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

Goal state before:
pp: case e'_4.e'_5
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' : ℝ≥0
hp'0 : ¬p' = 0
n : ℕ := finrank ℝ E
hn✝ : 0 < n
n' : ℝ≥0 := (↑n).conjExponent
h0n : 2 ≤ n
hn : (↑n).HolderConjugate n'
h1n : 1 ≤ ↑n
h2n : 0 < ↑n - 1
hp : 1 ≤ 1
hp' : (↑p')⁻¹ = ↑1⁻¹ - (↑n)⁻¹
h2p : ↑1 < ↑n
hnp : 0 < ↑(finrank ℝ E) - 1
⊢ eLpNormLESNormFDerivOneConst μ ↑(↑(finrank ℝ E)).conjExponent * Real.toNNReal 1 =
    eLpNormLESNormFDerivOneConst μ ↑(↑(finrank ℝ E)).conjExponent

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : E →ₗ[𝕜] E
hf : LipschitzWith 1 ⇑f
g : E →L[𝕜] ↥(f.eqLocus 1)
hg_proj : ∀ (x : ↥(f.eqLocus 1)), g ↑x = x
hg_ker : ↑(↑g).ker ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑(f - 1).range
x : E
⊢ x = x - ↑(g x) + ↑(g x)

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
f : E →L[𝕜] E
hf : ‖f‖ ≤ 1
x : E
⊢ Tendsto (fun x_1 ↦ birkhoffAverage 𝕜 (⇑f) id x_1 x) atTop (𝓝 ↑(((↑f).eqLocus ↑1).orthogonalProjectionOnto x))

Tactic:
apply (f : E →ₗ[𝕜] E).tendsto_birkhoffAverage_of_ker_subset_closure (f.lipschitz.weaken hf)
  · exact (f.eqLocus (1 : E →L[𝕜] E)).orthogonalProjectionOnto_mem_subspace_eq_self
  · clear x
    /- In other words, we need to verify that any vector that is orthogonal to the range of `f - 1`
    is a fixed point of `f`. -/
    rw [Submodule.ker_orthogonalProjectionOnto, ← Submodule.topologicalClosure_coe,
      SetLike.coe_subset_coe, ← Submodule.orthogonal_orthogonal_eq_closure]
    /- To verify this, we verify `‖f x‖ ≤ ‖x‖` (because `‖f‖ ≤ 1`) and `⟪f x, x⟫ = ‖x‖²`. -/
    refine Submodule.orthogonal_le fun x hx ↦ eq_of_norm_le_re_inner_eq_norm_sq (𝕜 := 𝕜) ?_ ?_
    · simpa using f.le_of_opNorm_le hf x
    · have : ∀ y, ⟪f y, x⟫ = ⟪y, x⟫ := by
        simpa [Submodule.mem_orthogonal, inner_sub_left, sub_eq_zero] using hx
      simp [this]

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

Goal state before:
pp: case mpr
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
h : ∀ (a : E) (a_1 : a ∈ T.domain), ⟪↑T ⟨a, ⋯⟩, x.1⟫ = ⟪a, x.2⟫
⊢ ∃ (x_1 : x.1 ∈ T†.domain), ↑T† ⟨x.1, ⋯⟩ = x.2

Tactic:
have hx : x.fst ∈ T†.domain := by
      apply mem_adjoint_domain_of_exists
      use x.snd
      rintro ⟨a, ha⟩
      rw [← inner_conj_symm, ← h a ha, inner_conj_symm]

Goal state after:
case mpr
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
h : ∀ (a : E) (a_1 : a ∈ T.domain), ⟪↑T ⟨a, ⋯⟩, x.1⟫ = ⟪a, x.2⟫
hx : x.1 ∈ T†.domain
⊢ ∃ (x_1 : x.1 ∈ T†.domain), ↑T† ⟨x.1, ⋯⟩ = x.2
Module: Mathlib.Analysis.InnerProductSpace.SingularValues

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : FiniteDimensional 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
inst✝ : FiniteDimensional 𝕜 F
T : E →ₗ[𝕜] F
i : ℕ
hi : finrank 𝕜 E ≤ i
⊢ T.singularValues i = 0

Tactic:
apply Finsupp.embDomain_of_notMem_range
  simp [hi]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : InnerProductSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace 𝕜 H
f : E →ₗᵢ[𝕜] G
g : F →ₗᵢ[𝕜] H
x y : E ⊗[𝕜] F
⊢ ∀ (x : E) (y : F) (x_1 : E) (y_1 : F),
    inner 𝕜 ((map f.toLinearMap g.toLinearMap) (x_1 ⊗ₜ[𝕜] y_1)) ((map f.toLinearMap g.toLinearMap) (x ⊗ₜ[𝕜] y)) =
      inner 𝕜 (x_1 ⊗ₜ[𝕜] y_1) (x ⊗ₜ[𝕜] y)

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : FiniteDimensional 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
inst✝ : FiniteDimensional 𝕜 F
T : E →ₗ[𝕜] F
this : (adjoint T ∘ₗ T).ker = ⊥ ↔ ¬End.HasEigenvalue (adjoint T ∘ₗ T) 0
i : ℕ
h : i < finrank 𝕜 E
hz : T.singularValues i ≤ 0
⊢ 0 = ↑(0 ^ 2)

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : FiniteDimensional 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
inst✝ : FiniteDimensional 𝕜 F
i : ℕ
⊢ (singularValues 0) i = 0 i

Tactic:
Finsupp.zero_apply,

Goal state after:
𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : FiniteDimensional 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
inst✝ : FiniteDimensional 𝕜 F
i : ℕ
⊢ (singularValues 0) i = 0
Module: Mathlib.Analysis.InnerProductSpace.SingularValues

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : FiniteDimensional 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
inst✝ : FiniteDimensional 𝕜 F
i : ℕ
⊢ finrank 𝕜 ↥⊥ ≤ i

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : InnerProductSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace 𝕜 H
x y : E ⊗[𝕜] F
⊢ ∀ (x : E) (y : F) (x_1 : E) (y_1 : F),
    (starRingEnd 𝕜) (inner 𝕜 (x ⊗ₜ[𝕜] y) (x_1 ⊗ₜ[𝕜] y_1)) = inner 𝕜 (x_1 ⊗ₜ[𝕜] y_1) (x ⊗ₜ[𝕜] y)

Tactic:
simp

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

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
x : E
y : F
⊢ ‖x ⊗ₜ[𝕜] y‖₊ = ‖x‖₊ * ‖y‖₊

Tactic:
simp

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

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
x x' : E
y y' : F
⊢ ‖x ⊗ₜ[𝕜] y‖ + ‖x' ⊗ₜ[𝕜] y'‖ ≤ ‖x‖ * ‖y‖ + ‖x'‖ * ‖y'‖

Tactic:
simp

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

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
x x' : E
y y' : F
⊢ ‖x ⊗ₜ[𝕜] y‖₊ + ‖x' ⊗ₜ[𝕜] y'‖₊ ≤ ‖x‖₊ * ‖y‖₊ + ‖x'‖₊ * ‖y'‖₊

Tactic:
simp

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

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
x x' : E
y y' : F
⊢ ‖x ⊗ₜ[𝕜] y‖ₑ + ‖x' ⊗ₜ[𝕜] y'‖ₑ ≤ ‖x‖ₑ * ‖y‖ₑ + ‖x'‖ₑ * ‖y'‖ₑ

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : RCLike 𝕜
a b c d : 𝕜
⊢ inner 𝕜 (a ⊗ₜ[𝕜] b) (c ⊗ₜ[𝕜] d) = inner 𝕜 (a * b) (c * d)

Tactic:
simp

Goal state after:
𝕜 : Type u_1
inst✝ : RCLike 𝕜
a b c d : 𝕜
⊢ c * (starRingEnd 𝕜) a * (d * (starRingEnd 𝕜) b) = c * d * ((starRingEnd 𝕜) a * (starRingEnd 𝕜) b)
Module: Mathlib.Analysis.InnerProductSpace.TensorProduct

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : InnerProductSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace 𝕜 H
x✝¹ : E
x✝ : F
⊢ ‖((mk 𝕜 E F) x✝¹) x✝‖ ≤ 1 * ‖x✝¹‖ * ‖x✝‖

Tactic:
simp

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

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
x✝ : E ⊗[𝕜] F
⊢ (mapIsometry LinearIsometry.id LinearIsometry.id) x✝ = LinearIsometry.id x✝

Tactic:
simp

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

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
x✝¹ : E
x✝ : F
⊢ ((AlgebraTensorModule.curry
          ↑(congrIsometry (LinearIsometryEquiv.refl 𝕜 E) (LinearIsometryEquiv.refl 𝕜 F)).toLinearEquiv)
        x✝¹)
      x✝ =
    ((AlgebraTensorModule.curry ↑(LinearIsometryEquiv.refl 𝕜 (E ⊗[𝕜] F)).toLinearEquiv) x✝¹) x✝

Tactic:
simp

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

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
x y : E ⊗[𝕜] F
⊢ inner 𝕜 ((TensorProduct.comm 𝕜 E F) 0) ((TensorProduct.comm 𝕜 E F) y) = inner 𝕜 0 y

Tactic:
simp

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

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
x y : E ⊗[𝕜] F
x✝¹ : E
x✝ : F
⊢ inner 𝕜 ((TensorProduct.comm 𝕜 E F) (x✝¹ ⊗ₜ[𝕜] x✝)) ((TensorProduct.comm 𝕜 E F) 0) = inner 𝕜 (x✝¹ ⊗ₜ[𝕜] x✝) 0

Tactic:
simp

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

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 : 𝕜 ⊗[𝕜] E
⊢ inner 𝕜 ((TensorProduct.lid 𝕜 E) 0) ((TensorProduct.lid 𝕜 E) y) = inner 𝕜 0 y

Tactic:
simp

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

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 : 𝕜 ⊗[𝕜] E
x✝¹ : 𝕜
x✝ : E
⊢ inner 𝕜 ((TensorProduct.lid 𝕜 E) (x✝¹ ⊗ₜ[𝕜] x✝)) ((TensorProduct.lid 𝕜 E) 0) = inner 𝕜 (x✝¹ ⊗ₜ[𝕜] x✝) 0

Tactic:
simp

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

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✝ : E ⊗[𝕜] 𝕜
⊢ ((commIsometry 𝕜 E 𝕜).trans (lidIsometry 𝕜 E)) x✝ = (ridIsometry 𝕜 E) x✝

Tactic:
simp

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

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✝ : 𝕜 ⊗[𝕜] E
⊢ ((commIsometry 𝕜 𝕜 E).trans (ridIsometry 𝕜 E)) x✝ = (lidIsometry 𝕜 E) x✝

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : InnerProductSpace 𝕜 G
x y : E ⊗[𝕜] F ⊗[𝕜] G
⊢ inner 𝕜 ((TensorProduct.assoc 𝕜 E F G) 0) ((TensorProduct.assoc 𝕜 E F G) y) = inner 𝕜 0 y

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : InnerProductSpace 𝕜 G
x y : E ⊗[𝕜] F ⊗[𝕜] G
a : E ⊗[𝕜] F
x✝ : G
⊢ inner 𝕜 ((TensorProduct.assoc 𝕜 E F G) (a ⊗ₜ[𝕜] x✝)) ((TensorProduct.assoc 𝕜 E F G) 0) = inner 𝕜 (a ⊗ₜ[𝕜] x✝) 0

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : InnerProductSpace 𝕜 G
x y : E ⊗[𝕜] F ⊗[𝕜] G
a : E ⊗[𝕜] F
x✝¹ : G
c : E ⊗[𝕜] F
x✝ : G
⊢ inner 𝕜 ((TensorProduct.assoc 𝕜 E F G) (0 ⊗ₜ[𝕜] x✝¹)) ((TensorProduct.assoc 𝕜 E F G) (c ⊗ₜ[𝕜] x✝)) =
    inner 𝕜 (0 ⊗ₜ[𝕜] x✝¹) (c ⊗ₜ[𝕜] x✝)

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : InnerProductSpace 𝕜 G
x y : E ⊗[𝕜] F ⊗[𝕜] G
a : E ⊗[𝕜] F
x✝³ : G
c : E ⊗[𝕜] F
x✝² : G
x✝¹ : E
x✝ : F
⊢ inner 𝕜 ((TensorProduct.assoc 𝕜 E F G) (x✝¹ ⊗ₜ[𝕜] x✝ ⊗ₜ[𝕜] x✝³)) ((TensorProduct.assoc 𝕜 E F G) (0 ⊗ₜ[𝕜] x✝²)) =
    inner 𝕜 (x✝¹ ⊗ₜ[𝕜] x✝ ⊗ₜ[𝕜] x✝³) (0 ⊗ₜ[𝕜] x✝²)

Tactic:
simp

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

Goal state before:
pp: case empty
ι : Type u_1
X : Type u_2
inst✝ : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
s : ι → Set X
hI : ∀ i ∈ ∅, ∀ j ∈ ∅, i ≠ j → AreSeparated (s i) (s j)
⊢ μ (⋃ i ∈ ∅, s i) = ∑ i ∈ ∅, μ (s i)

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : InnerProductSpace 𝕜 G
f₁ f₂ : E →L[𝕜] F
x✝ : E ⊗[𝕜] G
⊢ (rTensor G (f₁ + f₂)) x✝ = (rTensor G f₁ + rTensor G f₂) x✝

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : InnerProductSpace 𝕜 G
r : 𝕜
f : E →L[𝕜] F
x✝ : E ⊗[𝕜] G
⊢ (rTensor G (r • f)) x✝ = (r • rTensor G f) x✝

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
G : Type u_4
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup G
inst✝ : InnerProductSpace 𝕜 G
x✝ : E ⊗[𝕜] G
⊢ (rTensor G (ContinuousLinearMap.id 𝕜 E)) x✝ = (ContinuousLinearMap.id 𝕜 (E ⊗[𝕜] G)) x✝

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : InnerProductSpace 𝕜 G
x✝ : E ⊗[𝕜] G
⊢ (rTensor G 0) x✝ = 0 x✝

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : InnerProductSpace 𝕜 G
f : E →L[𝕜] F
x✝ : E ⊗[𝕜] G
⊢ (rTensor G (-f)) x✝ = (-rTensor G f) x✝

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : InnerProductSpace 𝕜 G
f₁ f₂ : E →L[𝕜] F
x✝ : E ⊗[𝕜] G
⊢ (rTensor G (f₁ - f₂)) x✝ = (rTensor G f₁ - rTensor G f₂) x✝

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : InnerProductSpace 𝕜 G
g : E →L[𝕜] F
x✝¹ : G
x✝ : E
⊢ ((AlgebraTensorModule.curry ↑(lTensor G g)) x✝¹) x✝ = ((AlgebraTensorModule.curry (LinearMap.lTensor G ↑g)) x✝¹) x✝

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : InnerProductSpace 𝕜 G
g : E →ₗᵢ[𝕜] F
x✝ : G ⊗[𝕜] E
⊢ (LinearIsometry.lTensor G g).toContinuousLinearMap x✝ = (lTensor G g.toContinuousLinearMap) x✝

Tactic:
simp

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

Goal state before:
pp: F : Type u_3
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
E : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
μ : Measure E
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : FiniteDimensional ℝ F
u : E → F
s : Set E
hu : ContDiff ℝ 1 u
h2u : support u ⊆ s
p q : ℝ≥0
hp : 1 ≤ p
h2p : p < ↑(finrank ℝ E)
hpq : ↑p⁻¹ - (↑(finrank ℝ E))⁻¹ ≤ (↑q)⁻¹
hs : Bornology.IsBounded s
hq0 : ¬q = 0
p' : ℝ≥0 := (p⁻¹ - (↑(finrank ℝ E))⁻¹)⁻¹
⊢ ↑p⁻¹ - ↑(↑(finrank ℝ E))⁻¹ = ↑p⁻¹ - (↑(finrank ℝ E))⁻¹

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : InnerProductSpace 𝕜 G
f₁ f₂ : E →L[𝕜] F
x✝ : G ⊗[𝕜] E
⊢ (lTensor G (f₁ + f₂)) x✝ = (lTensor G f₁ + lTensor G f₂) x✝

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : InnerProductSpace 𝕜 G
r : 𝕜
f : E →L[𝕜] F
x✝ : G ⊗[𝕜] E
⊢ (lTensor G (r • f)) x✝ = (r • lTensor G f) x✝

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
G : Type u_4
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup G
inst✝ : InnerProductSpace 𝕜 G
x✝ : G ⊗[𝕜] E
⊢ (lTensor G (ContinuousLinearMap.id 𝕜 E)) x✝ = (ContinuousLinearMap.id 𝕜 (G ⊗[𝕜] E)) x✝

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : InnerProductSpace 𝕜 G
x✝ : G ⊗[𝕜] E
⊢ (lTensor G 0) x✝ = 0 x✝

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : InnerProductSpace 𝕜 G
f : E →L[𝕜] F
x✝ : G ⊗[𝕜] E
⊢ (lTensor G (-f)) x✝ = (-lTensor G f) x✝

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : InnerProductSpace 𝕜 G
f₁ f₂ : E →L[𝕜] F
x✝ : G ⊗[𝕜] E
⊢ (lTensor G (f₁ - f₂)) x✝ = (lTensor G f₁ - lTensor G f₂) x✝

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : InnerProductSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace 𝕜 H
f : E →L[𝕜] F
g : G →L[𝕜] H
x✝¹ : E
x✝ : G
⊢ ((AlgebraTensorModule.curry ↑(mapL f g)) x✝¹) x✝ = ((AlgebraTensorModule.curry (map ↑f ↑g)) x✝¹) x✝

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : InnerProductSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace 𝕜 H
f : E →ₗᵢ[𝕜] F
g : G →ₗᵢ[𝕜] H
x✝ : E ⊗[𝕜] G
⊢ (mapIsometry f g).toContinuousLinearMap x✝ = (mapL f.toContinuousLinearMap g.toContinuousLinearMap) x✝

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝¹⁰ : RCLike 𝕜
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : InnerProductSpace 𝕜 E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : InnerProductSpace 𝕜 F
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : InnerProductSpace 𝕜 G
inst✝³ : NormedAddCommGroup H
inst✝² : InnerProductSpace 𝕜 H
A : Type u_6
inst✝¹ : NormedAddCommGroup A
inst✝ : InnerProductSpace 𝕜 A
f₁ : E →L[𝕜] F
f₂ : A →L[𝕜] E
g : G →L[𝕜] H
x✝ : A ⊗[𝕜] G
⊢ (mapL f₁ g ∘SL ContinuousLinearMap.rTensor G f₂) x✝ = (mapL (f₁ ∘SL f₂) g) x✝

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝¹⁰ : RCLike 𝕜
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : InnerProductSpace 𝕜 E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : InnerProductSpace 𝕜 F
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : InnerProductSpace 𝕜 G
inst✝³ : NormedAddCommGroup H
inst✝² : InnerProductSpace 𝕜 H
A : Type u_6
inst✝¹ : NormedAddCommGroup A
inst✝ : InnerProductSpace 𝕜 A
f : E →L[𝕜] F
g₁ : G →L[𝕜] H
g₂ : A →L[𝕜] G
x✝ : E ⊗[𝕜] A
⊢ (mapL f g₁ ∘SL ContinuousLinearMap.lTensor E g₂) x✝ = (mapL f (g₁ ∘SL g₂)) x✝

Tactic:
simp

Goal state after: