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

Goal state before:
pp: case e'_3
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf_int : ∀ (n : ℕ), Integrable (f n) μ
hF_int : Integrable F μ
hf_tendsto : Tendsto (fun i ↦ ∫ (a : α), f i a ∂μ) atTop (𝓝 (∫ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Antitone fun i ↦ f i a
hf_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), F a ≤ f i a
f' : ℕ → α → ℝ := fun i a ↦ -f i a
F' : α → ℝ := fun a ↦ -F a
x✝ : ℕ
⊢ ∫ (a : α), f' x✝ a ∂μ = -∫ (a : α), f x✝ a ∂μ

pp: case e'_5
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf_int : ∀ (n : ℕ), Integrable (f n) μ
hF_int : Integrable F μ
hf_tendsto : Tendsto (fun i ↦ ∫ (a : α), f i a ∂μ) atTop (𝓝 (∫ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Antitone fun i ↦ f i a
hf_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), F a ≤ f i a
f' : ℕ → α → ℝ := fun i a ↦ -f i a
F' : α → ℝ := fun a ↦ -F a
⊢ ∫ (a : α), F' a ∂μ = -∫ (a : α), F a ∂μ

Tactic:
· rw [integral_neg]

Goal state after:
case e'_5
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf_int : ∀ (n : ℕ), Integrable (f n) μ
hF_int : Integrable F μ
hf_tendsto : Tendsto (fun i ↦ ∫ (a : α), f i a ∂μ) atTop (𝓝 (∫ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Antitone fun i ↦ f i a
hf_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), F a ≤ f i a
f' : ℕ → α → ℝ := fun i a ↦ -f i a
F' : α → ℝ := fun a ↦ -F a
⊢ ∫ (a : α), F' a ∂μ = -∫ (a : α), F a ∂μ
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: case e'_5
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf_int : ∀ (n : ℕ), Integrable (f n) μ
hF_int : Integrable F μ
hf_tendsto : Tendsto (fun i ↦ ∫ (a : α), f i a ∂μ) atTop (𝓝 (∫ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Antitone fun i ↦ f i a
hf_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), F a ≤ f i a
f' : ℕ → α → ℝ := fun i a ↦ -f i a
F' : α → ℝ := fun a ↦ -F a
⊢ ∫ (a : α), F' a ∂μ = -∫ (a : α), F a ∂μ

Tactic:
· rw [integral_neg]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.SumMeasure

Goal state before:
pp: X : Type u_2
E : Type u_3
mX : MeasurableSpace X
inst✝⁴ : NormedAddCommGroup E
f : X → E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : MeasurableSingletonClass X
μ : Measure X
inst✝ : Fintype X
hf : Integrable f μ
⊢ ∫ (x : X), f x ∂μ = ∫ (x : X) in univ, f x ∂μ

pp: X : Type u_2
E : Type u_3
mX : MeasurableSpace X
inst✝⁴ : NormedAddCommGroup E
f : X → E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : MeasurableSingletonClass X
μ : Measure X
inst✝ : Fintype X
hf : Integrable f μ
⊢ IntegrableOn f (↑Finset.univ) μ

Tactic:
Measure.restrict_univ

Goal state after:
X : Type u_2
E : Type u_3
mX : MeasurableSpace X
inst✝⁴ : NormedAddCommGroup E
f : X → E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : MeasurableSingletonClass X
μ : Measure X
inst✝ : Fintype X
hf : Integrable f μ
⊢ ∫ (x : X), f x ∂μ = ∫ (x : X), f x ∂μ

X : Type u_2
E : Type u_3
mX : MeasurableSpace X
inst✝⁴ : NormedAddCommGroup E
f : X → E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : MeasurableSingletonClass X
μ : Measure X
inst✝ : Fintype X
hf : Integrable f μ
⊢ IntegrableOn f (↑Finset.univ) μ
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
F : Type w
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : CompleteSpace 𝕜
PE : Type u_1
PF : Type u_2
inst✝⁴ : MetricSpace PE
inst✝³ : NormedAddTorsor E PE
inst✝² : MetricSpace PF
inst✝¹ : NormedAddTorsor F PF
inst✝ : FiniteDimensional 𝕜 E
f : PE →ᵃ[𝕜] PF
fL : E →L[𝕜] F := LinearMap.toContinuousLinearMap f.linear
x y : PE
⊢ dist (f x) (f y) ≤ ↑‖fL‖₊ * dist x y

Tactic:
rw [NormedAddTorsor.dist_eq_norm', NormedAddTorsor.dist_eq_norm', ← f.linearMap_vsub]

Goal state after:
𝕜 : Type u
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
F : Type w
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : CompleteSpace 𝕜
PE : Type u_1
PF : Type u_2
inst✝⁴ : MetricSpace PE
inst✝³ : NormedAddTorsor E PE
inst✝² : MetricSpace PF
inst✝¹ : NormedAddTorsor F PF
inst✝ : FiniteDimensional 𝕜 E
f : PE →ᵃ[𝕜] PF
fL : E →L[𝕜] F := LinearMap.toContinuousLinearMap f.linear
x y : PE
⊢ ‖f.linear (x -ᵥ y)‖ ≤ ↑‖fL‖₊ * ‖x -ᵥ y‖
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
F : Type w
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : CompleteSpace 𝕜
inst✝¹ : FiniteDimensional 𝕜 E
inst✝ : SecondCountableTopology F
d : ℕ := finrank 𝕜 E
⊢ SecondCountableTopology (E →L[𝕜] F)

Tactic:
let e₁ : E ≃L[𝕜] Fin d → 𝕜 :=
    ContinuousLinearEquiv.ofFinrankEq (finrank_fin_fun 𝕜).symm

Goal state after:
𝕜 : Type u
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
F : Type w
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : CompleteSpace 𝕜
inst✝¹ : FiniteDimensional 𝕜 E
inst✝ : SecondCountableTopology F
d : ℕ := finrank 𝕜 E
e₁ : E ≃L[𝕜] Fin d → 𝕜 := ContinuousLinearEquiv.ofFinrankEq ⋯
⊢ SecondCountableTopology (E →L[𝕜] F)
Module: Mathlib.Topology.Order.LeftRightLim

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

Tactic:
filter_upwards [Ioo_mem_nhdsLT au] with c hc

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
s : Set β
s_mem : s ∈ 𝓝 (leftLim f a)
s_closed : IsClosed[inst✝³] s
h' : (𝓝[<] a).NeBot
b : α
hb : b ∈ Iio a
u : α
au : u < a
hu : Ioo u a ⊆ {x | f x ∈ s}
c : α
hc : c ∈ Ioo u a
⊢ leftLim f c ∈ id s
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))
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}
⊢ ∀ᶠ (x : α) in 𝓝[<] a, rightLim f x ∈ id s

Tactic:
filter_upwards [Ioo_mem_nhdsLT au] with c hc

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))
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
⊢ rightLim f c ∈ id 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
μ : Measure X
⊢ ∫ (x : X) in univ, f x ∂μ = ∫ (x : X), f x ∂μ

Tactic:
Measure.restrict_univ

Goal state after:
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
μ : Measure X
⊢ ∫ (x : X), f x ∂μ = ∫ (x : X), f x ∂μ
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
inst✝¹ : CompleteSpace E
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
inst✝ : Countable ι
f : ι → α → E
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hf' : ∑' (i : ι), ∫⁻ (a : α), ‖f i a‖ₑ ∂μ ≠ ∞
⊢ (setToFun μ T hT fun a ↦ ∑' (i : ι), f i a) = ∑' (i : ι), setToFun μ T hT (f i)

Tactic:
by_cases hF : CompleteSpace F; swap
  · simp [setToFun, hF]
  have hf'' i : AEMeasurable (‖f i ·‖ₑ) μ := (hf i).enorm
  have hhh : ∀ᵐ a : α ∂μ, Summable fun n => (‖f n a‖₊ : ℝ) := by
    rw [← lintegral_tsum hf''] at hf'
    refine (ae_lt_top' (AEMeasurable.tsum hf'') hf').mono ?_
    intro x hx
    rw [← ENNReal.tsum_coe_ne_top_iff_summable_coe]
    exact hx.ne
  convert!
    (MeasureTheory.hasSum_setToFun_of_dominated_convergence hT (fun i a => ‖f i a‖₊) hf _ hhh ⟨_, _⟩
        _).tsum_eq.symm
  · intro n
    filter_upwards with x
    rfl
  · fun_prop
  · dsimp [HasFiniteIntegral]
    have : ∫⁻ a, ∑' n, ‖f n a‖ₑ ∂μ < ⊤ := by rwa [lintegral_tsum hf'', lt_top_iff_ne_top]
    convert! this using 1
    apply lintegral_congr_ae
    simp_rw [← coe_nnnorm, ← NNReal.coe_tsum, enorm_eq_nnnorm, NNReal.nnnorm_eq]
    filter_upwards [hhh] with a ha
    exact ENNReal.coe_tsum (NNReal.summable_coe.mp ha)
  · filter_upwards [hhh] with x hx
    exact hx.of_norm.hasSum

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

Goal state before:
pp: R : Type u_1
inst✝⁴ : LinearOrder R
inst✝³ : TopologicalSpace R
f : StieltjesFunction R
inst✝² : OrderTopology R
inst✝¹ : CompactIccSpace R
inst✝ : DenselyOrdered R
a b : R
hab : a < b
s : ℕ → Set R
hs : Ioc a b ⊆ ⋃ i, s i
ε : ℝ≥0
εpos : 0 < ε
h : ∑' (i : ℕ), f.length (s i) < ∞
δ : ℝ≥0 := ε / 2
δpos : 0 < ↑δ
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑δ
a' : R
ha' : ↑f a' - ↑f a < ↑δ
aa' : a < a'
this : Nonempty R
⊢ ofReal (↑f b - ↑f a) ≤ ∑' (i : ℕ), f.length (s i) + ↑ε

Tactic:
have : ∀ i, ∃ p : R × R, Icc a' b ∩ s i ⊆ Iotop p.1 p.2 ∧
      (ofReal (f p.2 - f p.1) : ℝ≥0∞) < f.length (s i) + ε' i := by
    intro i
    have hl :=
      ENNReal.lt_add_right ((ENNReal.le_tsum i).trans_lt h).ne (ENNReal.coe_ne_zero.2 (ε'0 i).ne')
    conv at hl =>
      lhs
      rw [length_eq]
    simp only [iInf_lt_iff, exists_prop] at hl
    rcases hl with ⟨p, q', spq, hq'⟩
    have A : Icc a' b ∩ s i ⊆ Ioc p q' := by
      rintro x ⟨hx, h'x⟩
      apply spq
      simp [h'x, notMem_botSet_of_lt (aa'.trans_le hx.1)]
    by_cases htq' : IsTop q'
    · refine ⟨(p, q'), ?_, hq'⟩
      rintro x hx
      simp only [Iotop, htq', ↓reduceIte, mem_Ioc]
      exact ⟨(A hx).1, htq' _⟩
    have : (𝓝[>] q').NeBot := by simp [Filter.neBot_iff, nhdsGT_eq_bot_iff, htq', not_covBy]
    have : ContinuousWithinAt (fun r => ofReal (f r - f p)) (Ioi q') q' := by
      apply ENNReal.continuous_ofReal.continuousAt.comp_continuousWithinAt
      refine ContinuousWithinAt.sub ?_ continuousWithinAt_const
      exact (f.right_continuous q').mono Ioi_subset_Ici_self
    rcases (((tendsto_order.1 this).2 _ hq').and self_mem_nhdsWithin).exists with ⟨q, hq, q'q⟩
    exact ⟨⟨p, q⟩, A.trans ((Ioc_subset_Ioo_right q'q).trans Ioo_subset_Iotop), hq⟩

Goal state after:
R : Type u_1
inst✝⁴ : LinearOrder R
inst✝³ : TopologicalSpace R
f : StieltjesFunction R
inst✝² : OrderTopology R
inst✝¹ : CompactIccSpace R
inst✝ : DenselyOrdered R
a b : R
hab : a < b
s : ℕ → Set R
hs : Ioc a b ⊆ ⋃ i, s i
ε : ℝ≥0
εpos : 0 < ε
h : ∑' (i : ℕ), f.length (s i) < ∞
δ : ℝ≥0 := ε / 2
δpos : 0 < ↑δ
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑δ
a' : R
ha' : ↑f a' - ↑f a < ↑δ
aa' : a < a'
this✝ : Nonempty R
this : ∀ (i : ℕ), ∃ p, Icc a' b ∩ s i ⊆ Iotop p.1 p.2 ∧ ofReal (↑f p.2 - ↑f p.1) < f.length (s i) + ↑(ε' i)
⊢ ofReal (↑f b - ↑f a) ≤ ∑' (i : ℕ), f.length (s i) + ↑ε
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:
by rfl

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
hfi : Integrable f μ
h_meas : NullMeasurableSet {x | 0 ≤ f x} μ
x : X
hx : x ∈ {x | 0 ≤ f x}
⊢ 0 ≤ f x

Tactic:
exact hx

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

Goal state before:
pp: G : Type u_1
α : Type u_3
inst✝⁵ : Group G
inst✝⁴ : MulAction G α
inst✝³ : MeasurableSpace α
s t : Set α
μ : Measure α
inst✝² : MeasurableConstSMul G α
inst✝¹ : SMulInvariantMeasure G α μ
inst✝ : Countable G
hs : IsFundamentalDomain G s μ
htm : NullMeasurableSet t μ
ht : ∀ x ∈ t, ∀ y ∈ t, ∀ (g : G), g ≠ 1 → g • x ≠ y
g₁ g₂ : G
hne : g₁ ≠ g₂
x : α
hx : x ∈ t
y : α
hy : y ∈ t
hxy : g₂ • y = g₁ • x
⊢ g₂⁻¹ • g₁ • x = y

Tactic:
← hxy,

Goal state after:
G : Type u_1
α : Type u_3
inst✝⁵ : Group G
inst✝⁴ : MulAction G α
inst✝³ : MeasurableSpace α
s t : Set α
μ : Measure α
inst✝² : MeasurableConstSMul G α
inst✝¹ : SMulInvariantMeasure G α μ
inst✝ : Countable G
hs : IsFundamentalDomain G s μ
htm : NullMeasurableSet t μ
ht : ∀ x ∈ t, ∀ y ∈ t, ∀ (g : G), g ≠ 1 → g • x ≠ y
g₁ g₂ : G
hne : g₁ ≠ g₂
x : α
hx : x ∈ t
y : α
hy : y ∈ t
hxy : g₂ • y = g₁ • x
⊢ g₂⁻¹ • g₂ • y = y
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case refine_2
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
⊢ ∃ i, f.measure (Ioc (u i) a) ≠ ∞

Tactic:
· exact ⟨0, by simpa only [measure_Ioc] using ENNReal.ofReal_ne_top⟩

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
this : ∀ K ⊆ ⋃ g ∈ s, (fun x ↦ g * x) ⁻¹' V, index K V ≤ {g ∈ s | ((fun x ↦ g * x) ⁻¹' V ∩ K).Nonempty}.card
g₁ : G
a✝ : g₁ ∈ s
g₂ : G
h2g₂ : g₂ ∈ K₁.carrier
g₃ : G
h2g₃ : g₃ ∈ K₂.carrier
h1g₃ : g₁ * g₃ ∈ V
h1g₂ : g₁ * g₂ ∈ V
⊢ False

Tactic:
refine h.le_bot (?_ : g₁⁻¹ ∈ _)

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
g₁ : G
a✝ : g₁ ∈ s
g₂ : G
h2g₂ : g₂ ∈ K₁.carrier
g₃ : G
h2g₃ : g₃ ∈ K₂.carrier
h1g₃ : g₁ * g₃ ∈ V
h1g₂ : g₁ * g₂ ∈ V
⊢ g₁⁻¹ ∈ K₁.carrier * V⁻¹ ⊓ K₂.carrier * V⁻¹
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:
by
  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.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : IsOrderedModule ℝ E
μ : Measure X
f : X → E
s : Set X
inst✝¹ : ClosedIciTopology E
inst✝ : CompleteSpace E
c : E
hs : MeasurableSet s
hμs : μ s ≠ ∞
hf : ∀ x ∈ s, c ≤ f x
hfint : IntegrableOn (fun x ↦ f x) s μ
⊢ μ.real s • c ≤ ∫ (x : X) in s, f x ∂μ

Tactic:
rw [← setIntegral_const c]

Goal state after:
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : IsOrderedModule ℝ E
μ : Measure X
f : X → E
s : Set X
inst✝¹ : ClosedIciTopology E
inst✝ : CompleteSpace E
c : E
hs : MeasurableSet s
hμs : μ s ≠ ∞
hf : ∀ x ∈ s, c ≤ f x
hfint : IntegrableOn (fun x ↦ f x) s μ
⊢ ∫ (x : X) in s, c ∂μ ≤ ∫ (x : X) in s, f x ∂μ
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
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
hf : Tendsto (↑f) atBot atBot
x : R
this✝ : Nonempty R
val✝ : NoBotOrder R
this : NoMinOrder R
⊢ f.measure (Iio x) = ∞

Tactic:
obtain ⟨y, hy⟩ : ∃ y, y < x := exists_lt x

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
hf : Tendsto (↑f) atBot atBot
x : R
this✝ : Nonempty R
val✝ : NoBotOrder R
this : NoMinOrder R
y : R
hy : y < x
⊢ f.measure (Iio x) = ∞
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.carrier K₂.carrier
h₂ : IsClosed[inst✝¹] K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen[inst✝¹] U₁
h1U₂ : IsOpen[inst✝¹] U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
⊢ chaar K₀ (K₁ ⊔ K₂) = chaar K₀ K₁ + chaar K₀ K₂

Tactic:
rcases compact_open_separated_mul_right K₁.2 h1U₁ h2U₁ with ⟨L₁, h1L₁, h2L₁⟩

Goal state after:
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.carrier K₂.carrier
h₂ : IsClosed[inst✝¹] K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen[inst✝¹] U₁
h1U₂ : IsOpen[inst✝¹] U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
h2L₁ : K₁.carrier * L₁ ⊆ U₁
⊢ chaar K₀ (K₁ ⊔ K₂) = chaar K₀ K₁ + chaar K₀ K₂
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
h₀ h₀' : x ≠ 0
⊢ ⟪x, y⟫ = ↑‖x‖ * ↑‖y‖ ↔ (↑‖y‖ / ↑‖x‖) • x = y

Tactic:
← norm_ne_zero_iff,

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
h₀ : x ≠ 0
h₀' : ‖x‖ ≠ 0
⊢ ⟪x, y⟫ = ↑‖x‖ * ↑‖y‖ ↔ (↑‖y‖ / ↑‖x‖) • x = y
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
⊢ (⟪b, b⟫_𝕜 / ⟪b, b⟫_𝕜) • a = (1 / ⟪b, b⟫_𝕜) • ⟪b, b⟫_𝕜 • a

Tactic:
simp only [smul_smul]; ring_nf

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
h₃ : a = (⟪d, b⟫_𝕜 / ⟪b, b⟫_𝕜) • c
⊢ (⟪a, a⟫_𝕜 / ⟪a, a⟫_𝕜) • b = (1 / ⟪a, a⟫_𝕜) • ⟪a, a⟫_𝕜 • b

Tactic:
simp only [smul_smul]; ring_nf

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
inst✝ : FiniteDimensional 𝕜 ↥K
⊢ LinearMap.det ↑K.reflection.toLinearEquiv = (-1) ^ finrank 𝕜 ↥Kᗮ

Tactic:
by
  by_cases hK : FiniteDimensional 𝕜 Kᗮ
  swap
  · rw [finrank_of_infinite_dimensional hK, pow_zero, LinearMap.det_eq_one_of_finrank_eq_zero]
    exact finrank_of_infinite_dimensional fun h ↦ hK (h.finiteDimensional_submodule _)
  let e := K.prodEquivOfIsCompl _ K.isCompl_orthogonal
  let b := (finBasis 𝕜 K).prod (finBasis 𝕜 Kᗮ)
  have : LinearMap.toMatrix b b (e.symm ∘ₗ K.reflection.toLinearMap ∘ₗ e.symm.symm) =
      Matrix.fromBlocks 1 0 0 (-1) := by
    ext (_ | _) (_ | _) <;>
    simp [LinearMap.toMatrix_apply, b, Matrix.one_apply, Finsupp.single_apply, e, eq_comm,
      reflection_mem_subspace_eq_self, reflection_mem_subspace_orthogonalComplement_eq_neg]
  rw [← LinearMap.det_conj _ e.symm, ← LinearMap.det_toMatrix b, this, Matrix.det_fromBlocks_zero₂₁,
    Matrix.det_one, one_mul, Matrix.det_neg, Fintype.card_fin, Matrix.det_one, mul_one]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
inst✝ : K.HasOrthogonalProjection
v : E
⊢ ‖K.orthogonalProjectionOnto‖ * ‖v‖ ≤ 1 * ‖v‖

Tactic:
by gcongr; exact orthogonalProjectionOnto_norm_le K

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

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NontriviallyNormedField 𝕜'
σ : 𝕜 →+* 𝕜'
E : Type u_3
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜' F
f : E →SL[σ] F
σ' : 𝕜' →+* 𝕜
inst✝⁴ : RingHomInvPair σ σ'
inst✝³ : RingHomIsometric σ
inst✝² : RingHomIsometric σ'
inst✝¹ : CompleteSpace F
inst✝ : CompleteSpace E
surj : Surjective ⇑f
C : ℝ
C0 : C ≥ 0
g : F → E
hg : ∀ (y : F), dist (f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖
h : F → F := fun y ↦ y - f (g y)
y : F
⊢ ‖h y‖ ≤ 1 / 2 * ‖y‖

Tactic:
rw [← dist_eq_norm, dist_comm]

Goal state after:
𝕜 : Type u_1
𝕜' : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NontriviallyNormedField 𝕜'
σ : 𝕜 →+* 𝕜'
E : Type u_3
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜' F
f : E →SL[σ] F
σ' : 𝕜' →+* 𝕜
inst✝⁴ : RingHomInvPair σ σ'
inst✝³ : RingHomIsometric σ
inst✝² : RingHomIsometric σ'
inst✝¹ : CompleteSpace F
inst✝ : CompleteSpace E
surj : Surjective ⇑f
C : ℝ
C0 : C ≥ 0
g : F → E
hg : ∀ (y : F), dist (f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖
h : F → F := fun y ↦ y - f (g y)
y : F
⊢ dist (f (g y)) y ≤ 1 / 2 * ‖y‖
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✝¹ : PseudoEMetricSpace α
inst✝ : PseudoEMetricSpace β
f g h : WithLp p (α × β)
hp : 1 ≤ p.toReal
⊢ edist f h ≤ edist f g + edist g h

Tactic:
simp only [prod_edist_eq_add (zero_lt_one.trans_le hp)]

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✝¹ : PseudoEMetricSpace α
inst✝ : PseudoEMetricSpace β
f g h : WithLp p (α × β)
hp : 1 ≤ p.toReal
⊢ (edist f.fst h.fst ^ p.toReal + edist f.snd h.snd ^ p.toReal) ^ (1 / p.toReal) ≤
    (edist f.fst g.fst ^ p.toReal + edist f.snd g.snd ^ p.toReal) ^ (1 / p.toReal) +
      (edist g.fst h.fst ^ p.toReal + edist g.snd h.snd ^ p.toReal) ^ (1 / p.toReal)