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

Goal state before:
pp: F : Type u_3
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
E : Type u_4
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
u : E → F
hu : ContDiff ℝ 1 u
h2u : HasCompactSupport u
p : ℝ≥0
hp : (↑(finrank ℝ E)).HolderConjugate p
h0p : 0 < ↑p
⊢ eLpNorm u (↑p) μ ^ ↑p ≤
    ↑((lintegralPowLePowLIntegralFDerivConst μ ↑p ^ (↑p)⁻¹) ^ ↑p) * (∫⁻ (x : E), ‖fderiv ℝ u x‖ₑ ∂μ) ^ ↑p

Tactic:
← NNReal.rpow_mul,

Goal state after:
F : Type u_3
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
E : Type u_4
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
u : E → F
hu : ContDiff ℝ 1 u
h2u : HasCompactSupport u
p : ℝ≥0
hp : (↑(finrank ℝ E)).HolderConjugate p
h0p : 0 < ↑p
⊢ eLpNorm u (↑p) μ ^ ↑p ≤
    ↑(lintegralPowLePowLIntegralFDerivConst μ ↑p ^ ((↑p)⁻¹ * ↑p)) * (∫⁻ (x : E), ‖fderiv ℝ u x‖ₑ ∂μ) ^ ↑p
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)
⊢ ↑n - ↑p < ↑p * ↑n - ↑p

Tactic:
sub_lt_sub_iff_right,

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)
⊢ ↑n < ↑p * ↑n
Module: Mathlib.Analysis.FunctionalSpaces.SobolevInequality

Goal state before:
pp: case neg.inr
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
⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ

Tactic:
by_cases h3u : ∫⁻ x, ‖u x‖ₑ ^ (p' : ℝ) ∂μ = 0

Goal state after:
case pos
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
⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ

case neg
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
⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ
Module: Mathlib.Geometry.Euclidean.Volume.Measure

Goal state before:
pp: X : Type u_1
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
d : ℕ
⊢ μHE[d] = volume.addHaarScalarFactor μH[↑d] • μH[↑d]

Tactic:
rfl

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
bu : OrthonormalBasis ι 𝕜 U
bv : OrthonormalBasis ι 𝕜 V
this : FiniteDimensional 𝕜 V
hrank : finrank 𝕜 U = finrank 𝕜 ↥f.range
h : f.range = ⊤
⊢ f.normDet = ‖((toMatrix bu.toBasis bv.toBasis) f).det‖

Tactic:
let bv' : OrthonormalBasis ι 𝕜 f.range := bv.map (LinearIsometryEquiv.ofTop _ _ h).symm

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
bu : OrthonormalBasis ι 𝕜 U
bv : OrthonormalBasis ι 𝕜 V
this : FiniteDimensional 𝕜 V
hrank : finrank 𝕜 U = finrank 𝕜 ↥f.range
h : f.range = ⊤
bv' : OrthonormalBasis ι 𝕜 ↥f.range := bv.map (LinearIsometryEquiv.ofTop V f.range h).symm
⊢ f.normDet = ‖((toMatrix bu.toBasis bv.toBasis) f).det‖
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
bu : OrthonormalBasis ι 𝕜 U
bv : OrthonormalBasis ι 𝕜 V
this : FiniteDimensional 𝕜 V
hrank : finrank 𝕜 U = finrank 𝕜 ↥f.range
h : f.range = ⊤
bv' : OrthonormalBasis ι 𝕜 ↥f.range := bv.map (LinearIsometryEquiv.ofTop V f.range h).symm
⊢ ‖((toMatrix bu.toBasis bv'.toBasis) f.rangeRestrict).det‖ = ‖((toMatrix bu.toBasis bv.toBasis) f).det‖

Tactic:
rfl

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✝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
c : 𝕜
hc : ¬c = 0
h : f.ker = ⊥
bv : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 ↥f.range
bu : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 U := stdOrthonormalBasis 𝕜 U
bv' : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 ↥(c • f).range := bv.map (LinearIsometryEquiv.ofEq f.range (c • f).range ⋯)
this : finrank 𝕜 U = Fintype.card (Fin (finrank 𝕜 U))
⊢ ‖((toMatrix bu.toBasis bv'.toBasis) (c • f).rangeRestrict).det‖ =
    ‖((toMatrix bu.toBasis bv.toBasis) (c • f.rangeRestrict)).det‖

Tactic:
rfl

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

Goal state before:
pp: V : Type u_3
P : Type u_4
inst✝⁸ : NormedAddCommGroup V
inst✝⁷ : InnerProductSpace ℝ V
inst✝⁶ : MeasurableSpace V
inst✝⁵ : BorelSpace V
inst✝⁴ : FiniteDimensional ℝ V
inst✝³ : MetricSpace P
inst✝² : MeasurableSpace P
inst✝¹ : BorelSpace P
inst✝ : NormedAddTorsor V P
p : P
v : V
hv : v ≠ 0
t : Set P
ht : MeasurableSet t
hrank : finrank ℝ ↥(AffineSubspace.mk' p (ℝ ∙ v)).direction = 1
hrank' : finrank ℝ ↥(AffineSubspace.mk' p (ℝ ∙ v)).directionᗮ = finrank ℝ V - 1
f : ℝ ≃L[ℝ] ↥(AffineSubspace.mk' p (ℝ ∙ v)).direction :=
  (ContinuousLinearEquiv.toSpanNonzeroSingleton ℝ v hv).trans
    (ContinuousLinearEquiv.ofEq (ℝ ∙ v) (AffineSubspace.mk' p (ℝ ∙ v)).direction ⋯)
hf : MeasurableEmbedding ⇑f
p' : ↥(AffineSubspace.mk' p (ℝ ∙ v)) := ⟨p, ⋯⟩
g : ℝ → ↥(AffineSubspace.mk' p (ℝ ∙ v)) := ⇑(IsometryEquiv.vaddConst p') ∘ ⇑f
hadd : MeasurableEmbedding ⇑(IsometryEquiv.vaddConst p')
hg : MeasurableEmbedding g
hm : μHE[finrank ℝ ↥(AffineSubspace.mk' p (ℝ ∙ v)).direction] = ‖v‖ₑ • map g volume
x : ℝ
⊢ x • v +ᵥ p = ↑(g x)

Tactic:
rfl

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

Goal state before:
pp: 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 = ⊥
hrank : finrank 𝕜 ↥f.range < finrank 𝕜 U
⊢ (Matrix.gram 𝕜 fun x ↦ f.rangeRestrict (b x)).det = 0

Tactic:
contrapose! hrank with h0

Goal state after:
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 = ⊥
h0 : (Matrix.gram 𝕜 fun x ↦ f.rangeRestrict (b x)).det ≠ 0
⊢ finrank 𝕜 U ≤ finrank 𝕜 ↥f.range
Module: Mathlib.Analysis.InnerProductSpace.NormDet

Goal state before:
pp: case pos
𝕜 : Type u_1
U : Type u_2
V : Type u_3
W : Type u_4
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
inst✝¹ : NormedAddCommGroup W
inst✝ : InnerProductSpace 𝕜 W
f : U →ₗ[𝕜] V
g : V →ₗ[𝕜] W
hf : f.ker = ⊥
bu : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 U := stdOrthonormalBasis 𝕜 U
bv : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 ↥f.range
hgf : (g ∘ₗ f).ker = ⊥
bw : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 ↥(g ∘ₗ f).range
bw' : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 ↥(g.domRestrict f.range).range :=
  bw.map (LinearIsometryEquiv.ofEq (g ∘ₗ f).range (g.domRestrict f.range).range ⋯)
⊢ ‖((toMatrix bu.toBasis bw.toBasis) (g ∘ₗ f).rangeRestrict).det‖ =
    ‖((toMatrix bu.toBasis bw'.toBasis) ((g.domRestrict f.range).rangeRestrict ∘ₗ f.rangeRestrict)).det‖

Tactic:
rfl

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

Goal state before:
pp: case e'_3
X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
x : X
this :
  1 ≤
    ⨅ t, ⨅ (_ : {x} ⊆ ⋃ n, t n), ⨅ (_ : ∀ (n : ℕ), ediam (t n) ≤ 1), ∑' (n : ℕ), ⨆ (_ : (t n).Nonempty), ediam (t n) ^ 0
⊢ ⨅ t, ⨅ (_ : {x} ⊆ ⋃ n, t n), ⨅ (_ : ∀ (n : ℕ), ediam (t n) ≤ 1), ∑' (n : ℕ), ⨆ (_ : (t n).Nonempty), ediam (t n) ^ 0 =
    ⨅ t, ⨅ (_ : {x} ⊆ ⋃ n, t n), ⨅ (_ : ∀ (n : ℕ), ediam (t n) ≤ 1), ∑' (n : ℕ), ⨆ (_ : (t n).Nonempty), ediam (t n) ^ 0

Tactic:
rfl

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

Goal state before:
pp: case inr.inl
X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
s : Set X
hs : s.Subsingleton
x : X
hx : x ∈ s
hd : 0 ≤ 0
⊢ μH[0] {x} ≤ 1

pp: case inr.inr
X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
s : Set X
hs : s.Subsingleton
d : ℝ
hd : 0 ≤ d
x : X
hx : x ∈ s
dpos : 0 < d
⊢ μH[d] {x} ≤ 1

Tactic:
· simp only [le_refl, hausdorffMeasure_zero_singleton]

Goal state after:
case inr.inr
X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
s : Set X
hs : s.Subsingleton
d : ℝ
hd : 0 ≤ d
x : X
hx : x ∈ s
dpos : 0 < d
⊢ μH[d] {x} ≤ 1
Module: Mathlib.Analysis.InnerProductSpace.StandardSubspace

Goal state before:
pp: H : Type u_1
inst✝ : NormedAddCommGroup H
ipc : InnerProductSpace ℂ H
x y : H
⊢ ⟪y, x⟫.re = ((starRingEnd ℂ) ⟪y, x⟫).re

Tactic:
rfl

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

Goal state before:
pp: 𝕜 : Type u_1
H : Type u_2
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : InnerProductSpace 𝕜 H
inst✝³ : CompleteSpace H
inst✝² : Algebra ℝ (H →L[𝕜] H)
inst✝¹ : IsScalarTower ℝ 𝕜 (H →L[𝕜] H)
inst✝ : ContinuousFunctionalCalculus ℝ (H →L[𝕜] H) IsSelfAdjoint
f g : H →L[𝕜] H
⊢ f ≤ g ↔ ∃ p ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s), g = f + p

Tactic:
by
    constructor
    · intro h
      rw [le_def] at h
      obtain ⟨p, hp₁, -, hp₃⟩ := CFC.exists_sqrt_of_isSelfAdjoint_of_quasispectrumRestricts
        h.isSelfAdjoint h.spectrumRestricts
      refine ⟨p ^ 2, ?_, by symm; rwa [add_comm, ← eq_sub_iff_add_eq]⟩
      exact AddSubmonoid.subset_closure ⟨p, by simp only [hp₁.star_eq, sq]⟩
    · rintro ⟨p, hp, rfl⟩
      rw [le_def, add_sub_cancel_left]
      induction hp using AddSubmonoid.closure_induction with
      | mem _ hf =>
        obtain ⟨f, rfl⟩ := hf
        simpa using! ContinuousLinearMap.IsPositive.adjoint_conj isPositive_one f
      | zero => exact isPositive_zero
      | add f g _ _ hf hg => exact hf.add hg

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : RCLike 𝕜
inst✝⁶ : AddCommGroup E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : Module 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : CompleteSpace F
α : Type u_4
inst✝ : TopologicalSpace α
f : α → E →WOT[𝕜] F
s : Set α
⊢ (∀ x ∈ s, ContinuousWithinAt f s x) ↔ ∀ b ∈ s, ∀ (b_2 : E) (a : F), ContinuousWithinAt (fun x ↦ ⟪a, (f x) b_2⟫_𝕜) s b

Tactic:
continuousWithinAt_iff

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

Goal state before:
pp: case «0»
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
⊢ ![x, y] ((fun i ↦ i) ⟨0, ⋯⟩) = (![y, x] ∘ ⇑(Equiv.swap 0 1)) ((fun i ↦ i) ⟨0, ⋯⟩)

Tactic:
rfl

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

Goal state before:
pp: case «1»
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
⊢ ![x, y] ((fun i ↦ i) ⟨1, ⋯⟩) = (![y, x] ∘ ⇑(Equiv.swap 0 1)) ((fun i ↦ i) ⟨1, ⋯⟩)

Tactic:
rfl

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

Goal state before:
pp: case «0»
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
hF : Fact (finrank ℝ F = 2)
φ : E ≃ₗᵢ[ℝ] F
x y : F
⊢ (⇑φ.symm ∘ ![x, y]) ((fun i ↦ i) ⟨0, ⋯⟩) = ![φ.symm x, φ.symm y] ((fun i ↦ i) ⟨0, ⋯⟩)

Tactic:
rfl

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

Goal state before:
pp: case «1»
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
hF : Fact (finrank ℝ F = 2)
φ : E ≃ₗᵢ[ℝ] F
x y : F
⊢ (⇑φ.symm ∘ ![x, y]) ((fun i ↦ i) ⟨1, ⋯⟩) = ![φ.symm x, φ.symm y] ((fun i ↦ i) ⟨1, ⋯⟩)

Tactic:
rfl

Goal state after:
Module: Mathlib.Analysis.Matrix.LDL

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : WellFoundedLT n
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
this✝ : NormedAddCommGroup (n → 𝕜) := Sᵀ.toNormedAddCommGroup ⋯
this : InnerProductSpace 𝕜 (n → 𝕜) := Sᵀ.toInnerProductSpace ⋯
i j : n
⊢ gramSchmidt 𝕜 (⇑(Pi.basisFun 𝕜 n)) i j = ((Pi.basisFun 𝕜 n).toMatrix ⇑(gramSchmidtBasis (Pi.basisFun 𝕜 n)))ᵀ i j

Tactic:
Basis.coePiBasisFun.toMatrix_eq_transpose,

Goal state after:
𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : WellFoundedLT n
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
this✝ : NormedAddCommGroup (n → 𝕜) := Sᵀ.toNormedAddCommGroup ⋯
this : InnerProductSpace 𝕜 (n → 𝕜) := Sᵀ.toInnerProductSpace ⋯
i j : n
⊢ gramSchmidt 𝕜 (⇑(Pi.basisFun 𝕜 n)) i j = (⇑(gramSchmidtBasis (Pi.basisFun 𝕜 n)))ᵀᵀ i j
Module: Mathlib.Analysis.Matrix.LDL

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : WellFoundedLT n
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
this✝ : NormedAddCommGroup (n → 𝕜) := Sᵀ.toNormedAddCommGroup ⋯
this : InnerProductSpace 𝕜 (n → 𝕜) := Sᵀ.toInnerProductSpace ⋯
i j : n
⊢ gramSchmidt 𝕜 (⇑(Pi.basisFun 𝕜 n)) i j = (gramSchmidt 𝕜 ⇑(Pi.basisFun 𝕜 n))ᵀᵀ i j

Tactic:
rfl

Goal state after:
Module: Mathlib.Analysis.Matrix.LDL

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : WellFoundedLT n
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
i j : n
hij : i = j
⊢ lowerInv hS j ⬝ᵥ S *ᵥ star (lowerInv hS j) = (lowerInv hS * (S * (lowerInv hS)ᴴ)) j j

Tactic:
rfl

Goal state after:
Module: Mathlib.Analysis.Matrix.LDL

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : WellFoundedLT n
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
i j : n
hij : ¬i = j
⊢ lowerInv hS i ᵥ* S ⬝ᵥ star (lowerInv hS j) = lowerInv hS i ᵥ* S ⬝ᵥ (lowerInv hS).map star j

Tactic:
rfl

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 : (o.kahler x) y = 0
this : ‖x‖ * ‖y‖ = 0
⊢ x = 0 ∨ y = 0

Tactic:
rcases eq_zero_or_eq_zero_of_mul_eq_zero this with h | h

Goal state after:
case inl
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 : (o.kahler x) y = 0
this : ‖x‖ * ‖y‖ = 0
h : ‖x‖ = 0
⊢ x = 0 ∨ y = 0

case inr
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 : (o.kahler x) y = 0
this : ‖x‖ * ‖y‖ = 0
h : ‖y‖ = 0
⊢ x = 0 ∨ y = 0
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s a : ℂ
⊢ mellin (fun t ↦ ↑t ^ a • f t) s = mellin f (s + a)

Tactic:
by
  refine setIntegral_congr_fun measurableSet_Ioi fun t ht => ?_
  simp_rw [← sub_add_eq_add_sub, cpow_add _ _ (ofReal_ne_zero.2 <| ne_of_gt ht), mul_smul]

Goal state after:
Module: Mathlib.Analysis.MellinInversion

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
σ : ℝ
f : ℝ → E
x : ℝ
hx : 0 < x
hfx : ContinuousAt f x
g : ℝ → E := fun u ↦ rexp (-σ * u) • f (rexp (-u))
hf : Integrable g volume
hFf : Integrable (𝓕 g) volume
⊢ ContinuousAt f (rexp (- -Real.log x))

Tactic:
simpa [Real.exp_log hx] using hfx

Goal state after:
Module: Mathlib.Analysis.MellinInversion

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
σ : ℝ
f : ℝ → E
x : ℝ
hx : 0 < x
g : ℝ → E := fun u ↦ rexp (-σ * u) • f (rexp (-u))
hf : Integrable g volume
hFf : Integrable (𝓕 g) volume
hfx : ContinuousAt g (-Real.log x)
⊢ ↑x ^ ↑(-σ) • rexp (σ * Real.log x) • f (rexp (Real.log x)) = f x

Tactic:
rw [mul_comm σ, ← rpow_def_of_pos hx, Real.exp_log hx, ← Complex.ofReal_cpow hx.le]

Goal state after:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
σ : ℝ
f : ℝ → E
x : ℝ
hx : 0 < x
g : ℝ → E := fun u ↦ rexp (-σ * u) • f (rexp (-u))
hf : Integrable g volume
hFf : Integrable (𝓕 g) volume
hfx : ContinuousAt g (-Real.log x)
⊢ ↑(x ^ (-σ)) • x ^ σ • f x = f x
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Deriv

Goal state before:
pp: s✝ : ℂ
hs✝ : ∀ (m : ℕ), s✝ ≠ -↑m
n : ℕ
s : ℂ
IH : DifferentiableAt ℂ Gamma (s + 1)
hsre : -↑(n + 1) < s.re
hs : ∀ (m : ℕ), s ≠ -↑m
hsne : s ≠ 0
this : DifferentiableAt ℂ (Gamma ∘ fun s ↦ s + 1) s
⊢ ∀ (a : ℂ), a ≠ 0 → Gamma a = (Gamma ∘ fun s ↦ s + 1) a / id a

Tactic:
grind

Goal state after:
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
t : ℝ
ht : t ∈ Set.Icc 0 1
⊢ HasDerivAt f (f' (↑t * z)) (↑t * z)

Tactic:
exact hasDerivAt_log_sub_logTaylor n <|
      StarConvex.add_smul_mem starConvex_one_slitPlane (mem_slitPlane_of_norm_lt_one hz) ht.1 ht.2

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

Goal state before:
pp: s : ℝ
hs : ∀ (m : ℕ), s ≠ -↑m
⊢ ∀ (m : ℕ), ¬↑s = ↑(-↑m)

Tactic:
Complex.ofReal_inj

Goal state after:
s : ℝ
hs : ∀ (m : ℕ), s ≠ -↑m
⊢ ∀ (m : ℕ), ¬s = -↑m
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: s t a b : ℝ
hs : 0 < s
ht : 0 < t
ha : 0 < a
hb : 0 < b
hab : a + b = 1
f : ℝ → ℝ → ℝ → ℝ := fun c u x ↦ rexp (-c * x) * x ^ (c * (u - 1))
e : (1 / a).HolderConjugate (1 / b)
hab' : b = 1 - a
hst : 0 < a * s + b * t
posf : ∀ (c u x : ℝ), x ∈ Ioi 0 → 0 ≤ f c u x
posf' : ∀ (c u : ℝ), ∀ᵐ (x : ℝ) ∂volume.restrict (Ioi 0), 0 ≤ f c u x
⊢ Γ (a * s + b * t) ≤ Γ s ^ a * Γ t ^ b

Tactic:
have fpow :
    ∀ {c x : ℝ} (_ : 0 < c) (u : ℝ) (_ : 0 < x), exp (-x) * x ^ (u - 1) = f c u x ^ (1 / c) := by
    intro c x hc u hx
    dsimp only [f]
    rw [mul_rpow (exp_pos _).le ((rpow_nonneg hx.le) _), ← exp_mul, ← rpow_mul hx.le]
    congr 2 <;> field

Goal state after:
s t a b : ℝ
hs : 0 < s
ht : 0 < t
ha : 0 < a
hb : 0 < b
hab : a + b = 1
f : ℝ → ℝ → ℝ → ℝ := fun c u x ↦ rexp (-c * x) * x ^ (c * (u - 1))
e : (1 / a).HolderConjugate (1 / b)
hab' : b = 1 - a
hst : 0 < a * s + b * t
posf : ∀ (c u x : ℝ), x ∈ Ioi 0 → 0 ≤ f c u x
posf' : ∀ (c u : ℝ), ∀ᵐ (x : ℝ) ∂volume.restrict (Ioi 0), 0 ≤ f c u x
fpow : ∀ {c x : ℝ}, 0 < c → ∀ (u : ℝ), 0 < x → rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c)
⊢ Γ (a * s + b * t) ≤ Γ s ^ a * Γ t ^ b
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
hab : b < a
hf : f =O[atTop] fun x ↦ x ^ (-a)
t : ℝ
⊢ (fun x ↦ log x • f x) t = (fun t ↦ log t • f t) t

Tactic:
rfl

Goal state after: