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
G' : Type u_7
G'' : Type u_8
inst✝⁷ : NormedAddCommGroup G''
inst✝⁶ : PartialOrder G''
inst✝⁵ : IsOrderedAddMonoid G''
inst✝⁴ : NormedSpace ℝ G''
inst✝³ : NormedAddCommGroup G'
inst✝² : PartialOrder G'
inst✝¹ : NormedSpace ℝ G'
inst✝ : IsOrderedAddMonoid G'
T : Set α → G' →L[ℝ] G''
h_add : FinMeasAdditive μ T
hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∀ (x : G'), 0 ≤ x → 0 ≤ (T s) x
f g : α →ₛ G'
hfi : Integrable (⇑f) μ
hgi : Integrable (⇑g) μ
hfg : f ≤ g
⊢ setToSimpleFunc T f ≤ setToSimpleFunc T g

Tactic:
by
  rw [← sub_nonneg, ← setToSimpleFunc_sub T h_add hgi hfi]
  refine setToSimpleFunc_nonneg' T hT_nonneg _ ?_ (hgi.sub hfi)
  intro x
  simp only [coe_sub, sub_nonneg, coe_zero, Pi.zero_apply, Pi.sub_apply]
  exact hfg x

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : NormedField β
u v : α → β
l : Filter α
⊢ (u - v) =o[l] v ↔ ∃ φ, ∃ (_ : Tendsto φ l (𝓝 1)), u =ᶠ[l] φ * v

Tactic:
isLittleO_iff_exists_eq_mul

Goal state after:
α : Type u_1
β : Type u_2
inst✝ : NormedField β
u v : α → β
l : Filter α
⊢ (∃ φ, Tendsto φ l (𝓝 0) ∧ u - v =ᶠ[l] φ * v) ↔ ∃ φ, ∃ (_ : Tendsto φ l (𝓝 1)), u =ᶠ[l] φ * v
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
h_mono : Monotone fun i ↦ f i a
⊢ 0 a ≤ f i a - f 0 a

Tactic:
simp [h_mono zero_le]

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

Goal state before:
pp: case indicatorConst
α : Type u_1
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
G'' : Type u_8
inst✝⁵ : NormedAddCommGroup G''
inst✝⁴ : PartialOrder G''
inst✝³ : IsOrderedAddMonoid G''
inst✝² : NormedSpace ℝ G''
inst✝¹ : CompleteSpace G''
inst✝ : OrderClosedTopology G''
T T' : Set α → E →L[ℝ] G''
C C' : ℝ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ T' C'
hTT' : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∀ (x : E), (T s) x ≤ (T' s) x
c : E
s : Set α
hs : MeasurableSet s
hμs : μ s < ∞
⊢ (T s) c ≤ (T' s) c

Tactic:
exact hTT' s hs hμs c

Goal state after:
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 T' : Set α → E →L[ℝ] F
C C' : ℝ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ T' C'
h : T = T'
f : α → E
hF : CompleteSpace F
hf : ¬Integrable f μ
⊢ setToFun μ T hT f = setToFun μ T' hT' f

Tactic:
simp_rw [setToFun_undef _ hf]

Goal state after:
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 T' : Set α → E →L[ℝ] F
C C' : ℝ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ T' C'
h : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → T s = T' s
f : α → E
hF : CompleteSpace F
hf : ¬Integrable f μ
⊢ setToFun μ T hT f = setToFun μ T' hT' f

Tactic:
simp_rw [setToFun_undef _ hf]

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

Goal state before:
pp: case pos
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 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 : α → E
hF : CompleteSpace F
hf : Integrable f μ
⊢ setToFun μ T' hT' f = c • setToFun μ T hT f

Tactic:
simp_rw [setToFun_eq _ hf, L1.setToL1_smul_left' hT hT' c h_smul]

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

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

Tactic:
· rintro -
      convert! C0
      apply le_antisymm _ dist_nonneg'
      rw [dist_eq]
      exact csInf_le ⟨0, fun C => And.left⟩ ⟨le_rfl, fun x => False.elim (h (Nonempty.intro x))⟩

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

Goal state before:
pp: case insert
α : Type u
inst✝² : TopologicalSpace α
R : Type u_1
inst✝¹ : NonUnitalSeminormedRing R
inst✝ : IsCancelMulZero R
ι : Type u_2
f : ι → α →ᵇ 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‖₊
⊢ f j * ∑ i ∈ s, f i = 0

Tactic:
simpa [Finset.mul_sum] using Finset.sum_eq_zero fun i hi ↦ h (by grind)

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

Goal state before:
pp: case refine_1.refine_2
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), ‖g a‖ₑ ^ q ∂μ) ^ (1 / q) ≠ ∞

Tactic:
convert! hg.eLpNorm_ne_top
      rw [eLpNorm_eq_lintegral_rpow_enorm_toReal]
      · rw [ENNReal.toReal_ofReal hpq.symm.nonneg]
      · rw [Ne, ENNReal.ofReal_eq_zero, not_le]
        exact hpq.symm.pos
      · finiteness

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
hf : Summable f
this : ∀ {N : ℕ} {g : α → Fin N → ℝ}, Summable g → Summable fun x ↦ ‖g x‖
⊢ Summable fun x ↦ ‖f x‖

Tactic:
obtain v := Module.finBasis ℝ E
    set e := v.equivFunL
    have H : Summable fun x => ‖e (f x)‖ := this (e.summable.2 hf)
    refine .of_norm_bounded (H.mul_left ↑‖(e.symm : (Fin (finrank ℝ E) → ℝ) →L[ℝ] E)‖₊) fun i ↦ ?_
    simpa using (e.symm : (Fin (finrank ℝ E) → ℝ) →L[ℝ] E).le_opNorm (e <| f i)

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

Goal state before:
pp: case insert
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‖₊
⊢ f j * ∑ i ∈ s, f i = 0

Tactic:
simpa [Finset.mul_sum] using Finset.sum_eq_zero fun i hi ↦ h (by grind)

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'
h_eLpNorm_ne_top : eLpNorm (↑↑g - ↑↑f) 1 μ ≠ ∞
⊢ (eLpNorm (↑↑g - ↑↑f) 1 μ').toReal ≤ (c' * eLpNorm (↑↑g - ↑↑f) 1 μ).toReal

Tactic:
by
      refine toReal_mono (ENNReal.mul_ne_top hc' h_eLpNorm_ne_top) ?_
      refine (eLpNorm_mono_measure (⇑g - ⇑f) hμ'_le).trans_eq ?_
      rw [eLpNorm_smul_measure_of_ne_zero hc'0, smul_eq_mul]
      simp

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

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

Tactic:
suffices f =ᵐ[μ] 0 by
      rw [eLpNorm_congr_ae this, eLpNorm_zero, hr, ENNReal.coe_zero, mul_zero]

Goal state after:
case pos
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
r : ℝ≥0
hfint : Integrable f μ
hfint' : 0 ≤ ∫ (x : α), f x ∂μ
hf : ∀ᵐ (ω : α) ∂μ, f ω ≤ ↑r
hr : r = 0
⊢ f =ᵐ[μ] 0
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: case pos
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 c' : ℝ≥0∞
hc : c ≠ ∞
hc' : c' ≠ ∞
hμ_le : μ ≤ c • μ'
hμ'_le : μ' ≤ c' • μ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ' T C'
f : α → E
hf : Integrable f μ
⊢ setToFun μ T hT f = setToFun μ' T hT' f

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
μ' : Measure α
c c' : ℝ≥0∞
hc : c ≠ ∞
hc' : c' ≠ ∞
hμ_le : μ ≤ c • μ'
hμ'_le : μ' ≤ c' • μ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ' T C'
f : α → E
hf : ¬Integrable f μ
⊢ setToFun μ T hT f = setToFun μ' T hT' f

Tactic:
· exact setToFun_congr_measure_of_integrable c' hc' hμ'_le hT hT' f hf

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 C' : ℝ
μ' : Measure α
c c' : ℝ≥0∞
hc : c ≠ ∞
hc' : c' ≠ ∞
hμ_le : μ ≤ c • μ'
hμ'_le : μ' ≤ c' • μ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ' T C'
f : α → E
hf : ¬Integrable f μ
⊢ setToFun μ T hT f = setToFun μ' T hT' f
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 y : α
h : infEDist x E ≤ infEDist y E
h_le : infEDist y E * (↑δ.toNNReal)⁻¹ ≤ 1
⊢ infEDist y E * (↑δ.toNNReal)⁻¹ - infEDist x E * (↑δ.toNNReal)⁻¹ ≤ (↑δ.toNNReal)⁻¹ * edist x y

Tactic:
rw [← ENNReal.sub_mul (by simp [δ_pos]), mul_comm, edist_comm]
      gcongr
      simp only [tsub_le_iff_right]
      exact infEDist_le_edist_add_infEDist

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

Goal state before:
pp: case pos
R : Type u_1
inst✝² : LinearOrder R
inst✝¹ : TopologicalSpace R
inst✝ : OrderTopology R
a b : R
h : IsTop b
⊢ IsOpen[inst✝¹] (Ioc a b)

Tactic:
have : Ioc a b = Ioi a := Subset.antisymm (fun x hx ↦ hx.1) (fun x hx ↦ by exact ⟨hx, h _⟩)

Goal state after:
case pos
R : Type u_1
inst✝² : LinearOrder R
inst✝¹ : TopologicalSpace R
inst✝ : OrderTopology R
a b : R
h : IsTop b
this : Ioc a b = Ioi a
⊢ IsOpen[inst✝¹] (Ioc a b)
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case neg
R : Type u_1
inst✝² : LinearOrder R
inst✝¹ : TopologicalSpace R
inst✝ : OrderTopology R
a b : R
h : ¬IsTop b
⊢ IsOpen[inst✝¹] (Ioo a b)

Tactic:
simp [isOpen_Ioo]

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✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : X → E
μ : Measure X
ι : Type u_5
inst✝ : Fintype ι
s : ι → Set X
hs : ∀ (i : ι), MeasurableSet (s i)
h's : Pairwise (Disjoint on s)
hf : ∀ (i : ι), IntegrableOn f (s i) μ
⊢ (↑Finset.univ).Pairwise (Disjoint on s)

Tactic:
· simp [pairwise_univ, h's]

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

Goal state before:
pp: case inl
R : Type u_1
inst✝² : LinearOrder R
inst✝¹ : TopologicalSpace R
f✝ : StieltjesFunction R
inst✝ : OrderTopology R
f : R → ℝ
hf : Monotone f
x : R
s : Set ℝ
hs : s ∈ 𝓝 (rightLim f x)
l u : ℝ
hlu : rightLim f x ∈ Ioo l u
lus : Ioo l u ⊆ s
y₀ : R
hy₀ : x < y₀
y : R
xy : x < y
h'y : Ioo x y ⊆ f ⁻¹' Ioo l u
hz : x ∈ Ico x y
⊢ rightLim f x < u

Tactic:
exact hlu.2

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

Goal state before:
pp: case pos
R : Type u_1
inst✝⁴ : LinearOrder R
inst✝³ : TopologicalSpace R
f : StieltjesFunction R
inst✝² : OrderTopology R
inst✝¹ : CompactIccSpace R
inst✝ : DenselyOrdered R
a b : R
hab : a < b
s : ℕ → Set R
hs : Ioc a b ⊆ ⋃ i, s i
ε : ℝ≥0
εpos : 0 < ε
h : ∑' (i : ℕ), f.length (s i) < ∞
δ : ℝ≥0 := ε / 2
δpos : 0 < ↑δ
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑δ
a' : R
ha' : ↑f a' - ↑f a < ↑δ
aa' : a < a'
this : Nonempty R
i : ℕ
p q' : R
spq : s i \ botSet ⊆ Ioc p q'
hq' : ofReal (↑f q' - ↑f p) < f.length (s i) + ↑(ε' i)
A : Icc a' b ∩ s i ⊆ Ioc p q'
htq' : IsTop q'
x : R
hx : x ∈ Icc a' b ∩ s i
⊢ p < x ∧ x ≤ q'

Tactic:
exact ⟨(A hx).1, htq' _⟩

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✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
s : Set X
μ : Measure X
inst✝ : CompleteSpace E
c : E
⊢ ∫ (x : X) in s, c ∂μ = μ.real s • c

Tactic:
by
  rw [integral_const, measureReal_restrict_apply_univ]

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

Goal state before:
pp: E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
X : Type u_5
m m0 : MeasurableSpace X
μ : Measure X
hm : m ≤ m0
f : X → E
hf_meas : StronglyMeasurable f
s : Set X
hs : MeasurableSet s
⊢ ∫ (x : X) in s, f x ∂μ = ∫ (x : X) in s, f x ∂μ.trim hm

Tactic:
rwa [integral_trim hm hf_meas, restrict_trim hm μ]

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
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
this : Quotient.mk α_mod_G ⁻¹' Quotient.mk α_mod_G '' A n = ⋃ g, (fun x ↦ g • x) '' A n
measπAn : MeasurableSet (Quotient.mk α_mod_G '' A n)
⊢ ν (⋃ i, (fun x ↦ i • x) '' A n ∩ s) ≤ ∑' (g : G), ν (g • A n ∩ s)

Tactic:
exact measure_iUnion_le _

Goal state after:
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
K₀ : PositiveCompacts G
U : Set G
K : Compacts G
hU : (interior U).Nonempty
⊢ index (↑K) U ≤ index ↑K ↑K₀ * index (↑K₀) U

Tactic:
apply le_index_mul K₀ K hU

Goal state after:
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
l : ℝ
x : R
this✝ : Nonempty R
val✝ : OrderTop R
hf : Tendsto (↑f) (pure ⊤) (𝓝 l)
this : Ici x = Icc x ⊤
⊢ f.measure (Ici x) = ofReal (l - leftLim (↑f) x)

Tactic:
rw [this, measure_Icc, tendsto_nhds_unique hf (tendsto_pure_nhds f ⊤)]

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

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x y : F
t : ℝ
⊢ normSq x * t * t + t * re ⟪x, y⟫ + re (⟪y, x⟫ * ↑t) + re ⟪y, y⟫ =
    normSq x * t * t + re ⟪x, y⟫ * t + re ⟪y, x⟫ * t + re ⟪y, y⟫

Tactic:
mul_comm t _,

Goal state after:
𝕜 : Type u_1
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x y : F
t : ℝ
⊢ normSq x * t * t + re ⟪x, y⟫ * t + re (⟪y, x⟫ * ↑t) + re ⟪y, y⟫ =
    normSq x * t * t + re ⟪x, y⟫ * t + re ⟪y, x⟫ * t + re ⟪y, y⟫
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
h : ‖y‖ = 0
⊢ ⟪x, y⟫ = 0

Tactic:
rw [inner_eq_zero_symm, inner_eq_zero_of_left _ h]

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

Goal state before:
pp: case e_a
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
δ : ℝ
δpos : 0 < δ
hδ : ∫ (x : Y) in k, ‖L‖ * ‖g x‖ * δ ∂μ < ε
v : Set X
v_mem : v ∈ 𝓝[s] q
hv : ∀ p ∈ v, ∀ x ∈ k, ‖f p x - f q x‖ < δ
I : ∀ p ∈ s, IntegrableOn (fun y ↦ (L (g y)) (f p y)) k μ
p : X
hp : p ∈ v
h'p : p ∈ s
y : Y
hy : y ∉ k
⊢ (L (g y)) (f p y) = 0

Tactic:
simp [hfs p y h'p hy]

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

Goal state before:
pp: x y : ℝ
⊢ inner ℝ x y = x * y

Tactic:
rw [mul_comm]; rfl

Goal state after:
Module: Mathlib.Topology.Algebra.Module.ClosedSubmodule

Goal state before:
pp: R : Type u_2
M : Type u_3
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : Module R M
x : M
S : Set (ClosedSubmodule R M)
⊢ x ∈ sInf S ↔ ∀ s ∈ S, x ∈ s

Tactic:
by
  simp [← SetLike.mem_coe]

Goal state after:
Module: Mathlib.Topology.Algebra.Module.ClosedSubmodule

Goal state before:
pp: ι : Sort u_1
R : Type u_2
M : Type u_3
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : Module R M
x : M
f : ι → ClosedSubmodule R M
⊢ x ∈ ⨅ i, f i ↔ ∀ (i : ι), x ∈ f i

Tactic:
by
  simp [← SetLike.mem_coe]

Goal state after:
Module: Mathlib.Topology.Algebra.Module.ClosedSubmodule

Goal state before:
pp: ι : Sort u_1
R : Type u_2
N : Type u_4
inst✝⁵ : Semiring R
inst✝⁴ : AddCommMonoid N
inst✝³ : TopologicalSpace N
inst✝² : Module R N
inst✝¹ : ContinuousAdd N
inst✝ : ContinuousConstSMul R N
x : N
f : ι → ClosedSubmodule R N
⊢ x ∈ ⨆ i, f i ↔ x ∈ closure (⨆ i, ↑(f i)).carrier

Tactic:
by
  simp [← SetLike.mem_coe]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : SeminormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_4
G : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
inst✝ : CompleteSpace E
f : (i : ι) → G i
hf : ∀ ε > 0, ∃ N, ∀ (m : Finset ι), N ⊆ m → ∀ (n : Finset ι), N ⊆ n → |∑ i ∈ m, ‖f i‖ ^ 2 - ∑ i ∈ n, ‖f i‖ ^ 2| < ε
ε : ℝ
hε : ε > 0
hε' : 0 < ε ^ 2 / 2
a : Finset ι
H : ∀ (m : Finset ι), a ⊆ m → ∀ (n : Finset ι), a ⊆ n → |∑ i ∈ m, ‖f i‖ ^ 2 - ∑ i ∈ n, ‖f i‖ ^ 2| < ε ^ 2 / 2
s₁ : Finset ι
hs₁ : a ⊆ s₁
s₂ : Finset ι
hs₂ : a ⊆ s₂
has : a ⊆ s₁ ⊓ s₂
Hs₁ : ∑ x ∈ s₁ \ s₂, ‖f x‖ ^ 2 < ε ^ 2 / 2
⊢ ∑ x ∈ s₂ \ s₁, ‖f x‖ ^ 2 < ε ^ 2 / 2

Tactic:
convert! H _ hs₂ _ has

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : SeminormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_4
G : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
inst✝ : CompleteSpace E
f : (i : ι) → G i
hf : ∀ ε > 0, ∃ N, ∀ (m : Finset ι), N ⊆ m → ∀ (n : Finset ι), N ⊆ n → |∑ i ∈ m, ‖f i‖ ^ 2 - ∑ i ∈ n, ‖f i‖ ^ 2| < ε
ε : ℝ
hε : ε > 0
hε' : 0 < ε ^ 2 / 2
a : Finset ι
H : ∀ (m : Finset ι), a ⊆ m → ∀ (n : Finset ι), a ⊆ n → |∑ i ∈ m, ‖f i‖ ^ 2 - ∑ i ∈ n, ‖f i‖ ^ 2| < ε ^ 2 / 2
s₁ : Finset ι
hs₁ : a ⊆ s₁
s₂ : Finset ι
hs₂ : a ⊆ s₂
has : a ⊆ s₁ ⊓ s₂
Hs₁ : ∑ x ∈ s₁ \ s₂, ‖f x‖ ^ 2 < ε ^ 2 / 2
⊢ ∑ x ∈ s₂ \ s₁, ‖f x‖ ^ 2 = |∑ i ∈ s₂, ‖f i‖ ^ 2 - ∑ i ∈ s₁ ⊓ s₂, ‖f i‖ ^ 2|
Module: Mathlib.Analysis.InnerProductSpace.Projection.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
U V : Submodule 𝕜 E
inst✝ : U.HasOrthogonalProjection
h : U.orthogonalProjectionOnto ∘SL V.subtypeL = 0
u : E
hu : u ∈ U
v : E
hv : v ∈ V
⊢ ⟪v, u⟫ = 0

Tactic:
convert starProjection_inner_eq_zero v u hu

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
U V : Submodule 𝕜 E
inst✝ : U.HasOrthogonalProjection
h : U.orthogonalProjectionOnto ∘SL V.subtypeL = 0
u : E
hu : u ∈ U
v : E
hv : v ∈ V
⊢ v = v - U.starProjection v
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: p : ℝ≥0∞
ι : Type u_2
β : ι → Type u_4
inst✝² : Fact (1 ≤ p)
inst✝¹ : (i : ι) → PseudoEMetricSpace (β i)
inst✝ : Fintype ι
x y : WithLp p ((i : ι) → β i)
⊢ edist x.ofLp y.ofLp ≤ edist x y

Tactic:
edist_pi_def,

Goal state after:
p : ℝ≥0∞
ι : Type u_2
β : ι → Type u_4
inst✝² : Fact (1 ≤ p)
inst✝¹ : (i : ι) → PseudoEMetricSpace (β i)
inst✝ : Fintype ι
x y : WithLp p ((i : ι) → β i)
⊢ (Finset.univ.sup fun b ↦ edist (x.ofLp b) (y.ofLp b)) ≤ edist x y
Module: Mathlib.Analysis.Normed.Operator.Banach

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
inst✝¹⁴ : NontriviallyNormedField 𝕜
inst✝¹³ : NontriviallyNormedField 𝕜'
σ : 𝕜 →+* 𝕜'
E : Type u_3
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace 𝕜 E
F✝ : Type u_4
inst✝¹⁰ : NormedAddCommGroup F✝
inst✝⁹ : NormedSpace 𝕜' F✝
f✝ : E →SL[σ] F✝
σ' : 𝕜' →+* 𝕜
inst✝⁸ : RingHomInvPair σ σ'
inst✝⁷ : RingHomIsometric σ
inst✝⁶ : RingHomIsometric σ'
inst✝⁵ : CompleteSpace F✝
inst✝⁴ : CompleteSpace E
F : Type u_5
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : CompleteSpace F
f : E →L[𝕜] F
G : Submodule 𝕜 F
h : IsCompl (↑f).range G
inst✝ : CompleteSpace ↥G
hker : (↑f).ker = ⊥
⊢ Disjoint (↑f).range (↑G.subtypeL).range

Tactic:
· simp [h.disjoint]

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: ι : Type u_2
β : ι → Type u_4
inst✝¹ : Fintype ι
inst✝ : (i : ι) → SeminormedAddCommGroup (β i)
x y : PiLp 2 β
⊢ ↑(nndist x y) = ↑(NNReal.sqrt (∑ i, nndist (x.ofLp i) (y.ofLp i) ^ 2))

Tactic:
by
    push_cast
    exact dist_eq_of_L2 _ _

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
u : ℕ → E
hu : Bornology.IsBounded (range u)
hs : Pairwise (Disjoint on fun n ↦ {u n} + s)
h's : MeasurableSet s
H : ∀ (R : ℝ), μ (s ∩ closedBall 0 R) = 0
⊢ μ (⋃ n, s ∩ closedBall 0 ↑n) ≤ ∑' (n : ℕ), μ (s ∩ closedBall 0 ↑n)

Tactic:
exact measure_iUnion_le _

Goal state after:
Module: Mathlib.Analysis.BoxIntegral.Box.Basic

Goal state before:
pp: ι : Type u_1
I : Box ι
⊢ Option.isSome ↑I = true ↔ (↑↑I).Nonempty

Tactic:
unfold Option.isSome
    simp [I.nonempty_coe]

Goal state after:
Module: Mathlib.Analysis.BoxIntegral.Box.SubboxInduction

Goal state before:
pp: case pos
ι : Type u_1
I✝ J I : Box ι
s : Set ι
i : ι
h✝ : i ∈ s
⊢ (I.lower i + I.upper i) / 2 < I.upper i

Tactic:
simp only [left_lt_add_div_two, add_div_two_lt_right, I.lower_lt_upper]

Goal state after: