mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: case refine_2
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝ : AddCommMonoid β
T : Set α → β
T_empty : T ∅ = 0
h_add : FinMeasAdditive μ T
ι : Type u_8
S : ι → Set α
sι : Finset ι
hS_meas : ∀ (i : ι), MeasurableSet (S i)
⊢ ∀ (a : ι) (s : Finset ι),
    a ∉ s →
      ((∀ i ∈ s, μ (S i) ≠ ∞) →
          (∀ i ∈ s, ∀ j ∈ s, i ≠ j → Disjoint (S i) (S j)) → T (⋃ i ∈ s, S i) = ∑ i ∈ s, T (S i)) →
        (∀ i ∈ insert a s, μ (S i) ≠ ∞) →
          (∀ i ∈ insert a s, ∀ j ∈ insert a s, i ≠ j → Disjoint (S i) (S j)) →
            T (⋃ i ∈ insert a s, S i) = ∑ i ∈ insert a s, T (S i)

Tactic:
intro a

Goal state after:
case refine_2
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝ : AddCommMonoid β
T : Set α → β
T_empty : T ∅ = 0
h_add : FinMeasAdditive μ T
ι : Type u_8
S : ι → Set α
sι : Finset ι
hS_meas : ∀ (i : ι), MeasurableSet (S i)
a : ι
⊢ ∀ (s : Finset ι),
    a ∉ s →
      ((∀ i ∈ s, μ (S i) ≠ ∞) →
          (∀ i ∈ s, ∀ j ∈ s, i ≠ j → Disjoint (S i) (S j)) → T (⋃ i ∈ s, S i) = ∑ i ∈ s, T (S i)) →
        (∀ i ∈ insert a s, μ (S i) ≠ ∞) →
          (∀ i ∈ insert a s, ∀ j ∈ insert a s, i ≠ j → Disjoint (S i) (S j)) →
            T (⋃ i ∈ insert a s, S i) = ∑ i ∈ insert a s, T (S i)
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: case refine_2
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝ : AddCommMonoid β
T : Set α → β
T_empty : T ∅ = 0
h_add : FinMeasAdditive μ T
ι : Type u_8
S : ι → Set α
sι : Finset ι
hS_meas : ∀ (i : ι), MeasurableSet (S i)
a : ι
s : Finset ι
has : a ∉ s
h : (∀ i ∈ s, μ (S i) ≠ ∞) → (∀ i ∈ s, ∀ j ∈ s, i ≠ j → Disjoint (S i) (S j)) → T (⋃ i ∈ s, S i) = ∑ i ∈ s, T (S i)
hps : ∀ i ∈ insert a s, μ (S i) ≠ ∞
h_disj : ∀ i ∈ insert a s, ∀ j ∈ insert a s, i ≠ j → Disjoint (S i) (S j)
⊢ T (⋃ i ∈ insert a s, S i) = T (S a) + T (⋃ i ∈ s, S i)

Tactic:
←
    h_add (S a) (⋃ i ∈ s, S i) (hS_meas a) (measurableSet_biUnion _ fun i _ => hS_meas i)
      (hps a (Finset.mem_insert_self a s))

Goal state after:
case refine_2
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝ : AddCommMonoid β
T : Set α → β
T_empty : T ∅ = 0
h_add : FinMeasAdditive μ T
ι : Type u_8
S : ι → Set α
sι : Finset ι
hS_meas : ∀ (i : ι), MeasurableSet (S i)
a : ι
s : Finset ι
has : a ∉ s
h : (∀ i ∈ s, μ (S i) ≠ ∞) → (∀ i ∈ s, ∀ j ∈ s, i ≠ j → Disjoint (S i) (S j)) → T (⋃ i ∈ s, S i) = ∑ i ∈ s, T (S i)
hps : ∀ i ∈ insert a s, μ (S i) ≠ ∞
h_disj : ∀ i ∈ insert a s, ∀ j ∈ insert a s, i ≠ j → Disjoint (S i) (S j)
⊢ T (⋃ i ∈ insert a s, S i) = T (S a ∪ ⋃ i ∈ s, S i)

case refine_2.a
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝ : AddCommMonoid β
T : Set α → β
T_empty : T ∅ = 0
h_add : FinMeasAdditive μ T
ι : Type u_8
S : ι → Set α
sι : Finset ι
hS_meas : ∀ (i : ι), MeasurableSet (S i)
a : ι
s : Finset ι
has : a ∉ s
h : (∀ i ∈ s, μ (S i) ≠ ∞) → (∀ i ∈ s, ∀ j ∈ s, i ≠ j → Disjoint (S i) (S j)) → T (⋃ i ∈ s, S i) = ∑ i ∈ s, T (S i)
hps : ∀ i ∈ insert a s, μ (S i) ≠ ∞
h_disj : ∀ i ∈ insert a s, ∀ j ∈ insert a s, i ≠ j → Disjoint (S i) (S j)
⊢ μ (⋃ i ∈ s, S i) ≠ ∞

case refine_2.a
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝ : AddCommMonoid β
T : Set α → β
T_empty : T ∅ = 0
h_add : FinMeasAdditive μ T
ι : Type u_8
S : ι → Set α
sι : Finset ι
hS_meas : ∀ (i : ι), MeasurableSet (S i)
a : ι
s : Finset ι
has : a ∉ s
h : (∀ i ∈ s, μ (S i) ≠ ∞) → (∀ i ∈ s, ∀ j ∈ s, i ≠ j → Disjoint (S i) (S j)) → T (⋃ i ∈ s, S i) = ∑ i ∈ s, T (S i)
hps : ∀ i ∈ insert a s, μ (S i) ≠ ∞
h_disj : ∀ i ∈ insert a s, ∀ j ∈ insert a s, i ≠ j → Disjoint (S i) (S j)
⊢ Disjoint (S a) (⋃ i ∈ s, S i)
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: α : Type u_1
F : Type u_3
F' : Type u_4
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedAddCommGroup F'
inst✝¹ : NormedSpace ℝ F'
m : MeasurableSpace α
inst✝ : DecidablePred fun x ↦ x ≠ 0
T : Set α → F →L[ℝ] F'
hT : T ∅ = 0
f : α →ₛ F
s : Finset F
hs : {x ∈ f.range | x ≠ 0} ⊆ s
⊢ setToSimpleFunc T f = ∑ x ∈ s, (T (⇑f ⁻¹' {x})) x

Tactic:
rw [setToSimpleFunc_eq_sum_filter, Finset.sum_subset hs]

Goal state after:
α : Type u_1
F : Type u_3
F' : Type u_4
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedAddCommGroup F'
inst✝¹ : NormedSpace ℝ F'
m : MeasurableSpace α
inst✝ : DecidablePred fun x ↦ x ≠ 0
T : Set α → F →L[ℝ] F'
hT : T ∅ = 0
f : α →ₛ F
s : Finset F
hs : {x ∈ f.range | x ≠ 0} ⊆ s
⊢ ∀ x ∈ s, x ∉ {x ∈ f.range | x ≠ 0} → (T (⇑f ⁻¹' {x})) x = 0
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:
pp: α : Type u_1
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ ν : Measure α
s : Set α
hμs : μ s ≠ ∞
hνs : ν s ≠ ∞
x : F
⊢ (weightedSMul (μ + ν) s) x = (⇑(weightedSMul μ s) + ⇑(weightedSMul ν s)) x

Tactic:
simp_rw [Pi.add_apply, weightedSMul_apply]

Goal state after:
α : Type u_1
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ ν : Measure α
s : Set α
hμs : μ s ≠ ∞
hνs : ν s ≠ ∞
x : F
⊢ (μ + ν).real s • x = μ.real s • x + ν.real s • x
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:
pp: α : Type u_1
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
s : Set α
h_zero : μ s = 0
x : F
⊢ (weightedSMul μ s) x = 0 x

Tactic:
rw [weightedSMul_apply, measureReal_def, h_zero]

Goal state after:
α : Type u_1
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
s : Set α
h_zero : μ s = 0
x : F
⊢ ENNReal.toReal 0 • x = 0 x
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ↥(α →₁ₛ[μ] ℝ)
⊢ ∀ (a : α),
    (toSimpleFunc (posPart (-f))) a = (toSimpleFunc (-f)).posPart a →
      (toSimpleFunc (-f)) a = (-⇑(toSimpleFunc f)) a → (toSimpleFunc (posPart (-f))) a = (-toSimpleFunc f).posPart a

Tactic:
intro a

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ↥(α →₁ₛ[μ] ℝ)
a : α
⊢ (toSimpleFunc (posPart (-f))) a = (toSimpleFunc (-f)).posPart a →
    (toSimpleFunc (-f)) a = (-⇑(toSimpleFunc f)) a → (toSimpleFunc (posPart (-f))) a = (-toSimpleFunc f).posPart a
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
G' : Type u_7
G'' : Type u_8
inst✝⁷ : NormedAddCommGroup G''
inst✝⁶ : PartialOrder G''
inst✝⁵ : IsOrderedAddMonoid G''
inst✝⁴ : NormedSpace ℝ G''
inst✝³ : NormedAddCommGroup G'
inst✝² : PartialOrder G'
inst✝¹ : NormedSpace ℝ G'
inst✝ : IsOrderedAddMonoid G'
T : Set α → G' →L[ℝ] G''
h_add : FinMeasAdditive μ T
hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∀ (x : G'), 0 ≤ x → 0 ≤ (T s) x
f g : α →ₛ G'
hfi : Integrable (⇑f) μ
hgi : Integrable (⇑g) μ
hfg : f ≤ g
x : α
⊢ 0 x ≤ (g - f) x

Tactic:
simp only [coe_sub, sub_nonneg, coe_zero, Pi.zero_apply, Pi.sub_apply]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
G' : Type u_7
G'' : Type u_8
inst✝⁷ : NormedAddCommGroup G''
inst✝⁶ : PartialOrder G''
inst✝⁵ : IsOrderedAddMonoid G''
inst✝⁴ : NormedSpace ℝ G''
inst✝³ : NormedAddCommGroup G'
inst✝² : PartialOrder G'
inst✝¹ : NormedSpace ℝ G'
inst✝ : IsOrderedAddMonoid G'
T : Set α → G' →L[ℝ] G''
h_add : FinMeasAdditive μ T
hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∀ (x : G'), 0 ≤ x → 0 ≤ (T s) x
f g : α →ₛ G'
hfi : Integrable (⇑f) μ
hgi : Integrable (⇑g) μ
hfg : f ≤ g
x : α
⊢ f x ≤ g x
Module: Mathlib.Analysis.Asymptotics.AsymptoticEquivalent

Goal state before:
pp: case inr
α : Type u_1
β : Type u_2
inst✝ : NormedAddCommGroup β
u : α → β
l : Filter α
c : β
hu : u ~[l] const α c
h : ¬c = 0
⊢ Tendsto u l (𝓝 c)

Tactic:
· exact (isEquivalent_const_iff_tendsto h).mp hu

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : 0 ≤ᵐ[μ] f
hfm : AEStronglyMeasurable f μ
hfi : Integrable f μ
⊢ ∀ (x : α), 0 ≤ f x → ENNReal.ofReal (-f x) = 0

Tactic:
intro a

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : 0 ≤ᵐ[μ] f
hfm : AEStronglyMeasurable f μ
hfi : Integrable f μ
a : α
⊢ 0 ≤ f a → ENNReal.ofReal (-f a) = 0
Module: Mathlib.Analysis.Asymptotics.AsymptoticEquivalent

Goal state before:
pp: α : Type u_1
E : Type u_2
𝕜 : Type u_3
inst✝² : NormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
a b : α → 𝕜
u v : α → E
l : Filter α
hab : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : α) in l, ‖(a - b) x‖ ≤ c * ‖b x‖
huv : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : α) in l, ‖(u - v) x‖ ≤ c * ‖v x‖
φ : α → 𝕜
habφ : a =ᶠ[l] φ * b
this : ((fun x ↦ a x • u x) - fun x ↦ b x • v x) =ᶠ[l] fun x ↦ b x • (φ x • u x - v x)
C : ℝ
hC : C > 0
hCuv : ∀ᶠ (x : α) in l, ‖u x‖ ≤ C * ‖v x‖
hφ : ∀ ε > 0, ∀ᶠ (x : α) in l, ‖φ x - 1‖ < ε
c : ℝ
hc : 0 < c
⊢ ∀ᶠ (x : α) in l, ‖φ x • u x - v x‖ ≤ c * ‖v x‖

Tactic:
specialize hφ (c / 2 / C) (div_pos (div_pos hc zero_lt_two) hC)

Goal state after:
α : Type u_1
E : Type u_2
𝕜 : Type u_3
inst✝² : NormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
a b : α → 𝕜
u v : α → E
l : Filter α
hab : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : α) in l, ‖(a - b) x‖ ≤ c * ‖b x‖
huv : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : α) in l, ‖(u - v) x‖ ≤ c * ‖v x‖
φ : α → 𝕜
habφ : a =ᶠ[l] φ * b
this : ((fun x ↦ a x • u x) - fun x ↦ b x • v x) =ᶠ[l] fun x ↦ b x • (φ x • u x - v x)
C : ℝ
hC : C > 0
hCuv : ∀ᶠ (x : α) in l, ‖u x‖ ≤ C * ‖v x‖
c : ℝ
hφ : ∀ᶠ (x : α) in l, ‖φ x - 1‖ < c / 2 / C
hc : 0 < c
⊢ ∀ᶠ (x : α) in l, ‖φ x • u x - v x‖ ≤ c * ‖v x‖
Module: Mathlib.Analysis.Asymptotics.AsymptoticEquivalent

Goal state before:
pp: case zero
α : Type u_1
β : Type u_3
inst✝ : NormedField β
t u : α → β
l : Filter α
h : t ~[l] u
⊢ t ^ 0 ~[l] u ^ 0

Tactic:
| zero =>

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

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : IsOrderedModule ℝ E
inst✝¹ : ClosedIciTopology E
β : Type u_6
inst✝ : Preorder β
f : α → β → E
s : Set β
hf_mono : ∀ᵐ (x : α) ∂μ, MonotoneOn (f x) s
hf_int : ∀ a ∈ s, Integrable (fun x ↦ f x a) μ
⊢ MonotoneOn (fun b ↦ ∫ (x : α), f x b ∂μ) s

Tactic:
intro a

Goal state after:
α : Type u_1
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : IsOrderedModule ℝ E
inst✝¹ : ClosedIciTopology E
β : Type u_6
inst✝ : Preorder β
f : α → β → E
s : Set β
hf_mono : ∀ᵐ (x : α) ∂μ, MonotoneOn (f x) s
hf_int : ∀ a ∈ s, Integrable (fun x ↦ f x a) μ
a : β
⊢ a ∈ s → ∀ ⦃b : β⦄, b ∈ s → a ≤ b → (fun b ↦ ∫ (x : α), f x b ∂μ) a ≤ (fun b ↦ ∫ (x : α), f x b ∂μ) b
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : IsOrderedModule ℝ E
inst✝¹ : ClosedIciTopology E
β : Type u_6
inst✝ : Preorder β
f : α → β → E
s : Set β
hf_anti : ∀ᵐ (x : α) ∂μ, AntitoneOn (f x) s
hf_int : ∀ a ∈ s, Integrable (fun x ↦ f x a) μ
⊢ AntitoneOn (fun b ↦ ∫ (x : α), f x b ∂μ) s

Tactic:
intro a

Goal state after:
α : Type u_1
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : IsOrderedModule ℝ E
inst✝¹ : ClosedIciTopology E
β : Type u_6
inst✝ : Preorder β
f : α → β → E
s : Set β
hf_anti : ∀ᵐ (x : α) ∂μ, AntitoneOn (f x) s
hf_int : ∀ a ∈ s, Integrable (fun x ↦ f x a) μ
a : β
⊢ a ∈ s → ∀ ⦃b : β⦄, b ∈ s → a ≤ b → (fun b ↦ ∫ (x : α), f x b ∂μ) b ≤ (fun b ↦ ∫ (x : α), f x b ∂μ) a
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
inst✝⁵ : PartialOrder E
inst✝⁴ : IsOrderedAddMonoid E
inst✝³ : IsOrderedModule ℝ E
inst✝² : ClosedIciTopology E
β : Type u_6
inst✝¹ : AddCommMonoid β
inst✝ : Module ℝ β
f : α → β → E
s : Set β
hs : Convex ℝ s
hf_conv : ∀ᵐ (x : α) ∂μ, ConvexOn ℝ s (f x)
hf_int : ∀ a ∈ s, Integrable (fun x ↦ f x a) μ
⊢ ∀ ⦃x : β⦄,
    x ∈ s →
      ∀ ⦃y : β⦄,
        y ∈ s →
          ∀ ⦃a b : ℝ⦄,
            0 ≤ a →
              0 ≤ b →
                a + b = 1 →
                  (fun b ↦ ∫ (x : α), f x b ∂μ) (a • x + b • y) ≤
                    a • (fun b ↦ ∫ (x : α), f x b ∂μ) x + b • (fun b ↦ ∫ (x : α), f x b ∂μ) y

Tactic:
intro a

Goal state after:
α : Type u_1
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
inst✝⁵ : PartialOrder E
inst✝⁴ : IsOrderedAddMonoid E
inst✝³ : IsOrderedModule ℝ E
inst✝² : ClosedIciTopology E
β : Type u_6
inst✝¹ : AddCommMonoid β
inst✝ : Module ℝ β
f : α → β → E
s : Set β
hs : Convex ℝ s
hf_conv : ∀ᵐ (x : α) ∂μ, ConvexOn ℝ s (f x)
hf_int : ∀ a ∈ s, Integrable (fun x ↦ f x a) μ
a : β
⊢ a ∈ s →
    ∀ ⦃y : β⦄,
      y ∈ s →
        ∀ ⦃a_3 b : ℝ⦄,
          0 ≤ a_3 →
            0 ≤ b →
              a_3 + b = 1 →
                (fun b ↦ ∫ (x : α), f x b ∂μ) (a_3 • a + b • y) ≤
                  a_3 • (fun b ↦ ∫ (x : α), f x b ∂μ) a + b • (fun b ↦ ∫ (x : α), f x b ∂μ) y
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g : α → ℝ
hf : Integrable f μ
hg : Integrable g μ
hfg : f ≤ᵐ[μ] g
⊢ ∫ (a : α), f a ∂μ = ∫ (a : α), g a ∂μ ↔ f =ᵐ[μ] g

Tactic:
by
  refine ⟨fun h_le ↦ EventuallyEq.symm ?_, fun h ↦ integral_congr_ae h⟩
  rw [← sub_ae_eq_zero,
    ← integral_eq_zero_iff_of_nonneg_ae ((sub_nonneg_ae _ _).mpr hfg) (hg.sub hf)]
  simpa [Pi.sub_apply, integral_sub hg hf, sub_eq_zero, eq_comm]

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf : ∀ (n : ℕ), Integrable (f n) μ
hF : Integrable F μ
h_mono : ∀ᵐ (x : α) ∂μ, Monotone fun n ↦ f n x
h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (F x))
f' : ℕ → α → ℝ := fun n x ↦ f n x - f 0 x
hf'_nonneg : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), 0 ≤ f' n x
hf'_meas : ∀ (n : ℕ), Integrable (f' n) μ
this : Tendsto (fun n ↦ ∫ (x : α), f' n x ∂μ) atTop (𝓝 (∫ (x : α), (F - f 0) x ∂μ))
⊢ Tendsto (fun n ↦ ∫ (x : α), f n x ∂μ) atTop (𝓝 (∫ (x : α), F x ∂μ))

Tactic:
f',

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf : ∀ (n : ℕ), Integrable (f n) μ
hF : Integrable F μ
h_mono : ∀ᵐ (x : α) ∂μ, Monotone fun n ↦ f n x
h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (F x))
f' : ℕ → α → ℝ := fun n x ↦ f n x - f 0 x
hf'_nonneg : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), 0 ≤ f' n x
hf'_meas : ∀ (n : ℕ), Integrable (f' n) μ
this : Tendsto (fun n ↦ ∫ (x : α), f n x - f 0 x ∂μ) atTop (𝓝 (∫ (x : α), (F - f 0) x ∂μ))
⊢ Tendsto (fun n ↦ ∫ (x : α), f n x ∂μ) atTop (𝓝 (∫ (x : α), F x ∂μ))
Module: Mathlib.MeasureTheory.Integral.Bochner.SumMeasure

Goal state before:
pp: X : Type u_2
E : Type u_3
mX : MeasurableSpace X
inst✝⁴ : NormedAddCommGroup E
f : X → E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : MeasurableSingletonClass X
μ : Measure X
inst✝ : Countable X
hf : Integrable f μ
⊢ ∫ (x : X), f x ∂μ = ∑' (x : X), μ.real {x} • f x

Tactic:
rw [← Measure.sum_smul_dirac μ] at hf

Goal state after:
X : Type u_2
E : Type u_3
mX : MeasurableSpace X
inst✝⁴ : NormedAddCommGroup E
f : X → E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : MeasurableSingletonClass X
μ : Measure X
inst✝ : Countable X
hf : Integrable f (Measure.sum fun a ↦ μ {a} • Measure.dirac a)
⊢ ∫ (x : X), f x ∂μ = ∑' (x : X), μ.real {x} • f x
Module: Mathlib.Topology.Bornology.BoundedOperation

Goal state before:
pp: case zero
R : Type u_2
inst✝² : Bornology R
inst✝¹ : Monoid R
inst✝ : BoundedMul R
s : Set R
s_bdd : Bornology.IsBounded s
⊢ Bornology.IsBounded ((fun x ↦ x ^ 0) '' s)

Tactic:
| zero =>

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
H : Type u_6
inst✝ : NormedAddCommGroup H
f : ↥(Lp H 1 μ)
⊢ (∫⁻ (a : α), ‖↑↑f a‖ₑ ∂μ).toReal = ∫ (a : α), ‖↑↑f a‖ ∂μ

Tactic:
integral_eq_lintegral_of_nonneg_ae (Eventually.of_forall (by simp [norm_nonneg]))
      (Lp.aestronglyMeasurable f).norm

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
H : Type u_6
inst✝ : NormedAddCommGroup H
f : ↥(Lp H 1 μ)
⊢ (∫⁻ (a : α), ‖↑↑f a‖ₑ ∂μ).toReal = (∫⁻ (a : α), ENNReal.ofReal ‖↑↑f a‖ ∂μ).toReal
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
f : α → E
hT : DominatedFinMeasAdditive μ T C
h : μ = 0
⊢ setToFun μ T hT f = 0

Tactic:
have : f =ᵐ[μ] 0 := by simp [h, EventuallyEq]

Goal state after:
α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
f : α → E
hT : DominatedFinMeasAdditive μ T C
h : μ = 0
this : f =ᵐ[μ] 0
⊢ setToFun μ T hT f = 0
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
inst✝ : CompleteSpace F
hT : DominatedFinMeasAdditive μ T C
f : SimpleFunc α E
hf : Integrable (⇑f) μ
h'f : MemLp (⇑f) 1 μ
g : ↥(α →₁ₛ[μ] E) := ⋯
A : ⇑f =ᵐ[μ] ↑↑↑g
⊢ ⇑f =ᵐ[μ] ⇑(Lp.simpleFunc.toSimpleFunc g)

Tactic:
grw [A, Lp.simpleFunc.toSimpleFunc_eq_toFun]

Goal state after:
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
F : Type w
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : CompleteSpace 𝕜
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
g : ι → E
t : ι → 𝕜
⊢ ‖((⇑↑LinearMap.toContinuousLinearMap ∘ ⇑↑(LinearEquiv.piRing 𝕜 E ι 𝕜).symm) g) t‖ ≤ Fintype.card ι • ‖g‖ * ‖t‖

Tactic:
LinearEquiv.coe_toLinearMap,

Goal state after:
𝕜 : Type u
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
F : Type w
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : CompleteSpace 𝕜
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
g : ι → E
t : ι → 𝕜
⊢ ‖((⇑LinearMap.toContinuousLinearMap ∘ ⇑(LinearEquiv.piRing 𝕜 E ι 𝕜).symm) g) t‖ ≤ Fintype.card ι • ‖g‖ * ‖t‖
Module: Mathlib.Topology.ContinuousMap.Bounded.Normed

Goal state before:
pp: case insert
α : Type u
inst✝² : TopologicalSpace α
R : Type u_1
inst✝¹ : NonUnitalSeminormedRing R
inst✝ : IsCancelMulZero R
ι : Type u_2
f : ι → α →ᵇ R
h : Pairwise ((fun x1 x2 ↦ x1 * x2 = 0) on f)
j : ι
s : Finset ι
hj : j ∉ s
ih : ‖∑ i ∈ s, f i‖₊ = s.sup fun x ↦ ‖f x‖₊
⊢ ‖∑ i ∈ insert j s, f i‖₊ = (insert j s).sup fun x ↦ ‖f x‖₊

Tactic:
suffices f j * ∑ i ∈ s, f i = 0 by simpa [hj, ← ih] using nnnorm_add_eq_max this

Goal state after:
case insert
α : Type u
inst✝² : TopologicalSpace α
R : Type u_1
inst✝¹ : NonUnitalSeminormedRing R
inst✝ : IsCancelMulZero R
ι : Type u_2
f : ι → α →ᵇ R
h : Pairwise ((fun x1 x2 ↦ x1 * x2 = 0) on f)
j : ι
s : Finset ι
hj : j ∉ s
ih : ‖∑ i ∈ s, f i‖₊ = s.sup fun x ↦ ‖f x‖₊
⊢ f j * ∑ i ∈ s, f i = 0
Module: Mathlib.Topology.ContinuousMap.Compact

Goal state before:
pp: case insert
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
R : Type u_4
inst✝¹ : NonUnitalSeminormedRing R
inst✝ : IsCancelMulZero R
ι : Type u_5
f : ι → C(α, R)
h : Pairwise ((fun x1 x2 ↦ x1 * x2 = 0) on f)
j : ι
s : Finset ι
hj : j ∉ s
ih : ‖∑ i ∈ s, f i‖₊ = s.sup fun x ↦ ‖f x‖₊
⊢ ‖∑ i ∈ insert j s, f i‖₊ = (insert j s).sup fun x ↦ ‖f x‖₊

Tactic:
suffices f j * ∑ i ∈ s, f i = 0 by simpa [hj, ← ih] using nnnorm_add_eq_max this

Goal state after:
case insert
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
R : Type u_4
inst✝¹ : NonUnitalSeminormedRing R
inst✝ : IsCancelMulZero R
ι : Type u_5
f : ι → C(α, R)
h : Pairwise ((fun x1 x2 ↦ x1 * x2 = 0) on f)
j : ι
s : Finset ι
hj : j ∉ s
ih : ‖∑ i ∈ s, f i‖₊ = s.sup fun x ↦ ‖f x‖₊
⊢ f j * ∑ i ∈ s, f i = 0
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
E : Set α
⊢ (E.indicator fun x ↦ 1) ≤ thickenedIndicatorAux δ E

Tactic:
intro a

Goal state after:
α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
E : Set α
a : α
⊢ E.indicator (fun x ↦ 1) a ≤ thickenedIndicatorAux δ E a
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
⊢ (E.indicator fun x ↦ 1) ≤ ⇑(thickenedIndicator δ_pos E)

Tactic:
intro a

Goal state after:
α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
a : α
⊢ E.indicator (fun x ↦ 1) a ≤ (thickenedIndicator δ_pos E) a
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: case pos.h_ae
α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C C' : ℝ
μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ∞
hμ'_le : μ' ≤ c' • μ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ' T C'
f : α → E
hfμ : Integrable f μ
hF : CompleteSpace F
h_int : ∀ (g : α → E), Integrable g μ → Integrable g μ'
f₂ g₂ : α → E
hfg : f₂ =ᵐ[μ] g₂
a✝ : Integrable f₂ μ
hf_eq : setToFun μ T hT f₂ = setToFun μ' T hT' f₂
hfg' : f₂ =ᵐ[μ'] g₂
⊢ setToFun μ T hT g₂ = setToFun μ' T hT' g₂

Tactic:
← setToFun_congr_ae hT hfg,

Goal state after:
case pos.h_ae
α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C C' : ℝ
μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ∞
hμ'_le : μ' ≤ c' • μ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ' T C'
f : α → E
hfμ : Integrable f μ
hF : CompleteSpace F
h_int : ∀ (g : α → E), Integrable g μ → Integrable g μ'
f₂ g₂ : α → E
hfg : f₂ =ᵐ[μ] g₂
a✝ : Integrable f₂ μ
hf_eq : setToFun μ T hT f₂ = setToFun μ' T hT' f₂
hfg' : f₂ =ᵐ[μ'] g₂
⊢ setToFun μ T hT f₂ = setToFun μ' T hT' g₂
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: R : Type u_1
inst✝ : LinearOrder R
a b : R
⊢ Disjoint (Ioc a b) botSet

Tactic:
disjoint_iff_forall_ne

Goal state after:
R : Type u_1
inst✝ : LinearOrder R
a b : R
⊢ ∀ ⦃a_1 : R⦄, a_1 ∈ Ioc a b → ∀ ⦃b : R⦄, b ∈ botSet → a_1 ≠ b
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: R : Type u_1
inst✝⁴ : LinearOrder R
inst✝³ : TopologicalSpace R
inst✝² : OrderTopology R
inst✝¹ : DenselyOrdered R
inst✝ : NoMaxOrder R
f : StieltjesFunction R
x : R
⊢ ⨅ r, ↑f ↑r = ↑f x

Tactic:
by
  suffices Function.rightLim f x = ⨅ r : Ioi x, f r by rw [← this, f.rightLim_eq]
  rw [f.mono.rightLim_eq_sInf, sInf_image']

Goal state after: