mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.InnerProductSpace.SingularValues

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : FiniteDimensional 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
inst✝ : FiniteDimensional 𝕜 F
T : E →ₗ[𝕜] F
hS : ∀ m ∈ T.singularValues.support, m < finrank 𝕜 E
hT : (adjoint T ∘ₗ T).IsSymmetric
this : T.singularValues.support.attachFin hS = {i | ↑(hT.eigenvalues ⋯ i) = 0}ᶜ
⊢ finrank 𝕜 E - finrank 𝕜 ↥(adjoint T ∘ₗ T).ker = finrank 𝕜 ↥T.range

Tactic:
← (T.adjoint ∘ₗ T).finrank_range_add_finrank_ker,

Goal state after:
𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : FiniteDimensional 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
inst✝ : FiniteDimensional 𝕜 F
T : E →ₗ[𝕜] F
hS : ∀ m ∈ T.singularValues.support, m < finrank 𝕜 E
hT : (adjoint T ∘ₗ T).IsSymmetric
this : T.singularValues.support.attachFin hS = {i | ↑(hT.eigenvalues ⋯ i) = 0}ᶜ
⊢ finrank 𝕜 ↥(adjoint T ∘ₗ T).range + finrank 𝕜 ↥(adjoint T ∘ₗ T).ker - finrank 𝕜 ↥(adjoint T ∘ₗ T).ker =
    finrank 𝕜 ↥T.range
Module: Mathlib.Analysis.InnerProductSpace.SingularValues

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : FiniteDimensional 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
inst✝ : FiniteDimensional 𝕜 F
T : E →ₗ[𝕜] F
⊢ T.singularValues (finrank 𝕜 ↥T.range) = 0

Tactic:
← Finsupp.notMem_support_iff,

Goal state after:
𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : FiniteDimensional 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
inst✝ : FiniteDimensional 𝕜 F
T : E →ₗ[𝕜] F
⊢ finrank 𝕜 ↥T.range ∉ T.singularValues.support
Module: Mathlib.Analysis.InnerProductSpace.SingularValues

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : FiniteDimensional 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
inst✝ : FiniteDimensional 𝕜 F
T : E →ₗ[𝕜] F
n : ℕ
⊢ T.singularValues n = 0 ↔ finrank 𝕜 ↥T.range ≤ n

Tactic:
← Finsupp.notMem_support_iff,

Goal state after:
𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : FiniteDimensional 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
inst✝ : FiniteDimensional 𝕜 F
T : E →ₗ[𝕜] F
n : ℕ
⊢ n ∉ T.singularValues.support ↔ finrank 𝕜 ↥T.range ≤ n
Module: Mathlib.Analysis.InnerProductSpace.TensorProduct

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : 𝕜 ⊗[𝕜] E
x✝¹ : 𝕜
x✝ : E
⊢ ∀ (x : 𝕜) (y : E),
    inner 𝕜 ((TensorProduct.lid 𝕜 E) (x✝¹ ⊗ₜ[𝕜] x✝)) ((TensorProduct.lid 𝕜 E) (x ⊗ₜ[𝕜] y)) =
      inner 𝕜 (x✝¹ ⊗ₜ[𝕜] x✝) (x ⊗ₜ[𝕜] y)

Tactic:
simp [inner_smul_left, inner_smul_right, mul_assoc]

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.TensorProduct

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : InnerProductSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace 𝕜 H
f : E →L[𝕜] F
x : E ⊗[𝕜] G
n : ℕ
e : Fin n → E
g : Fin n → G
hx : x = ∑ j, e j ⊗ₜ[𝕜] g j
c : E ⊗[𝕜] G →₀ 𝕜
hc_supp : ↑c.support ⊆ {t | ∃ m n, m ⊗ₜ[𝕜] n = t}
hc : (c.sum fun mi r ↦ r • mi) = x
m : ℕ
A : Fin m → Fin n → 𝕜
hA :
  ∀ (i j : Fin n),
    ‖f‖ ^ 2 • inner 𝕜 (e i) (e j) - (∑ i, Matrix.vecMulVec (A i) (star (A i))) i j = inner 𝕜 (f (e i)) (f (e j))
⊢ ‖∑ x, f (e x) ⊗ₜ[𝕜] g x‖ ^ 2 ≤ ‖f‖ ^ 2 * ‖∑ j, e j ⊗ₜ[𝕜] g j‖ ^ 2

Tactic:
← inner_self_eq_norm_sq (𝕜 := 𝕜),

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : InnerProductSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace 𝕜 H
f : E →L[𝕜] F
x : E ⊗[𝕜] G
n : ℕ
e : Fin n → E
g : Fin n → G
hx : x = ∑ j, e j ⊗ₜ[𝕜] g j
c : E ⊗[𝕜] G →₀ 𝕜
hc_supp : ↑c.support ⊆ {t | ∃ m n, m ⊗ₜ[𝕜] n = t}
hc : (c.sum fun mi r ↦ r • mi) = x
m : ℕ
A : Fin m → Fin n → 𝕜
hA :
  ∀ (i j : Fin n),
    ‖f‖ ^ 2 • inner 𝕜 (e i) (e j) - (∑ i, Matrix.vecMulVec (A i) (star (A i))) i j = inner 𝕜 (f (e i)) (f (e j))
⊢ RCLike.re (inner 𝕜 (∑ x, f (e x) ⊗ₜ[𝕜] g x) (∑ x, f (e x) ⊗ₜ[𝕜] g x)) ≤
    ‖f‖ ^ 2 * RCLike.re (inner 𝕜 (∑ j, e j ⊗ₜ[𝕜] g j) (∑ j, e j ⊗ₜ[𝕜] g j))
Module: Mathlib.Analysis.InnerProductSpace.TensorProduct

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : InnerProductSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace 𝕜 H
f : E →L[𝕜] F
x : E ⊗[𝕜] G
n : ℕ
e : Fin n → E
g : Fin n → G
hx : x = ∑ j, e j ⊗ₜ[𝕜] g j
c : E ⊗[𝕜] G →₀ 𝕜
hc_supp : ↑c.support ⊆ {t | ∃ m n, m ⊗ₜ[𝕜] n = t}
hc : (c.sum fun mi r ↦ r • mi) = x
m : ℕ
A : Fin m → Fin n → 𝕜
hA :
  ∀ (i j : Fin n),
    ‖f‖ ^ 2 • inner 𝕜 (e i) (e j) - (∑ i, Matrix.vecMulVec (A i) (star (A i))) i j = inner 𝕜 (f (e i)) (f (e j))
⊢ 0 ≤
    RCLike.re
      (∑ x,
        ∑ x_1,
          ∑ x_2,
            inner 𝕜 ((starRingEnd 𝕜) (A x x_2) • g x_2) (g x_1) * star ((starRingEnd 𝕜) ((starRingEnd 𝕜) (A x x_1))))

Tactic:
simp [mul_comm, ← inner_smul_right, ← sum_inner, ← inner_sum, Finset.sum_nonneg]

Goal state after:
Module: Mathlib.Analysis.FunctionalSpaces.SobolevInequality

Goal state before:
pp: case e'_4.h
F : Type u_3
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
E : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
μ : Measure E
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : FiniteDimensional ℝ F
u : E → F
s : Set E
hu : ContDiff ℝ 1 u
h2u : support u ⊆ s
p q : ℝ≥0
hp : 1 ≤ p
h2p : p < ↑(finrank ℝ E)
hpq : ↑p⁻¹ - (↑(finrank ℝ E))⁻¹ ≤ (↑q)⁻¹
hs : Bornology.IsBounded s
hq0 : ¬q = 0
p' : ℝ≥0 := (p⁻¹ - (↑(finrank ℝ E))⁻¹)⁻¹
hp' : ↑p'⁻¹ = ↑p⁻¹ - (↑(finrank ℝ E))⁻¹
this : ↑q ≤ ↑p'
t : ℝ≥0 := (μ s).toNNReal ^ (1 / ↑q - 1 / ↑p')
C : ℝ≥0 := SNormLESNormFDerivOfEqConst F μ ↑p
⊢ 0 ≤ ↑q⁻¹ - ↑p'⁻¹

Tactic:
hp'

Goal state after:
case e'_4.h
F : Type u_3
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
E : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
μ : Measure E
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : FiniteDimensional ℝ F
u : E → F
s : Set E
hu : ContDiff ℝ 1 u
h2u : support u ⊆ s
p q : ℝ≥0
hp : 1 ≤ p
h2p : p < ↑(finrank ℝ E)
hpq : ↑p⁻¹ - (↑(finrank ℝ E))⁻¹ ≤ (↑q)⁻¹
hs : Bornology.IsBounded s
hq0 : ¬q = 0
p' : ℝ≥0 := (p⁻¹ - (↑(finrank ℝ E))⁻¹)⁻¹
hp' : ↑p'⁻¹ = ↑p⁻¹ - (↑(finrank ℝ E))⁻¹
this : ↑q ≤ ↑p'
t : ℝ≥0 := (μ s).toNNReal ^ (1 / ↑q - 1 / ↑p')
C : ℝ≥0 := SNormLESNormFDerivOfEqConst F μ ↑p
⊢ 0 ≤ ↑q⁻¹ - (↑p⁻¹ - (↑(finrank ℝ E))⁻¹)
Module: Mathlib.Topology.MetricSpace.HausdorffDimension

Goal state before:
pp: ι : Type u_1
X : Type u_2
Y : Type u_3
inst✝¹ : EMetricSpace X
inst✝ : EMetricSpace Y
s : Set X
this✝¹ : MeasurableSpace X := borel X
this✝ : BorelSpace X
⊢ ℝ≥0∞

Tactic:
exact ⨆ (d : ℝ≥0) (_ : @hausdorffMeasure X _ _ ⟨rfl⟩ d s = ∞), d

Goal state after:
Module: Mathlib.Topology.MetricSpace.HausdorffDimension

Goal state before:
pp: 𝕜 : Type u_4
E : Type u_5
F : Type u_6
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
e : E ≃L[𝕜] F
s : Set F
⊢ dimH (⇑e ⁻¹' s) = dimH s

Tactic:
by
  rw [← e.image_symm_eq_preimage, e.symm.dimH_image]

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.OfNorm

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
⊢ ‖x‖ ^ 2 = re (inner_ 𝕜 x x)

Tactic:
by
  simp only [inner_, normSq_apply, ofNat_re, ofNat_im, map_sub, map_add,
    ofReal_re, ofReal_im, mul_re, inv_re, mul_im, I_re, inv_im]
  have h₁ : ‖x - x‖ = 0 := by simp
  have h₂ : ‖x + x‖ = 2 • ‖x‖ := by convert norm_nsmul 𝕜 2 x; module
  rw [h₁, h₂]
  ring

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.NormDet

Goal state before:
pp: case pos
𝕜 : Type u_1
U : Type u_2
V : Type u_3
inst✝⁷ : RCLike 𝕜
inst✝⁶ : NormedAddCommGroup U
inst✝⁵ : InnerProductSpace 𝕜 U
inst✝⁴ : FiniteDimensional 𝕜 U
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace 𝕜 V
ι : Type u_5
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
f : U →ₗ[𝕜] V
b : OrthonormalBasis ι 𝕜 U
h : f.ker = ⊥
bv : OrthonormalBasis ι 𝕜 ↥f.range := orthonormalBasis_range h b
⊢ ↑(‖((toMatrix b.toBasis bv.toBasis) f.rangeRestrict).det‖ ^ 2) =
    ↑‖(Matrix.of fun i j ↦ (bv.repr (f.rangeRestrict (b j))).ofLp i).det‖ ^ 2

Tactic:
simp only [map_pow]

Goal state after:
case pos
𝕜 : Type u_1
U : Type u_2
V : Type u_3
inst✝⁷ : RCLike 𝕜
inst✝⁶ : NormedAddCommGroup U
inst✝⁵ : InnerProductSpace 𝕜 U
inst✝⁴ : FiniteDimensional 𝕜 U
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace 𝕜 V
ι : Type u_5
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
f : U →ₗ[𝕜] V
b : OrthonormalBasis ι 𝕜 U
h : f.ker = ⊥
bv : OrthonormalBasis ι 𝕜 ↥f.range := orthonormalBasis_range h b
⊢ (algebraMap ℝ 𝕜) ‖((toMatrix b.toBasis bv.toBasis) f.rangeRestrict).det‖ ^ 2 =
    ↑‖(Matrix.of fun i j ↦ (bv.repr (f.rangeRestrict (b j))).ofLp i).det‖ ^ 2
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
d₁ d₂ : ℝ
h : d₁ < d₂
s : Set X
H : μH[d₂] s ≠ 0 ∧ μH[d₁] s ≠ ∞
c : ℝ≥0
hc : c ≠ 0
⊢ 0 < ↑(c ^ (d₂ - d₁)⁻¹)

Tactic:
pos_iff_ne_zero,

Goal state after:
X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
d₁ d₂ : ℝ
h : d₁ < d₂
s : Set X
H : μH[d₂] s ≠ 0 ∧ μH[d₁] s ≠ ∞
c : ℝ≥0
hc : c ≠ 0
⊢ ↑(c ^ (d₂ - d₁)⁻¹) ≠ 0
Module: Mathlib.Analysis.InnerProductSpace.StandardSubspace

Goal state before:
pp: H : Type u_1
inst✝ : NormedAddCommGroup H
ipc : InnerProductSpace ℂ H
S : ClosedSubmodule ℝ H
x : H
⊢ UnitI⁻¹ • x ∈ Sᗮ ↔ ∀ y ∈ S, ⟪y, x⟫.im = 0

Tactic:
Units.inv_mk,

Goal state after:
H : Type u_1
inst✝ : NormedAddCommGroup H
ipc : InnerProductSpace ℂ H
S : ClosedSubmodule ℝ H
x : H
⊢ { val := UnitI.inv, inv := ↑UnitI, val_inv := ⋯, inv_val := ⋯ } • x ∈ Sᗮ ↔ ∀ y ∈ S, ⟪y, x⟫.im = 0
Module: Mathlib.Analysis.InnerProductSpace.Reproducing

Goal state before:
pp: case single_add
𝕜 : Type u_1
inst✝⁸ : RCLike 𝕜
X : Type u_2
V : Type u_3
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace 𝕜 V
H : Type u_4
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : InnerProductSpace 𝕜 H
inst✝³ : RKHS 𝕜 H X V
inst✝² : CompleteSpace H
inst✝¹ : CompleteSpace V
K : Matrix X X (V →L[𝕜] V)
inst✝ : Fact K.PosSemidef
f : OfKernel K
h : (ContinuousLinearMap.pi fun x ↦ adjoint (kerFun K x)) f = 0
i : X × V
a : 𝕜
f✝ : X × V →₀ 𝕜
a✝² : i ∉ f✝.support
a✝¹ : a ≠ 0
a✝ : ⟪↑f✝, f⟫_𝕜 = 0
⊢ ⟪UniformSpace.Completion.toComplL (Finsupp.single i a), f⟫_𝕜 = 0

Tactic:
have := (ext_iff_inner_left 𝕜).mp (congrFun h i.1) i.2

Goal state after:
case single_add
𝕜 : Type u_1
inst✝⁸ : RCLike 𝕜
X : Type u_2
V : Type u_3
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace 𝕜 V
H : Type u_4
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : InnerProductSpace 𝕜 H
inst✝³ : RKHS 𝕜 H X V
inst✝² : CompleteSpace H
inst✝¹ : CompleteSpace V
K : Matrix X X (V →L[𝕜] V)
inst✝ : Fact K.PosSemidef
f : OfKernel K
h : (ContinuousLinearMap.pi fun x ↦ adjoint (kerFun K x)) f = 0
i : X × V
a : 𝕜
f✝ : X × V →₀ 𝕜
a✝² : i ∉ f✝.support
a✝¹ : a ≠ 0
a✝ : ⟪↑f✝, f⟫_𝕜 = 0
this : ⟪i.2, (ContinuousLinearMap.pi fun x ↦ adjoint (kerFun K x)) f i.1⟫_𝕜 = ⟪i.2, 0 i.1⟫_𝕜
⊢ ⟪UniformSpace.Completion.toComplL (Finsupp.single i a), f⟫_𝕜 = 0
Module: Mathlib.Analysis.LocallyConvex.ContinuousOfBounded

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
E : Type u_3
F : Type u_4
inst✝¹¹ : AddCommGroup E
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : IsTopologicalAddGroup E
inst✝⁸ : AddCommGroup F
inst✝⁷ : TopologicalSpace F
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : Module 𝕜 E
inst✝⁴ : ContinuousSMul 𝕜 E
inst✝³ : NormedField 𝕜'
inst✝² : Module 𝕜' F
σ : 𝕜 →+* 𝕜'
inst✝¹ : RingHomIsometric σ
inst✝ : FirstCountableTopology E
f : E →ₛₗ[σ] F
hf : ∀ (s : Set E), IsVonNBounded 𝕜 s → IsVonNBounded 𝕜' (⇑f '' s)
c : 𝕜
hc0 : 0 < ‖c‖
hc1 : ‖c‖ < 1
c_ne : c ≠ 0
b : ℕ → Set E
bE1 : ∀ (i : ℕ), b i ∈ 𝓝 0 ∧ Balanced 𝕜 (b i)
bE : (𝓝 0).HasAntitoneBasis fun i ↦ b i
n : ℕ
a✝ : True
⊢ ‖c ^ n‖ ≤ 1

Tactic:
by grw [norm_pow, hc1, one_pow]

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.TwoDim

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x y : E
⊢ (o.areaForm x) y = -(o.areaForm y) x

Tactic:
simp only [areaForm_to_volumeForm]
  convert! o.volumeForm.map_swap ![y, x] (_ : (0 : Fin 2) ≠ 1)
  · ext i
    fin_cases i <;> rfl
  · simp

Goal state after:
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
ha : 0 < a
t : ℝ
ht : t ∈ Ioi 0
⊢ (fun t ↦ ↑t ^ (s - 1) • f (a * t)) t = (fun t ↦ ↑a ^ (1 - s) • (fun u ↦ ↑u ^ (s - 1) • f u) (a * t)) t

Tactic:
dsimp only

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
ha : 0 < a
t : ℝ
ht : t ∈ Ioi 0
⊢ ↑t ^ (s - 1) • f (a * t) = ↑a ^ (1 - s) • ↑(a * t) ^ (s - 1) • f (a * t)
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f g : ℝ → E
s : ℂ
hf : MellinConvergent f s
hg : MellinConvergent g s
⊢ MellinConvergent (fun t ↦ f t - g t) s

Tactic:
by simpa only [MellinConvergent, smul_sub] using! hf.sub hg

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Deriv

Goal state before:
pp: n : ℕ
⊢ ¬ContinuousAt Gamma (-↑n)

Tactic:
induction n

Goal state after:
case zero
⊢ ¬ContinuousAt Gamma (-↑0)

case succ
n✝ : ℕ
a✝ : ¬ContinuousAt Gamma (-↑n✝)
⊢ ¬ContinuousAt Gamma (-↑(n✝ + 1))
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: n : ℕ
z : ℂ
hz : ‖z‖ < 1
help : IntervalIntegrable (fun t ↦ t ^ n * (1 - ‖z‖)⁻¹) MeasureTheory.volume 0 1
f : ℂ → ℂ := fun z ↦ log (1 + z) - logTaylor (n + 1) z
f' : ℂ → ℂ := fun z ↦ (-z) ^ n * (1 + z)⁻¹
hderiv : ∀ t ∈ Set.Icc 0 1, HasDerivAt f (f' (0 + ↑t * z)) (0 + ↑t * z)
hcont : ContinuousOn (fun t ↦ f' (0 + ↑t * z)) (Set.Icc 0 1)
H : log (1 + z) - logTaylor (n + 1) z = z * ∫ (t : ℝ) in 0..1, (-(↑t * z)) ^ n * (1 + ↑t * z)⁻¹
⊢ ‖z‖ ^ (n + 1) * ‖∫ (x : ℝ) in 0..1, ↑x ^ n * (1 + ↑x * z)⁻¹‖ ≤ ‖z‖ ^ (n + 1) * ((1 - ‖z‖)⁻¹ / (↑n + 1))

Tactic:
gcongr _ * ?_

Goal state after:
case hbc
n : ℕ
z : ℂ
hz : ‖z‖ < 1
help : IntervalIntegrable (fun t ↦ t ^ n * (1 - ‖z‖)⁻¹) MeasureTheory.volume 0 1
f : ℂ → ℂ := fun z ↦ log (1 + z) - logTaylor (n + 1) z
f' : ℂ → ℂ := fun z ↦ (-z) ^ n * (1 + z)⁻¹
hderiv : ∀ t ∈ Set.Icc 0 1, HasDerivAt f (f' (0 + ↑t * z)) (0 + ↑t * z)
hcont : ContinuousOn (fun t ↦ f' (0 + ↑t * z)) (Set.Icc 0 1)
H : log (1 + z) - logTaylor (n + 1) z = z * ∫ (t : ℝ) in 0..1, (-(↑t * z)) ^ n * (1 + ↑t * z)⁻¹
⊢ ‖∫ (x : ℝ) in 0..1, ↑x ^ n * (1 + ↑x * z)⁻¹‖ ≤ (1 - ‖z‖)⁻¹ / (↑n + 1)
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: case e'_3
z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
der1 : ∀ x ∈ uIcc 0 (π / 2), HasDerivAt (fun y ↦ ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x
⊢ ↑n / (2 * z) * ∫ (x : ℝ) in 0..π / 2, Complex.sin (2 * z * ↑x) * ↑(sin x) * ↑(cos x) ^ (n - 1) =
    ↑(cos (π / 2)) ^ n * (Complex.sin (2 * z * ↑(π / 2)) / (2 * z)) - ↑(cos 0) ^ n * (0 / (2 * z)) -
      ∫ (x : ℝ) in 0..π / 2, -↑n * ↑(sin x) * ↑(cos x) ^ (n - 1) * (Complex.sin (2 * z * ↑x) / (2 * z))

Tactic:
zero_div,

Goal state after:
case e'_3
z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
der1 : ∀ x ∈ uIcc 0 (π / 2), HasDerivAt (fun y ↦ ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x
⊢ ↑n / (2 * z) * ∫ (x : ℝ) in 0..π / 2, Complex.sin (2 * z * ↑x) * ↑(sin x) * ↑(cos x) ^ (n - 1) =
    ↑(cos (π / 2)) ^ n * (Complex.sin (2 * z * ↑(π / 2)) / (2 * z)) - ↑(cos 0) ^ n * 0 -
      ∫ (x : ℝ) in 0..π / 2, -↑n * ↑(sin x) * ↑(cos x) ^ (n - 1) * (Complex.sin (2 * z * ↑x) / (2 * z))
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: n : ℕ
z : ℂ
hz1 : ‖z - 0‖ < 1
⊢ ‖z - 0‖ < 1 / 2 → ‖log (1 + z) - logTaylor (n + 1) z‖ ≤ 2 / (↑n + 1) * ‖z ^ (n + 1)‖

Tactic:
hz12

Goal state after:
n : ℕ
z : ℂ
hz1 : ‖z - 0‖ < 1
hz12 : ‖z - 0‖ < 1 / 2
⊢ ‖log (1 + z) - logTaylor (n + 1) z‖ ≤ 2 / (↑n + 1) * ‖z ^ (n + 1)‖
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: u : ℂ
hu : 0 < u.re
⊢ ∫ (x : ℝ) in 0..1, ↑x ^ (u - 1) = 1 / u

Tactic:
rw [integral_cpow (Or.inl _)]

Goal state after:
u : ℂ
hu : 0 < u.re
⊢ (↑1 ^ (u - 1 + 1) - ↑0 ^ (u - 1 + 1)) / (u - 1 + 1) = 1 / u

u : ℂ
hu : 0 < u.re
⊢ -1 < (u - 1).re
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
nne : ↑n ≠ 0
this :
  (∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n) -
      ↑n / (2 * z) * (↑n / (2 * z) * ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n) =
    ↑n / (2 * z) * -((↑n - 1) / (2 * z) * ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (n - 2))
⊢ (1 - 4 * z ^ 2 / ↑n ^ 2) * ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n =
    (↑n - 1) / ↑n * ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (n - 2)

Tactic:
convert! congr_arg (fun u : ℂ => -u * (2 * z) ^ 2 / n ^ 2) this using 1 <;> field

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

Goal state before:
pp: case e_a
n : ℕ
L : IntervalIntegrable (fun i ↦ sin i ^ n) volume 0 (π / 2)
R : IntervalIntegrable (fun i ↦ sin i ^ n) volume (π / 2) π
⊢ ∫ (x : ℝ) in 0..π / 2, cos x ^ n = ∫ (x : ℝ) in 0..π / 2, sin x ^ n

pp: case e_a
n : ℕ
L : IntervalIntegrable (fun i ↦ sin i ^ n) volume 0 (π / 2)
R : IntervalIntegrable (fun i ↦ sin i ^ n) volume (π / 2) π
⊢ ∫ (x : ℝ) in 0..π / 2, cos x ^ n = ∫ (x : ℝ) in π / 2..π, sin x ^ n

Tactic:
· nth_rw 1 [(by ring : 0 = π / 2 - π / 2)]
    nth_rw 3 [(by ring : π / 2 = π / 2 - 0)]
    rw [← integral_comp_sub_left]
    refine integral_congr fun x _ => ?_
    rw [cos_pi_div_two_sub]

Goal state after:
case e_a
n : ℕ
L : IntervalIntegrable (fun i ↦ sin i ^ n) volume 0 (π / 2)
R : IntervalIntegrable (fun i ↦ sin i ^ n) volume (π / 2) π
⊢ ∫ (x : ℝ) in 0..π / 2, cos x ^ n = ∫ (x : ℝ) in π / 2..π, sin x ^ n
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: case inr.zero
z : ℂ
hz : z ≠ 0
⊢ Complex.sin (↑π * z) = ↑π * z * (Complex.sin (2 * z * ↑(π / 2)) / (2 * z) - 0 / (2 * z)) / ↑(π / 2)

Tactic:
zero_div,

Goal state after:
case inr.zero
z : ℂ
hz : z ≠ 0
⊢ Complex.sin (↑π * z) = ↑π * z * (Complex.sin (2 * z * ↑(π / 2)) / (2 * z) - 0) / ↑(π / 2)
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: z : ℂ
hz : z ≠ 0
n : ℕ
A : ℂ := ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)
B : ℂ := ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)
C : ℝ := ∫ (x : ℝ) in 0..π / 2, cos x ^ (2 * n)
hn : Complex.sin (↑π * z) = ↑π * z * A * B / ↑C
aux' : 2 * n.succ = 2 * n + 2
⊢ 1 / 2 * (0 / (↑(2 * n) + 2) + (↑(2 * n) + 1) / (↑(2 * n) + 2) * ∫ (x : ℝ) in 0..π, sin x ^ (2 * n)) =
    (2 * ↑n + 1) / (2 * ↑n + 2) * (1 / 2 * ∫ (x : ℝ) in 0..π, sin x ^ (2 * n))

Tactic:
zero_div,

Goal state after:
z : ℂ
hz : z ≠ 0
n : ℕ
A : ℂ := ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)
B : ℂ := ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)
C : ℝ := ∫ (x : ℝ) in 0..π / 2, cos x ^ (2 * n)
hn : Complex.sin (↑π * z) = ↑π * z * A * B / ↑C
aux' : 2 * n.succ = 2 * n + 2
⊢ 1 / 2 * (0 + (↑(2 * n) + 1) / (↑(2 * n) + 2) * ∫ (x : ℝ) in 0..π, sin x ^ (2 * n)) =
    (2 * ↑n + 1) / (2 * ↑n + 2) * (1 / 2 * ∫ (x : ℝ) in 0..π, sin x ^ (2 * n))
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: s : ℂ
hs : 0 < s.re
n : ℕ
hn : n ≠ 0
this : ∀ (x : ℝ), x = x / ↑n * ↑n
| fun x ↦ ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)

Tactic:
x,

Goal state after:
s : ℂ
hs : 0 < s.re
n : ℕ
hn : n ≠ 0
this : ∀ (x : ℝ), x = x / ↑n * ↑n
x : ℝ
| ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: s : ℂ
hs : 0 < s.re
n : ℕ
hn : n ≠ 0
this✝ : ∀ (x : ℝ), x = x / ↑n * ↑n
this :
  ∫ (x : ℝ) in 0..↑n, ↑((1 - x / ↑n) ^ n) * ↑(x / ↑n * ↑n) ^ (s - 1) =
    ↑n • ∫ (x : ℝ) in 0 / ↑n..↑n / ↑n, ↑((1 - x) ^ n) * ↑(x * ↑n) ^ (s - 1)
⊢ ↑n ^ s * ∫ (x : ℝ) in 0..1, ↑x ^ (s - 1) * (1 - ↑x) ^ (↑n + 1 - 1) =
    ↑↑n * ∫ (x : ℝ) in 0 / ↑n..↑n / ↑n, ↑((1 - x) ^ n) * ↑(x * ↑n) ^ (s - 1)

Tactic:
zero_div,

Goal state after:
s : ℂ
hs : 0 < s.re
n : ℕ
hn : n ≠ 0
this✝ : ∀ (x : ℝ), x = x / ↑n * ↑n
this :
  ∫ (x : ℝ) in 0..↑n, ↑((1 - x / ↑n) ^ n) * ↑(x / ↑n * ↑n) ^ (s - 1) =
    ↑n • ∫ (x : ℝ) in 0 / ↑n..↑n / ↑n, ↑((1 - x) ^ n) * ↑(x * ↑n) ^ (s - 1)
⊢ ↑n ^ s * ∫ (x : ℝ) in 0..1, ↑x ^ (s - 1) * (1 - ↑x) ^ (↑n + 1 - 1) =
    ↑↑n * ∫ (x : ℝ) in 0..↑n / ↑n, ↑((1 - x) ^ n) * ↑(x * ↑n) ^ (s - 1)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n ↦ (Ioc 0 ↑n).indicator fun x ↦ ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
f_ible : ∀ (n : ℕ), Integrable (f n) (volume.restrict (Ioi 0))
f_tends : ∀ x ∈ Ioi 0, Tendsto (fun n ↦ f n x) atTop (𝓝 (↑(rexp (-x)) * ↑x ^ (s - 1)))
n : ℕ
⊢ ∀ᵐ (a : ℝ) ∂volume.restrict (Ioi 0), ‖f n a‖ ≤ rexp (-a) * a ^ (s.re - 1)

Tactic:
rw [ae_restrict_iff' measurableSet_Ioi]

Goal state after:
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n ↦ (Ioc 0 ↑n).indicator fun x ↦ ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
f_ible : ∀ (n : ℕ), Integrable (f n) (volume.restrict (Ioi 0))
f_tends : ∀ x ∈ Ioi 0, Tendsto (fun n ↦ f n x) atTop (𝓝 (↑(rexp (-x)) * ↑x ^ (s - 1)))
n : ℕ
⊢ ∀ᵐ (x : ℝ), x ∈ Ioi 0 → ‖f n x‖ ≤ rexp (-x) * x ^ (s.re - 1)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: case zero
s : ℂ
hs : ⌊1 - s.re⌋₊ = 0
⊢ Tendsto s.GammaSeq atTop (𝓝 (Gamma s))

Tactic:
rw [Nat.floor_eq_zero, sub_lt_self_iff] at hs

Goal state after:
case zero
s : ℂ
hs : 0 < s.re
⊢ Tendsto s.GammaSeq atTop (𝓝 (Gamma s))
Module: Mathlib.Topology.Algebra.AsymptoticCone

Goal state before:
pp: k : Type u_1
V : Type u_2
inst✝⁹ : Field k
inst✝⁸ : LinearOrder k
inst✝⁷ : IsStrictOrderedRing k
inst✝⁶ : TopologicalSpace k
inst✝⁵ : OrderTopology k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : TopologicalSpace V
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul k V
s : Set V
c : k
v p : V
hs : Convex k s
hc : 0 ≤ c
hp : s ∈ 𝓝 p
hv : ∃ᶠ (a : k × V) in atTop ×ˢ 𝓝 v, a.1 • a.2 +ᵥ c • v +ᵥ p ∈ s
this : Tendsto (fun u ↦ -(c • u) +ᵥ c • v +ᵥ p) (𝓝 v) (𝓝 p)
t : k
u : V
ht : 0 < t
hu : -(c • u) +ᵥ c • v +ᵥ p ∈ s
h : t • u +ᵥ c • v +ᵥ p ∈ s
⊢ c • v +ᵥ p ∈ s

Tactic:
apply hs.segment_subset hu h

Goal state after:
k : Type u_1
V : Type u_2
inst✝⁹ : Field k
inst✝⁸ : LinearOrder k
inst✝⁷ : IsStrictOrderedRing k
inst✝⁶ : TopologicalSpace k
inst✝⁵ : OrderTopology k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : TopologicalSpace V
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul k V
s : Set V
c : k
v p : V
hs : Convex k s
hc : 0 ≤ c
hp : s ∈ 𝓝 p
hv : ∃ᶠ (a : k × V) in atTop ×ˢ 𝓝 v, a.1 • a.2 +ᵥ c • v +ᵥ p ∈ s
this : Tendsto (fun u ↦ -(c • u) +ᵥ c • v +ᵥ p) (𝓝 v) (𝓝 p)
t : k
u : V
ht : 0 < t
hu : -(c • u) +ᵥ c • v +ᵥ p ∈ s
h : t • u +ᵥ c • v +ᵥ p ∈ s
⊢ c • v +ᵥ p ∈ [-(c • u) +ᵥ c • v +ᵥ p -[k] t • u +ᵥ c • v +ᵥ p]