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

Goal state before:
pp: case insert
δ : Type u_1
X : δ → Type u_3
inst✝² : (i : δ) → MeasurableSpace (X i)
μ : (i : δ) → Measure (X i)
inst✝¹ : DecidableEq δ
s✝ : Finset δ
inst✝ : ∀ (i : δ), SigmaFinite (μ i)
i : δ
f : ((i : δ) → X i) → ℝ≥0∞
hf : Measurable f
y : X i
i' : δ
s : Finset δ
hi' : i' ∉ s
ih : i ∉ s → ∀ (x : (i : δ) → X i), (∫⋯∫⁻_s, f ∂μ) (update x i y) = (∫⋯∫⁻_s, f ∘ fun x ↦ update x i y ∂μ) x
hi : i ∉ insert i' s
x : (i : δ) → X i
⊢ ∫⁻ (xᵢ : X i'), (∫⋯∫⁻_s, f ∂μ) (update (update x i y) i' xᵢ) ∂μ i' = (∫⋯∫⁻_insert i' s, f ∘ fun x ↦ update x i y ∂μ) x

Tactic:
lmarginal_insert _ (hf.comp measurable_update_left) hi'

Goal state after:
case insert
δ : Type u_1
X : δ → Type u_3
inst✝² : (i : δ) → MeasurableSpace (X i)
μ : (i : δ) → Measure (X i)
inst✝¹ : DecidableEq δ
s✝ : Finset δ
inst✝ : ∀ (i : δ), SigmaFinite (μ i)
i : δ
f : ((i : δ) → X i) → ℝ≥0∞
hf : Measurable f
y : X i
i' : δ
s : Finset δ
hi' : i' ∉ s
ih : i ∉ s → ∀ (x : (i : δ) → X i), (∫⋯∫⁻_s, f ∂μ) (update x i y) = (∫⋯∫⁻_s, f ∘ fun x ↦ update x i y ∂μ) x
hi : i ∉ insert i' s
x : (i : δ) → X i
⊢ ∫⁻ (xᵢ : X i'), (∫⋯∫⁻_s, f ∂μ) (update (update x i y) i' xᵢ) ∂μ i' =
    ∫⁻ (xᵢ : X i'), (∫⋯∫⁻_s, f ∘ fun x ↦ update x i y ∂μ) (update x i' xᵢ) ∂μ 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
s t : Set X
μ : Measure X
hst : s =ᵐ[μ] t
⊢ ∫ (x : X) in s, f x ∂μ = ∫ (x : X) in t, f x ∂μ

Tactic:
Measure.restrict_congr_set hst

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
hst : s =ᵐ[μ] t
⊢ ∫ (x : X) in t, f x ∂μ = ∫ (x : X) in t, f x ∂μ
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: R : Type u_1
inst✝² : LinearOrder R
inst✝¹ : TopologicalSpace R
inst✝ : OrderTopology R
f : StieltjesFunction R
x : R
⊢ rightLim (↑f) x = ↑f x

Tactic:
by
  rw [← f.mono.continuousWithinAt_Ioi_iff_rightLim_eq, continuousWithinAt_Ioi_iff_Ici]
  exact f.right_continuous' x

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
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
l : Filter ι
inst✝¹ : l.IsCountablyGenerated
F : ι → α → E
inst✝ : IsFiniteMeasure μ
f : α → E
h_meas : ∀ᶠ (n : ι) in l, AEStronglyMeasurable (F n) μ
h_bound : ∃ C, ∀ᶠ (n : ι) in l, ∀ᵐ (ω : α) ∂μ, ‖F n ω‖ ≤ C
h_lim : ∀ᵐ (ω : α) ∂μ, Tendsto (fun n ↦ F n ω) l (𝓝 (f ω))
⊢ Tendsto (fun n ↦ setToFun μ T hT (F n)) l (𝓝 (setToFun μ T hT f))

Tactic:
obtain ⟨c, h_boundc⟩ := h_bound
  let C : α → ℝ := (fun _ => c)
  exact tendsto_setToFun_filter_of_dominated_convergence hT
    C h_meas h_boundc (integrable_const c) h_lim

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

Goal state before:
pp: R : Type u_1
inst✝³ : LinearOrder R
inst✝² : TopologicalSpace R
f : StieltjesFunction R
inst✝¹ : OrderTopology R
inst✝ : CompactIccSpace R
a b : R
c d : ℕ → R
ss : Icc a b ⊆ ⋃ i, Iotop (c i) (d i)
this :
  ∀ (s : Finset ℕ) (b : R),
    Icc a b ⊆ ⋃ i ∈ ↑s, Iotop (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i ∈ s, ofReal (↑f (d i) - ↑f (c i))
⊢ ofReal (↑f b - ↑f a) ≤ ∑' (i : ℕ), ofReal (↑f (d i) - ↑f (c i))

Tactic:
rcases isCompact_Icc.elim_finite_subcover_image
        (fun (i : ℕ) (_ : i ∈ univ) => @isOpen_Iotop _ _ _ _ (c i) (d i)) (by simpa using ss) with
      ⟨s, _, hf, hs⟩

Goal state after:
R : Type u_1
inst✝³ : LinearOrder R
inst✝² : TopologicalSpace R
f : StieltjesFunction R
inst✝¹ : OrderTopology R
inst✝ : CompactIccSpace R
a b : R
c d : ℕ → R
ss : Icc a b ⊆ ⋃ i, Iotop (c i) (d i)
this :
  ∀ (s : Finset ℕ) (b : R),
    Icc a b ⊆ ⋃ i ∈ ↑s, Iotop (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i ∈ s, ofReal (↑f (d i) - ↑f (c i))
s : Set ℕ
left✝ : s ⊆ univ
hf : s.Finite
hs : Icc a b ⊆ ⋃ i ∈ s, Iotop (c i) (d i)
⊢ ofReal (↑f b - ↑f a) ≤ ∑' (i : ℕ), ofReal (↑f (d i) - ↑f (c i))
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
h3 : ∀ (g : G) {K : Compacts G}, μ (Compacts.map (fun x ↦ g * x) ⋯ K) = μ K
K : Compacts G
hK : μ K ≠ 0
U : Opens G
hU : (↑U).Nonempty
this✝ : (interior ↑U).Nonempty
s : Finset G
hs : K.carrier ⊆ ⋃ g ∈ s, (fun x ↦ g * x) ⁻¹' ↑U
this : ↑K ⊆ ↑(⨆ g ∈ s, (Opens.comap ↑(Homeomorph.mulLeft g)) U)
⊢ μ.innerContent (⨆ g ∈ s, (Opens.comap ↑(Homeomorph.mulLeft g)) U) ≤ ↑s.card * μ.innerContent U

Tactic:
refine
    (rel_iSup_sum μ.innerContent μ.innerContent_bot (· ≤ ·) μ.innerContent_iSup_nat _ _).trans ?_

Goal state after:
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
h3 : ∀ (g : G) {K : Compacts G}, μ (Compacts.map (fun x ↦ g * x) ⋯ K) = μ K
K : Compacts G
hK : μ K ≠ 0
U : Opens G
hU : (↑U).Nonempty
this✝ : (interior ↑U).Nonempty
s : Finset G
hs : K.carrier ⊆ ⋃ g ∈ s, (fun x ↦ g * x) ⁻¹' ↑U
this : ↑K ⊆ ↑(⨆ g ∈ s, (Opens.comap ↑(Homeomorph.mulLeft g)) U)
⊢ ∑ d ∈ s, μ.innerContent ((Opens.comap ↑(Homeomorph.mulLeft d)) U) ≤ ↑s.card * μ.innerContent U
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case inr.inl
R : Type u_1
inst✝¹ : LinearOrder R
inst✝ : TopologicalSpace R
f : StieltjesFunction R
c : R
t : Set R
this : Nonempty R
a b : R
h : t \ botSet ⊆ Ioc a b
hac : c ≤ a
hbc : b ≤ c
⊢ f.length (Ioc a b ∩ Ioi c) + f.length (Ioc a b \ Ioi c) ≤ ofReal (↑f b - ↑f a)

pp: case inr.inr
R : Type u_1
inst✝¹ : LinearOrder R
inst✝ : TopologicalSpace R
f : StieltjesFunction R
c : R
t : Set R
this : Nonempty R
a b : R
h : t \ botSet ⊆ Ioc a b
hac : c ≤ a
hbc : c ≤ b
⊢ f.length (Ioc a b ∩ Ioi c) + f.length (Ioc a b \ Ioi c) ≤ ofReal (↑f b - ↑f a)

Tactic:
· simp only [hbc, le_refl, Ioc_eq_empty, Ioc_inter_Ioi, min_eq_left, Ioc_sdiff_Ioi,
      f.length_empty, zero_add, or_true, le_sup_iff, f.length_Ioc, not_lt]

Goal state after:
case inr.inr
R : Type u_1
inst✝¹ : LinearOrder R
inst✝ : TopologicalSpace R
f : StieltjesFunction R
c : R
t : Set R
this : Nonempty R
a b : R
h : t \ botSet ⊆ Ioc a b
hac : c ≤ a
hbc : c ≤ b
⊢ f.length (Ioc a b ∩ Ioi c) + f.length (Ioc a b \ Ioi c) ≤ ofReal (↑f b - ↑f a)
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
hfi : Integrable f μ
h_meas : NullMeasurableSet {x | 0 ≤ f x} μ
x : X
hx : x ∈ {x | 0 ≤ f x}ᶜ
⊢ f x ≤ 0

Tactic:
Set.mem_compl_iff,

Goal state after:
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
hfi : Integrable f μ
h_meas : NullMeasurableSet {x | 0 ≤ f x} μ
x : X
hx : x ∉ {x | 0 ≤ f x}
⊢ f x ≤ 0
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
a b : R
⊢ f.measure (Ioc a b) = ofReal (↑f b - ↑f a)

Tactic:
StieltjesFunction.measure

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
a b : R
⊢ (let __OuterMeasure := f.outer;
      { toOuterMeasure := __OuterMeasure, m_iUnion := ⋯, trim_le := ⋯ })
      (Ioc a b) =
    ofReal (↑f b - ↑f a)
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case pos
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 : R
ha : IsBot a
this : leftLim (↑f) a = ↑f a
⊢ f.measure {a} ≤ ⊥

Tactic:
StieltjesFunction.measure

Goal state after:
case pos
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 : R
ha : IsBot a
this : leftLim (↑f) a = ↑f a
⊢ (let __OuterMeasure := f.outer;
      { toOuterMeasure := __OuterMeasure, m_iUnion := ⋯, trim_le := ⋯ })
      {a} ≤
    ⊥
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: case refine_1
G : Type u_1
α : Type u_3
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
a b : G
hab : a ≠ b
x : α
hx : x ∈ s ∧ ∀ (g : G), g ≠ 1 → x ∉ g • s
y : α
hy : y ∈ s ∧ ∀ (g : G), g ≠ 1 → y ∉ g • s
hxy : (fun x ↦ b • x) y = (fun x ↦ a • x) x
⊢ ¬b = a

Tactic:
eq_comm

Goal state after:
case refine_1
G : Type u_1
α : Type u_3
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
a b : G
hab : a ≠ b
x : α
hx : x ∈ s ∧ ∀ (g : G), g ≠ 1 → x ∉ g • s
y : α
hy : y ∈ s ∧ ∀ (g : G), g ≠ 1 → y ∉ g • s
hxy : (fun x ↦ b • x) y = (fun x ↦ a • x) x
⊢ ¬a = b
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
inst✝ : IsFiniteMeasure μ
C : ℝ
hf : ∀ᵐ (x : X) ∂μ, ‖f x‖ ≤ C
s : Set X
hs : MeasurableSet s
hf1 : Integrable f μ
⊢ ∫ (x : X), f x ∂μ = ∫ (x : X) in sᶜ, f x ∂μ + ∫ (x : X) in s, f x ∂μ

Tactic:
add_comm,

Goal state after:
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
inst✝ : IsFiniteMeasure μ
C : ℝ
hf : ∀ᵐ (x : X) ∂μ, ‖f x‖ ≤ C
s : Set X
hs : MeasurableSet s
hf1 : Integrable f μ
⊢ ∫ (x : X), f x ∂μ = ∫ (x : X) in s, f x ∂μ + ∫ (x : X) in sᶜ, f x ∂μ
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) atBot (𝓝 l)
x : R
this : Nonempty R
⊢ ofReal (↑f x - l) - ofReal (↑f x - leftLim (↑f) x) = ofReal (leftLim (↑f) x - l)

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) atBot (𝓝 l)
x : R
this : Nonempty 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) atBot (𝓝 l)
x : R
this : Nonempty R
⊢ {x} ⊆ Iic x

Tactic:
← ofReal_sub _ (sub_nonneg.mpr <| Monotone.leftLim_le f.mono' le_rfl)

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) atBot (𝓝 l)
x : R
this : Nonempty R
⊢ ofReal (↑f x - l - (↑f x - leftLim (↑f) x)) = ofReal (leftLim (↑f) x - l)

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) atBot (𝓝 l)
x : R
this : Nonempty 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) atBot (𝓝 l)
x : R
this : Nonempty R
⊢ {x} ⊆ Iic x
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
⊢ ofReal (l - leftLim (↑f) 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:
← ofReal_sub _ (sub_nonneg.mpr <| Monotone.leftLim_le f.mono' le_rfl)

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 - (↑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.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 : F
⊢ x = 0 → ⟪x, x⟫ = 0

Tactic:
rintro rfl
  exact inner_zero_left _

Goal state after:
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
g : StieltjesFunction R
l : ℝ
hfg : f.measure = g.measure
hfl : Tendsto (↑f) atBot (𝓝 l)
hgl : Tendsto (↑g) atBot (𝓝 l)
x : R
hf : ↑g x - l = ↑f x - l
⊢ ↑f x = ↑g x

pp: case 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
g : StieltjesFunction R
l : ℝ
hfg : f.measure = g.measure
hfl : Tendsto (↑f) atBot (𝓝 l)
hgl : Tendsto (↑g) atBot (𝓝 l)
x : R
hf : ofReal (↑g x - l) = ofReal (↑f x - l)
⊢ 0 ≤ ↑g x - l

pp: case hq
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
g : StieltjesFunction R
l : ℝ
hfg : f.measure = g.measure
hfl : Tendsto (↑f) atBot (𝓝 l)
hgl : Tendsto (↑g) atBot (𝓝 l)
x : R
hf : ofReal (↑g x - l) = ofReal (↑f x - l)
⊢ 0 ≤ ↑f x - l

Tactic:
eq_comm

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
g : StieltjesFunction R
l : ℝ
hfg : f.measure = g.measure
hfl : Tendsto (↑f) atBot (𝓝 l)
hgl : Tendsto (↑g) atBot (𝓝 l)
x : R
hf : ↑f x - l = ↑g x - l
⊢ ↑f x = ↑g x

case 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
g : StieltjesFunction R
l : ℝ
hfg : f.measure = g.measure
hfl : Tendsto (↑f) atBot (𝓝 l)
hgl : Tendsto (↑g) atBot (𝓝 l)
x : R
hf : ofReal (↑g x - l) = ofReal (↑f x - l)
⊢ 0 ≤ ↑g x - l

case hq
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
g : StieltjesFunction R
l : ℝ
hfg : f.measure = g.measure
hfl : Tendsto (↑f) atBot (𝓝 l)
hgl : Tendsto (↑g) atBot (𝓝 l)
x : R
hf : ofReal (↑g x - l) = ofReal (↑f x - l)
⊢ 0 ≤ ↑f x - l
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 + re ⟪x, y⟫ * t + re ⟪y, x⟫ * t + re ⟪y, y⟫ =
    normSq x * t * t + re ⟪x, y⟫ * t + re ⟪y, x⟫ * t + normSq y

Tactic:
rw [← normSq]

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
⊢ ↑‖⟪x, y⟫‖ ^ 2 * ↑(normSq x) - ↑(normSq x) * ((starRingEnd 𝕜) ⟪x, y⟫ * ⟪x, y⟫) - ⟪x, y⟫ * (↑(normSq x) * ⟪y, x⟫) +
      ↑(normSq x) * (↑(normSq x) * ↑(normSq y)) =
    ↑(normSq x * (normSq x * normSq y) - normSq x * ‖⟪x, y⟫‖ ^ 2)

Tactic:
RCLike.conj_mul,

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
⊢ ↑‖⟪x, y⟫‖ ^ 2 * ↑(normSq x) - ↑(normSq x) * ↑‖⟪x, y⟫‖ ^ 2 - ⟪x, y⟫ * (↑(normSq x) * ⟪y, x⟫) +
      ↑(normSq x) * (↑(normSq x) * ↑(normSq y)) =
    ↑(normSq x * (normSq x * normSq y) - normSq x * ‖⟪x, y⟫‖ ^ 2)
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
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
h₂ : IsClosed[inst✝¹] K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen[inst✝¹] U₁
h1U₂ : IsOpen[inst✝¹] U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
h2L₁ : K₁.carrier * L₁ ⊆ U₁
⊢ chaar K₀ (K₁ ⊔ K₂) = chaar K₀ K₁ + chaar K₀ K₂

Tactic:
rcases mem_nhds_iff.mp h1L₁ with ⟨V₁, h1V₁, h2V₁, h3V₁⟩

Goal state after:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
h₂ : IsClosed[inst✝¹] K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen[inst✝¹] U₁
h1U₂ : IsOpen[inst✝¹] U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
h2L₁ : K₁.carrier * L₁ ⊆ U₁
V₁ : Set G
h1V₁ : V₁ ⊆ L₁
h2V₁ : IsOpen[inst✝¹] V₁
h3V₁ : 1 ∈ V₁
⊢ chaar K₀ (K₁ ⊔ K₂) = chaar K₀ K₁ + chaar K₀ K₂
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 : E
⊢ ⟪x, x⟫ = ↑‖x‖ ^ 2

Tactic:
rw [← inner_self_ofReal_re, ← norm_sq_eq_re_inner, ofReal_pow]

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
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
RCLike.conj_mul,

Goal state after:
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
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
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
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
h₂ : IsClosed[inst✝¹] K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen[inst✝¹] U₁
h1U₂ : IsOpen[inst✝¹] U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
V₁ : Set G
h1V₁ : V₁ ⊆ L₁
h2V₁ : IsOpen[inst✝¹] V₁
h3V₁ : 1 ∈ V₁
h2L₁ : K₁.carrier * V₁ ⊆ U₁
L₂ : Set G
h1L₂ : L₂ ∈ 𝓝 1
V₂ : Set G
h1V₂ : V₂ ⊆ L₂
h2V₂ : IsOpen[inst✝¹] V₂
h3V₂ : 1 ∈ V₂
h2L₂ : K₂.carrier * V₂ ⊆ U₂
eval : (Compacts G → ℝ) → ℝ := fun f ↦ f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
V : Set G := V₁ ∩ V₂
U : Set G
h1U : U ⊆ ↑{ carrier := V⁻¹, is_open' := ⋯, mem' := ⋯ }.toOpens
h2U : IsOpen[inst✝¹] U
h3U : 1 ∈ U
⊢ prehaar (↑K₀) U K₁ + prehaar (↑K₀) U K₂ = prehaar (↑K₀) U (K₁ ⊔ K₂)

Tactic:
eq_comm

Goal state after:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
h₂ : IsClosed[inst✝¹] K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen[inst✝¹] U₁
h1U₂ : IsOpen[inst✝¹] U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
V₁ : Set G
h1V₁ : V₁ ⊆ L₁
h2V₁ : IsOpen[inst✝¹] V₁
h3V₁ : 1 ∈ V₁
h2L₁ : K₁.carrier * V₁ ⊆ U₁
L₂ : Set G
h1L₂ : L₂ ∈ 𝓝 1
V₂ : Set G
h1V₂ : V₂ ⊆ L₂
h2V₂ : IsOpen[inst✝¹] V₂
h3V₂ : 1 ∈ V₂
h2L₂ : K₂.carrier * V₂ ⊆ U₂
eval : (Compacts G → ℝ) → ℝ := fun f ↦ f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
V : Set G := V₁ ∩ V₂
U : Set G
h1U : U ⊆ ↑{ carrier := V⁻¹, is_open' := ⋯, mem' := ⋯ }.toOpens
h2U : IsOpen[inst✝¹] U
h3U : 1 ∈ U
⊢ prehaar (↑K₀) U (K₁ ⊔ K₂) = prehaar (↑K₀) U K₁ + prehaar (↑K₀) U K₂
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
v w : E
⊢ ⟪v, w⟫ = ↑‖v‖ ^ 2 ↔ ↑‖v‖ ^ 2 = ⟪v, w⟫

Tactic:
eq_comm

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
v w : E
⊢ ↑‖v‖ ^ 2 = ⟪v, w⟫ ↔ ↑‖v‖ ^ 2 = ⟪v, w⟫
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
v w : E
⊢ ⟪v, w⟫ = ↑‖w‖ ^ 2 ↔ ⟪v, w⟫ = ↑‖w‖ ^ 2

Tactic:
eq_comm

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
v w : E
⊢ ↑‖w‖ ^ 2 = ⟪v, w⟫ ↔ ↑‖w‖ ^ 2 = ⟪v, w⟫