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

Goal state before:
pp: case neg
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
β : Type u_6
m m0 : MeasurableSpace β
μ : Measure β
hm : m ≤ m0
f : β → G
hf : StronglyMeasurable f
hG : CompleteSpace G
this✝¹ : MeasurableSpace G := borel G
this✝ : BorelSpace G
hf_int : ¬Integrable f μ
⊢ ∫ (x : β), f x ∂μ = ∫ (x : β), f x ∂μ.trim hm

pp: case pos
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
β : Type u_6
m m0 : MeasurableSpace β
μ : Measure β
hm : m ≤ m0
f : β → G
hf : StronglyMeasurable f
hG : CompleteSpace G
this✝¹ : MeasurableSpace G := borel G
this✝ : BorelSpace G
hf_int : Integrable f μ
⊢ ∫ (x : β), f x ∂μ = ∫ (x : β), f x ∂μ.trim hm

Tactic:
· have hf_int_m : ¬Integrable f (μ.trim hm) := fun hf_int_m =>
      hf_int (integrable_of_integrable_trim hm hf_int_m)
    rw [integral_undef hf_int, integral_undef hf_int_m]

Goal state after:
case pos
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
β : Type u_6
m m0 : MeasurableSpace β
μ : Measure β
hm : m ≤ m0
f : β → G
hf : StronglyMeasurable f
hG : CompleteSpace G
this✝¹ : MeasurableSpace G := borel G
this✝ : BorelSpace G
hf_int : Integrable f μ
⊢ ∫ (x : β), f x ∂μ = ∫ (x : β), f x ∂μ.trim hm
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
β : Type u_6
m m0 : MeasurableSpace β
μ : Measure β
hm : m ≤ m0
f : β → G
hf : StronglyMeasurable f
⊢ ∫ (x : β), f x ∂μ = ∫ (x : β), f x ∂μ.trim hm

Tactic:
by_cases hG : CompleteSpace G; swap
  · simp [integral, hG]
  borelize G
  by_cases hf_int : Integrable f μ
  swap
  · have hf_int_m : ¬Integrable f (μ.trim hm) := fun hf_int_m =>
      hf_int (integrable_of_integrable_trim hm hf_int_m)
    rw [integral_undef hf_int, integral_undef hf_int_m]
  have : SeparableSpace (range f ∪ {0} : Set G) := hf.separableSpace_range_union_singleton
  let f_seq := @SimpleFunc.approxOn G β _ _ _ m _ hf.measurable (range f ∪ {0}) 0 (by simp) _
  have hf_seq_meas : ∀ n, StronglyMeasurable[m] (f_seq n) := fun n =>
    @SimpleFunc.stronglyMeasurable β G m _ (f_seq n)
  have hf_seq_int : ∀ n, Integrable (f_seq n) μ :=
    SimpleFunc.integrable_approxOn_range (hf.mono hm).measurable hf_int
  have hf_seq_int_m : ∀ n, Integrable (f_seq n) (μ.trim hm) := fun n =>
    (hf_seq_int n).trim hm (hf_seq_meas n)
  have hf_seq_eq : ∀ n, ∫ x, f_seq n x ∂μ = ∫ x, f_seq n x ∂μ.trim hm := fun n =>
    integral_trim_simpleFunc hm (f_seq n) (hf_seq_int n)
  have h_lim_1 : atTop.Tendsto (fun n => ∫ x, f_seq n x ∂μ) (𝓝 (∫ x, f x ∂μ)) := by
    refine tendsto_integral_of_L1 f hf_int.1 (Eventually.of_forall hf_seq_int) ?_
    exact SimpleFunc.tendsto_approxOn_range_L1_enorm (hf.mono hm).measurable hf_int
  have h_lim_2 : atTop.Tendsto (fun n => ∫ x, f_seq n x ∂μ) (𝓝 (∫ x, f x ∂μ.trim hm)) := by
    simp_rw [hf_seq_eq]
    refine @tendsto_integral_of_L1 β G _ _ m (μ.trim hm) _ f (hf_int.trim hm hf).1 _ _
      (Eventually.of_forall hf_seq_int_m) ?_
    exact @SimpleFunc.tendsto_approxOn_range_L1_enorm β G m _ _ _ f _ _ hf.measurable
      (hf_int.trim hm hf)
  exact tendsto_nhds_unique h_lim_1 h_lim_2

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

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

Tactic:
rw [← setToFun_congr_ae hT hfg, hf_eq, setToFun_congr_ae hT' hfg']

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

Goal state before:
pp: case neg
α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C C' : ℝ
μ' : Measure α
c c' : ℝ≥0∞
hc : c ≠ ∞
hc' : c' ≠ ∞
hμ_le : μ ≤ c • μ'
hμ'_le : μ' ≤ c' • μ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ' T C'
f : α → E
hf : ¬Integrable f μ
⊢ setToFun μ T hT f = setToFun μ' T hT' f

Tactic:
have h_int : ∀ g : α → E, ¬Integrable g μ → ¬Integrable g μ' := fun g =>
      mt fun h => h.of_measure_le_smul hc hμ_le

Goal state after:
case neg
α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C C' : ℝ
μ' : Measure α
c c' : ℝ≥0∞
hc : c ≠ ∞
hc' : c' ≠ ∞
hμ_le : μ ≤ c • μ'
hμ'_le : μ' ≤ c' • μ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ' T C'
f : α → E
hf : ¬Integrable f μ
h_int : ∀ (g : α → E), ¬Integrable g μ → ¬Integrable g μ'
⊢ setToFun μ T hT f = setToFun μ' T hT' f
Module: Mathlib.Topology.Order.LeftRightLim

Goal state before:
pp: case inr.refine_2
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x y : α
h : x ≤ y
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
h' : (𝓝[<] x).NeBot
⊢ ∀ b ∈ f '' Iio x, b ≤ f y

Tactic:
· simp only [mem_image, mem_Iio, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂]
    intro z hz
    exact hf (hz.le.trans h)

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

Goal state before:
pp: case inl
R : Type u_1
inst✝³ : LinearOrder R
inst✝² : TopologicalSpace R
f : StieltjesFunction R
inst✝¹ : OrderTopology R
inst✝ : CompactIccSpace R
a : R
c d : ℕ → R
s✝ s : Finset ℕ
IH :
  ∀ t ⊂ s,
    ∀ (b : R), Icc a b ⊆ ⋃ i ∈ ↑t, Iotop (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i ∈ t, ofReal (↑f (d i) - ↑f (c i))
b : R
cv : Icc a b ⊆ ⋃ i ∈ ↑s, Iotop (c i) (d i)
ab : b ≤ a
⊢ ofReal (↑f b - ↑f a) ≤ ∑ i ∈ s, ofReal (↑f (d i) - ↑f (c i))

pp: case inr
R : Type u_1
inst✝³ : LinearOrder R
inst✝² : TopologicalSpace R
f : StieltjesFunction R
inst✝¹ : OrderTopology R
inst✝ : CompactIccSpace R
a : R
c d : ℕ → R
s✝ s : Finset ℕ
IH :
  ∀ t ⊂ s,
    ∀ (b : R), Icc a b ⊆ ⋃ i ∈ ↑t, Iotop (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i ∈ t, ofReal (↑f (d i) - ↑f (c i))
b : R
cv : Icc a b ⊆ ⋃ i ∈ ↑s, Iotop (c i) (d i)
ab : a ≤ b
⊢ ofReal (↑f b - ↑f a) ≤ ∑ i ∈ s, ofReal (↑f (d i) - ↑f (c i))

Tactic:
· rw [ENNReal.ofReal_eq_zero.2 (sub_nonpos.2 (f.mono ab))]
    exact zero_le

Goal state after:
case inr
R : Type u_1
inst✝³ : LinearOrder R
inst✝² : TopologicalSpace R
f : StieltjesFunction R
inst✝¹ : OrderTopology R
inst✝ : CompactIccSpace R
a : R
c d : ℕ → R
s✝ s : Finset ℕ
IH :
  ∀ t ⊂ s,
    ∀ (b : R), Icc a b ⊆ ⋃ i ∈ ↑t, Iotop (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i ∈ t, ofReal (↑f (d i) - ↑f (c i))
b : R
cv : Icc a b ⊆ ⋃ i ∈ ↑s, Iotop (c i) (d i)
ab : a ≤ b
⊢ ofReal (↑f b - ↑f a) ≤ ∑ i ∈ s, ofReal (↑f (d i) - ↑f (c i))
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case a
R : Type u_1
inst✝³ : LinearOrder R
inst✝² : TopologicalSpace R
f : StieltjesFunction R
inst✝¹ : OrderTopology R
inst✝ : CompactIccSpace R
a : R
c d : ℕ → R
s✝ s : Finset ℕ
IH :
  ∀ t ⊂ s,
    ∀ (b : R), Icc a b ⊆ ⋃ i ∈ ↑t, Iotop (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i ∈ t, ofReal (↑f (d i) - ↑f (c i))
b : R
ab : a ≤ b
i : ℕ
cv : Icc a b ⊆ Iotop (c i) (d i) ∪ ⋃ x ∈ ↑(s.erase i), Iotop (c x) (d x)
is : i ∈ s
bcd : b ∈ Ioc (c i) (d i)
x : R
h₁ : a ≤ x
h₂ : x ≤ c i
⊢ x ∈ ⋃ i_1 ∈ ↑(s.erase i), Iotop (c i_1) (d i_1)

Tactic:
apply (cv ⟨h₁, le_trans h₂ (le_of_lt bcd.1)⟩).resolve_left (fun h ↦ ?_)

Goal state after:
R : Type u_1
inst✝³ : LinearOrder R
inst✝² : TopologicalSpace R
f : StieltjesFunction R
inst✝¹ : OrderTopology R
inst✝ : CompactIccSpace R
a : R
c d : ℕ → R
s✝ s : Finset ℕ
IH :
  ∀ t ⊂ s,
    ∀ (b : R), Icc a b ⊆ ⋃ i ∈ ↑t, Iotop (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i ∈ t, ofReal (↑f (d i) - ↑f (c i))
b : R
ab : a ≤ b
i : ℕ
cv : Icc a b ⊆ Iotop (c i) (d i) ∪ ⋃ x ∈ ↑(s.erase i), Iotop (c x) (d x)
is : i ∈ s
bcd : b ∈ Ioc (c i) (d i)
x : R
h₁ : a ≤ x
h₂ : x ≤ c i
h : x ∈ Iotop (c i) (d i)
⊢ False
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
A : Set G
hA : μ.outerMeasure A ≠ ∞
ε : ℝ≥0
hε : ε ≠ 0
⊢ ∃ U, A ⊆ ↑U ∧ μ.outerMeasure ↑U ≤ μ.outerMeasure A + ↑ε

Tactic:
rcases inducedOuterMeasure_exists_set _ μ.innerContent_iUnion_nat μ.innerContent_mono hA
      (ENNReal.coe_ne_zero.2 hε) with
    ⟨U, hU, h2U, h3U⟩

Goal state after:
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
A : Set G
hA : μ.outerMeasure A ≠ ∞
ε : ℝ≥0
hε : ε ≠ 0
U : Set G
hU : IsOpen U
h2U : A ⊆ U
h3U :
  (inducedOuterMeasure (fun s₁ hs₁ ↦ μ.innerContent { carrier := s₁, is_open' := hs₁ }) ⋯ ⋯) U ≤
    (inducedOuterMeasure (fun s₁ hs₁ ↦ μ.innerContent { carrier := s₁, is_open' := hs₁ }) ⋯ ⋯) A + ↑ε
⊢ ∃ U, A ⊆ ↑U ∧ μ.outerMeasure ↑U ≤ μ.outerMeasure A + ↑ε
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: case refine_1
G : Type w
inst✝³ : TopologicalSpace G
μ : Content G
inst✝² : MeasurableSpace G
inst✝¹ : R1Space G
inst✝ : BorelSpace G
H : μ.ContentRegular
K : Compacts G
⊢ μ.measure ↑K ≤ μ K

Tactic:
apply ENNReal.le_of_forall_pos_le_add

Goal state after:
case refine_1
G : Type w
inst✝³ : TopologicalSpace G
μ : Content G
inst✝² : MeasurableSpace G
inst✝¹ : R1Space G
inst✝ : BorelSpace G
H : μ.ContentRegular
K : Compacts G
⊢ ∀ (ε : ℝ≥0), 0 < ε → μ K < ∞ → μ.measure ↑K ≤ μ K + ↑ε
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
K : Compacts G
V : Set G
hV : (interior V).Nonempty
s : Finset G
h1s : ↑K ⊆ ⋃ g ∈ s, (fun h ↦ g * h) ⁻¹' ↑K₀
h2s : s.card = index ↑K ↑K₀
t : Finset G
h1t : ↑K₀ ⊆ ⋃ g ∈ t, (fun h ↦ g * h) ⁻¹' V
h2t : t.card = index (↑K₀) V
⊢ (t * s).card ∈ Finset.card '' {t | ↑K ⊆ ⋃ g ∈ t, (fun h ↦ g * h) ⁻¹' V}

Tactic:
refine ⟨_, ?_, rfl⟩

Goal state after:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
K : Compacts G
V : Set G
hV : (interior V).Nonempty
s : Finset G
h1s : ↑K ⊆ ⋃ g ∈ s, (fun h ↦ g * h) ⁻¹' ↑K₀
h2s : s.card = index ↑K ↑K₀
t : Finset G
h1t : ↑K₀ ⊆ ⋃ g ∈ t, (fun h ↦ g * h) ⁻¹' V
h2t : t.card = index (↑K₀) V
⊢ t * s ∈ {t | ↑K ⊆ ⋃ g ∈ t, (fun h ↦ g * h) ⁻¹' V}
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
⊢ 0 < index (↑K) V

Tactic:
index,

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
⊢ 0 < sInf (Finset.card '' {t | ↑K ⊆ ⋃ g ∈ t, (fun h ↦ g * h) ⁻¹' V})
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case pos
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
⊢ f.measure {a} = ofReal (↑f a - leftLim (↑f) a)

Tactic:
have : leftLim f a = f a := by
      apply leftLim_eq_of_eq_bot
      simp [nhdsLT_eq_bot_iff, ha]
    simp only [this, sub_self, ofReal_zero]
    apply eq_bot_iff.2
    rw [StieltjesFunction.measure]
    apply (outer_le_length _ _).trans
    rw [← length_sdiff_botSet]
    simp [subsingleton_botSet.eq_singleton_of_mem ha]

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 ∈ s | ((fun x ↦ g * x) ⁻¹' V ∩ K).Nonempty}.card ∈ Finset.card '' {t | K ⊆ ⋃ g ∈ t, (fun h ↦ g * h) ⁻¹' V}

Tactic:
refine ⟨_, ?_, rfl⟩

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 ∈ s | ((fun x ↦ g * x) ⁻¹' V ∩ K).Nonempty} ∈ {t | K ⊆ ⋃ g ∈ t, (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]

Goal state after:
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
⊢ 0 = ofReal (leftLim (↑f) b - ↑f a)
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
a b : R
hab : a < b
⊢ f.measure (Ioo a b) = ofReal (leftLim (↑f) b - ↑f a)

Tactic:
have A : Disjoint (Ioo a b) {b} := by simp

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
a b : R
hab : a < b
A : Disjoint (Ioo a b) {b}
⊢ f.measure (Ioo a b) = ofReal (leftLim (↑f) b - ↑f a)
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case hf
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
R : ℝ
f : X → ℝ
hR : 0 ≤ R
hfint : IntegrableOn f {x | R < f x} μ
hμ : μ {x | R < f x} ≠ 0
this : IntegrableOn (fun x ↦ R) {x | R < f x} μ
⊢ (fun x ↦ 0) ≤ᵐ[μ.restrict {x | R < f x}] fun a ↦ f a - R

Tactic:
EventuallyLE,

Goal state after:
case hf
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
R : ℝ
f : X → ℝ
hR : 0 ≤ R
hfint : IntegrableOn f {x | R < f x} μ
hμ : μ {x | R < f x} ≠ 0
this : IntegrableOn (fun x ↦ R) {x | R < f x} μ
⊢ ∀ᵐ (x : X) ∂μ.restrict {x | R < f x}, 0 ≤ f x - R
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : IsOrderedModule ℝ E
μ : Measure X
f g : X → E
s : Set X
inst✝ : ClosedIciTopology E
hf : IntegrableOn f s μ
hg : IntegrableOn g s μ
hs : MeasurableSet s
h : ∀ᵐ (x : X) ∂μ, x ∈ s → f x ≤ g x
⊢ f ≤ᵐ[μ.restrict s] g

Tactic:
EventuallyLE,

Goal state after:
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : IsOrderedModule ℝ E
μ : Measure X
f g : X → E
s : Set X
inst✝ : ClosedIciTopology E
hf : IntegrableOn f s μ
hg : IntegrableOn g s μ
hs : MeasurableSet s
h : ∀ᵐ (x : X) ∂μ, x ∈ s → f x ≤ g x
⊢ ∀ᵐ (x : X) ∂μ.restrict s, f x ≤ g 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:
EventuallyLE,

Goal state after:
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
⊢ ∀ᵐ (x : X) ∂μ.restrict s, 0 x ≤ f 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
U : Set G
K₁ K₂ : Compacts G
hU : (interior U).Nonempty
h : Disjoint (K₁.carrier * U⁻¹) (K₂.carrier * U⁻¹)
⊢ ↑(index (↑(K₁ ⊔ K₂)) U) = ↑(index (↑K₁) U) + ↑(index (↑K₂) U)

Tactic:
exact mod_cast index_union_eq K₁ K₂ hU h

Goal state after:
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
hf : Tendsto (↑f) atBot atBot
x : R
this : Nonempty R
r : ℝ≥0
N : R
hN : ∀ b ≤ N, ↑f b ≤ ↑f x - ↑r
⊢ ↑r ≤ f.measure (Iic x)

Tactic:
exact (f.measure_Ioc (min x N) x ▸ ENNReal.coe_nnreal_eq r ▸ (ENNReal.ofReal_le_ofReal <|
    le_sub_comm.mp <| hN _ (min_le_right x N))).trans (measure_mono Ioc_subset_Iic_self)

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:
EventuallyLE,

Goal state after:
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
⊢ ∀ᵐ (x : X) ∂μ.restrict s, f x ≤ 0 x
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_6
f : X → ℝ
m m0 : MeasurableSpace X
μ : Measure X
hm : m ≤ m0
g : SimpleFunc X ℝ
hf : Integrable f μ
⊢ Integrable (⇑g * f) μ

Tactic:
rw [← SimpleFunc.coe_toLargerSpace_eq hm g]; exact hf.simpleFunc_mul (g.toLargerSpace hm)

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 : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x : E
⊢ re ⟪0, x⟫ = 0

Tactic:
by
  simp only [inner_zero_left, map_zero]

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

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

Tactic:
have hδ : ∀ n : ℕ, δ < δ + 1 / (n + 1) := fun n =>
      lt_add_of_le_of_pos le_rfl Nat.one_div_pos_of_nat

Goal state after:
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Set F
ne : K.Nonempty
h₁ : IsComplete K
h₂ : Convex ℝ K
u : F
δ : ℝ := ⨅ w, ‖u - ↑w‖
this : Nonempty ↑K := Set.Nonempty.to_subtype ne
zero_le_δ : 0 ≤ δ
δ_le : ∀ (w : ↑K), δ ≤ ‖u - ↑w‖
δ_le' : ∀ w ∈ K, δ ≤ ‖u - w‖
hδ : ∀ (n : ℕ), δ < δ + 1 / (↑n + 1)
⊢ ∃ w, ∀ (n : ℕ), ‖u - ↑(w n)‖ < δ + 1 / (↑n + 1)
Module: Mathlib.MeasureTheory.Measure.Haar.Basic

Goal state before:
pp: G : Type u_1
inst✝⁴ : Group G
inst✝³ : TopologicalSpace G
inst✝² : IsTopologicalGroup G
inst✝¹ : MeasurableSpace G
inst✝ : BorelSpace G
K₀ : PositiveCompacts G
⊢ (haarMeasure K₀).IsHaarMeasure

Tactic:
by
  apply
    isHaarMeasure_of_isCompact_nonempty_interior (haarMeasure K₀) K₀ K₀.isCompact
      K₀.interior_nonempty
  · simp only [haarMeasure_self]; exact one_ne_zero
  · simp only [haarMeasure_self, ne_eq, ENNReal.one_ne_top, not_false_eq_true]

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

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
h : 0 < ‖x‖
⊢ ‖x‖ ^ 2 = ‖↑‖x‖‖ ^ 2

Tactic:
norm_ofReal,

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
h : 0 < ‖x‖
⊢ ‖x‖ ^ 2 = |‖x‖| ^ 2
Module: Mathlib.Analysis.InnerProductSpace.LinearMap

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
⊢ ‖(innerSL 𝕜) x‖ = ‖x‖

Tactic:
by
  refine
    le_antisymm ((innerSL 𝕜 x).opNorm_le_bound (norm_nonneg _) fun y => norm_inner_le_norm _ _) ?_
  rcases (norm_nonneg x).eq_or_lt' with (h | h)
  · simp [h]
  · refine (mul_le_mul_iff_left₀ h).mp ?_
    calc
      ‖x‖ * ‖x‖ = ‖(⟪x, x⟫ : 𝕜)‖ := by
        rw [← sq, inner_self_eq_norm_sq_to_K, norm_pow, norm_ofReal, abs_norm]
      _ ≤ ‖innerSL 𝕜 x‖ * ‖x‖ := (innerSL 𝕜 x).le_opNorm _

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

Goal state before:
pp: case mpr
V : Type u_3
inst✝¹ : SeminormedAddCommGroup V
inst✝ : InnerProductSpace ℂ V
T : V →ₗ[ℂ] V
h : ∀ (v : V), (starRingEnd ℂ) ⟪T v, v⟫_ℂ = ⟪T v, v⟫_ℂ
x y : V
⊢ ⟪T x, y⟫_ℂ =
    (⟪T (x + y), x + y⟫_ℂ - (starRingEnd ℂ) ⟪T (x - y), x - y⟫_ℂ +
          (starRingEnd ℂ) ⟪T (x + Complex.I • y), x + Complex.I • y⟫_ℂ * (starRingEnd ℂ) Complex.I -
        (starRingEnd ℂ) ⟪T (x - Complex.I • y), x - Complex.I • y⟫_ℂ * (starRingEnd ℂ) Complex.I) /
      (starRingEnd ℂ) 4

Tactic:
h (x - y),

Goal state after:
case mpr
V : Type u_3
inst✝¹ : SeminormedAddCommGroup V
inst✝ : InnerProductSpace ℂ V
T : V →ₗ[ℂ] V
h : ∀ (v : V), (starRingEnd ℂ) ⟪T v, v⟫_ℂ = ⟪T v, v⟫_ℂ
x y : V
⊢ ⟪T x, y⟫_ℂ =
    (⟪T (x + y), x + y⟫_ℂ - ⟪T (x - y), x - y⟫_ℂ +
          (starRingEnd ℂ) ⟪T (x + Complex.I • y), x + Complex.I • y⟫_ℂ * (starRingEnd ℂ) Complex.I -
        (starRingEnd ℂ) ⟪T (x - Complex.I • y), x - Complex.I • y⟫_ℂ * (starRingEnd ℂ) Complex.I) /
      (starRingEnd ℂ) 4
Module: Mathlib.Analysis.InnerProductSpace.Symmetric

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
inst✝ : CompleteSpace E
T : E →ₗ[𝕜] E
hT : T.IsSymmetric
u : ℕ → E
x y : E
hu : Filter.Tendsto u Filter.atTop (nhds x)
hTu : Filter.Tendsto (⇑T ∘ u) Filter.atTop (nhds y)
hlhs : ∀ (k : ℕ), ⟪T (u k) - T x, y - T x⟫ = ⟪u k - x, T (y - T x)⟫
⊢ Filter.Tendsto (fun t ↦ ⟪u t - x, T (y - T x)⟫) Filter.atTop (nhds 0)

Tactic:
← inner_zero_left (T (y - T x))

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
T : E →ₗ[𝕜] E
hT : T.IsSymmetric
u : ℕ → E
x y : E
hu : Filter.Tendsto u Filter.atTop (nhds x)
hTu : Filter.Tendsto (⇑T ∘ u) Filter.atTop (nhds y)
hlhs : ∀ (k : ℕ), ⟪T (u k) - T x, y - T x⟫ = ⟪u k - x, T (y - T x)⟫
⊢ Filter.Tendsto (fun t ↦ ⟪u t - x, T (y - T x)⟫) Filter.atTop (nhds ⟪0, T (y - T x)⟫)
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
u v z : E
hv : v ∈ K
hz : z ∈ Kᗮ
hu : u = v + z
⊢ u - v ∈ Kᗮ

Tactic:
simpa [hu]

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

Goal state before:
pp: case refine_1
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
h : K.orthogonalProjectionOnto v = 0
⊢ v ∈ Kᗮ

Tactic:
rw [← sub_zero v, ← coe_zero (p := K), ← h]

Goal state after:
case refine_1
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
h : K.orthogonalProjectionOnto v = 0
⊢ v - ↑(K.orthogonalProjectionOnto v) ∈ Kᗮ