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

Goal state before:
pp: 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
hi : i ∉ s
x : (i : δ) → X i
y : X i
⊢ (∫⋯∫⁻_s, f ∂μ) (update x i y) = (∫⋯∫⁻_s, f ∘ fun x ↦ update x i y ∂μ) x

Tactic:
by
  induction s using Finset.induction generalizing x with
  | empty => simp
  | insert i' s hi' ih =>
    rw [lmarginal_insert _ hf hi', lmarginal_insert _ (hf.comp measurable_update_left) hi']
    have hii' : i ≠ i' := mt (by rintro rfl; exact mem_insert_self i s) hi
    simp_rw [update_comm hii', ih (mt Finset.mem_insert_of_mem hi)]

Goal state after:
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))
this : 𝓝[≤] a = 𝓝[<] a ⊔ pure a
⊢ ContinuousWithinAt (leftLim f) (Iic a) a

Tactic:
ContinuousWithinAt,

Goal state after:
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))
this : 𝓝[≤] a = 𝓝[<] a ⊔ pure a
⊢ Tendsto (leftLim f) (𝓝[≤] a) (𝓝 (leftLim f a))
Module: Mathlib.Topology.Order.LeftRightLim

Goal state before:
pp: case neg
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))
h' : (𝓝[<] a).NeBot
b : α
hb : b ∈ Iio a
s : Set β
s_mem : s ∈ 𝓝 (leftLim f a)
s_closed : IsClosed[inst✝³] s
u : α
au : u < a
hu : Ioo u a ⊆ {x | f x ∈ s}
c : α
hc : c ∈ Ioo u a
h'c : (𝓝[>] c).NeBot
h''c : ∃ y, Tendsto f (𝓝[>] c) (𝓝 y)
⊢ rightLim f c ∈ id s

Tactic:
apply s_closed.mem_of_tendsto (tendsto_rightLim_of_tendsto h''c)

Goal state after:
case neg
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))
h' : (𝓝[<] a).NeBot
b : α
hb : b ∈ Iio a
s : Set β
s_mem : s ∈ 𝓝 (leftLim f a)
s_closed : IsClosed[inst✝³] s
u : α
au : u < a
hu : Ioo u a ⊆ {x | f x ∈ s}
c : α
hc : c ∈ Ioo u a
h'c : (𝓝[>] c).NeBot
h''c : ∃ y, Tendsto f (𝓝[>] c) (𝓝 y)
⊢ ∀ᶠ (x : α) in 𝓝[>] c, f x ∈ s
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
⊢ ∫ (x : X) in sᶜ, f x ∂μ = ∫ (x : X), f x ∂μ - ∫ (x : X) in s, f x ∂μ

Tactic:
by
  rw [← integral_add_compl₀ (μ := μ) hs hfi, add_sub_cancel_left]

Goal state after:
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
hx : x ∈ {x | leftLim (↑f) x ≠ ↑f x}
⊢ x ∈ {x | ¬ContinuousAt (↑f) x}

Tactic:
h'x

Goal state after:
R : Type u_1
inst✝² : LinearOrder R
inst✝¹ : TopologicalSpace R
inst✝ : OrderTopology R
f : StieltjesFunction R
x : R
hx : x ∈ {x | leftLim (↑f) x ≠ ↑f x}
h'x : ContinuousAt (↑f) x
⊢ False
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
t : Set X
μ : Measure X
ht_eq : ∀ᵐ (x : X) ∂μ, x ∈ t → f x = 0
hf : AEStronglyMeasurable f (μ.restrict t)
x : X
hx : x ∈ t → f x = AEStronglyMeasurable.mk f hf x
⊢ (x ∈ t → f x = 0) → x ∈ t → AEStronglyMeasurable.mk f hf x = 0 x

Tactic:
h'x

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
t : Set X
μ : Measure X
ht_eq : ∀ᵐ (x : X) ∂μ, x ∈ t → f x = 0
hf : AEStronglyMeasurable f (μ.restrict t)
x : X
hx : x ∈ t → f x = AEStronglyMeasurable.mk f hf x
h'x : x ∈ t → f x = 0
⊢ x ∈ t → AEStronglyMeasurable.mk f hf x = 0 x
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
ht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0
haux : StronglyMeasurable f
H : IntegrableOn f (s ∪ t) μ
k : Set X := f ⁻¹' {0}
hk : MeasurableSet k
h's : IntegrableOn f s μ
A : ∀ (u : Set X), ∫ (x : X) in u ∩ k, f x ∂μ = 0
x : X
hx : x ∈ t → f x = 0
⊢ x ∉ t \ k

Tactic:
h'x

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
ht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0
haux : StronglyMeasurable f
H : IntegrableOn f (s ∪ t) μ
k : Set X := f ⁻¹' {0}
hk : MeasurableSet k
h's : IntegrableOn f s μ
A : ∀ (u : Set X), ∫ (x : X) in u ∩ k, f x ∂μ = 0
x : X
hx : x ∈ t → f x = 0
h'x : x ∈ t \ k
⊢ False
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
ht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0
ht : IntegrableOn f t μ
H : IntegrableOn f (s ∪ t) μ
f' : X → E := AEStronglyMeasurable.mk f ⋯
x : X
hx : f x = 0
⊢ f x = AEStronglyMeasurable.mk f ⋯ x → AEStronglyMeasurable.mk f ⋯ x = 0

Tactic:
h'x

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
ht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0
ht : IntegrableOn f t μ
H : IntegrableOn f (s ∪ t) μ
f' : X → E := AEStronglyMeasurable.mk f ⋯
x : X
hx : f x = 0
h'x : f x = AEStronglyMeasurable.mk f ⋯ x
⊢ AEStronglyMeasurable.mk f ⋯ x = 0
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
ht : NullMeasurableSet t μ
hts : s ⊆ t
h't : ∀ᵐ (x : X) ∂μ, x ∈ t \ s → f x = 0
h : IntegrableOn f t μ
f' : X → E := AEStronglyMeasurable.mk f ⋯
x : X
hx : x ∈ t \ s → f x = 0
⊢ (x ∈ t → f x = AEStronglyMeasurable.mk f ⋯ x) → x ∈ t \ s → AEStronglyMeasurable.mk f ⋯ x = 0

Tactic:
h'x

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
ht : NullMeasurableSet t μ
hts : s ⊆ t
h't : ∀ᵐ (x : X) ∂μ, x ∈ t \ s → f x = 0
h : IntegrableOn f t μ
f' : X → E := AEStronglyMeasurable.mk f ⋯
x : X
hx : x ∈ t \ s → f x = 0
h'x : x ∈ t → f x = AEStronglyMeasurable.mk f ⋯ x
⊢ x ∈ t \ s → AEStronglyMeasurable.mk f ⋯ x = 0
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case inr
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)
⊢ ∃ s, t i ⊆ s ∧ MeasurableSet s ∧ f.outer s ≤ f.length (t i) + ofReal ↑(ε' i)

Tactic:
rw [sdiff_subset_iff] at h₁

Goal state after:
case inr
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)
⊢ ∃ s, t i ⊆ s ∧ MeasurableSet s ∧ f.outer s ≤ f.length (t i) + ofReal ↑(ε' 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 : Set X
μ : Measure X
h : ∀ᵐ (x : X) ∂μ, x ∉ s → f x = 0
x : X
hx : x ∉ s → f x = 0
⊢ x ∈ univ \ s → f x = 0

Tactic:
h'x

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
h : ∀ᵐ (x : X) ∂μ, x ∉ s → f x = 0
x : X
hx : x ∉ s → f x = 0
h'x : x ∈ univ \ s
⊢ 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 : R
ha : ¬IsBot a
b : R
hb : b < a
u : ℕ → R
u_mono : StrictMono u
u_lim : Tendsto u atTop (𝓝 a)
u_lt_a : ∀ (n : ℕ), u n < a
A : {a} = ⋂ n, Ioc (u n) a
L1 : Tendsto (fun n ↦ f.measure (Ioc (u n) a)) atTop (𝓝 (f.measure {a}))
⊢ Tendsto (fun n ↦ ofReal (↑f a - ↑f (u n))) atTop (𝓝 (ofReal (↑f a - leftLim (↑f) a)))

Tactic:
have : Tendsto (fun n => f (u n)) atTop (𝓝 (leftLim f a)) := by
      apply (f.mono.tendsto_leftLim a).comp
      exact
        tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _ u_lim
          (Eventually.of_forall fun n => u_lt_a n)

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 : R
ha : ¬IsBot a
b : R
hb : b < a
u : ℕ → R
u_mono : StrictMono u
u_lim : Tendsto u atTop (𝓝 a)
u_lt_a : ∀ (n : ℕ), u n < a
A : {a} = ⋂ n, Ioc (u n) a
L1 : Tendsto (fun n ↦ f.measure (Ioc (u n) a)) atTop (𝓝 (f.measure {a}))
this : Tendsto (fun n ↦ ↑f (u n)) atTop (𝓝 (leftLim (↑f) a))
⊢ Tendsto (fun n ↦ ofReal (↑f a - ↑f (u n))) atTop (𝓝 (ofReal (↑f a - leftLim (↑f) a)))
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) ≤ ν (A n)

Tactic:
fund_dom_s.measure_eq_tsum (A n)

Goal state after:
case refine_1
G : Type u_1
α : Type u_3
inst✝⁶ : Group G
inst✝⁵ : MulAction G α
inst✝⁴ : MeasurableSpace α
ν : Measure α
inst✝³ : SMulInvariantMeasure G α ν
inst✝² : Countable G
inst✝¹ : MeasurableConstSMul G α
i : SigmaFinite ν
μ : Measure (Quotient α_mod_G)
inst✝ : QuotientMeasureEqMeasurePreimage ν μ
A : ℕ → Set α
hA_meas : ∀ (i : ℕ), MeasurableSet (A i)
hA : ∀ (i : ℕ), ν (A i) < ∞
hA' : ⋃ i, A i = univ
n : ℕ
s : Set α
fund_dom_s : IsFundamentalDomain G s ν
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)
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
⊢ re ((starRingEnd 𝕜) ⟪y, x⟫) = re ⟪y, x⟫

Tactic:
conj_re

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
⊢ re ⟪y, x⟫ = re ⟪y, x⟫
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
C : ℝ
h : ∀ (x : R), |↑f x| ≤ C
h✝ : Nonempty R
H : Tendsto (↑f) atTop atTop
⊢ ∃ u, Tendsto (↑f) atTop (𝓝 u)

Tactic:
obtain ⟨x, hx⟩ : ∃ x, C + 1 ≤ f x := (tendsto_atTop.1 H (C + 1)).exists

Goal state after:
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
C : ℝ
h : ∀ (x : R), |↑f x| ≤ C
h✝ : Nonempty R
H : Tendsto (↑f) atTop atTop
x : R
hx : C + 1 ≤ ↑f x
⊢ ∃ u, Tendsto (↑f) atTop (𝓝 u)
Module: Mathlib.MeasureTheory.Measure.Haar.Basic

Goal state before:
pp: case h
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
this : IsCompact (haarProduct ↑K₀)
t : Finset (OpenNhdsOf 1)
V₀ : Set G := ⋯
h1V₀ : IsOpen[inst✝¹] V₀
h2V₀ : 1 ∈ V₀
⊢ 1 ∈ interior V₀

Tactic:
h1V₀.interior_eq

Goal state after:
case h
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
this : IsCompact (haarProduct ↑K₀)
t : Finset (OpenNhdsOf 1)
V₀ : Set G := ⋂ V ∈ t, V.carrier
h1V₀ : IsOpen[inst✝¹] V₀
h2V₀ : 1 ∈ V₀
⊢ 1 ∈ V₀
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:
by rw [← normSq]

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

Goal state before:
pp: case e'_4.e'_5
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
hzero : ¬⟪x, y⟫ = 0
hzero' : ‖⟪x, y⟫‖ ≠ 0
⊢ normSq x * (t * t) = normSq (⟪x, y⟫ • x) * (t / ‖⟪x, y⟫‖) * (t / ‖⟪x, y⟫‖)

Tactic:
field_simp

Goal state after:
case e'_4.e'_5
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
hzero : ¬⟪x, y⟫ = 0
hzero' : ‖⟪x, y⟫‖ ≠ 0
⊢ normSq x * t ^ 2 * ‖⟪x, y⟫‖ ^ 2 = t ^ 2 * normSq (⟪x, y⟫ • x)
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
⊢ re ((starRingEnd 𝕜) ⟪y, x⟫) = re ⟪y, x⟫

Tactic:
conj_re

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
⊢ re ⟪y, x⟫ = re ⟪y, x⟫
Module: Mathlib.Analysis.InnerProductSpace.Defs

Goal state before:
pp: case e'_4.e'_6
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
hzero : ¬⟪x, y⟫ = 0
hzero' : ‖⟪x, y⟫‖ ≠ 0
⊢ 2 * ‖⟪x, y⟫‖ * t = 2 * re ⟪⟪x, y⟫ • x, y⟫ * (t / ‖⟪x, y⟫‖)

Tactic:
field_simp

Goal state after:
case e'_4.e'_6
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
hzero : ¬⟪x, y⟫ = 0
hzero' : ‖⟪x, y⟫‖ ≠ 0
⊢ ‖⟪x, y⟫‖ ^ 2 * t = t * re ⟪⟪x, y⟫ • x, y⟫
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⟫‖ * ‖⟪x, y⟫‖ = ‖⟪x, y⟫‖ * ‖⟪y, x⟫‖

Tactic:
norm_inner_symm

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
⊢ ‖⟪y, x⟫‖ * ‖⟪y, x⟫‖ = ‖⟪y, x⟫‖ * ‖⟪y, x⟫‖
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
x y : F
h₁ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
h₂ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
h₃ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
conj_re

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
x y : F
h₁ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
h₂ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
h₃ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
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
cd : Core 𝕜 F
x y : F
h₁ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
h₂ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
h₃ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
conj_re

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
cd : Core 𝕜 F
x y : F
h₁ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
h₂ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
h₃ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
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
⊢ re ((starRingEnd 𝕜) ⟪x, y⟫) = re ⟪x, y⟫

Tactic:
conj_re

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
⊢ re ⟪x, y⟫ = re ⟪x, y⟫
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
⊢ 0 < (haarContent K₀).outerMeasure ↑K₀

Tactic:
refine zero_lt_one.trans_le ?_

Goal state after:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
⊢ 1 ≤ (haarContent K₀).outerMeasure ↑K₀
Module: Mathlib.Analysis.InnerProductSpace.Continuous

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedSpace ℝ E
inst✝ : IsScalarTower ℝ 𝕜 E
r : ℝ
x y : E
⊢ ⟪(r • x, y).1, (r • x, y).2⟫ = r • ⟪(x, y).1, (x, y).2⟫

Tactic:
simp only [← algebraMap_smul 𝕜 r x, algebraMap_eq_ofReal, inner_smul_real_left]

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

Goal state before:
pp: case mpr
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x : E
hx : x ≠ 0
r : 𝕜
hr : r ≠ 0
⊢ ‖⟪x, r • x⟫ / (↑‖x‖ * ↑‖r • x‖)‖ = 1

Tactic:
simp only [norm_div, norm_mul, norm_ofReal, abs_norm]

Goal state after:
case mpr
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x : E
hx : x ≠ 0
r : 𝕜
hr : r ≠ 0
⊢ ‖⟪x, r • x⟫‖ / (‖x‖ * ‖r • x‖) = 1
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
hle : ‖x‖ ≤ ‖y‖
h : re ⟪x, y⟫ = ‖y‖ ^ 2
H₁ : ‖x‖ ^ 2 ≤ ‖y‖ ^ 2
⊢ re ((starRingEnd 𝕜) ⟪x, y⟫) = ‖y‖ ^ 2

Tactic:
conj_re

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
hle : ‖x‖ ≤ ‖y‖
h : re ⟪x, y⟫ = ‖y‖ ^ 2
H₁ : ‖x‖ ^ 2 ≤ ‖y‖ ^ 2
⊢ re ⟪x, y⟫ = ‖y‖ ^ 2
Module: Mathlib.Analysis.InnerProductSpace.LinearMap

Goal state before:
pp: V : Type u_4
inst✝¹ : SeminormedAddCommGroup V
inst✝ : InnerProductSpace ℂ V
T : V →ₗ[ℂ] V
x y : V
⊢ ⟪T y, x⟫_ℂ =
    (⟪T (x + y), x + y⟫_ℂ - ⟪T (x - y), x - y⟫_ℂ + Complex.I * ⟪T (x + Complex.I • y), x + Complex.I • y⟫_ℂ -
        Complex.I * ⟪T (x - Complex.I • y), x - Complex.I • y⟫_ℂ) /
      4

Tactic:
simp only [map_add, map_sub, inner_add_left, inner_add_right, map_smul, inner_smul_left,
    inner_smul_right, Complex.conj_I, ← pow_two, Complex.I_sq, inner_sub_left, inner_sub_right,
    mul_add, ← mul_assoc, mul_neg, neg_neg, one_mul, neg_one_mul, mul_sub, sub_sub]
  ring

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

Goal state before:
pp: V : Type u_4
inst✝¹ : SeminormedAddCommGroup V
inst✝ : InnerProductSpace ℂ V
T : V →ₗ[ℂ] V
x y : V
⊢ ⟪T x, y⟫_ℂ =
    (⟪T (x + y), x + y⟫_ℂ - ⟪T (x - y), x - y⟫_ℂ - Complex.I * ⟪T (x + Complex.I • y), x + Complex.I • y⟫_ℂ +
        Complex.I * ⟪T (x - Complex.I • y), x - Complex.I • y⟫_ℂ) /
      4

Tactic:
simp only [map_add, map_sub, inner_add_left, inner_add_right, map_smul, inner_smul_left,
    inner_smul_right, Complex.conj_I, ← pow_two, Complex.I_sq, inner_sub_left, inner_sub_right,
    mul_add, ← mul_assoc, mul_neg, neg_neg, one_mul, neg_one_mul, mul_sub, sub_sub]
  ring

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
s : Set (Set E)
hs : DirectedOn (fun x1 x2 ↦ x1 ⊆ x2) s
h : ∀ a ∈ s, Orthonormal 𝕜 fun x ↦ ↑x
⊢ Orthonormal 𝕜 fun x ↦ ↑x

Tactic:
Set.sUnion_eq_iUnion

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
s : Set (Set E)
hs : DirectedOn (fun x1 x2 ↦ x1 ⊆ x2) s
h : ∀ a ∈ s, Orthonormal 𝕜 fun x ↦ ↑x
⊢ Orthonormal 𝕜 fun x ↦ ↑x
Module: Mathlib.Analysis.InnerProductSpace.Subspace

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
ι : Type u_4
G : ι → Type u_5
inst✝¹ : (i : ι) → NormedAddCommGroup (G i)
inst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
α : ι → Type u_6
v_family : (i : ι) → α i → G i
hv_family : ∀ (i : ι), Orthonormal 𝕜 (v_family i)
i : ι
v w : α i
hvw : ⟨i, v⟩ ≠ ⟨i, w⟩
h : v = w
⊢ False

Tactic:
by
      subst h
      exact hvw rfl

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

Goal state before:
pp: R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁸ : Semiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : TopologicalSpace M
inst✝⁵ : Module R M
inst✝⁴ : AddCommMonoid N
inst✝³ : TopologicalSpace N
inst✝² : Module R N
inst✝¹ : ContinuousAdd N
inst✝ : ContinuousConstSMul R N
f : M →L[R] N
s : ClosedSubmodule R M
t : ClosedSubmodule R N
⊢ map f s ≤ t ↔ s ≤ comap f t

Tactic:
by
  simp [map, Submodule.map_le_iff_le_comap]; simp [← toSubmodule_le_toSubmodule]

Goal state after:
Module: Mathlib.Analysis.RCLike.Lemmas

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
x : K
⊢ re x • 1 + im x • I = x

Tactic:
by simp [real_smul_eq_coe_mul]

Goal state after: