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

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
inst✝¹ : CompleteSpace F
inst✝ : IsFiniteMeasure μ
hT : DominatedFinMeasAdditive μ T C
x : E
⊢ (setToFun μ T hT fun x_1 ↦ x) = (T univ) x

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
F : Type w
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : CompleteSpace 𝕜
PE : Type u_1
PF : Type u_2
inst✝⁴ : MetricSpace PE
inst✝³ : NormedAddTorsor E PE
inst✝² : MetricSpace PF
inst✝¹ : NormedAddTorsor F PF
inst✝ : FiniteDimensional 𝕜 E
f : PE →ᵃ[𝕜] PF
hf : Function.Injective ⇑f
⊢ ∃ K, AntilipschitzWith K ⇑f

Tactic:
by

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

Goal state before:
pp: α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
m : MeasurableSpace α
μ : Measure α
f : α → G
c : ℝ≥0∞
⊢ ∫ (x : α), f x ∂c • μ = c.toReal • ∫ (x : α), f x ∂μ

Tactic:
by

Goal state after:
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
T : Set α → E →L[ℝ] F
C : ℝ
inst✝ : CompleteSpace F
hT : DominatedFinMeasAdditive μ T C
f : SimpleFunc α E
hf : Integrable (⇑f) μ
⊢ setToFun μ T hT ⇑f = ∑ x ∈ f.range, (T (⇑f ⁻¹' {x})) x

Tactic:
by

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
inst✝ : FiniteDimensional 𝕜 E
⊢ IsOpen {L | Injective ⇑L}

Tactic:
by

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

Goal state before:
pp: α : Type u_1
G : Type u_5
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
m : MeasurableSpace α
μ : Measure α
β : Type u_6
inst✝ : MeasurableSpace β
φ : α → β
hφ : Measurable φ
f : β → G
hfm : StronglyMeasurable f
⊢ ∫ (y : β), f y ∂Measure.map φ μ = ∫ (x : α), f (φ x) ∂μ

Tactic:
by

Goal state after:
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
T : Set α → E →L[ℝ] F
C : ℝ
inst✝ : CompleteSpace F
hT : DominatedFinMeasAdditive μ T C
f : SimpleFunc α E
hf : Integrable (⇑f) μ
⊢ setToFun μ T hT ⇑f = SimpleFunc.setToSimpleFunc T f

Tactic:
by

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

Goal state before:
pp: α : Type u_1
G : Type u_5
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
m : MeasurableSpace α
μ : Measure α
β : Type u_6
inst✝ : MeasurableSpace β
φ : α → β
hφ : AEMeasurable φ μ
f : β → G
hfm : AEStronglyMeasurable f (Measure.map φ μ)
g : β → G := AEStronglyMeasurable.mk f hfm
⊢ ∫ (y : β), g y ∂Measure.map φ μ = ∫ (y : β), g y ∂Measure.map (AEMeasurable.mk φ hφ) μ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
m : MeasurableSpace α
μ : Measure α
β : Type u_6
x✝ : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
g : β → G
⊢ ∫ (y : β), g y ∂Measure.map f μ = ∫ (x : α), g (f x) ∂μ

Tactic:
by

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

Goal state before:
pp: G : Type u_5
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
α : Type u_6
inst✝ : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
f : α → G
⊢ ∫ (x : ↑s), f ↑x ∂Measure.comap Subtype.val μ = ∫ (x : α) in s, f x ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: F : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝² : TopologicalSpace α
inst✝¹ : PseudoMetricSpace β
inst✝ : PseudoMetricSpace γ
f g✝ : α →ᵇ β
x : α
C : ℝ
g : α →ᵇ β
toFun✝ : α → β
continuous_toFun✝ : Continuous[inst✝², PseudoMetricSpace.toUniformSpace.toTopologicalSpace] toFun✝
map_bounded'✝ :
  ∃ C,
    ∀ (x y : α),
      dist ({ toFun := toFun✝, continuous_toFun := continuous_toFun✝ }.toFun x)
          ({ toFun := toFun✝, continuous_toFun := continuous_toFun✝ }.toFun y) ≤
        C
h :
  (fun f ↦ f.toFun) { toFun := toFun✝, continuous_toFun := continuous_toFun✝, map_bounded' := map_bounded'✝ } =
    (fun f ↦ f.toFun) g
⊢ { toFun := toFun✝, continuous_toFun := continuous_toFun✝, map_bounded' := map_bounded'✝ } = g

Tactic:
obtain ⟨⟨_, _⟩, _⟩ := g

Goal state after:
F : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝² : TopologicalSpace α
inst✝¹ : PseudoMetricSpace β
inst✝ : PseudoMetricSpace γ
f g : α →ᵇ β
x : α
C : ℝ
toFun✝¹ : α → β
continuous_toFun✝¹ : Continuous[inst✝², PseudoMetricSpace.toUniformSpace.toTopologicalSpace] toFun✝¹
map_bounded'✝¹ :
  ∃ C,
    ∀ (x y : α),
      dist ({ toFun := toFun✝¹, continuous_toFun := continuous_toFun✝¹ }.toFun x)
          ({ toFun := toFun✝¹, continuous_toFun := continuous_toFun✝¹ }.toFun y) ≤
        C
toFun✝ : α → β
continuous_toFun✝ : Continuous[inst✝², PseudoMetricSpace.toUniformSpace.toTopologicalSpace] toFun✝
map_bounded'✝ :
  ∃ C,
    ∀ (x y : α),
      dist ({ toFun := toFun✝, continuous_toFun := continuous_toFun✝ }.toFun x)
          ({ toFun := toFun✝, continuous_toFun := continuous_toFun✝ }.toFun y) ≤
        C
h :
  (fun f ↦ f.toFun) { toFun := toFun✝¹, continuous_toFun := continuous_toFun✝¹, map_bounded' := map_bounded'✝¹ } =
    (fun f ↦ f.toFun) { toFun := toFun✝, continuous_toFun := continuous_toFun✝, map_bounded' := map_bounded'✝ }
⊢ { toFun := toFun✝¹, continuous_toFun := continuous_toFun✝¹, map_bounded' := map_bounded'✝¹ } =
    { toFun := toFun✝, continuous_toFun := continuous_toFun✝, map_bounded' := map_bounded'✝ }
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝ : MeasurableSpace α
f : α → E
a : α
hfm : StronglyMeasurable f
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
⊢ ∫ (x : α), f a ∂Measure.dirac a = f a

Tactic:
by

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

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝ : MeasurableSpace α
f : α → E
a : α
hfm : StronglyMeasurable f
⊢ ∫ (x : α), f x ∂Measure.dirac a = f a

Tactic:
by

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

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
hE : CompleteSpace E
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSingletonClass α
f : α → E
a : α
⊢ ∫ (x : α), f a ∂Measure.dirac a = f a

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: F : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝² : TopologicalSpace α
inst✝¹ : PseudoMetricSpace β
inst✝ : PseudoMetricSpace γ
f✝ g✝ : α →ᵇ β
x : α
C : ℝ
f g : α →ᵇ β
h : (fun f ↦ f.toFun) f = (fun f ↦ f.toFun) g
⊢ f = g

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : PseudoMetricSpace β
f g : α →ᵇ β
⊢ ∃ C, 0 ≤ C ∧ ∀ (x : α), dist (f x) (g x) ≤ C

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: α : Type u
β : Type v
inst✝³ : TopologicalSpace α
inst✝² : PseudoMetricSpace β
f g : α →ᵇ β
C : ℝ
inst✝¹ : Nonempty α
inst✝ : CompactSpace α
w : ∀ (x : α), dist (f x) (g x) < C
⊢ Continuous[inst✝³, _] fun x ↦ dist (f x) (g x)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: α : Type u
β : Type v
inst✝³ : TopologicalSpace α
inst✝² : PseudoMetricSpace β
f g : α →ᵇ β
C : ℝ
inst✝¹ : Nonempty α
inst✝ : CompactSpace α
w : ∀ (x : α), dist (f x) (g x) < C
⊢ dist f g < C

Tactic:
by

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

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
hE : CompleteSpace E
mα : MeasurableSpace α
f : α → E
hf : StronglyMeasurable f
a : α
s : Set α
hs : MeasurableSet s
inst✝ : Decidable (a ∈ s)
⊢ ∫ (x : α) in s, f x ∂Measure.dirac a = if a ∈ s then f a else 0

Tactic:
by

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

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
G'' : Type u_8
inst✝⁴ : NormedAddCommGroup G''
inst✝³ : PartialOrder G''
inst✝² : IsOrderedAddMonoid G''
inst✝¹ : NormedSpace ℝ G''
inst✝ : OrderClosedTopology G''
T T' : Set α → E →L[ℝ] G''
C C' : ℝ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ T' C'
hTT' : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∀ (x : E), (T s) x ≤ (T' s) x
f : α → E
⊢ setToFun μ T hT f ≤ setToFun μ T' hT' f

Tactic:
by

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

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSingletonClass α
f : α → E
a : α
s : Set α
inst✝ : Decidable (a ∈ s)
⊢ ∫ (x : α) in s, f x ∂Measure.dirac a = if a ∈ s then f a else 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: α : Type u
β : Type v
inst✝² : TopologicalSpace α
inst✝¹ : PseudoMetricSpace β
f g : α →ᵇ β
C : ℝ
inst✝ : CompactSpace α
C0 : 0 < C
⊢ dist f g < C ↔ ∀ (x : α), dist (f x) (g x) < C

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: F : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝² : TopologicalSpace α
inst✝¹ : PseudoMetricSpace β
inst✝ : PseudoMetricSpace γ
f✝ g : α →ᵇ β
x✝ : α
C : ℝ
f : α →ᵇ β
x : α
⊢ dist (f x) (f x) ≤ 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: F : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝² : TopologicalSpace α
inst✝¹ : PseudoMetricSpace β
inst✝ : PseudoMetricSpace γ
f✝ g✝ : α →ᵇ β
x : α
C : ℝ
f g : α →ᵇ β
⊢ dist f g = dist g f

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: F : Type u_1
α : Type u
β✝ : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : PseudoMetricSpace β✝
inst✝¹ : PseudoMetricSpace γ
f g : α →ᵇ β✝
x : α
C : ℝ
β : Type ?u.21
inst✝ : MetricSpace β
x✝ y✝ : α →ᵇ β
hfg : dist x✝ y✝ = 0
⊢ x✝ = y✝

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : PseudoMetricSpace β
f g : α →ᵇ β
⊢ sInf {C | 0 ≤ C ∧ ∀ (x : α), dist (f x) (g x) ≤ C} = ↑(sInf {C | ∀ (x : α), nndist (f x) (g x) ≤ C})

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: α : Type u
β : Type v
inst✝² : TopologicalSpace α
inst✝¹ : PseudoMetricSpace β
f g : α →ᵇ β
inst✝ : IsEmpty α
⊢ dist f g = 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : PseudoMetricSpace β
f g : α →ᵇ β
⊢ dist f g = ⨆ x, dist (f x) (g x)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : PseudoMetricSpace β
f g : α →ᵇ β
⊢ ⨆ x, dist (f x) (g x) = ↑(⨆ x, nndist (f x) (g x))

Tactic:
by

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_nonneg : 0 ≤ᵐ[μ] f
hf_int : Integrable f μ
ε : ℝ
hμ : μ {x | ε ≤ f x} < ∞
this : Fact (μ {x | ε ≤ f x} < ∞)
⊢ ε * μ.real {x | ε ≤ f x} = ∫ (x : α) in {x | ε ≤ f x}, ε ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : PseudoMetricSpace β
f g : α →ᵇ β
⊢ edist f g = ⨆ x, edist (f x) (g x)

Tactic:
by

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_nonneg : 0 ≤ᵐ[μ] f
hf_int : Integrable f μ
ε : ℝ
⊢ ε * μ.real {x | ε ≤ f x} ≤ ∫ (x : α), f x ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : PseudoMetricSpace β
⊢ IsInducing (⇑UniformFun.ofFun ∘ DFunLike.coe)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: F : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝² : TopologicalSpace α
inst✝¹ : PseudoMetricSpace β
inst✝ : PseudoMetricSpace γ
f g : α →ᵇ β
x : α
C : ℝ
b : β
⊢ ∀ (x y : α), dist ((ContinuousMap.const α b).toFun x) ((ContinuousMap.const α b).toFun y) ≤ 0

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
G' : Type u_7
G'' : Type u_8
inst✝⁷ : NormedAddCommGroup G''
inst✝⁶ : PartialOrder G''
inst✝⁵ : IsOrderedAddMonoid G''
inst✝⁴ : NormedSpace ℝ G''
inst✝³ : NormedAddCommGroup G'
inst✝² : PartialOrder G'
inst✝¹ : NormedSpace ℝ G'
inst✝ : ClosedIciTopology G''
T : Set α → G' →L[ℝ] G''
C : ℝ
hT : DominatedFinMeasAdditive μ T C
hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∀ (x : G'), 0 ≤ x → 0 ≤ (T s) x
f : α → G'
hf : 0 ≤ᵐ[μ] f
⊢ 0 ≤ setToFun μ T hT f

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
F : Type w
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : CompleteSpace 𝕜
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
⊢ Continuous[Pi.topologicalSpace, topologicalSpace]
    (LinearMap.toContinuousLinearMap.symm ≪≫ₗ LinearEquiv.piRing 𝕜 E ι 𝕜).invFun

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: F✝ : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : PseudoMetricSpace β
inst✝¹ : PseudoMetricSpace γ
f✝ g : α →ᵇ β
x : α
C : ℝ
inst✝ : CompleteSpace β
f : ℕ → α →ᵇ β
hf : CauchySeq f
b : ℕ → ℝ
b0 : ∀ (n : ℕ), 0 ≤ b n
b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N
b_lim : Tendsto b atTop (𝓝 0)
f_bdd : ∀ (x : α) (n m N : ℕ), N ≤ n → N ≤ m → dist ((f n) x) ((f m) x) ≤ b N
fx_cau : ∀ (x : α), CauchySeq fun n ↦ (f n) x
F : α → β
hF : ∀ (x : α), Tendsto (fun n ↦ (f n) x) atTop (𝓝 (F x))
fF_bdd : ∀ (x : α) (N : ℕ), dist ((f N) x) (F x) ≤ b N
⊢ TendstoUniformly (fun n x ↦ (f n) x) F atTop

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_6
inst✝ : NormedAddCommGroup E
f g : α → E
p q : ℝ
hpq : p.HolderConjugate q
hf : MemLp f (ENNReal.ofReal p) μ
hg : MemLp g (ENNReal.ofReal q) μ
x : α
⊢ 0 x ≤ (fun a ↦ ‖g a‖ ^ q) x

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
G' : Type u_7
G'' : Type u_8
inst✝⁸ : NormedAddCommGroup G''
inst✝⁷ : PartialOrder G''
inst✝⁶ : IsOrderedAddMonoid G''
inst✝⁵ : NormedSpace ℝ G''
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : PartialOrder G'
inst✝² : NormedSpace ℝ G'
inst✝¹ : ClosedIciTopology G''
inst✝ : IsOrderedAddMonoid G'
T : Set α → G' →L[ℝ] G''
C : ℝ
hT : DominatedFinMeasAdditive μ T C
hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∀ (x : G'), 0 ≤ x → 0 ≤ (T s) x
f g : α → G'
hf : Integrable f μ
hg : Integrable g μ
hfg : f ≤ᵐ[μ] g
⊢ setToFun μ T hT f ≤ setToFun μ T hT g

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: F : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : PseudoMetricSpace β
inst✝¹ : PseudoMetricSpace γ
f✝ g : α →ᵇ β
x : α
C : ℝ
inst✝ : CompleteSpace β
f : ℕ → α →ᵇ β
hf : CauchySeq f
⊢ ∃ a, Tendsto f atTop (𝓝 a)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: F : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝² : TopologicalSpace α
inst✝¹ : PseudoMetricSpace β
inst✝ : PseudoMetricSpace γ
f✝ g : α →ᵇ β
x✝ : α
C✝ : ℝ
G : β → γ
C : ℝ≥0
H : LipschitzWith C G
f : α →ᵇ β
D : ℝ
hD : ∀ (x y : α), dist (f x) (f y) ≤ D
x y : α
⊢ ↑C * dist (f x) (f y) ≤ ↑(max C 0) * dist (f x) (f y)

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type v
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : CompleteSpace 𝕜
ι : Type u_1
inst✝ : Finite ι
f : ι → E
val✝ : Fintype ι
hf : ((LinearMap.lsum 𝕜 (fun x ↦ 𝕜) ℕ) fun i ↦ LinearMap.id.smulRight (f i)).ker = ⊥
K : ℝ≥0
K0 : K > 0
hK : AntilipschitzWith K ⇑((LinearMap.lsum 𝕜 (fun x ↦ 𝕜) ℕ) fun i ↦ LinearMap.id.smulRight (f i))
this : Tendsto (fun g ↦ ∑ i, ‖g i - f i‖) (𝓝 f) (𝓝 0)
g : ι → E
hg : ∑ i, ‖g i - f i‖ < (↑K)⁻¹
⊢ ∑ i, ‖g i - f i‖₊ < K⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: F : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝² : TopologicalSpace α
inst✝¹ : PseudoMetricSpace β
inst✝ : PseudoMetricSpace γ
f✝ g : α →ᵇ β
x✝ : α
C✝ : ℝ
G : β → γ
C : ℝ≥0
H : LipschitzWith C G
f : α →ᵇ β
D : ℝ
hD : ∀ (x y : α), dist (f x) (f y) ≤ D
x y : α
⊢ ↑(max C 0) * dist (f x) (f y) ≤ ↑(max C 0) * D

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_6
inst✝ : NormedAddCommGroup E
f g : α → E
p q : ℝ
hpq : p.HolderConjugate q
hf : MemLp f (ENNReal.ofReal p) μ
hg : MemLp g (ENNReal.ofReal q) μ
x : α
⊢ 0 x ≤ (fun a ↦ ‖f a‖ ^ p) x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: α : Type u
β : Type v
γ : Type w
inst✝² : TopologicalSpace α
inst✝¹ : PseudoMetricSpace β
inst✝ : PseudoMetricSpace γ
G : β → γ
C : ℝ≥0
H : LipschitzWith C G
f g : α →ᵇ β
x : α
⊢ ↑C * dist (f x) (g x) ≤ ↑C * dist f g

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_6
inst✝ : NormedAddCommGroup E
f g : α → E
p q : ℝ
hpq : p.HolderConjugate q
hf : MemLp f (ENNReal.ofReal p) μ
hg : MemLp g (ENNReal.ofReal q) μ
x : α
⊢ 0 x ≤ (fun a ↦ ‖f a‖ * ‖g a‖) x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: F : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : PseudoMetricSpace β
inst✝² : PseudoMetricSpace γ
f✝ g✝ : α →ᵇ β
x : α
C : ℝ
δ : Type u_2
inst✝¹ : TopologicalSpace δ
inst✝ : DiscreteTopology δ
f : α ↪ δ
g : α →ᵇ β
h : δ →ᵇ β
⊢ ∃ C, ∀ (x y : δ), dist (Function.extend (⇑f) (⇑g) (⇑h) x) (Function.extend (⇑f) (⇑g) (⇑h) y) ≤ C

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_6
inst✝ : NormedAddCommGroup E
f g : α → E
p q : ℝ
hpq : p.HolderConjugate q
hf : MemLp f (ENNReal.ofReal p) μ
hg : MemLp g (ENNReal.ofReal q) μ
⊢ ∫⁻ (a : α), ENNReal.ofReal (‖f a‖ * ‖g a‖) ∂μ = ∫⁻ (a : α), ((fun x ↦ ‖f x‖ₑ) * fun x ↦ ‖g x‖ₑ) a ∂μ

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type v
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : CompleteSpace 𝕜
ι : Type u_1
inst✝ : Finite ι
f : ι → E
hf : LinearIndependent 𝕜 f
⊢ ∀ᶠ (g : ι → E) in 𝓝 f, LinearIndependent 𝕜 g

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_6
inst✝ : NormedAddCommGroup E
f g : α → E
p q : ℝ
hpq : p.HolderConjugate q
hf : MemLp f (ENNReal.ofReal p) μ
hg : MemLp g (ENNReal.ofReal q) μ
h_left : ∫⁻ (a : α), ENNReal.ofReal (‖f a‖ * ‖g a‖) ∂μ = ∫⁻ (a : α), ((fun x ↦ ‖f x‖ₑ) * fun x ↦ ‖g x‖ₑ) a ∂μ
⊢ ∫⁻ (a : α), ENNReal.ofReal (‖f a‖ ^ p) ∂μ = ∫⁻ (a : α), ‖f a‖ₑ ^ p ∂μ

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type w
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace 𝕜
n : ℕ
⊢ IsOpen {f | ↑n ≤ (↑f).rank}

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: α : Type u
β : Type v
inst✝³ : TopologicalSpace α
inst✝² : PseudoMetricSpace β
δ : Type u_2
inst✝¹ : TopologicalSpace δ
inst✝ : DiscreteTopology δ
f : α ↪ δ
g₁ g₂ : α →ᵇ β
h₁ h₂ : δ →ᵇ β
x : ↑(range ⇑f)ᶜ
⊢ dist (h₁ ↑x) (h₂ ↑x) = dist ((extend f g₁ h₁) ↑x) ((extend f g₂ h₂) ↑x)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: α : Type u
β : Type v
inst✝³ : TopologicalSpace α
inst✝² : PseudoMetricSpace β
δ : Type u_2
inst✝¹ : TopologicalSpace δ
inst✝ : DiscreteTopology δ
f : α ↪ δ
g₁ g₂ : α →ᵇ β
h₁ h₂ : δ →ᵇ β
⊢ dist (extend f g₁ h₁) (extend f g₂ h₂) =
    max (dist g₁ g₂) (dist (h₁.domRestrict (range ⇑f)ᶜ) (h₂.domRestrict (range ⇑f)ᶜ))

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: α : Type u
β : Type v
inst✝³ : TopologicalSpace α
inst✝² : PseudoMetricSpace β
δ : Type u_2
inst✝¹ : TopologicalSpace δ
inst✝ : DiscreteTopology δ
f : α ↪ δ
h : δ →ᵇ β
g₁ g₂ : α →ᵇ β
⊢ dist (extend f g₁ h) (extend f g₂ h) = dist g₁ g₂

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: F : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝² : TopologicalSpace α
inst✝¹ : PseudoMetricSpace β
inst✝ : PseudoMetricSpace γ
f g : α →ᵇ β
x : α
C : ℝ
s : Set α
hs : IsClopen s
⊢ ∀ a ∈ frontier s, 1 a = 0

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_6
inst✝ : NormedAddCommGroup E
f g : α → E
p q : ℝ
hpq : p.HolderConjugate q
hf : MemLp f (ENNReal.ofReal p) μ
hg : MemLp g (ENNReal.ofReal q) μ
h_left : ∫⁻ (a : α), ENNReal.ofReal (‖f a‖ * ‖g a‖) ∂μ = ∫⁻ (a : α), ((fun x ↦ ‖f x‖ₑ) * fun x ↦ ‖g x‖ₑ) a ∂μ
h_right_f : ∫⁻ (a : α), ENNReal.ofReal (‖f a‖ ^ p) ∂μ = ∫⁻ (a : α), ‖f a‖ₑ ^ p ∂μ
⊢ ∫⁻ (a : α), ENNReal.ofReal (‖g a‖ ^ q) ∂μ = ∫⁻ (a : α), ‖g a‖ₑ ^ q ∂μ

Tactic:
by

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

Goal state before:
pp: 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 : ℝ
hT : DominatedFinMeasAdditive μ T C
⊢ Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun f ↦ setToFun μ T hT ↑↑f

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type v
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : CompleteSpace 𝕜
ι : Type u_1
inst✝ : Finite ι
i₀ : ι
ι' : Type (max 0 u_1) := { x // x ≠ i₀ }
val✝ : Fintype ι
this : Fintype ι'
⊢ Continuous[Pi.topologicalSpace, Pi.topologicalSpace] fun p i ↦ p ↑i -ᵥ p i₀

Tactic:
by

Goal state after: