mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.FunctionalSpaces.SobolevInequality

Goal state before:
pp: E : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F' : Type u_5
inst✝¹ : NormedAddCommGroup F'
inst✝ : InnerProductSpace ℝ F'
u : E → F'
hu : ContDiff ℝ 1 u
h2u : HasCompactSupport u
p p' : ℝ≥0
hp✝ : 1 ≤ p
hp'0 : ¬p' = 0
n : ℕ := finrank ℝ E
hn✝ : 0 < n
hp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹
n' : ℝ≥0 := (↑n).conjExponent
h2p : ↑p < ↑n
h0n : 2 ≤ n
hn : (↑n).HolderConjugate n'
h1n : 1 ≤ ↑n
h2n : 0 < ↑n - 1
hnp : 0 < ↑n - ↑p
hp : 1 < p
q : ℝ := (↑p).conjExponent
hq : (↑p).HolderConjugate q
h0p : p ≠ 0
h1p : ↑p ≠ 1
h3p : ↑p - 1 ≠ 0
h2q : 1 / ↑n' - 1 / q = 1 / ↑p'
γ : ℝ≥0 := NNReal.mk (↑p * (↑n - 1) / (↑n - ↑p)) ⋯
h0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)
h1γ : 1 < ↑γ
h2γ : γ * n' = p'
h3γ : (↑γ - 1) * q = ↑p'
h4γ : ↑γ ≠ 0
h3u : ¬∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ = 0
h4u : ∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ ≠ ∞
h5u : (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / q) ≠ 0
h6u : (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / q) ≠ ∞
h7u : Continuous u
v : E → ℝ := fun x ↦ ‖u x‖ ^ ↑γ
hv : ContDiff ℝ 1 v
h2v : HasCompactSupport v
C : ℝ≥0 := eLpNormLESNormFDerivOneConst μ ↑n'
this :
  (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / ↑n') ≤
    ↑C * ↑γ * (∫⁻ (x : E), ‖fderiv ℝ u x‖ₑ ^ ↑p ∂μ) ^ (1 / ↑p) * (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / q)
⊢ ↑C * ↑(NNReal.mk (↑p * (↑n - 1) / (↑n - ↑p)) ⋯) = ↑C * ↑(↑p * (↑n - 1) / (↑n - ↑p)).toNNReal

Tactic:
NNReal.coe_mk,

Goal state after:
E : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F' : Type u_5
inst✝¹ : NormedAddCommGroup F'
inst✝ : InnerProductSpace ℝ F'
u : E → F'
hu : ContDiff ℝ 1 u
h2u : HasCompactSupport u
p p' : ℝ≥0
hp✝ : 1 ≤ p
hp'0 : ¬p' = 0
n : ℕ := finrank ℝ E
hn✝ : 0 < n
hp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹
n' : ℝ≥0 := (↑n).conjExponent
h2p : ↑p < ↑n
h0n : 2 ≤ n
hn : (↑n).HolderConjugate n'
h1n : 1 ≤ ↑n
h2n : 0 < ↑n - 1
hnp : 0 < ↑n - ↑p
hp : 1 < p
q : ℝ := (↑p).conjExponent
hq : (↑p).HolderConjugate q
h0p : p ≠ 0
h1p : ↑p ≠ 1
h3p : ↑p - 1 ≠ 0
h2q : 1 / ↑n' - 1 / q = 1 / ↑p'
γ : ℝ≥0 := NNReal.mk (↑p * (↑n - 1) / (↑n - ↑p)) ⋯
h0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)
h1γ : 1 < ↑γ
h2γ : γ * n' = p'
h3γ : (↑γ - 1) * q = ↑p'
h4γ : ↑γ ≠ 0
h3u : ¬∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ = 0
h4u : ∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ ≠ ∞
h5u : (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / q) ≠ 0
h6u : (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / q) ≠ ∞
h7u : Continuous u
v : E → ℝ := fun x ↦ ‖u x‖ ^ ↑γ
hv : ContDiff ℝ 1 v
h2v : HasCompactSupport v
C : ℝ≥0 := eLpNormLESNormFDerivOneConst μ ↑n'
this :
  (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / ↑n') ≤
    ↑C * ↑γ * (∫⁻ (x : E), ‖fderiv ℝ u x‖ₑ ^ ↑p ∂μ) ^ (1 / ↑p) * (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / q)
⊢ ↑C * (↑p * (↑n - 1) / (↑n - ↑p)) = ↑C * ↑(↑p * (↑n - 1) / (↑n - ↑p)).toNNReal
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))
⊢ RCLike.re (∑ x, ∑ x_1, ‖f‖ ^ 2 • (inner 𝕜 (e x_1) (e x) * inner 𝕜 (g x_1) (g x))) -
      RCLike.re (∑ x, ∑ x_1, (∑ i, Matrix.vecMulVec (A i) (star (A i))) x_1 x * inner 𝕜 (g x_1) (g x)) ≤
    RCLike.re (∑ x, ∑ x_1, ‖f‖ ^ 2 • (inner 𝕜 (e x_1) (e x) * inner 𝕜 (g x_1) (g x)))

Tactic:
sub_le_self_iff,

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))
⊢ 0 ≤ RCLike.re (∑ x, ∑ x_1, (∑ i, Matrix.vecMulVec (A i) (star (A i))) x_1 x * inner 𝕜 (g x_1) (g x))
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))
⊢ ‖(LinearMap.rTensor G ↑f) x‖ ^ 2 ≤ (‖f‖ * ‖x‖) ^ 2

Tactic:
simp_rw [mul_pow, hx, map_sum, LinearMap.rTensor_tmul, coe_coe,
      ← inner_self_eq_norm_sq (𝕜 := 𝕜), inner_sum, sum_inner, inner_tmul, ← hA, sub_mul,
      Finset.sum_sub_distrib, map_sub, ← RCLike.smul_re, Finset.smul_sum, smul_mul_assoc,
      sub_le_self_iff, Matrix.sum_apply, mul_comm, Finset.mul_sum]

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))
⊢ 0 ≤ RCLike.re (∑ x, ∑ x_1, ∑ i, inner 𝕜 (g x_1) (g x) * Matrix.vecMulVec (A i) (star (A i)) x_1 x)
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
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : InnerProductSpace 𝕜 G
f : E →L[𝕜] F
x✝ : E ⊗[𝕜] G
⊢ (↑↑(commIsometry 𝕜 G F) ∘SL lTensor G f ∘SL ↑↑(commIsometry 𝕜 E G)) x✝ = (rTensor G f) x✝

Tactic:
simp [lTensor]

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
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : InnerProductSpace 𝕜 G
f : E →L[𝕜] F
x✝ : E ⊗[𝕜] G
⊢ (lTensor G f ∘SL ↑↑(commIsometry 𝕜 E G)) x✝ = (↑↑(commIsometry 𝕜 F G) ∘SL rTensor G f) x✝

Tactic:
simp [lTensor]

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
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : InnerProductSpace 𝕜 G
g : E →L[𝕜] F
x✝ : G ⊗[𝕜] E
⊢ (rTensor G g ∘SL ↑↑(commIsometry 𝕜 G E)) x✝ = (↑↑(commIsometry 𝕜 G F) ∘SL lTensor G g) x✝

Tactic:
simp [lTensor]

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
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : InnerProductSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : InnerProductSpace 𝕜 F
inst✝⁴ : NormedAddCommGroup G
inst✝³ : InnerProductSpace 𝕜 G
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : FiniteDimensional 𝕜 F
inst✝ : FiniteDimensional 𝕜 G
f : E →ₗ[𝕜] F
⊢ adjoint (lTensor G f) = lTensor G (adjoint f)

Tactic:
simp [lTensor]

Goal state after:
Module: Mathlib.Geometry.Euclidean.Volume.Measure

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹⁰ : EMetricSpace X
inst✝⁹ : MeasurableSpace X
inst✝⁸ : BorelSpace X
inst✝⁷ : EMetricSpace Y
inst✝⁶ : MeasurableSpace Y
inst✝⁵ : BorelSpace Y
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
⊢ μHE[finrank ℝ E].IsAddHaarMeasure

Tactic:
rw [euclideanHausdorffMeasure_def, ENNReal.smul_def]

Goal state after:
X : Type u_1
Y : Type u_2
inst✝¹⁰ : EMetricSpace X
inst✝⁹ : MeasurableSpace X
inst✝⁸ : BorelSpace X
inst✝⁷ : EMetricSpace Y
inst✝⁶ : MeasurableSpace Y
inst✝⁵ : BorelSpace Y
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
⊢ (↑(volume.addHaarScalarFactor μH[↑(finrank ℝ E)]) • μH[↑(finrank ℝ E)]).IsAddHaarMeasure
Module: Mathlib.Analysis.InnerProductSpace.TensorProduct

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 F
ι₁ : Type u_6
ι₂ : Type u_7
inst✝¹ : Fintype ι₁
inst✝ : Fintype ι₂
b₁ : OrthonormalBasis ι₁ 𝕜 E
b₂ : OrthonormalBasis ι₂ 𝕜 F
x : E
y : F
i : ι₁
j : ι₂
⊢ ((b₁.tensorProduct b₂).repr (x ⊗ₜ[𝕜] y)).ofLp (i, j) = (b₂.repr y).ofLp j * (b₁.repr x).ofLp i

Tactic:
by
  simp [OrthonormalBasis.tensorProduct]

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

Goal state before:
pp: 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
⊢ ↑C * eLpNorm (fderiv ℝ u) (↑p) μ * ↑t = ↑(eLpNormLESNormFDerivOfLeConst F μ s p q) * eLpNorm (fderiv ℝ u) (↑p) μ

Tactic:
eLpNormLESNormFDerivOfLeConst,

Goal state after:
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
⊢ ↑C * eLpNorm (fderiv ℝ u) (↑p) μ * ↑t =
    ↑((μ s).toNNReal ^ (1 / ↑q - 1 / ↑(p⁻¹ - (↑(finrank ℝ E))⁻¹)⁻¹) * SNormLESNormFDerivOfEqConst F μ ↑p) *
      eLpNorm (fderiv ℝ u) (↑p) μ
Module: Mathlib.Analysis.InnerProductSpace.TensorProduct

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 F
ι₁ : Type u_6
ι₂ : Type u_7
inst✝¹ : Fintype ι₁
inst✝ : Fintype ι₂
b₁ : OrthonormalBasis ι₁ 𝕜 E
b₂ : OrthonormalBasis ι₂ 𝕜 F
⊢ (b₁.tensorProduct b₂).toBasis = b₁.toBasis.tensorProduct b₂.toBasis

Tactic:
by
  simp [OrthonormalBasis.tensorProduct]

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

Goal state before:
pp: case refine_1
X : Type u_2
inst✝¹ : EMetricSpace X
inst✝ : SecondCountableTopology X
s : Set X
x : X
x✝ : x ∈ s
⊢ limsup dimH (𝓝[s] x).smallSets ≤ dimH s

pp: case refine_2
X : Type u_2
inst✝¹ : EMetricSpace X
inst✝ : SecondCountableTopology X
s : Set X
⊢ dimH s ≤ ⨆ x ∈ s, limsup dimH (𝓝[s] x).smallSets

Tactic:
· refine limsup_le_of_le isCobounded_le_of_bot ?_
    exact eventually_smallSets.2 ⟨s, self_mem_nhdsWithin, fun t => dimH_mono⟩

Goal state after:
case refine_2
X : Type u_2
inst✝¹ : EMetricSpace X
inst✝ : SecondCountableTopology X
s : Set X
⊢ dimH s ≤ ⨆ x ∈ s, limsup dimH (𝓝[s] x).smallSets
Module: Mathlib.Topology.MetricSpace.HausdorffDimension

Goal state before:
pp: case refine_1
X : Type u_2
inst✝¹ : EMetricSpace X
inst✝ : SecondCountableTopology X
s : Set X
x : X
⊢ limsup dimH (𝓝[s] x).smallSets ≤ dimH s

pp: case refine_2
X : Type u_2
inst✝¹ : EMetricSpace X
inst✝ : SecondCountableTopology X
s : Set X
⊢ dimH s ≤ ⨆ x, limsup dimH (𝓝[s] x).smallSets

Tactic:
· refine limsup_le_of_le isCobounded_le_of_bot ?_
    exact eventually_smallSets.2 ⟨s, self_mem_nhdsWithin, fun t => dimH_mono⟩

Goal state after:
case refine_2
X : Type u_2
inst✝¹ : EMetricSpace X
inst✝ : SecondCountableTopology X
s : Set X
⊢ dimH s ≤ ⨆ x, limsup dimH (𝓝[s] x).smallSets
Module: Mathlib.Analysis.InnerProductSpace.NormDet

Goal state before:
pp: 𝕜 : Type u_1
U : Type u_2
V : Type u_3
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup U
inst✝³ : InnerProductSpace 𝕜 U
inst✝² : FiniteDimensional 𝕜 U
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace 𝕜 V
f : U →ₗ[𝕜] V
tfae_1_iff_2 : f.normDet ≠ 0 ↔ f.ker = ⊥
tfae_1_iff_3 : f.normDet ≠ 0 ↔ finrank 𝕜 ↥f.range = finrank 𝕜 U
tfae_3_to_4 : finrank 𝕜 ↥f.range = finrank 𝕜 U → Nonempty (OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 ↥f.range)
⊢ Nonempty (OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 ↥f.range) → finrank 𝕜 ↥f.range = finrank 𝕜 U

Tactic:
rintro ⟨b⟩

Goal state after:
𝕜 : Type u_1
U : Type u_2
V : Type u_3
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup U
inst✝³ : InnerProductSpace 𝕜 U
inst✝² : FiniteDimensional 𝕜 U
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace 𝕜 V
f : U →ₗ[𝕜] V
tfae_1_iff_2 : f.normDet ≠ 0 ↔ f.ker = ⊥
tfae_1_iff_3 : f.normDet ≠ 0 ↔ finrank 𝕜 ↥f.range = finrank 𝕜 U
tfae_3_to_4 : finrank 𝕜 ↥f.range = finrank 𝕜 U → Nonempty (OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 ↥f.range)
b : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 ↥f.range
⊢ finrank 𝕜 ↥f.range = finrank 𝕜 U
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: X : Type u_2
inst✝ : EMetricSpace X
m : Set X → ℝ≥0∞
⊢ (mkMetric' m).IsMetric

Tactic:
rintro s t ⟨r, r0, hr⟩

Goal state after:
X : Type u_2
inst✝ : EMetricSpace X
m : Set X → ℝ≥0∞
s t : Set X
r : ℝ≥0∞
r0 : r ≠ 0
hr : ∀ x ∈ s, ∀ y ∈ t, r ≤ edist x y
⊢ (mkMetric' m) (s ∪ t) = (mkMetric' m) s + (mkMetric' m) t
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
m₁ m₂ : ℝ≥0∞ → ℝ≥0∞
hle : m₁ ≤ᶠ[𝓝 0] m₂
⊢ mkMetric m₁ ≤ mkMetric m₂

Tactic:
by
  convert! @mkMetric_mono_smul X _ _ _ _ m₂ _ ENNReal.one_ne_top one_ne_zero _ <;> simp [*]

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✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup U
inst✝⁴ : InnerProductSpace 𝕜 U
inst✝³ : FiniteDimensional 𝕜 U
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace 𝕜 V
inst✝ : CompleteSpace V
f : U →L[𝕜] V
this✝ : CompleteSpace U
this : CompleteSpace ↥(↑f).range
bu : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 U := stdOrthonormalBasis 𝕜 U
h : (↑f).ker = ⊥
⊢ ↑((↑f).normDet ^ 2) = (ContinuousLinearMap.adjoint f ∘SL f).det

Tactic:
obtain ⟨b⟩ := (f.normDet_ne_zero_tfae.out 1 3).mp h
    have hf : f = f.range.subtypeₗᵢ.toContinuousLinearMap ∘L f.rangeRestrict := rfl
    conv_rhs => rw [hf]
    rw [ContinuousLinearMap.adjoint_comp, ← ContinuousLinearMap.comp_assoc,
      ContinuousLinearMap.comp_assoc (ContinuousLinearMap.adjoint _),
      f.range.subtypeₗᵢ.adjoint_comp_self, ContinuousLinearMap.one_def, ContinuousLinearMap.comp_id,
      ContinuousLinearMap.det, ContinuousLinearMap.toLinearMap_comp, ← det_toMatrix bu.toBasis,
      toMatrix_comp bu.toBasis b.toBasis bu.toBasis, ← ContinuousLinearMap.adjoint_toLinearMap,
      toMatrix_adjoint, f.toLinearMap.normDet_eq_norm_det_toMatrix_rangeRestrict bu b]
    simp [RCLike.conj_mul]

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

Goal state before:
pp: H : Type u_1
inst✝² : NormedAddCommGroup H
inst✝¹ : InnerProductSpace ℂ H
inst✝ : CompleteSpace H
S : StandardSubspace H
⊢ S.toClosedSubmodule.symplComp ⊔ S.toClosedSubmodule.symplComp.mulI = ⊤

Tactic:
by
    simp [mulI_symplComp, ClosedSubmodule.sup_orthogonal, inf_comm, S.IsSeparating]

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

Goal state before:
pp: X : Type u_2
Y : Type u_3
inst✝⁵ : EMetricSpace X
inst✝⁴ : EMetricSpace Y
inst✝³ : MeasurableSpace X
inst✝² : BorelSpace X
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
f : X → Y
d : ℝ
hf : Isometry f
hd : 0 ≤ d ∨ Surjective f
s : Set X
⊢ 0 ≤ d → Monotone fun r ↦ r ^ d

Tactic:
exact ENNReal.monotone_rpow_of_nonneg

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.StrongTopology

Goal state before:
pp: R : Type u_1
𝕜₁ : Type u_2
𝕜₂ : Type u_3
E : Type u_4
F : Type u_5
inst✝¹⁴ : AddCommGroup E
inst✝¹³ : TopologicalSpace E
inst✝¹² : AddCommGroup F
inst✝¹¹ : TopologicalSpace F
inst✝¹⁰ : IsTopologicalAddGroup F
inst✝⁹ : Semiring R
inst✝⁸ : PartialOrder R
inst✝⁷ : NormedField 𝕜₁
inst✝⁶ : NormedField 𝕜₂
inst✝⁵ : Module 𝕜₁ E
inst✝⁴ : Module 𝕜₂ F
σ : 𝕜₁ →+* 𝕜₂
inst✝³ : Module R F
inst✝² : ContinuousConstSMul R F
inst✝¹ : LocallyConvexSpace R F
inst✝ : SMulCommClass 𝕜₂ R F
𝔖 : Set (Set E)
h𝔖₁ : 𝔖.Nonempty
h𝔖₂ : DirectedOn (fun x1 x2 ↦ x1 ⊆ x2) 𝔖
⊢ LocallyConvexSpace R (E →SLᵤ[σ, 𝔖] F)

Tactic:
by
  apply LocallyConvexSpace.ofBasisZero _ _ _ _
    (UniformConvergenceCLM.hasBasis_nhds_zero_of_basis _ _ _ h𝔖₁ h𝔖₂
      (LocallyConvexSpace.convex_basis_zero R F)) _
  rintro ⟨S, V⟩ ⟨_, _, hVconvex⟩ f hf g hg a b ha hb hab x hx
  exact hVconvex (hf x hx) (hg x hx) ha hb hab

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
hx : x ≠ 0
hy : y ≠ 0
⊢ (o.kahler x) y ≠ 0

Tactic:
apply mt o.eq_zero_or_eq_zero_of_kahler_eq_zero

Goal state after:
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
hx : x ≠ 0
hy : y ≠ 0
⊢ ¬(x = 0 ∨ y = 0)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Deriv

Goal state before:
pp: ⊢ Tendsto (fun z ↦ z * Gamma z) (𝓝[≠] 0) (𝓝 1)

Tactic:
by
  rw [show 𝓝 (1 : ℂ) = 𝓝 (Gamma (0 + 1)) by simp only [zero_add, Complex.Gamma_one]]
  refine tendsto_nhdsWithin_congr Gamma_add_one (continuousAt_iff_punctured_nhds.mp ?_)
  exact ContinuousAt.comp' (by simp [continuousAt_Gamma_one]) (continuous_add_const 1).continuousAt

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: n : ℕ
⊢ (fun z ↦ log (1 + z) - logTaylor (n + 1) z) =O[𝓝 0] fun z ↦ z ^ (n + 1)

Tactic:
rw [Asymptotics.isBigO_iff]
  use 2 / (n + 1)
  filter_upwards [
    eventually_norm_sub_lt 0 one_pos,
    eventually_norm_sub_lt 0 (show 0 < 1 / 2 by simp)] with z hz1 hz12
  rw [sub_zero] at hz1 hz12
  have : (1 - ‖z‖)⁻¹ ≤ 2 := by rw [inv_le_comm₀ (sub_pos_of_lt hz1) two_pos]; linarith
  apply (norm_log_sub_logTaylor_le n hz1).trans
  rw [mul_div_assoc, mul_comm, norm_pow]
  gcongr

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

Goal state before:
pp: case e'_2
z : ℂ
n : ℕ
hz : z ≠ 0
⊢ z ^ 2 / (↑n + 1) ^ 2 = 4 * z ^ 2 / ↑(2 * n + 2) ^ 2

Tactic:
simp only [Nat.cast_add, Nat.cast_mul, Nat.cast_two]
    nth_rw 2 [← mul_one (2 : ℂ)]
    rw [← mul_add, mul_pow, ← div_div]
    ring

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

Goal state before:
pp: s t : ℂ
hs : 0 < s.re
ht : 0 < t.re
conv_int :
  ∫ (x : ℝ) in Ioi 0,
      ∫ (t_1 : ℝ) in 0..x, ↑(rexp (-t_1)) * ↑t_1 ^ (s - 1) * (↑(rexp (-(x - t_1))) * ↑(x - t_1) ^ (t - 1)) ∂volume =
    (∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (s - 1) ∂volume) * ∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (t - 1) ∂volume
hst : 0 < (s + t).re
⊢ Gamma s * Gamma t = Gamma (s + t) * s.betaIntegral t

Tactic:
rw [Gamma_eq_integral hs, Gamma_eq_integral ht, Gamma_eq_integral hst, GammaIntegral,
    GammaIntegral, GammaIntegral, ← conv_int, ← MeasureTheory.integral_mul_const (betaIntegral _ _)]

Goal state after:
s t : ℂ
hs : 0 < s.re
ht : 0 < t.re
conv_int :
  ∫ (x : ℝ) in Ioi 0,
      ∫ (t_1 : ℝ) in 0..x, ↑(rexp (-t_1)) * ↑t_1 ^ (s - 1) * (↑(rexp (-(x - t_1))) * ↑(x - t_1) ^ (t - 1)) ∂volume =
    (∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (s - 1) ∂volume) * ∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (t - 1) ∂volume
hst : 0 < (s + t).re
⊢ ∫ (x : ℝ) in Ioi 0,
      ∫ (t_1 : ℝ) in 0..x, ↑(rexp (-t_1)) * ↑t_1 ^ (s - 1) * (↑(rexp (-(x - t_1))) * ↑(x - t_1) ^ (t - 1)) ∂volume =
    ∫ (a : ℝ) in Ioi 0, ↑(rexp (-a)) * ↑a ^ (s + t - 1) * s.betaIntegral t
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
F : ℝ → ℂ := fun x ↦ ↑x ^ u * (1 - ↑x) ^ v
⊢ u * u.betaIntegral (v + 1) = v * (u + 1).betaIntegral v

Tactic:
have hu' : 0 < re (u + 1) := by rw [add_re, one_re]; positivity

Goal state after:
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
F : ℝ → ℂ := fun x ↦ ↑x ^ u * (1 - ↑x) ^ v
hu' : 0 < (u + 1).re
⊢ u * u.betaIntegral (v + 1) = v * (u + 1).betaIntegral v
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: case refine_2
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
F : ℝ → ℂ := fun x ↦ ↑x ^ u * (1 - ↑x) ^ v
hu' : 0 < (u + 1).re
hv' : 0 < (v + 1).re
x : ℝ
hx : x ∈ Icc 0 1
⊢ ContinuousAt (fun x ↦ (1 - ↑x) ^ v) x

Tactic:
· refine (continuousAt_cpow_const_of_re_pos (Or.inl ?_) hv).comp (by fun_prop)
      rw [sub_re, one_re, ofReal_re, sub_nonneg]
      exact hx.2

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

Goal state before:
pp: case e'_11
⊢ Ioi (-1) = ((fun x ↦ x + 1 / 2) ∘ fun x ↦ 1 / 2 * x) ⁻¹' Ioi 0

Tactic:
preimage_comp,

Goal state after:
case e'_11
⊢ Ioi (-1) = (fun x ↦ 1 / 2 * x) ⁻¹' (fun x ↦ x + 1 / 2) ⁻¹' Ioi 0
Module: Mathlib.Topology.Algebra.AsymptoticCone

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝¹² : Field k
inst✝¹¹ : LinearOrder k
inst✝¹⁰ : AddCommGroup V
inst✝⁹ : Module k V
inst✝⁸ : AddTorsor V P
inst✝⁷ : TopologicalSpace V
inst✝⁶ : TopologicalSpace k
inst✝⁵ : OrderTopology k
inst✝⁴ : IsStrictOrderedRing k
inst✝³ : IsTopologicalAddGroup V
inst✝² : ContinuousSMul k V
inst✝¹ : TopologicalSpace P
inst✝ : IsTopologicalAddTorsor P
v : V
⊢ (asymptoticNhds k P v).bind 𝓝 = asymptoticNhds k P v

Tactic:
refine le_antisymm (fun s h => ?_) (bind_mono le_rfl (.of_forall pure_le_nhds))

Goal state after:
k : Type u_1
V : Type u_2
P : Type u_3
inst✝¹² : Field k
inst✝¹¹ : LinearOrder k
inst✝¹⁰ : AddCommGroup V
inst✝⁹ : Module k V
inst✝⁸ : AddTorsor V P
inst✝⁷ : TopologicalSpace V
inst✝⁶ : TopologicalSpace k
inst✝⁵ : OrderTopology k
inst✝⁴ : IsStrictOrderedRing k
inst✝³ : IsTopologicalAddGroup V
inst✝² : ContinuousSMul k V
inst✝¹ : TopologicalSpace P
inst✝ : IsTopologicalAddTorsor P
v : V
s : Set P
h : s ∈ asymptoticNhds k P v
⊢ s ∈ (asymptoticNhds k P v).bind 𝓝