awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.MeasureTheory.Integral.DominatedConvergence
Goal:
E : Type u_1
X : Type u_2
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : TopologicalSpace X
μ : Measure ℝ
inst_3 : NoAtoms μ
inst_4 : 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 (SProd.sprod (α := Set X) {q} (Icc a b))
M : ℝ
hM :
  Membership.mem (γ := Set ℝ)
    (upperBounds ((fun (x : X × ℝ) ↦ ‖Function.uncurry f x‖) '' SProd.sprod (α := Set X) {q} (Icc a b))) M
A : ∀ᶠ (δ : ℝ) in 𝓝[>] 0, Membership.mem (γ := Set ℝ) (Ioo 0 1) δ
B : ∀ᶠ (δ : ℝ) in 𝓝 0, Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b
C : ∀ᶠ (δ : ℝ) in 𝓝 0, (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b) < ε
δ : ℝ
hδ : Membership.mem (γ := Set ℝ) (Ioo 0 1) δ
h'δ : Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b
h''δ : (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b) < ε
⊢ ∃ (δ : ℝ),
    0 < δ ∧
      δ < 1 ∧ Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b ∧ (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b) < ε

Tactic:
⟨δ, hδ.1, hδ.2, h'δ, h''δ⟩

Premises used: And.right, And.left, And.intro, Exists.intro
-- Module: Mathlib.MeasureTheory.Integral.DominatedConvergence
Goal:
E : Type u_1
X : Type u_2
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : TopologicalSpace X
μ : Measure ℝ
inst_3 : NoAtoms μ
inst_4 : 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 (SProd.sprod (α := Set X) {q} (Icc a b))
M : ℝ
hM :
  Membership.mem (γ := Set ℝ)
    (upperBounds ((fun (x : X × ℝ) ↦ ‖Function.uncurry f x‖) '' SProd.sprod (α := Set X) {q} (Icc a b))) M
A : ∀ᶠ (δ : ℝ) in 𝓝[>] 0, Membership.mem (γ := Set ℝ) (Ioo 0 1) δ
B : ∀ᶠ (δ : ℝ) in 𝓝 0, Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b
C : ∀ᶠ (δ : ℝ) in 𝓝 0, (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b) < ε
δ : ℝ
hδ : Membership.mem (γ := Set ℝ) (Ioo 0 1) δ
h'δ : Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b
h''δ : (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b) < ε
⊢ ∃ (δ : ℝ),
    0 < δ ∧
      δ < 1 ∧ Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b ∧ (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b) < ε

Tactic:
exact ⟨δ, hδ.1, hδ.2, h'δ, h''δ⟩
-- Module: Mathlib.MeasureTheory.Integral.DominatedConvergence
Goal:
E : Type u_1
X : Type u_2
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : TopologicalSpace X
μ : Measure ℝ
inst_3 : NoAtoms μ
inst_4 : 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 (SProd.sprod (α := Set X) {q} (Icc a b))
M : ℝ
hM :
  Membership.mem (γ := Set ℝ)
    (upperBounds ((fun (x : X × ℝ) ↦ ‖Function.uncurry f x‖) '' SProd.sprod (α := Set X) {q} (Icc a b))) M
obtain :
  ∀ (δ : ℝ) (δpos : 0 < δ) (hδ : δ < 1) (h'δ : Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b)
    (h''δ : (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b) < ε),
    ∀ᶠ (x : X × ℝ) in 𝓝 (q, b₀),
      dist.{u_1} (α := E) (∫ (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
⊢ ∀ᶠ (x : X × ℝ) in 𝓝 (q, b₀),
    dist.{u_1} (α := E) (∫ (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 ⟨δ, δpos, hδ, h'δ, h''δ⟩ : ∃ (δ : ℝ), 0 < δ ∧ δ < 1 ∧ Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b ∧
      (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b) < ε := by
    have A : ∀ᶠ δ in 𝓝[>] (0 : ℝ), δ ∈ Ioo 0 1 := Ioo_mem_nhdsGT zero_lt_one
    have B : ∀ᶠ δ in 𝓝 0, Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b := by
      have I : Tendsto (fun δ ↦ b₀ - δ) (𝓝 0) (𝓝 (b₀ - 0)) := tendsto_const_nhds.sub tendsto_id
      have J : Tendsto (fun δ ↦ b₀ + δ) (𝓝 0) (𝓝 (b₀ + 0)) := tendsto_const_nhds.add tendsto_id
      simp only [sub_zero, add_zero] at I J
      filter_upwards [(tendsto_order.1 I).1 _ a_lt.2, (tendsto_order.1 J).2 _ lt_b.2] with δ hδ h'δ
      exact Icc_subset_Icc hδ.le h'δ.le
    have C : ∀ᶠ δ in 𝓝 0,
        (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b) < ε := by
      suffices Tendsto
        (fun δ ↦ (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b))
          (𝓝 0) (𝓝 ((M + 1) * (0 : ℝ≥0∞).toReal + 0 * μ.real (Icc a b))) by
        simp only [toReal_zero, mul_zero, zero_mul, add_zero] at this
        exact (tendsto_order.1 this).2 _ εpos
      apply Tendsto.add (Tendsto.mul tendsto_const_nhds _)
        (Tendsto.mul tendsto_id tendsto_const_nhds)
      exact (tendsto_toReal zero_ne_top).comp (tendsto_measure_Icc _ _)
    rcases (A.and ((B.and C).filter_mono nhdsWithin_le_nhds)).exists with ⟨δ, hδ, h'δ, h''δ⟩
    exact ⟨δ, hδ.1, hδ.2, h'δ, h''δ⟩

Premises used: MeasureTheory.Measure.real, Set.Icc, HAdd.hAdd, OfNat.ofNat
-- Module: Mathlib.MeasureTheory.Integral.DominatedConvergence
Goal:
E : Type u_1
X : Type u_2
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : TopologicalSpace X
μ : Measure ℝ
inst_3 : NoAtoms μ
inst_4 : 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 (SProd.sprod (α := Set X) {q} (Icc a b))
M : ℝ
hM :
  Membership.mem (γ := Set ℝ)
    (upperBounds ((fun (x : X × ℝ) ↦ ‖Function.uncurry f x‖) '' SProd.sprod (α := Set X) {q} (Icc a b))) M
δ : ℝ
δpos : 0 < δ
hδ : δ < 1
h'δ : Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b
h''δ : (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b) < ε
⊢ ∃ v ∈ 𝓝[univ] q, ∀ p ∈ v, ∀ x ∈ Icc a b, dist (f p x) (f q x) < δ

Tactic:
IsCompact.mem_uniformity_of_prod isCompact_Icc hf.continuousOn (mem_univ _)
      (dist_mem_uniformity δpos)

Premises used: Continuous.continuousOn, IsCompact.mem_uniformity_of_prod, Set.Mem, CompactIccSpace.isCompact_Icc, Set.mem_univ, Metric.dist_mem_uniformity
-- Module: Mathlib.MeasureTheory.Integral.DominatedConvergence
Goal:
E : Type u_1
X : Type u_2
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : TopologicalSpace X
μ : Measure ℝ
inst_3 : NoAtoms μ
inst_4 : 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 (SProd.sprod (α := Set X) {q} (Icc a b))
M : ℝ
hM :
  Membership.mem (γ := Set ℝ)
    (upperBounds ((fun (x : X × ℝ) ↦ ‖Function.uncurry f x‖) '' SProd.sprod (α := Set X) {q} (Icc a b))) M
δ : ℝ
δpos : 0 < δ
hδ : δ < 1
h'δ : Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b
h''δ : (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b) < ε
obtain :
  ∀ (v : Set X) (v_mem : v ∈ 𝓝[univ] q) (hv : ∀ p ∈ v, ∀ x ∈ Icc a b, dist (f p x) (f q x) < δ),
    ∀ᶠ (x : X × ℝ) in 𝓝 (q, b₀),
      dist.{u_1} (α := E) (∫ (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
⊢ ∀ᶠ (x : X × ℝ) in 𝓝 (q, b₀),
    dist.{u_1} (α := E) (∫ (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 ⟨v, v_mem, hv⟩ : ∃ v ∈ 𝓝[univ] q, ∀ p ∈ v, ∀ x ∈ Icc a b, dist (f p x) (f q x) < δ :=
    IsCompact.mem_uniformity_of_prod isCompact_Icc hf.continuousOn (mem_univ _)
      (dist_mem_uniformity δpos)

Premises used: Continuous.continuousOn, IsCompact.mem_uniformity_of_prod, Set.univ, CompactIccSpace.isCompact_Icc, Set.Icc, Set.mem_univ, Metric.dist_mem_uniformity
-- Module: Mathlib.MeasureTheory.Integral.DominatedConvergence
Goal:
E : Type u_1
X : Type u_2
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : TopologicalSpace X
μ : Measure ℝ
inst_3 : NoAtoms μ
inst_4 : 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 (SProd.sprod (α := Set X) {q} (Icc a b))
M : ℝ
hM :
  Membership.mem (γ := Set ℝ)
    (upperBounds ((fun (x : X × ℝ) ↦ ‖Function.uncurry f x‖) '' SProd.sprod (α := Set X) {q} (Icc a b))) M
δ : ℝ
δpos : 0 < δ
hδ : δ < 1
h'δ : Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b
h''δ : (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b) < ε
v : Set X
v_mem : v ∈ 𝓝[univ] q
hv : ∀ p ∈ v, ∀ x ∈ Icc a b, dist (f p x) (f q x) < δ
⊢ v ∈ 𝓝 q

Tactic:
rw [nhdsWithin_univ] at v_mem

Premises used: nhdsWithin_univ
-- Module: Mathlib.MeasureTheory.Integral.DominatedConvergence
Goal:
E : Type u_1
X : Type u_2
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : TopologicalSpace X
μ : Measure ℝ
inst_3 : NoAtoms μ
inst_4 : 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 (SProd.sprod (α := Set X) {q} (Icc a b))
M : ℝ
hM :
  Membership.mem (γ := Set ℝ)
    (upperBounds ((fun (x : X × ℝ) ↦ ‖Function.uncurry f x‖) '' SProd.sprod (α := Set X) {q} (Icc a b))) M
δ : ℝ
δpos : 0 < δ
hδ : δ < 1
h'δ : Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b
h''δ : (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b) < ε
v : Set X
v_mem : v ∈ 𝓝[univ] q
hv : ∀ p ∈ v, ∀ x ∈ Icc a b, dist (f p x) (f q x) < δ
v_mem₁ : v ∈ 𝓝 q
⊢ ∀ p ∈ v, ∀ x ∈ Icc a b, dist (f p x) (f q x) < δ

Tactic:
rw [nhdsWithin_univ] at v_mem

Premises used: nhdsWithin_univ
-- Module: Mathlib.MeasureTheory.Integral.DominatedConvergence
Goal:
E : Type u_1
X : Type u_2
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : TopologicalSpace X
μ : Measure ℝ
inst_3 : NoAtoms μ
inst_4 : 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 (SProd.sprod (α := Set X) {q} (Icc a b))
M : ℝ
hM :
  Membership.mem (γ := Set ℝ)
    (upperBounds ((fun (x : X × ℝ) ↦ ‖Function.uncurry f x‖) '' SProd.sprod (α := Set X) {q} (Icc a b))) M
δ : ℝ
δpos : 0 < δ
hδ : δ < 1
h'δ : Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b
h''δ : (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b) < ε
v : Set X
v_mem : v ∈ 𝓝 q
hv : ∀ p ∈ v, ∀ x ∈ Icc a b, dist (f p x) (f q x) < δ
simp : Ioo (b₀ - δ) (b₀ + δ) ∈ 𝓝 b₀
⊢ v ×ˢ Ioo (b₀ - δ) (b₀ + δ) ∈ 𝓝 (q, b₀)

Tactic:
simp only [prod_mem_nhds_iff, v_mem, true_and]

Premises used: prod_mem_nhds_iff, true_and
-- Module: Mathlib.MeasureTheory.Integral.DominatedConvergence
Goal:
E : Type u_1
X : Type u_2
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : TopologicalSpace X
μ : Measure ℝ
inst_3 : NoAtoms μ
inst_4 : 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 (SProd.sprod (α := Set X) {q} (Icc a b))
M : ℝ
hM :
  Membership.mem (γ := Set ℝ)
    (upperBounds ((fun (x : X × ℝ) ↦ ‖Function.uncurry f x‖) '' SProd.sprod (α := Set X) {q} (Icc a b))) M
δ : ℝ
δpos : 0 < δ
hδ : δ < 1
h'δ : Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b
h''δ : (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b) < ε
v : Set X
v_mem : v ∈ 𝓝 q
hv : ∀ p ∈ v, ∀ x ∈ Icc a b, dist (f p x) (f q x) < δ
ha : b₀ - δ < b₀
hb : b₀ < b₀ + δ
⊢ Ioo (b₀ - δ) (b₀ + δ) ∈ 𝓝 b₀

Tactic:
apply Ioo_mem_nhds

Premises used: Ioo_mem_nhds
-- Module: Mathlib.MeasureTheory.Integral.DominatedConvergence
Goal:
E : Type u_1
X : Type u_2
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : TopologicalSpace X
μ : Measure ℝ
inst_3 : NoAtoms μ
inst_4 : 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 (SProd.sprod (α := Set X) {q} (Icc a b))
M : ℝ
hM :
  Membership.mem (γ := Set ℝ)
    (upperBounds ((fun (x : X × ℝ) ↦ ‖Function.uncurry f x‖) '' SProd.sprod (α := Set X) {q} (Icc a b))) M
δ : ℝ
δpos : 0 < δ
hδ : δ < 1
h'δ : Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b
h''δ : (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b) < ε
v : Set X
v_mem : v ∈ 𝓝 q
hv : ∀ p ∈ v, ∀ x ∈ Icc a b, dist (f p x) (f q x) < δ
linarith : b₀ ≤ b₀ - δ → 0 - δ + (b₀ - (b₀ - δ)) = 0
⊢ b₀ - δ < b₀

Tactic:
linarith
-- Module: Mathlib.MeasureTheory.Integral.DominatedConvergence
Goal:
E : Type u_1
X : Type u_2
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : TopologicalSpace X
μ : Measure ℝ
inst_3 : NoAtoms μ
inst_4 : 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 (SProd.sprod (α := Set X) {q} (Icc a b))
M : ℝ
hM :
  Membership.mem (γ := Set ℝ)
    (upperBounds ((fun (x : X × ℝ) ↦ ‖Function.uncurry f x‖) '' SProd.sprod (α := Set X) {q} (Icc a b))) M
δ : ℝ
δpos : 0 < δ
hδ : δ < 1
h'δ : Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b
h''δ : (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b) < ε
v : Set X
v_mem : v ∈ 𝓝 q
hv : ∀ p ∈ v, ∀ x ∈ Icc a b, dist (f p x) (f q x) < δ
linarith : b₀ + δ ≤ b₀ → 0 - δ + (b₀ + δ - b₀) = 0
⊢ b₀ < b₀ + δ

Tactic:
linarith
-- Module: Mathlib.MeasureTheory.Integral.DominatedConvergence
Goal:
E : Type u_1
X : Type u_2
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : TopologicalSpace X
μ : Measure ℝ
inst_3 : NoAtoms μ
inst_4 : 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 (SProd.sprod (α := Set X) {q} (Icc a b))
M : ℝ
hM :
  Membership.mem (γ := Set ℝ)
    (upperBounds ((fun (x : X × ℝ) ↦ ‖Function.uncurry f x‖) '' SProd.sprod (α := Set X) {q} (Icc a b))) M
δ : ℝ
δpos : 0 < δ
hδ : δ < 1
h'δ : Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b
h''δ : (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b) < ε
v : Set X
v_mem : v ∈ 𝓝[univ] q
hv : ∀ p ∈ v, ∀ x ∈ Icc a b, dist (f p x) (f q x) < δ
⊢ v ×ˢ Ioo (b₀ - δ) (b₀ + δ) ∈ 𝓝 (q, b₀)

Tactic:
have : v ×ˢ (Ioo (b₀ - δ) (b₀ + δ)) ∈ 𝓝 (q, b₀) := by
    rw [nhdsWithin_univ] at v_mem
    simp only [prod_mem_nhds_iff, v_mem, true_and]
    apply Ioo_mem_nhds <;> linarith
-- Module: Mathlib.MeasureTheory.Integral.DominatedConvergence
Goal:
E : Type u_1
X : Type u_2
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : TopologicalSpace X
μ : Measure ℝ
inst_3 : NoAtoms μ
inst_4 : 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_1 : IsCompact (SProd.sprod (α := Set X) {q} (Icc a b))
M : ℝ
hM :
  Membership.mem (γ := Set ℝ)
    (upperBounds ((fun (x : X × ℝ) ↦ ‖Function.uncurry f x‖) '' SProd.sprod (α := Set X) {q} (Icc a b))) M
δ : ℝ
δpos : 0 < δ
hδ : δ < 1
h'δ : Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b
h''δ : (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b) < ε
v : Set X
v_mem : v ∈ 𝓝[univ] q
hv : ∀ p ∈ v, ∀ x ∈ Icc a b, dist (f p x) (f q x) < δ
this : v ×ˢ Ioo (b₀ - δ) (b₀ + δ) ∈ 𝓝 (q, b₀)
h :
  ∀ a ∈ v ×ˢ Ioo (b₀ - δ) (b₀ + δ),
    dist.{u_1} (α := E) (∫ (t : ℝ) in a₀..a.2, f a.1 t ∂μ) (∫ (t : ℝ) in a₀..b₀, f q t ∂μ) < ε
⊢ ∀ᶠ (x : X × ℝ) in 𝓝 (q, b₀),
    dist.{u_1} (α := E) (∫ (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:
filter_upwards [this]
-- Module: Mathlib.MeasureTheory.Integral.DominatedConvergence
Goal:
E : Type u_1
X : Type u_2
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : TopologicalSpace X
μ : Measure ℝ
inst_3 : NoAtoms μ
inst_4 : 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_1 : IsCompact (SProd.sprod (α := Set X) {q} (Icc a b))
M : ℝ
hM :
  Membership.mem (γ := Set ℝ)
    (upperBounds ((fun (x : X × ℝ) ↦ ‖Function.uncurry f x‖) '' SProd.sprod (α := Set X) {q} (Icc a b))) M
δ : ℝ
δpos : 0 < δ
hδ : δ < 1
h'δ : Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b
h''δ : (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b) < ε
v : Set X
v_mem : v ∈ 𝓝[univ] q
hv : ∀ p ∈ v, ∀ x ∈ Icc a b, dist (f p x) (f q x) < δ
this : v ×ˢ Ioo (b₀ - δ) (b₀ + δ) ∈ 𝓝 (q, b₀)
p : X
s : ℝ
hp : p ∈ v
hs : s ∈ Ioo (b₀ - δ) (b₀ + δ)
h :
  ∀ (hv : ∀ p ∈ v, ∀ x ∈ Icc a b, ‖f p x - f q x‖ < δ),
    ‖HSub.hSub (α := E) (∫ (t : ℝ) in a₀..s, f p t ∂μ) (∫ (t : ℝ) in a₀..b₀, f q t ∂μ)‖ < ε
⊢ dist.{u_1} (α := E) (∫ (t : ℝ) in a₀..(p, s).2, f (p, s).1 t ∂μ) (∫ (t : ℝ) in a₀..b₀, f q t ∂μ) < ε

Tactic:
simp only [dist_eq_norm] at hv ⊢

Premises used: dist_eq_norm
-- Module: Mathlib.MeasureTheory.Integral.DominatedConvergence
Goal:
E : Type u_1
X : Type u_2
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : TopologicalSpace X
μ : Measure ℝ
inst_3 : NoAtoms μ
inst_4 : 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_1 : IsCompact (SProd.sprod (α := Set X) {q} (Icc a b))
M : ℝ
hM :
  Membership.mem (γ := Set ℝ)
    (upperBounds ((fun (x : X × ℝ) ↦ ‖Function.uncurry f x‖) '' SProd.sprod (α := Set X) {q} (Icc a b))) M
δ : ℝ
δpos : 0 < δ
hδ : δ < 1
h'δ : Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b
h''δ : (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b) < ε
v : Set X
v_mem : v ∈ 𝓝[univ] q
this : v ×ˢ Ioo (b₀ - δ) (b₀ + δ) ∈ 𝓝 (q, b₀)
p : X
s : ℝ
hp : p ∈ v
hs : s ∈ Ioo (b₀ - δ) (b₀ + δ)
hv : ∀ p ∈ v, ∀ x ∈ Icc a b, ‖f p x - f q x‖ < δ
⊢ ∀ (r : X) (u v : ℝ), IntervalIntegrable (f r) μ u v

Tactic:
have J r u v : IntervalIntegrable (f r) μ u v := (hf.uncurry_left _).intervalIntegrable _ _
-- Module: Mathlib.MeasureTheory.Integral.DominatedConvergence
Goal:
E : Type u_1
X : Type u_2
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : TopologicalSpace X
μ : Measure ℝ
inst_3 : NoAtoms μ
inst_4 : 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_1 : IsCompact (SProd.sprod (α := Set X) {q} (Icc a b))
M : ℝ
hM :
  Membership.mem (γ := Set ℝ)
    (upperBounds ((fun (x : X × ℝ) ↦ ‖Function.uncurry f x‖) '' SProd.sprod (α := Set X) {q} (Icc a b))) M
δ : ℝ
δpos : 0 < δ
hδ : δ < 1
h'δ : Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b
h''δ : (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b) < ε
v : Set X
v_mem : v ∈ 𝓝[univ] q
this : v ×ˢ Ioo (b₀ - δ) (b₀ + δ) ∈ 𝓝 (q, b₀)
p : X
s : ℝ
hp : p ∈ v
hs : s ∈ Ioo (b₀ - δ) (b₀ + δ)
hv : ∀ p ∈ v, ∀ x ∈ Icc a b, ‖f p x - f q x‖ < δ
J : ∀ (r : X) (u v : ℝ), IntervalIntegrable (f r) μ u v
e_a :
  HSub.hSub (α := E) (∫ (t : ℝ) in a₀..s, f p t ∂μ) (∫ (t : ℝ) in a₀..b₀, f q t ∂μ) =
    HSub.hSub (α := E) (∫ (t : ℝ) in a₀..s, f p t ∂μ) (∫ (t : ℝ) in a₀..b₀, f p t ∂μ) +
      HSub.hSub (α := E) (∫ (t : ℝ) in a₀..b₀, f p t ∂μ) (∫ (t : ℝ) in a₀..b₀, f q t ∂μ)
⊢ ‖HSub.hSub (α := E) (∫ (t : ℝ) in a₀..s, f p t ∂μ) (∫ (t : ℝ) in a₀..b₀, f q t ∂μ)‖ =
    ‖HSub.hSub (α := E) (∫ (t : ℝ) in a₀..s, f p t ∂μ) (∫ (t : ℝ) in a₀..b₀, f p t ∂μ) +
        HSub.hSub (α := E) (∫ (t : ℝ) in a₀..b₀, f p t ∂μ) (∫ (t : ℝ) in a₀..b₀, f q t ∂μ)‖

Tactic:
congr 1
-- Module: Mathlib.MeasureTheory.Integral.DominatedConvergence
Goal:
E : Type u_1
X : Type u_2
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : TopologicalSpace X
μ : Measure ℝ
inst_3 : NoAtoms μ
inst_4 : 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_1 : IsCompact (SProd.sprod (α := Set X) {q} (Icc a b))
M : ℝ
hM :
  Membership.mem (γ := Set ℝ)
    (upperBounds ((fun (x : X × ℝ) ↦ ‖Function.uncurry f x‖) '' SProd.sprod (α := Set X) {q} (Icc a b))) M
δ : ℝ
δpos : 0 < δ
hδ : δ < 1
h'δ : Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b
h''δ : (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b) < ε
v : Set X
v_mem : v ∈ 𝓝[univ] q
this : v ×ˢ Ioo (b₀ - δ) (b₀ + δ) ∈ 𝓝 (q, b₀)
p : X
s : ℝ
hp : p ∈ v
hs : s ∈ Ioo (b₀ - δ) (b₀ + δ)
hv : ∀ p ∈ v, ∀ x ∈ Icc a b, ‖f p x - f q x‖ < δ
J : ∀ (r : X) (u v : ℝ), IntervalIntegrable (f r) μ u v
⊢ HSub.hSub (α := E) (∫ (t : ℝ) in a₀..s, f p t ∂μ) (∫ (t : ℝ) in a₀..b₀, f q t ∂μ) =
    HSub.hSub (α := E) (∫ (t : ℝ) in a₀..s, f p t ∂μ) (∫ (t : ℝ) in a₀..b₀, f p t ∂μ) +
      HSub.hSub (α := E) (∫ (t : ℝ) in a₀..b₀, f p t ∂μ) (∫ (t : ℝ) in a₀..b₀, f q t ∂μ)

Tactic:
abel
-- Module: Mathlib.MeasureTheory.Integral.DominatedConvergence
Goal:
E : Type u_1
X : Type u_2
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : TopologicalSpace X
μ : Measure ℝ
inst_3 : NoAtoms μ
inst_4 : 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_1 : IsCompact (SProd.sprod (α := Set X) {q} (Icc a b))
M : ℝ
hM :
  Membership.mem (γ := Set ℝ)
    (upperBounds ((fun (x : X × ℝ) ↦ ‖Function.uncurry f x‖) '' SProd.sprod (α := Set X) {q} (Icc a b))) M
δ : ℝ
δpos : 0 < δ
hδ : δ < 1
h'δ : Icc (b₀ - δ) (b₀ + δ) ⊆ Icc a b
h''δ : (M + 1) * μ.real (Icc (b₀ - δ) (b₀ + δ)) + δ * μ.real (Icc a b) < ε
v : Set X
v_mem : v ∈ 𝓝[univ] q
this : v ×ˢ Ioo (b₀ - δ) (b₀ + δ) ∈ 𝓝 (q, b₀)
p : X
s : ℝ
hp : p ∈ v
hs : s ∈ Ioo (b₀ - δ) (b₀ + δ)
hv : ∀ p ∈ v, ∀ x ∈ Icc a b, ‖f p x - f q x‖ < δ
J : ∀ (r : X) (u v : ℝ), IntervalIntegrable (f r) μ u v
⊢ ‖HSub.hSub (α := E) (∫ (t : ℝ) in a₀..s, f p t ∂μ) (∫ (t : ℝ) in a₀..b₀, f p t ∂μ) +
        HSub.hSub (α := E) (∫ (t : ℝ) in a₀..b₀, f p t ∂μ) (∫ (t : ℝ) in a₀..b₀, f q t ∂μ)‖ ≤
    ‖HSub.hSub (α := E) (∫ (t : ℝ) in a₀..s, f p t ∂μ) (∫ (t : ℝ) in a₀..b₀, f p t ∂μ)‖ +
      ‖HSub.hSub (α := E) (∫ (t : ℝ) in a₀..b₀, f p t ∂μ) (∫ (t : ℝ) in a₀..b₀, f q t ∂μ)‖

Tactic:
norm_add_le _ _

Premises used: norm_add_le