mathlib-initiative/mathlib-tactics huggingface dataset:
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
this : ∀ K ⊆ ⋃ g ∈ s, (fun x ↦ g * x) ⁻¹' V, index K V ≤ {g ∈ s | ((fun x ↦ g * x) ⁻¹' V ∩ K).Nonempty}.card
⊢ {g ∈ s | ((fun x ↦ g * x) ⁻¹' V ∩ K₁.carrier).Nonempty}.card +
      {g ∈ s | ((fun x ↦ g * x) ⁻¹' V ∩ K₂.carrier).Nonempty}.card ≤
    s.card

Tactic:
rw [← Finset.card_union_of_disjoint, Finset.filter_union_right]

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
this : ∀ K ⊆ ⋃ g ∈ s, (fun x ↦ g * x) ⁻¹' V, index K V ≤ {g ∈ s | ((fun x ↦ g * x) ⁻¹' V ∩ K).Nonempty}.card
⊢ {x ∈ s | ((fun x_1 ↦ x * x_1) ⁻¹' V ∩ K₁.carrier).Nonempty ∨ ((fun x_1 ↦ x * x_1) ⁻¹' V ∩ K₂.carrier).Nonempty}.card ≤
    s.card

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
this : ∀ K ⊆ ⋃ g ∈ s, (fun x ↦ g * x) ⁻¹' V, index K V ≤ {g ∈ s | ((fun x ↦ g * x) ⁻¹' V ∩ K).Nonempty}.card
⊢ Disjoint ({g ∈ s | ((fun x ↦ g * x) ⁻¹' V ∩ K₁.carrier).Nonempty})
    ({g ∈ s | ((fun x ↦ g * x) ⁻¹' V ∩ K₂.carrier).Nonempty})
Module: Mathlib.MeasureTheory.Measure.Stieltjes

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

Tactic:
have : Nonempty R := ⟨x⟩
  cases botOrderOrNoBotOrder R
  · 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 ⊥)]
  have : NoMinOrder R := NoBotOrder.to_noMinOrder R
  refine tendsto_nhds_unique (tendsto_measure_Ioc_atBot _ _) ?_
  simp_rw [measure_Ioc]
  exact ENNReal.tendsto_ofReal (Tendsto.const_sub _ hf)

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 : Set G
hK : IsCompact K
V : Set G
hV : (interior V).Nonempty
g : G
s : Finset G
h1s : K ⊆ ⋃ g ∈ s, (fun h ↦ g * h) ⁻¹' V
h2s : s.card = index K V
⊢ index ((fun h ↦ g * h) '' K) V ≤ s.card

Tactic:
apply Nat.sInf_le

Goal state after:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K : Set G
hK : IsCompact K
V : Set G
hV : (interior V).Nonempty
g : G
s : Finset G
h1s : K ⊆ ⋃ g ∈ s, (fun h ↦ g * h) ⁻¹' V
h2s : s.card = index K V
⊢ s.card ∈ Finset.card '' {t | (fun h ↦ g * h) '' K ⊆ ⋃ g ∈ t, (fun h ↦ g * h) ⁻¹' V}
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: R : Type u_1
inst✝⁸ : LinearOrder R
inst✝⁷ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁶ : OrderTopology R
inst✝⁵ : CompactIccSpace R
inst✝⁴ : MeasurableSpace R
inst✝³ : BorelSpace R
inst✝² : SecondCountableTopology R
inst✝¹ : DenselyOrdered R
inst✝ : Nonempty R
hf : Tendsto (↑f) atTop atTop
inhabited_h : Inhabited R
⊢ f.measure univ = ∞

Tactic:
rw [← top_le_iff, ← f.measure_Ioi_of_tendsto_atTop_atTop hf default]

Goal state after:
R : Type u_1
inst✝⁸ : LinearOrder R
inst✝⁷ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁶ : OrderTopology R
inst✝⁵ : CompactIccSpace R
inst✝⁴ : MeasurableSpace R
inst✝³ : BorelSpace R
inst✝² : SecondCountableTopology R
inst✝¹ : DenselyOrdered R
inst✝ : Nonempty R
hf : Tendsto (↑f) atTop atTop
inhabited_h : Inhabited R
⊢ f.measure (Ioi default) ≤ f.measure univ
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
⊢ ⟪x, 0⟫ = 0

Tactic:
rw [← inner_conj_symm, inner_zero_left]; simp only [map_zero]

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
⊢ ∀ᵐ (x : X) ∂μ.restrict s,
    ↑↑(f + g) x = (↑↑f + ↑↑g) x → ↑↑(MemLp.toLp ↑↑(f + g) ⋯) x = ↑↑(MemLp.toLp ↑↑f ⋯ + MemLp.toLp ↑↑g ⋯) x

Tactic:
refine
    (Lp.coeFn_add (MemLp.toLp f ((Lp.memLp f).restrict s))
          (MemLp.toLp g ((Lp.memLp g).restrict s))).mp ?_

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
⊢ ∀ᵐ (x : X) ∂μ.restrict s,
    ↑↑(MemLp.toLp ↑↑f ⋯ + MemLp.toLp ↑↑g ⋯) x = (↑↑(MemLp.toLp ↑↑f ⋯) + ↑↑(MemLp.toLp ↑↑g ⋯)) x →
      ↑↑(f + g) x = (↑↑f + ↑↑g) x → ↑↑(MemLp.toLp ↑↑(f + g) ⋯) x = ↑↑(MemLp.toLp ↑↑f ⋯ + MemLp.toLp ↑↑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 : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x : E
⊢ ⟪x, x⟫ = ↑‖x‖ ^ 2

Tactic:
← inner_self_ofReal_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 : E
⊢ ↑(re ⟪x, x⟫) = ↑‖x‖ ^ 2
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
⊢ ⟪x + y, x + y⟫ + ⟪x - y, x - y⟫ = 2 * (⟪x, x⟫ + ⟪y, y⟫)

Tactic:
simp only [inner_add_add_self, inner_sub_sub_self]
  ring

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

Goal state before:
pp: E : Type u_4
𝕜 : Type u_7
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x : E
S : Set E
hS : Dense S
h : ∀ v ∈ S, ⟪x, v⟫ = 0
K : Submodule 𝕜 E := span 𝕜 S
hK : Dense ↑K
⊢ x = 0

Tactic:
have : (⟪x, ·⟫) = 0 := (continuous_const.inner continuous_id).ext_on
    hK continuous_const fun v ↦ Submodule.span_induction h (by simp)
      (by simp +contextual [inner_add_right]) (by simp +contextual [inner_smul_right])

Goal state after:
E : Type u_4
𝕜 : Type u_7
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x : E
S : Set E
hS : Dense S
h : ∀ v ∈ S, ⟪x, v⟫ = 0
K : Submodule 𝕜 E := span 𝕜 S
hK : Dense ↑K
this : (fun x_1 ↦ ⟪x, x_1⟫) = 0
⊢ x = 0
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case e_a
Y : Type u_2
E : Type u_3
F : Type u_4
X : Type u_5
G : Type u_6
𝕜 : Type u_7
inst✝¹¹ : TopologicalSpace X
inst✝¹⁰ : TopologicalSpace Y
inst✝⁹ : MeasurableSpace Y
inst✝⁸ : OpensMeasurableSpace Y
μ : Measure Y
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : NormedSpace 𝕜 E
L : F →L[𝕜] G →L[𝕜] E
f : X → Y → G
s : Set X
k : Set Y
g : Y → F
hk : IsCompact k
hf : ContinuousOn (uncurry f) (s ×ˢ univ)
hfs : ∀ (p : X) (x : Y), p ∈ s → x ∉ k → f p x = 0
hg : IntegrableOn g k μ
A : ∀ p ∈ s, Continuous[inst✝¹⁰, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (f p)
q : X
hq : q ∈ s
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : 0 < δ
hδ : ∫ (x : Y) in k, ‖L‖ * ‖g x‖ * δ ∂μ < ε
v : Set X
v_mem : v ∈ 𝓝[s] q
hv : ∀ p ∈ v, ∀ x ∈ k, ‖f p x - f q x‖ < δ
I : ∀ p ∈ s, IntegrableOn (fun y ↦ (L (g y)) (f p y)) k μ
p : X
hp : p ∈ v
h'p : p ∈ s
⊢ ∫ (x : Y), (L (g x)) (f p x) ∂μ = ∫ (x : Y) in k, (L (g x)) (f p x) ∂μ

Tactic:
refine (setIntegral_eq_integral_of_forall_compl_eq_zero (fun y hy ↦ ?_)).symm
        simp [hfs p y h'p hy]

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 y : E
h₀ : x ≠ 0
h₀' : ¬↑‖x‖ = 0
h : (↑‖y‖ / ↑‖x‖) • x = y
⊢ ⟪x, y⟫ = ↑‖x‖ * ↑‖y‖

Tactic:
· conv_lhs => rw [← h, inner_smul_right, inner_self_eq_norm_sq_to_K]
    field

Goal state after:
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 = 1 • ((rankOne 𝕜) x) x → ‖x‖ = 1

Tactic:
rw [← sub_eq_zero, ← sub_smul]

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 - 1) • ((rankOne 𝕜) x) x = 0 → ‖x‖ = 1
Module: Mathlib.Topology.Baire.CompleteMetrizable

Goal state before:
pp: 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∞), δ ≠ 0 → ∃ y r, 0 < r ∧ r ≤ B (n + 1) ∧ closedEBall y r ⊆ closedEBall x δ ∩ f n

Tactic:
δ

Goal state after:
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∞
⊢ δ ≠ 0 → ∃ y r, 0 < r ∧ r ≤ B (n + 1) ∧ closedEBall y r ⊆ closedEBall x δ ∩ f n
Module: Mathlib.Analysis.InnerProductSpace.Subspace

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 → Submodule 𝕜 E
hV : OrthogonalFamily 𝕜 (fun i ↦ ↥(V i)) fun i ↦ (V i).subtypeₗᵢ
v : Π₀ (i : ι), ↥(V i)
hv : v ∈ ((DFinsupp.lsum ℕ) fun i ↦ (V i).subtype).ker
⊢ v = 0

Tactic:
LinearMap.mem_ker

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 → Submodule 𝕜 E
hV : OrthogonalFamily 𝕜 (fun i ↦ ↥(V i)) fun i ↦ (V i).subtypeₗᵢ
v : Π₀ (i : ι), ↥(V i)
hv : ((DFinsupp.lsum ℕ) fun i ↦ (V i).subtype) v = 0
⊢ v = 0
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
y : F
x✝ : y ∈ Set.univ
⊢ y ∈ ⋃ n, closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (⇑f '' ball 0 ↑n)

Tactic:
rcases surj y with ⟨x, hx⟩

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
y : F
x✝ : y ∈ Set.univ
x : E
hx : f x = y
⊢ y ∈ ⋃ n, closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (⇑f '' ball 0 ↑n)
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: case inr.inr
p : ℝ≥0∞
ι : Type u_2
β : ι → Type u_4
inst✝¹ : Fintype ι
inst✝ : (i : ι) → PseudoEMetricSpace (β i)
f : PiLp p β
h : 0 < p.toReal
⊢ edist f f = 0

Tactic:
· simp [edist_eq_sum h, ENNReal.zero_rpow_of_pos h, ENNReal.zero_rpow_of_pos (inv_pos.2 <| h)]

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

Goal state before:
pp: case inr
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 : NormSMulClass 𝕜 α
inst✝ : NormSMulClass 𝕜 β
c : 𝕜
f : WithLp p (α × β)
hp : 1 ≤ p.toReal
hp0 : 0 < p.toReal
hpt : p ≠ ∞
⊢ ‖c • f.fst‖₊ ^ p.toReal + ‖c • f.snd‖₊ ^ p.toReal = (‖c‖₊ * ‖f.fst‖₊) ^ p.toReal + (‖c‖₊ * ‖f.snd‖₊) ^ p.toReal

Tactic:
nnnorm_smul,

Goal state after:
case inr
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 : NormSMulClass 𝕜 α
inst✝ : NormSMulClass 𝕜 β
c : 𝕜
f : WithLp p (α × β)
hp : 1 ≤ p.toReal
hp0 : 0 < p.toReal
hpt : p ≠ ∞
⊢ (‖c‖₊ * ‖f.fst‖₊) ^ p.toReal + ‖c • f.snd‖₊ ^ p.toReal = (‖c‖₊ * ‖f.fst‖₊) ^ p.toReal + (‖c‖₊ * ‖f.snd‖₊) ^ p.toReal
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: ι : Type u_2
β : ι → Type u_4
inst✝¹ : Fintype ι
inst✝ : (i : ι) → SeminormedAddCommGroup (β i)
x : PiLp 2 β
⊢ ↑‖x‖₊ = ↑(NNReal.sqrt (∑ i, ‖x.ofLp i‖₊ ^ 2))

Tactic:
by
    push_cast
    exact norm_eq_of_L2 x

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

Goal state before:
pp: case inr
a : ℝ
r : ℝ≥0
⊢ volume (closedEBall a ↑r) = 2 * ↑r

Tactic:
Metric.closedEBall_coe,

Goal state after:
case inr
a : ℝ
r : ℝ≥0
⊢ volume (closedBall a ↑r) = 2 * ↑r
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: case inl
𝕜 : Type u_1
ι : Type u_2
α : ι → Type u_3
β : ι → Type u_4
inst✝⁸ : Fintype ι
inst✝⁷ : Semiring 𝕜
inst✝⁶ : (i : ι) → SeminormedAddCommGroup (α i)
inst✝⁵ : (i : ι) → SeminormedAddCommGroup (β i)
inst✝⁴ : (i : ι) → Module 𝕜 (α i)
inst✝³ : (i : ι) → Module 𝕜 (β i)
c : 𝕜
ι' : Type u_5
inst✝² : Fintype ι'
E : Type u_6
inst✝¹ : SeminormedAddCommGroup E
inst✝ : Module 𝕜 E
e : ι ≃ ι'
hp : Fact (1 ≤ ∞)
__LinearEquiv✝ : (PiLp ∞ fun x ↦ E) ≃ₗ[𝕜] PiLp ∞ fun x ↦ E :=
  WithLp.linearEquiv ∞ 𝕜 (ι → E) ≪≫ₗ
    (LinearEquiv.piCongrLeft' 𝕜 (fun x ↦ E) e ≪≫ₗ (WithLp.linearEquiv ∞ 𝕜 (ι' → E)).symm)
x' : PiLp ∞ fun x ↦ E
⊢ ‖(WithLp.linearEquiv ∞ 𝕜 (ι → E) ≪≫ₗ
          (LinearEquiv.piCongrLeft' 𝕜 (fun x ↦ E) e ≪≫ₗ (WithLp.linearEquiv ∞ 𝕜 (ι' → E)).symm))
        x'‖ =
    ‖x'‖

Tactic:
simp_rw [norm_eq_ciSup]
      exact e.symm.iSup_congr fun _ => rfl

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

Goal state before:
pp: ι : Type u_1
inst✝ : Fintype ι
f : (ι → ℝ) →ₗ[ℝ] ι → ℝ
hf : LinearMap.det f ≠ 0
M : Matrix ι ι ℝ := LinearMap.toMatrix' f
⊢ Measure.map (⇑f) volume = ofReal |(LinearMap.det f)⁻¹| • volume

Tactic:
have A : LinearMap.det f = det M := by simp only [M, LinearMap.det_toMatrix']

Goal state after:
ι : Type u_1
inst✝ : Fintype ι
f : (ι → ℝ) →ₗ[ℝ] ι → ℝ
hf : LinearMap.det f ≠ 0
M : Matrix ι ι ℝ := LinearMap.toMatrix' f
A : LinearMap.det f = M.det
⊢ Measure.map (⇑f) volume = ofReal |(LinearMap.det f)⁻¹| • volume
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Basic

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
f g : α → ℝ
s : Set α
hf : Measurable f
hg : Measurable g
hs : MeasurableSet s
⊢ MeasurableSet {a | a.1 ∈ s}

Tactic:
exact measurable_fst hs

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Submodule ℝ E
hs : s ≠ ⊤
x : E
hx : x ∉ s
c : ℝ
cpos : 0 < c
cone : c < 1
A : Bornology.IsBounded (range fun n ↦ c ^ n • x)
m n : ℕ
⊢ m ≠ n → (Disjoint on fun n ↦ {c ^ n • x} + ↑s) m n

Tactic:
hmn

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Submodule ℝ E
hs : s ≠ ⊤
x : E
hx : x ∉ s
c : ℝ
cpos : 0 < c
cone : c < 1
A : Bornology.IsBounded (range fun n ↦ c ^ n • x)
m n : ℕ
hmn : m ≠ n
⊢ (Disjoint on fun n ↦ {c ^ n • x} + ↑s) m n
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁶ : NormedField K
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace K E
b : Basis ι K E
inst✝³ : LinearOrder K
inst✝² : IsStrictOrderedRing K
inst✝¹ : FloorRing K
inst✝ : Fintype ι
m : E
i : ι
⊢ (b.repr ↑(floor b m)) i = ↑⌊(b.repr m) i⌋

Tactic:
by
  classical simp only [floor, ← Int.cast_smul_eq_zsmul K, b.repr.map_smul, Finsupp.single_apply,
    Finset.sum_apply', Basis.repr_self, Finsupp.smul_single', mul_one, Finset.sum_ite_eq', coe_sum,
    Finset.mem_univ, if_true, coe_smul_of_tower, Basis.restrictScalars_apply, map_sum]

Goal state after:
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁶ : NormedField K
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace K E
b : Basis ι K E
inst✝³ : LinearOrder K
inst✝² : IsStrictOrderedRing K
inst✝¹ : FloorRing K
inst✝ : Fintype ι
m : E
h : m ∈ span ℤ (Set.range ⇑b)
i : ι
z : ℤ
hz : (algebraMap ℤ K) z = (b.repr m) i
⊢ ↑⌈(algebraMap ℤ K) z⌉ = (algebraMap ℤ K) z

Tactic:
exact congr_arg (Int.cast : ℤ → K) (Int.ceil_intCast z)

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

Goal state before:
pp: μ : Measure ℝ
inst✝ : NullSingletonClass μ
s : Set ℝ
p : ℝ → Prop
h : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ, x ∈ s ∩ Ioo a b → p x
⊢ ∀ᵐ (x : ℝ) ∂μ, x ∈ s → p x

Tactic:
let T : s × s → Set ℝ := fun p => Ioo p.1 p.2
  let u := ⋃ i : ↥s × ↥s, T i
  have hfinite : (s \ u).Finite := s.finite_sdiff_iUnion_Ioo'
  obtain ⟨A, A_count, hA⟩ :
    ∃ A : Set (↥s × ↥s), A.Countable ∧ ⋃ i ∈ A, T i = ⋃ i : ↥s × ↥s, T i :=
    isOpen_iUnion_countable _ fun p => isOpen_Ioo
  have M : ∀ᵐ x ∂μ, x ∉ s \ u := hfinite.countable.ae_notMem _
  have M' : ∀ᵐ x ∂μ, ∀ (i : ↥s × ↥s), i ∈ A → x ∈ s ∩ T i → p x := by
    rw [ae_ball_iff A_count]
    rintro ⟨⟨a, as⟩, ⟨b, bs⟩⟩ -
    change ∀ᵐ x : ℝ ∂μ, x ∈ s ∩ Ioo a b → p x
    rcases le_or_gt b a with (hba | hab)
    · simp only [Ioo_eq_empty_of_le hba, inter_empty, IsEmpty.forall_iff, eventually_true,
        mem_empty_iff_false]
    · exact h a b as bs hab
  filter_upwards [M, M'] with x hx h'x
  intro xs
  by_cases Hx : x ∈ ⋃ i : ↥s × ↥s, T i
  · rw [← hA] at Hx
    obtain ⟨p, pA, xp⟩ : ∃ p : ↥s × ↥s, p ∈ A ∧ x ∈ T p := by
      simpa only [mem_iUnion, exists_prop, SetCoe.exists, exists_and_right] using Hx
    apply h'x p pA ⟨xs, xp⟩
  · exact False.elim (hx ⟨xs, Hx⟩)

Goal state after:
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: K : Type u_1
inst✝⁹ : NormedField K
inst✝⁸ : LinearOrder K
inst✝⁷ : IsStrictOrderedRing K
inst✝⁶ : HasSolidNorm K
inst✝⁵ : FloorRing K
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace K E
inst✝² : FiniteDimensional K E
inst✝¹ : ProperSpace E
L : Submodule ℤ E
inst✝ : DiscreteTopology ↥L
hs : IsZLattice K L
s : Set E
h_incl : s ⊆ ↑L
h_span : span K s = span K ↑L
h_lind : LinearIndependent K Subtype.val
b : Basis { x // x ∈ s } K E := Basis.mk h_lind ⋯
this✝ : Fintype ↑s
this : DiscreteTopology ↥L.toAddSubgroup
⊢ (fundamentalDomain b ∩ ↑L.toAddSubgroup).Finite

Tactic:
exact Metric.finite_isBounded_inter_isClosed
        DiscreteTopology.isDiscrete (fundamentalDomain_isBounded b) inferInstance

Goal state after:
Module: Mathlib.Analysis.BoxIntegral.Partition.Basic

Goal state before:
pp: ι : Type u_1
I J : Box ι
π₁ π₂ : Prepartition I
⊢ J ∈ π₁ ⊓ π₂ ↔ ∃ J₁ ∈ π₁, ∃ J₂ ∈ π₂, ↑J = ↑J₁ ⊓ ↑J₂

Tactic:
by
  simp only [inf_def, mem_biUnion, mem_restrict]

Goal state after:
Module: Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction

Goal state before:
pp: ι : Type u_1
inst✝ : Fintype ι
I J : Box ι
⊢ J ∈ splitCenter I ↔ ∃ s, I.splitCenterBox s = J

Tactic:
simp [splitCenter]

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.PiL2

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_3
inst✝³ : RCLike 𝕜
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : Fintype ι
b : OrthonormalBasis ι 𝕜 E
x y : E
⊢ ∑ i, ⟪x, b i⟫ * ⟪b i, y⟫ = ⟪x, y⟫

Tactic:
have := congr_arg (innerSL 𝕜 x) (b.sum_repr y)

Goal state after:
ι : Type u_1
𝕜 : Type u_3
inst✝³ : RCLike 𝕜
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : Fintype ι
b : OrthonormalBasis ι 𝕜 E
x y : E
this : ((innerSL 𝕜) x) (∑ i, (b.repr y).ofLp i • b i) = ((innerSL 𝕜) x) y
⊢ ∑ i, ⟪x, b i⟫ * ⟪b i, y⟫ = ⟪x, y⟫
Module: Mathlib.Analysis.BoxIntegral.Partition.Split

Goal state before:
pp: ι : Type u_1
I : Box ι
i : ι
x : ℝ
⊢ ↑(I.splitLower i x) = ↑I ∩ {y | y i ≤ x}

Tactic:
by
  rw [splitLower, coe_mk']
  ext y
  simp only [mem_univ_pi, mem_Ioc, mem_inter_iff, mem_coe, mem_ofPred_eq, forall_and, ← Pi.le_def,
    le_update_iff, le_min_iff, and_assoc, and_forall_ne (p := fun j => y j ≤ upper I j) i, mem_def]
  rw [and_comm (a := y i ≤ x)]

Goal state after:
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: K : Type u_1
inst✝⁹ : NormedField K
inst✝⁸ : LinearOrder K
inst✝⁷ : IsStrictOrderedRing K
inst✝⁶ : HasSolidNorm K
inst✝⁵ : FloorRing K
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace K E
inst✝² : FiniteDimensional K E
inst✝¹ : ProperSpace E
L : Submodule ℤ E
inst✝ : DiscreteTopology ↥L
ι : Type u_3
hs : IsZLattice K L
b : Basis ι ℤ ↥L
x : ↥L
i✝ i : ι
⊢ Finsupp.single i 1 = (Finsupp.mapRange.linearMap (Algebra.linearMap ℤ K)) (Finsupp.single i 1)

Tactic:
Finsupp.mapRange.linearMap_apply,

Goal state after:
K : Type u_1
inst✝⁹ : NormedField K
inst✝⁸ : LinearOrder K
inst✝⁷ : IsStrictOrderedRing K
inst✝⁶ : HasSolidNorm K
inst✝⁵ : FloorRing K
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace K E
inst✝² : FiniteDimensional K E
inst✝¹ : ProperSpace E
L : Submodule ℤ E
inst✝ : DiscreteTopology ↥L
ι : Type u_3
hs : IsZLattice K L
b : Basis ι ℤ ↥L
x : ↥L
i✝ i : ι
⊢ Finsupp.single i 1 = Finsupp.mapRange ⇑(Algebra.linearMap ℤ K) ⋯ (Finsupp.single i 1)
Module: Mathlib.Analysis.BoxIntegral.Partition.Split

Goal state before:
pp: case refine_1
ι : Type u_1
I : Box ι
inst✝ : Finite ι
π : Prepartition I
t : Finset (ι × ℝ)
ht : ∀ (I_1 J : Box ι), J ∈ π.boxes → ∀ J' ∈ splitMany I_1 t, ¬Disjoint ↑J ↑J' → J' ≤ J
J : Box ι
hJ : J ∈ π
J₁ : Box ι
h₁ : J₁ ∈ splitMany I t
hne : ↑J ⊓ ↑J₁ ≠ ⊥
⊢ ∃ J', (∃ a, (a ∈ splitMany I t ∧ ↑a ⊆ π.iUnion) ∧ ↑J ⊓ ↑a = J') ∧ ↑J ⊓ ↑J₁ ≤ J'

Tactic:
refine ⟨_, ⟨J₁, ⟨h₁, Subset.trans ?_ (π.subset_iUnion hJ)⟩, rfl⟩, le_rfl⟩

Goal state after:
case refine_1
ι : Type u_1
I : Box ι
inst✝ : Finite ι
π : Prepartition I
t : Finset (ι × ℝ)
ht : ∀ (I_1 J : Box ι), J ∈ π.boxes → ∀ J' ∈ splitMany I_1 t, ¬Disjoint ↑J ↑J' → J' ≤ J
J : Box ι
hJ : J ∈ π
J₁ : Box ι
h₁ : J₁ ∈ splitMany I t
hne : ↑J ⊓ ↑J₁ ≠ ⊥
⊢ ↑J₁ ⊆ ↑J
Module: Mathlib.Analysis.InnerProductSpace.PiL2

Goal state before:
pp: 𝕜 : Type u_3
inst✝³ : RCLike 𝕜
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
v : Set E
inst✝ : FiniteDimensional 𝕜 E
hv : Orthonormal 𝕜 Subtype.val
u₀ : Set E
hu₀s : u₀ ⊇ v
hu₀ : Orthonormal 𝕜 Subtype.val
hu₀_max : ∀ u ⊇ u₀, Orthonormal 𝕜 Subtype.val → u = u₀
⊢ ∃ u b, v ⊆ ↑u ∧ ⇑b = Subtype.val

Tactic:
maximal_orthonormal_iff_orthogonalComplement_eq_bot hu₀

Goal state after:
𝕜 : Type u_3
inst✝³ : RCLike 𝕜
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
v : Set E
inst✝ : FiniteDimensional 𝕜 E
hv : Orthonormal 𝕜 Subtype.val
u₀ : Set E
hu₀s : u₀ ⊇ v
hu₀ : Orthonormal 𝕜 Subtype.val
hu₀_max : (span 𝕜 u₀)ᗮ = ⊥
⊢ ∃ u b, v ⊆ ↑u ∧ ⇑b = Subtype.val