mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Algebra.Module.FiniteDimension

Goal state before:
pp: case hxy
𝕜 : Type u_4
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : CompleteSpace 𝕜
Eᵤ : Type u_6
inst✝⁵ : AddCommGroup Eᵤ
inst✝⁴ : Module 𝕜 Eᵤ
inst✝³ : UniformSpace Eᵤ
inst✝² : T2Space Eᵤ
inst✝¹ : IsUniformAddGroup Eᵤ
inst✝ : ContinuousSMul 𝕜 Eᵤ
U : Set Eᵤ
hU_nhds : U ∈ 𝓝 0
hU_tb : TotallyBounded U
F : Set Eᵤ
hF_finite : F.Finite
M : Submodule 𝕜 Eᵤ := Submodule.span 𝕜 F
this : FiniteDimensional 𝕜 ↥M := Finite.span_of_finite 𝕜 hF_finite
n : ℕ
c : 𝕜ˣ
hc0 : 0 < ‖↑c‖
hc1 : ‖↑c‖ < 1
hc_ne : ↑c ≠ 0
hF_cover : U ⊆ ⋃ y ∈ F, y +ᵥ ↑c • U
h_cover : U ⊆ ↑M + ↑c • U
ih : U ⊆ ↑M + ↑c ^ n • U
⊢ ↑M + ↑c ^ n • ↑M = ↑M

Tactic:
simp [← Units.val_pow_eq_pow_val, ← Units.smul_def]

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: 𝕜 : Type u
ι : Type v
E₁ : ι → Type wE₁
G : Type wG
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁵ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁴ : SeminormedAddCommGroup G
inst✝³ : NormedSpace 𝕜 G
inst✝² : Fintype ι
α : Type u_1
inst✝¹ : Fintype α
f : ContinuousMultilinearMap 𝕜 E₁ G
s : Set ι
e : α ≃ ↑s
inst✝ : DecidablePred fun x ↦ x ∈ s
x : (i : ι) → E₁ i
⊢ ‖f‖ * ∏ _i, ‖x‖ = ‖f‖ * ‖x‖ ^ Fintype.card { a // a ∉ s }

Tactic:
by rw [prod_const, card_univ]

Goal state after:
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: case add
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
hp_pos : p ≠ 0
hp_ne_top : p ≠ ∞
P : ↥(simpleFunc E p μ) → Prop
indicatorConst : ∀ (c : E) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ∞), P (simpleFunc.indicatorConst p hs ⋯ c)
add :
  ∀ ⦃f g : α →ₛ E⦄ (hf : MemLp (⇑f) p μ) (hg : MemLp (⇑g) p μ),
    Disjoint (support ⇑f) (support ⇑g) → P (f.toLp hf) → P (g.toLp hg) → P (f.toLp hf + g.toLp hg)
⊢ ∀ ⦃f g : α →ₛ E⦄,
    Disjoint (support ⇑f) (support ⇑g) →
      (∀ (hf : MemLp (⇑f) p μ), P (f.toLp hf)) →
        (∀ (hf : MemLp (⇑g) p μ), P (g.toLp hf)) → ∀ (hf : MemLp (⇑(f + g)) p μ), P ((f + g).toLp hf)

Tactic:
intro f g hfg hf hg hfg'
    obtain ⟨hf', hg'⟩ : MemLp f p μ ∧ MemLp g p μ :=
      (memLp_add_of_disjoint hfg f.stronglyMeasurable g.stronglyMeasurable).mp hfg'
    exact add hf' hg' hfg (hf hf') (hg hg')

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:
pp: case pos
α : Type u_1
F : Type u_3
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝² : PartialOrder F
inst✝¹ : IsOrderedAddMonoid F
inst✝ : IsOrderedModule ℝ F
ν : Measure α
f : α →ₛ F
hf : 0 ≤ᵐ[ν] ⇑f
hμν : μ ≤ ν
hfν : Integrable (⇑f) ν
x : α
hx : 0 ≤ f x
⊢ μ.real (⇑f ⁻¹' {f x}) • f x ≤ ν.real (⇑f ⁻¹' {f x}) • f x

Tactic:
obtain (hx | hx) := hx.eq_or_lt
    · simp [← hx]
    simp only [measureReal_def]
    gcongr
    exact integrable_iff.mp hfν (f x) hx.ne' |>.ne

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ↥(α →₁ₛ[μ] ℝ)
⊢ integral f = ‖posPart f‖ - ‖negPart f‖

Tactic:
have ae_eq₁ : (toSimpleFunc f).posPart =ᵐ[μ] (toSimpleFunc (posPart f)).map norm := by
    filter_upwards [posPart_toSimpleFunc f] with _ h
    rw [SimpleFunc.map_apply, h]
    conv_lhs => rw [← SimpleFunc.posPart_map_norm, SimpleFunc.map_apply]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ↥(α →₁ₛ[μ] ℝ)
ae_eq₁ : ⇑(toSimpleFunc f).posPart =ᵐ[μ] ⇑(SimpleFunc.map norm (toSimpleFunc (posPart f)))
⊢ integral f = ‖posPart f‖ - ‖negPart f‖
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: 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
h_add : FinMeasAdditive μ T
f g : α →ₛ E
hf : Integrable (⇑f) μ
hg : Integrable (⇑g) μ
hp_pair : Integrable (⇑(f.pair g)) μ
⊢ setToSimpleFunc T (f + g) = ∑ x ∈ (f.pair g).range, (T (⇑(f.pair g) ⁻¹' {x})) (x.1 + x.2)

Tactic:
rw [add_eq_map₂, map_setToSimpleFunc T h_add hp_pair]

Goal state after:
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
h_add : FinMeasAdditive μ T
f g : α →ₛ E
hf : Integrable (⇑f) μ
hg : Integrable (⇑g) μ
hp_pair : Integrable (⇑(f.pair g)) μ
⊢ 0.1 + 0.2 = 0
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_6
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝⁵ : NormedRing 𝕜
inst✝⁴ : Module 𝕜 E
inst✝³ : Module 𝕜 F
inst✝² : IsBoundedSMul 𝕜 E
inst✝¹ : IsBoundedSMul 𝕜 F
inst✝ : CompleteSpace F
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
h_smul : ∀ (c : 𝕜) (s : Set α) (x : E), (T s) (c • x) = c • (T s) x
f : ↥(α →₁ₛ[μ] E)
⊢ (setToL1' 𝕜 hT h_smul) ↑f = (setToL1SCLM α E μ hT) f

Tactic:
by
  apply ContinuousLinearMap.extend_eq _ _ simpleFunc.isUniformInducing
  · exact simpleFunc.denseRange one_ne_top

Goal state after:
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: case neg
𝕜 : Type u
inst✝³ : NontriviallyNormedField 𝕜
E : Type v
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CompleteSpace 𝕜
h : ¬∃ s, Nonempty (Basis (↥s) 𝕜 E)
⊢ Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun f ↦ LinearMap.det ↑f

Tactic:
LinearMap.det

Goal state after:
case neg
𝕜 : Type u
inst✝³ : NontriviallyNormedField 𝕜
E : Type v
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CompleteSpace 𝕜
h : ¬∃ s, Nonempty (Basis (↥s) 𝕜 E)
⊢ Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun f ↦
    (if H : ∃ s, Nonempty (Basis (↥s) 𝕜 E) then LinearMap.detAux (Trunc.mk ⋯.some) else 1) ↑f
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : PseudoMetricSpace β
f g : α →ᵇ β
⊢ sInf {C | 0 ≤ C ∧ ∀ (x : α), dist (f x) (g x) ≤ C} =
    sInf {x | ∃ (h : 0 ≤ x), NNReal.mk x h ∈ {C | ∀ (x : α), nndist (f x) (g x) ≤ C}}

Tactic:
mem_ofPred_eq,

Goal state after:
α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : PseudoMetricSpace β
f g : α →ᵇ β
⊢ sInf {C | 0 ≤ C ∧ ∀ (x : α), dist (f x) (g x) ≤ C} =
    sInf {x | ∃ (h : 0 ≤ x), ∀ (x_1 : α), nndist (f x_1) (g x_1) ≤ NNReal.mk x ⋯}
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: case neg
α : Type u_1
G : Type u_5
inst✝ : NormedAddCommGroup G
m : MeasurableSpace α
μ μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ∞
hμ'_le : μ' ≤ c' • μ
hc'0 : ¬c' = 0
⊢ Continuous fun f ↦ Integrable.toL1 ↑↑f ⋯

Tactic:
Metric.continuous_iff

Goal state after:
case neg
α : Type u_1
G : Type u_5
inst✝ : NormedAddCommGroup G
m : MeasurableSpace α
μ μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ∞
hμ'_le : μ' ≤ c' • μ
hc'0 : ¬c' = 0
⊢ ∀ (b : ↥(Lp G 1 μ)),
    ∀ ε > 0, ∃ δ > 0, ∀ (a : ↥(Lp G 1 μ)), dist a b < δ → dist (Integrable.toL1 ↑↑a ⋯) (Integrable.toL1 ↑↑b ⋯) < ε
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
x : α
x_out : x ∉ {x | infEDist x E < ENNReal.ofReal δ}
⊢ thickenedIndicatorAux δ E x = 0

Tactic:
mem_ofPred_eq,

Goal state after:
α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
x : α
x_out : ¬infEDist x E < ENNReal.ofReal δ
⊢ thickenedIndicatorAux δ E x = 0
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
f : X → E
μ : Measure X
ι : Type u_5
inst✝¹ : Preorder ι
inst✝ : atTop.IsCountablyGenerated
s : ι → Set X
hsm : ∀ (i : ι), NullMeasurableSet (s i) μ
h_mono : Monotone s
hne : atTop.NeBot
this : IsDirectedOrder ι
S : Set X := ⋃ i, s i
hfi : IntegrableOn f S μ
hfi' : ∫⁻ (x : X) in S, ↑‖f x‖₊ ∂μ < ∞
⊢ Tendsto (fun i ↦ ∫ (x : X) in s i, f x ∂μ) atTop (𝓝 (∫ (x : X) in S, f x ∂μ))

Tactic:
have hSm : NullMeasurableSet S μ := MeasurableSet.iUnion_of_monotone h_mono hsm

Goal state after:
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
f : X → E
μ : Measure X
ι : Type u_5
inst✝¹ : Preorder ι
inst✝ : atTop.IsCountablyGenerated
s : ι → Set X
hsm : ∀ (i : ι), NullMeasurableSet (s i) μ
h_mono : Monotone s
hne : atTop.NeBot
this : IsDirectedOrder ι
S : Set X := ⋃ i, s i
hfi : IntegrableOn f S μ
hfi' : ∫⁻ (x : X) in S, ↑‖f x‖₊ ∂μ < ∞
hSm : NullMeasurableSet S μ
⊢ Tendsto (fun i ↦ ∫ (x : X) in s i, f x ∂μ) atTop (𝓝 (∫ (x : X) in S, f x ∂μ))
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: G : Type w
inst✝ : TopologicalSpace G
μ : Content G
U : Opens G
hU : μ.innerContent U ≠ ∞
ε : ℝ≥0
hε : ε ≠ 0
h'ε : ↑ε ≠ 0
⊢ ∃ K, ↑K ⊆ ↑U ∧ μ.innerContent U ≤ μ K + ↑ε

Tactic:
rcases le_or_gt (μ.innerContent U) ε with h | h

Goal state after:
case inl
G : Type w
inst✝ : TopologicalSpace G
μ : Content G
U : Opens G
hU : μ.innerContent U ≠ ∞
ε : ℝ≥0
hε : ε ≠ 0
h'ε : ↑ε ≠ 0
h : μ.innerContent U ≤ ↑ε
⊢ ∃ K, ↑K ⊆ ↑U ∧ μ.innerContent U ≤ μ K + ↑ε

case inr
G : Type w
inst✝ : TopologicalSpace G
μ : Content G
U : Opens G
hU : μ.innerContent U ≠ ∞
ε : ℝ≥0
hε : ε ≠ 0
h'ε : ↑ε ≠ 0
h : ↑ε < μ.innerContent U
⊢ ∃ K, ↑K ⊆ ↑U ∧ μ.innerContent U ≤ μ K + ↑ε
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: case inr
G : Type w
inst✝ : TopologicalSpace G
μ : Content G
U : Opens G
hU : μ.innerContent U ≠ ∞
ε : ℝ≥0
hε : ε ≠ 0
h'ε : ↑ε ≠ 0
h : ↑ε < μ.innerContent U
h₂ : ∃ i, ∃ (_ : ↑i ⊆ ↑U), μ.innerContent U - ↑ε < μ i
⊢ ∃ K, ↑K ⊆ ↑U ∧ μ.innerContent U ≤ μ K + ↑ε

Tactic:
rcases h₂ with ⟨U, h1U, h2U⟩

Goal state after:
case inr
G : Type w
inst✝ : TopologicalSpace G
μ : Content G
U✝ : Opens G
hU : μ.innerContent U✝ ≠ ∞
ε : ℝ≥0
hε : ε ≠ 0
h'ε : ↑ε ≠ 0
h : ↑ε < μ.innerContent U✝
U : Compacts G
h1U : ↑U ⊆ ↑U✝
h2U : μ.innerContent U✝ - ↑ε < μ U
⊢ ∃ K, ↑K ⊆ ↑U✝ ∧ μ.innerContent U✝ ≤ μ K + ↑ε
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
s t : Set X
μ : Measure X
ht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0
haux : StronglyMeasurable f
H : IntegrableOn f (s ∪ t) μ
⊢ ∫ (x : X) in s ∪ t, f x ∂μ = ∫ (x : X) in s, f x ∂μ

Tactic:
let k := f ⁻¹' {0}

Goal state after:
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
s t : Set X
μ : Measure X
ht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0
haux : StronglyMeasurable f
H : IntegrableOn f (s ∪ t) μ
k : Set X := f ⁻¹' {0}
⊢ ∫ (x : X) in s ∪ t, f x ∂μ = ∫ (x : X) in s, f x ∂μ
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
hts : s ⊆ t
h't : ∀ᵐ (x : X) ∂μ, x ∈ t \ s → f x = 0
haux : StronglyMeasurable f
h'aux : IntegrableOn f t μ
⊢ ∫ (x : X) in t, f x ∂μ = ∫ (x : X) in s, f x ∂μ

Tactic:
let k := f ⁻¹' {0}

Goal state after:
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
s t : Set X
μ : Measure X
hts : s ⊆ t
h't : ∀ᵐ (x : X) ∂μ, x ∈ t \ s → f x = 0
haux : StronglyMeasurable f
h'aux : IntegrableOn f t μ
k : Set X := f ⁻¹' {0}
⊢ ∫ (x : X) in t, f x ∂μ = ∫ (x : X) in s, f x ∂μ
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
inst✝ : WeaklyLocallyCompactSpace G
K : Set G
hK : IsCompact K
F : Set G
h1F : IsCompact F
h2F : K ⊆ interior F
⊢ μ.outerMeasure (interior F) ≤ μ { carrier := F, isCompact' := h1F }

Tactic:
by
      apply μ.outerMeasure_le ⟨interior F, isOpen_interior⟩ ⟨F, h1F⟩ interior_subset

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 : Set α
μ : Measure α
inst✝² : MeasurableConstSMul G α
inst✝¹ : SMulInvariantMeasure G α μ
inst✝ : Countable G
ν : Measure α
h : IsFundamentalDomain G s μ
hν : ν ≪ μ
t : Set α
⊢ ν t = ∑' (g : G), ν (t ∩ g • s)

Tactic:
by
  have H : ν.restrict t ≪ μ := Measure.restrict_le_self.absolutelyContinuous.trans hν
  simpa only [setLIntegral_one, Pi.one_def,
    Measure.restrict_apply₀ ((h.nullMeasurableSet_smul _).mono_ac H), inter_comm] using
    h.lintegral_eq_tsum_of_ac H 1

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

Goal state before:

Tactic:
∫ x in t, f x ∂μ = ∫ x in t ∩ k, f x ∂μ + ∫ x in t \ k, f x ∂μ := by
      rw [integral_inter_add_sdiff hk h'aux]
    _ = ∫ x in t \ k, f x ∂μ := by
      rw [setIntegral_eq_zero_of_forall_eq_zero fun x hx => ?_, zero_add]; exact hx.2
    _ = ∫ x in s \ k, f x ∂μ := by
      apply setIntegral_congr_set
      filter_upwards [h't] with x hx
      change (x ∈ t \ k) = (x ∈ s \ k)
      simp only [eq_iff_iff, and_congr_left_iff, Set.mem_sdiff]
      intro h'x
      by_cases xs : x ∈ s
      · simp only [xs, hts xs]
      · simp only [xs, iff_false]
        intro xt
        exact h'x (hx ⟨xt, xs⟩)
    _ = ∫ x in s ∩ k, f x ∂μ + ∫ x in s \ k, f x ∂μ := by
      have : ∀ x ∈ s ∩ k, f x = 0 := fun x hx => hx.2
      rw [setIntegral_eq_zero_of_forall_eq_zero this, zero_add]
    _ = ∫ x in s, f x ∂μ := by rw [integral_inter_add_sdiff hk (h'aux.mono hts le_rfl)]

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
⊢ (Disjoint on fun g ↦ g • t ∩ s) g₁ g₂

Tactic:
dsimp [Function.onFun]

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
⊢ Disjoint (g₁ • t ∩ s) (g₂ • t ∩ s)
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 : PositiveCompacts G
V : Set G
hV : (interior V).Nonempty
t : Finset G
h1t : t ∈ {t | ↑K ⊆ ⋃ g ∈ t, (fun h ↦ g * h) ⁻¹' V}
h2t : t = ∅
⊢ False

Tactic:
subst h2t

Goal state after:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K : PositiveCompacts G
V : Set G
hV : (interior V).Nonempty
h1t : ∅ ∈ {t | ↑K ⊆ ⋃ g ∈ t, (fun h ↦ g * h) ⁻¹' V}
⊢ False
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
s : Finset G
h1s : K₁.carrier ⊆ ⋃ g ∈ s, (fun h ↦ g * h) ⁻¹' V
h2s : s.card = index K₁.carrier V
t : Finset G
h1t : K₂.carrier ⊆ ⋃ g ∈ t, (fun h ↦ g * h) ⁻¹' V
h2t : t.card = index K₂.carrier V
⊢ s ∪ t ∈ {t | K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ t, (fun h ↦ g * h) ⁻¹' V}

Tactic:
mem_ofPred_eq,

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
s : Finset G
h1s : K₁.carrier ⊆ ⋃ g ∈ s, (fun h ↦ g * h) ⁻¹' V
h2s : s.card = index K₁.carrier V
t : Finset G
h1t : K₂.carrier ⊆ ⋃ g ∈ t, (fun h ↦ g * h) ⁻¹' V
h2t : t.card = index K₂.carrier V
⊢ K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s ∪ t, (fun h ↦ g * h) ⁻¹' V
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
l : ℝ
hf : Tendsto (↑f) atBot (𝓝 l)
x : R
this✝ : Nonempty R
val✝ : NoBotOrder R
this : NoMinOrder R
⊢ Tendsto (fun x_1 ↦ f.measure (Ioc x_1 x)) atBot (𝓝 (ofReal (↑f x - l)))

Tactic:
simp_rw [measure_Ioc]

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
this : NoMinOrder R
⊢ Tendsto (fun x_1 ↦ ofReal (↑f x - ↑f x_1)) atBot (𝓝 (ofReal (↑f x - l)))
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
l : ℝ
hf : Tendsto (↑f) atTop (𝓝 l)
x : R
this : Nonempty R
val✝ : NoTopOrder R
⊢ f.measure (Ici x) = ofReal (l - leftLim (↑f) x)

Tactic:
have : NoMaxOrder R := NoTopOrder.to_noMaxOrder R

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) atTop (𝓝 l)
x : R
this✝ : Nonempty R
val✝ : NoTopOrder R
this : NoMaxOrder R
⊢ f.measure (Ici x) = ofReal (l - leftLim (↑f) x)
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
s : Set X
hs : MeasurableSet s
hf : ∀ᵐ (x : X) ∂μ, x ∈ s → 0 ≤ f x
⊢ 0 ≤ᵐ[μ.restrict s] f

Tactic:
by rwa [EventuallyLE, ae_restrict_iff' hs]

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
s : Set X
hs : MeasurableSet s
hf : ∀ᵐ (x : X) ∂μ, x ∈ s → f x ≤ 0
⊢ f ≤ᵐ[μ.restrict s] 0

Tactic:
by rwa [EventuallyLE, ae_restrict_iff' hs]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : RCLike 𝕜
inst✝ : SeminormedAddCommGroup E
c : InnerProductSpace 𝕜 E
x : E
⊢ 0 ≤ ‖x‖ ^ 2

Tactic:
apply sq_nonneg

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : RCLike 𝕜
inst✝ : NormedAddCommGroup E
c : InnerProductSpace 𝕜 E
x : E
⊢ 0 ≤ ‖x‖ ^ 2

Tactic:
apply sq_nonneg

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

Goal state before:
pp: case pos.hf
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
s : Set X
hs : ∀ x ∈ s, f x ≤ 1
h's : ∀ x ∈ sᶜ, f x ≤ 0
H : Integrable f μ
g : X → ℝ := ⋯
g_int : Integrable g μ
this : ENNReal.ofReal (∫ (x : X), f x ∂μ) ≤ ENNReal.ofReal (∫ (x : X), g x ∂μ)
⊢ ∀ (a : X), ENNReal.ofReal (g a) ≤ 1

pp: case pos.h'f
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
s : Set X
hs : ∀ x ∈ s, f x ≤ 1
h's : ∀ x ∈ sᶜ, f x ≤ 0
H : Integrable f μ
g : X → ℝ := ⋯
g_int : Integrable g μ
this : ENNReal.ofReal (∫ (x : X), f x ∂μ) ≤ ENNReal.ofReal (∫ (x : X), g x ∂μ)
⊢ ∀ a ∈ sᶜ, ENNReal.ofReal (g a) = 0

Tactic:
· intro x
    apply ENNReal.ofReal_le_of_le_toReal
    by_cases H : x ∈ s
    · simpa [g] using hs x H
    · apply le_trans _ zero_le_one
      simpa [g] using h's x H

Goal state after:
case pos.h'f
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
s : Set X
hs : ∀ x ∈ s, f x ≤ 1
h's : ∀ x ∈ sᶜ, f x ≤ 0
H : Integrable f μ
g : X → ℝ := ⋯
g_int : Integrable g μ
this : ENNReal.ofReal (∫ (x : X), f x ∂μ) ≤ ENNReal.ofReal (∫ (x : X), g x ∂μ)
⊢ ∀ a ∈ sᶜ, ENNReal.ofReal (g a) = 0
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁸ : RCLike 𝕜
inst✝⁷ : SeminormedAddCommGroup E
inst✝⁶ : InnerProductSpace 𝕜 E
𝕝 : Type u_4
inst✝⁵ : CommSemiring 𝕝
inst✝⁴ : StarRing 𝕝
inst✝³ : Algebra 𝕝 𝕜
inst✝² : Module 𝕝 E
inst✝¹ : IsScalarTower 𝕝 𝕜 E
inst✝ : StarModule 𝕝 𝕜
x y : E
r : 𝕝
⊢ ⟪r • x, y⟫ = (starRingEnd 𝕝) r • ⟪x, y⟫

Tactic:
rw [← algebraMap_smul 𝕜 r, InnerProductSpace.smul_left, starRingEnd_apply, starRingEnd_apply,
    ← algebraMap_star_comm, ← smul_eq_mul, algebraMap_smul]

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

Goal state before:
pp: X : Type u_1
M : Type u_5
inst✝² : NormedAddCommGroup M
inst✝¹ : NormedSpace ℝ M
mX : MeasurableSpace X
ν : Measure X
F : X → M
inst✝ : TopologicalSpace X
⊢ ∫ (x : X) in tsupport F, F x ∂ν = ∫ (x : X) in univ, F x ∂ν

Tactic:
setIntegral_eq_of_subset_of_forall_sdiff_eq_zero MeasurableSet.univ
    (subset_univ (tsupport F))

Goal state after:
X : Type u_1
M : Type u_5
inst✝² : NormedAddCommGroup M
inst✝¹ : NormedSpace ℝ M
mX : MeasurableSpace X
ν : Measure X
F : X → M
inst✝ : TopologicalSpace X
⊢ ∫ (x : X) in tsupport F, F x ∂ν = ∫ (x : X) in tsupport F, F x ∂ν

X : Type u_1
M : Type u_5
inst✝² : NormedAddCommGroup M
inst✝¹ : NormedSpace ℝ M
mX : MeasurableSpace X
ν : Measure X
F : X → M
inst✝ : TopologicalSpace X
⊢ ∀ x ∈ univ \ tsupport F, F x = 0
Module: Mathlib.Analysis.InnerProductSpace.Defs

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x y : F
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
by
        have h₁ : ‖⟪x, y⟫‖ ≤ ‖x‖ * ‖y‖ := norm_inner_le_norm _ _
        have h₂ : re ⟪x, y⟫ ≤ ‖⟪x, y⟫‖ := re_le_norm _
        have h₃ : re ⟪x, y⟫ ≤ ‖x‖ * ‖y‖ := h₂.trans h₁
        have h₄ : re ⟪y, x⟫ ≤ ‖x‖ * ‖y‖ := by rwa [← inner_conj_symm, conj_re]
        have : ‖x + y‖ * ‖x + y‖ ≤ (‖x‖ + ‖y‖) * (‖x‖ + ‖y‖) := by
          simp only [← inner_self_eq_norm_mul_norm, inner_add_add_self, mul_add, mul_comm, map_add]
          linarith
        exact nonneg_le_nonneg_of_sq_le_sq (add_nonneg (sqrt_nonneg _) (sqrt_nonneg _)) this

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
cd : Core 𝕜 F
x y : F
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
by
        have h₁ : ‖⟪x, y⟫‖ ≤ ‖x‖ * ‖y‖ := norm_inner_le_norm _ _
        have h₂ : re ⟪x, y⟫ ≤ ‖⟪x, y⟫‖ := re_le_norm _
        have h₃ : re ⟪x, y⟫ ≤ ‖x‖ * ‖y‖ := h₂.trans h₁
        have h₄ : re ⟪y, x⟫ ≤ ‖x‖ * ‖y‖ := by rwa [← inner_conj_symm, conj_re]
        have : ‖x + y‖ * ‖x + y‖ ≤ (‖x‖ + ‖y‖) * (‖x‖ + ‖y‖) := by
          simp only [← inner_self_eq_norm_mul_norm, inner_add_add_self, mul_add, mul_comm, map_add]
          linarith
        exact nonneg_le_nonneg_of_sq_le_sq (add_nonneg (sqrt_nonneg _) (sqrt_nonneg _)) this

Goal state after: