mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
_i : Fact (1 ≤ p)
hp_ne_top : p ≠ ∞
motive : (α → E) → Prop
indicator : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → μ s < ∞ → motive (s.indicator fun x ↦ c)
add :
  ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → MemLp f p μ → MemLp g p μ → motive f → motive g → motive (f + g)
closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {f | motive ↑↑f}
ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → MemLp f p μ → motive f → motive g
⊢ ∀ (f : α →ₛ E), MemLp (⇑f) p μ → motive ⇑f

Tactic:
apply SimpleFunc.induction

Goal state after:
case const
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
_i : Fact (1 ≤ p)
hp_ne_top : p ≠ ∞
motive : (α → E) → Prop
indicator : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → μ s < ∞ → motive (s.indicator fun x ↦ c)
add :
  ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → MemLp f p μ → MemLp g p μ → motive f → motive g → motive (f + g)
closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {f | motive ↑↑f}
ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → MemLp f p μ → motive f → motive g
⊢ ∀ (c : E) {s : Set α} (hs : MeasurableSet s),
    MemLp (⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p μ →
      motive ⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))

case add
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
_i : Fact (1 ≤ p)
hp_ne_top : p ≠ ∞
motive : (α → E) → Prop
indicator : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → μ s < ∞ → motive (s.indicator fun x ↦ c)
add :
  ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → MemLp f p μ → MemLp g p μ → motive f → motive g → motive (f + g)
closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {f | motive ↑↑f}
ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → MemLp f p μ → motive f → motive g
⊢ ∀ ⦃f g : α →ₛ E⦄,
    Disjoint (support ⇑f) (support ⇑g) →
      (MemLp (⇑f) p μ → motive ⇑f) → (MemLp (⇑g) p μ → motive ⇑g) → MemLp (⇑(f + g)) p μ → motive ⇑(f + g)
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:
pp: case refine_1
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ↥(Lp ℝ 1 μ)
⊢ Continuous integral

Tactic:
simp only [integral]
    exact cont _

Goal state after:
Module: Mathlib.Analysis.Asymptotics.AsymptoticEquivalent

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : NormedAddCommGroup β
u v : α → β
l : Filter α
c : β
huv : u ~[l] v
hu : Tendsto u l (𝓝 c)
⊢ Tendsto v l (𝓝 c)

Tactic:
by_cases h : c = 0
  · subst c
    rw [← isLittleO_one_iff ℝ] at hu ⊢
    simpa using (huv.symm.isLittleO.trans hu).add hu
  · rw [← isEquivalent_const_iff_tendsto h] at hu ⊢
    exact huv.symm.trans hu

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

Goal state before:
pp: case inr
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
hp_ne_top : p ≠ ∞
P : (α → E) → Prop
h0P :
  ∀ (c : E) ⦃s : Set α⦄,
    MeasurableSet s → μ s < ∞ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, eLpNorm (g - s.indicator fun x ↦ c) p μ ≤ ε ∧ P g
h1P : ∀ (f g : α → E), P f → P g → P (f + g)
h2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ
f : α → E
hf : MemLp f p μ
ε : ℝ≥0∞
hε : ε ≠ 0
hp_pos : p ≠ 0
⊢ ∀ (f' : α →ₛ E) (δ : ℝ≥0∞), δ ≠ 0 → MemLp (⇑f') p μ → ∃ g, eLpNorm (⇑f' - g) p μ ≤ δ ∧ P g

Tactic:
apply SimpleFunc.induction

Goal state after:
case inr.const
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
hp_ne_top : p ≠ ∞
P : (α → E) → Prop
h0P :
  ∀ (c : E) ⦃s : Set α⦄,
    MeasurableSet s → μ s < ∞ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, eLpNorm (g - s.indicator fun x ↦ c) p μ ≤ ε ∧ P g
h1P : ∀ (f g : α → E), P f → P g → P (f + g)
h2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ
f : α → E
hf : MemLp f p μ
ε : ℝ≥0∞
hε : ε ≠ 0
hp_pos : p ≠ 0
⊢ ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (δ : ℝ≥0∞),
    δ ≠ 0 →
      MemLp (⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p μ →
        ∃ g, eLpNorm (⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0)) - g) p μ ≤ δ ∧ P g

case inr.add
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
hp_ne_top : p ≠ ∞
P : (α → E) → Prop
h0P :
  ∀ (c : E) ⦃s : Set α⦄,
    MeasurableSet s → μ s < ∞ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, eLpNorm (g - s.indicator fun x ↦ c) p μ ≤ ε ∧ P g
h1P : ∀ (f g : α → E), P f → P g → P (f + g)
h2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ
f : α → E
hf : MemLp f p μ
ε : ℝ≥0∞
hε : ε ≠ 0
hp_pos : p ≠ 0
⊢ ∀ ⦃f g : α →ₛ E⦄,
    Disjoint (support ⇑f) (support ⇑g) →
      (∀ (δ : ℝ≥0∞), δ ≠ 0 → MemLp (⇑f) p μ → ∃ g, eLpNorm (⇑f - g) p μ ≤ δ ∧ P g) →
        (∀ (δ : ℝ≥0∞), δ ≠ 0 → MemLp (⇑g) p μ → ∃ g_1, eLpNorm (⇑g - g_1) p μ ≤ δ ∧ P g_1) →
          ∀ (δ : ℝ≥0∞), δ ≠ 0 → MemLp (⇑(f + g)) p μ → ∃ g_1, eLpNorm (⇑(f + g) - g_1) p μ ≤ δ ∧ P g_1
Module: Mathlib.Topology.Algebra.ContinuousAffineEquiv

Goal state before:
pp: k : Type u_1
P₁ : Type u_2
P₂ : Type u_3
V₁ : Type u_6
V₂ : Type u_7
inst✝⁸ : Ring k
inst✝⁷ : AddCommGroup V₁
inst✝⁶ : Module k V₁
inst✝⁵ : AddTorsor V₁ P₁
inst✝⁴ : TopologicalSpace P₁
inst✝³ : AddCommGroup V₂
inst✝² : Module k V₂
inst✝¹ : AddTorsor V₂ P₂
inst✝ : TopologicalSpace P₂
e : P₁ ≃ᴬ[k] P₂
s : Set P₂
⊢ ⇑e.symm '' s = ⇑e ⁻¹' s

Tactic:
rw [e.symm.image_eq_preimage_symm, e.symm_symm]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
hc : 1 < ‖c‖
R : ℝ
hR : ‖c‖ < R
F : Subspace 𝕜 E
hFc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑F
hF : ∃ x, x ∉ F
Rpos : 0 < R
this : ‖c‖ / R < 1
x : E
xF : x ∉ F
hx : ∀ y ∈ F, ‖c‖ / R * ‖x‖ ≤ ‖x - y‖
x0 : x ≠ 0
d : 𝕜
d0 : d ≠ 0
dxlt : ‖d • x‖ < R
ledx : R / ‖c‖ ≤ ‖d • x‖
y : E
hy : y ∈ F
y' : E := d⁻¹ • y
yy' : y = d • y'
⊢ ‖d‖ * (‖c‖ / R * ‖x‖) ≤ ‖d‖ * ‖x - y'‖

Tactic:
gcongr; exact hx y' (by simp [y', Submodule.smul_mem _ _ hy])

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
inst✝ : CompleteSpace F
T T' : Set α → E →L[ℝ] F
C C' : ℝ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ T' C'
c : ℝ
h_smul : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → T' s = c • T s
f : ↥(Lp E 1 μ)
⊢ (setToL1 hT') f = c • (setToL1 hT) f

Tactic:
by
  apply setToL1_unique hT' (A := c • setToL1 hT) _ f
  simp [setToL1_eq_setToL1SCLM, setToL1SCLM_smul_left' c hT hT' h_smul]

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.Isometry

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
P : Type u_10
inst✝⁴ : NormedField 𝕜
inst✝³ : SeminormedAddCommGroup V
inst✝² : NormedSpace 𝕜 V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
x y : P
⊢ dist ((pointReflection 𝕜 x) y) x = dist ((pointReflection 𝕜 x) y) ((pointReflection 𝕜 x) x)

Tactic:
pointReflection_self

Goal state after:
𝕜 : Type u_1
V : Type u_2
P : Type u_10
inst✝⁴ : NormedField 𝕜
inst✝³ : SeminormedAddCommGroup V
inst✝² : NormedSpace 𝕜 V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
x y : P
⊢ dist ((pointReflection 𝕜 x) y) x = dist ((pointReflection 𝕜 x) y) 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 : ℝ
hT : DominatedFinMeasAdditive μ T C
f : α → E
hF : CompleteSpace F
hf : ¬Integrable f μ
⊢ setToFun μ T hT (-f) = -0

Tactic:
setToFun_undef hT,

Goal state after:
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 : ℝ
hT : DominatedFinMeasAdditive μ T C
f : α → E
hF : CompleteSpace F
hf : ¬Integrable f μ
⊢ 0 = -0

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 : ℝ
hT : DominatedFinMeasAdditive μ T C
f : α → E
hF : CompleteSpace F
hf : ¬Integrable f μ
⊢ ¬Integrable (-f) μ
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 ν : Measure α
f : α → G
h₁ : Integrable f μ
h₂ : Integrable f ν
⊢ ↑(nndist (∫ (x : α), f x ∂μ) (∫ (x : α), f x ∂(μ + ν))) ≤ ∫⁻ (x : α), ‖f x‖ₑ ∂ν

Tactic:
integral_add_measure h₁ h₂,

Goal state after:
α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
m : MeasurableSpace α
μ ν : Measure α
f : α → G
h₁ : Integrable f μ
h₂ : Integrable f ν
⊢ ↑(nndist (∫ (x : α), f x ∂μ) (∫ (x : α), f x ∂μ + ∫ (x : α), f x ∂ν)) ≤ ∫⁻ (x : α), ‖f x‖ₑ ∂ν
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:
coe_nsmul,

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.Bounded.Normed

Goal state before:
pp: α : Type u
β : Type v
γ : Type w
inst✝² : TopologicalSpace α
inst✝¹ : SeminormedAddCommGroup β
inst✝ : TopologicalSpace γ
f : α →ᵇ β
g : C(γ, α)
⊢ ↑1 * dist f 0 ≤ ‖f‖

Tactic:
rw [NNReal.coe_one, one_mul, dist_zero_right]

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

Goal state before:
pp: case pos.h_closed
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'
⊢ IsClosed {f | setToFun μ T hT ↑↑f = setToFun μ' T hT' ↑↑f}

pp: case pos.h_ae
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 g : α → E⦄,
    f =ᵐ[μ] g → Integrable f μ → setToFun μ T hT f = setToFun μ' T hT' f → setToFun μ T hT g = setToFun μ' T hT' g

Tactic:
· refine isClosed_eq (continuous_setToFun hT) ?_
    have :
      (fun f : α →₁[μ] E => setToFun μ' T hT' f) = fun f : α →₁[μ] E =>
        setToFun μ' T hT' ((h_int f (L1.integrable_coeFn f)).toL1 f) := by
      ext1 f; exact setToFun_congr_ae hT' (Integrable.coeFn_toL1 _).symm
    rw [this]
    exact (continuous_setToFun hT').comp (continuous_L1_toL1 c' hc' hμ'_le)

Goal state after:
case pos.h_ae
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 g : α → E⦄,
    f =ᵐ[μ] g → Integrable f μ → setToFun μ T hT f = setToFun μ' T hT' f → setToFun μ T hT g = setToFun μ' T hT' g
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
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
l : Filter ι
inst✝ : l.IsCountablyGenerated
fs : ι → α → E
f : α → E
bound : α → ℝ
hfs_meas : ∀ᶠ (n : ι) in l, AEStronglyMeasurable (fs n) μ
h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a
bound_integrable : Integrable bound μ
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ fs n a) l (𝓝 (f a))
x : ℕ → ι
xl : Tendsto x atTop l
⊢ ∀ s ∈ l, ∃ a, ∀ b ≥ a, x b ∈ s

Tactic:
tendsto_atTop'

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
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
l : Filter ι
inst✝ : l.IsCountablyGenerated
fs : ι → α → E
f : α → E
bound : α → ℝ
hfs_meas : ∀ᶠ (n : ι) in l, AEStronglyMeasurable (fs n) μ
h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a
bound_integrable : Integrable bound μ
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ fs n a) l (𝓝 (f a))
x : ℕ → ι
xl : ∀ s ∈ l, ∃ a, ∀ (b : ℕ), a ≤ b → x b ∈ s
⊢ ∀ s ∈ l, ∃ a, ∀ b ≥ a, x b ∈ s
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✝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
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
l : Filter ι
inst✝ : l.IsCountablyGenerated
fs : ι → α → E
f : α → E
bound : α → ℝ
hfs_meas : ∀ᶠ (n : ι) in l, AEStronglyMeasurable (fs n) μ
h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a
bound_integrable : Integrable bound μ
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ fs n a) l (𝓝 (f a))
x : ℕ → ι
xl : Tendsto x atTop l
hxl : ∀ s ∈ l, ∃ a, ∀ b ≥ a, x b ∈ s
h✝ : {x | (fun n ↦ AEStronglyMeasurable (fs n) μ) x} ∩ {x | (fun n ↦ ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a) x} ∈ l
k : ℕ
h : ∀ b ≥ k, x b ∈ {x | (fun n ↦ AEStronglyMeasurable (fs n) μ) x} ∩ {x | (fun n ↦ ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a) x}
⊢ ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖fs (x (n + k)) a‖ ≤ bound a

Tactic:
exact fun n => (h _ (self_le_add_left _ _)).2

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✝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 E
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
inst✝ : Countable ι
f : ι → α → E
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hf' : ∫⁻ (a : α), ∑' (i : ι), ‖f i a‖ₑ ∂μ ≠ ∞
hF : CompleteSpace F
hf'' : ∀ (i : ι), AEMeasurable (fun x ↦ ‖f i x‖ₑ) μ
x : α
hx : ∑' (i : ι), ‖f i x‖ₑ < ∞
⊢ ∑' (a : ι), ↑‖f a x‖₊ ≠ ∞

Tactic:
exact hx.ne

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

Goal state before:
pp: G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
⊢ μ.innerContent (⨆ i, U i) ≤ ∑' (i : ℕ), μ.innerContent (U i)

Tactic:
have h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), μ (t.sup K) ≤ t.sum fun i => μ (K i) := by
    intro t K
    refine Finset.induction_on t ?_ ?_
    · simp only [μ.empty, nonpos_iff_eq_zero, Finset.sum_empty, Finset.sup_empty]
    · intro n s hn ih
      grw [Finset.sup_insert, Finset.sum_insert hn, μ.sup_le, ih]

Goal state after:
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), μ (t.sup K) ≤ ∑ i ∈ t, μ (K i)
⊢ μ.innerContent (⨆ i, U i) ≤ ∑' (i : ℕ), μ.innerContent (U i)
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
t : Set X
μ : Measure X
hU : ∫ (x : X) in t, f x ∂μ ≠ 0
⊢ ∃ x ∈ t, f x ≠ 0

Tactic:
by
  contrapose! hU; exact setIntegral_eq_zero_of_forall_eq_zero hU

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

Goal state before:
pp: G : Type u_1
α : Type u_3
inst✝⁴ : Group G
inst✝³ : MulAction G α
inst✝² : MeasurableSpace α
s : Set α
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : Countable G
h_meas : NullMeasurableSet s μ
h_ae_disjoint : ∀ (g : G), g ≠ 1 → AEDisjoint μ (g • s) s
h_qmp : ∀ (g : G), QuasiMeasurePreserving (fun x ↦ g • x) μ μ
h_measure_univ_le : μ univ ≤ ∑' (g : G), μ (g • s)
aedisjoint : Pairwise (AEDisjoint μ on fun g ↦ g • s)
g : G
⊢ NullMeasurableSet (g • s) μ

Tactic:
rw [← inv_inv g, ← preimage_smul]; exact h_meas.preimage (h_qmp g⁻¹)

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

Goal state before:
pp: G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
A : Set G
hA : MeasurableSet A
r : ℝ≥0∞
hr : ⨅ U, ⨅ (hU : IsOpen U), ⨅ (_ : A ⊆ U), μ.innerContent { carrier := U, is_open' := hU } < r
⊢ ∃ U ⊇ A, IsOpen U ∧ μ.measure U < r

Tactic:
simp only [iInf_lt_iff] at hr

Goal state after:
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
A : Set G
hA : MeasurableSet A
r : ℝ≥0∞
hr : ∃ i, ∃ (h : IsOpen i) (_ : A ⊆ i), μ.innerContent { carrier := i, is_open' := ⋯ } < r
⊢ ∃ U ⊇ A, IsOpen U ∧ μ.measure U < r
Module: Mathlib.MeasureTheory.Measure.Haar.Basic

Goal state before:
pp: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K V : Set G
hK : IsCompact K
hV : (interior V).Nonempty
this :
  sInf (Finset.card '' {t | K ⊆ ⋃ g ∈ t, (fun h ↦ g * h) ⁻¹' V}) ∈
    Finset.card '' {t | K ⊆ ⋃ g ∈ t, (fun h ↦ g * h) ⁻¹' V}
⊢ ∃ t, K ⊆ ⋃ g ∈ t, (fun h ↦ g * h) ⁻¹' V ∧ t.card = index K V

Tactic:
mem_image

Goal state after:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K V : Set G
hK : IsCompact K
hV : (interior V).Nonempty
this :
  ∃ x ∈ {t | K ⊆ ⋃ g ∈ t, (fun h ↦ g * h) ⁻¹' V},
    x.card = sInf (Finset.card '' {t | K ⊆ ⋃ g ∈ t, (fun h ↦ g * h) ⁻¹' V})
⊢ ∃ t, K ⊆ ⋃ g ∈ t, (fun h ↦ g * h) ⁻¹' V ∧ t.card = index K V
Module: Mathlib.MeasureTheory.Measure.Haar.Basic

Goal state before:
pp: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K : PositiveCompacts G
V : Set G
hV : (interior V).Nonempty
⊢ 0 ∉ Finset.card '' {t | ↑K ⊆ ⋃ g ∈ t, (fun h ↦ g * h) ⁻¹' V}

pp: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K : PositiveCompacts G
V : Set G
hV : (interior V).Nonempty
⊢ (Finset.card '' {t | ↑K ⊆ ⋃ g ∈ t, (fun h ↦ g * h) ⁻¹' V}).Nonempty

Tactic:
mem_image

Goal state after:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K : PositiveCompacts G
V : Set G
hV : (interior V).Nonempty
⊢ ¬∃ x ∈ {t | ↑K ⊆ ⋃ g ∈ t, (fun h ↦ g * h) ⁻¹' V}, x.card = 0

G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K : PositiveCompacts G
V : Set G
hV : (interior V).Nonempty
⊢ (Finset.card '' {t | ↑K ⊆ ⋃ g ∈ t, (fun h ↦ g * h) ⁻¹' V}).Nonempty
Module: Mathlib.MeasureTheory.Measure.Haar.Basic

Goal state before:
pp: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K K' V : Set G
hK' : IsCompact K'
h : K ⊆ K'
hV : (interior V).Nonempty
s : Finset G
h1s : K' ⊆ ⋃ g ∈ s, (fun h ↦ g * h) ⁻¹' V
h2s : s.card = index K' V
⊢ index K' V ∈ Finset.card '' {t | K ⊆ ⋃ g ∈ t, (fun h ↦ g * h) ⁻¹' V}

Tactic:
mem_image

Goal state after:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K K' V : Set G
hK' : IsCompact K'
h : K ⊆ K'
hV : (interior V).Nonempty
s : Finset G
h1s : K' ⊆ ⋃ g ∈ s, (fun h ↦ g * h) ⁻¹' V
h2s : s.card = index K' V
⊢ ∃ x ∈ {t | K ⊆ ⋃ g ∈ t, (fun h ↦ g * h) ⁻¹' V}, x.card = index K' V
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case inl
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 : b ≤ a
⊢ ofReal (leftLim (↑f) b - leftLim (↑f) a) = 0

Tactic:
simp [ENNReal.ofReal_eq_zero, f.mono.leftLim hab]

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

Goal state before:
pp: case refine_1
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
A : ℕ → Set α
hA_meas : ∀ (i : ℕ), MeasurableSet (A i)
hA : ∀ (i : ℕ), ν (A i) < ∞
hA' : ⋃ i, A i = univ
n : ℕ
⊢ μ (Quotient.mk α_mod_G '' A n) < ∞

Tactic:
obtain ⟨s, fund_dom_s⟩ := i'

Goal state after:
case refine_1
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
μ : Measure (Quotient α_mod_G)
inst✝ : QuotientMeasureEqMeasurePreimage ν μ
A : ℕ → Set α
hA_meas : ∀ (i : ℕ), MeasurableSet (A i)
hA : ∀ (i : ℕ), ν (A i) < ∞
hA' : ⋃ i, A i = univ
n : ℕ
s : Set α
fund_dom_s : IsFundamentalDomain G s ν
⊢ μ (Quotient.mk α_mod_G '' A n) < ∞
Module: Mathlib.MeasureTheory.Measure.Haar.Basic

Goal state before:
pp: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K : Set G
hK : IsCompact K
V : Set G
hV : (interior V).Nonempty
g : G
s : Finset G
h1s : K ⊆ ⋃ g ∈ s, (fun h ↦ g * h) ⁻¹' V
h2s : s.card = index K V
⊢ s.card ∈ Finset.card '' {t | (fun h ↦ g * h) '' K ⊆ ⋃ g ∈ t, (fun h ↦ g * h) ⁻¹' V}

Tactic:
mem_image

Goal state after:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K : Set G
hK : IsCompact K
V : Set G
hV : (interior V).Nonempty
g : G
s : Finset G
h1s : K ⊆ ⋃ g ∈ s, (fun h ↦ g * h) ⁻¹' V
h2s : s.card = index K V
⊢ ∃ x ∈ {t | (fun h ↦ g * h) '' K ⊆ ⋃ g ∈ t, (fun h ↦ g * h) ⁻¹' V}, x.card = s.card
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: F : Type u_3
inst✝¹ : SeminormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x : F
⊢ ⟪x, x⟫_ℝ = ‖x‖ * ‖x‖

Tactic:
have h := @inner_self_eq_norm_mul_norm ℝ F _ _ _ x

Goal state after:
F : Type u_3
inst✝¹ : SeminormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x : F
h : re ⟪x, x⟫_ℝ = ‖x‖ * ‖x‖
⊢ ⟪x, x⟫_ℝ = ‖x‖ * ‖x‖