mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Order.LeftRightLim

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : TopologicalSpace β
f : α → β
a : α
ha : IsBot a
A : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
⊢ Iio a = ∅

Tactic:
simp

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : TopologicalSpace β
f : α → β
a : α
ha : IsBot a
A : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
⊢ IsMin a
Module: Mathlib.Topology.Order.LeftRightLim

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : T3Space β
f : α → β
a : α
h : Tendsto f (𝓝[<] a) (𝓝 (leftLim f a))
⊢ 𝓝[<] a ⊔ 𝓝[Icc a a] a = 𝓝[<] a ⊔ pure a

Tactic:
simp

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

Goal state before:
pp: case empty
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
μ : Measure X
ι : Type u_5
s : ι → Set X
hs : ∀ i ∈ ∅, MeasurableSet (s i)
h's : (↑∅).Pairwise (Disjoint on s)
hf : ∀ i ∈ ∅, IntegrableOn f (s i) μ
⊢ ∫ (x : X) in ⋃ i ∈ ∅, s i, f x ∂μ = ∑ i ∈ ∅, ∫ (x : X) in s i, f x ∂μ

Tactic:
simp

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
x✝ : ι
⊢ s x✝ = ⋃ (_ : x✝ ∈ Finset.univ), s x✝

Tactic:
simp

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

Goal state before:
pp: R : Type u_1
inst✝¹ : LinearOrder R
inst✝ : TopologicalSpace R
f : StieltjesFunction R
c : ℝ
x✝¹ x✝ : R
⊢ x✝¹ ≤ x✝ → (fun x ↦ c) x✝¹ ≤ (fun x ↦ c) x✝

Tactic:
simp

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

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
s : Set X
μ : Measure X
hs : MeasurableSet s
hfi : IntegrableOn f s μ
⊢ ∫ (x : X) in s, f x ∂μ + ∫ (x : X) in sᶜ, 0 ∂μ = ∫ (x : X) in s, f x ∂μ

Tactic:
simp

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

Goal state before:
pp: R : Type u_1
inst✝¹ : LinearOrder R
inst✝ : TopologicalSpace R
f : StieltjesFunction R
hR : Nonempty R
inhabited_h : Inhabited R
⊢ ⨅ (_ : ∅ \ botSet ⊆ Ioc default default), ofReal (↑f default - ↑f default) ≤ 0

Tactic:
simp

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
T : Set α → E →L[ℝ] F
C : ℝ
β : Type u_7
mβ : MeasurableSpace β
inst✝ : SFinite μ
hT : DominatedFinMeasAdditive μ T C
h'T : ∀ (s : Set (β × α)), MeasurableSet s → StronglyMeasurable fun x ↦ T (Prod.mk x ⁻¹' s)
f : β → α → E
hf : StronglyMeasurable (Function.uncurry f)
hF : CompleteSpace F
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
this : SeparableSpace ↑(range (Function.uncurry f) ∪ {0})
⊢ 0 ∈ range (Function.uncurry f) ∪ {0}

Tactic:
simp

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

Goal state before:
pp: X : Type u_5
m : MeasurableSpace X
μ : Measure X
s : Set X
⊢ ∫ (x : X) in s, 1 ∂μ = μ.real s

Tactic:
simp

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

Goal state before:
pp: case pos.refine_1.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
β : Type u_7
mβ : MeasurableSpace β
inst✝ : SFinite μ
hT : DominatedFinMeasAdditive μ T C
h'T : ∀ (s : Set (β × α)), MeasurableSet s → StronglyMeasurable fun x ↦ T (Prod.mk x ⁻¹' s)
f : β → α → E
hf : StronglyMeasurable (Function.uncurry f)
hF : CompleteSpace F
this✝² : MeasurableSpace E := borel E
this✝¹ : BorelSpace E
this✝ : SeparableSpace ↑(range (Function.uncurry f) ∪ {0})
s : ℕ → SimpleFunc (β × α) E := SimpleFunc.approxOn (Function.uncurry f) ⋯ (range (Function.uncurry f) ∪ {0}) 0 ⋯
s' : ℕ → β → SimpleFunc α E := fun n x ↦ (s n).comp (Prod.mk x) ⋯
f' : ℕ → β → F := fun n ↦ {x | Integrable (f x) μ}.indicator fun x ↦ SimpleFunc.setToSimpleFunc T (s' n x)
hf' : ∀ (n : ℕ), StronglyMeasurable (f' n)
x : β
hfx : Integrable (f x) μ
this : ∀ (n : ℕ), Integrable (⇑(s' n x)) μ
n : ℕ
y : α
⊢ 0 ∈ range (Function.uncurry f) ∪ {0}

Tactic:
simp

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

Goal state before:
pp: case pos.refine_2.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
β : Type u_7
mβ : MeasurableSpace β
inst✝ : SFinite μ
hT : DominatedFinMeasAdditive μ T C
h'T : ∀ (s : Set (β × α)), MeasurableSet s → StronglyMeasurable fun x ↦ T (Prod.mk x ⁻¹' s)
f : β → α → E
hf : StronglyMeasurable (Function.uncurry f)
hF : CompleteSpace F
this✝² : MeasurableSpace E := borel E
this✝¹ : BorelSpace E
this✝ : SeparableSpace ↑(range (Function.uncurry f) ∪ {0})
s : ℕ → SimpleFunc (β × α) E := SimpleFunc.approxOn (Function.uncurry f) ⋯ (range (Function.uncurry f) ∪ {0}) 0 ⋯
s' : ℕ → β → SimpleFunc α E := fun n x ↦ (s n).comp (Prod.mk x) ⋯
f' : ℕ → β → F := fun n ↦ {x | Integrable (f x) μ}.indicator fun x ↦ SimpleFunc.setToSimpleFunc T (s' n x)
hf' : ∀ (n : ℕ), StronglyMeasurable (f' n)
x : β
hfx : Integrable (f x) μ
this : ∀ (n : ℕ), Integrable (⇑(s' n x)) μ
y : α
⊢ 0 ∈ range (Function.uncurry f) ∪ {0}

Tactic:
simp

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

Goal state before:
pp: G : Type w
inst✝ : TopologicalSpace G
⊢ ∀ (K₁ K₂ : Compacts G), ↑K₁ ⊆ ↑K₂ → 0 ≤ 0

Tactic:
simp

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

Goal state before:
pp: G : Type w
inst✝ : TopologicalSpace G
⊢ ∀ (K₁ K₂ : Compacts G), Disjoint ↑K₁ ↑K₂ → IsClosed ↑K₁ → IsClosed ↑K₂ → 0 = 0 + 0

Tactic:
simp

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

Goal state before:
pp: G : Type w
inst✝ : TopologicalSpace G
⊢ ∀ (K₁ K₂ : Compacts G), 0 ≤ 0 + 0

Tactic:
simp

Goal state after:
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
i : ℕ
hR : IsEmpty R
⊢ t i ⊆ ∅

Tactic:
simp

Goal state after:
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
i : ℕ
hR : IsEmpty R
⊢ f.outer ∅ ≤ f.length (t i) + ofReal ↑(ε' i)

Tactic:
simp

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

Goal state before:
pp: G : Type w
inst✝³ : TopologicalSpace G
μ : Content G
inst✝² : R1Space G
S : MeasurableSpace G
inst✝¹ : BorelSpace G
inst✝ : WeaklyLocallyCompactSpace G
K : Set G
hK : IsCompact K
⊢ μ.measure K < ∞

Tactic:
apply (measure_mono subset_closure).trans_lt _

Goal state after:
G : Type w
inst✝³ : TopologicalSpace G
μ : Content G
inst✝² : R1Space G
S : MeasurableSpace G
inst✝¹ : BorelSpace G
inst✝ : WeaklyLocallyCompactSpace G
K : Set G
hK : IsCompact K
⊢ μ.measure (closure K) < ∞
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: case hf
G : Type u_1
α : Type u_3
E : Type u_5
inst✝⁶ : Group G
inst✝⁵ : MulAction G α
inst✝⁴ : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
s t : Set α
μ : Measure α
inst✝² : MeasurableConstSMul G α
inst✝¹ : SMulInvariantMeasure G α μ
inst✝ : Countable G
hs : IsFundamentalDomain G s μ
ht : IsFundamentalDomain G t μ
f : α → E
hf : ∀ (g : G) (x : α), f (g • x) = f x
g : G
x : α
⊢ ‖f (g • x)‖ₑ = ‖f x‖ₑ

Tactic:
hf

Goal state after:
case hf
G : Type u_1
α : Type u_3
E : Type u_5
inst✝⁶ : Group G
inst✝⁵ : MulAction G α
inst✝⁴ : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
s t : Set α
μ : Measure α
inst✝² : MeasurableConstSMul G α
inst✝¹ : SMulInvariantMeasure G α μ
inst✝ : Countable G
hs : IsFundamentalDomain G s μ
ht : IsFundamentalDomain G t μ
f : α → E
hf : ∀ (g : G) (x : α), f (g • x) = f x
g : G
x : α
⊢ ‖f x‖ₑ = ‖f x‖ₑ
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
i : ℕ
hR : Nonempty R
a b : R
h₁ : t i ⊆ botSet ∪ Ioc a b
h₂ : f.outer (Ioc a b) < f.length (t i) + ↑(ε' i)
⊢ 0 + f.outer (Ioc a b) = f.outer (Ioc a b)

Tactic:
simp

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

Goal state before:
pp: case neg
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
C : ℝ
hs : μ s < ∞
hC : ∀ᵐ (x : X) ∂μ, x ∈ s → ‖f x‖ ≤ C
hfm : ¬AEStronglyMeasurable f (μ.restrict s)
⊢ ‖∫ (x : X) in s, f x ∂μ‖ ≤ C * μ.real s

Tactic:
integral_non_aestronglyMeasurable hfm

Goal state after:
case neg
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
C : ℝ
hs : μ s < ∞
hC : ∀ᵐ (x : X) ∂μ, x ∈ s → ‖f x‖ ≤ C
hfm : ¬AEStronglyMeasurable f (μ.restrict s)
⊢ ‖0‖ ≤ C * μ.real s
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
hab : a ≤ b
⊢ Disjoint {a} (Ioc a b)

Tactic:
simp

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
a b : R
hab : a < b
⊢ Disjoint (Ioo a b) {b}

Tactic:
simp

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

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
R : ℝ
f : X → ℝ
hR : 0 ≤ R
hfint : IntegrableOn f {x | R < f x} μ
hμ : μ {x | R < f x} ≠ 0
this : IntegrableOn (fun x ↦ R) {x | R < f x} μ
⊢ μ.real {x | R < f x} * R < ∫ (x : X) in {x | R < f x}, f x ∂μ

Tactic:
rw [← sub_pos, ← smul_eq_mul, ← setIntegral_const, ← integral_sub hfint this,
    setIntegral_pos_iff_support_of_nonneg_ae]

Goal state after:
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
R : ℝ
f : X → ℝ
hR : 0 ≤ R
hfint : IntegrableOn f {x | R < f x} μ
hμ : μ {x | R < f x} ≠ 0
this : IntegrableOn (fun x ↦ R) {x | R < f x} μ
⊢ 0 < μ ((support fun a ↦ f a - R) ∩ {x | R < f x})

case hf
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
R : ℝ
f : X → ℝ
hR : 0 ≤ R
hfint : IntegrableOn f {x | R < f x} μ
hμ : μ {x | R < f x} ≠ 0
this : IntegrableOn (fun x ↦ R) {x | R < f x} μ
⊢ 0 ≤ᵐ[μ.restrict {x | R < f x}] fun a ↦ f a - R

case hfi
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
R : ℝ
f : X → ℝ
hR : 0 ≤ R
hfint : IntegrableOn f {x | R < f x} μ
hμ : μ {x | R < f x} ≠ 0
this : IntegrableOn (fun x ↦ R) {x | R < f x} μ
⊢ IntegrableOn (fun a ↦ f a - R) {x | R < 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
a b : R
hab : a < b
⊢ Disjoint {a} (Ioo a b)

Tactic:
simp

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
A : ℕ → Set α
hA_meas : ∀ (i : ℕ), MeasurableSet (A i)
hA : ∀ (i : ℕ), ν (A i) < ∞
hA' : ⋃ i, A i = univ
⊢ ∀ (i : ℕ), Quotient.mk α_mod_G '' A i ∈ univ

Tactic:
simp

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
l : ℝ
hf : Tendsto (↑f) atBot (𝓝 l)
x : R
this : Nonempty R
val✝ : OrderBot R
⊢ Iic x = Icc ⊥ x

Tactic:
simp

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
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)

Tactic:
simp

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
l : ℝ
hf : Tendsto (↑f) atBot (𝓝 l)
x : R
this : Nonempty R
⊢ f.measure {x} ≠ ∞

Tactic:
simp

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
l : ℝ
hf : Tendsto (↑f) atBot (𝓝 l)
x : R
this : Nonempty R
⊢ {x} ⊆ Iic x

Tactic:
simp

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
l : ℝ
hf : Tendsto (↑f) atTop (𝓝 l)
x : R
this : Nonempty R
val✝ : OrderTop R
⊢ Ici x = Icc x ⊤

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.Basic

Goal state before:
pp: case e'_3
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K : Set G
hK : IsCompact K
g : G
V : Set G
hV : (interior V).Nonempty
⊢ K = (fun x ↦ g⁻¹ * (g * x)) '' K

Tactic:
simp

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
l : ℝ
hf : Tendsto (↑f) atTop (𝓝 l)
x : R
⊢ ofReal (l - leftLim (↑f) x - (↑f x - leftLim (↑f) x)) = ofReal (l - ↑f x)

Tactic:
simp

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
l : ℝ
hf : Tendsto (↑f) atTop (𝓝 l)
x : R
⊢ f.measure {x} ≠ ∞

Tactic:
simp

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
l : ℝ
hf : Tendsto (↑f) atTop (𝓝 l)
x : R
⊢ {x} ⊆ Ici x

Tactic:
simp

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
hU : (interior U).Nonempty
K : Set G
hK : IsCompact K
a✝ : { carrier := K, isCompact' := hK } ∈ univ
⊢ prehaar (↑K₀) U { carrier := K, isCompact' := hK } ∈
    (fun K ↦ Icc 0 ↑(index ↑K ↑K₀)) { carrier := K, isCompact' := hK }

Tactic:
mem_Icc

Goal state after:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
hU : (interior U).Nonempty
K : Set G
hK : IsCompact K
a✝ : { carrier := K, isCompact' := hK } ∈ univ
⊢ 0 ≤ prehaar (↑K₀) U { carrier := K, isCompact' := hK } ∧
    prehaar (↑K₀) U { carrier := K, isCompact' := hK } ≤ ↑(index ↑{ carrier := K, isCompact' := hK } ↑K₀)
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
K : Compacts G
⊢ IsCompact (Icc 0 ↑(index ↑K ↑K₀))

Tactic:
apply isCompact_Icc

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
x : R
⊢ univ ∈ 𝓝 x

Tactic:
simp

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
x b c : R
h : Icc b c ∈ 𝓝 x
⊢ f.measure (Icc b c) < ∞

Tactic:
simp

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
⊢ (starRingEnd 𝕜) (-1) * ⟪x, y⟫ = -⟪x, y⟫

Tactic:
simp

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
K : Compacts G
this : chaar K₀ K ∈ (fun K ↦ Icc 0 ↑(index ↑K ↑K₀)) K
⊢ 0 ≤ chaar K₀ K

Tactic:
mem_Icc

Goal state after:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
K : Compacts G
this : 0 ≤ chaar K₀ K ∧ chaar K₀ K ≤ ↑(index ↑K ↑K₀)
⊢ 0 ≤ chaar K₀ K
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
c : ℝ
a b : R
hab : a ≤ b
⊢ c ≤ ↑(StieltjesFunction.const R c) a

Tactic:
simp

Goal state after:
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
⊢ I * ((starRingEnd 𝕜) ⟪x, x⟫ - ⟪x, x⟫) / 2 = ↑0

Tactic:
simp

Goal state after:
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
⊢ (starRingEnd 𝕜) (-1) * ⟪x, y⟫ = -⟪x, y⟫

Tactic:
simp

Goal state after:
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⟫

Tactic:
simp

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 : WithSeminorms fun i ↦ p
⊢ ⨅ i, PseudoMetricSpace.toUniformSpace.toTopologicalSpace = PseudoMetricSpace.toUniformSpace.toTopologicalSpace

Tactic:
simp

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
v : F
⊢ 1 = √1

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x : E
⊢ re ⟪x, x⟫ ≤ 0 ↔ x = 0

Tactic:
simp

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✝ : RCLike 𝕜
⊢ ∀ (x : PUnit.{?u.7 + 1}), ‖x‖ ^ 2 = re 0

Tactic:
simp

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✝ : RCLike 𝕜
⊢ ∀ (x y : PUnit.{?u.7 + 1}), (starRingEnd 𝕜) 0 = 0

Tactic:
simp

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✝ : RCLike 𝕜
⊢ ∀ (x y z : PUnit.{?u.7 + 1}), 0 = 0 + 0

Tactic:
simp

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✝ : RCLike 𝕜
⊢ ∀ (x y : PUnit.{?u.7 + 1}) (r : 𝕜), 0 = (starRingEnd 𝕜) r * 0

Tactic:
simp

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

Goal state before:
pp: case convert_3
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
v : ι → E
hz : ∀ (i : ι), v i ≠ 0
ho : Pairwise fun i j ↦ ⟪v i, v j⟫ = 0
s : Finset ι
g : ι → 𝕜
hg : ∑ i ∈ s, g i • v i = 0
i : ι
hi : i ∈ s
⊢ ∀ b ∈ s, b ≠ i → ⟪v i, g b • v b⟫ = 0

Tactic:
intro j

Goal state after:
case convert_3
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
v : ι → E
hz : ∀ (i : ι), v i ≠ 0
ho : Pairwise fun i j ↦ ⟪v i, v j⟫ = 0
s : Finset ι
g : ι → 𝕜
hg : ∑ i ∈ s, g i • v i = 0
i : ι
hi : i ∈ s
j : ι
⊢ j ∈ s → j ≠ i → ⟪v i, g j • v j⟫ = 0