mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
f : α → ℝ≥0 × ℝ≥0∞ := fun x ↦ (1, infEDist x E / ENNReal.ofReal δ)
sub : ℝ≥0 × ℝ≥0∞ → ℝ≥0∞ := fun p ↦ ↑p.1 - p.2
⊢ (fun x ↦ 1 - infEDist x E / ENNReal.ofReal δ) = sub ∘ f

Tactic:
rfl

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

Goal state before:
pp: case pos.h_add
α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C C' : ℝ
μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ∞
hμ'_le : μ' ≤ c' • μ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ' T C'
f : α → E
hfμ : Integrable f μ
hF : CompleteSpace F
h_int : ∀ (g : α → E), Integrable g μ → Integrable g μ'
f₂ g₂ : α → E
a✝ : Disjoint (Function.support f₂) (Function.support g₂)
hf₂ : Integrable f₂ μ
hg₂ : Integrable g₂ μ
h_eq_f : setToFun μ T hT f₂ = setToFun μ' T hT' f₂
h_eq_g : setToFun μ T hT g₂ = setToFun μ' T hT' g₂
⊢ setToFun μ T hT (f₂ + g₂) = setToFun μ' T hT' (f₂ + g₂)

Tactic:
setToFun_add hT hf₂ hg₂,

Goal state after:
case pos.h_add
α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C C' : ℝ
μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ∞
hμ'_le : μ' ≤ c' • μ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ' T C'
f : α → E
hfμ : Integrable f μ
hF : CompleteSpace F
h_int : ∀ (g : α → E), Integrable g μ → Integrable g μ'
f₂ g₂ : α → E
a✝ : Disjoint (Function.support f₂) (Function.support g₂)
hf₂ : Integrable f₂ μ
hg₂ : Integrable g₂ μ
h_eq_f : setToFun μ T hT f₂ = setToFun μ' T hT' f₂
h_eq_g : setToFun μ T hT g₂ = setToFun μ' T hT' g₂
⊢ setToFun μ T hT f₂ + setToFun μ T hT g₂ = setToFun μ' T hT' (f₂ + g₂)
Module: Mathlib.MeasureTheory.Integral.Marginal

Goal state before:
pp: δ : Type u_1
X : δ → Type u_3
inst✝¹ : (i : δ) → MeasurableSpace (X i)
μ : (i : δ) → Measure (X i)
inst✝ : DecidableEq δ
f : ((i : δ) → X i) → ℝ≥0∞
i : δ
α : Type u_1 := ↥{i}
e : X ↑default ≃ᵐ ((i_1 : α) → X ↑i_1) := (MeasurableEquiv.piUnique fun j ↦ X ↑j).symm
x : (i : δ) → X i
⊢ ∫⁻ (y : X ↑default), f (updateFinset x {i} (e y)) ∂μ ↑default =
    ∫⁻ (xᵢ : X i), f (updateFinset x {i} (uniqueElim xᵢ)) ∂μ i

Tactic:
rfl

Goal state after:
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 t : Finset δ
inst✝ : ∀ (i : δ), SigmaFinite (μ i)
f : ((i : δ) → X i) → ℝ≥0∞
hf : Measurable f
hst : Disjoint s t
x : (i : δ) → X i
e : ((i : ↥s) → X ↑i) × ((i : ↥t) → X ↑i) ≃ᵐ ((i : ↥(s ∪ t)) → X ↑i) := MeasurableEquiv.piFinsetUnion X hst
⊢ (∫⁻ (y : (i : ↥s) → X ↑i),
      ∫⁻ (z : (j : ↥t) → X ↑j),
        f (updateFinset x (s ∪ t) (e (y, z))) ∂Measure.pi fun j ↦ μ ↑j ∂Measure.pi fun i ↦ μ ↑i) =
    ∫⁻ (y : (i : ↥s) → X ↑i),
      ∫⁻ (y_1 : (i : ↥t) → X ↑i),
        f (updateFinset x (s ∪ t) ((piFinsetUnion X hst) (y, y_1))) ∂Measure.pi fun j ↦ μ ↑j ∂Measure.pi fun i ↦ μ ↑i

Tactic:
rfl

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

Goal state before:
pp: δ : Type u_1
X : δ → Type u_3
inst✝³ : (i : δ) → MeasurableSpace (X i)
μ : (i : δ) → Measure (X i)
inst✝² : DecidableEq δ
inst✝¹ : ∀ (i : δ), SigmaFinite (μ i)
inst✝ : Fintype δ
f : ((i : δ) → X i) → ℝ≥0∞
e : ↥Finset.univ ≃ δ := subtypeUnivEquiv ⋯
x : (i : δ) → X i
⊢ (∫⁻ (y : (i : ↥Finset.univ) → X ↑i), f fun i ↦ y ⟨i, ⋯⟩ ∂Measure.pi fun i ↦ μ ↑i) =
    ∫⁻ (a : (b : ↥Finset.univ) → X (e b)), f ((MeasurableEquiv.piCongrLeft X e) a) ∂Measure.pi fun i' ↦ μ (e i')

Tactic:
rfl

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

Goal state before:
pp: δ : Type u_1
δ' : Type u_2
X : δ → Type u_3
inst✝³ : (i : δ) → MeasurableSpace (X i)
μ : (i : δ) → Measure (X i)
inst✝² : DecidableEq δ
inst✝¹ : ∀ (i : δ), SigmaFinite (μ i)
inst✝ : DecidableEq δ'
e : δ' → δ
he : Injective e
s : Finset δ'
f : ((i : δ') → X (e i)) → ℝ≥0∞
hf : Measurable f
x : (i : δ) → X i
h : Measurable fun x ↦ (fun {x_1} ↦ x) ∘' e
⊢ (∫⋯∫⁻_Finset.image e s, f ∘ fun x ↦ (fun {x_1} ↦ x) ∘' e ∂μ) x =
    (∫⋯∫⁻_s, f ∂(fun {x} ↦ μ) ∘' e) ((fun {x_1} ↦ x) ∘' e)

Tactic:
induction s using Finset.induction generalizing x with

Goal state after:
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:
induction s using Finset.induction generalizing x with

Goal state after:
Module: Mathlib.Topology.Order.LeftRightLim

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace α
inst✝² : OrderTopology α
inst✝¹ : T3Space β
inst✝ : NoTopOrder α
f g : α → β
b : β
h : Tendsto f atTop (𝓝 b)
h' : ∀ᶠ (x : α) in atTop, MapClusterPt (g x) (𝓝 x) f
⊢ Tendsto g atTop (𝓝 b)

Tactic:
rcases isEmpty_or_nonempty α with hα | hα
  · simp [filter_eq_bot_of_isEmpty atTop]
  apply (closed_nhds_basis b).tendsto_right_iff.2
  rintro s ⟨s_mem, s_closed⟩
  obtain ⟨u, hu⟩ : ∃ a, ∀ (b : α), a ≤ b → MapClusterPt (g b) (𝓝 b) f ∧ f b ∈ s := by
    simpa [eventually_atTop] using h'.and (h s_mem)
  filter_upwards [Ioi_mem_atTop u] with a (ha : u < a)
  apply s_closed.mem_of_mapClusterPt (hu a ha.le).1
  filter_upwards [Ici_mem_nhds ha] with y hy using (hu y hy).2

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

Goal state before:
pp: case pos.convert_2
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‖₊
n : ι
x : α
⊢ ‖f n x‖ ≤ ↑‖f n x‖₊

Tactic:
rfl

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
μ : Measure X
ι : Type u_5
t : Finset ι
s : ι → Set X
hs : ∀ i ∈ t, MeasurableSet (s i)
hf : ∀ i ∈ t, IntegrableOn f (s i) μ
u : Finset ι
hu : u ∈ {x ∈ t.powerset | x.Nonempty}
i : ι
hi : i ∈ u
⊢ i ∈ t

Tactic:
grind

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

Goal state before:
pp: case convert_2
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), μ (t.sup K) ≤ ∑ i ∈ t, μ (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
t : Finset ℕ
ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : ↑K = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun n ↦ { carrier := K' n, isCompact' := ⋯ }
i : ℕ
a✝ : i ∈ t
⊢ μ (L i) ≤ μ (L i)

Tactic:
rfl

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 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) in s ∪ t, f' x ∂μ = ∫ (x : X) in s, f' x ∂μ

Tactic:
by
      apply
        integral_union_eq_left_of_ae_aux _ H.1.stronglyMeasurable_mk (H.congr_fun_ae H.1.ae_eq_mk)
      filter_upwards [ht_eq,
        ae_mono (Measure.restrict_mono subset_union_right le_rfl) H.1.ae_eq_mk] with x hx h'x
      rw [← h'x, hx]

Goal state after:
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: G : Type u_1
α : Type u_3
E : Type u_5
inst✝⁷ : Group G
inst✝⁶ : MulAction G α
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
s : Set α
μ : Measure α
inst✝³ : MeasurableConstSMul G α
inst✝² : SMulInvariantMeasure G α μ
inst✝¹ : Countable G
inst✝ : NormedSpace ℝ E
h : IsFundamentalDomain G s μ
f : α → E
hf : Integrable f μ
⊢ μ ≪ μ

Tactic:
rfl

Goal state after:
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: G : Type u_1
α : Type u_3
inst✝⁵ : Countable G
inst✝⁴ : Group G
inst✝³ : MulAction G α
inst✝² : MeasurableSpace α
μ : Measure α
s : Set α
hs : IsFundamentalDomain G s μ
inst✝¹ : MeasurableConstSMul G α
inst✝ : SMulInvariantMeasure G α μ
⊢ ⋃ g, g⁻¹ • s ⊆ ⋃ g, g⁻¹ • s

Tactic:
rfl

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

Goal state before:
pp: case inl
R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
l : ℝ
hf : Tendsto (↑f) atBot (𝓝 l)
x : R
this : Nonempty R
val✝ : OrderBot R
⊢ f.measure (Iic x) = ofReal (↑f x - l)

pp: case inr
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✝ : NoBotOrder R
⊢ f.measure (Iic x) = ofReal (↑f x - l)

Tactic:
· have : Iic x = Icc ⊥ x := by simp
    rw [atBot_eq_pure_of_isBot isBot_bot] at hf
    rw [this, measure_Icc, leftLim_eq_of_isBot isBot_bot,
      tendsto_nhds_unique hf (tendsto_pure_nhds f ⊥)]

Goal state after:
case inr
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✝ : NoBotOrder R
⊢ f.measure (Iic x) = ofReal (↑f x - l)
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
x : R
hx : C + 1 ≤ ↑f x
⊢ ∃ u, Tendsto (↑f) atTop (𝓝 u)

Tactic:
grind

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

Goal state before:
pp: case inl
R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
C : ℝ
h : ∀ (x : R), |↑f x| ≤ C
h✝ : Nonempty R
u : ℝ
hu : Tendsto (↑f) atTop (𝓝 u)
H : Tendsto (↑f) atBot atBot
x : R
hx : ↑f x ≤ -C - 1
⊢ ∃ l, Tendsto (↑f) atBot (𝓝 l)

Tactic:
grind

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

Tactic:
mul_conj

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 - ↑(normSq x) * ↑‖⟪x, y⟫‖ ^ 2 +
      ↑(normSq x) * (↑(normSq x) * ↑(normSq y)) =
    ↑(normSq x * (normSq x * normSq y) - normSq x * ‖⟪x, y⟫‖ ^ 2)
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case inl
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≥0
f : StieltjesFunction R
a b : R
h : a ≤ b
ha : 𝓝[<] a = ⊥
⊢ ↑(c • f) a = ↑c * ↑f a

Tactic:
rfl

Goal state after:
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≥0
f : StieltjesFunction R
a b : R
h : a ≤ b
this : leftLim (↑(c • f)) a = ↑c * leftLim (↑f) a
⊢ ↑c • ofReal (↑f b - leftLim (↑f) a) = c • ofReal (↑f b - leftLim (↑f) a)

Tactic:
rfl

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
⊢ re ⟪x, x⟫ * t ^ 2 * ‖⟪x, y⟫‖ ^ 2 = t ^ 2 * re (⟪x, y⟫ * (starRingEnd 𝕜) ⟪x, y⟫ * ⟪x, x⟫)

Tactic:
mul_conj

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
⊢ re ⟪x, x⟫ * t ^ 2 * ‖⟪x, y⟫‖ ^ 2 = t ^ 2 * re (↑‖⟪x, y⟫‖ ^ 2 * ⟪x, x⟫)
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: F : Type u_3
inst✝¹ : SeminormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y : F
⊢ (starRingEnd ℝ) ⟪x, y⟫_ℝ = ⟪x, y⟫_ℝ

Tactic:
rfl

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Set F
ne : K.Nonempty
h₁ : IsComplete K
h₂ : Convex ℝ K
u : F
δ : ℝ := ⨅ w, ‖u - ↑w‖
this : Nonempty ↑K := Set.Nonempty.to_subtype ne
zero_le_δ : 0 ≤ δ
δ_le : ∀ (w : ↑K), δ ≤ ‖u - ↑w‖
δ_le' : ∀ w ∈ K, δ ≤ ‖u - w‖
hδ : ∀ (n : ℕ), δ < δ + 1 / (↑n + 1)
h : ∀ (n : ℕ), ∃ i, ‖u - ↑i‖ < δ + 1 / (↑n + 1)
w : ℕ → ↑K := fun n ↦ Classical.choose ⋯
⊢ ∃ w, ∀ (n : ℕ), ‖u - ↑(w n)‖ < δ + 1 / (↑n + 1)

Tactic:
exact ⟨w, fun n => Classical.choose_spec (h n)⟩

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

Goal state before:
pp: case e'_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
⊢ g i * ⟪v i, v i⟫ = ⟪v i, g i • v i⟫

Tactic:
inner_smul_right

Goal state after:
case e'_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
⊢ g i * ⟪v i, v i⟫ = g i * ⟪v i, v i⟫
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
h : ‖x‖ = 0
⊢ ‖⟪x, y⟫‖ ≤ 0

Tactic:
exact norm_inner_le_norm _ _ |>.trans <| by simp [h]

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
K₁ K₂ : Compacts G
h : Disjoint ↑K₁ ↑K₂
_h₁ : IsClosed[inst✝¹] ↑K₁
h₂ : IsClosed[inst✝¹] ↑K₂
⊢ ⟨chaar K₀ K₁ + chaar K₀ K₂, ⋯⟩ = ⟨chaar K₀ K₁, ⋯⟩ + ⟨chaar K₀ K₂, ⋯⟩

Tactic:
rfl

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
⊢ ⟨1, ⋯⟩ = 1

Tactic:
rfl

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

Goal state before:
pp: F : Type u_3
inst✝¹ : SeminormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y : F
⊢ ⟪x, x⟫_ℝ + ⟪y, x⟫_ℝ = ⟪y, x⟫_ℝ + ⟪y, y⟫_ℝ ↔ ⟪x, x⟫_ℝ = ⟪y, y⟫_ℝ

Tactic:
grind

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
n : ℕ
v : E
vs : Fin n → E
⊢ ((∀ (i : Fin n.succ), ‖Matrix.vecCons v vs i‖ = 1) ∧
      Pairwise fun i j ↦ ⟪Matrix.vecCons v vs i, Matrix.vecCons v vs j⟫ = 0) ↔
    ‖v‖ = 1 ∧ (∀ (i : Fin n), ⟪v, vs i⟫ = 0) ∧ (∀ (i : Fin n), ‖vs i‖ = 1) ∧ Pairwise fun i j ↦ ⟪vs i, vs j⟫ = 0

Tactic:
pairwise_fin_succ_iff_of_isSymm,

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
n : ℕ
v : E
vs : Fin n → E
⊢ ((∀ (i : Fin n.succ), ‖Matrix.vecCons v vs i‖ = 1) ∧
      (∀ (j : Fin n), ⟪Matrix.vecCons v vs 0, Matrix.vecCons v vs j.succ⟫ = 0) ∧
        Pairwise fun i j ↦ ⟪Matrix.vecCons v vs i.succ, Matrix.vecCons v vs j.succ⟫ = 0) ↔
    ‖v‖ = 1 ∧ (∀ (i : Fin n), ⟪v, vs i⟫ = 0) ∧ (∀ (i : Fin n), ‖vs i‖ = 1) ∧ Pairwise fun i j ↦ ⟪vs i, vs j⟫ = 0
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y : F
⊢ ‖x + y‖ = ‖x‖ + ‖y‖ ↔ ‖y‖ • x = ‖x‖ • y

Tactic:
rw [← pow_left_inj₀ (norm_nonneg _) (Left.add_nonneg (norm_nonneg _) (norm_nonneg _)) two_ne_zero,
    norm_add_sq (𝕜 := ℝ), add_pow_two, add_left_inj, add_right_inj, re_to_real, mul_assoc,
    mul_right_inj' two_ne_zero, ← inner_eq_norm_mul_iff_real]

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

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

Tactic:
rfl

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝ : RCLike 𝕜
⊢ instInnerProductSpaceRealComplex.toSMul = Complex.instRCLike.toSMul

Tactic:
rfl

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

Goal state before:
pp: case pos
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⟫ = ⟪(V i) v, (V j) w⟫

Tactic:
rfl

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
u v : E
hv : v ∈ K
this : InnerProductSpace ℝ E := rclikeToReal 𝕜 E
K' : Submodule ℝ E := restrictScalars ℝ K
H : ‖u - v‖ = ⨅ w, ‖u - ↑w‖
A : ∀ w ∈ K, re ⟪u - v, w⟫_𝕜 = 0
w : E
hw : w ∈ K
⊢ re ⟪u - v, -I • w⟫_𝕜 = re (-I * ⟪u - v, w⟫_𝕜)

Tactic:
inner_smul_right

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
u v : E
hv : v ∈ K
this : InnerProductSpace ℝ E := rclikeToReal 𝕜 E
K' : Submodule ℝ E := restrictScalars ℝ K
H : ‖u - v‖ = ⨅ w, ‖u - ↑w‖
A : ∀ w ∈ K, re ⟪u - v, w⟫_𝕜 = 0
w : E
hw : w ∈ K
⊢ re (-I * ⟪u - v, w⟫_𝕜) = re (-I * ⟪u - v, w⟫_𝕜)
Module: Mathlib.Analysis.InnerProductSpace.LinearMap

Goal state before:
pp: 𝕜 : Type u_4
inst✝² : RCLike 𝕜
F : Type u_8
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
x : F
hx : x ≠ 0
⊢ ↑‖x‖ ^ 2 • ((rankOne 𝕜) x) x = ((rankOne 𝕜) x) x → ‖x‖ = 1

Tactic:
nth_rw 2 [← one_smul 𝕜 (rankOne 𝕜 x x)]

Goal state after:
𝕜 : Type u_4
inst✝² : RCLike 𝕜
F : Type u_8
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
x : F
hx : x ≠ 0
⊢ ↑‖x‖ ^ 2 • ((rankOne 𝕜) x) x = 1 • ((rankOne 𝕜) x) x → ‖x‖ = 1
Module: Mathlib.Analysis.InnerProductSpace.LinearMap

Goal state before:
pp: 𝕜 : Type u_4
inst✝² : RCLike 𝕜
F : Type u_8
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
x : F
hx : x ≠ 0
⊢ ↑(-1) = -1

Tactic:
grind

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

Goal state before:
pp: 𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
F : Type u_8
H : Type u_9
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace 𝕜 H
a c : F
b d : H
ha : a ≠ 0
hb : b ≠ 0
h : ∀ (x : H), ⟪b, x⟫_𝕜 • a = ⟪d, x⟫_𝕜 • c
h₂ : ∀ (x : F), ⟪a, x⟫_𝕜 • b = ⟪c, x⟫_𝕜 • d
⊢ (1 / ⟪b, b⟫_𝕜) • ⟪b, b⟫_𝕜 • a = (⟪d, b⟫_𝕜 / ⟪b, b⟫_𝕜) • c

Tactic:
simp only [h, smul_smul]; ring_nf

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

Goal state before:
pp: R : Type u_2
M : Type u_3
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : Module R M
x : M
s : ClosedSubmodule R M
⊢ x ∈ ↑s ↔ x ∈ s

Tactic:
rfl

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

Goal state before:
pp: R : Type u_2
M : Type u_3
inst✝⁵ : Semiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : TopologicalSpace M
inst✝² : Module R M
inst✝¹ : ContinuousAdd M
inst✝ : ContinuousConstSMul R M
s : ClosedSubmodule R M
x✝ : M
⊢ x✝ ∈ ↑s ↔ x✝ ∈ s

Tactic:
rfl

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

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

Tactic:
rfl

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
K : Submodule 𝕜 E
f : E ≃ₗᵢ[𝕜] F
this : f.toLinearIsometry.range = ⊤
⊢ (map f.toLinearIsometry.toLinearMap K)ᗮ = (map (↑f.toLinearEquiv) K)ᗮ

Tactic:
rfl

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

Goal state before:
pp: case refine_2
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
U V : Submodule 𝕜 E
hUV : IsCompl U V
h : U ⟂ V
x y : E
⊢ ⟪(U.subtype ∘ₗ U.projectionOnto V hUV) x, y⟫ = ⟪x, (U.subtype ∘ₗ U.projectionOnto V hUV) y⟫

Tactic:
· nth_rw 2 [← projection_add_projection_eq_self hUV x]
    nth_rw 1 [← projection_add_projection_eq_self hUV y]
    rw [isOrtho_iff_inner_eq] at h
    simp [inner_add_right, inner_add_left, h, inner_eq_zero_symm]

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Projection.Submodule

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_4
inst✝² : Preorder ι
U : ι → Submodule 𝕜 E
inst✝¹ : ∀ (i : ι), (U i).HasOrthogonalProjection
inst✝ : (⨆ i, U i).topologicalClosure.HasOrthogonalProjection
hU : Monotone U
x : E
h : atTop.NeBot
left✝ : Nonempty ι
right✝ : IsDirectedOrder ι
y : E := (⨆ i, U i).topologicalClosure.starProjection x
proj_x : ∀ (i : ι), (U i).orthogonalProjectionOnto x = (U i).orthogonalProjectionOnto y
this : ∀ ε > 0, ∃ I, ∀ i ≥ I, ‖(U i).starProjection y - y‖ < ε
⊢ Tendsto (fun i ↦ (U i).starProjection x) atTop (𝓝 ((⨆ i, U i).topologicalClosure.starProjection x))

Tactic:
by
    simpa only [starProjection_apply, proj_x, NormedAddCommGroup.tendsto_atTop] using! this

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

Goal state before:
pp: case hvo
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
v w : E
⊢ ∀ w_1 ∈ 𝕜 ∙ v, ⟪↑‖v‖ ^ 2 • w - ⟪v, w⟫ • v, w_1⟫ = 0

Tactic:
· rw [← Submodule.mem_orthogonal', Submodule.mem_orthogonal_singleton_iff_inner_left]
    simp [inner_sub_left, inner_smul_left, inner_self_eq_norm_sq_to_K, mul_comm]

Goal state after: