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
⊢ 1 / ↑n' - 1 / q = 1 / ↑p'

Tactic:
one_div,

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
⊢ (↑n')⁻¹ - q⁻¹ = (↑p')⁻¹
Module: Mathlib.Analysis.InnerProductSpace.LinearPMap

Goal state before:
pp: case mpr
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
g : Submodule 𝕜 (E × F)
x : F × E
h : ∀ (a : E) (b : F), (a, b) ∈ g → ⟪b, x.1⟫ - ⟪a, x.2⟫ = 0
u : WithLp 2 (F × E)
⊢ ∀ (x_1 : E) (x_2 : F), (x_1, x_2) ∈ g → toLp 2 (x_2, -x_1) = u → ⟪u.fst, (toLp 2 x).fst⟫ + ⟪u.snd, (toLp 2 x).snd⟫ = 0

Tactic:
a

Goal state after:
case mpr
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
g : Submodule 𝕜 (E × F)
x : F × E
h : ∀ (a : E) (b : F), (a, b) ∈ g → ⟪b, x.1⟫ - ⟪a, x.2⟫ = 0
u : WithLp 2 (F × E)
a : E
⊢ ∀ (x_1 : F), (a, x_1) ∈ g → toLp 2 (x_1, -a) = u → ⟪u.fst, (toLp 2 x).fst⟫ + ⟪u.snd, (toLp 2 x).snd⟫ = 0
Module: Mathlib.Analysis.InnerProductSpace.LinearPMap

Goal state before:
pp: case mp
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
T : E →ₗ.[𝕜] F
inst✝ : CompleteSpace E
hT : Dense ↑T.domain
x : F × E
hx : x.1 ∈ T†.domain
h : ↑T† ⟨x.1, ⋯⟩ = x.2
a : E
ha : a ∈ T.domain
⊢ ⟪↑⟨a, ha⟩, ↑T† ⟨x.1, hx⟩⟫ - ⟪a, ↑T† ⟨x.1, ⋯⟩⟫ = 0

Tactic:
sub_self

Goal state after:
case mp
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
T : E →ₗ.[𝕜] F
inst✝ : CompleteSpace E
hT : Dense ↑T.domain
x : F × E
hx : x.1 ∈ T†.domain
h : ↑T† ⟨x.1, ⋯⟩ = x.2
a : E
ha : a ∈ T.domain
⊢ 0 = 0
Module: Mathlib.Analysis.InnerProductSpace.LinearPMap

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
T : E →ₗ.[𝕜] F
inst✝ : CompleteSpace E
hT : Dense ↑T.domain
⊢ _root_.IsClosed
    (⇑(WithLp.linearEquiv 2 𝕜 (F × E)) ''
      ↑(Submodule.map (↑((LinearEquiv.skewSwap 𝕜 F E).symm.trans (WithLp.linearEquiv 2 𝕜 (F × E)).symm)) T.graph)ᗮ)

Tactic:
LinearEquiv.image_eq_preimage_symm

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
T : E →ₗ.[𝕜] F
inst✝ : CompleteSpace E
hT : Dense ↑T.domain
⊢ _root_.IsClosed
    (⇑(WithLp.linearEquiv 2 𝕜 (F × E)).symm ⁻¹'
      ↑(Submodule.map (↑((LinearEquiv.skewSwap 𝕜 F E).symm.trans (WithLp.linearEquiv 2 𝕜 (F × E)).symm)) T.graph)ᗮ)
Module: Mathlib.Analysis.InnerProductSpace.LinearPMap

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
T : E →ₗ.[𝕜] F
inst✝ : CompleteSpace E
hT : Dense ↑T.domain
⊢ T†.IsClosed

Tactic:
by
  rw [IsClosed, adjoint_graph_eq_graph_adjoint hT, Submodule.adjoint]
  simp only [Submodule.map_coe]
  rw [LinearEquiv.coe_coe, LinearEquiv.image_eq_preimage_symm]
  exact (Submodule.isClosed_orthogonal _).preimage (WithLp.prod_continuous_toLp _ _ _)

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
hu : ContDiff ℝ 1 u
h2u : HasCompactSupport u
p p' : ℝ≥0
hp : 1 ≤ p
hn : 0 < finrank ℝ E
hp' : (↑p')⁻¹ = ↑p⁻¹ - (↑(finrank ℝ E))⁻¹
F' : Type := EuclideanSpace ℝ (Fin (finrank ℝ F))
e : F ≃L[ℝ] F' := toEuclidean
C₁ : ℝ≥0 := ‖↑e.symm‖₊
C : ℝ≥0 := eLpNormLESNormFDerivOfEqInnerConst μ ↑p
C₂ : ℝ≥0 := ‖↑e‖₊
v : E → F' := ⇑e ∘ u
hv : ContDiff ℝ 1 v
h2v : HasCompactSupport v
this : eLpNorm v (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ v) (↑p) μ
h4v : ∀ (x : E), ‖fderiv ℝ v x‖ ≤ ↑C₂ * ‖fderiv ℝ u x‖
⊢ eLpNorm u (↑p') μ = eLpNorm (fun x ↦ e.symm (e (u x))) (↑p') μ

Tactic:
e.symm_apply_apply

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

Goal state before:
pp: case pos
X : Type u_2
inst✝ : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
t : Set X
ht : t ∈ {s | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s}
s : Set X
S : ℕ → Set X := fun n ↦ {x | x ∈ s ∧ (↑n)⁻¹ ≤ infEDist x t}
Ssep : ∀ (n : ℕ), AreSeparated (S n) t
Ssep' : ∀ (n : ℕ), AreSeparated (S n) (s ∩ t)
S_sub : ∀ (n : ℕ), S n ⊆ s \ t
hSs : ∀ (n : ℕ), μ (s ∩ t) + μ (S n) ≤ μ s
iUnion_S : ⋃ n, S n = s \ t
htop : μ (s \ t) = ∞
⊢ μ (s ∩ t) + μ (s \ t) ≤ μ s

pp: case neg
X : Type u_2
inst✝ : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
t : Set X
ht : t ∈ {s | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s}
s : Set X
S : ℕ → Set X := fun n ↦ {x | x ∈ s ∧ (↑n)⁻¹ ≤ infEDist x t}
Ssep : ∀ (n : ℕ), AreSeparated (S n) t
Ssep' : ∀ (n : ℕ), AreSeparated (S n) (s ∩ t)
S_sub : ∀ (n : ℕ), S n ⊆ s \ t
hSs : ∀ (n : ℕ), μ (s ∩ t) + μ (S n) ≤ μ s
iUnion_S : ⋃ n, S n = s \ t
htop : ¬μ (s \ t) = ∞
⊢ μ (s ∩ t) + μ (s \ t) ≤ μ s

Tactic:
· rw [htop, add_top, ← htop]
    exact μ.mono sdiff_subset

Goal state after:
case neg
X : Type u_2
inst✝ : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
t : Set X
ht : t ∈ {s | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s}
s : Set X
S : ℕ → Set X := fun n ↦ {x | x ∈ s ∧ (↑n)⁻¹ ≤ infEDist x t}
Ssep : ∀ (n : ℕ), AreSeparated (S n) t
Ssep' : ∀ (n : ℕ), AreSeparated (S n) (s ∩ t)
S_sub : ∀ (n : ℕ), S n ⊆ s \ t
hSs : ∀ (n : ℕ), μ (s ∩ t) + μ (S n) ≤ μ s
iUnion_S : ⋃ n, S n = s \ t
htop : ¬μ (s \ t) = ∞
⊢ μ (s ∩ t) + μ (s \ t) ≤ μ s
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 ≤ 1 / ↑q - 1 / ↑p'

Tactic:
one_div,

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 - 1 / ↑p'
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: case neg
X : Type u_2
inst✝ : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
t : Set X
ht : t ∈ {s | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s}
s : Set X
S : ℕ → Set X := fun n ↦ {x | x ∈ s ∧ (↑n)⁻¹ ≤ infEDist x t}
Ssep : ∀ (n : ℕ), AreSeparated (S n) t
Ssep' : ∀ (n : ℕ), AreSeparated (S n) (s ∩ t)
S_sub : ∀ (n : ℕ), S n ⊆ s \ t
hSs : ∀ (n : ℕ), μ (s ∩ t) + μ (S n) ≤ μ s
iUnion_S : ⋃ n, S n = s \ t
htop : ¬μ (s \ t) = ∞
r : ℕ
⊢ ∑' (k : ℕ), μ (S (2 * k + 1 + r) \ S (2 * k + r)) ≤ μ (s \ t)

Tactic:
rw [← iUnion_S, ENNReal.tsum_eq_iSup_nat, iSup_le_iff]

Goal state after:
case neg
X : Type u_2
inst✝ : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
t : Set X
ht : t ∈ {s | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s}
s : Set X
S : ℕ → Set X := fun n ↦ {x | x ∈ s ∧ (↑n)⁻¹ ≤ infEDist x t}
Ssep : ∀ (n : ℕ), AreSeparated (S n) t
Ssep' : ∀ (n : ℕ), AreSeparated (S n) (s ∩ t)
S_sub : ∀ (n : ℕ), S n ⊆ s \ t
hSs : ∀ (n : ℕ), μ (s ∩ t) + μ (S n) ≤ μ s
iUnion_S : ⋃ n, S n = s \ t
htop : ¬μ (s \ t) = ∞
r : ℕ
⊢ ∀ (i : ℕ), ∑ a ∈ Finset.range i, μ (S (2 * a + 1 + r) \ S (2 * a + r)) ≤ μ (⋃ n, S n)
Module: Mathlib.Geometry.Euclidean.Volume.Measure

Goal state before:
pp: V : Type u_3
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
inst✝ : FiniteDimensional ℝ V
⊢ map ⇑(stdOrthonormalBasis ℝ V).repr.toIsometryEquiv.symm μHE[finrank ℝ V] =
    map (⇑(stdOrthonormalBasis ℝ V).repr.symm) volume

Tactic:
EuclideanSpace.euclideanHausdorffMeasure_eq_volume

Goal state after:
V : Type u_3
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
inst✝ : FiniteDimensional ℝ V
⊢ map (⇑(stdOrthonormalBasis ℝ V).repr.toIsometryEquiv.symm) volume = map (⇑(stdOrthonormalBasis ℝ V).repr.symm) volume
Module: Mathlib.Topology.MetricSpace.HausdorffDimension

Goal state before:
pp: X : Type u_2
inst✝ : EMetricSpace X
s t : Set X
⊢ max (dimH s) (dimH (bif false then s else t)) = max (dimH s) (dimH t)

Tactic:
cond

Goal state after:
X : Type u_2
inst✝ : EMetricSpace X
s t : Set X
⊢ max (dimH s) (dimH t) = max (dimH s) (dimH t)
Module: Mathlib.Topology.MetricSpace.HausdorffDimension

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

Tactic:
by
  refine le_antisymm (iSup_le fun x => ?_) ?_
  · refine limsup_le_of_le isCobounded_le_of_bot ?_
    exact eventually_smallSets.2 ⟨s, self_mem_nhdsWithin, fun t => dimH_mono⟩
  · rw [← bsupr_limsup_dimH]; exact iSup₂_le_iSup _ _

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

Goal state before:
pp: X : Type u_2
inst✝² : EMetricSpace X
μ : OuterMeasure X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
hm : μ.IsMetric
⊢ borel X ≤ μ.caratheodory

Tactic:
exact hm.borel_le_caratheodory

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

Goal state before:
pp: E : Type u_4
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
d : ℝ≥0
hd : ↑(finrank ℝ E) < ↑d
⊢ dimH univ < ↑d

Tactic:
dimH_univ_eq_finrank

Goal state after:
E : Type u_4
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
d : ℝ≥0
hd : ↑(finrank ℝ E) < ↑d
⊢ ↑(finrank ℝ E) < ↑d
Module: Mathlib.Analysis.InnerProductSpace.Reproducing

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : RCLike 𝕜
X : Type u_2
V : Type u_3
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace 𝕜 V
inst✝ : CompleteSpace V
K : Matrix X X (V →L[𝕜] V)
h p1 p2 p3 : Prop
htfae : h → [p1, p2, p3].TFAE
⊢ [h ∧ p1, h ∧ p2, h ∧ p3].TFAE

Tactic:
tfae_have 1 → 2 := fun ⟨h, t⟩ ↦ ⟨h, ((htfae h).out 0 1).mp t⟩

Goal state after:
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
X : Type u_2
V : Type u_3
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace 𝕜 V
inst✝ : CompleteSpace V
K : Matrix X X (V →L[𝕜] V)
h p1 p2 p3 : Prop
htfae : h → [p1, p2, p3].TFAE
tfae_1_to_2 : h ∧ p1 → h ∧ p2
⊢ [h ∧ p1, h ∧ p2, h ∧ p3].TFAE
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: case inl.inr.inl
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
r : ℝ≥0
f : X → Y
s : Set X
hr : 0 < r
h : HolderOnWith 0 r f s
x : X
hx : x ∈ s
this : f '' s = {f x}
hd : 0 ≤ 0
⊢ μH[0] {f x} ≤ μH[0] s

Tactic:
hausdorffMeasure_zero_singleton

Goal state after:
case inl.inr.inl
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
r : ℝ≥0
f : X → Y
s : Set X
hr : 0 < r
h : HolderOnWith 0 r f s
x : X
hx : x ∈ s
this : f '' s = {f x}
hd : 0 ≤ 0
⊢ 1 ≤ μH[0] s
Module: Mathlib.Analysis.InnerProductSpace.Reproducing

Goal state before:
pp: 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 g : H₀ K
x✝³ : X × V
x✝² : 𝕜
x✝¹ : X × V
x✝ : 𝕜
⊢ (starRingEnd 𝕜) (star x✝ * x✝² * ⟪(K x✝³.1 x✝¹.1) x✝¹.2, x✝³.2⟫_𝕜) = star x✝² * x✝ * ⟪(K x✝¹.1 x✝³.1) x✝³.2, x✝¹.2⟫_𝕜

Tactic:
← (Fact.out : K.PosSemidef).isHermitian.apply

Goal state after:
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 g : H₀ K
x✝³ : X × V
x✝² : 𝕜
x✝¹ : X × V
x✝ : 𝕜
⊢ (starRingEnd 𝕜) (star x✝ * x✝² * ⟪(star (K x✝¹.1 x✝³.1)) x✝¹.2, x✝³.2⟫_𝕜) =
    star x✝² * x✝ * ⟪(K x✝¹.1 x✝³.1) x✝³.2, x✝¹.2⟫_𝕜
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: case inl.inr.inl
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
s : Set Y
hf : AntilipschitzWith 0 f
x : X
hx : x ∈ f ⁻¹' s
this : f ⁻¹' s = {x}
hd : 0 ≤ 0
⊢ μH[0] {x} ≤ μH[0] s

Tactic:
hausdorffMeasure_zero_singleton

Goal state after:
case inl.inr.inl
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
s : Set Y
hf : AntilipschitzWith 0 f
x : X
hx : x ∈ f ⁻¹' s
this : f ⁻¹' s = {x}
hd : 0 ≤ 0
⊢ 1 ≤ μH[0] s
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✝ : IsTopologicalAddGroup F
f : E →ₛₗ[σ] F
U : Set F
hU : U ∈ 𝓝 0
V : Set E
hV : V ∈ 𝓝 0
h✝ : IsVonNBounded 𝕜' (⇑f '' V)
r : ℝ
hr : r > 0
h : ∀ (c : 𝕜'), r ≤ ‖c‖ → ⇑f '' V ⊆ c • U
x : 𝕜
hx : r < ‖x‖
⊢ r ≤ ‖σ x‖

Tactic:
simpa using hx.le

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)
z : E [⋀^Fin 0]→ₗ[ℝ] ℝ ≃ₗ[ℝ] ℝ := AlternatingMap.constLinearEquivOfIsEmpty.symm
⊢ E →ₗ[ℝ] E →ₗ[ℝ] ℝ

Tactic:
let y : E [⋀^Fin 1]→ₗ[ℝ] ℝ →ₗ[ℝ] E →ₗ[ℝ] ℝ :=
    LinearMap.llcomp ℝ E (E [⋀^Fin 0]→ₗ[ℝ] ℝ) ℝ z ∘ₗ AlternatingMap.curryLeftLinearMap

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)
z : E [⋀^Fin 0]→ₗ[ℝ] ℝ ≃ₗ[ℝ] ℝ := AlternatingMap.constLinearEquivOfIsEmpty.symm
y : E [⋀^Fin 1]→ₗ[ℝ] ℝ →ₗ[ℝ] E →ₗ[ℝ] ℝ :=
  (LinearMap.llcomp ℝ E (E [⋀^Fin 0]→ₗ[ℝ] ℝ) ℝ) ↑z ∘ₗ AlternatingMap.curryLeftLinearMap
⊢ E →ₗ[ℝ] E →ₗ[ℝ] ℝ
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: case inl
E : Type u_5
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s : Set ℝ
⊢ μH[1] ((fun r ↦ r • 0) '' s) = ‖0‖₊ • μH[1] s

Tactic:
have := nullSingletonClass_hausdorff E one_pos

Goal state after:
case inl
E : Type u_5
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s : Set ℝ
this : NullSingletonClass μH[1]
⊢ μH[1] ((fun r ↦ r • 0) '' s) = ‖0‖₊ • μH[1] s
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: E : Type u_7
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
x y : E
⊢ μH[1] (segment ℝ x y) = edist x y

Tactic:
rw [← affineSegment_eq_segment, hausdorffMeasure_affineSegment]

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

Goal state before:
pp: 𝕜₁ : Type u_5
𝕜₂ : Type u_6
𝕜₃ : Type u_7
E : Type u_9
F : Type u_10
G : Type u_11
inst✝¹⁸ : NormedField 𝕜₁
inst✝¹⁷ : NormedField 𝕜₂
inst✝¹⁶ : NormedField 𝕜₃
σ₁₂ : 𝕜₁ →+* 𝕜₂
σ₁₃ : 𝕜₁ →+* 𝕜₃
σ₂₃ : 𝕜₂ →+* 𝕜₃
inst✝¹⁵ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
inst✝¹⁴ : AddCommGroup E
inst✝¹³ : TopologicalSpace E
inst✝¹² : Module 𝕜₁ E
inst✝¹¹ : AddCommGroup F
inst✝¹⁰ : TopologicalSpace F
inst✝⁹ : Module 𝕜₂ F
inst✝⁸ : AddCommGroup G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : Module 𝕜₃ G
inst✝⁵ : IsTopologicalAddGroup F
inst✝⁴ : ContinuousConstSMul 𝕜₂ F
inst✝³ : IsTopologicalAddGroup G
inst✝² : ContinuousConstSMul 𝕜₃ G
inst✝¹ : RingHomSurjective σ₂₃
inst✝ : RingHomIsometric σ₂₃
g : F →SWOT[σ₂₃] G
x : E
z : StrongDual 𝕜₃ G
σ_bij : Function.Bijective ⇑σ₂₃
σ_equiv : 𝕜₂ ≃+* 𝕜₃ := RingEquiv.ofBijective σ₂₃ σ_bij
invPair : RingHomInvPair σ₂₃ ↑σ_equiv.symm := RingHomInvPair.of_ringEquiv σ_equiv
invPair_symm : RingHomInvPair (↑σ_equiv.symm) σ₂₃ := RingHomInvPair.symm σ₂₃ ↑σ_equiv.symm
σ_li : 𝕜₂ ≃ₛₗᵢ[σ₂₃] 𝕜₃ :=
  let __LinearEquiv := LinearEquiv.ofBijective σ₂₃.toSemilinearMap σ_bij;
  { toLinearEquiv := __LinearEquiv, norm_map' := ⋯ }
| fun a ↦ z ((g.comp a) x)

Tactic:
a

Goal state after:
𝕜₁ : Type u_5
𝕜₂ : Type u_6
𝕜₃ : Type u_7
E : Type u_9
F : Type u_10
G : Type u_11
inst✝¹⁸ : NormedField 𝕜₁
inst✝¹⁷ : NormedField 𝕜₂
inst✝¹⁶ : NormedField 𝕜₃
σ₁₂ : 𝕜₁ →+* 𝕜₂
σ₁₃ : 𝕜₁ →+* 𝕜₃
σ₂₃ : 𝕜₂ →+* 𝕜₃
inst✝¹⁵ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
inst✝¹⁴ : AddCommGroup E
inst✝¹³ : TopologicalSpace E
inst✝¹² : Module 𝕜₁ E
inst✝¹¹ : AddCommGroup F
inst✝¹⁰ : TopologicalSpace F
inst✝⁹ : Module 𝕜₂ F
inst✝⁸ : AddCommGroup G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : Module 𝕜₃ G
inst✝⁵ : IsTopologicalAddGroup F
inst✝⁴ : ContinuousConstSMul 𝕜₂ F
inst✝³ : IsTopologicalAddGroup G
inst✝² : ContinuousConstSMul 𝕜₃ G
inst✝¹ : RingHomSurjective σ₂₃
inst✝ : RingHomIsometric σ₂₃
g : F →SWOT[σ₂₃] G
x : E
z : StrongDual 𝕜₃ G
σ_bij : Function.Bijective ⇑σ₂₃
σ_equiv : 𝕜₂ ≃+* 𝕜₃ := RingEquiv.ofBijective σ₂₃ σ_bij
invPair : RingHomInvPair σ₂₃ ↑σ_equiv.symm := RingHomInvPair.of_ringEquiv σ_equiv
invPair_symm : RingHomInvPair (↑σ_equiv.symm) σ₂₃ := RingHomInvPair.symm σ₂₃ ↑σ_equiv.symm
σ_li : 𝕜₂ ≃ₛₗᵢ[σ₂₃] 𝕜₃ :=
  let __LinearEquiv := LinearEquiv.ofBijective σ₂₃.toSemilinearMap σ_bij;
  { toLinearEquiv := __LinearEquiv, norm_map' := ⋯ }
a : E →SWOT[σ₁₂] F
| z ((g.comp a) x)
Module: Mathlib.Analysis.MellinInversion

Goal state before:
pp: case e_f
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
u : ℝ
⊢ cexp (-s * ↑u) • f (rexp (-u)) = cexp (↑(-2 * π * (u * (s.im / (2 * π)))) * I) • rexp (-s.re * u) • f (rexp (-u))

Tactic:
trans Complex.exp (-s.im * u * I) • (Real.exp (-s.re * u) • f (Real.exp (-u)))

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
u : ℝ
⊢ cexp (-s * ↑u) • f (rexp (-u)) = cexp (-↑s.im * ↑u * I) • rexp (-s.re * u) • f (rexp (-u))

E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
u : ℝ
⊢ cexp (-↑s.im * ↑u * I) • rexp (-s.re * u) • f (rexp (-u)) =
    cexp (↑(-2 * π * (u * (s.im / (2 * π)))) * I) • rexp (-s.re * u) • f (rexp (-u))
Module: Mathlib.Analysis.MellinInversion

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
σ : ℝ
f : ℂ → E
x : ℝ
hx : 0 < x
⊢ (1 / (2 * π)) • ∫ (y : ℝ), ↑x ^ (-(↑σ + ↑y * I)) • f (↑σ + ↑y * I) =
    ↑x ^ (-↑σ) • ∫ (y : ℝ), cexp (2 * ↑π * (↑y * -↑(Real.log x)) * I) • f (↑σ + 2 * ↑π * ↑y * I)

Tactic:
one_div,

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
σ : ℝ
f : ℂ → E
x : ℝ
hx : 0 < x
⊢ (2 * π)⁻¹ • ∫ (y : ℝ), ↑x ^ (-(↑σ + ↑y * I)) • f (↑σ + ↑y * I) =
    ↑x ^ (-↑σ) • ∫ (y : ℝ), cexp (2 * ↑π * (↑y * -↑(Real.log x)) * I) • f (↑σ + 2 * ↑π * ↑y * I)
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: case h₁
z : ℂ
hz : ‖z‖ ≤ 1 / 2
hz3 : (1 - ‖z‖)⁻¹ ≤ 2
⊢ ‖z‖ ^ 2 ≤ ‖z‖ / 2

Tactic:
rw [sq, div_eq_mul_one_div]
      gcongr

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

Goal state before:
pp: z : ℂ
hz : ‖z‖ < 1
⊢ HasSum (fun n ↦ z ^ n / ↑n) (-log (1 - z) + ∑ i ∈ Finset.range 1, z ^ i / ↑i)

Tactic:
simpa using hasSum_taylorSeries_neg_log hz

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

Goal state before:
pp: case e'_5
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
this✝ : ∫ (x : ℝ) in 0..π / 2, cos x ^ (2 * n.succ) = (2 * ↑n + 1) / (2 * ↑n + 2) * C
this :
  ↑π * z * (A * (1 - z ^ 2 / (↑n + 1) ^ 2)) *
      ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n.succ) =
    ↑π * z * A * ((1 - z ^ 2 / ↑n.succ ^ 2) * ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n.succ))
⊢ ↑n.succ = ↑n + 1

Tactic:
rw [Nat.cast_succ]

Goal state after:
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: a s : ℂ
hs : 0 < s.re + a.re
this : HasMellin ((Ioc 0 1).indicator fun x ↦ 1) (s + a) (1 / (s + a))
⊢ HasMellin ((Ioc 0 1).indicator fun t ↦ ↑t ^ a) s (1 / (s + a))

Tactic:
simp_rw [HasMellin, ← MellinConvergent.cpow_smul, ← mellin_cpow_smul, ← indicator_smul,
    smul_eq_mul, mul_one] at this

Goal state after:
a s : ℂ
hs : 0 < s.re + a.re
this :
  MellinConvergent ((Ioc 0 1).indicator fun a_1 ↦ ↑a_1 ^ a) s ∧
    mellin ((Ioc 0 1).indicator fun a_1 ↦ ↑a_1 ^ a) s = 1 / (s + a)
⊢ HasMellin ((Ioc 0 1).indicator fun t ↦ ↑t ^ a) s (1 / (s + a))