mathlib-initiative/mathlib-tactics huggingface dataset:
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
⊢ a ^ p = ∫ (t : ℝ) in Ioi 0, cfcₙ ((↑p).rpowIntegrand₀₁ t) a ∂μ

Tactic:
case =>

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
⊢ IntegrableOn (fun t ↦ cfcₙ ((↑p).rpowIntegrand₁₂ t) a) (Ioi 0) μ

Tactic:
case =>

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
⊢ a ^ p = ∫ (t : ℝ) in Ioi 0, cfcₙ ((↑p).rpowIntegrand₁₂ t) a ∂μ

Tactic:
case =>

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

Goal state before:
pp: ε : ℝ
hε : 0 < ε
x : ℝ
heq : rexp (-(ε * x * x)) + x * (rexp (-(ε * x * x)) * (-2 * ε * x)) = rexp (-(ε * x * x)) * (1 - 2 * (ε * x * x))
⊢ |rexp (-(ε * x * x)) + x * (rexp (-(ε * x * x)) * (-2 * ε * x))| ≤ 1

Tactic:
heq,

Goal state after:
ε : ℝ
hε : 0 < ε
x : ℝ
heq : rexp (-(ε * x * x)) + x * (rexp (-(ε * x * x)) * (-2 * ε * x)) = rexp (-(ε * x * x)) * (1 - 2 * (ε * x * x))
⊢ |rexp (-(ε * x * x)) * (1 - 2 * (ε * x * x))| ≤ 1
Module: Mathlib.Analysis.SpecialFunctions.Log.Base

Goal state before:
pp: b a c : ℝ
n : ℕ
ha : a ≠ 0
⊢ Tendsto (fun x ↦ logb b x ^ n / (a * x + c)) atTop (𝓝 0)

Tactic:
cases eq_or_ne (log b) 0 with
  | inl h => simpa [logb, h] using! ((tendsto_mul_add_inv_atTop_nhds_zero _ _ ha).const_mul _)
  | inr h => apply (tendsto_pow_log_div_mul_add_atTop (a * (log b) ^ n) (c * (log b) ^ n) n
                (by positivity)).congr fun x ↦ by simp [field, div_pow, logb]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.RegularityCompacts

Goal state before:
pp: case mp
α : Type u_1
inst✝² : MeasurableSpace α
μ : Measure α
inst✝¹ : TopologicalSpace α
inst✝ : R1Space α
h : μ.InnerRegularWRT (IsCompact ∘ closure) IsClosed
A : Set α
hA : IsClosed A
r : ℝ≥0∞
hr : r < μ A
⊢ ∃ K ⊆ A, IsCompact K ∧ r < μ K

Tactic:
rcases h hA r hr with ⟨K, ⟨hK1, hK2, hK3⟩⟩

Goal state after:
case mp
α : Type u_1
inst✝² : MeasurableSpace α
μ : Measure α
inst✝¹ : TopologicalSpace α
inst✝ : R1Space α
h : μ.InnerRegularWRT (IsCompact ∘ closure) IsClosed
A : Set α
hA : IsClosed A
r : ℝ≥0∞
hr : r < μ A
K : Set α
hK1 : K ⊆ A
hK2 : (IsCompact ∘ closure) K
hK3 : r < μ K
⊢ ∃ K ⊆ A, IsCompact K ∧ r < μ K
Module: Mathlib.MeasureTheory.Measure.RegularityCompacts

Goal state before:
pp: case mpr
α : Type u_1
inst✝² : MeasurableSpace α
μ : Measure α
inst✝¹ : TopologicalSpace α
inst✝ : R1Space α
h : μ.InnerRegularWRT IsCompact IsClosed
A : Set α
hA : IsClosed A
r : ℝ≥0∞
hr : r < μ A
⊢ ∃ K ⊆ A, (IsCompact ∘ closure) K ∧ r < μ K

Tactic:
rcases h hA r hr with ⟨K, ⟨hK1, hK2, hK3⟩⟩

Goal state after:
case mpr
α : Type u_1
inst✝² : MeasurableSpace α
μ : Measure α
inst✝¹ : TopologicalSpace α
inst✝ : R1Space α
h : μ.InnerRegularWRT IsCompact IsClosed
A : Set α
hA : IsClosed A
r : ℝ≥0∞
hr : r < μ A
K : Set α
hK1 : K ⊆ A
hK2 : IsCompact K
hK3 : r < μ K
⊢ ∃ K ⊆ A, (IsCompact ∘ closure) K ∧ r < μ K
Module: Mathlib.MeasureTheory.Measure.RegularityCompacts

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p q : Set α → Prop
hpq : ∀ (A B : Set α), p A → q B → p (A ∩ B)
hμ : ∀ (ε : ℝ≥0∞), 0 < ε → ∃ K, p K ∧ μ Kᶜ < ε
A : Set α
hA : q A
r : ℝ≥0∞
hr : r < μ A
K' : Set α
hpK' : p K'
hK'_lt : μ K'ᶜ < μ A - r
⊢ μ (A \ (K' ∩ A)) < μ A - r

Tactic:
refine (measure_mono fun x ↦ ?_).trans_lt hK'_lt

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p q : Set α → Prop
hpq : ∀ (A B : Set α), p A → q B → p (A ∩ B)
hμ : ∀ (ε : ℝ≥0∞), 0 < ε → ∃ K, p K ∧ μ Kᶜ < ε
A : Set α
hA : q A
r : ℝ≥0∞
hr : r < μ A
K' : Set α
hpK' : p K'
hK'_lt : μ K'ᶜ < μ A - r
x : α
⊢ x ∈ A \ (K' ∩ A) → x ∈ K'ᶜ
Module: Mathlib.MeasureTheory.Measure.RegularityCompacts

Goal state before:
pp: α : Type u_1
inst✝⁵ : MeasurableSpace α
inst✝⁴ : TopologicalSpace α
inst✝³ : SecondCountableTopology α
inst✝² : IsCompletelyPseudoMetrizableSpace α
inst✝¹ : OpensMeasurableSpace α
P : Measure α
inst✝ : IsFiniteMeasure P
ε : ℝ≥0∞
hε : 0 < ε
this : UpgradedIsCompletelyPseudoMetrizableSpace α := upgradeIsCompletelyPseudoMetrizable α
h✝ : IsEmpty α
⊢ ∃ K, IsCompact (closure K) ∧ P Kᶜ < ε

Tactic:
case =>

Goal state after:
Module: Mathlib.MeasureTheory.Measure.RegularityCompacts

Goal state before:
pp: α : Type u_1
inst✝⁵ : MeasurableSpace α
inst✝⁴ : TopologicalSpace α
inst✝³ : SecondCountableTopology α
inst✝² : IsCompletelyPseudoMetrizableSpace α
inst✝¹ : OpensMeasurableSpace α
P : Measure α
inst✝ : IsFiniteMeasure P
ε : ℝ≥0∞
hε : 0 < ε
this : UpgradedIsCompletelyPseudoMetrizableSpace α := upgradeIsCompletelyPseudoMetrizable α
h✝ : Nonempty α
⊢ ∃ K, IsCompact (closure K) ∧ P Kᶜ < ε

Tactic:
case =>

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

Goal state before:
pp: n : ℕ
hn : n ≠ 0
⊢ ConvexOn ℝ Set.univ ((Set.Ici (↑n - 1)).piecewise (fun x ↦ Polynomial.eval x (descPochhammer ℝ n)) 0)

Tactic:
rw [← Nat.pos_iff_ne_zero] at hn

Goal state after:
n : ℕ
hn : 0 < n
⊢ ConvexOn ℝ Set.univ ((Set.Ici (↑n - 1)).piecewise (fun x ↦ Polynomial.eval x (descPochhammer ℝ n)) 0)
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:
case =>

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
⊢ guess ^ (n + 2) ≤ a

Tactic:
case =>

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

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

Tactic:
case =>

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

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

Tactic:
case =>

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

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

Tactic:
h'

Goal state after:
case h
a b : Multiset ℂ
h : ∀ j ∈ a, ∀ (k : ℕ), j ≠ -↑k
n : ℕ
hn : (b.toFinset.sup fun x ↦ ⌈-x.re⌉₊) + 1 ≤ n
h' : regularizedHGFunCoeff a b n = 0
⊢ False
Module: Mathlib.Analysis.SpecialFunctions.Sigmoid

Goal state before:
pp: case refine_1
⊢ range sigmoid ⊆ Ioo 0 1

pp: case refine_2
x : ℝ
hx : x ∈ Ioo 0 1
⊢ x ∈ range sigmoid

Tactic:
· rintro - ⟨x, rfl⟩
    push _ ∈ _
    bound

Goal state after:
case refine_2
x : ℝ
hx : x ∈ Ioo 0 1
⊢ x ∈ range sigmoid
Module: Mathlib.Analysis.SpecialFunctions.Sigmoid

Goal state before:
pp: x : ℝ
hx : x ∈ Ioo 0 1
⊢ 1 < x⁻¹

Tactic:
one_lt_inv_iff₀

Goal state after:
x : ℝ
hx : x ∈ Ioo 0 1
⊢ 0 < x ∧ x < 1
Module: Mathlib.Analysis.SpecialFunctions.Stirling

Goal state before:
pp: n : ℕ
⊢ log (stirlingSeq n) - log (stirlingSeq (n + 1)) ≤ 1 / (4 * ↑n ^ 2)

Tactic:
grw [log_stirlingSeq_sdiff_le]

Goal state after:
n : ℕ
⊢ 1 / (12 * ↑n * (↑n + 1)) ≤ 1 / (4 * ↑n ^ 2)
Module: Mathlib.Analysis.SpecialFunctions.Stirling

Goal state before:
pp: n : ℕ
f : ℕ → ℝ := fun k ↦ log (stirlingSeq (k + 1))
g : ℕ → ℝ := fun k ↦ 1 / (12 * (↑k + 1))
k : ℕ
hk : k ∈ range n
⊢ f k - f (k + 1) ≤ g k - g (k + 1)

Tactic:
grw [log_stirlingSeq_sdiff_le]

Goal state after:
n : ℕ
f : ℕ → ℝ := fun k ↦ log (stirlingSeq (k + 1))
g : ℕ → ℝ := fun k ↦ 1 / (12 * (↑k + 1))
k : ℕ
hk : k ∈ range n
⊢ 1 / (12 * ↑(k + 1) * (↑(k + 1) + 1)) ≤ g k - g (k + 1)
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Orthogonality

Goal state before:
pp: n : ℤ
hn : n ≠ 0
hn' : ↑n ≠ 0
h✝ : 0 ≤ n
⊢ ∫ (θ : ℝ) in 0..↑n * π, deriv sin θ = 0

Tactic:
case =>

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

Goal state before:
pp: n : ℤ
hn✝ : n ≠ 0
hn' : ↑n ≠ 0
hn : n < 0
⊢ ∫ (θ : ℝ) in 0..↑n * π, deriv sin θ = 0

Tactic:
case =>

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

Goal state before:
pp: case e'_3
n : ℕ
k : ℤ
hn : n ≠ 0
hk : ¬2 * ↑n ∣ k
⊢ (-1) ^ k - 1 = Complex.exp (↑k / ↑n * ↑π * I) ^ n - 1

Tactic:
rw [← Complex.exp_nat_mul,
      show (n * (k / n * π * I)) = k * (π * I) by field [show (n : ℂ) ≠ 0 by aesop],
      Complex.exp_int_mul, Complex.exp_pi_mul_I]

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

Goal state before:
pp: n : ℕ
hn : n ≠ 0
c : ℕ → ℝ
hP : (∑ i ∈ range (2 * n), c i • ↑(chebyshevTsequence ℝ) i).degree < 2 * ↑n
i : ℕ
hrange : i ∈ range (2 * n)
hi : i = 0
⊢ ∫ (a : ℝ), eval a (T ℝ ↑i) ∂measureT = sumZeroes n (T ℝ ↑i)

Tactic:
case =>

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

Goal state before:
pp: n : ℕ
hn : n ≠ 0
c : ℕ → ℝ
hP : (∑ i ∈ range (2 * n), c i • ↑(chebyshevTsequence ℝ) i).degree < 2 * ↑n
i : ℕ
hrange : i ∈ range (2 * n)
hi : ¬i = 0
⊢ ∫ (a : ℝ), eval a (T ℝ ↑i) ∂measureT = sumZeroes n (T ℝ ↑i)

Tactic:
case =>

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

Goal state before:
pp: n : ℕ
P : ℝ[X]
hPdeg : P.degree ≤ ↑n
hPbnd : ∀ x ∈ Set.Icc (-1) 1, |eval x P| ≤ 1
hP : P = 0
⊢ P.leadingCoeff ≤ 2 ^ (n - 1)

Tactic:
case =>

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

Goal state before:
pp: n : ℕ
P : ℝ[X]
hPdeg : P.degree ≤ ↑n
hPbnd : ∀ x ∈ Set.Icc (-1) 1, |eval x P| ≤ 1
hP : ¬P = 0
⊢ P.leadingCoeff ≤ 2 ^ (n - 1)

Tactic:
case =>

Goal state after:
Module: Mathlib.Order.Interval.Finset.Box

Goal state before:
pp: x : ℤ × ℤ
⊢ x ∈ box 0 ↔ max x.1.natAbs x.2.natAbs = 0

Tactic:
by simp [Prod.ext_iff]

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

Goal state before:
pp: n k i : ℕ
hk₁ : 0 < k
hk₂ : k ≤ n
hi : i ≤ n
x : ℝ
hx : 1 ≤ x
h✝ : ¬(1 ≤ i ∧ i ≤ n - k)
⊢ ∃ s ⊆ (Finset.range (n + 1)).erase i, s.card = n - k ∧ 0 ∉ s

Tactic:
case =>

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

Goal state before:
pp: n k i : ℕ
hk₁ : 0 < k
hk₂ : k ≤ n
hi : i ≤ n
x : ℝ
hx : 1 ≤ x
h✝ : 1 ≤ i ∧ i ≤ n - k
⊢ ∃ s ⊆ (Finset.range (n + 1)).erase i, s.card = n - k ∧ 0 ∉ s

Tactic:
case =>

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.TsumUniformlyOn

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
F : Type u_3
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : IsRCLikeNormedField 𝕜
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set 𝕜
f : ι → 𝕜 → F
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
x : 𝕜
hx : x ∈ s
hf : ∀ y ∈ s, Summable fun n ↦ f n y
h : SummableLocallyUniformlyOn (fun n ↦ derivWithin (fun z ↦ f n z) s) s
hf2 : ∀ (n : ι), ∀ r ∈ s, DifferentiableAt 𝕜 (f n) r
⊢ ∀ᶠ (n : Finset ι) in (SummationFilter.unconditional ι).filter,
    ∀ x ∈ s, HasDerivAt (fun y ↦ ∑ b ∈ n, (fun n ↦ f n y) b) (∑ i ∈ n, derivWithin (fun z ↦ f i z) s x) x

Tactic:
filter_upwards with t r hr using HasDerivAt.fun_sum
      (fun q hq ↦ ((hf2 q r hr).differentiableWithinAt.hasDerivWithinAt.hasDerivAt)
      (hs.mem_nhds hr))

Goal state after:
Module: Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable

Goal state before:
pp: z : ℍ
a b : ℤ
h0 : z ∈ verticalStrip |z.re| z.im
⊢ ∃ c, ∀ᶠ (x : Fin 2 → ℤ) in cofinite, ‖((↑(x 0) + ↑a) * ↑z + ↑(x 1) + ↑b)⁻¹‖ ≤ c * ‖‖![x 0 + a, x 1 + b]‖⁻¹‖

Tactic:
use ‖r ⟨⟨|z.re|, z.im⟩, z.2⟩‖⁻¹

Goal state after:
case h
z : ℍ
a b : ℤ
h0 : z ∈ verticalStrip |z.re| z.im
⊢ ∀ᶠ (x : Fin 2 → ℤ) in cofinite,
    ‖((↑(x 0) + ↑a) * ↑z + ↑(x 1) + ↑b)⁻¹‖ ≤
      ‖r { coe := { re := |z.re|, im := z.im }, coe_im_pos := ⋯ }‖⁻¹ * ‖‖![x 0 + a, x 1 + b]‖⁻¹‖
Module: Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass

Goal state before:
pp: L : PeriodPair
this : 7 ≤ meromorphicOrderAt (L.relation * id ^ 6) 0
⊢ 1 ≤ meromorphicOrderAt L.relation 0

Tactic:
meromorphicOrderAt_mul (by fun_prop) (by fun_prop),

Goal state after:
L : PeriodPair
this : 7 ≤ meromorphicOrderAt L.relation 0 + meromorphicOrderAt (id ^ 6) 0
⊢ 1 ≤ meromorphicOrderAt L.relation 0
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema

Goal state before:
pp: n k : ℕ
hn : n ≠ 0
hk₀ : 0 < k
hk₁ : k < n
hk₂ : Even k
⊢ IsLocalExtr (fun x ↦ eval x (T ℝ ↑n)) (cos (↑k * π / ↑n))

Tactic:
case =>

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

Goal state before:
pp: n k : ℕ
hn : n ≠ 0
hk₀ : 0 < k
hk₁ : k < n
hk₂ : Odd k
⊢ IsLocalExtr (fun x ↦ eval x (T ℝ ↑n)) (cos (↑k * π / ↑n))

Tactic:
case =>

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

Goal state before:
pp: n k : ℕ
hn : n ≠ 0
hk : k ≤ n
hk₂ : Even k
⊢ IsExtrOn (fun x ↦ eval x (T ℝ ↑n)) (Set.Icc (-1) 1) (cos (↑k * π / ↑n))

Tactic:
case =>

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

Goal state before:
pp: n k : ℕ
hn : n ≠ 0
hk : k ≤ n
hk₂ : Odd k
⊢ IsExtrOn (fun x ↦ eval x (T ℝ ↑n)) (Set.Icc (-1) 1) (cos (↑k * π / ↑n))

Tactic:
case =>

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

Goal state before:
pp: z : ℂ
⊢ (↑π * z).cot = (cexp (2 * ↑π * I * z) + 1) / (I * (1 - cexp (2 * ↑π * I * z)))

Tactic:
by
  rw [cot_eq_exp_ratio (π * z)]
  ring_nf

Goal state after:
Module: Mathlib.Analysis.SpecificLimits.ArithmeticGeometric

Goal state before:

Tactic:
b / (1 - a)
    _ = a * (b / (1 - a)) + b := by grind
    _ < a * arithGeom a b u₀ n + b := by gcongr

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

Goal state before:
pp: x : ℂ
k : ℕ∞
hx : x ∈ ℂ_ℤ
⊢ ContDiffWithinAt ℂ (↑k) (fun x ↦ (↑π * x).cot) ℍₒ x

Tactic:
by
  simp_rw [Complex.cot, Complex.cos, Complex.sin]
  exact ContDiffWithinAt.div (by fun_prop) (by fun_prop) (sin_pi_mul_ne_zero hx)

Goal state after:
Module: Mathlib.Tactic.NormNum.Prime

Goal state before:
pp: case refine_1
n k k' : ℕ
e : k + 2 = k'
h : MinFacHelper n k
np : n.minFac ≠ k
⊢ k % 2 = 1

Tactic:
exact h.2.1

Goal state after:
Module: Mathlib.NumberTheory.Real.GoldenRatio

Goal state before:
pp: case inl
n : ℕ
⊢ ↑(Int.fib ↑n) = (φ ^ ↑n - ψ ^ ↑n) / √5

Tactic:
exact coe_fib_eq n

Goal state after:
Module: Mathlib.Analysis.Subadditive

Goal state before:
pp: case zero
u : ℕ → ℝ
h : Subadditive u
n r : ℕ
⊢ u (0 * n + r) ≤ ↑0 * u n + u r

Tactic:
simp only [Nat.cast_zero, zero_mul, zero_add]

Goal state after:
case zero
u : ℕ → ℝ
h : Subadditive u
n r : ℕ
⊢ u r ≤ u r
Module: Mathlib.Analysis.Subadditive

Goal state before:
pp: u : ℕ → ℝ
h : Subadditive u
k n r : ℕ
⊢ u (k * n + r) ≤ ↑k * u n + u r

Tactic:
induction k with
  | zero => simp only [Nat.cast_zero, zero_mul, zero_add]; rfl
  | succ k IH =>
    calc
      u ((k + 1) * n + r) = u (n + (k * n + r)) := by congr 1; ring
      _ ≤ u n + u (k * n + r) := h _ _
      _ ≤ u n + (k * u n + u r) := by grw [IH]
      _ = (k + 1 : ℕ) * u n + u r := by simp; ring

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.DiagramLemmas.KernelCokernelComp

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
X Y Z : C
f : X ⟶ Y
g : Y ⟶ Z
A : C
k : Y ⊞ Z ⟶ A
hk : φ f g ≫ k = 0
⊢ { l // π f g ≫ l = k }

Tactic:
by
    refine ⟨cokernel.desc _ (biprod.inr ≫ k) ?_, ?_⟩
    all_goals
      obtain ⟨k₁, k₂, rfl⟩ := biprod.decomp_hom_from k
      simp only [comp_add, φ_snd_assoc, biprod.ext_from_iff, inl_φ_assoc,
        BinaryBicone.inl_fst_assoc, BinaryBicone.inl_snd_assoc, zero_comp, add_zero, comp_zero,
        inr_φ_fst_assoc, neg_comp, id_comp, BinaryBicone.inr_snd_assoc, neg_add_eq_zero] at hk
      obtain ⟨hk, rfl⟩ := hk
      aesop

Goal state after:
Module: Mathlib.Analysis.SpecificLimits.FloorPow

Goal state before:
pp: case h.hb
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < 1 - c⁻¹
i : ℕ
a✝ : i ∈ {i ∈ range N | j < c ^ i}
⊢ 0 < ↑⌊c ^ i⌋₊ ^ 2

Tactic:
apply sq_pos_of_pos
        refine zero_lt_one.trans_le ?_
        simp only [Nat.le_floor, one_le_pow₀, hc.le, Nat.one_le_cast, Nat.cast_one]

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Indization.FilteredColimits

Goal state before:
pp: case refine_1
C : Type u
inst✝² : Category.{v, u} C
I : Type v
inst✝¹ : SmallCategory I
inst✝ : IsFiltered I
F : I ⥤ Cᵒᵖ ⥤ Type v
hF : ∀ (i : I), IsIndObject (F.obj i)
this : IsFiltered (CostructuredArrow yoneda (colimit F))
s : (i : I) → Set (CostructuredArrow yoneda (F.obj i))
hs : ∀ (i : I), Small.{v, max u v} ↑(s i)
j : (i : I) → CostructuredArrow yoneda (F.obj i) → CostructuredArrow yoneda (F.obj i)
hjs : ∀ (i : I) (i_1 : CostructuredArrow yoneda (F.obj i)), j i i_1 ∈ s i
hj : ∀ (i : I) (i_1 : CostructuredArrow yoneda (F.obj i)), Nonempty (i_1 ⟶ j i i_1)
A : CostructuredArrow yoneda (colimit F)
i : I
y : (F.obj i).obj (Opposite.op A.left)
hy : yonedaEquiv A.hom = (ConcreteCategory.hom ((colimit.ι F i).app (Opposite.op A.left))) y
y' : CostructuredArrow yoneda (F.obj i) := CostructuredArrow.mk (yonedaEquiv.symm y)
x : y' ⟶ j i y'
⊢ (CostructuredArrow.map (colimit.ι F i)).obj (j i y') ∈ ⋃ i, (CostructuredArrow.map (colimit.ι F i)).obj '' s i

pp: case refine_2
C : Type u
inst✝² : Category.{v, u} C
I : Type v
inst✝¹ : SmallCategory I
inst✝ : IsFiltered I
F : I ⥤ Cᵒᵖ ⥤ Type v
hF : ∀ (i : I), IsIndObject (F.obj i)
this : IsFiltered (CostructuredArrow yoneda (colimit F))
s : (i : I) → Set (CostructuredArrow yoneda (F.obj i))
hs : ∀ (i : I), Small.{v, max u v} ↑(s i)
j : (i : I) → CostructuredArrow yoneda (F.obj i) → CostructuredArrow yoneda (F.obj i)
hjs : ∀ (i : I) (i_1 : CostructuredArrow yoneda (F.obj i)), j i i_1 ∈ s i
hj : ∀ (i : I) (i_1 : CostructuredArrow yoneda (F.obj i)), Nonempty (i_1 ⟶ j i i_1)
A : CostructuredArrow yoneda (colimit F)
i : I
y : (F.obj i).obj (Opposite.op A.left)
hy : yonedaEquiv A.hom = (ConcreteCategory.hom ((colimit.ι F i).app (Opposite.op A.left))) y
y' : CostructuredArrow yoneda (F.obj i) := CostructuredArrow.mk (yonedaEquiv.symm y)
x : y' ⟶ j i y'
⊢ A ⟶ (CostructuredArrow.map (colimit.ι F i)).obj (j i y')

Tactic:
· simp only [Set.mem_iUnion, Set.mem_image]
    exact ⟨i, j i y', hjs _ _, rfl⟩

Goal state after:
case refine_2
C : Type u
inst✝² : Category.{v, u} C
I : Type v
inst✝¹ : SmallCategory I
inst✝ : IsFiltered I
F : I ⥤ Cᵒᵖ ⥤ Type v
hF : ∀ (i : I), IsIndObject (F.obj i)
this : IsFiltered (CostructuredArrow yoneda (colimit F))
s : (i : I) → Set (CostructuredArrow yoneda (F.obj i))
hs : ∀ (i : I), Small.{v, max u v} ↑(s i)
j : (i : I) → CostructuredArrow yoneda (F.obj i) → CostructuredArrow yoneda (F.obj i)
hjs : ∀ (i : I) (i_1 : CostructuredArrow yoneda (F.obj i)), j i i_1 ∈ s i
hj : ∀ (i : I) (i_1 : CostructuredArrow yoneda (F.obj i)), Nonempty (i_1 ⟶ j i i_1)
A : CostructuredArrow yoneda (colimit F)
i : I
y : (F.obj i).obj (Opposite.op A.left)
hy : yonedaEquiv A.hom = (ConcreteCategory.hom ((colimit.ι F i).app (Opposite.op A.left))) y
y' : CostructuredArrow yoneda (F.obj i) := CostructuredArrow.mk (yonedaEquiv.symm y)
x : y' ⟶ j i y'
⊢ A ⟶ (CostructuredArrow.map (colimit.ι F i)).obj (j i y')
Module: Mathlib.CategoryTheory.Limits.IndYoneda

Goal state before:
pp: C : Type u₁
inst✝² : Category.{u₂, u₁} C
I : Type v₁
inst✝¹ : Category.{v₂, v₁} I
F : I ⥤ C
inst✝ : HasColimit F
i : I
⊢ (coyonedaOpColimitIsoLimitCoyoneda F).hom ≫ limit.π (F.op ⋙ coyoneda) (op i) = coyoneda.map (colimit.ι F i).op

Tactic:
simp only [coyonedaOpColimitIsoLimitCoyoneda, Functor.mapIso_symm,
    Iso.trans_hom, Iso.symm_hom, Functor.mapIso_inv, Category.assoc, preservesLimitIso_hom_π,
    ← Functor.map_comp, limitOpIsoOpColimit_inv_comp_π]

Goal state after: