mathlib-initiative/mathlib-tactics huggingface dataset:
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 → ℝ≥0
int_f : ∫⁻ (x : α), ↑(f x) ∂μ ≠ ∞
ε : ℝ≥0∞
ε0 : ε ≠ 0
fs : α →ₛ ℝ≥0
fs_le_f : ∀ (x : α), fs x ≤ f x
int_fs : ∫⁻ (x : α), ↑(f x) ∂μ ≤ ∫⁻ (x : α), ↑(fs x) ∂μ + ε / 2
int_fs_lt_top : ∫⁻ (x : α), ↑(fs x) ∂μ ≠ ∞
g : α → ℝ≥0
g_le_fs : ∀ (x : α), g x ≤ fs x
gcont : UpperSemicontinuous g
gint : ∫⁻ (x : α), ↑(fs x) ∂μ ≤ ∫⁻ (x : α), ↑(g x) ∂μ + ε / 2
⊢ ∫⁻ (x : α), ↑(g x) ∂μ + ε / 2 + ε / 2 = ∫⁻ (x : α), ↑(g x) ∂μ + ε

Tactic:
by rw [add_assoc, ENNReal.add_halves]

Goal state after:
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
s : Set E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul ℝ E
x : E
ε : ℝ
hε : 0 < ε
hs₂ : IsOpen[inst✝¹] s
hx : x ∈ ε • s
this : ε⁻¹ • x ∈ s
h_gauge_lt : gauge s (ε⁻¹ • x) < 1
⊢ gauge s x < ε

Tactic:
gauge_smul_of_nonneg (inv_nonneg.2 hε.le),

Goal state after:
E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
s : Set E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul ℝ E
x : E
ε : ℝ
hε : 0 < ε
hs₂ : IsOpen[inst✝¹] s
hx : x ∈ ε • s
this : ε⁻¹ • x ∈ s
h_gauge_lt : ε⁻¹ • gauge s x < 1
⊢ gauge s x < ε
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
s : Set E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul ℝ E
hs : s ∈ 𝓝 0
⊢ ContinuousAt (gauge s) 0

Tactic:
rw [ContinuousAt, gauge_zero]
  exact tendsto_gauge_nhds_zero hs

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

Goal state before:
pp: R : Type u_4
V : Type u_5
W : Type u_6
inst✝¹³ : NormedField R
inst✝¹² : AddCommGroup V
inst✝¹¹ : AddCommGroup W
inst✝¹⁰ : TopologicalSpace R
inst✝⁹ : TopologicalSpace V
inst✝⁸ : TopologicalSpace W
inst✝⁷ : IsTopologicalRing R
inst✝⁶ : Module R V
inst✝⁵ : Module R W
inst✝⁴ : SeparatingDual R V
inst✝³ : IsTopologicalAddGroup V
inst✝² : IsTopologicalAddGroup W
inst✝¹ : ContinuousSMul R V
inst✝ : ContinuousSMul R W
f g : V ≃L[R] W
x✝¹ : ∃ α, ⇑f = ⇑(α • g)
y : Rˣ
h : ⇑f = ⇑(y • g)
x✝ : V
⊢ (↑y • 1) x✝ = (↑g.symm ∘SL ↑f) x✝

Tactic:
simp [h]

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

Goal state before:
pp: R : Type u_4
V : Type u_5
W : Type u_6
inst✝¹³ : NormedField R
inst✝¹² : AddCommGroup V
inst✝¹¹ : AddCommGroup W
inst✝¹⁰ : TopologicalSpace R
inst✝⁹ : TopologicalSpace V
inst✝⁸ : TopologicalSpace W
inst✝⁷ : IsTopologicalRing R
inst✝⁶ : Module R V
inst✝⁵ : Module R W
inst✝⁴ : SeparatingDual R V
inst✝³ : IsTopologicalAddGroup V
inst✝² : IsTopologicalAddGroup W
inst✝¹ : ContinuousSMul R V
inst✝ : ContinuousSMul R W
f g : V ≃L[R] W
x✝ : ∃ y, y • 1 = ↑g.symm ∘SL ↑f
y : R
h : y • 1 = ↑g.symm ∘SL ↑f
hy : ¬y = 0
x : V
⊢ f x = (Units.mk0 y hy • g) x

Tactic:
by simp [by simpa [eq_symm_apply] using congr($h x)]

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : CompleteSpace F
inst✝³ : MeasurableSpace 𝕜
inst✝² : OpensMeasurableSpace 𝕜
inst✝¹ : MeasurableSpace F
inst✝ : BorelSpace F
f : 𝕜 → F
⊢ Measurable (deriv f)

Tactic:
simpa only [fderiv_apply_one_eq_deriv] using measurable_fderiv_apply_const 𝕜 f 1

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

Goal state before:
pp: case h_ind
X : Type u_1
E : Type u_3
F : Type u_4
inst✝¹¹ : MeasurableSpace X
μ : Measure X
𝕜 : Type u_6
𝕜' : Type u_7
inst✝¹⁰ : RCLike 𝕜
inst✝⁹ : RCLike 𝕜'
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜' F
inst✝⁴ : NormedSpace ℝ F
σ : 𝕜 →+* 𝕜'
inst✝³ : RingHomIsometric σ
inst✝² : CompleteSpace F
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hσ : ∀ (r : ℝ) (x : 𝕜), σ (r • x) = r • σ x
L : E →SL[σ] F
φ : X → E
φ_int : Integrable φ μ
e : E
s : Set X
s_meas : MeasurableSet s
a✝ : μ s < ∞
⊢ ∫ (x : X), L (s.indicator (fun x ↦ e) x) ∂μ = L (∫ (x : X), s.indicator (fun x ↦ e) x ∂μ)

Tactic:
rw [integral_indicator_const e s_meas, ← @smul_one_smul E ℝ 𝕜 _ _ _ _ _ (μ.real s) e,
      map_smulₛₗ, hσ, map_one, smul_assoc, one_smul,
      ← integral_indicator_const (L e) s_meas]

Goal state after:
case h_ind
X : Type u_1
E : Type u_3
F : Type u_4
inst✝¹¹ : MeasurableSpace X
μ : Measure X
𝕜 : Type u_6
𝕜' : Type u_7
inst✝¹⁰ : RCLike 𝕜
inst✝⁹ : RCLike 𝕜'
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜' F
inst✝⁴ : NormedSpace ℝ F
σ : 𝕜 →+* 𝕜'
inst✝³ : RingHomIsometric σ
inst✝² : CompleteSpace F
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hσ : ∀ (r : ℝ) (x : 𝕜), σ (r • x) = r • σ x
L : E →SL[σ] F
φ : X → E
φ_int : Integrable φ μ
e : E
s : Set X
s_meas : MeasurableSet s
a✝ : μ s < ∞
⊢ ∫ (x : X), L (s.indicator (fun x ↦ e) x) ∂μ = ∫ (x : X), s.indicator (fun x ↦ L e) x ∂μ
Module: Mathlib.Topology.Algebra.IsUniformGroup.Order

Goal state before:
pp: α : Type u_1
ι : Type u_2
β : Type u_3
inst✝⁶ : UniformSpace β
inst✝⁵ : AddGroup β
inst✝⁴ : IsUniformAddGroup β
inst✝³ : PartialOrder β
inst✝² : OrderTopology β
inst✝¹ : AddLeftMono β
inst✝ : AddRightMono β
f : ι → α → β
g : α → β
K : Set α
p : Filter ι
u v : β
huv : u < v
hf : TendstoUniformlyOn f g p K
hg : ∀ x ∈ K, g x ≤ u
⊢ ∀ᶠ (i : ι) in p, ∀ x ∈ K, f i x < v

Tactic:
simp only [tendstoUniformlyOn_iff_tendsto, uniformity_eq_comap_neg_add_nhds_zero,
    tendsto_iff_eventually, eventually_comap, Prod.forall] at *
  conv at hf => enter [2]; rw [eventually_iff_exists_mem]
  have hf2 := hf (fun x ↦ -x.1 + x.2 < -u + v) ⟨_, (isOpen_gt' (-u + v)).mem_nhds (by simp [huv]),
    fun y hy a b hab ↦ (hab.symm ▸ hy :)⟩
  filter_upwards [eventually_prod_principal_iff.mp hf2] with i hi x hx
  simpa using add_lt_add_of_le_of_lt (hg x hx) (hi x hx)

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

Goal state before:
pp: X : Type u_1
E : Type u_3
F : Type u_4
inst✝⁷ : MeasurableSpace X
μ : Measure X
𝕜 : Type u_6
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : NormedSpace ℝ E
L : E ≃L[𝕜] F
φ : X → E
⊢ ∫ (x : X), L (φ x) ∂μ = L (∫ (x : X), φ x ∂μ)

Tactic:
have : CompleteSpace E ↔ CompleteSpace F :=
    completeSpace_congr (e := L.toEquiv) L.isUniformEmbedding

Goal state after:
X : Type u_1
E : Type u_3
F : Type u_4
inst✝⁷ : MeasurableSpace X
μ : Measure X
𝕜 : Type u_6
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : NormedSpace ℝ E
L : E ≃L[𝕜] F
φ : X → E
this : CompleteSpace E ↔ CompleteSpace F
⊢ ∫ (x : X), L (φ x) ∂μ = L (∫ (x : X), φ x ∂μ)
Module: Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap

Goal state before:
pp: case pos
X : Type u_1
E : Type u_3
inst✝² : MeasurableSpace X
μ : Measure X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → ℝ≥0
f_meas : Measurable f
g : X → E
hE : CompleteSpace E
⊢ (∫ (x : X), g x ∂μ.withDensity fun x ↦ ↑(f x)) = ∫ (x : X), f x • g x ∂μ

pp: case neg
X : Type u_1
E : Type u_3
inst✝² : MeasurableSpace X
μ : Measure X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → ℝ≥0
f_meas : Measurable f
g : X → E
hE : ¬CompleteSpace E
⊢ (∫ (x : X), g x ∂μ.withDensity fun x ↦ ↑(f x)) = ∫ (x : X), f x • g x ∂μ

Tactic:
swap

Goal state after:
case neg
X : Type u_1
E : Type u_3
inst✝² : MeasurableSpace X
μ : Measure X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → ℝ≥0
f_meas : Measurable f
g : X → E
hE : ¬CompleteSpace E
⊢ (∫ (x : X), g x ∂μ.withDensity fun x ↦ ↑(f x)) = ∫ (x : X), f x • g x ∂μ

case pos
X : Type u_1
E : Type u_3
inst✝² : MeasurableSpace X
μ : Measure X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → ℝ≥0
f_meas : Measurable f
g : X → E
hE : CompleteSpace E
⊢ (∫ (x : X), g x ∂μ.withDensity fun x ↦ ↑(f x)) = ∫ (x : X), f x • g x ∂μ
Module: Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap

Goal state before:
pp: case pos
X : Type u_1
E : Type u_3
inst✝² : MeasurableSpace X
μ : Measure X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → ℝ≥0
f_meas : Measurable f
g : X → E
hE : CompleteSpace E
hg : Integrable g (μ.withDensity fun x ↦ ↑(f x))
⊢ (∫ (x : X), g x ∂μ.withDensity fun x ↦ ↑(f x)) = ∫ (x : X), f x • g x ∂μ

pp: case neg
X : Type u_1
E : Type u_3
inst✝² : MeasurableSpace X
μ : Measure X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → ℝ≥0
f_meas : Measurable f
g : X → E
hE : CompleteSpace E
hg : ¬Integrable g (μ.withDensity fun x ↦ ↑(f x))
⊢ (∫ (x : X), g x ∂μ.withDensity fun x ↦ ↑(f x)) = ∫ (x : X), f x • g x ∂μ

Tactic:
swap

Goal state after:
case neg
X : Type u_1
E : Type u_3
inst✝² : MeasurableSpace X
μ : Measure X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → ℝ≥0
f_meas : Measurable f
g : X → E
hE : CompleteSpace E
hg : ¬Integrable g (μ.withDensity fun x ↦ ↑(f x))
⊢ (∫ (x : X), g x ∂μ.withDensity fun x ↦ ↑(f x)) = ∫ (x : X), f x • g x ∂μ

case pos
X : Type u_1
E : Type u_3
inst✝² : MeasurableSpace X
μ : Measure X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → ℝ≥0
f_meas : Measurable f
g : X → E
hE : CompleteSpace E
hg : Integrable g (μ.withDensity fun x ↦ ↑(f x))
⊢ (∫ (x : X), g x ∂μ.withDensity fun x ↦ ↑(f x)) = ∫ (x : X), f x • g x ∂μ
Module: Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap

Goal state before:
pp: case pos
X : Type u_1
E : Type u_3
inst✝² : MeasurableSpace X
μ : Measure X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → ℝ≥0
f_meas : Measurable f
g : X → E
hE : CompleteSpace E
hg : Integrable g (μ.withDensity fun x ↦ ↑(f x))
⊢ (∫ (x : X), g x ∂μ.withDensity fun x ↦ ↑(f x)) = ∫ (x : X), f x • g x ∂μ

Tactic:
refine Integrable.induction
    (P := fun g => ∫ x, g x ∂μ.withDensity (fun x => f x) = ∫ x, f x • g x ∂μ) ?_ ?_ ?_ ?_ hg

Goal state after:
case pos.refine_1
X : Type u_1
E : Type u_3
inst✝² : MeasurableSpace X
μ : Measure X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → ℝ≥0
f_meas : Measurable f
g : X → E
hE : CompleteSpace E
hg : Integrable g (μ.withDensity fun x ↦ ↑(f x))
⊢ ∀ (c : E) ⦃s : Set X⦄,
    MeasurableSet s →
      (μ.withDensity fun x ↦ ↑(f x)) s < ∞ →
        (∫ (x : X), s.indicator (fun x ↦ c) x ∂μ.withDensity fun x ↦ ↑(f x)) =
          ∫ (x : X), f x • s.indicator (fun x ↦ c) x ∂μ

case pos.refine_2
X : Type u_1
E : Type u_3
inst✝² : MeasurableSpace X
μ : Measure X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → ℝ≥0
f_meas : Measurable f
g : X → E
hE : CompleteSpace E
hg : Integrable g (μ.withDensity fun x ↦ ↑(f x))
⊢ ∀ ⦃f_1 g : X → E⦄,
    Disjoint (Function.support f_1) (Function.support g) →
      Integrable f_1 (μ.withDensity fun x ↦ ↑(f x)) →
        Integrable g (μ.withDensity fun x ↦ ↑(f x)) →
          (∫ (x : X), f_1 x ∂μ.withDensity fun x ↦ ↑(f x)) = ∫ (x : X), f x • f_1 x ∂μ →
            (∫ (x : X), g x ∂μ.withDensity fun x ↦ ↑(f x)) = ∫ (x : X), f x • g x ∂μ →
              (∫ (x : X), (f_1 + g) x ∂μ.withDensity fun x ↦ ↑(f x)) = ∫ (x : X), f x • (f_1 + g) x ∂μ

case pos.refine_3
X : Type u_1
E : Type u_3
inst✝² : MeasurableSpace X
μ : Measure X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → ℝ≥0
f_meas : Measurable f
g : X → E
hE : CompleteSpace E
hg : Integrable g (μ.withDensity fun x ↦ ↑(f x))
⊢ IsClosed {f_1 | (∫ (x : X), ↑↑f_1 x ∂μ.withDensity fun x ↦ ↑(f x)) = ∫ (x : X), f x • ↑↑f_1 x ∂μ}

case pos.refine_4
X : Type u_1
E : Type u_3
inst✝² : MeasurableSpace X
μ : Measure X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → ℝ≥0
f_meas : Measurable f
g : X → E
hE : CompleteSpace E
hg : Integrable g (μ.withDensity fun x ↦ ↑(f x))
⊢ ∀ ⦃f_1 g : X → E⦄,
    f_1 =ᵐ[μ.withDensity fun x ↦ ↑(f x)] g →
      Integrable f_1 (μ.withDensity fun x ↦ ↑(f x)) →
        (∫ (x : X), f_1 x ∂μ.withDensity fun x ↦ ↑(f x)) = ∫ (x : X), f x • f_1 x ∂μ →
          (∫ (x : X), g x ∂μ.withDensity fun x ↦ ↑(f x)) = ∫ (x : X), f x • g x ∂μ
Module: Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap

Goal state before:
pp: case pos.refine_1
X : Type u_1
E : Type u_3
inst✝² : MeasurableSpace X
μ : Measure X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → ℝ≥0
f_meas : Measurable f
g : X → E
hE : CompleteSpace E
hg : Integrable g (μ.withDensity fun x ↦ ↑(f x))
c : E
s : Set X
s_meas : MeasurableSet s
hs : (μ.withDensity fun x ↦ ↑(f x)) s < ∞
⊢ (ENNReal.ofReal (∫ (a : X) in s, ↑(f a) ∂μ)).toReal • c = ∫ (a : X) in s, f a • c ∂μ

pp: case pos.refine_1.hfi
X : Type u_1
E : Type u_3
inst✝² : MeasurableSpace X
μ : Measure X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → ℝ≥0
f_meas : Measurable f
g : X → E
hE : CompleteSpace E
hg : Integrable g (μ.withDensity fun x ↦ ↑(f x))
c : E
s : Set X
s_meas : MeasurableSet s
hs : (μ.withDensity fun x ↦ ↑(f x)) s < ∞
⊢ Integrable (fun x ↦ ↑(f x)) (μ.restrict s)

Tactic:
ENNReal.toReal_ofReal,

Goal state after:
case pos.refine_1
X : Type u_1
E : Type u_3
inst✝² : MeasurableSpace X
μ : Measure X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → ℝ≥0
f_meas : Measurable f
g : X → E
hE : CompleteSpace E
hg : Integrable g (μ.withDensity fun x ↦ ↑(f x))
c : E
s : Set X
s_meas : MeasurableSet s
hs : (μ.withDensity fun x ↦ ↑(f x)) s < ∞
⊢ (∫ (a : X) in s, ↑(f a) ∂μ) • c = ∫ (a : X) in s, f a • c ∂μ

case pos.refine_1
X : Type u_1
E : Type u_3
inst✝² : MeasurableSpace X
μ : Measure X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → ℝ≥0
f_meas : Measurable f
g : X → E
hE : CompleteSpace E
hg : Integrable g (μ.withDensity fun x ↦ ↑(f x))
c : E
s : Set X
s_meas : MeasurableSet s
hs : (μ.withDensity fun x ↦ ↑(f x)) s < ∞
⊢ 0 ≤ ∫ (a : X) in s, ↑(f a) ∂μ

case pos.refine_1.hfi
X : Type u_1
E : Type u_3
inst✝² : MeasurableSpace X
μ : Measure X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → ℝ≥0
f_meas : Measurable f
g : X → E
hE : CompleteSpace E
hg : Integrable g (μ.withDensity fun x ↦ ↑(f x))
c : E
s : Set X
s_meas : MeasurableSet s
hs : (μ.withDensity fun x ↦ ↑(f x)) s < ∞
⊢ Integrable (fun x ↦ ↑(f x)) (μ.restrict s)
Module: Mathlib.MeasureTheory.Integral.DominatedConvergence

Goal state before:
pp: case pos
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
inst✝ : Countable ι
f : ι → α → G
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hf' : ∑' (i : ι), ∫⁻ (a : α), ‖f i a‖ₑ ∂μ ≠ ∞
hG : CompleteSpace G
⊢ ∫ (a : α), ∑' (i : ι), f i a ∂μ = ∑' (i : ι), ∫ (a : α), f i a ∂μ

pp: case neg
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
inst✝ : Countable ι
f : ι → α → G
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hf' : ∑' (i : ι), ∫⁻ (a : α), ‖f i a‖ₑ ∂μ ≠ ∞
hG : ¬CompleteSpace G
⊢ ∫ (a : α), ∑' (i : ι), f i a ∂μ = ∑' (i : ι), ∫ (a : α), f i a ∂μ

Tactic:
swap

Goal state after:
case neg
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
inst✝ : Countable ι
f : ι → α → G
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hf' : ∑' (i : ι), ∫⁻ (a : α), ‖f i a‖ₑ ∂μ ≠ ∞
hG : ¬CompleteSpace G
⊢ ∫ (a : α), ∑' (i : ι), f i a ∂μ = ∑' (i : ι), ∫ (a : α), f i a ∂μ

case pos
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
inst✝ : Countable ι
f : ι → α → G
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hf' : ∑' (i : ι), ∫⁻ (a : α), ‖f i a‖ₑ ∂μ ≠ ∞
hG : CompleteSpace G
⊢ ∫ (a : α), ∑' (i : ι), f i a ∂μ = ∑' (i : ι), ∫ (a : α), f i a ∂μ
Module: Mathlib.MeasureTheory.Integral.DominatedConvergence

Goal state before:
pp: case pos
α : Type u_1
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
ι : Type u_4
inst✝ : Countable ι
F : ι → α → E
hF_int : ∀ (i : ι), Integrable (F i) μ
hF_sum : Summable fun i ↦ ∫ (a : α), ‖F i a‖ ∂μ
hE : CompleteSpace E
⊢ HasSum (fun x ↦ ∫ (a : α), F x a ∂μ) (∫ (a : α), ∑' (i : ι), F i a ∂μ)

pp: case neg
α : Type u_1
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
ι : Type u_4
inst✝ : Countable ι
F : ι → α → E
hF_int : ∀ (i : ι), Integrable (F i) μ
hF_sum : Summable fun i ↦ ∫ (a : α), ‖F i a‖ ∂μ
hE : ¬CompleteSpace E
⊢ HasSum (fun x ↦ ∫ (a : α), F x a ∂μ) (∫ (a : α), ∑' (i : ι), F i a ∂μ)

Tactic:
swap

Goal state after:
case neg
α : Type u_1
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
ι : Type u_4
inst✝ : Countable ι
F : ι → α → E
hF_int : ∀ (i : ι), Integrable (F i) μ
hF_sum : Summable fun i ↦ ∫ (a : α), ‖F i a‖ ∂μ
hE : ¬CompleteSpace E
⊢ HasSum (fun x ↦ ∫ (a : α), F x a ∂μ) (∫ (a : α), ∑' (i : ι), F i a ∂μ)

case pos
α : Type u_1
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
ι : Type u_4
inst✝ : Countable ι
F : ι → α → E
hF_int : ∀ (i : ι), Integrable (F i) μ
hF_sum : Summable fun i ↦ ∫ (a : α), ‖F i a‖ ∂μ
hE : CompleteSpace E
⊢ HasSum (fun x ↦ ∫ (a : α), F x a ∂μ) (∫ (a : α), ∑' (i : ι), F i a ∂μ)
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: case inr
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)
this : CauchySeq L0
f' : F
f'K : f' ∈ K
hf' : Tendsto L0 atTop (𝓝 f')
Lf' : ∀ (e p : ℕ), n e ≤ p → ‖L e (n e) p - f'‖ ≤ 12 * (1 / 2) ^ e
ε : ℝ
εpos : 0 < ε
e : ℕ
he : (1 / 2) ^ e < ε / 16
y : ℝ
hy : y ∈ Icc x (x + (1 / 2) ^ (n e + 1))
xy : x < y
yzero : 0 < y - x
y_le : y - x ≤ (1 / 2) ^ (n e + 1)
yone : y - x ≤ 1
k : ℕ
k_gt : n e < k
m : ℕ := k - 1
h'k : y - x ≤ (1 / 2) ^ (m + 1)
hk : (1 / 2) ^ (m + 1 + 1) < y - x
m_ge : n e ≤ m
km : k = m + 1
⊢ ‖f y - f x - (y - x) • f'‖ ≤ ε * ‖y - x‖

Tactic:
have J : ‖f y - f x - (y - x) • L e (n e) m‖ ≤ 4 * (1 / 2) ^ e * ‖y - x‖ :=
      calc
        ‖f y - f x - (y - x) • L e (n e) m‖ ≤ (1 / 2) ^ e * (1 / 2) ^ m := by
          apply le_of_mem_A (hn e (n e) m le_rfl m_ge).2.2
          · simp only [one_div, inv_pow, left_mem_Icc, le_add_iff_nonneg_right]
            positivity
          · simp only [pow_add, tsub_le_iff_left] at h'k
            simpa only [hy.1, mem_Icc, true_and, one_div, pow_one] using! h'k
        _ = 4 * (1 / 2) ^ e * (1 / 2) ^ (m + 2) := by ring
        _ ≤ 4 * (1 / 2) ^ e * (y - x) := by gcongr
        _ = 4 * (1 / 2) ^ e * ‖y - x‖ := by rw [Real.norm_of_nonneg yzero.le]

Goal state after:
case inr
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)
this : CauchySeq L0
f' : F
f'K : f' ∈ K
hf' : Tendsto L0 atTop (𝓝 f')
Lf' : ∀ (e p : ℕ), n e ≤ p → ‖L e (n e) p - f'‖ ≤ 12 * (1 / 2) ^ e
ε : ℝ
εpos : 0 < ε
e : ℕ
he : (1 / 2) ^ e < ε / 16
y : ℝ
hy : y ∈ Icc x (x + (1 / 2) ^ (n e + 1))
xy : x < y
yzero : 0 < y - x
y_le : y - x ≤ (1 / 2) ^ (n e + 1)
yone : y - x ≤ 1
k : ℕ
k_gt : n e < k
m : ℕ := k - 1
h'k : y - x ≤ (1 / 2) ^ (m + 1)
hk : (1 / 2) ^ (m + 1 + 1) < y - x
m_ge : n e ≤ m
km : k = m + 1
J : ‖f y - f x - (y - x) • L e (n e) m‖ ≤ 4 * (1 / 2) ^ e * ‖y - x‖
⊢ ‖f y - f x - (y - x) • f'‖ ≤ ε * ‖y - x‖
Module: Mathlib.MeasureTheory.Integral.DominatedConvergence

Goal state before:
pp: case pos
ι : Type u_1
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
inst✝ : Countable ι
f : ι → C(ℝ, E)
hf_sum : Summable fun i ↦ ‖ContinuousMap.restrict (↑{ carrier := [[a, b]], isCompact' := ⋯ }) (f i)‖
hE : CompleteSpace E
⊢ HasSum (fun i ↦ ∫ (x : ℝ) in a..b, (f i) x) (∫ (x : ℝ) in a..b, ∑' (i : ι), (f i) x)

pp: case neg
ι : Type u_1
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
inst✝ : Countable ι
f : ι → C(ℝ, E)
hf_sum : Summable fun i ↦ ‖ContinuousMap.restrict (↑{ carrier := [[a, b]], isCompact' := ⋯ }) (f i)‖
hE : ¬CompleteSpace E
⊢ HasSum (fun i ↦ ∫ (x : ℝ) in a..b, (f i) x) (∫ (x : ℝ) in a..b, ∑' (i : ι), (f i) x)

Tactic:
swap

Goal state after:
case neg
ι : Type u_1
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
inst✝ : Countable ι
f : ι → C(ℝ, E)
hf_sum : Summable fun i ↦ ‖ContinuousMap.restrict (↑{ carrier := [[a, b]], isCompact' := ⋯ }) (f i)‖
hE : ¬CompleteSpace E
⊢ HasSum (fun i ↦ ∫ (x : ℝ) in a..b, (f i) x) (∫ (x : ℝ) in a..b, ∑' (i : ι), (f i) x)

case pos
ι : Type u_1
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
inst✝ : Countable ι
f : ι → C(ℝ, E)
hf_sum : Summable fun i ↦ ‖ContinuousMap.restrict (↑{ carrier := [[a, b]], isCompact' := ⋯ }) (f i)‖
hE : CompleteSpace E
⊢ HasSum (fun i ↦ ∫ (x : ℝ) in a..b, (f i) x) (∫ (x : ℝ) in a..b, ∑' (i : ι), (f i) x)
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
⊢ (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

pp: case neg
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
⊢ (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:
swap

Goal state after:
case neg
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
⊢ (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

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
⊢ (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
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: E : Type u_5
inst✝ : NormedAddCommGroup E
f : ℝ → E
h₁f : ∀ (x : ℝ), f x = f (-x)
h₂f : ∀ (x : ℝ), 0 < x → IntervalIntegrable f volume 0 x
t : ℝ
ht : ‖f (min 0 t)‖ₑ ≠ ∞
h : t < 0
⊢ 0 < -t

Tactic:
simp [h]

Goal state after: