mathlib-initiative/mathlib-tactics huggingface dataset:
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
⊢ f.outer.trim s ≤ f.outer s

Tactic:
rw [OuterMeasure.trim_eq_iInf]

Goal state after:
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, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), f.outer t ≤ f.outer s
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
g : ℕ → Set R
hg : ∀ (i : ℕ), t i ⊆ g i ∧ MeasurableSet (g i) ∧ f.outer (g i) ≤ f.length (t i) + ↑(ε' i)
⊢ ⨅ (_ : MeasurableSet (iUnion g)), f.outer (iUnion g) ≤ ∑' (a : ℕ), (f.length (t a) + ↑(ε' a))

Tactic:
apply iInf_le_of_le (MeasurableSet.iUnion fun i => (hg i).2.1) _

Goal state after:
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
g : ℕ → Set R
hg : ∀ (i : ℕ), t i ⊆ g i ∧ MeasurableSet (g i) ∧ f.outer (g i) ≤ f.length (t i) + ↑(ε' i)
⊢ f.outer (iUnion g) ≤ ∑' (a : ℕ), (f.length (t a) + ↑(ε' a))
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: G : Type u_1
H : Type u_2
α : Type u_3
inst✝⁴ : Group G
inst✝³ : MulAction G α
s : Set α
inst✝² : Group H
inst✝¹ : MulAction H α
inst✝ : SMulCommClass H G α
g : H
⊢ g • s ∩ ⋃ g_1, ⋃ (_ : g_1 ≠ 1), g_1 • g • s = g • s ∩ ⋃ i, ⋃ (_ : i ≠ 1), g • i • s

Tactic:
smul_comm g (_ : G) (_ : Set α)

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

Goal state before:
pp: G : Type u_1
H : Type u_2
α : Type u_3
inst✝⁴ : Group G
inst✝³ : MulAction G α
s : Set α
inst✝² : Group H
inst✝¹ : MulAction H α
inst✝ : SMulCommClass H G α
g : H
⊢ g • s \ ⋃ g_1, ⋃ (_ : g_1 ≠ 1), g_1 • g • s = g • s \ ⋃ i, ⋃ (_ : i ≠ 1), g • i • s

Tactic:
smul_comm g (_ : G) (_ : Set α)

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
K₁ K₂ : Compacts G
V : Set G
hV : (interior V).Nonempty
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h ↦ g * h) ⁻¹' V
h2s : s.card = index (K₁.carrier ∪ K₂.carrier) V
K : Set G
hK : K ⊆ ⋃ g ∈ s, (fun x ↦ g * x) ⁻¹' V
g : G
⊢ g ∈ K → g ∈ ⋃ g ∈ {g ∈ s | ((fun x ↦ g * x) ⁻¹' V ∩ K).Nonempty}, (fun h ↦ g * h) ⁻¹' V

Tactic:
hg

Goal state after:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : (interior V).Nonempty
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h ↦ g * h) ⁻¹' V
h2s : s.card = index (K₁.carrier ∪ K₂.carrier) V
K : Set G
hK : K ⊆ ⋃ g ∈ s, (fun x ↦ g * x) ⁻¹' V
g : G
hg : g ∈ K
⊢ g ∈ ⋃ g ∈ {g ∈ s | ((fun x ↦ g * x) ⁻¹' V ∩ K).Nonempty}, (fun h ↦ g * h) ⁻¹' V
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
a b : R
hab : b ≤ a
⊢ f.measure (Ioo a b) = ofReal (leftLim (↑f) b - ↑f a)

Tactic:
simp only [hab, measure_empty, Ioo_eq_empty, not_lt]
    symm
    simp [ENNReal.ofReal_eq_zero, f.mono.leftLim_le hab]

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
inst✝ : IsFiniteMeasure μ
C : ℝ
hf : ∀ᵐ (x : X) ∂μ, ‖f x‖ ≤ C
s : Set X
hs : MeasurableSet s
hf1 : Integrable f μ
⊢ ‖∫ (x : X), f x ∂μ - ∫ (x : X) in s, f x ∂μ‖ ≤ μ.real sᶜ * C

Tactic:
have h0 : ∫ (x : X), f x ∂μ - ∫ x in s, f x ∂μ = ∫ x in sᶜ, f x ∂μ := by
    rw [sub_eq_iff_eq_add, add_comm, integral_add_compl hs hf1]
  have h1 : ∫ x in sᶜ, ‖f x‖ ∂μ ≤ ∫ _ in sᶜ, C ∂μ :=
    integral_mono_ae hf1.norm.restrict (integrable_const C) (ae_restrict_of_ae hf)
  have h2 : ∫ _ in sᶜ, C ∂μ = μ.real sᶜ * C := by
    rw [setIntegral_const C, smul_eq_mul]
  rw [h0, ← h2]
  exact le_trans (norm_integral_le_integral_norm f) h1

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 : F
⊢ ⟪0, x⟫ = 0

Tactic:
rw [← zero_smul 𝕜 (0 : F), inner_smul_left]
  simp only [zero_mul, map_zero]

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
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
⊢ ∑ i ∈ s, g i • v i = 0 → ∀ i ∈ s, g i = 0

Tactic:
hg

Goal state after:
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 ∈ s, g i = 0
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 : IsOpen[inst✝¹] U
hK₀ : ↑K₀ ⊆ U
⊢ 1 ≤ (haarContent K₀) K₀.toCompacts

Tactic:
exact haarContent_self.ge

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

Goal state before:
pp: G : Type u_1
inst✝⁷ : Group G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : μ.InnerRegularCompactLTTop
inst✝ : LocallyCompactSpace G
E : Set G
hE : MeasurableSet E
hEapprox : ∃ K ⊆ E, IsCompact K ∧ 0 < μ K
K : Set G
hKE : K ⊆ E
hK : IsCompact K
K_closed : IsClosed[inst✝⁶] K
hKpos : 0 < μ K
g : G
⊢ μ (g • K \ K) < μ K → g ∈ E / E

Tactic:
hg

Goal state after:
G : Type u_1
inst✝⁷ : Group G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : μ.InnerRegularCompactLTTop
inst✝ : LocallyCompactSpace G
E : Set G
hE : MeasurableSet E
hEapprox : ∃ K ⊆ E, IsCompact K ∧ 0 < μ K
K : Set G
hKE : K ⊆ E
hK : IsCompact K
K_closed : IsClosed[inst✝⁶] K
hKpos : 0 < μ K
g : G
hg : μ (g • K \ K) < μ K
⊢ g ∈ E / E
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
this : InnerProductSpace.Core 𝕜 E := InnerProductSpace.toCore
h : ⟪x, y⟫ • x = ⟪x, x⟫ • y
⊢ y = ⟪x, x⟫⁻¹ • ⟪x, x⟫ • y

Tactic:
inv_smul_smul₀

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
this : InnerProductSpace.Core 𝕜 E := InnerProductSpace.toCore
h : ⟪x, y⟫ • x = ⟪x, x⟫ • y
⊢ y = y

case ha
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
this : InnerProductSpace.Core 𝕜 E := InnerProductSpace.toCore
h : ⟪x, y⟫ • x = ⟪x, x⟫ • y
⊢ ⟪x, x⟫ ≠ 0
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: case mp
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y : F
⊢ ⟪x, y⟫_ℝ / (‖x‖ * ‖y‖) = 1 → x ≠ 0 ∧ ∃ r, 0 < r ∧ y = r • x

pp: case mpr
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y : F
⊢ (x ≠ 0 ∧ ∃ r, 0 < r ∧ y = r • x) → ⟪x, y⟫_ℝ / (‖x‖ * ‖y‖) = 1

Tactic:
· intro h
    have hx₀ : x ≠ 0 := fun h₀ => by simp [h₀] at h
    have hy₀ : y ≠ 0 := fun h₀ => by simp [h₀] at h
    refine ⟨hx₀, ‖y‖ / ‖x‖, div_pos (norm_pos_iff.2 hy₀) (norm_pos_iff.2 hx₀), ?_⟩
    exact ((inner_eq_norm_mul_iff_div hx₀).1 (eq_of_div_eq_one h)).symm

Goal state after:
case mpr
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y : F
⊢ (x ≠ 0 ∧ ∃ r, 0 < r ∧ y = r • x) → ⟪x, y⟫_ℝ / (‖x‖ * ‖y‖) = 1
Module: Mathlib.Analysis.InnerProductSpace.Orthonormal

Goal state before:
pp: case mpr.left
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
ι : Type u_4
inst✝ : DecidableEq ι
v : ι → E
h : ∀ (i j : ι), ⟪v i, v j⟫ = if i = j then 1 else 0
i : ι
h' : ‖v i‖ ^ 2 = 1 ^ 2
h₁ : 0 ≤ ‖v i‖
⊢ ‖v i‖ = 1

Tactic:
have h₂ : (0 : ℝ) ≤ 1 := zero_le_one

Goal state after:
case mpr.left
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
ι : Type u_4
inst✝ : DecidableEq ι
v : ι → E
h : ∀ (i j : ι), ⟪v i, v j⟫ = if i = j then 1 else 0
i : ι
h' : ‖v i‖ ^ 2 = 1 ^ 2
h₁ : 0 ≤ ‖v i‖
h₂ : 0 ≤ 1
⊢ ‖v i‖ = 1
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
ι : Type u_4
v : ι → E
hv : Orthonormal 𝕜 v
l : ι →₀ 𝕜
hl : (linearCombination 𝕜 v) l = 0
i : ι
⊢ l i = 0 i

Tactic:
have key : ⟪v i, Finsupp.linearCombination 𝕜 v l⟫ = ⟪v i, 0⟫ := by rw [hl]

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
v : ι → E
hv : Orthonormal 𝕜 v
l : ι →₀ 𝕜
hl : (linearCombination 𝕜 v) l = 0
i : ι
key : ⟪v i, (linearCombination 𝕜 v) l⟫ = ⟪v i, 0⟫
⊢ l i = 0 i
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
ι : Type u_4
ι' : Type u_5
v : ι → E
hv : ∀ (i j : ι), ⟪v i, v j⟫ = if i = j then 1 else 0
f : ι' → ι
hf : Function.Injective f
i j : ι'
⊢ ⟪(v ∘ f) i, (v ∘ f) j⟫ = if i = j then 1 else 0

Tactic:
convert! hv (f i) (f j) using 1

Goal state after:
case e'_3
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
ι' : Type u_5
v : ι → E
hv : ∀ (i j : ι), ⟪v i, v j⟫ = if i = j then 1 else 0
f : ι' → ι
hf : Function.Injective f
i j : ι'
⊢ (if i = j then 1 else 0) = if f i = f j then 1 else 0
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, T x⟫_ℂ = 0

Tactic:
inner_map_polarization

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 (T x + x), T x + x⟫_ℂ - ⟪T (T x - x), T x - x⟫_ℂ + Complex.I * ⟪T (T x + Complex.I • x), T x + Complex.I • x⟫_ℂ -
        Complex.I * ⟪T (T x - Complex.I • x), T x - Complex.I • x⟫_ℂ) /
      4 =
    0
Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal

Goal state before:
pp: case mpr.a
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Set F
h✝ : Convex ℝ K
u v : F
hv : v ∈ K
this✝ : Nonempty ↑K := Nonempty.intro ⟨v, hv⟩
h : ∀ w ∈ K, ⟪u - v, w - v⟫_ℝ ≤ 0
w : ↑K
this : ⟪u - v, ↑w - v⟫_ℝ ≤ 0
⊢ ‖u - v‖ * ‖u - v‖ ≤ ‖u - ↑w‖ * ‖u - ↑w‖

Tactic:
calc
        ‖u - v‖ * ‖u - v‖ ≤ ‖u - v‖ * ‖u - v‖ - 2 * ⟪u - v, w - v⟫_ℝ := by linarith
        _ ≤ ‖u - v‖ ^ 2 - 2 * ⟪u - v, w - v⟫_ℝ + ‖w - v‖ ^ 2 := by
          rw [sq]
          refine le_add_of_nonneg_right ?_
          exact sq_nonneg _
        _ = ‖u - v - (w - v)‖ ^ 2 := (@norm_sub_sq ℝ _ _ _ _ _ _).symm
        _ = ‖u - w‖ * ‖u - w‖ := by
          have : u - v - (w - v) = u - w := by abel
          rw [this, sq]

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

Goal state before:
pp: case refine_2
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
⊢ min (min (δ / 2) r) (B (n + 1)) ≤ B (n + 1)

Tactic:
exact min_le_right _ _

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

Goal state before:
pp: case zero
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
center : ℕ → X → ℝ≥0∞ → X
radius : ℕ → X → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ) (x : X) (δ : ℝ≥0∞), δ ≠ 0 → 0 < radius n x δ
HB : ∀ (n : ℕ) (x : X) (δ : ℝ≥0∞), δ ≠ 0 → radius n x δ ≤ B (n + 1)
Hball : ∀ (n : ℕ) (x : X) (δ : ℝ≥0∞), δ ≠ 0 → closedEBall (center n x δ) (radius n x δ) ⊆ closedEBall x δ ∩ f n
x : X
ε : ℝ≥0∞
εpos : 0 < ε
F : ℕ → X × ℝ≥0∞ := fun n ↦ Nat.recOn n (x, min ε (B 0)) fun n p ↦ (center n p.1 p.2, radius n p.1 p.2)
c : ℕ → X := fun n ↦ (F n).1
r : ℕ → ℝ≥0∞ := fun n ↦ (F n).2
rpos : ∀ (n : ℕ), 0 < r n
r0 : ∀ (n : ℕ), r n ≠ 0
⊢ r 0 ≤ B 0

Tactic:
exact min_le_right _ _

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✝ : CompleteSpace E
f : (i : ι) → G i
hf : ∀ ε > 0, ∃ N, ∀ (m : Finset ι), N ⊆ m → ∀ (n : Finset ι), N ⊆ n → |∑ i ∈ m, ‖f i‖ ^ 2 - ∑ i ∈ n, ‖f i‖ ^ 2| < ε
ε : ℝ
hε : ε > 0
hε' : 0 < ε ^ 2 / 2
a : Finset ι
H : ∀ (m : Finset ι), a ⊆ m → ∀ (n : Finset ι), a ⊆ n → |∑ i ∈ m, ‖f i‖ ^ 2 - ∑ i ∈ n, ‖f i‖ ^ 2| < ε ^ 2 / 2
s₁ : Finset ι
hs₁ : a ⊆ s₁
s₂ : Finset ι
hs₂ : a ⊆ s₂
has : a ⊆ s₁ ⊓ s₂
this : s₁ ⊓ s₂ ⊆ s₁
⊢ ∑ x ∈ s₁ \ s₂, ‖f x‖ ^ 2 = |∑ x ∈ s₁ \ (s₁ ⊓ s₂), ‖f x‖ ^ 2 + ∑ x ∈ s₁ ⊓ s₂, ‖f x‖ ^ 2 - ∑ i ∈ s₁ ⊓ s₂, ‖f i‖ ^ 2|

Tactic:
add_tsub_cancel_right,

Goal state after:
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✝ : CompleteSpace E
f : (i : ι) → G i
hf : ∀ ε > 0, ∃ N, ∀ (m : Finset ι), N ⊆ m → ∀ (n : Finset ι), N ⊆ n → |∑ i ∈ m, ‖f i‖ ^ 2 - ∑ i ∈ n, ‖f i‖ ^ 2| < ε
ε : ℝ
hε : ε > 0
hε' : 0 < ε ^ 2 / 2
a : Finset ι
H : ∀ (m : Finset ι), a ⊆ m → ∀ (n : Finset ι), a ⊆ n → |∑ i ∈ m, ‖f i‖ ^ 2 - ∑ i ∈ n, ‖f i‖ ^ 2| < ε ^ 2 / 2
s₁ : Finset ι
hs₁ : a ⊆ s₁
s₂ : Finset ι
hs₂ : a ⊆ s₂
has : a ⊆ s₁ ⊓ s₂
this : s₁ ⊓ s₂ ⊆ s₁
⊢ ∑ x ∈ s₁ \ s₂, ‖f x‖ ^ 2 = |∑ x ∈ s₁ \ (s₁ ⊓ s₂), ‖f x‖ ^ 2|
Module: Mathlib.Topology.Baire.CompleteMetrizable

Goal state before:
pp: case h.left
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
center : ℕ → X → ℝ≥0∞ → X
radius : ℕ → X → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ) (x : X) (δ : ℝ≥0∞), δ ≠ 0 → 0 < radius n x δ
HB : ∀ (n : ℕ) (x : X) (δ : ℝ≥0∞), δ ≠ 0 → radius n x δ ≤ B (n + 1)
Hball : ∀ (n : ℕ) (x : X) (δ : ℝ≥0∞), δ ≠ 0 → closedEBall (center n x δ) (radius n x δ) ⊆ closedEBall x δ ∩ f n
x : X
ε : ℝ≥0∞
εpos : 0 < ε
F : ℕ → X × ℝ≥0∞ := fun n ↦ Nat.recOn n (x, min ε (B 0)) fun n p ↦ (center n p.1 p.2, radius n p.1 p.2)
c : ℕ → X := fun n ↦ (F n).1
r : ℕ → ℝ≥0∞ := fun n ↦ (F n).2
rpos : ∀ (n : ℕ), 0 < r n
r0 : ∀ (n : ℕ), r n ≠ 0
rB : ∀ (n : ℕ), r n ≤ B n
incl : ∀ (n : ℕ), closedEBall (c (n + 1)) (r (n + 1)) ⊆ closedEBall (c n) (r n) ∩ f n
cdist : ∀ (n : ℕ), edist (c n) (c (n + 1)) ≤ B n
this✝ : CauchySeq c
y : X
ylim : Tendsto c atTop (nhds y)
I : ∀ (n m : ℕ), m ≥ n → closedEBall (c m) (r m) ⊆ closedEBall (c n) (r n)
yball : ∀ (n : ℕ), y ∈ closedEBall (c n) (r n)
n : ℕ
this : closedEBall (c (n + 1)) (r (n + 1)) ⊆ f n
⊢ y ∈ f n

Tactic:
exact this (yball (n + 1))

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✝ : CompleteSpace E
f : (i : ι) → G i
hf : ∀ ε > 0, ∃ N, ∀ (m : Finset ι), N ⊆ m → ∀ (n : Finset ι), N ⊆ n → |∑ i ∈ m, ‖f i‖ ^ 2 - ∑ i ∈ n, ‖f i‖ ^ 2| < ε
ε : ℝ
hε : ε > 0
hε' : 0 < ε ^ 2 / 2
a : Finset ι
H : ∀ (m : Finset ι), a ⊆ m → ∀ (n : Finset ι), a ⊆ n → |∑ i ∈ m, ‖f i‖ ^ 2 - ∑ i ∈ n, ‖f i‖ ^ 2| < ε ^ 2 / 2
s₁ : Finset ι
hs₁ : a ⊆ s₁
s₂ : Finset ι
hs₂ : a ⊆ s₂
has : a ⊆ s₁ ⊓ s₂
Hs₁ : ∑ x ∈ s₁ \ s₂, ‖f x‖ ^ 2 < ε ^ 2 / 2
this : s₁ ⊓ s₂ ⊆ s₂
⊢ ∑ x ∈ s₂ \ s₁, ‖f x‖ ^ 2 = |∑ x ∈ s₂ \ (s₁ ⊓ s₂), ‖f x‖ ^ 2 + ∑ x ∈ s₁ ⊓ s₂, ‖f x‖ ^ 2 - ∑ i ∈ s₁ ⊓ s₂, ‖f i‖ ^ 2|

Tactic:
add_tsub_cancel_right,

Goal state after:
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✝ : CompleteSpace E
f : (i : ι) → G i
hf : ∀ ε > 0, ∃ N, ∀ (m : Finset ι), N ⊆ m → ∀ (n : Finset ι), N ⊆ n → |∑ i ∈ m, ‖f i‖ ^ 2 - ∑ i ∈ n, ‖f i‖ ^ 2| < ε
ε : ℝ
hε : ε > 0
hε' : 0 < ε ^ 2 / 2
a : Finset ι
H : ∀ (m : Finset ι), a ⊆ m → ∀ (n : Finset ι), a ⊆ n → |∑ i ∈ m, ‖f i‖ ^ 2 - ∑ i ∈ n, ‖f i‖ ^ 2| < ε ^ 2 / 2
s₁ : Finset ι
hs₁ : a ⊆ s₁
s₂ : Finset ι
hs₂ : a ⊆ s₂
has : a ⊆ s₁ ⊓ s₂
Hs₁ : ∑ x ∈ s₁ \ s₂, ‖f x‖ ^ 2 < ε ^ 2 / 2
this : s₁ ⊓ s₂ ⊆ s₂
⊢ ∑ x ∈ s₂ \ s₁, ‖f x‖ ^ 2 = |∑ x ∈ s₂ \ (s₁ ⊓ s₂), ‖f x‖ ^ 2|
Module: Mathlib.Analysis.InnerProductSpace.Orthogonal

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
x y : E
h : ∀ (v : ↥K), ⟪x, ↑v⟫ = ⟪y, ↑v⟫
⊢ ∀ u ∈ K, ⟪x - y, u⟫ = 0

Tactic:
intro u hu

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
x y : E
h : ∀ (v : ↥K), ⟪x, ↑v⟫ = ⟪y, ↑v⟫
u : E
hu : u ∈ K
⊢ ⟪x - y, u⟫ = 0
Module: Mathlib.Analysis.InnerProductSpace.Orthogonal

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
x y : E
h : ∀ (v : ↥K), ⟪↑v, x⟫ = ⟪↑v, y⟫
⊢ x - y ∈ Kᗮ

Tactic:
intro u hu

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
x y : E
h : ∀ (v : ↥K), ⟪↑v, x⟫ = ⟪↑v, y⟫
u : E
hu : u ∈ K
⊢ ⟪u, x - y⟫ = 0
Module: Mathlib.Analysis.InnerProductSpace.Orthogonal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
⊢ ⊥ᗮ = ⊤

Tactic:
by
  rw [← top_orthogonal_eq_bot, eq_top_iff]
  exact le_orthogonal_orthogonal ⊤

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

Goal state before:
pp: case refine_1
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.subtype ∘ₗ U.projectionOnto V hUV).IsSymmetric
u : E
hu : u ∈ U
v : E
hv : v ∈ V
⊢ ⟪(U.subtype ∘ₗ U.projectionOnto V hUV) ↑⟨v, hv⟩, ↑⟨u, hu⟩⟫ = 0

Tactic:
comp_apply,

Goal state after:
case refine_1
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.subtype ∘ₗ U.projectionOnto V hUV).IsSymmetric
u : E
hu : u ∈ U
v : E
hv : v ∈ V
⊢ ⟪U.subtype ((U.projectionOnto V hUV) ↑⟨v, hv⟩), ↑⟨u, hu⟩⟫ = 0
Module: Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
v : Set E
hv : Orthonormal 𝕜 Subtype.val
h : ∀ x ∈ (span 𝕜 v)ᗮ, x = 0
u : Set E
huv : v ⊆ u
hu : Orthonormal 𝕜 Subtype.val
x : E
hxu : x ∈ u
hxv : x ∉ v
y : E
hy : y ∈ span 𝕜 v
hxv' : ⟨x, hxu⟩ ∉ Subtype.val ⁻¹' v
⊢ ∃ l ∈ supported 𝕜 𝕜 (Subtype.val ⁻¹' v), (linearCombination 𝕜 Subtype.val) l = y

Tactic:
rw [← Finsupp.mem_span_image_iff_linearCombination]
      simp [huv, inter_eq_self_of_subset_right, hy]

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.ProdLp

Goal state before:
pp: p : ℝ≥0∞
𝕜 : Type u_1
α : Type u_2
β : Type u_3
hp : Fact (1 ≤ p)
inst✝⁶ : SeminormedAddCommGroup α
inst✝⁵ : SeminormedAddCommGroup β
inst✝⁴ : SeminormedRing 𝕜
inst✝³ : Module 𝕜 α
inst✝² : Module 𝕜 β
inst✝¹ : IsBoundedSMul 𝕜 α
inst✝ : IsBoundedSMul 𝕜 β
c : 𝕜
f : WithLp p (α × β)
⊢ ‖c • f‖₊ ≤ ‖c‖₊ * ‖f‖₊

Tactic:
rcases p.dichotomy with (rfl | hp)
    · simp only [← prod_nnnorm_ofLp, ofLp_smul]
      exact norm_smul_le _ _
    · have hp0 : 0 < p.toReal := zero_lt_one.trans_le hp
      have hpt : p ≠ ⊤ := p.toReal_pos_iff_ne_top.mp hp0
      rw [prod_nnnorm_eq_add hpt, prod_nnnorm_eq_add hpt, one_div, NNReal.rpow_inv_le_iff hp0,
        NNReal.mul_rpow, ← NNReal.rpow_mul, inv_mul_cancel₀ hp0.ne', NNReal.rpow_one, mul_add,
        ← NNReal.mul_rpow, ← NNReal.mul_rpow]
      gcongr <;> exact nnnorm_smul_le _ _

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Banach

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CompleteSpace E
f : E →L[𝕜] E
⊢ IsUnit f ↔ Bijective ⇑f

Tactic:
by
  constructor
  · rintro ⟨f, rfl⟩
    exact ofUnit f |>.bijective
  · refine fun h ↦ ⟨toUnit <| .ofBijective f ?_ ?_, rfl⟩ <;>
    simp only [LinearMap.range_eq_top, LinearMap.ker_eq_bot, f.coe_coe, h.1, h.2]

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Banach

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
inst✝¹⁴ : NontriviallyNormedField 𝕜
inst✝¹³ : NontriviallyNormedField 𝕜'
σ : 𝕜 →+* 𝕜'
E : Type u_3
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace 𝕜 E
F✝ : Type u_4
inst✝¹⁰ : NormedAddCommGroup F✝
inst✝⁹ : NormedSpace 𝕜' F✝
f✝ : E →SL[σ] F✝
σ' : 𝕜' →+* 𝕜
inst✝⁸ : RingHomInvPair σ σ'
inst✝⁷ : RingHomIsometric σ
inst✝⁶ : RingHomIsometric σ'
inst✝⁵ : CompleteSpace F✝
inst✝⁴ : CompleteSpace E
F : Type u_5
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : CompleteSpace F
f : E →L[𝕜] F
G : Submodule 𝕜 F
h : IsCompl (↑f).range G
inst✝ : CompleteSpace ↥G
hker : (↑f).ker = ⊥
⊢ (↑(f.coprod G.subtypeL)).ker = ⊥

Tactic:
rw [ker_coprod_of_disjoint_range]
      · simp [hker]
      · simp [h.disjoint]

Goal state after: