mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Calculus.ContDiff.Defs

Goal state before:
pp: 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
x : E
n : ℕ∞ω
h : ContDiffAt 𝕜 n f x
h' : n ≠ ∞
⊢ ∀ᶠ (y : E) in 𝓝 x, ContDiffAt 𝕜 n f y

Tactic:
by
  simpa [nhdsWithin_univ] using! ContDiffWithinAt.eventually h h'

Goal state after:
Module: Mathlib.Topology.MetricSpace.CauSeqFilter

Goal state before:
pp: β : Type v
inst✝ : NormedField β
f : ℕ → β
hf : CauchySeq f
hf1 : (map f atTop).NeBot
hf2 : ∀ s ∈ uniformity β, ∃ t ∈ map f atTop, t ×ˢ t ⊆ s
ε : ℝ
hε : ε > 0
t : Set β
ht : t ∈ map f atTop
htsub : t ×ˢ t ⊆ {x | dist x.1 x.2 < ε}
⊢ ∃ i, ∀ j ≥ i, ‖f j - f i‖ < ε

Tactic:
simp only [mem_map, mem_atTop_sets, mem_preimage] at ht

Goal state after:
β : Type v
inst✝ : NormedField β
f : ℕ → β
hf : CauchySeq f
hf1 : (map f atTop).NeBot
hf2 : ∀ s ∈ uniformity β, ∃ t ∈ map f atTop, t ×ˢ t ⊆ s
ε : ℝ
hε : ε > 0
t : Set β
htsub : t ×ˢ t ⊆ {x | dist x.1 x.2 < ε}
ht : ∃ a, ∀ (b : ℕ), a ≤ b → f b ∈ t
⊢ ∃ i, ∀ j ≥ i, ‖f j - f i‖ < ε
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: case h₂
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
n : ℕ
c : OrderedFinpartition n
f₁ f₂ : F [×c.length]→L[𝕜] G
g₁ g₂ : (i : Fin c.length) → E [×c.partSize i]→L[𝕜] F
⊢ ‖c.compAlongOrderedFinpartition f₁ g₂ - c.compAlongOrderedFinpartition f₂ g₂‖ ≤ ‖f₁ - f₂‖ * ∏ i, ‖g₂ i‖

Tactic:
exact c.norm_compAlongOrderedFinpartition_le (f₁ - f₂) g₂

Goal state after:
Module: Mathlib.Topology.Order.ExtendFrom

Goal state before:
pp: case neg.inl
α : Type u_1
β : Type u_2
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : DenselyOrdered α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
f : α → β
a b : α
la lb : β
inst✝ : RegularSpace β
hf : ContinuousOn f (uIoo a b)
ha : Tendsto f (𝓝[uIoo a b] a) (𝓝 la)
hb : Tendsto f (𝓝[uIoo a b] b) (𝓝 lb)
hab : a ≠ b
hab' : a < b
⊢ ContinuousOn (extendFrom (uIoo a b) f) (uIcc a b)

Tactic:
simp only [hab', uIoo_of_lt, nhdsWithin_Ioo_eq_nhdsGT, nhdsWithin_Ioo_eq_nhdsLT,
      uIcc_of_lt] at ha hb hf ⊢

Goal state after:
case neg.inl
α : Type u_1
β : Type u_2
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : DenselyOrdered α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
f : α → β
a b : α
la lb : β
inst✝ : RegularSpace β
hab : a ≠ b
hab' : a < b
ha : Tendsto f (𝓝[>] a) (𝓝 la)
hb : Tendsto f (𝓝[<] b) (𝓝 lb)
hf : ContinuousOn f (Ioo a b)
⊢ ContinuousOn (extendFrom (Ioo a b) f) (Icc a b)
Module: Mathlib.Analysis.Calculus.Deriv.MeanValue

Goal state before:
pp: f : ℝ → ℝ
a : ℝ
hf : Tendsto (derivWithin f (Ioi a)) (𝓝[>] a) atTop
hcont_at_a : ContinuousWithinAt f (Ici a) a
hdiff : HasDerivWithinAt f (derivWithin f (Ioi a) a) (Ioi a) a
⊢ False

Tactic:
rw [hasDerivWithinAt_iff_tendsto_slope, Set.sdiff_singleton_eq_self self_notMem_Ioi] at hdiff

Goal state after:
f : ℝ → ℝ
a : ℝ
hf : Tendsto (derivWithin f (Ioi a)) (𝓝[>] a) atTop
hcont_at_a : ContinuousWithinAt f (Ici a) a
hdiff : Tendsto (slope f a) (𝓝[>] a) (𝓝 (derivWithin f (Ioi a) a))
⊢ False
Module: Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory

Goal state before:

Tactic:
∫⁻ x, g0 x + g1 x ∂μ = (∫⁻ x, g0 x ∂μ) + ∫⁻ x, g1 x ∂μ :=
        lintegral_add_left g0_cont.measurable _
      _ ≤ (∫⁻ x, f x ∂μ) + ε / 2 + (0 + ε / 2) := by
        refine add_le_add ?_ ?_
        · convert! g0_int using 2
          exact lintegral_congr_ae (fmeas.ae_eq_mk.fun_comp _)
        · convert! g1_int
          simp only [smeas, μs, lintegral_const, Set.univ_inter, MeasurableSet.univ,
            lintegral_indicator, mul_zero, restrict_apply]
      _ = (∫⁻ x, f x ∂μ) + ε := by simp only [add_assoc, ENNReal.add_halves, zero_add]

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: R : Type u_7
M : Type u_8
inst✝⁵ : Ring R
inst✝⁴ : LinearOrder R
inst✝³ : AddLeftStrictMono R
inst✝² : AddCommGroup M
inst✝¹ : Nontrivial M
inst✝ : Module R M
C : ConvexCone R M
h : Submodule.span R ↑C = ⊤
x : M
⊢ x ∈ ↑C - ↑C

Tactic:
have hne : (C : Set M).Nonempty := Set.nonempty_iff_ne_empty.2 fun h' => by simp [h'] at h

Goal state after:
R : Type u_7
M : Type u_8
inst✝⁵ : Ring R
inst✝⁴ : LinearOrder R
inst✝³ : AddLeftStrictMono R
inst✝² : AddCommGroup M
inst✝¹ : Nontrivial M
inst✝ : Module R M
C : ConvexCone R M
h : Submodule.span R ↑C = ⊤
x : M
hne : (↑C).Nonempty
⊢ x ∈ ↑C - ↑C
Module: Mathlib.Geometry.Convex.Cone.Pointed

Goal state before:
pp: R : Type u_1
E : Type u_2
inst✝⁴ : Ring R
inst✝³ : LinearOrder R
inst✝² : IsOrderedRing R
inst✝¹ : AddCommGroup E
inst✝ : Module R E
C : PointedCone R E
⊢ C.lineal = sSup {S | ↑S ≤ C}

Tactic:
by
  simp_rw [gc_ofSubmodule_lineal.le_iff_le, Set.Iic_def, csSup_Iic]

Goal state after:
Module: Mathlib.Analysis.Convex.Cone.Extension

Goal state before:
pp: case refine_1
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : PointedCone ℝ E
f : E →ₗ.[ℝ] ℝ
nonneg : ∀ (x : ↥f.domain), ↑x ∈ s → 0 ≤ ↑f x
dense : ∀ (y : E), ∃ x, ↑x + y ∈ s
hdom : f.domain ≠ ⊤
y : E
hy : y ∉ f.domain
Sp : Set ℝ := ↑f '' {x | ↑x + y ∈ s}
Sn : Set ℝ := ↑f '' {x | -↑x - y ∈ s}
x : ↥f.domain
hx : -↑(-x) + -y ∈ s
⊢ {x | -↑x - y ∈ s}.Nonempty

Tactic:
← sub_eq_add_neg

Goal state after:
case refine_1
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : PointedCone ℝ E
f : E →ₗ.[ℝ] ℝ
nonneg : ∀ (x : ↥f.domain), ↑x ∈ s → 0 ≤ ↑f x
dense : ∀ (y : E), ∃ x, ↑x + y ∈ s
hdom : f.domain ≠ ⊤
y : E
hy : y ∉ f.domain
Sp : Set ℝ := ↑f '' {x | ↑x + y ∈ s}
Sn : Set ℝ := ↑f '' {x | -↑x - y ∈ s}
x : ↥f.domain
hx : -↑(-x) - y ∈ s
⊢ {x | -↑x - y ∈ s}.Nonempty
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → E →L[𝕜] F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
M :
  ∀ (e p q e' p' q' : ℕ),
    n e ≤ p → n e ≤ q → n e' ≤ p' → n e' ≤ q' → e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
L0 : ℕ → E →L[𝕜] F := fun e ↦ L e (n e) (n e)
⊢ CauchySeq L0

Tactic:
rw [Metric.cauchySeq_iff']

Goal state after:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → E →L[𝕜] F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
M :
  ∀ (e p q e' p' q' : ℕ),
    n e ≤ p → n e ≤ q → n e' ≤ p' → n e' ≤ q' → e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
L0 : ℕ → E →L[𝕜] F := fun e ↦ L e (n e) (n e)
⊢ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (L0 n) (L0 N) < ε
Module: Mathlib.Analysis.SpecialFunctions.Log.Deriv

Goal state before:
pp: x : ℝ
h : |x| < 1
term : ℕ → ℝ := fun n ↦ -1 * ((-x) ^ (n + 1) / (↑n + 1)) + x ^ (n + 1) / (↑n + 1)
h_term_eq_goal : (term ∘ fun x ↦ 2 * x) = fun k ↦ 2 * (1 / (2 * ↑k + 1)) * x ^ (2 * k + 1)
⊢ ∀ x ∉ Set.range fun x ↦ 2 * x, term x = 0

Tactic:
intro m hm
    rw [range_two_mul, Set.mem_ofPred_eq, ← Nat.even_add_one] at hm
    dsimp [term]
    rw [Even.neg_pow hm, neg_one_mul, neg_add_cancel]

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

Goal state before:
pp: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
f : α →ₛ ℝ≥0
int_f : ∫⁻ (x : α), ↑(f x) ∂μ ≠ ∞
ε : ℝ≥0∞
ε0 : ε ≠ 0
⊢ ∃ g, (∀ (x : α), g x ≤ f x) ∧ UpperSemicontinuous g ∧ ∫⁻ (x : α), ↑(f x) ∂μ ≤ ∫⁻ (x : α), ↑(g x) ∂μ + ε

Tactic:
induction f using MeasureTheory.SimpleFunc.induction generalizing ε with
  | @const c s hs =>
    by_cases hc : c = 0
    · exact ⟨fun _ => 0, by simp [hc, upperSemicontinuous_const]⟩
    have μs_lt_top : μ s < ∞ := by simpa [hs, hc, ENNReal.mul_eq_top, lt_top_iff_ne_top] using int_f
    have : (0 : ℝ≥0∞) < ε / c := ENNReal.div_pos_iff.2 ⟨ε0, ENNReal.coe_ne_top⟩
    obtain ⟨F, Fs, F_closed, μF⟩ : ∃ (F : _), F ⊆ s ∧ IsClosed F ∧ μ s < μ F + ε / c :=
      hs.exists_isClosed_lt_add μs_lt_top.ne this.ne'
    refine
      ⟨Set.indicator F fun _ => c, fun x => ?_, F_closed.upperSemicontinuous_indicator zero_le,
        ?_⟩
    · simp only [SimpleFunc.coe_const, SimpleFunc.const_zero, SimpleFunc.coe_zero,
        Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise, ← Function.const_def]
      grw [Fs]
    · suffices (c : ℝ≥0∞) * μ s ≤ c * μ F + ε by
        simpa only [hs, F_closed.measurableSet, SimpleFunc.coe_const, Function.const_apply,
          lintegral_const, ENNReal.coe_indicator, Set.univ_inter, MeasurableSet.univ,
          SimpleFunc.const_zero, lintegral_indicator, SimpleFunc.coe_zero,
          Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise, Measure.restrict_apply]
      calc
        (c : ℝ≥0∞) * μ s ≤ c * (μ F + ε / c) := by grw [μF]
        _ = c * μ F + ε := by
          simp_rw [mul_add]
          rw [ENNReal.mul_div_cancel _ ENNReal.coe_ne_top]
          simpa using hc
  | @add f₁ f₂ _ h₁ h₂ =>
    have A : ((∫⁻ x : α, f₁ x ∂μ) + ∫⁻ x : α, f₂ x ∂μ) ≠ ⊤ := by
      rwa [← lintegral_add_left f₁.measurable.coe_nnreal_ennreal]
    rcases h₁ (ENNReal.add_ne_top.1 A).1 (ENNReal.half_pos ε0).ne' with
      ⟨g₁, f₁_le_g₁, g₁cont, g₁int⟩
    rcases h₂ (ENNReal.add_ne_top.1 A).2 (ENNReal.half_pos ε0).ne' with
      ⟨g₂, f₂_le_g₂, g₂cont, g₂int⟩
    refine
      ⟨fun x => g₁ x + g₂ x, fun x => add_le_add (f₁_le_g₁ x) (f₂_le_g₂ x), g₁cont.add g₂cont, ?_⟩
    simp only [SimpleFunc.coe_add, ENNReal.coe_add, Pi.add_apply]
    rw [lintegral_add_left f₁.measurable.coe_nnreal_ennreal,
      lintegral_add_left g₁cont.measurable.coe_nnreal_ennreal]
    convert! add_le_add g₁int g₂int using 1
    conv_lhs => rw [← ENNReal.add_halves ε]
    abel

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.Separation

Goal state before:
pp: case inr.inl
E : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
s : Set E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ContinuousSMul ℝ E
hs₁ : Convex ℝ s
hs₂ : IsOpen s
a₀ : E
ha₀ : a₀ ∈ s
ht : Convex ℝ ∅
disj : Disjoint s ∅
⊢ ∃ f u, (∀ a ∈ s, f a < u) ∧ ∀ b ∈ ∅, u ≤ f b

pp: case inr.inr
E : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
s t : Set E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ContinuousSMul ℝ E
hs₁ : Convex ℝ s
hs₂ : IsOpen s
ht : Convex ℝ t
disj : Disjoint s t
a₀ : E
ha₀ : a₀ ∈ s
b₀ : E
hb₀ : b₀ ∈ t
⊢ ∃ f u, (∀ a ∈ s, f a < u) ∧ ∀ b ∈ t, u ≤ f b

Tactic:
· exact ⟨0, 1, fun a _ha => zero_lt_one, by simp⟩

Goal state after:
case inr.inr
E : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
s t : Set E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ContinuousSMul ℝ E
hs₁ : Convex ℝ s
hs₂ : IsOpen s
ht : Convex ℝ t
disj : Disjoint s t
a₀ : E
ha₀ : a₀ ∈ s
b₀ : E
hb₀ : b₀ ∈ t
⊢ ∃ f u, (∀ a ∈ s, f a < u) ∧ ∀ b ∈ t, u ≤ f b
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
x : ℝ
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
M :
  ∀ (e p q e' p' q' : ℕ), n e ≤ p → n e ≤ q → n e' ≤ p' → n e' ≤ q' → e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * (1 / 2) ^ e
L0 : ℕ → F := fun e ↦ L e (n e) (n e)
⊢ CauchySeq L0

Tactic:
rw [Metric.cauchySeq_iff']

Goal state after:
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
x : ℝ
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
M :
  ∀ (e p q e' p' q' : ℕ), n e ≤ p → n e ≤ q → n e' ≤ p' → n e' ≤ q' → e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * (1 / 2) ^ e
L0 : ℕ → F := fun e ↦ L e (n e) (n e)
⊢ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (L0 n) (L0 N) < ε
Module: Mathlib.MeasureTheory.Integral.DominatedConvergence

Goal state before:
pp: α : Type u_1
G : Type u_3
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
m : MeasurableSpace α
μ : Measure α
ι : Type u_4
l : Filter ι
inst✝ : l.IsCountablyGenerated
F : ι → α → G
f : α → G
bound : α → ℝ
hF_meas : ∀ᶠ (n : ι) in l, AEStronglyMeasurable (F n) μ
h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a
bound_integrable : Integrable bound μ
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ F n a) l (𝓝 (f a))
⊢ Tendsto (fun n ↦ setToFun μ (weightedSMul μ) ⋯ (F n)) l (𝓝 (setToFun μ (weightedSMul μ) ⋯ f))

Tactic:
exact tendsto_setToFun_filter_of_dominated_convergence (dominatedFinMeasAdditive_weightedSMul μ)
    bound hF_meas h_bound bound_integrable h_lim

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

Goal state before:
pp: E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
μ : Measure ℝ
ι : Type u_3
l : Filter ι
inst✝ : l.IsCountablyGenerated
F : ι → ℝ → E
bound : ℝ → ℝ
hF_meas : ∀ᶠ (n : ι) in l, AEStronglyMeasurable (F n) (μ.restrict (Ι a b))
h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (x : ℝ) ∂μ, x ∈ Ι a b → ‖F n x‖ ≤ bound x
bound_integrable : IntervalIntegrable bound μ a b
h_lim : ∀ᵐ (x : ℝ) ∂μ, x ∈ Ι a b → Tendsto (fun n ↦ F n x) l (𝓝 (f x))
⊢ Tendsto (fun n ↦ ∫ (x : ℝ) in a..b, F n x ∂μ) l (𝓝 (∫ (x : ℝ) in a..b, f x ∂μ))

Tactic:
by
  simp only [intervalIntegrable_iff, intervalIntegral_eq_integral_uIoc,
    ← ae_restrict_iff' (α := ℝ) (μ := μ) measurableSet_uIoc] at *
  exact tendsto_const_nhds.smul <|
    tendsto_integral_filter_of_dominated_convergence bound hF_meas h_bound bound_integrable h_lim

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

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a : ℝ
⊢ FTCFilter a (𝓝[{a}] a) ⊥

Tactic:
nhdsWithin,

Goal state after:
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a : ℝ
⊢ FTCFilter a (𝓝 a ⊓ 𝓟 {a}) ⊥
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus

Goal state before:
pp: case pos
ι : Type u_1
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
f : ℝ → E
c : E
l l' : Filter ℝ
lt : Filter ι
μ : Measure ℝ
u v : ι → ℝ
inst✝¹ : l'.IsMeasurablyGenerated
inst✝ : TendstoIxxClass Ioc l l'
hfm : StronglyMeasurableAtFilter f l' μ
hf : Tendsto f (l' ⊓ ae μ) (𝓝 c)
hl : μ.FiniteAtFilter l'
hu : Tendsto u lt l
hv : Tendsto v lt l
hE : CompleteSpace E
A :
  (fun i ↦ ∫ (x : ℝ) in Ioc (u i) (v i), f x ∂μ - μ.real (Ioc (u i) (v i)) • c) =o[lt] fun i ↦ μ.real (Ioc (u i) (v i))
B :
  (fun i ↦ ∫ (x : ℝ) in Ioc (v i) (u i), f x ∂μ - μ.real (Ioc (v i) (u i)) • c) =o[lt] fun i ↦ μ.real (Ioc (v i) (u i))
⊢ (fun t ↦ ∫ (x : ℝ) in u t..v t, f x ∂μ - ∫ (x : ℝ) in u t..v t, c ∂μ) =o[lt] fun t ↦ ∫ (x : ℝ) in u t..v t, 1 ∂μ

Tactic:
integral_const',

Goal state after:
case pos
ι : Type u_1
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
f : ℝ → E
c : E
l l' : Filter ℝ
lt : Filter ι
μ : Measure ℝ
u v : ι → ℝ
inst✝¹ : l'.IsMeasurablyGenerated
inst✝ : TendstoIxxClass Ioc l l'
hfm : StronglyMeasurableAtFilter f l' μ
hf : Tendsto f (l' ⊓ ae μ) (𝓝 c)
hl : μ.FiniteAtFilter l'
hu : Tendsto u lt l
hv : Tendsto v lt l
hE : CompleteSpace E
A :
  (fun i ↦ ∫ (x : ℝ) in Ioc (u i) (v i), f x ∂μ - μ.real (Ioc (u i) (v i)) • c) =o[lt] fun i ↦ μ.real (Ioc (u i) (v i))
B :
  (fun i ↦ ∫ (x : ℝ) in Ioc (v i) (u i), f x ∂μ - μ.real (Ioc (v i) (u i)) • c) =o[lt] fun i ↦ μ.real (Ioc (v i) (u i))
⊢ (fun t ↦ ∫ (x : ℝ) in u t..v t, f x ∂μ - (μ.real (Ioc (u t) (v t)) - μ.real (Ioc (v t) (u t))) • c) =o[lt] fun t ↦
    (μ.real (Ioc (u t) (v t)) - μ.real (Ioc (v t) (u t))) • 1
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: F : Type u_1
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
f : ℝ → F
inst✝ : CompleteSpace F
⊢ StronglyMeasurable fun x ↦ derivWithin f (Ici x) x

Tactic:
borelize F
  apply stronglyMeasurable_iff_measurable_separable.2 ⟨measurable_derivWithin_Ici f, ?_⟩
  obtain ⟨t, t_count, ht⟩ : ∃ t : Set ℝ, t.Countable ∧ Dense t := exists_countable_dense ℝ
  suffices H : range (fun x ↦ derivWithin f (Ici x) x) ⊆ closure (Submodule.span ℝ (f '' t)) from
    IsSeparable.mono (t_count.image f).isSeparable.span.closure H
  rintro - ⟨x, rfl⟩
  suffices H' : range (fun y ↦ derivWithin f (Ici x) y) ⊆ closure (Submodule.span ℝ (f '' t)) from
    H' (mem_range_self _)
  apply range_derivWithin_subset_closure_span_image
  calc Ici x
    = closure (Ioi x ∩ closure t) := by simp [dense_iff_closure_eq.1 ht]
  _ ⊆ closure (closure (Ioi x ∩ t)) := by
      apply closure_mono
      simpa [inter_comm] using (isOpen_Ioi (a := x)).closure_inter (s := t)
  _ ⊆ closure (Ici x ∩ t) := by
      rw [closure_closure]
      exact closure_mono (inter_subset_inter_left _ Ioi_subset_Ici_self)

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

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : TopologicalSpace X
μ : Measure ℝ
inst✝¹ : NullSingletonClass μ
inst✝ : IsLocallyFiniteMeasure μ
f : X → ℝ → E
a₀ : ℝ
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Function.uncurry f)
q : X
b₀ ε : ℝ
εpos : ε > 0
a : ℝ
a_lt : a < a₀ ∧ a < b₀
b : ℝ
lt_b : a₀ < b ∧ b₀ < b
this : IsCompact ({q} ×ˢ Icc a b)
⊢ ∀ᶠ (x : X × ℝ) in 𝓝 (q, b₀),
    dist (∫ (t : ℝ) in a₀..x.2, f x.1 t ∂μ) (∫ (t : ℝ) in a₀..(q, b₀).2, f (q, b₀).1 t ∂μ) < ε

Tactic:
obtain ⟨M, hM⟩ := this.bddAbove_image hf.norm.continuousOn

Goal state after:
E : Type u_1
X : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : TopologicalSpace X
μ : Measure ℝ
inst✝¹ : NullSingletonClass μ
inst✝ : IsLocallyFiniteMeasure μ
f : X → ℝ → E
a₀ : ℝ
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Function.uncurry f)
q : X
b₀ ε : ℝ
εpos : ε > 0
a : ℝ
a_lt : a < a₀ ∧ a < b₀
b : ℝ
lt_b : a₀ < b ∧ b₀ < b
this : IsCompact ({q} ×ˢ Icc a b)
M : ℝ
hM : M ∈ upperBounds ((fun x ↦ ‖Function.uncurry f x‖) '' {q} ×ˢ Icc a b)
⊢ ∀ᶠ (x : X × ℝ) in 𝓝 (q, b₀),
    dist (∫ (t : ℝ) in a₀..x.2, f x.1 t ∂μ) (∫ (t : ℝ) in a₀..(q, b₀).2, f (q, b₀).1 t ∂μ) < ε
Module: Mathlib.Topology.Algebra.Order.Floor

Goal state before:
pp: α : Type u_1
inst✝⁵ : Ring α
inst✝⁴ : LinearOrder α
inst✝³ : FloorRing α
inst✝² : TopologicalSpace α
inst✝¹ : IsStrictOrderedRing α
inst✝ : OrderClosedTopology α
x : α
h₁ : ↑(⌈x⌉ - 1) < x
⊢ x ≤ ↑(⌈x⌉ - 1) + 1

Tactic:
rw [cast_sub, cast_one, sub_add_cancel]; exact le_ceil _

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

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
a : ℝ
h : Measurable ⇑(measurableEquivIoc T a)
s : Set ↑(Ioc a (a + T))
hs : MeasurableSet s
⊢ (Measure.map (⇑(equivIoc T a)) volume) s = (Measure.comap Subtype.val volume) s

Tactic:
comap_apply _ Subtype.val_injective (fun _ ↦ measurableSet_Ioc.subtype_image) _ hs,

Goal state after:
T : ℝ
hT : Fact (0 < T)
a : ℝ
h : Measurable ⇑(measurableEquivIoc T a)
s : Set ↑(Ioc a (a + T))
hs : MeasurableSet s
⊢ (Measure.map (⇑(equivIoc T a)) volume) s = volume (Subtype.val '' s)
Module: Mathlib.MeasureTheory.Measure.Haar.Quotient

Goal state before:
pp: case neZeroV.a
G : Type u_1
inst✝¹⁴ : Group G
inst✝¹³ : MeasurableSpace G
inst✝¹² : TopologicalSpace G
inst✝¹¹ : IsTopologicalGroup G
inst✝¹⁰ : BorelSpace G
inst✝⁹ : PolishSpace G
Γ : Subgroup G
inst✝⁸ : Γ.Normal
inst✝⁷ : T2Space (G ⧸ Γ)
inst✝⁶ : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
inst✝⁵ : Countable ↥Γ
ν : Measure G
inst✝⁴ : ν.IsHaarMeasure
inst✝³ : ν.IsMulRightInvariant
inst✝² : SigmaFinite ν
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥Γ.op) 𝓕 ν
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : SigmaFinite μ
V : Set (G ⧸ Γ)
hV : (interior V).Nonempty
meas_V : MeasurableSet V
hμK : μ V = ν (QuotientGroup.mk ⁻¹' V ∩ 𝓕)
neTopV : μ V ≠ ∞
c_eq_zero : ν (QuotientGroup.mk ⁻¹' V ∩ 𝓕) = 0
⊢ ν (interior (QuotientGroup.mk ⁻¹' V)) = 0

Tactic:
apply measure_mono_null (h := interior_subset)

Goal state after:
case neZeroV.a
G : Type u_1
inst✝¹⁴ : Group G
inst✝¹³ : MeasurableSpace G
inst✝¹² : TopologicalSpace G
inst✝¹¹ : IsTopologicalGroup G
inst✝¹⁰ : BorelSpace G
inst✝⁹ : PolishSpace G
Γ : Subgroup G
inst✝⁸ : Γ.Normal
inst✝⁷ : T2Space (G ⧸ Γ)
inst✝⁶ : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
inst✝⁵ : Countable ↥Γ
ν : Measure G
inst✝⁴ : ν.IsHaarMeasure
inst✝³ : ν.IsMulRightInvariant
inst✝² : SigmaFinite ν
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥Γ.op) 𝓕 ν
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : SigmaFinite μ
V : Set (G ⧸ Γ)
hV : (interior V).Nonempty
meas_V : MeasurableSet V
hμK : μ V = ν (QuotientGroup.mk ⁻¹' V ∩ 𝓕)
neTopV : μ V ≠ ∞
c_eq_zero : ν (QuotientGroup.mk ⁻¹' V ∩ 𝓕) = 0
⊢ ν (QuotientGroup.mk ⁻¹' V) = 0
Module: Mathlib.MeasureTheory.Measure.Haar.Quotient

Goal state before:
pp: G : Type u_1
inst✝¹² : Group G
inst✝¹¹ : MeasurableSpace G
inst✝¹⁰ : TopologicalSpace G
inst✝⁹ : IsTopologicalGroup G
inst✝⁸ : BorelSpace G
inst✝⁷ : PolishSpace G
Γ : Subgroup G
inst✝⁶ : Γ.Normal
inst✝⁵ : T2Space (G ⧸ Γ)
inst✝⁴ : SecondCountableTopology (G ⧸ Γ)
inst✝³ : Countable ↥Γ
ν : Measure G
inst✝² : ν.IsHaarMeasure
inst✝¹ : ν.IsMulRightInvariant
inst✝ : SigmaFinite ν
K : PositiveCompacts (G ⧸ Γ)
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥Γ.op) 𝓕 ν
h𝓕_finite : ν 𝓕 ≠ ∞
c : ℝ≥0
μ : Measure (G ⧸ Γ) := ↑c • haarMeasure K
hμK : μ ↑K = ↑c
⊢ SigmaFinite μ

Tactic:
exact SMul.sigmaFinite c

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

Goal state before:
pp: E : Type u_1
inst✝ : NormedAddCommGroup E
c : ℂ
R : ℝ
f₁ f₂ : ℂ → E
hf : f₁ =ᶠ[codiscreteWithin (sphere c |R|)] f₂
hf₁ : CircleIntegrable f₁ c R
hR : ¬R = 0
⊢ Ι 0 (2 * π) ⊆ univ

Tactic:
by simp only [Set.subset_univ]

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
ι : Type u_2
s : Finset ι
f : ι → ℂ → E
c : ℂ
R : ℝ
h : ∀ i ∈ s, CircleIntegrable (f i) c R
⊢ ∮ (z : ℂ) in C(c, R), ∑ i ∈ s, f i z = ∑ i ∈ s, ∮ (z : ℂ) in C(c, R), f i z

Tactic:
by
  simp only [circleIntegral, Finset.smul_sum,
    intervalIntegral.integral_finsetSum fun i hi ↦ (h i hi).out]

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

Goal state before:
pp: case inl.inr
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
a b : Fin (n + 1) → ℝ
hle : a ≤ b
f : (Fin (n + 1) → ℝ) → Fin (n + 1) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hc : ContinuousOn f (Set.Icc a b)
Hd : ∀ x ∈ (Set.univ.pi fun i ↦ Set.Ioo (a i) (b i)) \ s, HasFDerivAt f (f' x) x
Hi : IntegrableOn (fun x ↦ ∑ i, (f' x) (e i) i) (Set.Icc a b) volume
i : Fin (n + 1)
hi : a i = b i
hi' : Set.Ioc (a i) (b i) = ∅
this : (Set.univ.pi fun j ↦ Set.Ioc (a j) (b j)) = ∅
j : Fin (n + 1)
hne : i ≠ j
⊢ (∫ (x : Fin n → ℝ) in Set.Icc (a ∘ j.succAbove) (b ∘ j.succAbove), f (j.insertNth (b j) x) j) -
      ∫ (x : Fin n → ℝ) in Set.Icc (a ∘ j.succAbove) (b ∘ j.succAbove), f (j.insertNth (a j) x) j =
    0

Tactic:
rcases Fin.exists_succAbove_eq hne with ⟨i, rfl⟩
      have : Icc (a ∘ j.succAbove) (b ∘ j.succAbove) =ᵐ[volume] (∅ : Set ℝⁿ) := by
        rw [ae_eq_empty, Real.volume_Icc_pi, prod_eq_zero (Finset.mem_univ i)]
        simp [hi]
      rw [setIntegral_congr_set this, setIntegral_congr_set this, setIntegral_empty,
        setIntegral_empty, sub_self]

Goal state after: