μ) ↔
    Integrable (fun x ↦ f ↑x.2) (μ.toSphere.prod (Measure.volumeIoiPow (dim E - 1)))
⊢ MeasurableSet (Ioi 0)

E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : MeasurableSpace E
F : Type u_2
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : Nontrivial E
μ : Measure E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : BorelSpace E
inst✝ : μ.IsAddHaarMeasure
f : ℝ → F
this :
  Integrable (fun x ↦ f ‖↑x‖) (Measure.comap Subtype.val μ) ↔
    Integrable (fun x ↦ f ↑x.2) (μ.toSphere.prod (Measure.volumeIoiPow (dim E - 1)))
⊢ μ.toSphere ≠ 0
Module: Mathlib.Analysis.SpecialFunctions.RegularizedHypergeometric

Goal state before:
pp: a b c z : ℂ
hc : ∀ (k : ℕ), c ≠ -↑k
⊢ (Gamma c)⁻¹ • (ordinaryHypergeometricSeries ℂ a b c).sum z = (a.regularizedGaussHGFunSeries b c).sum z

Tactic:
FormalMultilinearSeries.const_smul_sum_apply

Goal state after:
a b c z : ℂ
hc : ∀ (k : ℕ), c ≠ -↑k
⊢ ((Gamma c)⁻¹ • ordinaryHypergeometricSeries ℂ a b c).sum z = (a.regularizedGaussHGFunSeries b c).sum z
Module: Mathlib.MeasureTheory.Constructions.HaarToSphere

Goal state before:
pp: E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : MeasurableSpace E
F : Type u_2
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : Nontrivial E
μ : Measure E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : BorelSpace E
inst✝ : μ.IsAddHaarMeasure
f : ℝ → F
this :
  Integrable (fun x ↦ f ‖↑x‖) (Measure.comap Subtype.val μ) ↔
    Integrable (fun x ↦ f ↑x.2) (μ.toSphere.prod (Measure.volumeIoiPow (dim E - 1)))
⊢ μ.toSphere ≠ 0

Tactic:
· simp

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

Goal state before:
pp: x : ℝ
h : DifferentiableWithinAt ℝ arcsin (Neg.neg '' Ici (-x)) (- -x)
⊢ x ≠ 1

Tactic:
have := (h.comp (-x) differentiableWithinAt_id.fun_neg (mapsTo_image _ _)).fun_neg

Goal state after:
x : ℝ
h : DifferentiableWithinAt ℝ arcsin (Neg.neg '' Ici (-x)) (- -x)
this : DifferentiableWithinAt ℝ (fun i ↦ -(arcsin ∘ Neg.neg) i) (Ici (-x)) (-x)
⊢ x ≠ 1
Module: Mathlib.Analysis.SpecialFunctions.Stirling

Goal state before:
pp: case zero
⊢ log (stirlingSeq 0) = log ↑0! - 1 / 2 * log (2 * ↑0) - ↑0 * log (↑0 / rexp 1)

pp: case succ
n✝ : ℕ
⊢ log (stirlingSeq (n✝ + 1)) = log ↑(n✝ + 1)! - 1 / 2 * log (2 * ↑(n✝ + 1)) - ↑(n✝ + 1) * log (↑(n✝ + 1) / rexp 1)

Tactic:
· simp

Goal state after:
case succ
n✝ : ℕ
⊢ log (stirlingSeq (n✝ + 1)) = log ↑(n✝ + 1)! - 1 / 2 * log (2 * ↑(n✝ + 1)) - ↑(n✝ + 1) * log (↑(n✝ + 1) / rexp 1)
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.ContDiff

Goal state before:
pp: case inl
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : ℝ → E
a : ℝ
inst✝ : CompleteSpace E
h : ContDiffOn ℝ 1 f (Icc a a)
hab : a ≤ a
⊢ ∫ (x : ℝ) in a..a, deriv f x = f a - f a

pp: case inr
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
inst✝ : CompleteSpace E
h : ContDiffOn ℝ 1 f (Icc a b)
hab : a ≤ b
h'ab : a < b
⊢ ∫ (x : ℝ) in a..b, deriv f x = f b - f a

Tactic:
· simp

Goal state after:
case inr
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
inst✝ : CompleteSpace E
h : ContDiffOn ℝ 1 f (Icc a b)
hab : a ≤ b
h'ab : a < b
⊢ ∫ (x : ℝ) in a..b, deriv f x = f b - f a
Module: Mathlib.Analysis.SpecialFunctions.Stirling

Goal state before:
pp: case succ
n✝ : ℕ
⊢ log ↑(n✝ + 1)! - (1 / 2 * log (2 * ↑(n✝ + 1)) + log ((↑(n✝ + 1) / rexp 1) ^ (n✝ + 1))) =
    log ↑(n✝ + 1)! - 1 / 2 * log (2 * ↑(n✝ + 1)) - ↑(n✝ + 1) * log (↑(n✝ + 1) / rexp 1)

pp: case succ.hx
n✝ : ℕ
⊢ 0 < 2 * ↑(n✝ + 1)

pp: case succ.hx
n✝ : ℕ
⊢ √(2 * ↑(n✝ + 1)) ≠ 0

pp: case succ.hy
n✝ : ℕ
⊢ (↑(n✝ + 1) / rexp 1) ^ (n✝ + 1) ≠ 0

pp: case succ.hx
n✝ : ℕ
⊢ ↑(n✝ + 1)! ≠ 0

pp: case succ.hy
n✝ : ℕ
⊢ √(2 * ↑(n✝ + 1)) * (↑(n✝ + 1) / rexp 1) ^ (n✝ + 1) ≠ 0

Tactic:
Real.log_pow,

Goal state after:
case succ
n✝ : ℕ
⊢ log ↑(n✝ + 1)! - (1 / 2 * log (2 * ↑(n✝ + 1)) + ↑(n✝ + 1) * log (↑(n✝ + 1) / rexp 1)) =
    log ↑(n✝ + 1)! - 1 / 2 * log (2 * ↑(n✝ + 1)) - ↑(n✝ + 1) * log (↑(n✝ + 1) / rexp 1)

case succ.hx
n✝ : ℕ
⊢ 0 < 2 * ↑(n✝ + 1)

case succ.hx
n✝ : ℕ
⊢ √(2 * ↑(n✝ + 1)) ≠ 0

case succ.hy
n✝ : ℕ
⊢ (↑(n✝ + 1) / rexp 1) ^ (n✝ + 1) ≠ 0

case succ.hx
n✝ : ℕ
⊢ ↑(n✝ + 1)! ≠ 0

case succ.hy
n✝ : ℕ
⊢ √(2 * ↑(n✝ + 1)) * (↑(n✝ + 1) / rexp 1) ^ (n✝ + 1) ≠ 0
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
x : ℂ
r : ℕ
hr : 2 < r
⊢ HasSum (fun l ↦ ((↑l - x) ^ r)⁻¹) (L.sumInvPow x r)

Tactic:
by
  refine Summable.hasSum (.of_norm_bounded (ZLattice.summable_norm_sub_zpow _
    (-r) (by simpa) x) fun l ↦ ?_)
  rw [← zpow_natCast, ← zpow_neg, ← norm_zpow]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Orthogonality

Goal state before:
pp: case hf'
f : ℝ → ℝ
⊢ ∀ x ∈ Ioo (min 1 (-1)) (max 1 (-1)), -(1 / √(1 - x ^ 2)) ≤ 0

Tactic:
· simp

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

Goal state before:
pp: case inl
⊢ √(2 * π * ↑0) * (↑0 / rexp 1) ^ 0 ≤ ↑0!

pp: case inr
n : ℕ
hn : n ≠ 0
⊢ √(2 * π * ↑n) * (↑n / rexp 1) ^ n ≤ ↑n !

Tactic:
· simp

Goal state after:
case inr
n : ℕ
hn : n ≠ 0
⊢ √(2 * π * ↑n) * (↑n / rexp 1) ^ n ≤ ↑n !
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Orthogonality

Goal state before:
pp: n : ℤ
hn✝ : n ≠ 0
hn' : ↑n ≠ 0
hn : n < 0
⊢ -(sin 0 - sin (↑n * π)) = 0

pp: n : ℤ
hn✝ : n ≠ 0
hn' : ↑n ≠ 0
hn : n < 0
⊢ ↑n * π ≤ 0

Tactic:
· simp

Goal state after:
n : ℤ
hn✝ : n ≠ 0
hn' : ↑n ≠ 0
hn : n < 0
⊢ ↑n * π ≤ 0
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Orthogonality

Goal state before:
pp: n m : ℕ
h : n ≠ m
⊢ ↑n + ↑m ≠ 0

Tactic:
by grind

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Orthogonality

Goal state before:
pp: n m : ℕ
h : n ≠ m
⊢ ↑n - ↑m ≠ 0

Tactic:
by grind

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Orthogonality

Goal state before:
pp: ⊢ ∫ (x : ℝ), eval x (T ℝ 0) * eval x (T ℝ 0) ∂measureT = π

Tactic:
← eval_mul,

Goal state after:
⊢ ∫ (x : ℝ), eval x (T ℝ 0 * T ℝ 0) ∂measureT = π
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Extremal

Goal state before:
pp: case inl
⊢ StrictAntiOn (fun x ↦ node 0 x) ↑(Finset.range (0 + 1))

pp: case inr
n : ℕ
hn : n ≠ 0
⊢ StrictAntiOn (fun x ↦ node n x) ↑(Finset.range (n + 1))

Tactic:
· simp

Goal state after:
case inr
n : ℕ
hn : n ≠ 0
⊢ StrictAntiOn (fun x ↦ node n x) ↑(Finset.range (n + 1))