mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.ContinuousMap.Bounded.Star

Goal state before:
pp: F : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : NonUnitalNormedRing β
inst✝¹ : StarRing β
inst✝ : CStarRing β
f : α →ᵇ β
⊢ ‖f‖ * ‖f‖ ≤ ‖star f * f‖

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Compact

Goal state before:
pp: α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
inst✝¹ : PseudoMetricSpace β
inst✝ : SeminormedAddCommGroup E
f : C(α, β)
⊢ (mkOfCompact f).toContinuousMap = f

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Compact

Goal state before:
pp: α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
inst✝¹ : PseudoMetricSpace β
inst✝ : SeminormedAddCommGroup E
f : α →ᵇ β
⊢ mkOfCompact f.toContinuousMap = f

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Compact

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : CompactSpace α
inst✝ : PseudoMetricSpace β
⊢ ∀ (s : Set (C(α, β) × C(α, β))),
    s ∈ uniformity C(α, β) ↔
      ∃ t ∈ uniformity (α →ᵇ β),
        ∀ (x y : C(α, β)), ((equivBoundedOfCompact α β) x, (equivBoundedOfCompact α β) y) ∈ t → (x, y) ∈ s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Compact

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : CompactSpace α
inst✝ : PseudoMetricSpace β
f g : C(α, β)
x : α
⊢ dist (f x) (g x) ≤ dist f g

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Compact

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : CompactSpace α
inst✝ : PseudoMetricSpace β
f g : C(α, β)
C : ℝ
C0 : 0 ≤ C
⊢ dist f g ≤ C ↔ ∀ (x : α), dist (f x) (g x) ≤ C

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Compact

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
inst✝¹ : PseudoMetricSpace β
f g : C(α, β)
C : ℝ
inst✝ : Nonempty α
⊢ dist f g ≤ C ↔ ∀ (x : α), dist (f x) (g x) ≤ C

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Compact

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
inst✝¹ : PseudoMetricSpace β
f g : C(α, β)
C : ℝ
inst✝ : Nonempty α
⊢ dist f g < C ↔ ∀ (x : α), dist (f x) (g x) < C

Tactic:
by

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

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
f : α → E
⊢ (Summable fun x ↦ ‖f x‖) ↔ Summable f

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Compact

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : CompactSpace α
inst✝ : PseudoMetricSpace β
f g : C(α, β)
C : ℝ
C0 : 0 < C
⊢ dist f g < C ↔ ∀ (x : α), dist (f x) (g x) < C

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 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 : AEStronglyMeasurable f (Measure.map φ μ)
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
⊢ setToFun μ' T' hT' f = setToFun μ T hT (f ∘ φ)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Compact

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : CompactSpace α
inst✝ : PseudoMetricSpace β
f g : C(α, β)
⊢ dist f g = ⨆ x, dist (f x) (g x)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Compact

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : CompactSpace α
inst✝ : PseudoMetricSpace β
f g : C(α, β)
⊢ nndist f g = ⨆ x, nndist (f x) (g x)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Compact

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : CompactSpace α
inst✝ : PseudoMetricSpace β
f g : C(α, β)
⊢ edist f g = ⨆ x, edist (f x) (g x)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Compact

Goal state before:
pp: α : Type u_1
β : Type u_2
E : Type u_3
inst✝⁸ : TopologicalSpace α
inst✝⁷ : CompactSpace α
inst✝⁶ : PseudoMetricSpace β
inst✝⁵ : SeminormedAddCommGroup E
f✝ g✝ : C(α, β)
C : ℝ
R : Type u_4
inst✝⁴ : Zero R
inst✝³ : Zero β
inst✝² : PseudoMetricSpace R
inst✝¹ : SMul R β
inst✝ : IsBoundedSMul R β
r : R
f g : C(α, β)
⊢ dist (r • f) (r • g) ≤ dist r 0 * dist f g

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Compact

Goal state before:
pp: α : Type u_1
β : Type u_2
E : Type u_3
inst✝⁸ : TopologicalSpace α
inst✝⁷ : CompactSpace α
inst✝⁶ : PseudoMetricSpace β
inst✝⁵ : SeminormedAddCommGroup E
f✝ g : C(α, β)
C : ℝ
R : Type u_4
inst✝⁴ : Zero R
inst✝³ : Zero β
inst✝² : PseudoMetricSpace R
inst✝¹ : SMul R β
inst✝ : IsBoundedSMul R β
r₁ r₂ : R
f : C(α, β)
⊢ dist (r₁ • f) (r₂ • f) ≤ dist r₁ r₂ * dist f 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Compact

Goal state before:
pp: α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
inst✝¹ : PseudoMetricSpace β
inst✝ : SeminormedAddCommGroup E
x y : C(α, E)
⊢ dist x y = ‖-x + y‖

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Compact

Goal state before:
pp: α : Type u_1
β : Type u_2
E✝ : Type u_3
inst✝⁶ : TopologicalSpace α
inst✝⁵ : CompactSpace α
inst✝⁴ : PseudoMetricSpace β
inst✝³ : SeminormedAddCommGroup E✝
inst✝² : Nonempty α
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : Nontrivial E
⊢ NontrivialTopology C(α, E)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Compact

Goal state before:
pp: α : Type u_1
β : Type u_2
E : Type u_3
inst✝⁶ : TopologicalSpace α
inst✝⁵ : CompactSpace α
inst✝⁴ : PseudoMetricSpace β
inst✝³ : SeminormedAddCommGroup E
inst✝² : Nonempty α
inst✝¹ : One E
inst✝ : NormOneClass E
⊢ ‖1‖ = 1

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Compact

Goal state before:
pp: α : Type u_1
E : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
inst✝¹ : SeminormedAddCommGroup E
f : C(α, E)
inst✝ : Fintype α
⊢ 0 ≤ ‖⇑f‖

Tactic:
by

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

Goal state before:
pp: F : Type u_1
inst✝² : NormedRing F
inst✝¹ : NormOneClass F
inst✝ : NormMulClass F
k : ℕ
r : F
hr : ‖r‖ < 1
u : ℕ → F
hu : u =O[atTop] fun n ↦ ↑(n ^ k)
r' : ℝ
hrr' : ‖r‖ < r'
h : r' < 1
⊢ (fun n ↦ ‖u n * r ^ n‖) =O[atTop] fun n ↦ ‖u n‖ * ‖r‖ ^ n

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Compact

Goal state before:
pp: α : Type u_1
E : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
inst✝¹ : SeminormedAddCommGroup E
f : C(α, E)
inst✝ : Fintype α
⊢ 0 ≤ ‖f‖

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Compact

Goal state before:
pp: α : Type u_1
E : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
inst✝¹ : SeminormedAddCommGroup E
f : C(α, E)
inst✝ : Fintype α
⊢ ‖f‖ = ‖⇑f‖

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Compact

Goal state before:
pp: α : Type u_1
β : Type u_2
E : Type u_3
inst✝⁶ : TopologicalSpace α
inst✝⁵ : CompactSpace α
inst✝⁴ : PseudoMetricSpace β
inst✝³ : SeminormedAddCommGroup E
𝕜 : Type u_4
inst✝² : NormedRing 𝕜
inst✝¹ : Module 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
c : 𝕜
f : C(α, E)
⊢ (addEquivBoundedOfCompact α E).toFun (c • f) = (RingHom.id 𝕜) c • (addEquivBoundedOfCompact α E).toFun f

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Compact

Goal state before:
pp: α : Type u_1
inst✝⁵ : TopologicalSpace α
inst✝⁴ : CompactSpace α
R : Type u_4
β : Type u_5
inst✝³ : SeminormedAddCommGroup β
inst✝² : SeminormedRing R
inst✝¹ : Module R β
inst✝ : NormSMulClass R β
f : C(α, R)
b : β
⊢ ‖f • const α b‖₊ = ‖f‖₊ * ‖b‖₊

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Compact

Goal state before:
pp: α : Type u_1
inst✝⁵ : TopologicalSpace α
inst✝⁴ : CompactSpace α
R : Type u_4
β : Type u_5
inst✝³ : SeminormedAddCommGroup β
inst✝² : SeminormedRing R
inst✝¹ : Module R β
inst✝ : NormSMulClass R β
f : C(α, R)
b : β
⊢ ‖f • const α b‖ = ‖f‖ * ‖b‖

Tactic:
by

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

Goal state before:
pp: F : Type u_1
inst✝² : NormedRing F
inst✝¹ : NormOneClass F
inst✝ : NormMulClass F
k : ℕ
r : F
hr : ‖r‖ < 1
u : ℕ → F
hu : u =O[atTop] fun n ↦ ↑(n ^ k)
r' : ℝ
hrr' : ‖r‖ < r'
h : r' < 1
⊢ (fun n ↦ ‖u n‖ * ‖r‖ ^ n) =O[atTop] fun n ↦ ‖↑n ^ k‖ * ‖r‖ ^ n

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Compact

Goal state before:
pp: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
R : Type u_4
inst✝¹ : NonUnitalSeminormedRing R
inst✝ : IsCancelMulZero R
f g : C(α, R)
h : f * g = 0
⊢ mkOfCompact f * mkOfCompact g = 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Compact

Goal state before:
pp: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
R : Type u_4
inst✝¹ : NonUnitalSeminormedRing R
inst✝ : IsCancelMulZero R
f g : C(α, R)
h : f * g = 0
⊢ ‖f + g‖ = max ‖f‖ ‖g‖

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Compact

Goal state before:
pp: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
R : Type u_4
inst✝¹ : NonUnitalSeminormedRing R
inst✝ : IsCancelMulZero R
f g : C(α, R)
h : f * g = 0
⊢ f * -g = 0

Tactic:
by

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

Goal state before:
pp: F : Type u_1
inst✝² : NormedRing F
inst✝¹ : NormOneClass F
inst✝ : NormMulClass F
k : ℕ
r : F
hr : ‖r‖ < 1
u : ℕ → F
hu : u =O[atTop] fun n ↦ ↑(n ^ k)
r' : ℝ
hrr' : ‖r‖ < r'
h : r' < 1
⊢ (fun n ↦ ‖↑n ^ k‖ * ‖r‖ ^ n) =O[atTop] fun n ↦ ‖r' ^ n‖

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Compact

Goal state before:
pp: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
R : Type u_4
inst✝¹ : NonUnitalSeminormedRing R
inst✝ : IsCancelMulZero R
f g : C(α, R)
h : f * g = 0
⊢ ‖f - g‖ = max ‖f‖ ‖g‖

Tactic:
by

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

Goal state before:
pp: F : Type u_1
inst✝² : NormedRing F
inst✝¹ : NormOneClass F
inst✝ : NormMulClass F
k : ℕ
r : F
hr : ‖r‖ < 1
u : ℕ → F
hu : u =O[atTop] fun n ↦ ↑(n ^ k)
⊢ Summable fun n ↦ ‖u n * r ^ n‖

Tactic:
by

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

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace ℝ F
inst✝¹ : FiniteDimensional ℝ E
inst✝ : FiniteDimensional ℝ F
f : ℕ → E
g : ℕ → F
h : f =Θ[atTop] g
⊢ f =Θ[cofinite] g

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Compact

Goal state before:
pp: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
R : Type u_4
inst✝¹ : NonUnitalSeminormedRing R
inst✝ : IsCancelMulZero R
ι : Type u_5
f : ι → C(α, R)
h : Pairwise ((fun x1 x2 ↦ x1 * x2 = 0) on f)
j : ι
s : Finset ι
hj : j ∉ s
ih : ‖∑ i ∈ s, f i‖₊ = s.sup fun x ↦ ‖f x‖₊
this : f j * ∑ i ∈ s, f i = 0
⊢ ‖∑ i ∈ insert j s, f i‖₊ = (insert j s).sup fun x ↦ ‖f x‖₊

Tactic:
by

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

Goal state before:
pp: α : Type u_1
G : Type u_5
inst✝ : NormedAddCommGroup G
m : MeasurableSpace α
μ μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ∞
hμ'_le : μ' ≤ c' • μ
hc'0 : c' = 0
⊢ μ' = 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Compact

Goal state before:
pp: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
R : Type u_4
inst✝¹ : NonUnitalSeminormedRing R
inst✝ : IsCancelMulZero R
ι : Type u_5
f : ι → C(α, R)
h : Pairwise ((fun x1 x2 ↦ x1 * x2 = 0) on f)
j : ι
s : Finset ι
hj : j ∉ s
ih : ‖∑ i ∈ s, f i‖₊ = s.sup fun x ↦ ‖f x‖₊
i : ι
hi : i ∈ s
⊢ j ≠ i

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
p q : ℝ
hpq : p.HolderConjugate q
f g : α → ℝ
hf_nonneg : 0 ≤ᵐ[μ] f
hg_nonneg : 0 ≤ᵐ[μ] g
hf : MemLp f (ENNReal.ofReal p) μ
hg : MemLp g (ENNReal.ofReal q) μ
⊢ ∫ (a : α), f a * g a ∂μ ≤ (∫ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫ (a : α), g a ^ q ∂μ) ^ (1 / q)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
m : MeasurableSpace α
hE : CompleteSpace E
μ : Measure α
f : α → E
hf : StronglyMeasurable f
a : α
⊢ ∫ (a : α) in {a}, f a ∂μ = μ.real {a} • f a

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Compact

Goal state before:
pp: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
R : Type u_4
inst✝¹ : NonUnitalSeminormedRing R
inst✝ : IsCancelMulZero R
ι : Type u_5
f : ι → C(α, R)
s : Finset ι
h : Pairwise ((fun x1 x2 ↦ x1 * x2 = 0) on f)
⊢ ‖∑ i ∈ s, f i‖₊ = s.sup fun x ↦ ‖f x‖₊

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
m : MeasurableSpace α
hE : CompleteSpace E
inst✝ : MeasurableSingletonClass α
μ : Measure α
f : α → E
a : α
⊢ ∫ (a : α) in {a}, f a ∂μ = μ.real {a} • f a

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Compact

Goal state before:
pp: X : Type u_1
inst✝³ : TopologicalSpace X
inst✝² : LocallyCompactSpace X
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
ι : Type u_3
F : ι → C(X, E)
hF : ∀ (K : Compacts X), Summable fun i ↦ ‖restrict (↑K) (F i)‖
K : Compacts X
⊢ ∀ (s : Finset ι), restrict (↑K) (∑ i ∈ s, F i) = ∑ i ∈ s, restrict (↑K) (F i)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Compact

Goal state before:
pp: X : Type u_1
inst✝³ : TopologicalSpace X
inst✝² : LocallyCompactSpace X
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
ι : Type u_3
F : ι → C(X, E)
hF : ∀ (K : Compacts X), Summable fun i ↦ ‖restrict (↑K) (F i)‖
⊢ Summable F

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Compact

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : SeminormedAddCommGroup β
inst✝² : StarAddMonoid β
inst✝¹ : NormedStarGroup β
inst✝ : CompactSpace α
f : C(α, β)
⊢ ‖star f‖ ≤ ‖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✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
hE : CompleteSpace E
inst✝ : Unique α
f : α → E
⊢ ∫ (x : α), f x ∂μ = ∫ (x : α), f default ∂μ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
G : Type u_5
inst✝ : NormedAddCommGroup G
m : MeasurableSpace α
μ μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ∞
hμ'_le : μ' ≤ c' • μ
hc'0 : c' = 0
hμ'0 : μ' = 0
⊢ (fun f ↦ Integrable.toL1 ↑↑f ⋯) = 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Compact

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : CompactSpace α
inst✝² : NonUnitalNormedRing β
inst✝¹ : StarRing β
inst✝ : CStarRing β
f : C(α, β)
⊢ ‖f‖ * ‖f‖ ≤ ‖star f * 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✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
hE : CompleteSpace E
inst✝ : Unique α
f : α → E
⊢ ∫ (x : α), f default ∂μ = μ.real univ • f default

Tactic:
by

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

Goal state before:
pp: F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
β : Type u_6
m m0 : MeasurableSpace β
μ : Measure β
hm : m ≤ m0
f : β →ₛ F
hf_int : Integrable (⇑f) μ
⊢ ∫ (x : β), f x ∂μ = ∑ x ∈ f.range, μ.real (⇑f ⁻¹' {x}) • x

Tactic:
by

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

Goal state before:
pp: F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
β : Type u_6
m m0 : MeasurableSpace β
μ : Measure β
hm : m ≤ m0
f : β →ₛ F
hf_int : Integrable (⇑f) μ
⊢ ∫ (x : β), f x ∂μ = ∫ (x : β), f x ∂μ.trim hm

Tactic:
by

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

Goal state before:
pp: G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
β : Type u_6
m m0 : MeasurableSpace β
μ : Measure β
hm : m ≤ m0
f : β → G
hf : StronglyMeasurable f
hG : CompleteSpace G
this✝¹ : MeasurableSpace G := borel G
this✝ : BorelSpace G
hf_int : Integrable f μ
this : SeparableSpace ↑(range f ∪ {0})
⊢ 0 ∈ range f ∪ {0}

Tactic:
by

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

Goal state before:
pp: case pos
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
β : Type u_6
m m0 : MeasurableSpace β
μ : Measure β
hm : m ≤ m0
f : β → G
hf : StronglyMeasurable f
hG : CompleteSpace G
this✝¹ : MeasurableSpace G := borel G
this✝ : BorelSpace G
hf_int : Integrable f μ
this : SeparableSpace ↑(range f ∪ {0})
f_seq : ℕ → β →ₛ G := SimpleFunc.approxOn f ⋯ (range f ∪ {0}) 0 ⋯
hf_seq_meas : ∀ (n : ℕ), StronglyMeasurable ⇑(f_seq n)
hf_seq_int : ∀ (n : ℕ), Integrable (⇑(f_seq n)) μ
⊢ ∫ (x : β), f x ∂μ = ∫ (x : β), f x ∂μ.trim hm

Tactic:
have hf_seq_int_m : ∀ n, Integrable (f_seq n) (μ.trim hm) := fun n =>
    (hf_seq_int n).trim hm (hf_seq_meas n)

Goal state after:
case pos
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
β : Type u_6
m m0 : MeasurableSpace β
μ : Measure β
hm : m ≤ m0
f : β → G
hf : StronglyMeasurable f
hG : CompleteSpace G
this✝¹ : MeasurableSpace G := borel G
this✝ : BorelSpace G
hf_int : Integrable f μ
this : SeparableSpace ↑(range f ∪ {0})
f_seq : ℕ → β →ₛ G := SimpleFunc.approxOn f ⋯ (range f ∪ {0}) 0 ⋯
hf_seq_meas : ∀ (n : ℕ), StronglyMeasurable ⇑(f_seq n)
hf_seq_int : ∀ (n : ℕ), Integrable (⇑(f_seq n)) μ
hf_seq_int_m : ∀ (n : ℕ), Integrable (⇑(f_seq n)) (μ.trim hm)
⊢ ∫ (x : β), f x ∂μ = ∫ (x : β), f x ∂μ.trim hm
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
⊢ Continuous[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace, _] (thickenedIndicatorAux δ E)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
E : Set α
x : α
⊢ thickenedIndicatorAux δ E x ≤ 1

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
E : Set α
⊢ thickenedIndicatorAux δ (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E) = thickenedIndicatorAux δ E

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
E : Set α
x : α
x_in_E : x ∈ E
⊢ thickenedIndicatorAux δ E x = 1

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
E : Set α
x : α
x_mem : x ∈ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E
⊢ thickenedIndicatorAux δ E x = 1

Tactic:
by

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

Goal state before:
pp: G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
β : Type u_6
m m0 : MeasurableSpace β
μ : Measure β
hm : m ≤ m0
f : β → G
hf : StronglyMeasurable f
hG : CompleteSpace G
this✝¹ : MeasurableSpace G := borel G
this✝ : BorelSpace G
hf_int : Integrable f μ
this : SeparableSpace ↑(range f ∪ {0})
f_seq : ℕ → β →ₛ G := SimpleFunc.approxOn f ⋯ (range f ∪ {0}) 0 ⋯
hf_seq_meas : ∀ (n : ℕ), StronglyMeasurable ⇑(f_seq n)
hf_seq_int : ∀ (n : ℕ), Integrable (⇑(f_seq n)) μ
hf_seq_int_m : ∀ (n : ℕ), Integrable (⇑(f_seq n)) (μ.trim hm)
hf_seq_eq : ∀ (n : ℕ), ∫ (x : β), (f_seq n) x ∂μ = ∫ (x : β), (f_seq n) x ∂μ.trim hm
⊢ Tendsto (fun n ↦ ∫ (x : β), (f_seq n) x ∂μ) atTop (𝓝 (∫ (x : β), f x ∂μ))

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
x : α
x_out : x ∉ thickening δ E
⊢ thickenedIndicatorAux δ E x = 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
E : Set α
⊢ (E.indicator fun x ↦ 1) ≤ thickenedIndicatorAux δ E

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
E : Set α
x y : α
h : infEDist x E ≤ infEDist y E
hle : infEDist x E / ENNReal.ofReal δ ≤ 1
⊢ 1 ≠ ∞

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
E : Set α
x y : α
h : infEDist x E ≤ infEDist y E
hle : 1 ≤ infEDist x E / ENNReal.ofReal δ
⊢ infEDist x E / ENNReal.ofReal δ ≤ infEDist y E / ENNReal.ofReal δ

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
E : Set α
x y : α
h : infEDist x E ≤ infEDist y E
⊢ thickenedIndicatorAux δ E y ≤ thickenedIndicatorAux δ E x

Tactic:
by

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

Goal state before:
pp: G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
β : Type u_6
m m0 : MeasurableSpace β
μ : Measure β
hm : m ≤ m0
f : β → G
hf : StronglyMeasurable f
hG : CompleteSpace G
this✝¹ : MeasurableSpace G := borel G
this✝ : BorelSpace G
hf_int : Integrable f μ
this : SeparableSpace ↑(range f ∪ {0})
f_seq : ℕ → β →ₛ G := SimpleFunc.approxOn f ⋯ (range f ∪ {0}) 0 ⋯
hf_seq_meas : ∀ (n : ℕ), StronglyMeasurable ⇑(f_seq n)
hf_seq_int : ∀ (n : ℕ), Integrable (⇑(f_seq n)) μ
hf_seq_int_m : ∀ (n : ℕ), Integrable (⇑(f_seq n)) (μ.trim hm)
hf_seq_eq : ∀ (n : ℕ), ∫ (x : β), (f_seq n) x ∂μ = ∫ (x : β), (f_seq n) x ∂μ.trim hm
h_lim_1 : Tendsto (fun n ↦ ∫ (x : β), (f_seq n) x ∂μ) atTop (𝓝 (∫ (x : β), f x ∂μ))
⊢ Tendsto (fun n ↦ ∫ (x : β), (f_seq n) x ∂μ) atTop (𝓝 (∫ (x : β), f x ∂μ.trim hm))

Tactic:
by

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

Goal state before:
pp: α : Type u_1
G : Type u_5
inst✝ : NormedAddCommGroup G
m : MeasurableSpace α
μ μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ∞
hμ'_le : μ' ≤ c' • μ
hc'0 : ¬c' = 0
f : ↥(Lp G 1 μ)
ε : ℝ
hε_pos : ε > 0
g : ↥(Lp G 1 μ)
hfg : (eLpNorm (↑↑g - ↑↑f) 1 μ).toReal < ε / 2 / c'.toReal
h_int : ∀ (f' : ↥(Lp G 1 μ)), Integrable (↑↑f') μ' :=
  fun f' ↦ Integrable.of_measure_le_smul hc' hμ'_le (L1.integrable_coeFn f')
this : eLpNorm (↑↑(Integrable.toL1 ↑↑g ⋯) - ↑↑(Integrable.toL1 ↑↑f ⋯)) 1 μ' = eLpNorm (↑↑g - ↑↑f) 1 μ'
⊢ eLpNorm (↑↑g - ↑↑f) 1 μ ≠ ∞

Tactic:
by

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

Goal state before:
pp: G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
β : Type u_6
m m0 : MeasurableSpace β
μ : Measure β
hm : m ≤ m0
f : β → G
hf : StronglyMeasurable f
⊢ ∫ (x : β), f x ∂μ = ∫ (x : β), f x ∂μ.trim hm

Tactic:
by

Goal state after: