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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝ : SeminormedAddCommGroup β
T : Set α → β
C : ℝ
c : ℝ≥0∞
hc_ne_top : c ≠ ∞
hT : DominatedFinMeasAdditive (c • μ) T C
⊢ DominatedFinMeasAdditive μ T (c.toReal * C)

Tactic:
by
  have h : ∀ s, MeasurableSet s → c • μ s = ∞ → μ s = ∞ := by
    intro s _ hcμs
    simp only [hc_ne_top, smul_eq_mul, ENNReal.mul_eq_top, or_false, Ne,
      false_and] at hcμs
    exact hcμs.2
  refine ⟨hT.1.of_eq_top_imp_eq_top (μ := c • μ) h, fun s hs hμs => ?_⟩
  have hcμs : c • μ s ≠ ∞ := mt (h s hs) hμs.ne
  rw [smul_eq_mul] at hcμs
  refine (hT.2 s hs hcμs.lt_top).trans (le_of_eq ?_)
  simp only [measureReal_ennreal_smul_apply]
  ring

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

Goal state before:
pp: α : Type u_1
F : Type u_3
F' : Type u_4
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedAddCommGroup F'
inst✝¹ : NormedSpace ℝ F'
m : MeasurableSpace α
inst✝ : DecidablePred fun x ↦ x ≠ 0
T : Set α → F →L[ℝ] F'
hT : T ∅ = 0
f : α →ₛ F
s : Finset F
hs : {x ∈ f.range | x ≠ 0} ⊆ s
⊢ ∑ x ∈ f.range with x ≠ 0, (T (⇑f ⁻¹' {x})) x = ∑ x ∈ s, (T (⇑f ⁻¹' {x})) x

Tactic:
Finset.sum_subset hs

Goal state after:
α : Type u_1
F : Type u_3
F' : Type u_4
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedAddCommGroup F'
inst✝¹ : NormedSpace ℝ F'
m : MeasurableSpace α
inst✝ : DecidablePred fun x ↦ x ≠ 0
T : Set α → F →L[ℝ] F'
hT : T ∅ = 0
f : α →ₛ F
s : Finset F
hs : {x ∈ f.range | x ≠ 0} ⊆ s
⊢ ∑ x ∈ s, (T (⇑f ⁻¹' {x})) x = ∑ x ∈ s, (T (⇑f ⁻¹' {x})) x

α : Type u_1
F : Type u_3
F' : Type u_4
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedAddCommGroup F'
inst✝¹ : NormedSpace ℝ F'
m : MeasurableSpace α
inst✝ : DecidablePred fun x ↦ x ≠ 0
T : Set α → F →L[ℝ] F'
hT : T ∅ = 0
f : α →ₛ F
s : Finset F
hs : {x ∈ f.range | x ≠ 0} ⊆ s
⊢ ∀ x ∈ s, x ∉ {x ∈ f.range | x ≠ 0} → (T (⇑f ⁻¹' {x})) x = 0
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: case add
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
hp_pos : p ≠ 0
hp_ne_top : p ≠ ∞
P : ↥(simpleFunc E p μ) → Prop
indicatorConst : ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ∞), P (simpleFunc.indicatorConst p hs ⋯ c)
add :
  ∀ ⦃f g : α →ₛ E⦄ (hf : MemLp (⇑f) p μ) (hg : MemLp (⇑g) p μ),
    Disjoint (support ⇑f) (support ⇑g) → P (f.toLp hf) → P (g.toLp hg) → P (f.toLp hf + g.toLp hg)
⊢ ∀ ⦃f g : α →ₛ E⦄,
    Disjoint (support ⇑f) (support ⇑g) →
      (∀ (hf : MemLp (⇑f) p μ), P (f.toLp hf)) →
        (∀ (hf : MemLp (⇑g) p μ), P (g.toLp hf)) → ∀ (hf : MemLp (⇑(f + g)) p μ), P ((f + g).toLp hf)

Tactic:
intro f g hfg hf hg hfg'

Goal state after:
case add
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
hp_pos : p ≠ 0
hp_ne_top : p ≠ ∞
P : ↥(simpleFunc E p μ) → Prop
indicatorConst : ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ∞), P (simpleFunc.indicatorConst p hs ⋯ c)
add :
  ∀ ⦃f g : α →ₛ E⦄ (hf : MemLp (⇑f) p μ) (hg : MemLp (⇑g) p μ),
    Disjoint (support ⇑f) (support ⇑g) → P (f.toLp hf) → P (g.toLp hg) → P (f.toLp hf + g.toLp hg)
f g : α →ₛ E
hfg : Disjoint (support ⇑f) (support ⇑g)
hf : ∀ (hf : MemLp (⇑f) p μ), P (f.toLp hf)
hg : ∀ (hf : MemLp (⇑g) p μ), P (g.toLp hf)
hfg' : MemLp (⇑(f + g)) p μ
⊢ P ((f + g).toLp hfg')
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:
pp: α : Type u_1
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝ : DecidablePred fun x ↦ x ≠ 0
f : α →ₛ F
s : Finset F
hs : {x ∈ f.range | x ≠ 0} ⊆ s
⊢ ∑ x ∈ f.range with x ≠ 0, μ.real (⇑f ⁻¹' {x}) • x = ∑ x ∈ s, μ.real (⇑f ⁻¹' {x}) • x

Tactic:
Finset.sum_subset hs

Goal state after:
α : Type u_1
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝ : DecidablePred fun x ↦ x ≠ 0
f : α →ₛ F
s : Finset F
hs : {x ∈ f.range | x ≠ 0} ⊆ s
⊢ ∑ x ∈ s, μ.real (⇑f ⁻¹' {x}) • x = ∑ x ∈ s, μ.real (⇑f ⁻¹' {x}) • x

α : Type u_1
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝ : DecidablePred fun x ↦ x ≠ 0
f : α →ₛ F
s : Finset F
hs : {x ∈ f.range | x ≠ 0} ⊆ s
⊢ ∀ x ∈ s, x ∉ {x ∈ f.range | x ≠ 0} → μ.real (⇑f ⁻¹' {x}) • x = 0
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ↥(α →₁ₛ[μ] ℝ)
⊢ ⇑(toSimpleFunc f).posPart =ᵐ[μ] ⇑(SimpleFunc.map norm (toSimpleFunc (posPart f)))

Tactic:
filter_upwards [posPart_toSimpleFunc f] with _ h

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ↥(α →₁ₛ[μ] ℝ)
a✝ : α
h : (toSimpleFunc (posPart f)) a✝ = (toSimpleFunc f).posPart a✝
⊢ (toSimpleFunc f).posPart a✝ = (SimpleFunc.map norm (toSimpleFunc (posPart f))) a✝
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:
pp: α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝⁶ : NormedAddCommGroup E
m : MeasurableSpace α
μ : Measure α
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedRing 𝕜
inst✝³ : Module 𝕜 E
inst✝² : IsBoundedSMul 𝕜 E
inst✝¹ : SMulCommClass ℝ 𝕜 E
inst✝ : CompleteSpace E
f✝ : ↥(Lp E 1 μ)
f : ↥(α →₁ₛ[μ] E)
⊢ integral ↑f = (integralCLM' 𝕜) ↑f

Tactic:
norm_cast

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

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
h_add : FinMeasAdditive μ T
c : ℝ
f : α →ₛ E
hf : Integrable (⇑f) μ
⊢ ∑ x ∈ f.range, c • (T (⇑f ⁻¹' {x})) x = c • setToSimpleFunc T f

Tactic:
by simp only [setToSimpleFunc, smul_sum]

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

Goal state before:
pp: α : Type u_1
F : Type u_3
𝕜 : Type u_6
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
E : Type u_7
inst✝³ : NormedAddCommGroup E
inst✝² : SMulZeroClass 𝕜 E
inst✝¹ : NormedSpace ℝ E
inst✝ : DistribSMul 𝕜 F
T : Set α → E →L[ℝ] F
h_add : FinMeasAdditive μ T
h_smul : ∀ (c : 𝕜) (s : Set α) (x : E), (T s) (c • x) = c • (T s) x
c : 𝕜
f : α →ₛ E
hf : Integrable (⇑f) μ
⊢ ∑ x ∈ f.range, c • (T (⇑f ⁻¹' {x})) x = c • setToSimpleFunc T f

Tactic:
by simp only [setToSimpleFunc, smul_sum]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:
pp: case refine_3
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ↥(Lp ℝ 1 μ)
s : ↥(α →₁ₛ[μ] ℝ)
⊢ integral ↑s = ‖Lp.posPart ↑s‖ - ‖Lp.negPart ↑s‖

Tactic:
norm_cast

Goal state after:
case refine_3
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ↥(Lp ℝ 1 μ)
s : ↥(α →₁ₛ[μ] ℝ)
⊢ SimpleFunc.integral s = ‖SimpleFunc.posPart s‖ - ‖SimpleFunc.negPart s‖
Module: Mathlib.Analysis.Asymptotics.AsymptoticEquivalent

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : NormedAddCommGroup β
u : α → β
l : Filter α
h : u =O[l] 0
⊢ ∃ s ∈ l, Set.EqOn u 0 s

Tactic:
exact ⟨{ x : α | u x = 0 }, isBigO_zero_right_iff.mp h, fun x hx ↦ hx⟩

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

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

Tactic:
integral_undef hfi

Goal state after:
case neg
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : 0 ≤ᵐ[μ] f
hfm : AEStronglyMeasurable f μ
hfi : ¬Integrable f μ
⊢ 0 = (∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ).toReal
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
inst✝ : CompleteSpace E
f : α →ₛ E
hfi : Integrable (⇑f) μ
⊢ integral μ f = ∫ (x : α), f x ∂μ

Tactic:
rw [MeasureTheory.integral_eq f hfi, ← L1.SimpleFunc.toLp_one_eq_toL1,
    L1.SimpleFunc.integral_L1_eq_integral, L1.SimpleFunc.integral_eq_integral]

Goal state after:
α : Type u_1
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace E
f : α →ₛ E
hfi : Integrable (⇑f) μ
⊢ integral μ f = integral μ (Lp.simpleFunc.toSimpleFunc (f.toLp ⋯))
Module: Mathlib.Analysis.Normed.Group.AddTorsor

Goal state before:
pp: V : Type u_2
P : Type u_3
inst✝² : SeminormedAddCommGroup V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
v v' : V
p p' : P
⊢ ↑(nndist (v +ᵥ p) (v' +ᵥ p')) ≤ ↑(nndist v v') + ↑(nndist p p')

Tactic:
norm_cast

Goal state after:
V : Type u_2
P : Type u_3
inst✝² : SeminormedAddCommGroup V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
v v' : V
p p' : P
⊢ nndist (v +ᵥ p) (v' +ᵥ p') ≤ nndist v v' + nndist p p'
Module: Mathlib.Analysis.Normed.Group.AddTorsor

Goal state before:
pp: V : Type u_2
P : Type u_3
inst✝² : SeminormedAddCommGroup V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
p₁ p₂ p₃ p₄ : P
⊢ ↑(nndist (p₁ -ᵥ p₂) (p₃ -ᵥ p₄)) ≤ ↑(nndist p₁ p₃) + ↑(nndist p₂ p₄)

Tactic:
norm_cast

Goal state after:
V : Type u_2
P : Type u_3
inst✝² : SeminormedAddCommGroup V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
p₁ p₂ p₃ p₄ : P
⊢ nndist (p₁ -ᵥ p₂) (p₃ -ᵥ p₄) ≤ nndist p₁ p₃ + nndist p₂ p₄
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf : ∀ (n : ℕ), Integrable (f n) μ
hF : Integrable F μ
h_mono : ∀ᵐ (x : α) ∂μ, Monotone fun n ↦ f n x
h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (F x))
f' : ℕ → α → ℝ := fun n x ↦ f n x - f 0 x
hf'_nonneg : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), 0 ≤ f' n x
⊢ Tendsto (fun n ↦ ∫ (x : α), f n x ∂μ) atTop (𝓝 (∫ (x : α), F x ∂μ))

Tactic:
have hf'_meas : ∀ n, Integrable (f' n) μ := fun n ↦ (hf n).sub (hf 0)

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf : ∀ (n : ℕ), Integrable (f n) μ
hF : Integrable F μ
h_mono : ∀ᵐ (x : α) ∂μ, Monotone fun n ↦ f n x
h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (F x))
f' : ℕ → α → ℝ := fun n x ↦ f n x - f 0 x
hf'_nonneg : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), 0 ≤ f' n x
hf'_meas : ∀ (n : ℕ), Integrable (f' n) μ
⊢ Tendsto (fun n ↦ ∫ (x : α), f n x ∂μ) atTop (𝓝 (∫ (x : α), F x ∂μ))
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf_int : ∀ (n : ℕ), Integrable (f n) μ
hF_int : Integrable F μ
hf_tendsto : Tendsto (fun i ↦ ∫ (a : α), f i a ∂μ) atTop (𝓝 (∫ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i ↦ f i a
hf_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
f' : ℕ → α → ℝ≥0∞ := fun n a ↦ ENNReal.ofReal (f n a - f 0 a)
F' : α → ℝ≥0∞ := fun a ↦ ENNReal.ofReal (F a - f 0 a)
⊢ ∀ (i : ℕ), ∫⁻ (a : α), f' i a ∂μ = ENNReal.ofReal (∫ (a : α), f i a ∂μ - ∫ (a : α), f 0 a ∂μ)

Tactic:
intro i

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf_int : ∀ (n : ℕ), Integrable (f n) μ
hF_int : Integrable F μ
hf_tendsto : Tendsto (fun i ↦ ∫ (a : α), f i a ∂μ) atTop (𝓝 (∫ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i ↦ f i a
hf_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
f' : ℕ → α → ℝ≥0∞ := fun n a ↦ ENNReal.ofReal (f n a - f 0 a)
F' : α → ℝ≥0∞ := fun a ↦ ENNReal.ofReal (F a - f 0 a)
i : ℕ
⊢ ∫⁻ (a : α), f' i a ∂μ = ENNReal.ofReal (∫ (a : α), f i a ∂μ - ∫ (a : α), f 0 a ∂μ)
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: case refine_2
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
ι : Type u_7
s : Finset ι
f : ι → α → E
⊢ ∀ (a : ι) (s : Finset ι),
    a ∉ s →
      ((∀ i ∈ s, Integrable (f i) μ) → setToFun μ T hT (∑ i ∈ s, f i) = ∑ i ∈ s, setToFun μ T hT (f i)) →
        (∀ i ∈ insert a s, Integrable (f i) μ) →
          setToFun μ T hT (∑ i ∈ insert a s, f i) = ∑ i ∈ insert a s, setToFun μ T hT (f i)

Tactic:
intro i

Goal state after:
case refine_2
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
ι : Type u_7
s : Finset ι
f : ι → α → E
i : ι
⊢ ∀ (s : Finset ι),
    i ∉ s →
      ((∀ i ∈ s, Integrable (f i) μ) → setToFun μ T hT (∑ i ∈ s, f i) = ∑ i ∈ s, setToFun μ T hT (f i)) →
        (∀ i_1 ∈ insert i s, Integrable (f i_1) μ) →
          setToFun μ T hT (∑ i ∈ insert i s, f i) = ∑ i ∈ insert i s, setToFun μ T hT (f i)
Module: Mathlib.MeasureTheory.Integral.Bochner.SumMeasure

Goal state before:
pp: X : Type u_2
E : Type u_3
mX : MeasurableSpace X
inst✝⁴ : NormedAddCommGroup E
f : X → E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : MeasurableSingletonClass X
μ : Measure X
inst✝ : Fintype X
hf : Integrable f μ
⊢ ∫ (x : X), f x ∂μ = ∑ x, μ.real {x} • f x

Tactic:
← setIntegral_finset .univ,

Goal state after:
X : Type u_2
E : Type u_3
mX : MeasurableSpace X
inst✝⁴ : NormedAddCommGroup E
f : X → E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : MeasurableSingletonClass X
μ : Measure X
inst✝ : Fintype X
hf : Integrable f μ
⊢ ∫ (x : X), f x ∂μ = ∫ (x : X) in ↑Finset.univ, f x ∂μ

X : Type u_2
E : Type u_3
mX : MeasurableSpace X
inst✝⁴ : NormedAddCommGroup E
f : X → E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : MeasurableSingletonClass X
μ : Measure X
inst✝ : Fintype X
hf : Integrable f μ
⊢ IntegrableOn f (↑Finset.univ) μ
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
⊢ ∃ K, LipschitzWith K ⇑f

Tactic:
by
  let fL : E →L[𝕜] F := f.linear.toContinuousLinearMap
  refine ⟨‖fL‖₊, LipschitzWith.of_dist_le_mul fun x y ↦ ?_⟩
  rw [NormedAddTorsor.dist_eq_norm', NormedAddTorsor.dist_eq_norm', ← f.linearMap_vsub]
  exact fL.le_opNorm _

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

Goal state before:
pp: case pos
𝕜 : Type u
inst✝³ : NontriviallyNormedField 𝕜
E : Type v
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CompleteSpace 𝕜
h : ∃ s, Nonempty (Basis (↥s) 𝕜 E)
⊢ Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun f ↦ LinearMap.det ↑f

Tactic:
rcases h with ⟨s, ⟨b⟩⟩
    have : FiniteDimensional 𝕜 E := b.finiteDimensional_of_finite
    classical
    simp_rw [LinearMap.det_eq_det_toMatrix_of_finset b]
    refine Continuous.matrix_det ?_
    exact
      ((LinearMap.toMatrix b b).toLinearMap.comp
          (ContinuousLinearMap.coeLM 𝕜)).continuous_of_finiteDimensional

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

Goal state before:
pp: case cons
α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
ι : Type u_6
m : MeasurableSpace α
f : α → G
μ : ι → Measure α
a✝ : ι
s✝ : Finset ι
h : a✝ ∉ s✝
ih : (∀ i ∈ s✝, Integrable f (μ i)) → ∫ (a : α), f a ∂∑ i ∈ s✝, μ i = ∑ i ∈ s✝, ∫ (a : α), f a ∂μ i
hf : ∀ i ∈ Finset.cons a✝ s✝ h, Integrable f (μ i)
⊢ ∫ (a : α), f a ∂∑ i ∈ Finset.cons a✝ s✝ h, μ i = ∑ i ∈ Finset.cons a✝ s✝ h, ∫ (a : α), f a ∂μ i

Tactic:
rw [Finset.forall_mem_cons] at hf

Goal state after:
case cons
α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
ι : Type u_6
m : MeasurableSpace α
f : α → G
μ : ι → Measure α
a✝ : ι
s✝ : Finset ι
h : a✝ ∉ s✝
ih : (∀ i ∈ s✝, Integrable f (μ i)) → ∫ (a : α), f a ∂∑ i ∈ s✝, μ i = ∑ i ∈ s✝, ∫ (a : α), f a ∂μ i
hf : Integrable f (μ a✝) ∧ ∀ x ∈ s✝, Integrable f (μ x)
⊢ ∫ (a : α), f a ∂∑ i ∈ Finset.cons a✝ s✝ h, μ i = ∑ i ∈ Finset.cons a✝ s✝ h, ∫ (a : α), f a ∂μ i
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: case mp
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
f : E →ₗ[𝕜] F
⊢ Injective ⇑f → ∃ K > 0, AntilipschitzWith K ⇑f

pp: case mpr
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
f : E →ₗ[𝕜] F
⊢ (∃ K > 0, AntilipschitzWith K ⇑f) → Injective ⇑f

Tactic:
· rw [← LinearMap.ker_eq_bot]
    exact f.exists_antilipschitzWith

Goal state after:
case mpr
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
f : E →ₗ[𝕜] F
⊢ (∃ K > 0, AntilipschitzWith K ⇑f) → Injective ⇑f
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:
simp_rw [edist_nndist, nndist_eq_iSup]
  refine ENNReal.coe_iSup ⟨nndist f g, ?_⟩
  rintro - ⟨x, hx, rfl⟩
  exact nndist_coe_le_nndist x

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

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : SeminormedAddCommGroup β
f : α →ᵇ β
n : ℕ
⊢ n • ⇑f = ↑n • ⇑f

Tactic:
natCast_zsmul

Goal state after:
α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : SeminormedAddCommGroup β
f : α →ᵇ β
n : ℕ
⊢ n • ⇑f = n • ⇑f
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)
x✝ : α
⊢ ((addEquivBoundedOfCompact α E).toFun (c • f)) x✝ = ((RingHom.id 𝕜) c • (addEquivBoundedOfCompact α E).toFun f) x✝

Tactic:
norm_cast

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

Goal state before:

Tactic:
fun n ↦ ‖(u n) * r ^ n‖
  _ =O[atTop] fun n ↦ ‖u n‖ * ‖r‖ ^ n := by
      apply (IsBigOWith.of_bound (c := ‖(1 : ℝ)‖) ?_).isBigO
      filter_upwards [eventually_norm_pow_le r] with n hn
      simp
  _ =O[atTop] fun n ↦ ‖((n : F) ^ k)‖ * ‖r‖ ^ n := by
      simpa [Nat.cast_pow] using
      (isBigO_norm_left.mpr (isBigO_norm_right.mpr hu)).mul (isBigO_refl (fun n ↦ (‖r‖ ^ n)) atTop)
  _ =O[atTop] fun n ↦ ‖r' ^ n‖ := by
      convert!
        isBigO_norm_right.mpr
          (isBigO_norm_left.mpr
            (isLittleO_pow_const_mul_const_pow_const_pow_of_norm_lt k hrr').isBigO)
      simp only [norm_pow, norm_mul]

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

Goal state before:
pp: case h.h₁
α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
x y : α
key : ∀ (x : α), thickenedIndicatorAux δ E x ≤ 1
⊢ ↑(thickenedIndicatorAux δ E x).toNNReal ≤ 1

Tactic:
norm_cast

Goal state after:
case h.h₁
α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
x y : α
key : ∀ (x : α), thickenedIndicatorAux δ E x ≤ 1
⊢ (thickenedIndicatorAux δ E x).toNNReal ≤ 1
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: case h₂
α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
x y : α
key : ∀ (x : α), thickenedIndicatorAux δ E x ≤ 1
⊢ ↑(thickenedIndicatorAux δ E y).toNNReal ≤ 1

Tactic:
norm_cast

Goal state after:
case h₂
α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
x y : α
key : ∀ (x : α), thickenedIndicatorAux δ E x ≤ 1
⊢ (thickenedIndicatorAux δ E y).toNNReal ≤ 1
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: case neg
α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
x y : α
h : infEDist x E ≤ infEDist y E
h_le : ¬infEDist y E * (↑δ.toNNReal)⁻¹ ≤ 1
⊢ 1 - infEDist x E * (↑δ.toNNReal)⁻¹ - (1 - infEDist y E * (↑δ.toNNReal)⁻¹) ≤ (↑δ.toNNReal)⁻¹ * edist x y

Tactic:
· simp only [tsub_le_iff_right]
    rw [tsub_eq_zero_of_le (not_le.mp h_le).le, add_zero, mul_comm]
    calc 1
    _ ≤ infEDist y E * (↑δ.toNNReal)⁻¹ := (not_le.mp h_le).le
    _ ≤ edist x y * (↑δ.toNNReal)⁻¹ + infEDist x E * (↑δ.toNNReal)⁻¹ := by
      rw [← add_mul, edist_comm]
      gcongr
      exact infEDist_le_edist_add_infEDist

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
r : ℝ
hfint : Integrable f μ
hfint' : 0 ≤ ∫ (x : α), f x ∂μ
hf : ∀ᵐ (ω : α) ∂μ, f ω ≤ r
⊢ eLpNorm f 1 μ ≤ 2 * μ univ * ENNReal.ofReal r

Tactic:
by
  refine eLpNorm_one_le_of_le hfint hfint' ?_
  simp only [Real.coe_toNNReal', le_max_iff]
  filter_upwards [hf] with ω hω using Or.inl hω

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

Goal state before:
pp: δ : Type u_1
X : δ → Type u_3
inst✝¹ : (i : δ) → MeasurableSpace (X i)
μ : (i : δ) → Measure (X i)
inst✝ : DecidableEq δ
s : Finset δ
i : δ
hi : i ∈ s
f : ((i : δ) → X i) → ℝ≥0∞
x : (i : δ) → X i
y : X i
⊢ (∫⋯∫⁻_s, f ∂μ) (update x i y) = (∫⋯∫⁻_s, f ∂μ) x

Tactic:
grind [MeasureTheory.lmarginal_congr]

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

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
s t : Set X
μ : Measure X
ht : NullMeasurableSet t μ
hfs : IntegrableOn f s μ
⊢ ∫ (x : X) in s ∩ t, f x ∂μ + ∫ (x : X) in s \ t, f x ∂μ = ∫ (x : X) in s, f x ∂μ

Tactic:
← Measure.restrict_inter_add_sdiff₀ s ht,

Goal state after:
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
s t : Set X
μ : Measure X
ht : NullMeasurableSet t μ
hfs : IntegrableOn f s μ
⊢ ∫ (x : X) in s ∩ t, f x ∂μ + ∫ (x : X) in s \ t, f x ∂μ = ∫ (x : X), f x ∂(μ.restrict (s ∩ t) + μ.restrict (s \ t))
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case a
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
μ : Measure X
ι : Type u_5
t : Finset ι
s : ι → Set X
hs : ∀ i ∈ t, MeasurableSet (s i)
hf : ∀ i ∈ t, IntegrableOn f (s i) μ
A : ∀ u ∈ {x ∈ t.powerset | x.Nonempty}, MeasurableSet (⋂ i ∈ u, s i)
this :
  ∑ x ∈ t.powerset with x.Nonempty, ∫ (a : X) in ⋂ i ∈ x, s i, (-1) ^ (#x + 1) • f a ∂μ =
    ∑ x ∈ t.powerset with x.Nonempty, ∫ (a : X), (⋂ i ∈ x, s i).indicator (fun a ↦ (-1) ^ (#x + 1) • f a) a ∂μ
x : X
u : Finset ι
hu : u ∈ {t ∈ t.powerset | t.Nonempty}
⊢ (-1) ^ (#u + 1) • (⋂ i ∈ u, s i).indicator f x = (-1) ^ (#u + 1) • (⋂ i ∈ u, s i).indicator f x

Tactic:
norm_cast

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

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
ι : Type u_5
t : Finset ι
s : ι → Set X
hs : ∀ i ∈ t, MeasurableSet (s i)
hf : ∀ i ∈ t, μ (s i) ≠ ∞
⊢ ∀ i ∈ t, IntegrableOn (fun x ↦ 1) (s i) μ

Tactic:
intro i

Goal state after:
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
ι : Type u_5
t : Finset ι
s : ι → Set X
hs : ∀ i ∈ t, MeasurableSet (s i)
hf : ∀ i ∈ t, μ (s i) ≠ ∞
i : ι
⊢ i ∈ t → IntegrableOn (fun x ↦ 1) (s i) μ