mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Probability.Independence.Kernel.Indep

Goal state before:
pp: case inl
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
π : ι → Set (Set Ω)
a : ι
S : Finset ι
hp_ind : iIndepSets π κ μ
haS : a ∉ S
t1 t2 : Set Ω
s : Finset ι
hs_mem : s ⊆ S
ft1 : ι → Set Ω
hft1_mem : ∀ x ∈ s, ft1 x ∈ π x
ht1_eq : t1 = ⋂ x ∈ s, ft1 x
ht2_mem_pia : t2 ∈ π a
f : ι → Set Ω := fun n ↦ if n = a then t2 else if n ∈ s then ft1 n else univ
n : ι
hn_mem_insert : n ∈ insert a s
hn_mem : n = a
⊢ (if n = a then t2 else if n ∈ s then ft1 n else univ) ∈ π n

pp: case inr
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
π : ι → Set (Set Ω)
a : ι
S : Finset ι
hp_ind : iIndepSets π κ μ
haS : a ∉ S
t1 t2 : Set Ω
s : Finset ι
hs_mem : s ⊆ S
ft1 : ι → Set Ω
hft1_mem : ∀ x ∈ s, ft1 x ∈ π x
ht1_eq : t1 = ⋂ x ∈ s, ft1 x
ht2_mem_pia : t2 ∈ π a
f : ι → Set Ω := fun n ↦ if n = a then t2 else if n ∈ s then ft1 n else univ
n : ι
hn_mem_insert : n ∈ insert a s
hn_mem : n ∈ s
⊢ (if n = a then t2 else if n ∈ s then ft1 n else univ) ∈ π n

Tactic:
· simp [hn_mem, ht2_mem_pia]

Goal state after:
case inr
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
π : ι → Set (Set Ω)
a : ι
S : Finset ι
hp_ind : iIndepSets π κ μ
haS : a ∉ S
t1 t2 : Set Ω
s : Finset ι
hs_mem : s ⊆ S
ft1 : ι → Set Ω
hft1_mem : ∀ x ∈ s, ft1 x ∈ π x
ht1_eq : t1 = ⋂ x ∈ s, ft1 x
ht2_mem_pia : t2 ∈ π a
f : ι → Set Ω := fun n ↦ if n = a then t2 else if n ∈ s then ft1 n else univ
n : ι
hn_mem_insert : n ∈ insert a s
hn_mem : n ∈ s
⊢ (if n = a then t2 else if n ∈ s then ft1 n else univ) ∈ π n
Module: Mathlib.Probability.Independence.Integration

Goal state before:
pp: case indicator
Ω : Type u_1
Mf mΩ : MeasurableSpace Ω
μ : Measure Ω
hMf : Mf ≤ mΩ
c : ℝ≥0∞
T : Set Ω
h_meas_T : MeasurableSet T
h_ind : IndepSets {s | MeasurableSet s} {T} μ
h_mul_indicator : ∀ (g : Ω → ℝ≥0∞), Measurable g → Measurable fun a ↦ g a * T.indicator (fun x ↦ c) a
⊢ ∀ (c_1 : ℝ≥0∞) ⦃s : Set Ω⦄,
    MeasurableSet s →
      ∫⁻ (ω : Ω), s.indicator (fun x ↦ c_1) ω * T.indicator (fun x ↦ c) ω ∂μ =
        (∫⁻ (ω : Ω), s.indicator (fun x ↦ c_1) ω ∂μ) * ∫⁻ (ω : Ω), T.indicator (fun x ↦ c) ω ∂μ

Tactic:
intro c' s' h_meas_s'
    simp_rw [← inter_indicator_mul]
    rw [lintegral_indicator (MeasurableSet.inter (hMf _ h_meas_s') h_meas_T),
      lintegral_indicator (hMf _ h_meas_s'), lintegral_indicator h_meas_T]
    simp only [lintegral_const, univ_inter,
      MeasurableSet.univ, Measure.restrict_apply]
    rw [IndepSets_iff] at h_ind
    rw [mul_mul_mul_comm, h_ind s' T h_meas_s' (Set.mem_singleton _)]

Goal state after:
Module: Mathlib.Probability.Independence.Kernel.Indep

Goal state before:
pp: α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
π : ι → Set (Set Ω)
a : ι
S : Finset ι
hp_ind : iIndepSets π κ μ
haS : a ∉ S
⊢ IndepSets (piiUnionInter π ↑S) (π a) κ μ

Tactic:
rintro t1 t2 ⟨s, hs_mem, ft1, hft1_mem, ht1_eq⟩ ht2_mem_pia
  rw [Finset.coe_subset] at hs_mem
  classical
  let f := fun n => ite (n = a) t2 (ite (n ∈ s) (ft1 n) Set.univ)
  have h_f_mem : ∀ n ∈ insert a s, f n ∈ π n := by
    intro n hn_mem_insert
    dsimp only [f]
    rcases Finset.mem_insert.mp hn_mem_insert with hn_mem | hn_mem
    · simp [hn_mem, ht2_mem_pia]
    · grind
  have h_f_mem_pi : ∀ n ∈ s, f n ∈ π n := fun x hxS => h_f_mem x (by simp [hxS])
  have h_t1 : t1 = ⋂ n ∈ s, f n := by
    suffices h_forall : ∀ n ∈ s, f n = ft1 n by grind
    intro n hnS
    have hn_ne_a : n ≠ a := by rintro rfl; exact haS (hs_mem hnS)
    simp_rw [f, if_pos hnS, if_neg hn_ne_a]
  have h_μ_t1 : ∀ᵐ a' ∂μ, κ a' t1 = ∏ n ∈ s, κ a' (f n) := by
    filter_upwards [hp_ind s h_f_mem_pi] with a' ha'
    rw [h_t1, ← ha']
  have h_t2 : t2 = f a := by simp [f]
  have h_μ_inter : ∀ᵐ a' ∂μ, κ a' (t1 ∩ t2) = ∏ n ∈ insert a s, κ a' (f n) := by
    have h_t1_inter_t2 : t1 ∩ t2 = ⋂ n ∈ insert a s, f n := by
      rw [h_t1, h_t2, Finset.set_biInter_insert, Set.inter_comm]
    filter_upwards [hp_ind (insert a s) h_f_mem] with a' ha'
    rw [h_t1_inter_t2, ← ha']
  have has : a ∉ s := fun has_mem => haS (hs_mem has_mem)
  filter_upwards [h_μ_t1, h_μ_inter] with a' ha1 ha2
  rw [ha2, Finset.prod_insert has, h_t2, mul_comm, ha1]

Goal state after:
Module: Mathlib.Probability.IdentDistrib

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
μ : Measure α
ν : Measure β
f : α → γ
g : β → γ
inst✝² : TopologicalSpace γ
inst✝¹ : PseudoMetrizableSpace γ
inst✝ : BorelSpace γ
h : IdentDistrib f g μ ν
hf : AEStronglyMeasurable f μ
⊢ AEStronglyMeasurable g ν

Tactic:
refine aestronglyMeasurable_iff_aemeasurable_separable.2 ⟨h.aemeasurable_snd, ?_⟩
  rcases (aestronglyMeasurable_iff_aemeasurable_separable.1 hf).2 with ⟨t, t_sep, ht⟩
  refine ⟨closure t, t_sep.closure, ?_⟩
  apply h.ae_mem_snd isClosed_closure.measurableSet
  filter_upwards [ht] with x hx using subset_closure hx

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

Goal state before:
pp: ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω'' : Type u_4
Ω : ι → Type u_5
m : (i : ι) → MeasurableSpace (Ω i)
μ : (i : ι) → Measure (Ω i)
inst✝⁶ : ∀ (i : ι), IsProbabilityMeasure (μ i)
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝⁵ : IsProbabilityMeasure μ'
m'' : MeasurableSpace Ω''
μ'' : Measure Ω''
inst✝⁴ : IsProbabilityMeasure μ''
mE : MeasurableSpace E
l : Filter ι
inst✝³ : TopologicalSpace E
inst✝² : HasOuterApproxClosed E
inst✝¹ : BorelSpace E
X : (i : ι) → Ω i → E
Z : Ω' → E
W : Ω'' → E
inst✝ : l.NeBot
h1 : TendstoInDistribution X l Z μ μ'
h2 : TendstoInDistribution X l W μ μ''
h_eq : ⟨Measure.map Z μ', ⋯⟩ = ⟨Measure.map W μ'', ⋯⟩
⊢ Measure.map Z μ' = Measure.map W μ''

Tactic:
rw [Subtype.ext_iff] at h_eq

Goal state after:
ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω'' : Type u_4
Ω : ι → Type u_5
m : (i : ι) → MeasurableSpace (Ω i)
μ : (i : ι) → Measure (Ω i)
inst✝⁶ : ∀ (i : ι), IsProbabilityMeasure (μ i)
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝⁵ : IsProbabilityMeasure μ'
m'' : MeasurableSpace Ω''
μ'' : Measure Ω''
inst✝⁴ : IsProbabilityMeasure μ''
mE : MeasurableSpace E
l : Filter ι
inst✝³ : TopologicalSpace E
inst✝² : HasOuterApproxClosed E
inst✝¹ : BorelSpace E
X : (i : ι) → Ω i → E
Z : Ω' → E
W : Ω'' → E
inst✝ : l.NeBot
h1 : TendstoInDistribution X l Z μ μ'
h2 : TendstoInDistribution X l W μ μ''
h_eq : ↑⟨Measure.map Z μ', ⋯⟩ = ↑⟨Measure.map W μ'', ⋯⟩
⊢ Measure.map Z μ' = Measure.map W μ''
Module: Mathlib.MeasureTheory.Function.ConvergenceInDistribution

Goal state before:
pp: ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω'' : Type u_4
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝⁵ : IsProbabilityMeasure μ'
m'' : MeasurableSpace Ω''
μ'' : Measure Ω''
inst✝⁴ : IsProbabilityMeasure μ''
mE : MeasurableSpace E
l : Filter ι
inst✝³ : SeminormedAddCommGroup E
inst✝² : SecondCountableTopology E
inst✝¹ : BorelSpace E
X : ι → Ω'' → E
inst✝ : l.IsCountablyGenerated
Y : ι → Ω'' → E
Z : Ω' → E
hXZ : TendstoInDistribution X l Z (fun x ↦ μ'') μ'
hXY : TendstoInMeasure μ'' (Y - X) l 0
hY : ∀ (i : ι), AEMeasurable (Y i) μ''
hZ : AEMeasurable Z μ'
hX : ∀ (i : ι), AEMeasurable (X i) μ''
hE : Nonempty E
x₀ : E := hE.some
F : E → ℝ
M : ℝ
hF_bounded : ∀ (x y : E), dist (F x) (F y) ≤ M
L : NNReal
hF_lip : LipschitzWith L F
hF_cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] F
hL : 0 < L
ε : ℝ
hε : ε > 0
n : ι
h_int_Y : Integrable (fun x ↦ F (Y n x)) μ''
h_int_X : Integrable (fun x ↦ F (X n x)) μ''
h_int_sub : Integrable (fun a ↦ ‖F (Y n a) - F (X n a)‖) μ''
⊢ ∫ (a : Ω'') in {x | ‖Y n x - X n x‖ < ε / 2}, ‖F (Y n a) - F (X n a)‖ ∂μ'' +
      ∫ (a : Ω'') in {x | ¬‖Y n x - X n x‖ < ε / 2}, ‖F (Y n a) - F (X n a)‖ ∂μ'' =
    ∫ (a : Ω''), ‖F (Y n a) - F (X n a)‖ ∂μ''

Tactic:
refine integral_add_compl₀ ?_ h_int_sub

Goal state after:
ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω'' : Type u_4
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝⁵ : IsProbabilityMeasure μ'
m'' : MeasurableSpace Ω''
μ'' : Measure Ω''
inst✝⁴ : IsProbabilityMeasure μ''
mE : MeasurableSpace E
l : Filter ι
inst✝³ : SeminormedAddCommGroup E
inst✝² : SecondCountableTopology E
inst✝¹ : BorelSpace E
X : ι → Ω'' → E
inst✝ : l.IsCountablyGenerated
Y : ι → Ω'' → E
Z : Ω' → E
hXZ : TendstoInDistribution X l Z (fun x ↦ μ'') μ'
hXY : TendstoInMeasure μ'' (Y - X) l 0
hY : ∀ (i : ι), AEMeasurable (Y i) μ''
hZ : AEMeasurable Z μ'
hX : ∀ (i : ι), AEMeasurable (X i) μ''
hE : Nonempty E
x₀ : E := hE.some
F : E → ℝ
M : ℝ
hF_bounded : ∀ (x y : E), dist (F x) (F y) ≤ M
L : NNReal
hF_lip : LipschitzWith L F
hF_cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] F
hL : 0 < L
ε : ℝ
hε : ε > 0
n : ι
h_int_Y : Integrable (fun x ↦ F (Y n x)) μ''
h_int_X : Integrable (fun x ↦ F (X n x)) μ''
h_int_sub : Integrable (fun a ↦ ‖F (Y n a) - F (X n a)‖) μ''
⊢ NullMeasurableSet {x | ‖Y n x - X n x‖ < ε / 2} μ''
Module: Mathlib.Probability.Process.Filtration

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝ : Preorder ι
x✝ : Set (Filtration ι m)
⊢ IsGLB x✝ (sInf x✝)

Tactic:
by
    dsimp +instances [instInfSet]
    split_ifs with hn
    · refine .of_image (f := seq) .rfl ?_
      simpa only [isGLB_pi, Set.image_image] using! fun _ ↦ isGLB_sInf _
    · rw [Set.not_nonempty_iff_eq_empty] at hn
      simpa [hn] using! Filtration.le

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Count

Goal state before:
pp: case refine_2
α : Type u_1
ε : Type u_2
inst✝³ : MeasurableSpace α
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
f : α → ε
p : ℝ≥0∞
inst✝ : Finite α
h : ∀ (i : α), ‖f i‖ₑ < ∞
this : Fintype α
⊢ eLpNorm (fun x ↦ Finset.univ.sup fun x ↦ ‖f x‖ₑ) p count < ∞

Tactic:
· exact (memLp_const_enorm <| by simp [h, LT.lt.ne]).eLpNorm_lt_top

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

Goal state before:
pp: case succ
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
hp_top : p ≠ ∞
n : ℕ
h : ∀ {f : Fin n → α → β}, (∀ (i : Fin n), MemLp (f i) p μ) → UnifTight f p μ
⊢ ∀ {f : Fin (n + 1) → α → β}, (∀ (i : Fin (n + 1)), MemLp (f i) p μ) → UnifTight f p μ

Tactic:
| succ n h =>

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

Goal state before:
pp: E : Type u_1
inst✝⁶ : MeasurableSpace E
μ : Measure E
F s : Set E
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : μ.IsAddHaarMeasure
L : AddSubgroup E
inst✝ : Countable ↥L
fund : IsAddFundamentalDomain (↥L) F μ
h_symm : ∀ x ∈ s, -x ∈ s
h_conv : Convex ℝ s
h✝ : μ F * 2 ^ finrank ℝ E < μ s
h_vol : μ F < μ (2⁻¹ • s)
x y : ↥L
hxy : x ≠ y
h : ¬Disjoint (x +ᵥ 2⁻¹ • s) (y +ᵥ 2⁻¹ • s)
v : E
hv : 2 • v ∈ s
w : E
hw : 2 • w ∈ s
hvw : y +ᵥ w = x +ᵥ v
⊢ ↑(x - y) ∈ s

Tactic:
AddSubgroup.vadd_def,

Goal state after:
E : Type u_1
inst✝⁶ : MeasurableSpace E
μ : Measure E
F s : Set E
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : μ.IsAddHaarMeasure
L : AddSubgroup E
inst✝ : Countable ↥L
fund : IsAddFundamentalDomain (↥L) F μ
h_symm : ∀ x ∈ s, -x ∈ s
h_conv : Convex ℝ s
h✝ : μ F * 2 ^ finrank ℝ E < μ s
h_vol : μ F < μ (2⁻¹ • s)
x y : ↥L
hxy : x ≠ y
h : ¬Disjoint (x +ᵥ 2⁻¹ • s) (y +ᵥ 2⁻¹ • s)
v : E
hv : 2 • v ∈ s
w : E
hw : 2 • w ∈ s
hvw : ↑y +ᵥ w = ↑x +ᵥ v
⊢ ↑(x - y) ∈ s
Module: Mathlib.MeasureTheory.Group.GeometryOfNumbers

Goal state before:
pp: E : Type u_1
inst✝⁶ : MeasurableSpace E
μ : Measure E
F s : Set E
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : μ.IsAddHaarMeasure
L : AddSubgroup E
inst✝ : Countable ↥L
fund : IsAddFundamentalDomain (↥L) F μ
h_symm : ∀ x ∈ s, -x ∈ s
h_conv : Convex ℝ s
h✝ : μ F * 2 ^ finrank ℝ E < μ s
h_vol : μ F < μ (2⁻¹ • s)
x y : ↥L
hxy : x ≠ y
h : ¬Disjoint (x +ᵥ 2⁻¹ • s) (y +ᵥ 2⁻¹ • s)
v : E
hv : 2 • v ∈ s
w : E
hw : 2 • w ∈ s
hvw : w - v = ↑(x - y)
⊢ ↑(x - y) ∈ s

Tactic:
← hvw,

Goal state after:
E : Type u_1
inst✝⁶ : MeasurableSpace E
μ : Measure E
F s : Set E
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : μ.IsAddHaarMeasure
L : AddSubgroup E
inst✝ : Countable ↥L
fund : IsAddFundamentalDomain (↥L) F μ
h_symm : ∀ x ∈ s, -x ∈ s
h_conv : Convex ℝ s
h✝ : μ F * 2 ^ finrank ℝ E < μ s
h_vol : μ F < μ (2⁻¹ • s)
x y : ↥L
hxy : x ≠ y
h : ¬Disjoint (x +ᵥ 2⁻¹ • s) (y +ᵥ 2⁻¹ • s)
v : E
hv : 2 • v ∈ s
w : E
hw : 2 • w ∈ s
hvw : w - v = ↑(x - y)
⊢ w - v ∈ s
Module: Mathlib.MeasureTheory.Group.ModularCharacter

Goal state before:
pp: G : Type u_1
inst✝⁶ : TopologicalSpace G
inst✝⁵ : Group G
inst✝⁴ : IsTopologicalGroup G
inst✝³ : LocallyCompactSpace G
inst✝² : MeasurableSpace G
inst✝¹ : BorelSpace G
μ : Measure G
inst✝ : μ.IsHaarMeasure
g : G
ν : Measure G := haar
f : G → ℝ
f_cont : Continuous f
f_comp : HasCompactSupport ⇑{ toFun := f, continuous_toFun := f_cont }
f_nonneg : 0 ≤ { toFun := f, continuous_toFun := f_cont }
f_one : { toFun := f, continuous_toFun := f_cont } 1 ≠ 0
int_f_ne_zero : ∀ (μ₀ : Measure G) [μ₀.IsHaarMeasure], ∫ (x : G), f x ∂μ₀ ≠ 0
t : ∫ (x : G), f (x * g) ∂ν = ∫ (x : G), f (x * g) ∂ν.haarScalarFactor μ • μ
r : ↑(ν.haarScalarFactor μ) / ↑(ν.haarScalarFactor μ) = 1
⊢ (∫ (x : G), f (x * g) ∂ν.haarScalarFactor μ • μ) / ∫ (x : G), f x ∂ν.haarScalarFactor μ • μ =
    (ν.haarScalarFactor μ • ∫ (x : G), f (x * g) ∂μ) / ν.haarScalarFactor μ • ∫ (x : G), f x ∂μ

Tactic:
rw [integral_smul_nnreal_measure, integral_smul_nnreal_measure]

Goal state after:
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: case inr
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝ : TopologicalSpace α
Λ : (α →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
a : ℝ
f : α →C_c ℝ
ha : a ≤ 0
⊢ ↑(Λ (a • f).nnrealPart) - ↑(Λ (-(a • f)).nnrealPart) = (RingHom.id ℝ) a • (↑(Λ f.nnrealPart) - ↑(Λ (-f).nnrealPart))

Tactic:
· simp only [RingHom.id_apply, smul_eq_mul, ← (smul_neg a f),
            nnrealPart_smul_neg f ha, nnrealPart_smul_neg (-f) ha, map_smul,
            NNReal.coe_mul, Real.coe_toNNReal', neg_neg, sup_of_le_left (neg_nonneg.mpr ha)]
          ring

Goal state after:
Module: Mathlib.MeasureTheory.Integral.CurveIntegral.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
a b c d : E
ω : E → E →L[𝕜] F
γ : Path a b
hc : c = a
hd : d = b
⊢ curveIntegralFun ω (γ.cast hc hd) = curveIntegralFun ω γ

Tactic:
ext t

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
a b c d : E
ω : E → E →L[𝕜] F
γ : Path a b
hc : c = a
hd : d = b
t : ℝ
⊢ curveIntegralFun ω (γ.cast hc hd) t = curveIntegralFun ω γ t
Module: Mathlib.MeasureTheory.Integral.CurveIntegral.Basic

Goal state before:
pp: case inl
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
a b c : E
t : ℝ
ω : E → E →L[𝕜] F
γab : Path a b
γbc : Path b c
ht : t < 1 / 2
instE : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
H₁ : ⇑(γab.trans γbc).extend =ᶠ[𝓝 t] fun s ↦ γab.extend (2 * s)
ht₀ : t < 0
⊢ 𝓝[Icc 0 2] (2 * t) = 𝓝[I] (2 * t)

Tactic:
rw [notMem_closure_iff_nhdsWithin_eq_bot.mp, notMem_closure_iff_nhdsWithin_eq_bot.mp] <;>
        simp_intro h <;> linarith

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

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
s : Set α
f g : α → ℝ
μ : Measure α
hs_conn : IsConnected s
hs_meas : MeasurableSet s
hf : ContinuousOn f s
hg : IntegrableOn g s μ
hfg : IntegrableOn (fun x ↦ f x * g x) s μ
hg0 : ∀ᵐ (x : α) ∂μ.restrict s, 0 ≤ g x
ρ : α → ENNReal := fun x ↦ ENNReal.ofReal (g x)
ν : Measure α := μ.withDensity ρ
hρ_ae : AEMeasurable ρ (μ.restrict s)
hρ_top : ∀ᵐ (x : α) ∂μ.restrict s, ρ x < ⊤
h_toReal_ae : (fun x ↦ (ρ x).toReal) =ᵐ[μ.restrict s] g
heq : ∫ (x : α) in s, f x * g x ∂μ = ∫ (x : α) in s, f x ∂ν
hg1 : ∫ (x : α) in s, g x ∂μ = ∫ (x : α) in s, 1 ∂ν
hν0 : ¬∫ (x : α) in s, 1 ∂ν = 0
hν0' : (ν s).toReal ≠ 0
hνfin : ν s ≠ ⊤
hν0'' : ν s ≠ 0
hmul_ae : (fun x ↦ (ρ x).toReal * f x) =ᵐ[μ.restrict s] fun x ↦ f x * g x
⊢ IntegrableOn (fun x ↦ (ρ x).toReal * f x) s μ

Tactic:
rwa [integrableOn_congr_fun_ae hmul_ae]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral

Goal state before:
pp: case bound
E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
a : E
r : ℝ
hr : 0 ≤ r
hdf : ∀ᶠ (x : E) in 𝓝 a, DifferentiableAt ℝ f x
hderiv : fderiv ℝ f =O[𝓝 a] fun x ↦ ‖x - a‖ ^ r
C : ℝ
hC₀ : C > 0
hC : ∀ᶠ (x : E) in 𝓝 a, ‖fderiv ℝ f x‖ ≤ C * ‖‖x - a‖ ^ r‖
ε : ℝ
hε₀ : ε > 0
hε : ∀ y ∈ ball a ε, DifferentiableAt ℝ f y ∧ ‖fderiv ℝ f y‖ ≤ C * ‖‖y - a‖ ^ r‖
y : E
hy : y ∈ ball a ε
hsub : closedBall a ‖y - a‖ ⊆ ball a ε
⊢ ∀ x ∈ closedBall a ‖y - a‖, ‖fderiv ℝ f x‖ ≤ C * ‖y - a‖ ^ r

Tactic:
intro z hz

Goal state after:
case bound
E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
a : E
r : ℝ
hr : 0 ≤ r
hdf : ∀ᶠ (x : E) in 𝓝 a, DifferentiableAt ℝ f x
hderiv : fderiv ℝ f =O[𝓝 a] fun x ↦ ‖x - a‖ ^ r
C : ℝ
hC₀ : C > 0
hC : ∀ᶠ (x : E) in 𝓝 a, ‖fderiv ℝ f x‖ ≤ C * ‖‖x - a‖ ^ r‖
ε : ℝ
hε₀ : ε > 0
hε : ∀ y ∈ ball a ε, DifferentiableAt ℝ f y ∧ ‖fderiv ℝ f y‖ ≤ C * ‖‖y - a‖ ^ r‖
y : E
hy : y ∈ ball a ε
hsub : closedBall a ‖y - a‖ ⊆ ball a ε
z : E
hz : z ∈ closedBall a ‖y - a‖
⊢ ‖fderiv ℝ f z‖ ≤ C * ‖y - a‖ ^ r
Module: Mathlib.Order.Interval.Set.Union

Goal state before:

Tactic:
_ ⊆ Ioc (a 0) (a N) ∪ Ioc (a N) (a (N + 1)) := Ioc_subset_Ioc_union_Ioc
    _ ⊆ _ := by simpa [Finset.range_add_one] using
                  union_subset_union_right (Ioc (a N) (a (N + 1))) ih

Goal state after:
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic

Goal state before:
pp: case h
X : Type u_1
inst✝² : TopologicalSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
K : Compacts X
V : Set X
hVcp : IsCompact V
hKsubintV : K.carrier ⊆ interior V
hIsCompact_closure_interior : IsCompact (closure (interior V))
f : C(X, ℝ)
hsuppfsubV : tsupport ⇑f ⊆ interior V
hfeq1onK : EqOn (⇑f) 1 K.carrier
hfinicc : ∀ (x : X), f x ∈ Icc 0 1
hfHasCompactSupport : HasCompactSupport ⇑f
x : X
hx : x ∈ K
⊢ Real.toNNReal 1 = (f x).toNNReal

Tactic:
Real.toNNReal_eq_toNNReal_iff (zero_le_one' ℝ) (hfinicc x).1

Goal state after:
case h
X : Type u_1
inst✝² : TopologicalSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
K : Compacts X
V : Set X
hVcp : IsCompact V
hKsubintV : K.carrier ⊆ interior V
hIsCompact_closure_interior : IsCompact (closure (interior V))
f : C(X, ℝ)
hsuppfsubV : tsupport ⇑f ⊆ interior V
hfeq1onK : EqOn (⇑f) 1 K.carrier
hfinicc : ∀ (x : X), f x ∈ Icc 0 1
hfHasCompactSupport : HasCompactSupport ⇑f
x : X
hx : x ∈ K
⊢ 1 = f x
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
K₁ K₂ : Compacts X
disj : Disjoint ↑K₁ ↑K₂
⊢ ∀ b ∈ ⇑Λ '' {f | ∀ x ∈ K₁ ⊔ K₂, 1 ≤ f x}, rieszContentAux Λ K₁ + rieszContentAux Λ K₂ ≤ b

Tactic:
intro b ⟨f, ⟨hf, Λf_eq_b⟩⟩

Goal state after:
X : Type u_1
inst✝² : TopologicalSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
K₁ K₂ : Compacts X
disj : Disjoint ↑K₁ ↑K₂
b : ℝ≥0
f : X →C_c ℝ≥0
hf : f ∈ {f | ∀ x ∈ K₁ ⊔ K₂, 1 ≤ f x}
Λf_eq_b : Λ f = b
⊢ rieszContentAux Λ K₁ + rieszContentAux Λ K₂ ≤ b
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule

Goal state before:
pp: case inr.inl
f : ℝ → ℝ
a b : ℝ
h_df : DifferentiableOn ℝ f [[a, b]]
h_ddf : DifferentiableOn ℝ (_root_.derivWithin f [[a, b]]) [[a, b]]
ζ : ℝ
fpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f [[a, b]] x| ≤ ζ
N : ℕ
N_nonzero : 0 < N
h_eq : a = b
⊢ |trapezoidal_error f N a b| ≤ |b - a| ^ 3 * ζ / (12 * ↑N ^ 2)

Tactic:
simp [h_eq]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule

Goal state before:
pp: case inl
f : ℝ → ℝ
a b : ℝ
h_f_c2 : ContDiffOn ℝ 2 f [[a, b]]
ζ : ℝ
fpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f [[a, b]] x| ≤ ζ
N : ℕ
N_nonzero : 0 < N
h_eq : a = b
⊢ |trapezoidal_error f N a b| ≤ |b - a| ^ 3 * ζ / (12 * ↑N ^ 2)

Tactic:
simp [h_eq]

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

Goal state before:
pp: n : ℕ
c : Fin n → ℂ
R θ : Fin n → ℝ
⊢ torusMap c R θ = c ↔ R = 0

Tactic:
simp [funext_iff, torusMap, exp_ne_zero]

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : (Fin 1 → ℂ) → E
c : Fin 1 → ℂ
R : Fin 1 → ℝ
H₁ : (⇑(MeasurableEquiv.funUnique (Fin 1) ℝ).symm ⁻¹' Icc 0 fun x ↦ 2 * π) = Icc 0 (2 * π)
θ : Fin 1 → ℝ
i : Fin 1
⊢ torusMap c R θ i = circleMap (c 0) (R 0) (θ 0)

Tactic:
Subsingleton.elim i 0

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : (Fin 1 → ℂ) → E
c : Fin 1 → ℂ
R : Fin 1 → ℝ
H₁ : (⇑(MeasurableEquiv.funUnique (Fin 1) ℝ).symm ⁻¹' Icc 0 fun x ↦ 2 * π) = Icc 0 (2 * π)
θ : Fin 1 → ℝ
i : Fin 1
⊢ torusMap c R θ 0 = circleMap (c 0) (R 0) (θ 0)
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : MeasurableSpace X
inst✝¹ : BorelSpace X
Λ : (X →C_c ℝ) →ₚ[ℝ] ℝ
inst✝ : LocallyCompactSpace X
f : X →C_c ℝ
μ : Measure X := rieszMeasure Λ
K : Set X := tsupport ⇑f
ε : ℝ
hε : 0 < ε
a b : ℝ
hab : a < b ∧ range ⇑f ⊆ Ioo a b
N : ℕ
hN : 0 < N
ε' : ℝ := (b - a) / ↑N
hε' : 0 < ε' ∧ ε' * (2 * μ.real K + |a| + b + ε') ≤ ε
E : Fin N → Set X
hE :
  tsupport ⇑f = ⋃ j, E j ∧
    univ.PairwiseDisjoint E ∧
      (∀ (n : Fin N), ∀ x ∈ E n, a + ε' * ↑↑n < f x ∧ f x ≤ a + ε' * (↑↑n + 1)) ∧ ∀ (n : Fin N), MeasurableSet (E n)
y : Fin N → ℝ := fun n ↦ a + ε' * (↑↑n + 1)
hE' : ∀ (n : Fin N), μ (E n) ≠ ∞
V : Fin N → Opens X
hV : ∀ (n : Fin N), E n ⊆ ↑(V n) ∧ (∀ x ∈ V n, f x < y n + ε') ∧ μ ↑(V n) ≤ μ (E n) + ENNReal.ofReal (ε' / ↑N)
⊢ Λ f ≤ ∫ (x : X), f x ∂rieszMeasure Λ + ε

Tactic:
obtain ⟨g, hg⟩ : ∃ g : Fin N → C_c(X, ℝ), (∀ n, tsupport (g n) ⊆ (V n).carrier) ∧
      EqOn (∑ n : Fin N, (g n)) 1 (tsupport f.toFun) ∧ (∀ n x, (g n) x ∈ Icc 0 1) ∧
      ∀ n, HasCompactSupport (g n) := by
    have : tsupport f ⊆ ⋃ n, (V n).carrier := calc
      _ = ⋃ j, E j := hE.1
      _ ⊆ _ := by gcongr with n; exact (hV n).1
    obtain ⟨g', hg⟩ := exists_continuous_sum_one_of_isOpen_isCompact (fun n ↦ (V n).2) f.2 this
    exact ⟨fun n ↦ ⟨g' n, hg.2.2.2 n⟩, hg⟩

Goal state after:
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : MeasurableSpace X
inst✝¹ : BorelSpace X
Λ : (X →C_c ℝ) →ₚ[ℝ] ℝ
inst✝ : LocallyCompactSpace X
f : X →C_c ℝ
μ : Measure X := rieszMeasure Λ
K : Set X := tsupport ⇑f
ε : ℝ
hε : 0 < ε
a b : ℝ
hab : a < b ∧ range ⇑f ⊆ Ioo a b
N : ℕ
hN : 0 < N
ε' : ℝ := (b - a) / ↑N
hε' : 0 < ε' ∧ ε' * (2 * μ.real K + |a| + b + ε') ≤ ε
E : Fin N → Set X
hE :
  tsupport ⇑f = ⋃ j, E j ∧
    univ.PairwiseDisjoint E ∧
      (∀ (n : Fin N), ∀ x ∈ E n, a + ε' * ↑↑n < f x ∧ f x ≤ a + ε' * (↑↑n + 1)) ∧ ∀ (n : Fin N), MeasurableSet (E n)
y : Fin N → ℝ := fun n ↦ a + ε' * (↑↑n + 1)
hE' : ∀ (n : Fin N), μ (E n) ≠ ∞
V : Fin N → Opens X
hV : ∀ (n : Fin N), E n ⊆ ↑(V n) ∧ (∀ x ∈ V n, f x < y n + ε') ∧ μ ↑(V n) ≤ μ (E n) + ENNReal.ofReal (ε' / ↑N)
g : Fin N → X →C_c ℝ
hg :
  (∀ (n : Fin N), tsupport ⇑(g n) ⊆ (V n).carrier) ∧
    EqOn (∑ n, ⇑(g n)) 1 (tsupport f.toFun) ∧
      (∀ (n : Fin N) (x : X), (g n) x ∈ Icc 0 1) ∧ ∀ (n : Fin N), HasCompactSupport ⇑(g n)
⊢ Λ f ≤ ∫ (x : X), f x ∂rieszMeasure Λ + ε