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
⊢ HasFiniteIntegral bound (μ.restrict (Ioi 0))

Tactic:
hasFiniteIntegral_norm_iff

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)
hbound : ∀ᵐ (t : ℝ) ∂μ.restrict (Ioi 0), ∀ z ∈ quasispectrum ℝ a, ‖f t z‖ ≤ bound t
⊢ HasFiniteIntegral (fun t ↦ f t maxr) (μ.restrict (Ioi 0))
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
⊢ HasFiniteIntegral bound (μ.restrict (Ioi 0))

Tactic:
hasFiniteIntegral_norm_iff

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)
hbound : ∀ᵐ (t : ℝ) ∂μ.restrict (Ioi 0), ∀ z ∈ quasispectrum ℝ a, ‖f t z‖ ≤ bound t
⊢ HasFiniteIntegral (fun t ↦ f t maxr) (μ.restrict (Ioi 0))
Module: Mathlib.Data.Int.Log

Goal state before:
pp: case inl
R : Type u_1
inst✝³ : Semifield R
inst✝² : LinearOrder R
inst✝¹ : IsStrictOrderedRing R
inst✝ : FloorSemiring R
b : ℕ
hb : 1 < b
r : R
hr : r ≤ 0
⊢ r ≤ ↑b ^ clog b r

Tactic:
rw [clog_of_right_le_zero _ hr, zpow_zero]

Goal state after:
case inl
R : Type u_1
inst✝³ : Semifield R
inst✝² : LinearOrder R
inst✝¹ : IsStrictOrderedRing R
inst✝ : FloorSemiring R
b : ℕ
hb : 1 < b
r : R
hr : r ≤ 0
⊢ r ≤ 1
Module: Mathlib.Data.Int.Log

Goal state before:
pp: R : Type u_1
inst✝³ : Semifield R
inst✝² : LinearOrder R
inst✝¹ : IsStrictOrderedRing R
inst✝ : FloorSemiring R
b : ℕ
⊢ clog b 1 = 0

Tactic:
clog_of_one_le_right _ le_rfl,

Goal state after:
R : Type u_1
inst✝³ : Semifield R
inst✝² : LinearOrder R
inst✝¹ : IsStrictOrderedRing R
inst✝ : FloorSemiring R
b : ℕ
⊢ ↑(Nat.clog b ⌈1⌉₊) = 0
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:
le_logb_iff_rpow_le hb1' hr,

Goal state after:
case a
b : ℕ
r : ℝ
hr✝ : 0 ≤ r
hr : 0 < r
hb : 1 < b
hb1' : 1 < ↑b
⊢ ↑b ^ ↑(Int.log b r) ≤ r
Module: Mathlib.Analysis.SpecialFunctions.Log.Base

Goal state before:
pp: b n : ℕ
⊢ ⌊logb ↑b ↑n⌋₊ = Nat.log b n

Tactic:
by
  obtain _ | _ | b := b
  · simp [Real.logb]
  · simp [Real.logb]
  obtain rfl | hn := eq_or_ne n 0
  · simp
  rw [← Nat.cast_inj (R := ℤ), Int.natCast_floor_eq_floor, floor_logb_natCast (by simp),
    Int.log_natCast]
  exact logb_nonneg (by simp [Nat.cast_add_one_pos]) (Nat.one_le_cast.2 (by lia))

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

Goal state before:
pp: X : Type u_1
inst✝³ : TopologicalSpace X
inst✝² : MeasurableSpace X
μ : Measure X
inst✝¹ : IsFiniteMeasure μ
inst✝ : OpensMeasurableSpace X
f : X →ᵇ ℝ≥0
⊢ Integrable (NNReal.toReal ∘ ⇑f) μ

Tactic:
refine ⟨(NNReal.continuous_coe.comp f.continuous).measurable.aestronglyMeasurable, ?_⟩
  simp only [hasFiniteIntegral_iff_enorm, Function.comp_apply, NNReal.enorm_eq]
  exact lintegral_lt_top_of_nnreal _ f

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 : BorelSpace α
P : Measure α
inst✝ : IsFiniteMeasure P
⊢ P.InnerRegularWRT (fun s ↦ IsCompact s ∧ IsClosed s) fun s ↦ MeasurableSet s ∧ P s ≠ ∞

Tactic:
refine Measure.InnerRegularWRT.measurableSet_of_isOpen ?_ ?_

Goal state after:
case refine_1
α : Type u_1
inst✝⁵ : MeasurableSpace α
inst✝⁴ : TopologicalSpace α
inst✝³ : SecondCountableTopology α
inst✝² : IsCompletelyPseudoMetrizableSpace α
inst✝¹ : BorelSpace α
P : Measure α
inst✝ : IsFiniteMeasure P
⊢ P.InnerRegularWRT (fun s ↦ IsCompact s ∧ IsClosed s) IsOpen

case refine_2
α : Type u_1
inst✝⁵ : MeasurableSpace 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 ∧ IsClosed s → IsOpen U → IsCompact (s \ U) ∧ IsClosed (s \ U)
Module: Mathlib.Analysis.SpecialFunctions.MulExpNegMulSqIntegral

Goal state before:
pp: E : Type u_1
inst✝⁴ : TopologicalSpace E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
P : Measure E
inst✝¹ : IsFiniteMeasure P
ε : ℝ
P' : Measure E
inst✝ : IsFiniteMeasure P'
A : Subalgebra ℝ (E →ᵇ ℝ)
hε : 0 < ε
heq : ∀ g ∈ A, ∫ (x : E), g x ∂P = ∫ (x : E), g x ∂P'
g : E →ᵇ ℝ
hgA : g ∈ A
one_add_inv_mul_mem : ∀ (n : ℕ), g * (1 + (↑n)⁻¹ • -(ε • g * g)) ^ n ∈ A
⊢ ∫ (x : E), ε.mulExpNegMulSq (g x) ∂P = ∫ (x : E), ε.mulExpNegMulSq (g x) ∂P'

Tactic:
have limP : Tendsto (fun n : ℕ => ∫ x, (g * (1 + (n : ℝ)⁻¹ • -(ε • g * g)) ^ n) x ∂P) atTop
      (𝓝 (∫ x, mulExpNegMulSq ε (g x) ∂P')) := by
    rw [funext fun n => heq _ (one_add_inv_mul_mem n)]
    exact tendsto_integral_mul_one_add_inv_smul_sq_pow g hε

Goal state after:
E : Type u_1
inst✝⁴ : TopologicalSpace E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
P : Measure E
inst✝¹ : IsFiniteMeasure P
ε : ℝ
P' : Measure E
inst✝ : IsFiniteMeasure P'
A : Subalgebra ℝ (E →ᵇ ℝ)
hε : 0 < ε
heq : ∀ g ∈ A, ∫ (x : E), g x ∂P = ∫ (x : E), g x ∂P'
g : E →ᵇ ℝ
hgA : g ∈ A
one_add_inv_mul_mem : ∀ (n : ℕ), g * (1 + (↑n)⁻¹ • -(ε • g * g)) ^ n ∈ A
limP :
  Tendsto (fun n ↦ ∫ (x : E), (g * (1 + (↑n)⁻¹ • -(ε • g * g)) ^ n) x ∂P) atTop
    (𝓝 (∫ (x : E), ε.mulExpNegMulSq (g x) ∂P'))
⊢ ∫ (x : E), ε.mulExpNegMulSq (g x) ∂P = ∫ (x : E), ε.mulExpNegMulSq (g x) ∂P'
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 • μ (Subtype.val '' ⇑(homeomorphUnitSphereProd E) ⁻¹' (Prod.fst ⁻¹' s ∩ univ ×ˢ Iio ⟨1, toSphere._proof_1⟩)) =
    ↑(dim E) * μ (Ioo 0 1 • Subtype.val '' s)

Tactic:
univ_prod,

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) ⁻¹' (Prod.fst ⁻¹' s ∩ Prod.snd ⁻¹' Iio ⟨1, toSphere._proof_1⟩)) =
    ↑(dim E) * μ (Ioo 0 1 • Subtype.val '' s)
Module: Mathlib.Analysis.SpecialFunctions.Pow.NthRootLemmas

Goal state before:
pp: n : ℕ
h : n ≠ 0
⊢ n.nthRoot 0 = 0

Tactic:
by
  rcases n with _ | _ | _ <;> grind [nthRoot, nthRoot.go]

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

Goal state before:
pp: L : PeriodPair
⊢ HasSumLocallyUniformly (fun l z ↦ 1 / (z - ↑l) ^ 2 - 1 / ↑l ^ 2) ℘[L]

Tactic:
convert! L.hasSumLocallyUniformly_weierstrassPExcept (L.ω₁ / 2) using 3 with l

Goal state after:
case e'_5
L : PeriodPair
l : ↥L.lattice
x✝ : ℂ
⊢ 1 / (x✝ - ↑l) ^ 2 - 1 / ↑l ^ 2 = if ↑l = L.ω₁ / 2 then 0 else 1 / (x✝ - ↑l) ^ 2 - 1 / ↑l ^ 2

case e'_6
L : PeriodPair
⊢ ℘[L] = ℘[L - L.ω₁ / 2]
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
r : ℝ
⊢ IntegrableOn (fun x ↦ f ‖x‖) (ball 0 r) μ ↔ Integrable (fun x ↦ (Iio r).indicator f ‖x‖) μ

Tactic:
← integrable_indicator_iff measurableSet_ball

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
r : ℝ
⊢ Integrable ((ball 0 r).indicator fun x ↦ f ‖x‖) μ ↔ Integrable (fun x ↦ (Iio r).indicator f ‖x‖) μ
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Extremal

Goal state before:
pp: α : ℝ
i : ℕ
hα : |α| ≤ 1
⊢ (-1) ^ i * α ≤ 1

Tactic:
apply le_of_abs_le

Goal state after:
α : ℝ
i : ℕ
hα : |α| ≤ 1
⊢ |(-1) ^ i * α| ≤ 1
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.ChebyshevGauss

Goal state before:
pp: case inr
n : ℕ
k : ℤ
hk : ¬2 * ↑n ∣ k
hn : n ≠ 0
⊢ ∑ x ∈ range n,
      (Complex.exp (↑k * ((2 * ↑x + 1) / (2 * ↑n) * ↑π) * I) +
        Complex.exp (↑(-k) * ((2 * ↑x + 1) / (2 * ↑n) * ↑π) * I)) =
    0

Tactic:
have : (-1 : ℂ) ^ (-k) = (-1) ^ k := by rw [← Int.cast_negOnePow, ← Int.cast_negOnePow]; simp

Goal state after:
case inr
n : ℕ
k : ℤ
hk : ¬2 * ↑n ∣ k
hn : n ≠ 0
this : (-1) ^ (-k) = (-1) ^ k
⊢ ∑ x ∈ range n,
      (Complex.exp (↑k * ((2 * ↑x + 1) / (2 * ↑n) * ↑π) * I) +
        Complex.exp (↑(-k) * ((2 * ↑x + 1) / (2 * ↑n) * ↑π) * I)) =
    0
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema

Goal state before:
pp: n k : ℕ
hk : k < n
⊢ Multiset.count (cos ((↑k + 1) * π / (↑n + 1))) (U ℝ ↑n).roots = 1

Tactic:
roots_U_real,

Goal state after:
n k : ℕ
hk : k < n
⊢ Multiset.count (cos ((↑k + 1) * π / (↑n + 1)))
      (Finset.image (fun k ↦ cos ((↑k + 1) * π / (↑n + 1))) (Finset.range n)).val =
    1
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema

Goal state before:
pp: case mp
n : ℕ
hn : 2 ≤ n
x : ℝ
hx : x ∈ (U ℝ ↑(n - 1)).roots
⊢ ∃ k ∈ Finset.Ioo 0 n, x = cos (↑k * π / ↑n)

Tactic:
roots_U_real,

Goal state after:
case mp
n : ℕ
hn : 2 ≤ n
x : ℝ
hx : x ∈ (Finset.image (fun k ↦ cos ((↑k + 1) * π / (↑(n - 1) + 1))) (Finset.range (n - 1))).val
⊢ ∃ k ∈ Finset.Ioo 0 n, x = cos (↑k * π / ↑n)
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent

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

Tactic:
Complex.cot,

Goal state after:
z : ℂ
⊢ cos z / sin z = (cexp (2 * I * z) + 1) / (I * (1 - cexp (2 * I * z)))
Module: Mathlib.Analysis.SpecificLimits.ArithmeticGeometric

Goal state before:
pp: case succ.e_a
R : Type u_1
a b u₀ : R
inst✝ : CommSemiring R
n : ℕ
hn : arithGeom a b u₀ n = a ^ n * u₀ + b * ∑ k ∈ Finset.range n, a ^ k
⊢ a * (b * ∑ k ∈ Finset.range n, a ^ k) + b = b * ∑ k ∈ Finset.range (n + 1), a ^ k

Tactic:
Finset.sum_range_succ',

Goal state after:
case succ.e_a
R : Type u_1
a b u₀ : R
inst✝ : CommSemiring R
n : ℕ
hn : arithGeom a b u₀ n = a ^ n * u₀ + b * ∑ k ∈ Finset.range n, a ^ k
⊢ a * (b * ∑ k ∈ Finset.range n, a ^ k) + b = b * (∑ k ∈ Finset.range n, a ^ (k + 1) + a ^ 0)
Module: Mathlib.Analysis.SpecificLimits.ArithmeticGeometric

Goal state before:
pp: R : Type u_1
a b u₀ : R
inst✝² : Field R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
ha : 1 < a
h0 : b / (1 - a) < u₀
n : ℕ
h_lt : b / (1 - a) < arithGeom a b u₀ n
⊢ arithGeom a b u₀ n < arithGeom a b u₀ (n + 1)

Tactic:
div_lt_iff_of_neg (sub_neg.mpr ha)

Goal state after:
R : Type u_1
a b u₀ : R
inst✝² : Field R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
ha : 1 < a
h0 : b / (1 - a) < u₀
n : ℕ
h_lt : arithGeom a b u₀ n * (1 - a) < b
⊢ arithGeom a b u₀ n < arithGeom a b u₀ (n + 1)
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:
Complex.cot,

Goal state after:
x : ℂ
k : ℕ∞
hx : x ∈ ℂ_ℤ
⊢ ContDiffWithinAt ℂ (↑k) (fun x ↦ Complex.cos (↑π * x) / Complex.sin (↑π * x)) ℍₒ x
Module: Mathlib.Tactic.NormNum.Prime

Goal state before:
pp: case inr.inl
k : ℕ
h✝ : MinFacHelper (succ 0) k
this : 2 < (succ 0).minFac
h : succ 0 > 0
⊢ 1 < succ 0

pp: case inr.inr
n k : ℕ
h✝¹ : MinFacHelper n k
this : 2 < n.minFac
h✝ : n > 0
h : succ 0 < n
⊢ 1 < n

Tactic:
· simp_all

Goal state after:
case inr.inr
n k : ℕ
h✝¹ : MinFacHelper n k
this : 2 < n.minFac
h✝ : n > 0
h : succ 0 < n
⊢ 1 < n
Module: Mathlib.CategoryTheory.Abelian.Projective.Basic

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
P : C
hP : PreservesFiniteColimits (preadditiveCoyonedaObj P)
⊢ (preadditiveCoyonedaObj P).PreservesEpimorphisms

Tactic:
have := Functor.preservesHomologyOfExact (preadditiveCoyonedaObj P)

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
P : C
hP : PreservesFiniteColimits (preadditiveCoyonedaObj P)
this : (preadditiveCoyonedaObj P).PreservesHomology
⊢ (preadditiveCoyonedaObj P).PreservesEpimorphisms
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
⊢ Epi
    ((kernelCokernelCompSequence f g).map' 4 5 instEpiMap'KernelCokernelCompSequenceOfNatNat._proof_2
      instEpiMap'KernelCokernelCompSequenceOfNatNat._proof_3)

Tactic:
by
  dsimp [ComposableArrows.Precomp.map]
  infer_instance

Goal state after:
Module: Mathlib.CategoryTheory.Filtered.Small

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFilteredOrEmpty C
α : Type w
f : α → C
j : C
h : filteredClosure f j
⊢ ∃ a, FilteredClosureSmall.abstractFilteredClosureRealization f a = { obj := j, property := h }.obj

Tactic:
induction h with
  | base x =>
      refine ⟨⟨0, ?_⟩, ?_⟩
      · simp only [FilteredClosureSmall.bundledAbstractFilteredClosure]
        exact ULift.up x
      · simp only [FilteredClosureSmall.abstractFilteredClosureRealization,
          FilteredClosureSmall.bundledAbstractFilteredClosure]
        rfl
  | max hj₁ hj₂ ih ih' =>
    rcases ih with ⟨⟨n, x⟩, rfl⟩
    rcases ih' with ⟨⟨m, y⟩, rfl⟩
    refine ⟨⟨(Max.max n m).succ, ?_⟩, ?_⟩
    · simp only [FilteredClosureSmall.bundledAbstractFilteredClosure]
      refine FilteredClosureSmall.InductiveStep.max ?_ ?_ x y
      all_goals apply Nat.lt_succ_of_le
      exacts [Nat.le_max_left _ _, Nat.le_max_right _ _]
    · simp only [FilteredClosureSmall.abstractFilteredClosureRealization]
      rw! [FilteredClosureSmall.bundledAbstractFilteredClosure]
      rfl
  | coeq hj₁ hj₂ g g' ih ih' =>
    rcases ih with ⟨⟨n, x⟩, rfl⟩
    rcases ih' with ⟨⟨m, y⟩, rfl⟩
    refine ⟨⟨(Max.max n m).succ, ?_⟩, ?_⟩
    · simp only [FilteredClosureSmall.bundledAbstractFilteredClosure]
      refine FilteredClosureSmall.InductiveStep.coeq ?_ ?_ x y g g'
      all_goals apply Nat.lt_succ_of_le
      exacts [Nat.le_max_left _ _, Nat.le_max_right _ _]
    · simp only [FilteredClosureSmall.abstractFilteredClosureRealization]
      rw! [FilteredClosureSmall.bundledAbstractFilteredClosure]
      rfl

Goal state after:
Module: Mathlib.CategoryTheory.Comma.Final

Goal state before:
pp: A : Type u₁
inst✝⁵ : Category.{v₁, u₁} A
B : Type u₂
inst✝⁴ : Category.{v₂, u₂} B
T : Type u₃
inst✝³ : Category.{v₃, u₃} T
L : A ⥤ T
R : B ⥤ T
inst✝² : IsCofiltered A
inst✝¹ : IsCofiltered B
inst✝ : ∀ (b : B), IsCofiltered (CostructuredArrow L (R.obj b))
⊢ IsCofilteredOrEmpty (Comma L R)

Tactic:
refine ⟨fun j₁ j₂ ↦ ?_, fun j₁ j₂ u v ↦ ?_⟩

Goal state after:
case refine_1
A : Type u₁
inst✝⁵ : Category.{v₁, u₁} A
B : Type u₂
inst✝⁴ : Category.{v₂, u₂} B
T : Type u₃
inst✝³ : Category.{v₃, u₃} T
L : A ⥤ T
R : B ⥤ T
inst✝² : IsCofiltered A
inst✝¹ : IsCofiltered B
inst✝ : ∀ (b : B), IsCofiltered (CostructuredArrow L (R.obj b))
j₁ j₂ : Comma L R
⊢ ∃ W x x, True

case refine_2
A : Type u₁
inst✝⁵ : Category.{v₁, u₁} A
B : Type u₂
inst✝⁴ : Category.{v₂, u₂} B
T : Type u₃
inst✝³ : Category.{v₃, u₃} T
L : A ⥤ T
R : B ⥤ T
inst✝² : IsCofiltered A
inst✝¹ : IsCofiltered B
inst✝ : ∀ (b : B), IsCofiltered (CostructuredArrow L (R.obj b))
j₁ j₂ : Comma L R
u v : j₁ ⟶ j₂
⊢ ∃ W h, h ≫ u = h ≫ v
Module: Mathlib.CategoryTheory.Limits.Indization.Category

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
J : Type v
inst✝¹ : SmallCategory J
inst✝ : IsFiltered J
⊢ ObjectProperty.strictColimitsOfShape IsIndObject J ≤ IsIndObject

Tactic:
rintro _ ⟨F, hF⟩

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
J : Type v
inst✝² : SmallCategory J
inst✝¹ : IsFiltered J
F : J ⥤ Cᵒᵖ ⥤ Type v
inst✝ : HasColimit F
hF : ∀ (j : J), IsIndObject (F.obj j)
⊢ IsIndObject (colimit F)
Module: Mathlib.CategoryTheory.Limits.Indization.Category

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
J : Type v
inst✝ : HasLimitsOfShape (Discrete J) C
⊢ ObjectProperty.strictLimitsOfShape IsIndObject (Discrete J) ≤ IsIndObject

Tactic:
rintro _ ⟨F, hF⟩

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
J : Type v
inst✝¹ : HasLimitsOfShape (Discrete J) C
F : Discrete J ⥤ Cᵒᵖ ⥤ Type v
inst✝ : HasLimit F
hF : ∀ (j : Discrete J), IsIndObject (F.obj j)
⊢ IsIndObject (limit F)
Module: Mathlib.CategoryTheory.Limits.FilteredColimitCommutesProduct

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
ι : Type u_2
inst✝ : HasProductsOfShape ι C
H :
  ∀ (J : ι → Type w) [inst : (i : ι) → SmallCategory (J i)] [∀ (i : ι), IsFiltered (J i)] (F : (i : ι) → J i ⥤ C)
    [inst_2 : ∀ (i : ι), HasColimit (F i)],
    ∃ (x : HasColimit (pointwiseProduct F)), IsIso (colimitPointwiseProductToProductColimit F)
⊢ IsIPCOfShape.{w, u_2, v_1, u_1} ι C

Tactic:
refine .of_forall_exists fun J _ _ F _ ↦ ?_

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
ι : Type u_2
inst✝ : HasProductsOfShape ι C
H :
  ∀ (J : ι → Type w) [inst : (i : ι) → SmallCategory (J i)] [∀ (i : ι), IsFiltered (J i)] (F : (i : ι) → J i ⥤ C)
    [inst_2 : ∀ (i : ι), HasColimit (F i)],
    ∃ (x : HasColimit (pointwiseProduct F)), IsIso (colimitPointwiseProductToProductColimit F)
J : ι → Type w
x✝² : (i : ι) → SmallCategory (J i)
x✝¹ : ∀ (i : ι), IsFiltered (J i)
F : (i : ι) → J i ⥤ C
x✝ : ∀ (i : ι), HasColimit (F i)
⊢ ∃ c x, Nonempty (IsColimit (coconePointwiseProduct c))
Module: Mathlib.CategoryTheory.Abelian.GrothendieckCategory.ModuleEmbedding.GabrielPopescu

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : IsGrothendieckAbelian.{v, v, u} C
G : C
hG : IsSeparator G
this✝¹ : (preadditiveCoyonedaObj G).PreservesInjectiveObjects
this✝ : (tensorObj G).PreservesMonomorphisms
this : PreservesBinaryBiproducts (tensorObj G)
⊢ PreservesFiniteLimits (tensorObj G)

Tactic:
have : (tensorObj G).Additive := Functor.additive_of_preservesBinaryBiproducts _

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : IsGrothendieckAbelian.{v, v, u} C
G : C
hG : IsSeparator G
this✝² : (preadditiveCoyonedaObj G).PreservesInjectiveObjects
this✝¹ : (tensorObj G).PreservesMonomorphisms
this✝ : PreservesBinaryBiproducts (tensorObj G)
this : (tensorObj G).Additive
⊢ PreservesFiniteLimits (tensorObj G)
Module: Mathlib.CategoryTheory.Abelian.Preradical.Radical

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
Φ : Preradical C
⊢ isRadical C Φ ↔ IsZero (Φ.quotient ⋙ Φ.r)

Tactic:
isRadical_iff_isIso,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
Φ : Preradical C
⊢ IsIso (Φ.toColon Φ) ↔ IsZero (Φ.quotient ⋙ Φ.r)
Module: Mathlib.CategoryTheory.Abelian.Injective.Resolution

Goal state before:
pp: case zero
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : EnoughInjectives C
Z : C
⊢ Injective ((ofCocomplex Z).X 0)

Tactic:
apply Injective.injective_under

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.Injective.Resolution

Goal state before:
pp: case succ.zero
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : EnoughInjectives C
Z : C
⊢ Injective ((ofCocomplex Z).X (0 + 1))

Tactic:
apply Injective.injective_under

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.Injective.Resolution

Goal state before:
pp: case succ.succ.zero
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : EnoughInjectives C
Z : C
⊢ Injective ((ofCocomplex Z).X (0 + 1 + 1))

Tactic:
apply Injective.injective_under

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.Injective.Resolution

Goal state before:
pp: case succ.succ.succ
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : EnoughInjectives C
Z : C
n : ℕ
⊢ Injective ((ofCocomplex Z).X (n + 1 + 1 + 1))

Tactic:
apply Injective.injective_under

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.Injective.Resolution

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : EnoughInjectives C
Z : C
⊢ Injective.ι Z ≫ (ofCocomplex Z).d 0 1 = 0

Tactic:
by rw [ofCocomplex_d_0_1, cokernel.condition_assoc, zero_comp]

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.SerreClass.Localization

Goal state before:
pp: case refine_1
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
P : ObjectProperty C
inst✝ : P.IsSerreClass
X Y : C
f : X ⟶ Y
Y' : C
s : Y ⟶ Y'
hs : P.isoModSerre s
eq : f ≫ s = 0
⊢ P (Abelian.image f)

Tactic:
← exists_comp_monoModSerre_eq_zero_iff P

Goal state after:
case refine_1
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
P : ObjectProperty C
inst✝ : P.IsSerreClass
X Y : C
f : X ⟶ Y
Y' : C
s : Y ⟶ Y'
hs : P.isoModSerre s
eq : f ≫ s = 0
⊢ ∃ Y' s, ∃ (_ : P.monoModSerre s), f ≫ s = 0