mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Bornology.BoundedOperation

Goal state before:
pp: 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
x y : R
x_in_s : x ∈ s
y_in_t : y ∈ t
⊢ ‖x‖ ≤ Af

pp: 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
x y : R
x_in_s : x ∈ s
y_in_t : y ∈ t
⊢ ‖y‖ ≤ Ag

Tactic:
· exact mem_closedBall_zero_iff.mp (hAf x_in_s)

Goal state after:
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
x y : R
x_in_s : x ∈ s
y_in_t : y ∈ t
⊢ ‖y‖ ≤ Ag
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
h'f : MemLp (⇑f) 1 μ
⊢ setToFun μ T hT ⇑f = ∑ x ∈ f.range, (T (⇑f ⁻¹' {x})) x

Tactic:
let g := f.toLp h'f

Goal state after:
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
h'f : MemLp (⇑f) 1 μ
g : ↥(α →₁ₛ[μ] E) := f.toLp h'f
⊢ setToFun μ T hT ⇑f = ∑ x ∈ f.range, (T (⇑f ⁻¹' {x})) x
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:
rcases eq_or_ne c ∞ with (rfl | hc)
  · rw [ENNReal.toReal_top, zero_smul, integral_eq_setToFun, setToFun_top_smul_measure]
  -- Main case: `c ≠ ∞`
  simp_rw [integral_eq_setToFun, ← setToFun_smul_left]
  have hdfma : DominatedFinMeasAdditive μ (weightedSMul (c • μ) : Set α → G →L[ℝ] G) c.toReal :=
    mul_one c.toReal ▸ (dominatedFinMeasAdditive_weightedSMul (c • μ)).of_smul_measure hc
  have hdfma_smul := dominatedFinMeasAdditive_weightedSMul (F := G) (c • μ)
  rw [← setToFun_congr_smul_measure c hc hdfma hdfma_smul f]
  exact setToFun_congr_left' _ _ (fun s _ _ => weightedSMul_smul_measure μ c) f

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, nndist (f x) (g x)) = ⨆ x, ↑(nndist (f x) (g x))

Tactic:
refine ENNReal.coe_iSup ⟨nndist f g, ?_⟩

Goal state after:
α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : PseudoMetricSpace β
f g : α →ᵇ β
⊢ nndist f g ∈ upperBounds (range fun x ↦ nndist (f x) (g x))
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
φ : E →L[𝕜] ι → 𝕜 := ↑v.equivFunL
⊢ ∑ i, ‖v.equivFun e i‖₊ * M = (∑ i, ‖v.equivFun e i‖₊) * M

Tactic:
rw [Finset.sum_mul]

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
h_right_g : ∫⁻ (a : α), ENNReal.ofReal (‖g a‖ ^ q) ∂μ = ∫⁻ (a : α), ‖g a‖ₑ ^ q ∂μ
⊢ (∫⁻ (a : α), ‖f a‖ₑ ^ p ∂μ) ^ (1 / p) =
    (∫⁻ (x : α), ‖f x‖ₑ ^ (ENNReal.ofReal p).toReal ∂μ) ^ (1 / (ENNReal.ofReal p).toReal)

pp: case hp_ne_zero
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
h_right_g : ∫⁻ (a : α), ENNReal.ofReal (‖g a‖ ^ q) ∂μ = ∫⁻ (a : α), ‖g a‖ₑ ^ q ∂μ
⊢ ENNReal.ofReal p ≠ 0

pp: case hp_ne_top
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
h_right_g : ∫⁻ (a : α), ENNReal.ofReal (‖g a‖ ^ q) ∂μ = ∫⁻ (a : α), ‖g a‖ₑ ^ q ∂μ
⊢ ENNReal.ofReal p ≠ ∞

Tactic:
· rw [ENNReal.toReal_ofReal hpq.nonneg]

Goal state after:
case hp_ne_zero
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
h_right_g : ∫⁻ (a : α), ENNReal.ofReal (‖g a‖ ^ q) ∂μ = ∫⁻ (a : α), ‖g a‖ₑ ^ q ∂μ
⊢ ENNReal.ofReal p ≠ 0

case hp_ne_top
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
h_right_g : ∫⁻ (a : α), ENNReal.ofReal (‖g a‖ ^ q) ∂μ = ∫⁻ (a : α), ‖g a‖ₑ ^ q ∂μ
⊢ ENNReal.ofReal p ≠ ∞
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
C : ℝ
C0 : 0 ≤ C
⊢ ‖f‖ ≤ C ↔ ∀ (x : α), ‖f x‖ ≤ C

Tactic:
simpa using! @dist_le _ _ _ _ f 0 _ C0

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
g : β → E := ⋯
A : setToFun μ' T' hT' f = setToFun μ' T' hT' g
⊢ f ∘ φ =ᵐ[μ] g ∘ φ

Tactic:
exact ae_of_ae_map hφ.aemeasurable hfm.ae_eq_mk

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
μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ∞
hμ'_le : μ' ≤ c' • μ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ' T C'
f✝ : α → E
hfμ : Integrable f✝ μ
hF : CompleteSpace F
h_int : ∀ (g : α → E), Integrable g μ → Integrable g μ'
f : ↥(Lp E 1 μ)
⊢ setToFun μ' T hT' ↑↑f = setToFun μ' T hT' ↑↑(Integrable.toL1 ↑↑f ⋯)

Tactic:
exact setToFun_congr_ae hT' (Integrable.coeFn_toL1 _).symm

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

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δseq : ℕ → ℝ
δseq_pos : ∀ (n : ℕ), 0 < δseq n
δseq_lim : Tendsto δseq atTop (𝓝 0)
E : Set α
key :
  ∀ (x : α),
    Tendsto (fun i ↦ thickenedIndicatorAux (δseq i) E x) atTop
      (𝓝 ((closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).indicator (fun x ↦ 1) x))
x : α
⊢ (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).indicator (fun x ↦ 1) x ≠ ∞

Tactic:
by_cases x_mem : x ∈ closure E

Goal state after:
case pos
α : Type u_1
inst✝ : PseudoEMetricSpace α
δseq : ℕ → ℝ
δseq_pos : ∀ (n : ℕ), 0 < δseq n
δseq_lim : Tendsto δseq atTop (𝓝 0)
E : Set α
key :
  ∀ (x : α),
    Tendsto (fun i ↦ thickenedIndicatorAux (δseq i) E x) atTop
      (𝓝 ((closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).indicator (fun x ↦ 1) x))
x : α
x_mem : x ∈ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E
⊢ (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).indicator (fun x ↦ 1) x ≠ ∞

case neg
α : Type u_1
inst✝ : PseudoEMetricSpace α
δseq : ℕ → ℝ
δseq_pos : ∀ (n : ℕ), 0 < δseq n
δseq_lim : Tendsto δseq atTop (𝓝 0)
E : Set α
key :
  ∀ (x : α),
    Tendsto (fun i ↦ thickenedIndicatorAux (δseq i) E x) atTop
      (𝓝 ((closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).indicator (fun x ↦ 1) x))
x : α
x_mem : x ∉ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E
⊢ (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).indicator (fun x ↦ 1) x ≠ ∞
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: case neg
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
c : ℝ≥0∞
hc_ne_top : c ≠ ∞
hT : DominatedFinMeasAdditive μ T C
hT_smul : DominatedFinMeasAdditive (c • μ) T C'
f : α → E
hc0 : ¬c = 0
⊢ setToFun μ T hT f = setToFun (c • μ) T hT_smul f

Tactic:
refine setToFun_congr_measure c⁻¹ c ?_ hc_ne_top (le_of_eq ?_) le_rfl hT hT_smul f

Goal state after:
case neg.refine_1
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
c : ℝ≥0∞
hc_ne_top : c ≠ ∞
hT : DominatedFinMeasAdditive μ T C
hT_smul : DominatedFinMeasAdditive (c • μ) T C'
f : α → E
hc0 : ¬c = 0
⊢ c⁻¹ ≠ ∞

case neg.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 C' : ℝ
c : ℝ≥0∞
hc_ne_top : c ≠ ∞
hT : DominatedFinMeasAdditive μ T C
hT_smul : DominatedFinMeasAdditive (c • μ) T C'
f : α → E
hc0 : ¬c = 0
⊢ μ = c⁻¹ • c • μ
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: R : Type u_1
inst✝⁴ : LinearOrder R
inst✝³ : TopologicalSpace R
inst✝² : OrderTopology R
inst✝¹ : DenselyOrdered R
inst✝ : NoMaxOrder R
f : StieltjesFunction R
x : R
⊢ ⨅ r, ↑f ↑r = ↑f x

Tactic:
suffices Function.rightLim f x = ⨅ r : Ioi x, f r by rw [← this, f.rightLim_eq]

Goal state after:
R : Type u_1
inst✝⁴ : LinearOrder R
inst✝³ : TopologicalSpace R
inst✝² : OrderTopology R
inst✝¹ : DenselyOrdered R
inst✝ : NoMaxOrder R
f : StieltjesFunction R
x : R
⊢ rightLim (↑f) x = ⨅ r, ↑f ↑r
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: R : Type u_1
inst✝⁶ : LinearOrder R
inst✝⁵ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁴ : OrderTopology R
inst✝³ : CompactIccSpace R
inst✝² : MeasurableSpace R
inst✝¹ : BorelSpace R
inst✝ : DenselyOrdered R
s : Set R
t : ℕ → Set R
ht : s ⊆ ⋃ i, t i
ε : ℝ≥0
ε0 : 0 < ε
h : ∑' (i : ℕ), f.length (t i) < ∞
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑ε
⊢ ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), f.outer t ≤ ∑' (i : ℕ), f.length (t i) + ↑ε

Tactic:
← hε

Goal state after:
R : Type u_1
inst✝⁶ : LinearOrder R
inst✝⁵ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁴ : OrderTopology R
inst✝³ : CompactIccSpace R
inst✝² : MeasurableSpace R
inst✝¹ : BorelSpace R
inst✝ : DenselyOrdered R
s : Set R
t : ℕ → Set R
ht : s ⊆ ⋃ i, t i
ε : ℝ≥0
ε0 : 0 < ε
h : ∑' (i : ℕ), f.length (t i) < ∞
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑ε
⊢ ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), f.outer t ≤ ∑' (i : ℕ), f.length (t i) + ∑' (i : ℕ), ↑(ε' i)
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: G : Type u_1
α : Type u_3
inst✝⁷ : Group G
inst✝⁶ : MulAction G α
inst✝⁵ : MeasurableSpace α
s t : Set α
μ : Measure α
inst✝⁴ : MeasurableConstSMul G α
inst✝³ : SMulInvariantMeasure G α μ
inst✝² : Countable G
β : Type u_6
inst✝¹ : TopologicalSpace β
inst✝ : PseudoMetrizableSpace β
hs : IsFundamentalDomain G s μ
ht : IsFundamentalDomain G t μ
f : α → β
hf : ∀ (g : G) (x : α), f (g • x) = f x
⊢ (∀ (g : G), AEStronglyMeasurable f (μ.restrict (g⁻¹ • (g • s ∩ t)))) ↔
    ∀ (g : G), AEStronglyMeasurable f (μ.restrict (g • s ∩ t))

Tactic:
refine forall_congr' fun g => ?_

Goal state after:
G : Type u_1
α : Type u_3
inst✝⁷ : Group G
inst✝⁶ : MulAction G α
inst✝⁵ : MeasurableSpace α
s t : Set α
μ : Measure α
inst✝⁴ : MeasurableConstSMul G α
inst✝³ : SMulInvariantMeasure G α μ
inst✝² : Countable G
β : Type u_6
inst✝¹ : TopologicalSpace β
inst✝ : PseudoMetrizableSpace β
hs : IsFundamentalDomain G s μ
ht : IsFundamentalDomain G t μ
f : α → β
hf : ∀ (g : G) (x : α), f (g • x) = f x
g : G
⊢ AEStronglyMeasurable f (μ.restrict (g⁻¹ • (g • s ∩ t))) ↔ AEStronglyMeasurable f (μ.restrict (g • s ∩ t))
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: case refine_1
G : Type w
inst✝³ : TopologicalSpace G
μ : Content G
inst✝² : MeasurableSpace G
inst✝¹ : R1Space G
inst✝ : BorelSpace G
H : μ.ContentRegular
K : Compacts G
⊢ ∀ (ε : ℝ≥0), 0 < ε → μ K < ∞ → μ.measure ↑K ≤ μ K + ↑ε

Tactic:
intro ε

Goal state after:
case refine_1
G : Type w
inst✝³ : TopologicalSpace G
μ : Content G
inst✝² : MeasurableSpace G
inst✝¹ : R1Space G
inst✝ : BorelSpace G
H : μ.ContentRegular
K : Compacts G
ε : ℝ≥0
⊢ 0 < ε → μ K < ∞ → μ.measure ↑K ≤ μ K + ↑ε
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 : Set X
μ : Measure X
h : ∀ᵐ (x : X) ∂μ, x ∉ s → f x = 0
⊢ ∫ (x : X) in univ, f x ∂μ = ∫ (x : X) in s, f x ∂μ

Tactic:
apply setIntegral_eq_of_subset_of_ae_sdiff_eq_zero nullMeasurableSet_univ (subset_univ _)

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 : Set X
μ : Measure X
h : ∀ᵐ (x : X) ∂μ, x ∉ s → f x = 0
⊢ ∀ᵐ (x : X) ∂μ, x ∈ univ \ s → f x = 0
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case inr.hp
R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
a b : R
hab : a < b
A : Disjoint (Ioo a b) {b}
D : ↑f b - ↑f a = ↑f b - leftLim (↑f) b + (leftLim (↑f) b - ↑f a)
this : f.measure (Ioo a b) + ofReal (↑f b - leftLim (↑f) b) = ofReal (↑f b - leftLim (↑f) b + (leftLim (↑f) b - ↑f a))
⊢ 0 ≤ ↑f b - leftLim (↑f) b

Tactic:
simp only [f.mono.leftLim_le le_rfl, sub_nonneg]

Goal state after:
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: G : Type u_1
α : Type u_3
inst✝⁶ : Group G
inst✝⁵ : MulAction G α
inst✝⁴ : MeasurableSpace α
ν : Measure α
inst✝³ : SMulInvariantMeasure G α ν
inst✝² : Countable G
inst✝¹ : MeasurableConstSMul G α
i : SigmaFinite ν
i' : HasFundamentalDomain G α ν
μ : Measure (Quotient α_mod_G)
inst✝ : QuotientMeasureEqMeasurePreimage ν μ
⊢ SigmaFinite μ

Tactic:
sigmaFinite_iff

Goal state after:
G : Type u_1
α : Type u_3
inst✝⁶ : Group G
inst✝⁵ : MulAction G α
inst✝⁴ : MeasurableSpace α
ν : Measure α
inst✝³ : SMulInvariantMeasure G α ν
inst✝² : Countable G
inst✝¹ : MeasurableConstSMul G α
i : SigmaFinite ν
i' : HasFundamentalDomain G α ν
μ : Measure (Quotient α_mod_G)
inst✝ : QuotientMeasureEqMeasurePreimage ν μ
⊢ Nonempty (μ.FiniteSpanningSetsIn univ)
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
l : ℝ
hf : Tendsto (↑f) atTop (𝓝 l)
x : R
⊢ f.measure (Ici x) - ofReal (↑f x - leftLim (↑f) x) = ofReal (l - ↑f x)

pp: R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
l : ℝ
hf : Tendsto (↑f) atTop (𝓝 l)
x : R
⊢ f.measure {x} ≠ ∞

pp: R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
l : ℝ
hf : Tendsto (↑f) atTop (𝓝 l)
x : R
⊢ {x} ⊆ Ici x

Tactic:
f.measure_Ici hf,

Goal state after:
R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
l : ℝ
hf : Tendsto (↑f) atTop (𝓝 l)
x : R
⊢ ofReal (l - leftLim (↑f) x) - ofReal (↑f x - leftLim (↑f) x) = ofReal (l - ↑f x)

R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
l : ℝ
hf : Tendsto (↑f) atTop (𝓝 l)
x : R
⊢ f.measure {x} ≠ ∞

R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
l : ℝ
hf : Tendsto (↑f) atTop (𝓝 l)
x : R
⊢ {x} ⊆ Ici x
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure X
s : Set X
f : ↥(Lp E p μ)
⊢ ‖MemLp.toLp ↑↑f ⋯‖ ≤ ‖f‖

Tactic:
by
  rw [Lp.norm_def, Lp.norm_def, eLpNorm_congr_ae (MemLp.coeFn_toLp _)]
  refine ENNReal.toReal_mono (Lp.eLpNorm_ne_top _) ?_
  exact eLpNorm_mono_measure _ Measure.restrict_le_self

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
r : 𝕜
x : F
⊢ ‖r • x‖ ≤ ‖r‖ * ‖x‖

Tactic:
by
    rw [norm_eq_sqrt_re_inner, inner_smul_left, inner_smul_right, ← mul_assoc]
    rw [RCLike.conj_mul, ← ofReal_pow, re_ofReal_mul, sqrt_mul, ← ofReal_normSq_eq_inner_self,
      ofReal_re]
    · simp [sqrt_normSq_eq_norm]
    · positivity

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

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜 F
cd : Core 𝕜 F
tF : TopologicalSpace F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul 𝕜 F
h : ContinuousAt (fun v ↦ ⟪v, v⟫) 0
h' : IsVonNBounded 𝕜 {v | re ⟪v, v⟫ < 1}
p : Seminorm 𝕜 F := normSeminorm 𝕜 F
this : p.ball 0 1 = {v | re ⟪v, v⟫ < 1}
⊢ {v | re ⟪v, v⟫ < 1} ∈ 𝓝 0 ∧ IsVonNBounded 𝕜 {v | re ⟪v, v⟫ < 1}

Tactic:
refine ⟨?_, h'⟩

Goal state after:
𝕜 : Type u_1
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜 F
cd : Core 𝕜 F
tF : TopologicalSpace F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul 𝕜 F
h : ContinuousAt (fun v ↦ ⟪v, v⟫) 0
h' : IsVonNBounded 𝕜 {v | re ⟪v, v⟫ < 1}
p : Seminorm 𝕜 F := normSeminorm 𝕜 F
this : p.ball 0 1 = {v | re ⟪v, v⟫ < 1}
⊢ {v | re ⟪v, v⟫ < 1} ∈ 𝓝 0
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: Y : Type u_2
E : Type u_3
F : Type u_4
X : Type u_5
G : Type u_6
𝕜 : Type u_7
inst✝¹¹ : TopologicalSpace X
inst✝¹⁰ : TopologicalSpace Y
inst✝⁹ : MeasurableSpace Y
inst✝⁸ : OpensMeasurableSpace Y
μ : Measure Y
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : NormedSpace 𝕜 E
L : F →L[𝕜] G →L[𝕜] E
f : X → Y → G
s : Set X
k : Set Y
g : Y → F
hk : IsCompact k
hf : ContinuousOn (uncurry f) (s ×ˢ univ)
hfs : ∀ (p : X) (x : Y), p ∈ s → x ∉ k → f p x = 0
hg : IntegrableOn g k μ
A : ∀ p ∈ s, Continuous[inst✝¹⁰, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (f p)
q : X
hq : q ∈ s
⊢ ContinuousWithinAt (fun x ↦ ∫ (y : Y), (L (g y)) (f x y) ∂μ) s q

Tactic:
apply Metric.continuousWithinAt_iff'.2 (fun ε εpos ↦ ?_)

Goal state after:
Y : Type u_2
E : Type u_3
F : Type u_4
X : Type u_5
G : Type u_6
𝕜 : Type u_7
inst✝¹¹ : TopologicalSpace X
inst✝¹⁰ : TopologicalSpace Y
inst✝⁹ : MeasurableSpace Y
inst✝⁸ : OpensMeasurableSpace Y
μ : Measure Y
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : NormedSpace 𝕜 E
L : F →L[𝕜] G →L[𝕜] E
f : X → Y → G
s : Set X
k : Set Y
g : Y → F
hk : IsCompact k
hf : ContinuousOn (uncurry f) (s ×ˢ univ)
hfs : ∀ (p : X) (x : Y), p ∈ s → x ∉ k → f p x = 0
hg : IntegrableOn g k μ
A : ∀ p ∈ s, Continuous[inst✝¹⁰, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (f p)
q : X
hq : q ∈ s
ε : ℝ
εpos : ε > 0
⊢ ∀ᶠ (x : X) in 𝓝[s] q, dist (∫ (y : Y), (L (g y)) (f x y) ∂μ) (∫ (y : Y), (L (g y)) (f q y) ∂μ) < ε
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
⊢ ‖x + y‖ * ‖x + y‖ + ‖x - y‖ * ‖x - y‖ = 2 * (‖x‖ * ‖x‖ + ‖y‖ * ‖y‖)

Tactic:
simp only [← @inner_self_eq_norm_mul_norm 𝕜]
  rw [← re.map_add, parallelogram_law, two_mul, two_mul]
  simp only [re.map_add]

Goal state after: