mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:

Tactic:
(eLpNorm (⇑g - ⇑f) 1 μ').toReal ≤ (c' * eLpNorm (⇑g - ⇑f) 1 μ).toReal := by
      refine toReal_mono (ENNReal.mul_ne_top hc' h_eLpNorm_ne_top) ?_
      refine (eLpNorm_mono_measure (⇑g - ⇑f) hμ'_le).trans_eq ?_
      rw [eLpNorm_smul_measure_of_ne_zero hc'0, smul_eq_mul]
      simp
    _ = c'.toReal * (eLpNorm (⇑g - ⇑f) 1 μ).toReal := toReal_mul
    _ ≤ c'.toReal * (ε / 2 / c'.toReal) := by gcongr
    _ = ε / 2 := by
      refine mul_div_cancel₀ (ε / 2) ?_; rw [Ne, toReal_eq_zero_iff]; simp [hc', hc'0]
    _ < ε := half_lt_self hε_pos

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case neg
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
s : Set X
μ : Measure X
hs : MeasurableSet s
hfi : ¬IntegrableOn f s μ
⊢ ¬Integrable (s.indicator f) μ

Tactic:
integrable_indicator_iff hs

Goal state after:
case neg
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
s : Set X
μ : Measure X
hs : MeasurableSet s
hfi : ¬IntegrableOn f s μ
⊢ ¬IntegrableOn f s μ
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
s t : Set X
μ : Measure X
ht : MeasurableSet t
⊢ ∫ (x : X) in s, t.indicator f x ∂μ = ∫ (x : X) in s ∩ t, f x ∂μ

Tactic:
by
  rw [integral_indicator ht, Measure.restrict_restrict ht, Set.inter_comm]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
f : X → E
μ : Measure X
ι : Type u_5
inst✝¹ : Preorder ι
inst✝ : atTop.IsCountablyGenerated
s : ι → Set X
hsm : ∀ (i : ι), NullMeasurableSet (s i) μ
h_mono : Monotone s
hne : atTop.NeBot
this✝ : IsDirectedOrder ι
S : Set X := ⋃ i, s i
hfi : IntegrableOn f S μ
hfi' : (μ.withDensity fun x ↦ ↑‖f x‖₊) S < ∞
hSm : NullMeasurableSet S μ
hsub : ∀ {i : ι}, s i ⊆ S
ν : Measure X := μ.withDensity fun x ↦ ‖f x‖ₑ
hν : ν = μ.withDensity fun x ↦ ‖f x‖ₑ
ε : ℝ≥0
ε0 : 0 < ↑ε
this : ∀ᶠ (i : ι) in atTop, ν (s i) ∈ Icc (ν S - ↑ε) (ν S + ↑ε)
i : ι
hi : ν (s i) ∈ Icc (ν S - ↑ε) (ν S + ↑ε)
hsm' : NullMeasurableSet (s i) ν
⊢ ν S - ν (s i) ≤ ↑ε

pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
f : X → E
μ : Measure X
ι : Type u_5
inst✝¹ : Preorder ι
inst✝ : atTop.IsCountablyGenerated
s : ι → Set X
hsm : ∀ (i : ι), NullMeasurableSet (s i) μ
h_mono : Monotone s
hne : atTop.NeBot
this✝ : IsDirectedOrder ι
S : Set X := ⋃ i, s i
hfi : IntegrableOn f S μ
hfi' : (μ.withDensity fun x ↦ ↑‖f x‖₊) S < ∞
hSm : NullMeasurableSet S μ
hsub : ∀ {i : ι}, s i ⊆ S
ν : Measure X := μ.withDensity fun x ↦ ‖f x‖ₑ
hν : ν = μ.withDensity fun x ↦ ‖f x‖ₑ
ε : ℝ≥0
ε0 : 0 < ↑ε
this : ∀ᶠ (i : ι) in atTop, ν (s i) ∈ Icc (ν S - ↑ε) (ν S + ↑ε)
i : ι
hi : ν (s i) ∈ Icc (ν S - ↑ε) (ν S + ↑ε)
hsm' : NullMeasurableSet (s i) ν
⊢ ν (s i) ≠ ∞

Tactic:
exacts [tsub_le_iff_tsub_le.mp hi.1,
    (hi.2.trans_lt <| ENNReal.add_lt_top.2 ⟨hfi', ENNReal.coe_lt_top⟩).ne]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
t : Set X
μ : Measure X
ht_eq : ∀ᵐ (x : X) ∂μ, x ∈ t → f x = 0
hf : AEStronglyMeasurable f (μ.restrict t)
x : X
hx : x ∈ t → f x = AEStronglyMeasurable.mk f hf x
h'x : x ∈ t → f x = 0
h''x : x ∈ t
⊢ AEStronglyMeasurable.mk f hf x = 0 x

Tactic:
← hx h''x

Goal state after:
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
t : Set X
μ : Measure X
ht_eq : ∀ᵐ (x : X) ∂μ, x ∈ t → f x = 0
hf : AEStronglyMeasurable f (μ.restrict t)
x : X
hx : x ∈ t → f x = AEStronglyMeasurable.mk f hf x
h'x : x ∈ t → f x = 0
h''x : x ∈ t
⊢ f x = 0 x
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
s t : Set X
μ : Measure X
hts : s ⊆ t
h't : ∀ᵐ (x : X) ∂μ, x ∈ t \ s → f x = 0
haux : StronglyMeasurable f
h'aux : IntegrableOn f t μ
k : Set X := f ⁻¹' {0}
hk : MeasurableSet k
⊢ ∫ (x : X) in t, f x ∂μ = ∫ (x : X) in t ∩ k, f x ∂μ + ∫ (x : X) in t \ k, f x ∂μ

Tactic:
rw [integral_inter_add_sdiff hk h'aux]

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

Goal state before:
pp: G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
A : Set G
hA : μ.outerMeasure A ≠ ∞
ε : ℝ≥0
hε : ε ≠ 0
⊢ ∃ U, A ⊆ ↑U ∧ μ.outerMeasure ↑U ≤ μ.outerMeasure A + ↑ε

Tactic:
by
  rcases inducedOuterMeasure_exists_set _ μ.innerContent_iUnion_nat μ.innerContent_mono hA
      (ENNReal.coe_ne_zero.2 hε) with
    ⟨U, hU, h2U, h3U⟩
  exact ⟨⟨U, hU⟩, h2U, h3U⟩

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : X → E
μ : Measure X
inst✝ : IsFiniteMeasure μ
C : ℝ
hf : ∀ᵐ (x : X) ∂μ, ‖f x‖ ≤ C
s : Set X
hs : MeasurableSet s
hf1 : Integrable f μ
h0 : ∫ (x : X), f x ∂μ - ∫ (x : X) in s, f x ∂μ = ∫ (x : X) in sᶜ, f x ∂μ
h1 : ∫ (x : X) in sᶜ, ‖f x‖ ∂μ ≤ ∫ (x : X) in sᶜ, C ∂μ
h2 : ∫ (x : X) in sᶜ, C ∂μ = μ.real sᶜ * C
⊢ ‖∫ (x : X), f x ∂μ - ∫ (x : X) in s, f x ∂μ‖ ≤ μ.real sᶜ * C

Tactic:
rw [h0, ← h2]

Goal state after:
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : X → E
μ : Measure X
inst✝ : IsFiniteMeasure μ
C : ℝ
hf : ∀ᵐ (x : X) ∂μ, ‖f x‖ ≤ C
s : Set X
hs : MeasurableSet s
hf1 : Integrable f μ
h0 : ∫ (x : X), f x ∂μ - ∫ (x : X) in s, f x ∂μ = ∫ (x : X) in sᶜ, f x ∂μ
h1 : ∫ (x : X) in sᶜ, ‖f x‖ ∂μ ≤ ∫ (x : X) in sᶜ, C ∂μ
h2 : ∫ (x : X) in sᶜ, C ∂μ = μ.real sᶜ * C
⊢ ‖∫ (x : X) in sᶜ, f x ∂μ‖ ≤ ∫ (x : X) in sᶜ, C ∂μ
Module: Mathlib.Analysis.InnerProductSpace.Defs

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x : F
⊢ im ↑(normSq x) = im ⟪x, x⟫

Tactic:
by simp only [inner_self_im, ofReal_im]

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

Goal state before:
pp: R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
c : ℝ≥0
f : StieltjesFunction R
a b : R
h : a ≤ b
⊢ ofReal (↑(c • f) b - leftLim (↑(c • f)) a) = c • ofReal (↑f b - leftLim (↑f) a)

Tactic:
change ofReal (c * f b - leftLim (c • f) a) = c • ofReal (f b - leftLim f a)

Goal state after:
R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
c : ℝ≥0
f : StieltjesFunction R
a b : R
h : a ≤ b
⊢ ofReal (↑c * ↑f b - leftLim (↑(c • f)) a) = c • ofReal (↑f b - leftLim (↑f) a)
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_6
f : X → ℝ
m0 : MeasurableSpace X
μ : Measure X
g : SimpleFunc X ℝ
hf : Integrable f μ
c : ℝ
s : Set X
hs : MeasurableSet s
this : s.indicator (const X c) * f = s.indicator (c • f)
⊢ Integrable (s.indicator (c • f)) μ

Tactic:
integrable_indicator_iff hs

Goal state after:
X : Type u_6
f : X → ℝ
m0 : MeasurableSpace X
μ : Measure X
g : SimpleFunc X ℝ
hf : Integrable f μ
c : ℝ
s : Set X
hs : MeasurableSet s
this : s.indicator (const X c) * f = s.indicator (c • f)
⊢ IntegrableOn (c • f) s μ
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case hbc
Y : Type u_2
E : Type u_3
F : Type u_4
X : Type u_5
G : Type u_6
𝕜 : Type u_7
inst✝¹¹ : TopologicalSpace X
inst✝¹⁰ : TopologicalSpace Y
inst✝⁹ : MeasurableSpace Y
inst✝⁸ : OpensMeasurableSpace Y
μ : Measure Y
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : NormedSpace 𝕜 E
L : F →L[𝕜] G →L[𝕜] E
f : X → Y → G
s : Set X
k : Set Y
g : Y → F
hk : IsCompact k
hf : ContinuousOn (uncurry f) (s ×ˢ univ)
hfs : ∀ (p : X) (x : Y), p ∈ s → x ∉ k → f p x = 0
hg : IntegrableOn g k μ
A : ∀ p ∈ s, Continuous[inst✝¹⁰, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (f p)
q : X
hq : q ∈ s
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : 0 < δ
hδ : ∫ (x : Y) in k, ‖L‖ * ‖g x‖ * δ ∂μ < ε
v : Set X
v_mem : v ∈ 𝓝[s] q
hv : ∀ p ∈ v, ∀ x ∈ k, ‖f p x - f q x‖ < δ
p : X
hp : p ∈ s
C : ℝ
hC : ∀ (y : Y), ‖f p y‖ ≤ C
this : IntegrableOn (fun y ↦ ‖L‖ * ‖g y‖ * C) k μ
y : Y
⊢ ‖f p y‖ ≤ C

Tactic:
apply hC

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

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 : E
hle : ‖x‖ ≤ ‖y‖
h : re ⟪x, y⟫ = ‖y‖ ^ 2
⊢ x = y

Tactic:
suffices H : re ⟪x - y, x - y⟫ ≤ 0 by rwa [re_inner_self_nonpos, sub_eq_zero] at H
  have H₁ : ‖x‖ ^ 2 ≤ ‖y‖ ^ 2 := by gcongr
  have H₂ : re ⟪y, x⟫ = ‖y‖ ^ 2 := by rwa [← inner_conj_symm, conj_re]
  simp only [inner_sub_left, inner_sub_right]
  simpa [h, H₂] using H₁

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
η : Type u_5
s : η → Set E
hs : Directed (fun x1 x2 ↦ x1 ⊆ x2) s
h : ∀ (i : η), Orthonormal 𝕜 fun x ↦ ↑x
x : E
i : η
hxi : x ∈ (fun i ↦ s i) i
y : E
j : η
hyj : y ∈ (fun i ↦ s i) j
⊢ ⟪x, y⟫ = if x = y then 1 else 0

Tactic:
obtain ⟨k, hik, hjk⟩ := hs i j

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
η : Type u_5
s : η → Set E
hs : Directed (fun x1 x2 ↦ x1 ⊆ x2) s
h : ∀ (i : η), Orthonormal 𝕜 fun x ↦ ↑x
x : E
i : η
hxi : x ∈ (fun i ↦ s i) i
y : E
j : η
hyj : y ∈ (fun i ↦ s i) j
k : η
hik : s i ⊆ s k
hjk : s j ⊆ s k
⊢ ⟪x, y⟫ = if x = y then 1 else 0
Module: Mathlib.Analysis.InnerProductSpace.Orthogonal

Goal state before:
pp: 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
f : E →ₗᵢ[𝕜] F
U V : Submodule 𝕜 F
h : U ⟂ V
⊢ Submodule.comap (↑f) U ⟂ Submodule.comap (↑f) V

Tactic:
isOrtho_iff_inner_eq

Goal state after:
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
f : E →ₗᵢ[𝕜] F
U V : Submodule 𝕜 F
h : ∀ u ∈ U, ∀ v ∈ V, ⟪u, v⟫ = 0
⊢ ∀ u ∈ Submodule.comap (↑f) U, ∀ v ∈ Submodule.comap (↑f) V, ⟪u, v⟫ = 0
Module: Mathlib.Analysis.InnerProductSpace.Symmetric

Goal state before:
pp: case refine_2
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
U V : Submodule 𝕜 E
hUV : IsCompl U V
h : U ⟂ V
x y : E
⊢ ⟪(U.subtype ∘ₗ U.projectionOnto V hUV) x, (U.projection V hUV) y + (V.projection U ⋯) y⟫ =
    ⟪(U.projection V hUV) x + (V.projection U ⋯) x, (U.subtype ∘ₗ U.projectionOnto V hUV) y⟫

Tactic:
isOrtho_iff_inner_eq

Goal state after:
case refine_2
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
U V : Submodule 𝕜 E
hUV : IsCompl U V
h : ∀ u ∈ U, ∀ v ∈ V, ⟪u, v⟫ = 0
x y : E
⊢ ⟪(U.subtype ∘ₗ U.projectionOnto V hUV) x, (U.projection V hUV) y + (V.projection U ⋯) y⟫ =
    ⟪(U.projection V hUV) x + (V.projection U ⋯) x, (U.subtype ∘ₗ U.projectionOnto V hUV) y⟫
Module: Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional

Goal state before:
pp: F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace ℝ F
inst✝ : FiniteDimensional ℝ F
n : ℕ
IH :
  ∀ (φ : F ≃ₗᵢ[ℝ] F),
    finrank ℝ ↥(↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).kerᗮ ≤ n →
      ∃ l, l.length ≤ n ∧ φ = (List.map (fun v ↦ (ℝ ∙ v)ᗮ.reflection) l).prod
φ : F ≃ₗᵢ[ℝ] F
hn : finrank ℝ ↥(↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).kerᗮ ≤ n + 1
W : Submodule ℝ F := (↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).ker
hW : ∀ w ∈ W, φ w = w
hn' : ¬finrank ℝ ↥Wᗮ ≤ n
this : Nontrivial ↥Wᗮ
v : ↥Wᗮ
hv : v ≠ 0
w : F
hw : w ∈ W
⊢ ⟪w, ↑v⟫_ℝ = 0

Tactic:
exact v.prop w hw

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional

Goal state before:
pp: F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace ℝ F
inst✝ : FiniteDimensional ℝ F
n : ℕ
IH :
  ∀ (φ : F ≃ₗᵢ[ℝ] F),
    finrank ℝ ↥(↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).kerᗮ ≤ n →
      ∃ l, l.length ≤ n ∧ φ = (List.map (fun v ↦ (ℝ ∙ v)ᗮ.reflection) l).prod
φ : F ≃ₗᵢ[ℝ] F
hn : finrank ℝ ↥(↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).kerᗮ ≤ n + 1
W : Submodule ℝ F := (↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).ker
hW : ∀ w ∈ W, φ w = w
hn' : ¬finrank ℝ ↥Wᗮ ≤ n
this : Nontrivial ↥Wᗮ
v : ↥Wᗮ
hv : v ≠ 0
hφv : φ ↑v ∈ Wᗮ
hv' : ↑v ∉ W
x : F := ↑v - φ ↑v
ρ : F ≃ₗᵢ[ℝ] F := (ℝ ∙ x)ᗮ.reflection
V : Submodule ℝ F := (↑(ContinuousLinearMap.id ℝ F - ↑↑(φ.trans ρ))).ker
w : F
hw : ρ (φ w) = w
⊢ w ∈ V

Tactic:
change w - ρ (φ w) = 0

Goal state after:
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace ℝ F
inst✝ : FiniteDimensional ℝ F
n : ℕ
IH :
  ∀ (φ : F ≃ₗᵢ[ℝ] F),
    finrank ℝ ↥(↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).kerᗮ ≤ n →
      ∃ l, l.length ≤ n ∧ φ = (List.map (fun v ↦ (ℝ ∙ v)ᗮ.reflection) l).prod
φ : F ≃ₗᵢ[ℝ] F
hn : finrank ℝ ↥(↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).kerᗮ ≤ n + 1
W : Submodule ℝ F := (↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).ker
hW : ∀ w ∈ W, φ w = w
hn' : ¬finrank ℝ ↥Wᗮ ≤ n
this : Nontrivial ↥Wᗮ
v : ↥Wᗮ
hv : v ≠ 0
hφv : φ ↑v ∈ Wᗮ
hv' : ↑v ∉ W
x : F := ↑v - φ ↑v
ρ : F ≃ₗᵢ[ℝ] F := (ℝ ∙ x)ᗮ.reflection
V : Submodule ℝ F := (↑(ContinuousLinearMap.id ℝ F - ↑↑(φ.trans ρ))).ker
w : F
hw : ρ (φ w) = w
⊢ w - ρ (φ w) = 0
Module: Mathlib.Analysis.Normed.Operator.Banach

Goal state before:
pp: case inr
𝕜 : Type u_1
𝕜' : Type u_2
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NontriviallyNormedField 𝕜'
σ : 𝕜 →+* 𝕜'
E : Type u_3
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜' F
f : E →SL[σ] F
σ' : 𝕜' →+* 𝕜
inst✝³ : RingHomInvPair σ σ'
inst✝² : RingHomIsometric σ
inst✝¹ : RingHomIsometric σ'
inst✝ : CompleteSpace F
surj : Surjective ⇑f
A : ⋃ n, closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (⇑f '' ball 0 ↑n) = Set.univ
n : ℕ
a : F
ε : ℝ
εpos : ε > 0
H : ball a ε ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (⇑f '' ball 0 ↑n)
c : 𝕜
hc : 1 < ‖c‖
y : F
hy : y ≠ 0
⊢ ∃ x, dist (f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖

Tactic:
· have hc' : 1 < ‖σ c‖ := by simp only [RingHomIsometric.norm_map, hc]
    rcases rescale_to_shell hc' (half_pos εpos) hy with ⟨d, hd, ydlt, -, dinv⟩
    let δ := ‖d‖ * ‖y‖ / 4
    have δpos : 0 < δ := by positivity
    have : a + d • y ∈ ball a ε := by
      simp [dist_eq_norm, lt_of_le_of_lt ydlt.le (half_lt_self εpos)]
    rcases Metric.mem_closure_iff.1 (H this) _ δpos with ⟨z₁, z₁im, h₁⟩
    rcases (mem_image _ _ _).1 z₁im with ⟨x₁, hx₁, xz₁⟩
    rw [← xz₁] at h₁
    rw [mem_ball, dist_eq_norm, sub_zero] at hx₁
    have : a ∈ ball a ε := by
      simp only [mem_ball, dist_self]
      exact εpos
    rcases Metric.mem_closure_iff.1 (H this) _ δpos with ⟨z₂, z₂im, h₂⟩
    rcases (mem_image _ _ _).1 z₂im with ⟨x₂, hx₂, xz₂⟩
    rw [← xz₂] at h₂
    rw [mem_ball, dist_eq_norm, sub_zero] at hx₂
    let x := x₁ - x₂
    have I : ‖f x - d • y‖ ≤ 2 * δ :=
      calc
        ‖f x - d • y‖ = ‖f x₁ - (a + d • y) - (f x₂ - a)‖ := by
          congr 1
          simp only [x, f.map_sub]
          abel
        _ ≤ ‖f x₁ - (a + d • y)‖ + ‖f x₂ - a‖ := norm_sub_le _ _
        _ ≤ 2 * δ := by grind [dist_eq_norm']
    have J : ‖f (σ' d⁻¹ • x) - y‖ ≤ 1 / 2 * ‖y‖ :=
      calc
        ‖f (σ' d⁻¹ • x) - y‖ = ‖d⁻¹ • f x - (d⁻¹ * d) • y‖ := by
          rwa [f.map_smulₛₗ _, inv_mul_cancel₀, one_smul, map_inv₀, map_inv₀,
            RingHomCompTriple.comp_apply, RingHom.id_apply]
        _ = ‖d⁻¹ • (f x - d • y)‖ := by rw [mul_smul, smul_sub]
        _ = ‖d‖⁻¹ * ‖f x - d • y‖ := by rw [norm_smul, norm_inv]
        _ ≤ ‖d‖⁻¹ * (2 * δ) := by gcongr
        _ = 1 / 2 * ‖y‖ := by simp [δ, field]; norm_num
    rw [← dist_eq_norm] at J
    have K : ‖σ' d⁻¹ • x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖ :=
      calc
        ‖σ' d⁻¹ • x‖ = ‖d‖⁻¹ * ‖x₁ - x₂‖ := by rw [norm_smul, RingHomIsometric.norm_map, norm_inv]
        _ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖ * (n + n) := by
          gcongr
          · simpa using dinv
          · exact le_trans (norm_sub_le _ _) (by gcongr)
        _ = (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖ := by ring
    exact ⟨σ' d⁻¹ • x, J, K⟩

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
v : Set E
hv : Orthonormal 𝕜 Subtype.val
x : E
hx' : x ∈ (span 𝕜 v)ᗮ
hx : x ≠ 0
e : E := (↑‖x‖)⁻¹ • x
he : ‖e‖ = 1
he' : e ∈ (span 𝕜 v)ᗮ
he'' : e ∉ v
a : E
ha : a ∈ v
⊢ ⟪a, e⟫_𝕜 = 0

Tactic:
exact he' a (Submodule.subset_span ha)

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.ProdLp

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝¹ : SeminormedAddCommGroup α
inst✝ : SeminormedAddCommGroup β
x : WithLp 2 (α × β)
⊢ ‖x‖ = √(‖x.fst‖ ^ 2 + ‖x.snd‖ ^ 2)

Tactic:
prod_norm_eq_of_nat 2 (by norm_cast) _,

Goal state after:
α : Type u_2
β : Type u_3
inst✝¹ : SeminormedAddCommGroup α
inst✝ : SeminormedAddCommGroup β
x : WithLp 2 (α × β)
⊢ (‖x.fst‖ ^ 2 + ‖x.snd‖ ^ 2) ^ (1 / ↑2) = √(‖x.fst‖ ^ 2 + ‖x.snd‖ ^ 2)
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: p : ℝ≥0∞
ι : Type u_2
β : ι → Type u_4
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → Zero (β i)
i i' : ι
h : i ≠ i'
a : β i
⊢ (single p i a).ofLp i' = 0

Tactic:
by
  rw [ofLp_single, Pi.single_eq_of_ne' h]

Goal state after:
Module: Mathlib.LinearAlgebra.UnitaryGroup

Goal state before:
pp: n : Type u
inst✝³ : DecidableEq n
inst✝² : Fintype n
α : Type v
inst✝¹ : CommRing α
inst✝ : StarRing α
A✝ : Matrix n n α
A : ↥(specialUnitaryGroup n α)
⊢ star ↑A ∈ ↑(unitaryGroup n α)

Tactic:
by simpa using A.prop.1

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Basic

Goal state before:
pp: a b : ℝ
⊢ volume (uIoc a b) = ofReal |b - a|

Tactic:
by
  simp [uIoc, volume_Ioc, max_sub_min_eq_abs]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Basic

Goal state before:
pp: ι : Type u_1
inst✝ : Fintype ι
a b : ι → ℝ
⊢ volume (Icc a b) = ∏ i, ofReal (b i - a i)

Tactic:
by
  rw [← pi_univ_Icc, volume_pi_pi]
  simp only [Real.volume_Icc]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Basic

Goal state before:
pp: ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
⊢ ∏ _i, ↑1 * ediam s = ediam s ^ Fintype.card ι

Tactic:
simp only [ENNReal.coe_one, one_mul, Finset.prod_const, Fintype.card]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Basic

Goal state before:
pp: ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
D : ι → ℝ
s : ι → Set ℝ
hs : ∀ (i : ι), MeasurableSet (s i)
this : (⇑(toLin' (Matrix.diagonal D)) ⁻¹' univ.pi fun i ↦ s i) = univ.pi fun i ↦ (fun x ↦ D i * x) ⁻¹' s i
i : ι
h : ¬∏ i, D i = 0
⊢ D i ≠ 0

Tactic:
exact Finset.prod_ne_zero_iff.1 h i (Finset.mem_univ i)

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: case inr
p✝ : ℝ≥0∞
𝕜 : Type u_1
ι✝ : Type u_2
α✝ : ι✝ → Type u_3
β : ι✝ → Type u_4
hp✝ : Fact (1 ≤ p✝)
inst✝¹⁰ : Fintype ι✝
inst✝⁹ : Semiring 𝕜
inst✝⁸ : (i : ι✝) → SeminormedAddCommGroup (α✝ i)
inst✝⁷ : (i : ι✝) → SeminormedAddCommGroup (β i)
inst✝⁶ : (i : ι✝) → Module 𝕜 (α✝ i)
inst✝⁵ : (i : ι✝) → Module 𝕜 (β i)
c : 𝕜
ι : Type u_5
κ : Type u_6
p : ℝ≥0∞
α : ι ⊕ κ → Type u_7
inst✝⁴ : Fintype ι
inst✝³ : Fintype κ
inst✝² : Fact (1 ≤ p)
inst✝¹ : (i : ι ⊕ κ) → SeminormedAddCommGroup (α i)
inst✝ : (i : ι ⊕ κ) → Module 𝕜 (α i)
x : (i : ι ⊕ κ) → α i
hp : 1 ≤ p.toReal
⊢ ‖(WithLp.linearEquiv p 𝕜 ((i : ι ⊕ κ) → α i) ≪≫ₗ LinearEquiv.sumPiEquivProdPi 𝕜 ι κ α ≪≫ₗ
            (WithLp.linearEquiv p 𝕜 ((s : ι) → α (Sum.inl s))).symm.prodCongr
              (WithLp.linearEquiv p 𝕜 ((t : κ) → α (Sum.inr t))).symm ≪≫ₗ
          (WithLp.linearEquiv p 𝕜 (WithLp p ((s : ι) → α (Sum.inl s)) × WithLp p ((t : κ) → α (Sum.inr t)))).symm)
        ((WithLp.linearEquiv p 𝕜 ((i : ι ⊕ κ) → α i)).symm x)‖ =
    ‖(WithLp.linearEquiv p 𝕜 ((i : ι ⊕ κ) → α i)).symm x‖

Tactic:
· have : 0 < p.toReal := by positivity
      have hpt : p ≠ ⊤ := (toReal_pos_iff_ne_top p).mp this
      simp_rw [← coe_nnnorm]; congr 1 -- convert to nnnorm to avoid needing positivity arguments
      simp [nnnorm_eq_sum hpt, WithLp.prod_nnnorm_eq_add hpt, NNReal.rpow_inv_rpow this.ne']

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar

Goal state before:
pp: E : Type u_1
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
s : Set E
u : ℕ → E
hu : Bornology.IsBounded (range u)
hs : Pairwise (Disjoint on fun n ↦ {u n} + s)
h's : MeasurableSet s
H : ∀ (R : ℝ), μ (s ∩ closedBall 0 R) = 0
⊢ μ s = 0

Tactic:
by
    rw [← nonpos_iff_eq_zero]
    calc
      μ s ≤ ∑' n : ℕ, μ (s ∩ closedBall 0 n) := by
        conv_lhs => rw [← iUnion_inter_closedBall_nat s 0]
        exact measure_iUnion_le _
      _ = 0 := by simp only [H, tsum_zero]

Goal state after:
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁶ : NormedField K
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace K E
b : Basis ι K E
inst✝³ : LinearOrder K
inst✝² : IsStrictOrderedRing K
inst✝¹ : FloorRing K
inst✝ : Fintype ι
m v : E
h : v ∈ span ℤ (Set.range ⇑b)
⊢ fract b (v + m) = fract b m

Tactic:
by
  refine (Basis.ext_elem_iff b).mpr fun i => ?_
  simp_rw [repr_fract_apply, Int.fract_eq_fract]
  use (b.restrictScalars ℤ).repr ⟨v, h⟩ i
  rw [map_add, Finsupp.coe_add, Pi.add_apply, add_tsub_cancel_right,
    ← eq_intCast (algebraMap ℤ K) _, Basis.restrictScalars_repr_apply, coe_mk]

Goal state after:
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁷ : NormedField K
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace K E
b : Basis ι K E
inst✝⁴ : LinearOrder K
inst✝³ : IsStrictOrderedRing K
inst✝² : FloorRing K
inst✝¹ : Fintype ι
inst✝ : HasSolidNorm K
m : E
i : ι
x✝ : i ∈ Finset.univ
⊢ Int.fract ((b.repr m) i) ≤ 1

Tactic:
exact le_of_lt (Int.fract_lt_one _)

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Basic

Goal state before:
pp: case pos
μ : Measure ℝ
inst✝ : NullSingletonClass μ
s : Set ℝ
p : ℝ → Prop
h : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ.restrict (s ∩ Ioo a b), p x
T : ↑s × ↑s → Set ℝ := fun p ↦ Ioo ↑p.1 ↑p.2
u : Set ℝ := ⋃ i, T i
hfinite : (s \ u).Finite
A : Set (↑s × ↑s)
A_count : A.Countable
hA : ⋃ i ∈ A, T i = ⋃ i, T i
x : ℝ
hx : x ∈ s
h'x : x ∈ ⋃ i ∈ A, T i
⊢ x ∈ s \ u ∪ ⋃ p ∈ A, s ∩ T p

Tactic:
obtain ⟨p, pA, xp⟩ : ∃ p : ↥s × ↥s, p ∈ A ∧ x ∈ T p := by
        simpa only [mem_iUnion, exists_prop, SetCoe.exists, exists_and_right] using h'x

Goal state after:
case pos
μ : Measure ℝ
inst✝ : NullSingletonClass μ
s : Set ℝ
p✝ : ℝ → Prop
h : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ.restrict (s ∩ Ioo a b), p✝ x
T : ↑s × ↑s → Set ℝ := fun p ↦ Ioo ↑p.1 ↑p.2
u : Set ℝ := ⋃ i, T i
hfinite : (s \ u).Finite
A : Set (↑s × ↑s)
A_count : A.Countable
hA : ⋃ i ∈ A, T i = ⋃ i, T i
x : ℝ
hx : x ∈ s
h'x : x ∈ ⋃ i ∈ A, T i
p : ↑s × ↑s
pA : p ∈ A
xp : x ∈ T p
⊢ x ∈ s \ u ∪ ⋃ p ∈ A, s ∩ T p
Module: Mathlib.Analysis.BoxIntegral.Box.Basic

Goal state before:
pp: case coe.bot
ι : Type u_1
a✝ : Box ι
⊢ ↑↑a✝ ⊆ ↑⊥ ↔ ↑a✝ ≤ ⊥

Tactic:
simp [subset_empty_iff]

Goal state after: