mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Constructions.Polish.StronglyMeasurable

Goal state before:
pp: ι : Type u_1
X : Type u_2
E : Type u_3
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace E
inst✝² : Countable ι
l : Filter ι
inst✝¹ : l.IsCountablyGenerated
f : ι → X → E
inst✝ : IsCompletelyPseudoMetrizableSpace E
hf : ∀ (i : ι), StronglyMeasurable (f i)
⊢ MeasurableSet {x | ∃ c, Tendsto (fun x_1 ↦ f x_1 x) l (𝓝 c)}

Tactic:
obtain rfl | hl := eq_or_neBot l

Goal state after:
case inl
ι : Type u_1
X : Type u_2
E : Type u_3
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace E
inst✝² : Countable ι
f : ι → X → E
inst✝¹ : IsCompletelyPseudoMetrizableSpace E
hf : ∀ (i : ι), StronglyMeasurable (f i)
inst✝ : ⊥.IsCountablyGenerated
⊢ MeasurableSet {x | ∃ c, Tendsto (fun x_1 ↦ f x_1 x) ⊥ (𝓝 c)}

case inr
ι : Type u_1
X : Type u_2
E : Type u_3
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace E
inst✝² : Countable ι
l : Filter ι
inst✝¹ : l.IsCountablyGenerated
f : ι → X → E
inst✝ : IsCompletelyPseudoMetrizableSpace E
hf : ∀ (i : ι), StronglyMeasurable (f i)
hl : l.NeBot
⊢ MeasurableSet {x | ∃ c, Tendsto (fun x_1 ↦ f x_1 x) l (𝓝 c)}
Module: Mathlib.MeasureTheory.Constructions.Polish.StronglyMeasurable

Goal state before:
pp: ι : Type u_1
X : Type u_2
E : Type u_3
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace E
inst✝² : Countable ι
l : Filter ι
inst✝¹ : l.IsCountablyGenerated
f : ι → X → E
inst✝ : IsCompletelyPseudoMetrizableSpace E
hf : ∀ (i : ι), StronglyMeasurable (f i)
⊢ MeasurableSet {x | ∃ c, Tendsto (fun x_1 ↦ f x_1 x) l (𝓝 c)}

Tactic:
obtain rfl | hl := eq_or_neBot l
  · simp_all
  borelize E
  let := upgradeIsCompletelyPseudoMetrizable E
  let s := closure (⋃ i, range (f i))
  have : SecondCountableTopology s := @UniformSpace.secondCountable_of_separable s _ _
    (IsSeparable.iUnion (fun i ↦ (hf i).isSeparable_range)).closure.separableSpace
  have : IsCompletelyPseudoMetrizableSpace s := isClosed_closure.isCompletelyPseudoMetrizableSpace
  let g i x : s := ⟨f i x, subset_closure <| mem_iUnion.2 ⟨i, ⟨x, rfl⟩⟩⟩
  have mg i : Measurable (g i) := (hf i).measurable.subtype_mk
  convert! MeasureTheory.measurableSet_exists_tendsto (l := l) mg with x
  refine ⟨fun ⟨c, hc⟩ ↦ ⟨⟨c, ?_⟩, tendsto_subtype_rng.2 hc⟩,
    fun ⟨c, hc⟩ ↦ ⟨c, tendsto_subtype_rng.1 hc⟩⟩
  exact mem_closure_of_tendsto hc (Eventually.of_forall fun i ↦ mem_iUnion.2 ⟨i, ⟨x, rfl⟩⟩)

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.Polish.StronglyMeasurable

Goal state before:
pp: ι : Type u_1
X : Type u_2
E : Type u_3
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace E
inst✝² : Countable ι
l : Filter ι
inst✝¹ : l.IsCountablyGenerated
f : ι → X → E
hE : Nonempty E
inst✝ : IsCompletelyMetrizableSpace E
hf : ∀ (i : ι), StronglyMeasurable (f i)
⊢ StronglyMeasurable fun x ↦ l.limUnder fun x_1 ↦ f x_1 x

Tactic:
obtain rfl | hl := eq_or_neBot l

Goal state after:
case inl
ι : Type u_1
X : Type u_2
E : Type u_3
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace E
inst✝² : Countable ι
f : ι → X → E
hE : Nonempty E
inst✝¹ : IsCompletelyMetrizableSpace E
hf : ∀ (i : ι), StronglyMeasurable (f i)
inst✝ : ⊥.IsCountablyGenerated
⊢ StronglyMeasurable fun x ↦ ⊥.limUnder fun x_1 ↦ f x_1 x

case inr
ι : Type u_1
X : Type u_2
E : Type u_3
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace E
inst✝² : Countable ι
l : Filter ι
inst✝¹ : l.IsCountablyGenerated
f : ι → X → E
hE : Nonempty E
inst✝ : IsCompletelyMetrizableSpace E
hf : ∀ (i : ι), StronglyMeasurable (f i)
hl : l.NeBot
⊢ StronglyMeasurable fun x ↦ l.limUnder fun x_1 ↦ f x_1 x
Module: Mathlib.Topology.Algebra.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u_4
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : CompleteSpace 𝕜
Eᵤ : Type u_6
inst✝⁵ : AddCommGroup Eᵤ
inst✝⁴ : Module 𝕜 Eᵤ
inst✝³ : UniformSpace Eᵤ
inst✝² : T2Space Eᵤ
inst✝¹ : IsUniformAddGroup Eᵤ
inst✝ : ContinuousSMul 𝕜 Eᵤ
h : ∃ x, ∃ U ∈ 𝓝 x, TotallyBounded U
⊢ FiniteDimensional 𝕜 Eᵤ

Tactic:
by
  rcases h with ⟨x, U, hU_nhds, hU_tb⟩
  exact FiniteDimensional.of_totallyBounded_nhds (𝕜 := 𝕜) hU_nhds hU_tb

Goal state after:
Module: Mathlib.Topology.Algebra.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : CompleteSpace 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : TopologicalSpace E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : Module 𝕜 E
inst✝ : ContinuousSMul 𝕜 E
A B : Submodule 𝕜 E
B_cofg : FiniteDimensional 𝕜 (E ⧸ B)
hB : IsClosed[inst✝³] ↑B
hAB : Disjoint A B
C : Submodule 𝕜 E
B_le_C : B ≤ C
C_compl_A : IsCompl C A
⊢ A.ClosedComplemented

Tactic:
have C_cofg : FiniteDimensional 𝕜 (E ⧸ C) := CoFG.of_le B_le_C B_cofg

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : CompleteSpace 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : TopologicalSpace E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : Module 𝕜 E
inst✝ : ContinuousSMul 𝕜 E
A B : Submodule 𝕜 E
B_cofg : FiniteDimensional 𝕜 (E ⧸ B)
hB : IsClosed[inst✝³] ↑B
hAB : Disjoint A B
C : Submodule 𝕜 E
B_le_C : B ≤ C
C_compl_A : IsCompl C A
C_cofg : FiniteDimensional 𝕜 (E ⧸ C)
⊢ A.ClosedComplemented
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: 𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹⁰ : Fintype ι'
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁷ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁶ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁵ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁴ : SeminormedAddCommGroup G
inst✝³ : NormedSpace 𝕜 G
inst✝² : SeminormedAddCommGroup G'
inst✝¹ : NormedSpace 𝕜 G'
inst✝ : Fintype ι
f : G →L[𝕜] ContinuousMultilinearMap 𝕜 E G'
m : (i : ι) → E i
x : G
⊢ ‖{ toFun := fun x ↦ (f x) m, map_add' := ⋯, map_smul' := ⋯ } x‖ ≤ (‖f‖ * ∏ i, ‖m i‖) * ‖x‖

Tactic:
by
          rw [mul_right_comm]
          exact (f x).le_of_opNorm_le (f.le_opNorm x) _

Goal state after:
Module: Mathlib.MeasureTheory.Function.L1Space.AEEqFun

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ↥(Lp β 1 μ)
⊢ ENNReal.ofReal ‖f - g‖ = ∫⁻ (x : α), ‖↑↑f x - ↑↑g x‖ₑ ∂μ

Tactic:
ofReal_norm_eq_lintegral,

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ↥(Lp β 1 μ)
⊢ ∫⁻ (x : α), ‖↑↑(f - g) x‖ₑ ∂μ = ∫⁻ (x : α), ‖↑↑f x - ↑↑g x‖ₑ ∂μ
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: 𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹⁰ : Fintype ι'
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁷ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁶ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁵ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁴ : SeminormedAddCommGroup G
inst✝³ : NormedSpace 𝕜 G
inst✝² : SeminormedAddCommGroup G'
inst✝¹ : NormedSpace 𝕜 G'
inst✝ : Fintype ι
g : ContinuousMultilinearMap 𝕜 E₁ G
h : DecidableEq ι
f : (i : ι) → E i →L[𝕜] E₁ i
i : ι
⊢ ∀ (x y : E i →L[𝕜] E₁ i),
    g.compContinuousLinearMap (update f i (x + y)) =
      g.compContinuousLinearMap (update f i x) + g.compContinuousLinearMap (update f i y)

Tactic:
f₁

Goal state after:
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹⁰ : Fintype ι'
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁷ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁶ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁵ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁴ : SeminormedAddCommGroup G
inst✝³ : NormedSpace 𝕜 G
inst✝² : SeminormedAddCommGroup G'
inst✝¹ : NormedSpace 𝕜 G'
inst✝ : Fintype ι
g : ContinuousMultilinearMap 𝕜 E₁ G
h : DecidableEq ι
f : (i : ι) → E i →L[𝕜] E₁ i
i : ι
f₁ : E i →L[𝕜] E₁ i
⊢ ∀ (y : E i →L[𝕜] E₁ i),
    g.compContinuousLinearMap (update f i (f₁ + y)) =
      g.compContinuousLinearMap (update f i f₁) + g.compContinuousLinearMap (update f i y)
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: β : Type u_2
E : Type u_4
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
inst✝¹ : OpensMeasurableSpace E
f : β → E
hf : Measurable f
s : Set E
y₀ : E
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
μ : Measure β
hμ : ∀ᵐ (x : β) ∂μ, f x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hi : HasFiniteIntegral (fun x ↦ f x - y₀) μ
⊢ Tendsto (fun n ↦ ∫⁻ (x : β), ‖(approxOn f hf s y₀ h₀ n) x - f x‖ₑ ∂μ) atTop (𝓝 0)

Tactic:
simpa [eLpNorm_one_eq_lintegral_enorm] using!
    tendsto_approxOn_Lp_eLpNorm hf h₀ one_ne_top hμ
      (by simpa [eLpNorm_one_eq_lintegral_enorm] using! hi)

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

Goal state before:
pp: case neg
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
p : ℝ≥0∞
f : α →ₛ E
hf : ∀ (y : E), y ≠ 0 → μ (⇑f ⁻¹' {y}) < ∞
hp0 : ¬p = 0
hp_top : ¬p = ∞
⊢ (∑ y ∈ f.range, ‖y‖ₑ ^ p.toReal * μ (⇑f ⁻¹' {y})) ^ (1 / p.toReal) < ∞

Tactic:
refine ENNReal.rpow_lt_top_of_nonneg (by simp) (ENNReal.sum_lt_top.mpr fun y _ => ?_).ne

Goal state after:
case neg
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
p : ℝ≥0∞
f : α →ₛ E
hf : ∀ (y : E), y ≠ 0 → μ (⇑f ⁻¹' {y}) < ∞
hp0 : ¬p = 0
hp_top : ¬p = ∞
y : E
x✝ : y ∈ f.range
⊢ ‖y‖ₑ ^ p.toReal * μ (⇑f ⁻¹' {y}) < ∞
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: case pos
α : Type u_1
F : Type u_3
F' : Type u_4
G : Type u_5
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedAddCommGroup F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
T : Set α → F →L[ℝ] F'
h_add : FinMeasAdditive μ T
f : α →ₛ G
hf : Integrable (⇑f) μ
g : G → F
hg : g 0 = 0
T_empty : T ∅ = 0
hfp : ∀ x ∈ f.range, x ≠ 0 → μ (⇑f ⁻¹' {x}) ≠ ∞
a : α
hb : f a ∈ f.range
h0 : g (f a) = 0
⊢ 0 = ∑ x ∈ f.range with g x = 0, (T (⇑f ⁻¹' {x})) (g x)

Tactic:
Finset.sum_eq_zero fun x hx => ?_

Goal state after:
case pos
α : Type u_1
F : Type u_3
F' : Type u_4
G : Type u_5
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedAddCommGroup F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
T : Set α → F →L[ℝ] F'
h_add : FinMeasAdditive μ T
f : α →ₛ G
hf : Integrable (⇑f) μ
g : G → F
hg : g 0 = 0
T_empty : T ∅ = 0
hfp : ∀ x ∈ f.range, x ≠ 0 → μ (⇑f ⁻¹' {x}) ≠ ∞
a : α
hb : f a ∈ f.range
h0 : g (f a) = 0
⊢ 0 = 0

α : Type u_1
F : Type u_3
F' : Type u_4
G : Type u_5
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedAddCommGroup F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
T : Set α → F →L[ℝ] F'
h_add : FinMeasAdditive μ T
f : α →ₛ G
hf : Integrable (⇑f) μ
g : G → F
hg : g 0 = 0
T_empty : T ∅ = 0
hfp : ∀ x ∈ f.range, x ≠ 0 → μ (⇑f ⁻¹' {x}) ≠ ∞
a : α
hb : f a ∈ f.range
h0 : g (f a) = 0
x : G
hx : x ∈ {x ∈ f.range | g x = 0}
⊢ (T (⇑f ⁻¹' {x})) (g x) = 0
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝¹ : NormedAddCommGroup E
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedSpace ℝ E
f : ↥(α →₁ₛ[μ] E)
⊢ ‖integral f‖ ≤ ‖f‖

Tactic:
by
  rw [integral, norm_eq_integral]
  exact (toSimpleFunc f).norm_integral_le_integral_norm (SimpleFunc.integrable f)

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f μ
⊢ Integrable (fun a ↦ ↑(-f a).toNNReal) μ

Tactic:
exact hf.neg.real_toNNReal

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

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
inst✝⁵ : PartialOrder E
inst✝⁴ : IsOrderedAddMonoid E
inst✝³ : IsOrderedModule ℝ E
inst✝² : ClosedIciTopology E
β : Type u_6
inst✝¹ : AddCommMonoid β
inst✝ : Module ℝ β
f : α → β → E
s : Set β
hs : Convex ℝ s
hf_conv : ∀ᵐ (x : α) ∂μ, ConvexOn ℝ s (f x)
hf_int : ∀ a ∈ s, Integrable (fun x ↦ f x a) μ
a : β
ha : a ∈ s
b : β
hb : b ∈ s
p q : ℝ
hp : 0 ≤ p
hq : 0 ≤ q
hpq : p + q = 1
⊢ Integrable (fun x ↦ f x (p • a + q • b)) μ

Tactic:
refine hf_int _ ?_

Goal state after:
α : Type u_1
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
inst✝⁵ : PartialOrder E
inst✝⁴ : IsOrderedAddMonoid E
inst✝³ : IsOrderedModule ℝ E
inst✝² : ClosedIciTopology E
β : Type u_6
inst✝¹ : AddCommMonoid β
inst✝ : Module ℝ β
f : α → β → E
s : Set β
hs : Convex ℝ s
hf_conv : ∀ᵐ (x : α) ∂μ, ConvexOn ℝ s (f x)
hf_int : ∀ a ∈ s, Integrable (fun x ↦ f x a) μ
a : β
ha : a ∈ s
b : β
hb : b ∈ s
p q : ℝ
hp : 0 ≤ p
hq : 0 ≤ q
hpq : p + q = 1
⊢ p • a + q • b ∈ s
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
inst✝⁵ : PartialOrder E
inst✝⁴ : IsOrderedAddMonoid E
inst✝³ : IsOrderedModule ℝ E
inst✝² : ClosedIciTopology E
β : Type u_6
inst✝¹ : AddCommMonoid β
inst✝ : Module ℝ β
f : α → β → E
s : Set β
hs : Convex ℝ s
hf_conv : ∀ᵐ (x : α) ∂μ, ConvexOn ℝ s (f x)
hf_int : ∀ a ∈ s, Integrable (fun x ↦ f x a) μ
a : β
ha : a ∈ s
b : β
hb : b ∈ s
p q : ℝ
hp : 0 ≤ p
hq : 0 ≤ q
hpq : p + q = 1
x : α
hx : ConvexOn ℝ s (f x)
⊢ f x (p • a + q • b) ≤ p • f x a + q • f x b

Tactic:
exact hx.2 ha hb hp hq hpq

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : 0 ≤ᵐ[μ] f
hfi : Integrable f μ
⊢ ∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ = 0 ∨ ¬HasFiniteIntegral f μ ↔ f =ᵐ[μ] 0

Tactic:
hfi.2,

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : 0 ≤ᵐ[μ] f
hfi : Integrable f μ
⊢ ∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ = 0 ∨ ¬True ↔ f =ᵐ[μ] 0
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
c : ℝ
f : ↥(Lp E 1 μ)
⊢ ∀ (f : ↥(α →₁ₛ[μ] E)), (setToL1SCLM α E μ ⋯) f = (c • setToL1 hT) ↑f

Tactic:
simp [setToL1_eq_setToL1SCLM, setToL1SCLM_smul_left c hT]

Goal state after:
Module: Mathlib.Topology.Bornology.BoundedOperation

Goal state before:
pp: case h
R : Type u_1
inst✝ : NonUnitalSeminormedRing R
s t : Set R
hs : ∃ C, ∀ ⦃x : R⦄, x ∈ s → ∀ ⦃y : R⦄, y ∈ s → dist x y ≤ C
ht : ∃ C, ∀ ⦃x : R⦄, x ∈ t → ∀ ⦃y : R⦄, y ∈ t → dist x y ≤ C
Af : ℝ
hAf : s ⊆ Metric.closedBall 0 Af
Ag : ℝ
hAg : t ⊆ Metric.closedBall 0 Ag
z : R
hz : z ∈ s * t
w : R
hw : w ∈ s * t
x₁ : R
hx₁ : x₁ ∈ s
y₁ : R
hy₁ : y₁ ∈ t
z_eq : x₁ * y₁ = z
x₂ : R
hx₂ : x₂ ∈ s
y₂ : R
hy₂ : y₂ ∈ t
w_eq : x₂ * y₂ = w
hAf' : 0 ≤ Af
aux : ∀ {x y : R}, x ∈ s → y ∈ t → ‖x * y‖ ≤ Af * Ag
⊢ ‖x₁ * y₁ - x₂ * y₂‖ ≤ 2 * Af * Ag

Tactic:
calc ‖x₁ * y₁ - x₂ * y₂‖
     _ ≤ ‖x₁ * y₁‖ + ‖x₂ * y₂‖ := norm_sub_le _ _
     _ ≤ Af * Ag + Af * Ag     := add_le_add (aux hx₁ hy₁) (aux hx₂ hy₂)
     _ = 2 * Af * Ag           := by simp [← two_mul, mul_assoc]

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

Goal state before:
pp: 𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type w
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : CompleteSpace 𝕜
ι : Type u_1
inst✝ : Fintype ι
v : Basis ι 𝕜 E
u : E →L[𝕜] F
M : ℝ≥0
hu : ∀ (i : ι), ‖u (v i)‖₊ ≤ M
e : E
⊢ ‖u e‖₊ ≤ Fintype.card ι • ‖↑v.equivFunL‖₊ * M * ‖e‖₊

Tactic:
set φ := v.equivFunL.toContinuousLinearMap

Goal state after:
𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type w
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : CompleteSpace 𝕜
ι : Type u_1
inst✝ : Fintype ι
v : Basis ι 𝕜 E
u : E →L[𝕜] F
M : ℝ≥0
hu : ∀ (i : ι), ‖u (v i)‖₊ ≤ M
e : E
φ : E →L[𝕜] ι → 𝕜 := ↑v.equivFunL
⊢ ‖u e‖₊ ≤ Fintype.card ι • ‖φ‖₊ * M * ‖e‖₊
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: F : Type u_1
α : Type u
β : Type v
γ : Type w
R : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : PseudoMetricSpace R
inst✝² : Monoid R
inst✝¹ : BoundedMul R
inst✝ : ContinuousMul R
⊢ ∀ (x y : α →ᵇ R), (x * y).toContinuousMap = x.toContinuousMap * y.toContinuousMap

Tactic:
intros

Goal state after:
F : Type u_1
α : Type u
β : Type v
γ : Type w
R : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : PseudoMetricSpace R
inst✝² : Monoid R
inst✝¹ : BoundedMul R
inst✝ : ContinuousMul R
x✝ y✝ : α →ᵇ R
⊢ (x✝ * y✝).toContinuousMap = x✝.toContinuousMap * y✝.toContinuousMap
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: case hbc
F : Type u_1
α : Type u
β : Type v
γ : Type w
𝕜 : Type u_2
inst✝⁶ : PseudoMetricSpace 𝕜
inst✝⁵ : TopologicalSpace α
inst✝⁴ : PseudoMetricSpace β
inst✝³ : Zero 𝕜
inst✝² : Zero β
inst✝¹ : SMul 𝕜 β
inst✝ : IsBoundedSMul 𝕜 β
c : 𝕜
f : α →ᵇ β
b : ℝ
hb : ∀ (x y : α), dist (f x) (f y) ≤ b
x y : α
⊢ dist (f x) (f y) ≤ b

Tactic:
apply hb

Goal state after:
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: case pos
α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C C' : ℝ
hT : DominatedFinMeasAdditive μ T C
β : Type u_7
x✝ : MeasurableSpace β
μ' : Measure β
φ : α → β
T' : Set β → E →L[ℝ] F
hT' : DominatedFinMeasAdditive μ' T' C'
f : β → E
hf : Integrable (f ∘ φ) μ
hfm : StronglyMeasurable f
hφ : Measurable φ
hμ' : μ' ≤ Measure.map φ μ
h : ∀ (s : Set β) (x : E), MeasurableSet s → (T' s) x = (T (φ ⁻¹' s)) x
hF : CompleteSpace F
hfi' : Integrable f μ'
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
this : SeparableSpace ↑(range f ∪ {0})
⊢ setToFun μ' T' hT' f = setToFun μ T hT (f ∘ φ)

Tactic:
refine tendsto_nhds_unique
    (tendsto_setToFun_approxOn_of_measurable_of_range_subset
      hT' hfm.measurable hfi' _ Subset.rfl) ?_

Goal state after:
case pos
α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C C' : ℝ
hT : DominatedFinMeasAdditive μ T C
β : Type u_7
x✝ : MeasurableSpace β
μ' : Measure β
φ : α → β
T' : Set β → E →L[ℝ] F
hT' : DominatedFinMeasAdditive μ' T' C'
f : β → E
hf : Integrable (f ∘ φ) μ
hfm : StronglyMeasurable f
hφ : Measurable φ
hμ' : μ' ≤ Measure.map φ μ
h : ∀ (s : Set β) (x : E), MeasurableSet s → (T' s) x = (T (φ ⁻¹' s)) x
hF : CompleteSpace F
hfi' : Integrable f μ'
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
this : SeparableSpace ↑(range f ∪ {0})
⊢ Tendsto (fun n ↦ setToFun μ' T' hT' ⇑(SimpleFunc.approxOn f ⋯ (range f ∪ {0}) 0 ⋯ n)) atTop
    (𝓝 (setToFun μ T hT (f ∘ φ)))
Module: Mathlib.Topology.ContinuousMap.Bounded.Normed

Goal state before:
pp: α : Type u
β : Type v
γ : Type w
𝕜✝ : Type u_1
inst✝⁶ : TopologicalSpace α
inst✝⁵ : SeminormedAddCommGroup β
f g✝ : α →ᵇ β
x : α
C : ℝ
inst✝⁴ : NontriviallyNormedField 𝕜✝
inst✝³ : NormedSpace 𝕜✝ β
𝕜 : Type u_2
inst✝² : TopologicalSpace γ
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 β
g : C(γ, α)
⊢ ∀ (x y : α →ᵇ β), (x + y).compContinuous g = x.compContinuous g + y.compContinuous g

Tactic:
intros

Goal state after:
α : Type u
β : Type v
γ : Type w
𝕜✝ : Type u_1
inst✝⁶ : TopologicalSpace α
inst✝⁵ : SeminormedAddCommGroup β
f g✝ : α →ᵇ β
x : α
C : ℝ
inst✝⁴ : NontriviallyNormedField 𝕜✝
inst✝³ : NormedSpace 𝕜✝ β
𝕜 : Type u_2
inst✝² : TopologicalSpace γ
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 β
g : C(γ, α)
x✝ y✝ : α →ᵇ β
⊢ (x✝ + y✝).compContinuous g = x✝.compContinuous g + y✝.compContinuous g
Module: Mathlib.Topology.ContinuousMap.Bounded.Normed

Goal state before:
pp: α : Type u
β : Type v
γ : Type w
𝕜✝ : Type u_1
inst✝⁶ : TopologicalSpace α
inst✝⁵ : SeminormedAddCommGroup β
f g✝ : α →ᵇ β
x : α
C : ℝ
inst✝⁴ : NontriviallyNormedField 𝕜✝
inst✝³ : NormedSpace 𝕜✝ β
𝕜 : Type u_2
inst✝² : TopologicalSpace γ
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 β
g : C(γ, α)
⊢ ∀ (m : 𝕜) (x : α →ᵇ β), (m • x).compContinuous g = (RingHom.id 𝕜) m • x.compContinuous g

Tactic:
intros

Goal state after:
α : Type u
β : Type v
γ : Type w
𝕜✝ : Type u_1
inst✝⁶ : TopologicalSpace α
inst✝⁵ : SeminormedAddCommGroup β
f g✝ : α →ᵇ β
x : α
C : ℝ
inst✝⁴ : NontriviallyNormedField 𝕜✝
inst✝³ : NormedSpace 𝕜✝ β
𝕜 : Type u_2
inst✝² : TopologicalSpace γ
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 β
g : C(γ, α)
m✝ : 𝕜
x✝ : α →ᵇ β
⊢ (m✝ • x✝).compContinuous g = (RingHom.id 𝕜) m✝ • x✝.compContinuous g
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
x : E
s : Set E
hx : x ∈ s
hs : s ≠ univ
y : E
hys : y ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] sᶜ
hyd : infDist x sᶜ = dist x y
⊢ ∃ y ∈ frontier s, infDist x sᶜ = dist x y

Tactic:
closure_compl

Goal state after:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
x : E
s : Set E
hx : x ∈ s
hs : s ≠ univ
y : E
hys : y ∈ (interior s)ᶜ
hyd : infDist x sᶜ = dist x y
⊢ ∃ y ∈ frontier s, infDist x sᶜ = dist x y
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : Nontrivial E
x : E
K : Set E
hK : IsCompact K
hx : x ∈ K
⊢ x ∈ interior K ∪ frontier K

Tactic:
← closure_eq_interior_union_frontier

Goal state after:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : Nontrivial E
x : E
K : Set E
hK : IsCompact K
hx : x ∈ K
⊢ x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] K
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: α : Type u_1
⊢ ∀ {N : ℕ} {g : α → Fin N → ℝ}, Summable g → Summable fun x ↦ ‖g x‖

Tactic:
intro N g hg

Goal state after:
α : Type u_1
N : ℕ
g : α → Fin N → ℝ
hg : Summable g
⊢ Summable fun x ↦ ‖g x‖
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: case neg
α : Type u_1
inst✝¹ : PseudoEMetricSpace α
β : Type u_2
inst✝ : One β
f : α → β
E : Set α
x : α
x_mem_closure : x ∉ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E
⊢ ∀ᶠ (δ : ℝ) in 𝓝[>] 0,
    (thickening δ E).mulIndicator f x =
      (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).mulIndicator f x

Tactic:
· have obs := eventually_notMem_thickening_of_infEDist_pos x_mem_closure
    filter_upwards [mem_nhdsWithin_of_mem_nhds obs, self_mem_nhdsWithin]
      with δ x_notin_thE _
    simp only [x_notin_thE, not_false_eq_true, mulIndicator_of_notMem, x_mem_closure]

Goal state after:
Module: Mathlib.Topology.Order.LeftRightLim

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
f : α → β
a : α
s : Set β
hs : s ∈ 𝓝 (f a)
s' : Set β
hs' : f ⁻¹' s' ∈ 𝓝[≤] a
⊢ f a ∈ s'

Tactic:
apply mem_of_mem_nhdsWithin self_mem_Iic hs'

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

Goal state before:
pp: G : Type w
inst✝ : TopologicalSpace G
μ : Content G
U : Opens G
hU : μ.innerContent U ≠ ∞
ε : ℝ≥0
hε : ε ≠ 0
h'ε : ↑ε ≠ 0
h : ↑ε < μ.innerContent U
h₂ : μ.innerContent U - ↑ε < μ.innerContent U
| μ.innerContent U

Tactic:
rw [innerContent]

Goal state after:
G : Type w
inst✝ : TopologicalSpace G
μ : Content G
U : Opens G
hU : μ.innerContent U ≠ ∞
ε : ℝ≥0
hε : ε ≠ 0
h'ε : ↑ε ≠ 0
h : ↑ε < μ.innerContent U
h₂ : μ.innerContent U - ↑ε < μ.innerContent U
| ⨆ K, ⨆ (_ : ↑K ⊆ ↑U), μ K