mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Distribution.ContDiffMapSupportedIn

Goal state before:
pp: case pos
𝕜 : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : SMulCommClass ℝ 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedSpace 𝕜 F'
inst✝ : SMulCommClass ℝ 𝕜 F'
n n₁ n₂ k : ℕ∞
K K₁ K₂ : Compacts E
c : 𝕜
f : 𝓓^{n₁}_{K₁}(E, F)
h✝ : n₂ ≤ n₁ ∧ K₁ ≤ K₂
a✝ : E
⊢ (ContDiffMapSupportedIn.of_support_subset ⋯ ⋯) a✝ =
    ((RingHom.id 𝕜) c • ContDiffMapSupportedIn.of_support_subset ⋯ ⋯) a✝

Tactic:
simp

Goal state after:
Module: Mathlib.Analysis.Distribution.ContDiffMapSupportedIn

Goal state before:
pp: case neg
𝕜 : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : SMulCommClass ℝ 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedSpace 𝕜 F'
inst✝ : SMulCommClass ℝ 𝕜 F'
n n₁ n₂ k : ℕ∞
K K₁ K₂ : Compacts E
c : 𝕜
f : 𝓓^{n₁}_{K₁}(E, F)
h✝ : ¬(n₂ ≤ n₁ ∧ K₁ ≤ K₂)
a✝ : E
⊢ 0 a✝ = ((RingHom.id 𝕜) c • 0) a✝

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
T : E →ₗ[𝕜] E
inst✝ : FiniteDimensional 𝕜 E
n : ℕ
hT : T.IsSymmetric
hn : finrank 𝕜 E = n
μ : 𝕜
⊢ ∀ (i : Fin n),
    i ∈ {i | ↑((hT.unsortedEigenvalues hn ∘ ⇑(Tuple.sort (hT.unsortedEigenvalues hn)) ∘ ⇑Fin.revPerm) i) = μ} ↔
      (Equiv.trans Fin.revPerm (Tuple.sort (hT.unsortedEigenvalues hn))) i ∈ {i | ↑(hT.unsortedEigenvalues hn i) = μ}

Tactic:
simp

Goal state after:
Module: Mathlib.Analysis.Distribution.ContDiffMapSupportedIn

Goal state before:
pp: case neg
𝕜 : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : SMulCommClass ℝ 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedSpace 𝕜 F'
inst✝ : SMulCommClass ℝ 𝕜 F'
n n₁ n₂ k : ℕ∞
K K₁ K₂ : Compacts E
f g : 𝓓^{n}_{K}(E, F)
hk : ¬k + 1 ≤ n
⊢ 0 = 0 + 0

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
T : E →ₗ[𝕜] E
inst✝ : FiniteDimensional 𝕜 E
n : ℕ
hT : T.IsSymmetric
hn : finrank 𝕜 E = n
⊢ (Finset.univ.val.bind fun i ↦ (Polynomial.X - Polynomial.C ((RCLike.ofReal ∘ hT.eigenvalues hn) i)).roots) =
    Multiset.map (RCLike.ofReal ∘ hT.eigenvalues hn) Finset.univ.val

Tactic:
simp

Goal state after:
Module: Mathlib.Analysis.Distribution.ContDiffMapSupportedIn

Goal state before:
pp: case neg
𝕜 : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : SMulCommClass ℝ 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedSpace 𝕜 F'
inst✝ : SMulCommClass ℝ 𝕜 F'
n n₁ n₂ k : ℕ∞
K K₁ K₂ : Compacts E
c : 𝕜
f : 𝓓^{n}_{K}(E, F)
hk : ¬k + 1 ≤ n
⊢ 0 = (RingHom.id 𝕜) c • 0

Tactic:
simp

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

Goal state before:
pp: case convert_4
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
T : E →ₗ[𝕜] E
inst✝ : FiniteDimensional 𝕜 E
n : ℕ
hT : T.IsSymmetric
hn : finrank 𝕜 E = n
this : n = finrank 𝕜 E
⊢ List.Pairwise (fun a b ↦ a ≥ b) (List.ofFn fun x ↦ hT.eigenvalues ⋯ x)

Tactic:
← List.sortedGE_iff_pairwise

Goal state after:
case convert_4
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
T : E →ₗ[𝕜] E
inst✝ : FiniteDimensional 𝕜 E
n : ℕ
hT : T.IsSymmetric
hn : finrank 𝕜 E = n
this : n = finrank 𝕜 E
⊢ (List.ofFn fun x ↦ hT.eigenvalues ⋯ x).SortedGE
Module: Mathlib.Analysis.Distribution.ContDiffMapSupportedIn

Goal state before:
pp: case neg
𝕜 : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : SMulCommClass ℝ 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedSpace 𝕜 F'
inst✝ : SMulCommClass ℝ 𝕜 F'
n n₁ n₂ k : ℕ∞
K K₁ K₂ : Compacts E
i : ℕ
f g : 𝓓^{n}_{K}(E, F)
hi : ¬k + ↑i ≤ n
⊢ 0 = 0 + 0

Tactic:
simp

Goal state after:
Module: Mathlib.Analysis.Distribution.ContDiffMapSupportedIn

Goal state before:
pp: case neg
𝕜 : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : SMulCommClass ℝ 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedSpace 𝕜 F'
inst✝ : SMulCommClass ℝ 𝕜 F'
n n₁ n₂ k : ℕ∞
K K₁ K₂ : Compacts E
i : ℕ
c : 𝕜
f : 𝓓^{n}_{K}(E, F)
hi : ¬k + ↑i ≤ n
⊢ 0 = (RingHom.id 𝕜) c • 0

Tactic:
simp

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

Goal state before:
pp: case inr.inr
E : Type u_1
n : Type u_2
𝕜 : Type u_4
inst✝³ : RCLike 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : DecidableEq n
i : n
x : E
⊢ gram 𝕜 (Pi.single i x) i i = single i i (⟪x, x⟫_𝕜) i i

Tactic:
simp

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

Goal state before:
pp: E : Type u_1
n : Type u_2
𝕜 : Type u_4
inst✝³ : RCLike 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : Finite n
v : n → E
this : Fintype n
x : n → 𝕜
⊢ re 0 ≤ re ⟪∑ i, x i • v i, ∑ i, x i • v i⟫_𝕜 ∧ im 0 = im ⟪∑ i, x i • v i, ∑ i, x i • v i⟫_𝕜

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
n : Type u_2
inst✝¹ : RCLike 𝕜
inst✝ : Fintype n
M : Matrix n n 𝕜
hM : M.PosSemidef
⊢ ∀ (x y : Matrix n n 𝕜) (r : 𝕜), (y * M * (r • x)ᴴ).trace = (starRingEnd 𝕜) r * (y * M * xᴴ).trace

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
n : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA✝ : IsSelfAdjoint A
hA : A.IsHermitian
f : C(↑(spectrum ℝ A), ℝ)
x✝ : n
⊢ (RCLike.ofReal ∘ 0) x✝ = 0 x✝

Tactic:
simp

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs

Goal state before:
pp: A : Type u_2
inst✝¹⁶ : NonUnitalRing A
inst✝¹⁵ : StarRing A
inst✝¹⁴ : TopologicalSpace A
inst✝¹³ : Module ℝ A
inst✝¹² : SMulCommClass ℝ A A
inst✝¹¹ : IsScalarTower ℝ A A
inst✝¹⁰ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁹ : PartialOrder A
inst✝⁸ : StarOrderedRing A
inst✝⁷ : NonnegSpectrumClass ℝ A
inst✝⁶ : IsTopologicalRing A
inst✝⁵ : T2Space A
inst✝⁴ : StarModule ℝ A
R : Type u_3
inst✝³ : Semiring R
inst✝² : SMulWithZero R ℝ≥0
inst✝¹ : SMul R A
inst✝ : IsScalarTower R ℝ≥0 A
r : R
a : A
⊢ abs (r • a) = r • abs a

Tactic:
by
  suffices ∀ r : ℝ≥0, abs (r • a) = r • abs a by simpa using this (r • 1)
  intro r
  rw [abs, sqrt_eq_iff _ _ (star_mul_self_nonneg _) (smul_nonneg (by positivity) (abs_nonneg _))]
  simp [mul_smul_comm, smul_mul_assoc, abs_mul_abs]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NonUnitalRing A
inst✝⁷ : TopologicalSpace A
inst✝⁶ : Module 𝕜 A
inst✝⁵ : StarRing A
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : NonUnitalContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
x✝¹ : 𝕜
x✝ : x✝¹ ∈ quasispectrum 𝕜 a
⊢ 0 ≤ ↑‖f x✝¹‖

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Function.L2Space

Goal state before:
pp: case e'_2.e'_6
α : Type u_1
F : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup F
f : α → F
hf : AEStronglyMeasurable f μ
x✝ : α
⊢ ‖f x✝‖ ^ 2 = ‖f x✝‖ ^ ENNReal.toReal 2

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Function.L2Space

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : AEStronglyMeasurable f μ
x✝ : α
⊢ f x✝ ^ 2 = ‖f x✝‖ ^ 2

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Function.L2Space

Goal state before:
pp: α : Type u_1
F : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup F
f : ↥(Lp F 2 μ)
⊢ ENNReal.ofReal 2 = 2

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Function.L2Space

Goal state before:
pp: α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝² : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f : ↥(Lp E 2 μ)
x : α
⊢ 2 = ↑2

Tactic:
simp

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs

Goal state before:
pp: A : Type u_2
inst✝⁹ : Ring A
inst✝⁸ : StarRing A
inst✝⁷ : TopologicalSpace A
inst✝⁶ : Algebra ℝ A
inst✝⁵ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsTopologicalRing A
inst✝ : T2Space A
⊢ abs 1 = 1

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Function.L2Space

Goal state before:
pp: α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝² : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f : ↥(Lp E 2 μ)
⊢ ENNReal.toReal 2 = 2

Tactic:
simp

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs

Goal state before:
pp: case ofNat
A : Type u_2
inst✝¹⁰ : Ring A
inst✝⁹ : StarRing A
inst✝⁸ : TopologicalSpace A
inst✝⁷ : Algebra ℝ A
inst✝⁶ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing A
inst✝³ : NonnegSpectrumClass ℝ A
inst✝² : IsTopologicalRing A
inst✝¹ : T2Space A
inst✝ : StarModule ℝ A
a✝ : ℕ
⊢ abs ↑(Int.ofNat a✝) = ↑|Int.ofNat a✝|

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Function.L2Space

Goal state before:
pp: α : Type u_1
𝕜 : Type u_4
inst✝ : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
hμs : μ s ≠ ∞
f : ↥(Lp 𝕜 2 μ)
⊢ ⟪1, ∫ (x : α) in s, ↑↑f x ∂μ⟫ = ∫ (x : α) in s, ↑↑f x ∂μ

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Function.L2Space

Goal state before:
pp: α : Type u_1
𝕜 : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
inst✝¹ : RCLike 𝕜
μ : Measure α
inst✝ : IsFiniteMeasure μ
f g : α →ᵇ 𝕜
hf_ae : ↑↑((BoundedContinuousFunction.toLp 2 μ 𝕜) f) =ᵐ[μ] ⇑f
hg_ae : ↑↑((BoundedContinuousFunction.toLp 2 μ 𝕜) g) =ᵐ[μ] ⇑g
a✝ : α
⊢ ↑↑((BoundedContinuousFunction.toLp 2 μ 𝕜) f) a✝ = f a✝ →
    ↑↑((BoundedContinuousFunction.toLp 2 μ 𝕜) g) a✝ = g a✝ →
      ⟪↑↑((BoundedContinuousFunction.toLp 2 μ 𝕜) f) a✝, ↑↑((BoundedContinuousFunction.toLp 2 μ 𝕜) g) a✝⟫ =
        g a✝ * (starRingEnd 𝕜) (f a✝)

Tactic:
hf

Goal state after:
α : Type u_1
𝕜 : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
inst✝¹ : RCLike 𝕜
μ : Measure α
inst✝ : IsFiniteMeasure μ
f g : α →ᵇ 𝕜
hf_ae : ↑↑((BoundedContinuousFunction.toLp 2 μ 𝕜) f) =ᵐ[μ] ⇑f
hg_ae : ↑↑((BoundedContinuousFunction.toLp 2 μ 𝕜) g) =ᵐ[μ] ⇑g
a✝ : α
hf : ↑↑((BoundedContinuousFunction.toLp 2 μ 𝕜) f) a✝ = f a✝
⊢ ↑↑((BoundedContinuousFunction.toLp 2 μ 𝕜) g) a✝ = g a✝ →
    ⟪↑↑((BoundedContinuousFunction.toLp 2 μ 𝕜) f) a✝, ↑↑((BoundedContinuousFunction.toLp 2 μ 𝕜) g) a✝⟫ =
      g a✝ * (starRingEnd 𝕜) (f a✝)
Module: Mathlib.Analysis.Distribution.ContDiffMapSupportedIn

Goal state before:
pp: case neg
𝕜 : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
inst✝²¹ : NontriviallyNormedField 𝕜
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
inst✝¹⁸ : NormedAddCommGroup F
inst✝¹⁷ : NormedSpace ℝ F
inst✝¹⁶ : NormedSpace 𝕜 F
inst✝¹⁵ : SMulCommClass ℝ 𝕜 F
inst✝¹⁴ : NormedAddCommGroup F'
inst✝¹³ : NormedSpace ℝ F'
inst✝¹² : NormedSpace 𝕜 F'
inst✝¹¹ : SMulCommClass ℝ 𝕜 F'
n n₁ n₂ k : ℕ∞
K K₁ K₂ : Compacts E
m : MeasurableSpace E
inst✝¹⁰ : OpensMeasurableSpace E
F₁ : Type u_5
F₂ : Type u_6
F₃ : Type u_7
inst✝⁹ : NormedAddCommGroup F₁
inst✝⁸ : NormedSpace 𝕜 F₁
inst✝⁷ : NormedSpace ℝ F₁
inst✝⁶ : NormedAddCommGroup F₂
inst✝⁵ : NormedSpace 𝕜 F₂
inst✝⁴ : NormedAddCommGroup F₃
inst✝³ : NormedSpace 𝕜 F₃
inst✝² : SMulCommClass ℝ 𝕜 F₁
inst✝¹ : NormedSpace ℝ F₃
inst✝ : SMulCommClass ℝ 𝕜 F₃
B : F₁ →L[𝕜] F₂ →L[𝕜] F₃
μ : Measure E
φ : E → F₂
f g : 𝓓^{n}_{K}(E, F₁)
hφ : ¬IntegrableOn φ (↑K) μ
⊢ 0 = 0 + 0

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Function.L2Space

Goal state before:
pp: α : Type u_1
𝕜 : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
inst✝¹ : RCLike 𝕜
μ : Measure α
inst✝ : IsFiniteMeasure μ
f g : α →ᵇ 𝕜
hf_ae : ↑↑((BoundedContinuousFunction.toLp 2 μ 𝕜) f) =ᵐ[μ] ⇑f
hg_ae : ↑↑((BoundedContinuousFunction.toLp 2 μ 𝕜) g) =ᵐ[μ] ⇑g
a✝ : α
hf : ↑↑((BoundedContinuousFunction.toLp 2 μ 𝕜) f) a✝ = f a✝
hg : ↑↑((BoundedContinuousFunction.toLp 2 μ 𝕜) g) a✝ = g a✝
⊢ ⟪f a✝, g a✝⟫ = g a✝ * (starRingEnd 𝕜) (f a✝)

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Function.L2Space

Goal state before:
pp: α : Type u_1
𝕜 : Type u_2
inst✝⁵ : TopologicalSpace α
inst✝⁴ : MeasurableSpace α
inst✝³ : BorelSpace α
inst✝² : RCLike 𝕜
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : CompactSpace α
f g : C(α, 𝕜)
hf_ae : ↑↑((ContinuousMap.toLp 2 μ 𝕜) f) =ᵐ[μ] ⇑f
hg_ae : ↑↑((ContinuousMap.toLp 2 μ 𝕜) g) =ᵐ[μ] ⇑g
a✝ : α
⊢ ↑↑((ContinuousMap.toLp 2 μ 𝕜) f) a✝ = f a✝ →
    ↑↑((ContinuousMap.toLp 2 μ 𝕜) g) a✝ = g a✝ →
      ⟪↑↑((ContinuousMap.toLp 2 μ 𝕜) f) a✝, ↑↑((ContinuousMap.toLp 2 μ 𝕜) g) a✝⟫ = g a✝ * (starRingEnd 𝕜) (f a✝)

Tactic:
hf

Goal state after:
α : Type u_1
𝕜 : Type u_2
inst✝⁵ : TopologicalSpace α
inst✝⁴ : MeasurableSpace α
inst✝³ : BorelSpace α
inst✝² : RCLike 𝕜
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : CompactSpace α
f g : C(α, 𝕜)
hf_ae : ↑↑((ContinuousMap.toLp 2 μ 𝕜) f) =ᵐ[μ] ⇑f
hg_ae : ↑↑((ContinuousMap.toLp 2 μ 𝕜) g) =ᵐ[μ] ⇑g
a✝ : α
hf : ↑↑((ContinuousMap.toLp 2 μ 𝕜) f) a✝ = f a✝
⊢ ↑↑((ContinuousMap.toLp 2 μ 𝕜) g) a✝ = g a✝ →
    ⟪↑↑((ContinuousMap.toLp 2 μ 𝕜) f) a✝, ↑↑((ContinuousMap.toLp 2 μ 𝕜) g) a✝⟫ = g a✝ * (starRingEnd 𝕜) (f a✝)
Module: Mathlib.MeasureTheory.Function.L2Space

Goal state before:
pp: α : Type u_1
𝕜 : Type u_2
inst✝⁵ : TopologicalSpace α
inst✝⁴ : MeasurableSpace α
inst✝³ : BorelSpace α
inst✝² : RCLike 𝕜
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : CompactSpace α
f g : C(α, 𝕜)
hf_ae : ↑↑((ContinuousMap.toLp 2 μ 𝕜) f) =ᵐ[μ] ⇑f
hg_ae : ↑↑((ContinuousMap.toLp 2 μ 𝕜) g) =ᵐ[μ] ⇑g
a✝ : α
hf : ↑↑((ContinuousMap.toLp 2 μ 𝕜) f) a✝ = f a✝
hg : ↑↑((ContinuousMap.toLp 2 μ 𝕜) g) a✝ = g a✝
⊢ ⟪f a✝, g a✝⟫ = g a✝ * (starRingEnd 𝕜) (f a✝)

Tactic:
simp

Goal state after:
Module: Mathlib.Analysis.Distribution.ContDiffMapSupportedIn

Goal state before:
pp: case neg
𝕜 : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
inst✝²¹ : NontriviallyNormedField 𝕜
inst✝²⁰ : NormedAddCommGroup E
inst✝¹⁹ : NormedSpace ℝ E
inst✝¹⁸ : NormedAddCommGroup F
inst✝¹⁷ : NormedSpace ℝ F
inst✝¹⁶ : NormedSpace 𝕜 F
inst✝¹⁵ : SMulCommClass ℝ 𝕜 F
inst✝¹⁴ : NormedAddCommGroup F'
inst✝¹³ : NormedSpace ℝ F'
inst✝¹² : NormedSpace 𝕜 F'
inst✝¹¹ : SMulCommClass ℝ 𝕜 F'
n n₁ n₂ k : ℕ∞
K K₁ K₂ : Compacts E
m : MeasurableSpace E
inst✝¹⁰ : OpensMeasurableSpace E
F₁ : Type u_5
F₂ : Type u_6
F₃ : Type u_7
inst✝⁹ : NormedAddCommGroup F₁
inst✝⁸ : NormedSpace 𝕜 F₁
inst✝⁷ : NormedSpace ℝ F₁
inst✝⁶ : NormedAddCommGroup F₂
inst✝⁵ : NormedSpace 𝕜 F₂
inst✝⁴ : NormedAddCommGroup F₃
inst✝³ : NormedSpace 𝕜 F₃
inst✝² : SMulCommClass ℝ 𝕜 F₁
inst✝¹ : NormedSpace ℝ F₃
inst✝ : SMulCommClass ℝ 𝕜 F₃
B : F₁ →L[𝕜] F₂ →L[𝕜] F₃
μ : Measure E
φ : E → F₂
c : 𝕜
f : 𝓓^{n}_{K}(E, F₁)
hφ : ¬IntegrableOn φ (↑K) μ
⊢ 0 = (RingHom.id 𝕜) c • 0

Tactic:
simp

Goal state after:
Module: Mathlib.Analysis.Distribution.ContDiffMapSupportedIn

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝¹³ : NontriviallyNormedField 𝕜
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
n : ℕ∞
K : Compacts E
m : MeasurableSpace E
inst✝¹⁰ : OpensMeasurableSpace E
F₁ : Type u_5
F₂ : Type u_6
F₃ : Type u_7
inst✝⁹ : NormedAddCommGroup F₁
inst✝⁸ : NormedSpace 𝕜 F₁
inst✝⁷ : NormedSpace ℝ F₁
inst✝⁶ : NormedAddCommGroup F₂
inst✝⁵ : NormedSpace 𝕜 F₂
inst✝⁴ : NormedAddCommGroup F₃
inst✝³ : NormedSpace 𝕜 F₃
inst✝² : SMulCommClass ℝ 𝕜 F₁
inst✝¹ : NormedSpace ℝ F₃
inst✝ : SMulCommClass ℝ 𝕜 F₃
B : F₁ →L[𝕜] F₂ →L[𝕜] F₃
μ : Measure E
φ : E → F₂
f : 𝓓^{n}_{K}(E, F₁)
hφ : IntegrableOn φ (↑K) μ
x : E
⊢ ‖B‖ * ‖f x‖ * ‖φ x‖ ≤ ‖φ x‖ * ‖B‖ * (ContDiffMapSupportedIn.seminorm 𝕜 E F₁ n K 0) f

Tactic:
norm_apply_le_seminorm 𝕜,

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝¹³ : NontriviallyNormedField 𝕜
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
n : ℕ∞
K : Compacts E
m : MeasurableSpace E
inst✝¹⁰ : OpensMeasurableSpace E
F₁ : Type u_5
F₂ : Type u_6
F₃ : Type u_7
inst✝⁹ : NormedAddCommGroup F₁
inst✝⁸ : NormedSpace 𝕜 F₁
inst✝⁷ : NormedSpace ℝ F₁
inst✝⁶ : NormedAddCommGroup F₂
inst✝⁵ : NormedSpace 𝕜 F₂
inst✝⁴ : NormedAddCommGroup F₃
inst✝³ : NormedSpace 𝕜 F₃
inst✝² : SMulCommClass ℝ 𝕜 F₁
inst✝¹ : NormedSpace ℝ F₃
inst✝ : SMulCommClass ℝ 𝕜 F₃
B : F₁ →L[𝕜] F₂ →L[𝕜] F₃
μ : Measure E
φ : E → F₂
f : 𝓓^{n}_{K}(E, F₁)
hφ : IntegrableOn φ (↑K) μ
x : E
⊢ ‖B‖ * (ContDiffMapSupportedIn.seminorm 𝕜 E F₁ n K 0) f * ‖φ x‖ ≤
    ‖φ x‖ * ‖B‖ * (ContDiffMapSupportedIn.seminorm 𝕜 E F₁ n K 0) f
Module: Mathlib.Analysis.Distribution.ContDiffMapSupportedIn

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
inst✝²⁰ : NontriviallyNormedField 𝕜
inst✝¹⁹ : NormedAddCommGroup E
inst✝¹⁸ : NormedSpace ℝ E
inst✝¹⁷ : NormedAddCommGroup F
inst✝¹⁶ : NormedSpace ℝ F
inst✝¹⁵ : NormedSpace 𝕜 F
inst✝¹⁴ : SMulCommClass ℝ 𝕜 F
inst✝¹³ : NormedAddCommGroup F'
inst✝¹² : NormedSpace ℝ F'
inst✝¹¹ : NormedSpace 𝕜 F'
inst✝¹⁰ : SMulCommClass ℝ 𝕜 F'
n n₁ n₂ k : ℕ∞
K K₁ K₂ : Compacts E
F₁ : Type u_5
F₂ : Type u_6
F₃ : Type u_7
G : Type u_8
inst✝⁹ : NormedAlgebra ℝ 𝕜
inst✝⁸ : NormedAddCommGroup F₁
inst✝⁷ : NormedSpace 𝕜 F₁
inst✝⁶ : NormedSpace ℝ F₁
inst✝⁵ : NormedAddCommGroup F₂
inst✝⁴ : NormedSpace 𝕜 F₂
inst✝³ : NormedSpace ℝ F₂
inst✝² : NormedAddCommGroup F₃
inst✝¹ : NormedSpace 𝕜 F₃
inst✝ : NormedSpace ℝ F₃
B : F₁ →L[𝕜] F₂ →L[𝕜] F₃
g : E → F₂
hg : ContDiff ℝ (↑n) g
f✝ g✝ : 𝓓^{n}_{K}(E, F₁)
x✝ : E
⊢ (B ((f✝ + g✝) x✝)) (g x✝) = (B (f✝ x✝)) (g x✝) + (B (g✝ x✝)) (g x✝)

Tactic:
simp

Goal state after:
Module: Mathlib.Analysis.Distribution.ContDiffMapSupportedIn

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
inst✝²⁰ : NontriviallyNormedField 𝕜
inst✝¹⁹ : NormedAddCommGroup E
inst✝¹⁸ : NormedSpace ℝ E
inst✝¹⁷ : NormedAddCommGroup F
inst✝¹⁶ : NormedSpace ℝ F
inst✝¹⁵ : NormedSpace 𝕜 F
inst✝¹⁴ : SMulCommClass ℝ 𝕜 F
inst✝¹³ : NormedAddCommGroup F'
inst✝¹² : NormedSpace ℝ F'
inst✝¹¹ : NormedSpace 𝕜 F'
inst✝¹⁰ : SMulCommClass ℝ 𝕜 F'
n n₁ n₂ k : ℕ∞
K K₁ K₂ : Compacts E
F₁ : Type u_5
F₂ : Type u_6
F₃ : Type u_7
G : Type u_8
inst✝⁹ : NormedAlgebra ℝ 𝕜
inst✝⁸ : NormedAddCommGroup F₁
inst✝⁷ : NormedSpace 𝕜 F₁
inst✝⁶ : NormedSpace ℝ F₁
inst✝⁵ : NormedAddCommGroup F₂
inst✝⁴ : NormedSpace 𝕜 F₂
inst✝³ : NormedSpace ℝ F₂
inst✝² : NormedAddCommGroup F₃
inst✝¹ : NormedSpace 𝕜 F₃
inst✝ : NormedSpace ℝ F₃
B : F₁ →L[𝕜] F₂ →L[𝕜] F₃
g : E → F₂
hg : ContDiff ℝ (↑n) g
c✝ : 𝕜
f✝ : 𝓓^{n}_{K}(E, F₁)
x✝ : E
⊢ (B ((c✝ • f✝) x✝)) (g x✝) = c✝ • (B (f✝ x✝)) (g x✝)

Tactic:
simp

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

Goal state before:
pp: E : Type u_3
F : Type u_4
inst✝⁷ : MeasurableSpace E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : BorelSpace E
inst✝¹ : NormedSpace ℝ F
μ : Measure E
inst✝ : IsFiniteMeasureOnCompacts μ
p : ℝ≥0∞
hp : p ≠ ∞
hp₂ : 1 ≤ p
f : E → F
hf : MemLp f p μ
ε : ℝ
hε : 0 < ε
hε₂ : 0 < ε / 2
hε₂' : 0 < ENNReal.ofReal (ε / 2)
g : E → F
hg₁ : HasCompactSupport g
hg₂ : eLpNorm (f - g) p μ ≤ ENNReal.ofReal (ε / 2)
hg₃ : Continuous g
hg₄ : MemLp g p μ
g' : E → F
hg'₁ : HasCompactSupport g'
hg'₂ : ContDiff ℝ ∞ g'
hg'₃ : eLpNorm (g - g') ?m.157 μ ≤ ENNReal.ofReal (ε / 2)
⊢ f - g' = f - g - (g' - g)

Tactic:
simp

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Bases

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
X : Type u_2
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
β : Type u_3
L : SummationFilter β
b : GeneralSchauderBasis β 𝕜 X L
A : Finset β
⊢ Module.finrank 𝕜 ↥(↑(b.proj A)).range = A.card

Tactic:
rw [range_proj_eq_span, Set.image_eq_range, finrank_span_eq_card]
  · exact Fintype.card_coe A
  · exact b.linearIndependent.comp (fun i : A ↦ i.val) Subtype.val_injective

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Bases

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
X : Type u_2
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
b : SchauderBasis 𝕜 X
n i : ℕ
⊢ (if i ∈ Finset.range n then ↑b i else 0) = if i < n then ↑b i else 0

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Function.ContinuousMapDense

Goal state before:
pp: α : Type u_1
inst✝⁸ : TopologicalSpace α
inst✝⁷ : NormalSpace α
inst✝⁶ : MeasurableSpace α
inst✝⁵ : BorelSpace α
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
μ : Measure α
inst✝³ : NormedSpace ℝ E
inst✝² : R1Space α
inst✝¹ : WeaklyLocallyCompactSpace α
inst✝ : μ.Regular
p : ℝ
hp : 0 < p
f : α → E
hf : MemLp f (ENNReal.ofReal p) μ
ε : ℝ
hε : 0 < ε
I : 0 < ε ^ (1 / p)
A : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0
B : ENNReal.ofReal p ≠ 0
g : α → E
g_support : HasCompactSupport g
hg : eLpNorm (f - g) (↑p.toNNReal) μ ≤ ENNReal.ofReal (ε ^ (1 / p))
g_cont : Continuous[inst✝⁸, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
g_mem : MemLp g (↑p.toNNReal) μ
⊢ ∃ g,
    HasCompactSupport g ∧
      ∫ (x : α), ‖f x - g x‖ ^ p ∂μ ≤ ε ∧
        Continuous[inst✝⁸, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g ∧ MemLp g (ENNReal.ofReal p) μ

Tactic:
change eLpNorm _ (ENNReal.ofReal p) _ ≤ _ at hg

Goal state after:
α : Type u_1
inst✝⁸ : TopologicalSpace α
inst✝⁷ : NormalSpace α
inst✝⁶ : MeasurableSpace α
inst✝⁵ : BorelSpace α
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
μ : Measure α
inst✝³ : NormedSpace ℝ E
inst✝² : R1Space α
inst✝¹ : WeaklyLocallyCompactSpace α
inst✝ : μ.Regular
p : ℝ
hp : 0 < p
f : α → E
hf : MemLp f (ENNReal.ofReal p) μ
ε : ℝ
hε : 0 < ε
I : 0 < ε ^ (1 / p)
A : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0
B : ENNReal.ofReal p ≠ 0
g : α → E
g_support : HasCompactSupport g
g_cont : Continuous[inst✝⁸, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
g_mem : MemLp g (↑p.toNNReal) μ
hg : eLpNorm (f - g) (ENNReal.ofReal p) μ ≤ ENNReal.ofReal (ε ^ (1 / p))
⊢ ∃ g,
    HasCompactSupport g ∧
      ∫ (x : α), ‖f x - g x‖ ^ p ∂μ ≤ ε ∧
        Continuous[inst✝⁸, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g ∧ MemLp g (ENNReal.ofReal p) μ
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Deriv

Goal state before:
pp: case zero
E : Type u_5
F : Type u_8
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : NormedSpace ℝ E
f : 𝓢(E, F)
m : Fin 0 → E
x : E
⊢ (∂^{m} f) x = (iteratedFDeriv ℝ 0 (⇑f) x) m

Tactic:
simp

Goal state after:
Module: Mathlib.Analysis.Distribution.SchwartzSpace.Deriv

Goal state before:
pp: case zero
E : Type u_5
F : Type u_8
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : NormedSpace ℝ E
f : 𝓢(E, F)
m : Fin 0 → E
⊢ tsupport ⇑(∂^{m} f) ⊆ tsupport ⇑f

Tactic:
simp

Goal state after: