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

Goal state before:
pp: case neg
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
hμ : ¬IsFiniteMeasure μ
this : μ univ = ∞
⊢ eLpNorm f 1 μ ≤ (if 2 = 0 then 0 else ∞) * ↑r

Tactic:
if_neg,

Goal state after:
case neg
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
hμ : ¬IsFiniteMeasure μ
this : μ univ = ∞
⊢ eLpNorm f 1 μ ≤ ∞ * ↑r

case neg.hnc
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
hμ : ¬IsFiniteMeasure μ
this : μ univ = ∞
⊢ ¬2 = 0
Module: Mathlib.Topology.Order.LeftRightLim

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
⊢ Monotone (leftLim f)

Tactic:
intro x y h

Goal state after:
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x y : α
h : x ≤ y
⊢ leftLim f x ≤ leftLim f y
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 : NullMeasurableSet s μ
hfi : Integrable f μ
this : ∫ (x : X) in s ∪ sᶜ, f x ∂μ = ∫ (x : X) in s, f x ∂μ + ∫ (x : X) in sᶜ, f x ∂μ
⊢ ∫ (x : X) in univ, f x ∂μ = ∫ (x : X), f x ∂μ

Tactic:
setIntegral_univ

Goal state after:
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
s : Set X
μ : Measure X
hs : NullMeasurableSet s μ
hfi : Integrable f μ
this : ∫ (x : X) in s ∪ sᶜ, f x ∂μ = ∫ (x : X) in s, f x ∂μ + ∫ (x : X) in sᶜ, f x ∂μ
⊢ ∫ (x : X), f x ∂μ = ∫ (x : X), f x ∂μ
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' : ∑' (i : ι), ∫⁻ (a : α), ‖f i a‖ₑ ∂μ ≠ ∞
hF : CompleteSpace F
hf'' : ∀ (i : ι), AEMeasurable (fun x ↦ ‖f i x‖ₑ) μ
hhh : ∀ᵐ (a : α) ∂μ, Summable fun n ↦ ↑‖f n a‖₊
this : ∫⁻ (a : α), ∑' (n : ι), ‖f n a‖ₑ ∂μ < ∞
⊢ (fun a ↦ ↑‖↑(∑' (a_1 : ι), ‖f a_1 a‖₊)‖₊) =ᵐ[μ] fun a ↦ ∑' (n : ι), ↑‖f n a‖₊

Tactic:
NNReal.nnnorm_eq

Goal state after:
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' : ∑' (i : ι), ∫⁻ (a : α), ‖f i a‖ₑ ∂μ ≠ ∞
hF : CompleteSpace F
hf'' : ∀ (i : ι), AEMeasurable (fun x ↦ ‖f i x‖ₑ) μ
hhh : ∀ᵐ (a : α) ∂μ, Summable fun n ↦ ↑‖f n a‖₊
this : ∫⁻ (a : α), ∑' (n : ι), ‖f n a‖ₑ ∂μ < ∞
⊢ (fun a ↦ ↑(∑' (a_1 : ι), ‖f a_1 a‖₊)) =ᵐ[μ] fun a ↦ ∑' (n : ι), ↑‖f n a‖₊
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})
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)
n : ℕ
x : β
y : E
⊢ y ∈ (s' n x).range → y ∈ (s n).range

Tactic:
simp_rw [SimpleFunc.mem_range]

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
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})
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)
n : ℕ
x : β
y : E
⊢ y ∈ range ⇑(s' n x) → y ∈ range ⇑(s n)
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'
⊢ ∃ p, Icc a' b ∩ s i ⊆ Iotop p.1 p.2 ∧ ofReal (↑f p.2 - ↑f p.1) < f.length (s i) + ↑(ε' i)

Tactic:
refine ⟨(p, q'), ?_, hq'⟩

Goal state after:
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'
⊢ Icc a' b ∩ s i ⊆ Iotop (p, q').1 (p, q').2
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : Group G
inst✝ : SeparatelyContinuousMul G
h : ∀ (g : G) {K : Compacts G}, μ (Compacts.map (fun x ↦ g * x) ⋯ K) = μ K
g : G
U : Opens G
⊢ μ.innerContent ((Opens.comap ↑(Homeomorph.mulLeft g)) U) = μ.innerContent U

Tactic:
convert! μ.innerContent_comap (Homeomorph.mulLeft g) (fun K => h g) U

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
μ : Measure X
inst✝ : PartialOrder E
f : X → E
hf : AEStronglyMeasurable f μ
⊢ {x | f x ≤ 0} = {x | f x < 0} ∪ {x | f x = 0}

Tactic:
simp_rw [le_iff_lt_or_eq, ofPred_or]

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
K : PositiveCompacts G
V : Set G
hV : (interior V).Nonempty
⊢ ¬∃ x ∈ {t | ↑K ⊆ ⋃ g ∈ t, (fun h ↦ g * h) ⁻¹' V}, x.card = 0

Tactic:
rintro ⟨t, h1t, h2t⟩; rw [Finset.card_eq_zero] at h2t; subst h2t
    obtain ⟨g, hg⟩ := K.interior_nonempty
    change g ∈ (∅ : Set G)
    convert! h1t (interior_subset hg); symm
    simp only [Finset.notMem_empty, iUnion_of_empty, iUnion_empty]

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
t : Set X
μ : Measure X
inst✝ : CompleteSpace E
p : ℝ≥0∞
ht : MeasurableSet t
hμt : μ t ≠ ∞
e : E
⊢ ∫ (x : X), ↑↑(indicatorConstLp p ht hμt e) x ∂μ = ∫ (x : X) in univ, ↑↑(indicatorConstLp p ht hμt e) x ∂μ

Tactic:
setIntegral_univ

Goal state after:
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
t : Set X
μ : Measure X
inst✝ : CompleteSpace E
p : ℝ≥0∞
ht : MeasurableSet t
hμt : μ t ≠ ∞
e : E
⊢ ∫ (x : X), ↑↑(indicatorConstLp p ht hμt e) x ∂μ = ∫ (x : X), ↑↑(indicatorConstLp p ht hμt e) x ∂μ
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
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ K₂.carrier
⊢ prehaar (↑K₀) U K₁ ≤ prehaar (↑K₀) U K₂

Tactic:
simp only [prehaar]

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
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ K₂.carrier
⊢ ↑(index (↑K₁) U) / ↑(index (↑K₀) U) ≤ ↑(index (↑K₂) U) / ↑(index (↑K₀) U)
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
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ K₂.carrier
⊢ 0 < ↑(index (↑K₀) U)

Tactic:
· exact mod_cast index_pos K₀ hU

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
K₁ K₂ : Compacts G
hU : (interior U).Nonempty
⊢ prehaar (↑K₀) U (K₁ ⊔ K₂) ≤ prehaar (↑K₀) U K₁ + prehaar (↑K₀) U K₂

Tactic:
simp only [prehaar]

Goal state after:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
K₁ K₂ : Compacts G
hU : (interior U).Nonempty
⊢ ↑(index (↑(K₁ ⊔ K₂)) U) / ↑(index (↑K₀) U) ≤ ↑(index (↑K₁) U) / ↑(index (↑K₀) U) + ↑(index (↑K₂) U) / ↑(index (↑K₀) U)
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
K₁ K₂ : Compacts G
hU : (interior U).Nonempty
⊢ 0 < ↑(index (↑K₀) U)

Tactic:
· exact mod_cast index_pos K₀ hU

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
K₁ K₂ : Compacts G
hU : (interior U).Nonempty
h : Disjoint (K₁.carrier * U⁻¹) (K₂.carrier * U⁻¹)
⊢ prehaar (↑K₀) U (K₁ ⊔ K₂) = prehaar (↑K₀) U K₁ + prehaar (↑K₀) U K₂

Tactic:
simp only [prehaar]

Goal state after:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
K₁ K₂ : Compacts G
hU : (interior U).Nonempty
h : Disjoint (K₁.carrier * U⁻¹) (K₂.carrier * U⁻¹)
⊢ ↑(index (↑(K₁ ⊔ K₂)) U) / ↑(index (↑K₀) U) = ↑(index (↑K₁) U) / ↑(index (↑K₀) U) + ↑(index (↑K₂) U) / ↑(index (↑K₀) U)
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
ι : Type u_5
inst✝⁵ : Countable ι
μ : Measure X
inst✝⁴ : NormedAddCommGroup E
inst✝³ : TopologicalSpace X
inst✝² : BorelSpace X
inst✝¹ : T2Space X
inst✝ : IsLocallyFiniteMeasure μ
f : C(X, E)
s : ι → Compacts X
hf : Summable fun i ↦ ‖ContinuousMap.restrict (↑(s i)) f‖ * μ.real ↑(s i)
i : ι
⊢ ‖∫ (x : X) in ↑(s i), ‖f x‖ ∂μ‖ ≤ ‖ContinuousMap.restrict (↑(s i)) f‖ * μ.real ↑(s i)

Tactic:
exact
    norm_setIntegral_le_of_norm_le_const (s i).isCompact.measure_lt_top
      fun x hx => (norm_norm (f x)).symm ▸ (f.restrict (s i : Set X)).norm_coe_le_norm ⟨x, hx⟩

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✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure X
f g : ↥(Lp E p μ)
s : Set X
x : X
hx1 : ↑↑(MemLp.toLp ↑↑(f + g) ⋯) x = ↑↑(f + g) x
hx2 : ↑↑(MemLp.toLp ↑↑g ⋯) x = ↑↑g x
hx3 : ↑↑(MemLp.toLp ↑↑f ⋯) x = ↑↑f x
hx4 : ↑↑(MemLp.toLp ↑↑f ⋯ + MemLp.toLp ↑↑g ⋯) x = (↑↑(MemLp.toLp ↑↑f ⋯) + ↑↑(MemLp.toLp ↑↑g ⋯)) x
hx5 : ↑↑(f + g) x = (↑↑f + ↑↑g) x
⊢ (↑↑f + ↑↑g) x = ↑↑f x + ↑↑g x

Tactic:
Pi.add_apply

Goal state after:
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure X
f g : ↥(Lp E p μ)
s : Set X
x : X
hx1 : ↑↑(MemLp.toLp ↑↑(f + g) ⋯) x = ↑↑(f + g) x
hx2 : ↑↑(MemLp.toLp ↑↑g ⋯) x = ↑↑g x
hx3 : ↑↑(MemLp.toLp ↑↑f ⋯) x = ↑↑f x
hx4 : ↑↑(MemLp.toLp ↑↑f ⋯ + MemLp.toLp ↑↑g ⋯) x = (↑↑(MemLp.toLp ↑↑f ⋯) + ↑↑(MemLp.toLp ↑↑g ⋯)) x
hx5 : ↑↑(f + g) x = (↑↑f + ↑↑g) x
⊢ ↑↑f x + ↑↑g x = ↑↑f x + ↑↑g x
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 y : E
hx₀ : ¬x = 0
this : ‖x‖ ^ 2 ≠ 0
h : re ⟪x, x⟫ * (re ⟪x, x⟫ * re ⟪y, y⟫ - ‖⟪x, y⟫‖ ^ 2) = 0
⊢ y = (⟪x, y⟫ / ⟪x, x⟫) • x

Tactic:
let : InnerProductSpace.Core 𝕜 E := InnerProductSpace.toCore

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
hx₀ : ¬x = 0
this✝ : ‖x‖ ^ 2 ≠ 0
h : re ⟪x, x⟫ * (re ⟪x, x⟫ * re ⟪y, y⟫ - ‖⟪x, y⟫‖ ^ 2) = 0
this : InnerProductSpace.Core 𝕜 E := InnerProductSpace.toCore
⊢ y = (⟪x, y⟫ / ⟪x, x⟫) • x
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 y : E
tfae_1_to_2 : ‖⟪x, y⟫‖ = ‖x‖ * ‖y‖ → x = 0 ∨ y = (⟪x, y⟫ / ⟪x, x⟫) • x
tfae_2_to_3 : x = 0 ∨ y = (⟪x, y⟫ / ⟪x, x⟫) • x → x = 0 ∨ ∃ r, y = r • x
⊢ (x = 0 ∨ ∃ r, y = r • x) → ‖⟪x, y⟫‖ = ‖x‖ * ‖y‖

Tactic:
rintro (rfl | ⟨r, rfl⟩) <;>
    simp [inner_smul_right, norm_smul, inner_self_eq_norm_sq_to_K,
      sq, mul_left_comm]

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

Goal state before:
pp: case neg
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✝ : DecidableEq ι
i j : ι
v : G i
w : G j
h : ¬i = j
⊢ ⟪(V i) v, (V j) w⟫ = 0

Tactic:
· exact hV h v w

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✝ : Fintype ι
l : (i : ι) → G i
i : ι
v : G i
⊢ (∑ j, if i = j then ⟪(V i) v, (V j) (l j)⟫ else 0) = ⟪v, l i⟫

Tactic:
by
      simp only [Finset.sum_ite_eq, Finset.mem_univ, (V i).inner_map_map, if_true]

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

Goal state before:
pp: case mp
V : Type u_4
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℂ V
T : V →ₗ[ℂ] V
hT : ∀ (x : V), ⟪T x, x⟫_ℂ = 0
x : V
⊢ T x = 0

Tactic:
← @inner_self_eq_zero ℂ V,

Goal state after:
case mp
V : Type u_4
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℂ V
T : V →ₗ[ℂ] V
hT : ∀ (x : V), ⟪T x, x⟫_ℂ = 0
x : V
⊢ ⟪T x, T x⟫_ℂ = 0
Module: Mathlib.Analysis.InnerProductSpace.LinearMap

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T : E →L[𝕜] E
x : E
c : 𝕜
⊢ T.reApplyInnerSelf (c • x) = ‖c‖ ^ 2 * T.reApplyInnerSelf x

Tactic:
simp only [map_smul, ContinuousLinearMap.reApplyInnerSelf_apply, inner_smul_left,
    inner_smul_right, ← mul_assoc, mul_conj, ← ofReal_pow, ← smul_re,
    Algebra.smul_def (‖c‖ ^ 2) ⟪T x, x⟫, algebraMap_eq_ofReal]

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

Goal state before:
pp: 𝕜 : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
inst✝⁶ : RCLike 𝕜
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : SeminormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : SeminormedAddCommGroup G
inst✝ : InnerProductSpace 𝕜 G
x : E
y z : F
w : G
⊢ ((rankOne 𝕜) x) y ∘SL ((rankOne 𝕜) z) w = ⟪y, z⟫_𝕜 • ((rankOne 𝕜) x) w

Tactic:
simp [comp_rankOne]

Goal state after:
Module: Mathlib.Topology.Baire.CompleteMetrizable

Goal state before:
pp: case refine_1
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : IsCompletelyPseudoMetrizableSpace X
x✝ : UpgradedIsCompletelyPseudoMetrizableSpace X := upgradeIsCompletelyPseudoMetrizable X
f : ℕ → Set X
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n ↦ 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
n : ℕ
x : X
δ : ℝ≥0∞
δpos : δ ≠ 0
this : x ∈ closure (f n)
y : X
ys : y ∈ f n
xy : edist y x < δ / 2
r : ℝ≥0∞
rpos : r > 0
hr : closedEBall y r ⊆ f n
⊢ 0 < min (min (δ / 2) r) (B (n + 1))

Tactic:
show 0 < min (min (δ / 2) r) (B (n + 1))

Goal state after:
case refine_1
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : IsCompletelyPseudoMetrizableSpace X
x✝ : UpgradedIsCompletelyPseudoMetrizableSpace X := upgradeIsCompletelyPseudoMetrizable X
f : ℕ → Set X
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n ↦ 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
n : ℕ
x : X
δ : ℝ≥0∞
δpos : δ ≠ 0
this : x ∈ closure (f n)
y : X
ys : y ∈ f n
xy : edist y x < δ / 2
r : ℝ≥0∞
rpos : r > 0
hr : closedEBall y r ⊆ f n
⊢ 0 < min (min (δ / 2) r) (B (n + 1))
Module: Mathlib.Analysis.Normed.Operator.Banach

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NontriviallyNormedField 𝕜'
σ : 𝕜 →+* 𝕜'
E : Type u_3
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜' F
f : E →SL[σ] F
σ' : 𝕜' →+* 𝕜
inst✝³ : RingHomInvPair σ σ'
inst✝² : RingHomIsometric σ
inst✝¹ : RingHomIsometric σ'
inst✝ : CompleteSpace F
surj : Surjective ⇑f
⊢ ∃ C ≥ 0, ∀ (y : F), ∃ x, dist (f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ C * ‖y‖

Tactic:
have A : ⋃ n : ℕ, closure (f '' ball 0 n) = Set.univ := by
    refine Subset.antisymm (subset_univ _) fun y _ => ?_
    rcases surj y with ⟨x, hx⟩
    rcases exists_nat_gt ‖x‖ with ⟨n, hn⟩
    refine mem_iUnion.2 ⟨n, subset_closure ?_⟩
    refine (mem_image _ _ _).2 ⟨x, ⟨?_, hx⟩⟩
    rwa [mem_ball, dist_eq_norm, sub_zero]

Goal state after:
𝕜 : Type u_1
𝕜' : Type u_2
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NontriviallyNormedField 𝕜'
σ : 𝕜 →+* 𝕜'
E : Type u_3
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜' F
f : E →SL[σ] F
σ' : 𝕜' →+* 𝕜
inst✝³ : RingHomInvPair σ σ'
inst✝² : RingHomIsometric σ
inst✝¹ : RingHomIsometric σ'
inst✝ : CompleteSpace F
surj : Surjective ⇑f
A : ⋃ n, closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (⇑f '' ball 0 ↑n) = Set.univ
⊢ ∃ C ≥ 0, ∀ (y : F), ∃ x, dist (f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ C * ‖y‖
Module: Mathlib.Analysis.Normed.Operator.Banach

Goal state before:
pp: case inr
𝕜 : Type u_1
𝕜' : Type u_2
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NontriviallyNormedField 𝕜'
σ : 𝕜 →+* 𝕜'
E : Type u_3
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜' F
f : E →SL[σ] F
σ' : 𝕜' →+* 𝕜
inst✝³ : RingHomInvPair σ σ'
inst✝² : RingHomIsometric σ
inst✝¹ : RingHomIsometric σ'
inst✝ : CompleteSpace F
surj : Surjective ⇑f
A : ⋃ n, closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (⇑f '' ball 0 ↑n) = Set.univ
n : ℕ
a : F
ε : ℝ
εpos : ε > 0
H : ball a ε ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (⇑f '' ball 0 ↑n)
c : 𝕜
hc : 1 < ‖c‖
y : F
hy : y ≠ 0
hc' : 1 < ‖σ c‖
d : 𝕜'
hd : d ≠ 0
ydlt : ‖d • y‖ < ε / 2
dinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖σ c‖ * ‖y‖
δ : ℝ := ‖d‖ * ‖y‖ / 4
⊢ ∃ x, dist (f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖

Tactic:
have δpos : 0 < δ := by positivity

Goal state after:
case inr
𝕜 : Type u_1
𝕜' : Type u_2
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NontriviallyNormedField 𝕜'
σ : 𝕜 →+* 𝕜'
E : Type u_3
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜' F
f : E →SL[σ] F
σ' : 𝕜' →+* 𝕜
inst✝³ : RingHomInvPair σ σ'
inst✝² : RingHomIsometric σ
inst✝¹ : RingHomIsometric σ'
inst✝ : CompleteSpace F
surj : Surjective ⇑f
A : ⋃ n, closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (⇑f '' ball 0 ↑n) = Set.univ
n : ℕ
a : F
ε : ℝ
εpos : ε > 0
H : ball a ε ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (⇑f '' ball 0 ↑n)
c : 𝕜
hc : 1 < ‖c‖
y : F
hy : y ≠ 0
hc' : 1 < ‖σ c‖
d : 𝕜'
hd : d ≠ 0
ydlt : ‖d • y‖ < ε / 2
dinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖σ c‖ * ‖y‖
δ : ℝ := ‖d‖ * ‖y‖ / 4
δpos : 0 < δ
⊢ ∃ x, dist (f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖
Module: Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional

Goal state before:
pp: case e'_2
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
K : Submodule 𝕜 E
⊢ finrank 𝕜 ↥K + finrank 𝕜 ↥Kᗮ = finrank 𝕜 ↥K + finrank 𝕜 ↥(Kᗮ ⊓ ⊤)

Tactic:
rw [inf_top_eq]

Goal state after:
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
K : Submodule 𝕜 E
inst✝ : K.HasOrthogonalProjection
v : ↥K
⊢ K.starProjection ↑v = ↑v

Tactic:
by simp [starProjection_apply]

Goal state after:
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
K : Submodule 𝕜 E
inst✝ : K.HasOrthogonalProjection
u : ↥K
v : E
⊢ ⟪K.starProjection v, ↑u⟫ = ⟪K.starProjection v, ↑u⟫ + ⟪v - K.starProjection v, ↑u⟫

Tactic:
starProjection_inner_eq_zero _ _ (Submodule.coe_mem _),

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
inst✝ : K.HasOrthogonalProjection
u : ↥K
v : E
⊢ ⟪K.starProjection v, ↑u⟫ = ⟪K.starProjection v, ↑u⟫ + 0
Module: Mathlib.Analysis.Normed.Lp.ProdLp

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
x y : WithLp ∞ (α × β)
⊢ ↑(nndist x y) = ↑(max (nndist x.fst y.fst) (nndist x.snd y.snd))

Tactic:
by
    push_cast
    exact prod_dist_eq_sup _ _

Goal state after:
Module: Mathlib.LinearAlgebra.UnitaryGroup

Goal state before:
pp: case left
n : Type u
inst✝⁵ : DecidableEq n
inst✝⁴ : Fintype n
R : Type u_1
m : Type u_2
inst✝³ : Semiring R
inst✝² : StarRing R
inst✝¹ : Fintype m
inst✝ : DecidableEq m
U₁ : Matrix n n R
U₂ : Matrix m m R
hU₁ : U₁ ∈ unitary (Matrix n n R)
hU₂ : U₂ ∈ unitary (Matrix m m R)
i✝ j✝ : n × m
⊢ ∑ x, ∑ x_1, star U₂ i✝.2 x_1 * (star U₁ i✝.1 x * U₁ x j✝.1) * U₂ x_1 j✝.2 = 1 i✝ j✝

Tactic:
rw [Finset.sum_comm]

Goal state after:
case left
n : Type u
inst✝⁵ : DecidableEq n
inst✝⁴ : Fintype n
R : Type u_1
m : Type u_2
inst✝³ : Semiring R
inst✝² : StarRing R
inst✝¹ : Fintype m
inst✝ : DecidableEq m
U₁ : Matrix n n R
U₂ : Matrix m m R
hU₁ : U₁ ∈ unitary (Matrix n n R)
hU₂ : U₂ ∈ unitary (Matrix m m R)
i✝ j✝ : n × m
⊢ ∑ y, ∑ x, star U₂ i✝.2 y * (star U₁ i✝.1 x * U₁ x j✝.1) * U₂ y j✝.2 = 1 i✝ j✝