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_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
⊢ Integrable (ε := ℝ) (fun (a : ℝ) ↦ ‖f p a - f q a‖) (μ.restrict (Icc a b))

Tactic:
exact ((hf.uncurry_left _).sub (hf.uncurry_left _)).norm.integrableOn_Icc

Premises used: Continuous.norm, Continuous.sub, Continuous.uncurry_left, Continuous.integrableOn_Icc
-- 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
⊢ Ι a₀ b₀ ⊆ Icc a b

Tactic:
uIoc_subset_uIcc.trans (uIcc_subset_Icc ⟨a_lt.1.le, lt_b.1.le⟩ ⟨a_lt.2.le, lt_b.2.le⟩)

Premises used: HasSubset.Subset.trans, Set.uIoc_subset_uIcc, LT.lt.le, Set.uIcc_subset_Icc, And.right, And.left, And.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_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
⊢ Ι a₀ b₀ ⊆ Icc a b

Tactic:
exact uIoc_subset_uIcc.trans (uIcc_subset_Icc ⟨a_lt.1.le, lt_b.1.le⟩ ⟨a_lt.2.le, lt_b.2.le⟩)

Premises used: HasSubset.Subset.trans, Set.uIoc_subset_uIcc, LT.lt.le, Set.uIcc_subset_Icc, And.right, And.left, And.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_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
h₁ : IntegrableOn (ε := ℝ) (fun (x : ℝ) ↦ ‖f p x‖) (Icc (b₀ - δ) (b₀ + δ)) μ
h₂ : IntegrableOn (ε := ℝ) (fun (x : ℝ) ↦ M + 1) (Icc (b₀ - δ) (b₀ + δ)) μ
h₃ : NullMeasurableSet (Icc (b₀ - δ) (b₀ + δ)) μ
h₄ : ∀ (x : ℝ) (hx : x ∈ Icc (b₀ - δ) (b₀ + δ)), ‖f p x‖ ≤ M + 1
h₅ : IntegrableOn (ε := ℝ) (fun (x : ℝ) ↦ ‖f p x - f q x‖) (Icc a b) μ
h₆ : IntegrableOn (ε := ℝ) (fun (x : ℝ) ↦ δ) (Icc a b) μ
h₇ : NullMeasurableSet (Icc a b) μ
h₈ : ∀ (x : ℝ) (hx : x ∈ Icc a b), ‖f p x - f q x‖ ≤ δ
⊢ HAdd.hAdd (α := ℝ) (∫ (t : ℝ) in Icc (b₀ - δ) (b₀ + δ), ‖f p t‖ ∂μ) (∫ (t : ℝ) in Icc a b, ‖f p t - f q t‖ ∂μ) ≤
    HAdd.hAdd (α := ℝ) (∫ (t : ℝ) in Icc (b₀ - δ) (b₀ + δ), M + 1 ∂μ) (∫ (_t : ℝ) in Icc a b, δ ∂μ)

Tactic:
gcongr with x hx x hx
-- 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
⊢ IntegrableOn (ε := ℝ) (fun (x : ℝ) ↦ ‖f p x‖) (Icc (b₀ - δ) (b₀ + δ)) μ

Tactic:
(hf.uncurry_left _).norm.integrableOn_Icc

Premises used: Continuous.norm, Continuous.uncurry_left, Continuous.integrableOn_Icc
-- 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
⊢ IntegrableOn (ε := ℝ) (fun (x : ℝ) ↦ ‖f p x‖) (Icc (b₀ - δ) (b₀ + δ)) μ

Tactic:
exact (hf.uncurry_left _).norm.integrableOn_Icc

Premises used: Continuous.norm, Continuous.uncurry_left, Continuous.integrableOn_Icc
-- 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
⊢ IntegrableOn (ε := ℝ) (fun (x : ℝ) ↦ M + 1) (Icc (b₀ - δ) (b₀ + δ)) μ

Tactic:
continuous_const.integrableOn_Icc

Premises used: continuous_const, Continuous.integrableOn_Icc
-- 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
⊢ IntegrableOn (ε := ℝ) (fun (x : ℝ) ↦ M + 1) (Icc (b₀ - δ) (b₀ + δ)) μ

Tactic:
exact continuous_const.integrableOn_Icc

Premises used: continuous_const, Continuous.integrableOn_Icc
-- 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
⊢ NullMeasurableSet (Icc (b₀ - δ) (b₀ + δ)) μ

Tactic:
nullMeasurableSet_Icc

Premises used: nullMeasurableSet_Icc
-- 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
⊢ NullMeasurableSet (Icc (b₀ - δ) (b₀ + δ)) μ

Tactic:
exact nullMeasurableSet_Icc

Premises used: nullMeasurableSet_Icc
-- 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
x : ℝ
hx : x ∈ Icc (b₀ - δ) (b₀ + δ)
e_a : f p x = f q x + (f p x - f q x)
⊢ ‖f p x‖ = ‖f q x + (f p x - f q x)‖

Tactic:
congr
-- 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
x : ℝ
hx : x ∈ Icc (b₀ - δ) (b₀ + δ)
⊢ f p x = f q x + (f p x - f q x)

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
x : ℝ
hx : x ∈ Icc (b₀ - δ) (b₀ + δ)
⊢ ‖f q x + (f p x - f q x)‖ ≤ ‖f q x‖ + ‖f p x - f q x‖

Tactic:
norm_add_le _ _

Premises used: norm_add_le
-- 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
x : ℝ
hx : x ∈ Icc (b₀ - δ) (b₀ + δ)
h₁ : ‖f q x‖ ≤ M
h₂ : ‖f p x - f q x‖ ≤ δ
⊢ ‖f q x‖ + ‖f p x - f q x‖ ≤ M + δ

Tactic:
gcongr
-- 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
x : ℝ
hx : x ∈ Icc (b₀ - δ) (b₀ + δ)
h₁ :
  Membership.mem (γ := Set ℝ) ((fun (x : X × ℝ) ↦ ‖Function.uncurry f x‖) '' SProd.sprod (α := Set X) {q} (Icc a b))
    ‖f q x‖
⊢ ‖f q x‖ ≤ M

Tactic:
apply hM
-- 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
x : ℝ
hx : x ∈ Icc (b₀ - δ) (b₀ + δ)
h₁ :
  Membership.mem (γ := Set ℝ) ((fun (x : X × ℝ) ↦ ‖Function.uncurry f x‖) '' SProd.sprod (α := Set X) {q} (Icc a b))
    ((fun (x : X × ℝ) ↦ ‖Function.uncurry f x‖) (q, x))
⊢ Membership.mem (γ := Set ℝ) ((fun (x : X × ℝ) ↦ ‖Function.uncurry f x‖) '' SProd.sprod (α := Set X) {q} (Icc a b))
    ‖f q x‖

Tactic:
change (fun x ↦ ‖Function.uncurry f x‖) (q, x) ∈ _

Premises used: Norm.norm, Prod.mk, Function.uncurry
-- 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
x : ℝ
hx : x ∈ Icc (b₀ - δ) (b₀ + δ)
h₁ : (q, x) ∈ SProd.sprod (α := Set X) {q} (Icc a b)
⊢ Membership.mem (γ := Set ℝ) ((fun (x : X × ℝ) ↦ ‖Function.uncurry f x‖) '' SProd.sprod (α := Set X) {q} (Icc a b))
    ((fun (x : X × ℝ) ↦ ‖Function.uncurry f x‖) (q, x))

Tactic:
apply mem_image_of_mem

Premises used: Set.mem_image_of_mem
-- 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
x : ℝ
hx : x ∈ Icc (b₀ - δ) (b₀ + δ)
h₁ : x ∈ Icc a b
⊢ (q, x) ∈ SProd.sprod (α := Set X) {q} (Icc a b)

Tactic:
simp only [singleton_prod, mem_image, Prod.mk.injEq, true_and, exists_eq_right]

Premises used: Set.mem_image, exists_eq_right, Set.singleton_prod, Prod.mk.injEq, 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_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
x : ℝ
hx : x ∈ Icc (b₀ - δ) (b₀ + δ)
⊢ x ∈ Icc a b

Tactic:
h'δ hx