ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.SimpleFunc.vadd_apply
@[to_additive (attr := simp)]
theorem smul_apply [SMul K β] (k : K) (f : α →ₛ β) (a : α) : (k • f) a = k • f a 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {K : Type u_5} [VAdd K β] (k : K) (f : MeasureTheory.SimpleFunc α β) (a : α)
Goal: ↑(k +ᵥ f) a = k +ᵥ ↑f a
-- MeasureTheory.SimpleFunc.smul_apply
@[to_additive (attr := simp)]
theorem smul_apply [SMul K β] (k : K) (f : α →ₛ β) (a : α) : (k • f) a = k • f a 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {K : Type u_5} [SMul K β] (k : K) (f : MeasureTheory.SimpleFunc α β) (a : α)
Goal: ↑(k • f) a = k • ↑f a
-- MeasureTheory.SimpleFunc.coe_pow
@[simp]
theorem coe_pow [Monoid β] (f : α →ₛ β) (n : ℕ) : ⇑(f ^ n) = (⇑f) ^ n 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [Monoid β] (f : MeasureTheory.SimpleFunc α β) (n : ℕ)
Goal: ↑(f ^ n) = ↑f ^ n
-- MeasureTheory.SimpleFunc.pow_apply
theorem pow_apply [Monoid β] (n : ℕ) (f : α →ₛ β) (a : α) : (f ^ n) a = f a ^ n 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [Monoid β] (n : ℕ) (f : MeasureTheory.SimpleFunc α β) (a : α)
Goal: ↑(f ^ n) a = ↑f a ^ n
-- MeasureTheory.SimpleFunc.coe_zpow
@[simp]
theorem coe_zpow [DivInvMonoid β] (f : α →ₛ β) (z : ℤ) : ⇑(f ^ z) = (⇑f) ^ z 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [DivInvMonoid β] (f : MeasureTheory.SimpleFunc α β) (z : ℤ)
Goal: ↑(f ^ z) = ↑f ^ z
-- MeasureTheory.SimpleFunc.zpow_apply
theorem zpow_apply [DivInvMonoid β] (z : ℤ) (f : α →ₛ β) (a : α) : (f ^ z) a = f a ^ z 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [DivInvMonoid β] (z : ℤ) (f : MeasureTheory.SimpleFunc α β) (a : α)
Goal: ↑(f ^ z) a = ↑f a ^ z
-- MeasureTheory.SimpleFunc.smul_eq_map
theorem smul_eq_map [SMul K β] (k : K) (f : α →ₛ β) : k • f = f.map (k • ·) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {K : Type u_5} [SMul K β] (k : K) (f : MeasureTheory.SimpleFunc α β)
Goal: k • f = MeasureTheory.SimpleFunc.map (fun x => k • x) f
-- MeasureTheory.SimpleFunc.finset_sup_apply
theorem finset_sup_apply [SemilatticeSup β] [OrderBot β] {f : γ → α →ₛ β} (s : Finset γ) (a : α) :
    s.sup f a = s.sup fun c => f c a 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [SemilatticeSup β] [OrderBot β] {f : γ → MeasureTheory.SimpleFunc α β} (s : Finset γ) (a : α)
Goal: ↑(s.sup f) a = s.sup fun c => ↑(f c) a
-- MeasureTheory.SimpleFunc.restrict_of_not_measurable
theorem restrict_of_not_measurable {f : α →ₛ β} {s : Set α} (hs : ¬MeasurableSet s) :
    restrict f s = 0 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [Zero β] {f : MeasureTheory.SimpleFunc α β} {s : Set α} (hs : ¬MeasurableSet s)
Goal: f.restrict s = 0
-- MeasureTheory.SimpleFunc.coe_restrict
@[simp]
theorem coe_restrict (f : α →ₛ β) {s : Set α} (hs : MeasurableSet s) :
    ⇑(restrict f s) = indicator s f 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [Zero β] (f : MeasureTheory.SimpleFunc α β) {s : Set α} (hs : MeasurableSet s)
Goal: ↑(f.restrict s) = s.indicator ↑f
-- MeasureTheory.SimpleFunc.restrict_univ
@[simp]
theorem restrict_univ (f : α →ₛ β) : restrict f univ = f 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [Zero β] (f : MeasureTheory.SimpleFunc α β)
Goal: f.restrict Set.univ = f
-- MeasureTheory.SimpleFunc.restrict_empty
@[simp]
theorem restrict_empty (f : α →ₛ β) : restrict f ∅ = 0 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [Zero β] (f : MeasureTheory.SimpleFunc α β)
Goal: f.restrict ∅ = 0
-- MeasureTheory.SimpleFunc.map_restrict_of_zero
theorem map_restrict_of_zero [Zero γ] {g : β → γ} (hg : g 0 = 0) (f : α →ₛ β) (s : Set α) :
    (f.restrict s).map g = (f.map g).restrict s 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [Zero β] [Zero γ] {g : β → γ} (hg : g 0 = 0) (f : MeasureTheory.SimpleFunc α β) (s : Set α)
Goal: MeasureTheory.SimpleFunc.map g (f.restrict s) = (MeasureTheory.SimpleFunc.map g f).restrict s
-- MeasureTheory.SimpleFunc.map_coe_ennreal_restrict
theorem map_coe_ennreal_restrict (f : α →ₛ ℝ≥0) (s : Set α) :
    (f.restrict s).map ((↑) : ℝ≥0 → ℝ≥0∞) = (f.map (↑)).restrict s 
Arguments: {α : Type u_1} [MeasurableSpace α] (f : MeasureTheory.SimpleFunc α ℝ≥0) (s : Set α)
Goal: MeasureTheory.SimpleFunc.map ENNReal.ofNNReal (f.restrict s) =
  (MeasureTheory.SimpleFunc.map ENNReal.ofNNReal f).restrict s
-- MeasureTheory.SimpleFunc.map_coe_nnreal_restrict
theorem map_coe_nnreal_restrict (f : α →ₛ ℝ≥0) (s : Set α) :
    (f.restrict s).map ((↑) : ℝ≥0 → ℝ) = (f.map (↑)).restrict s 
Arguments: {α : Type u_1} [MeasurableSpace α] (f : MeasureTheory.SimpleFunc α ℝ≥0) (s : Set α)
Goal: MeasureTheory.SimpleFunc.map NNReal.toReal (f.restrict s) = (MeasureTheory.SimpleFunc.map NNReal.toReal f).restrict s
-- MeasureTheory.SimpleFunc.restrict_apply
theorem restrict_apply (f : α →ₛ β) {s : Set α} (hs : MeasurableSet s) (a) :
    restrict f s a = indicator s f a 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [Zero β] (f : MeasureTheory.SimpleFunc α β) {s : Set α} (hs : MeasurableSet s) (a : α)
Goal: ↑(f.restrict s) a = s.indicator (↑f) a
-- MeasureTheory.SimpleFunc.restrict_preimage
theorem restrict_preimage (f : α →ₛ β) {s : Set α} (hs : MeasurableSet s) {t : Set β}
    (ht : (0 : β) ∉ t) : restrict f s ⁻¹' t = s ∩ f ⁻¹' t 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [Zero β] (f : MeasureTheory.SimpleFunc α β) {s : Set α} (hs : MeasurableSet s) {t : Set β} (ht : 0 ∉ t)
Goal: ↑(f.restrict s) ⁻¹' t = s ∩ ↑f ⁻¹' t
-- MeasureTheory.SimpleFunc.restrict_preimage_singleton
theorem restrict_preimage_singleton (f : α →ₛ β) {s : Set α} (hs : MeasurableSet s) {r : β}
    (hr : r ≠ 0) : restrict f s ⁻¹' {r} = s ∩ f ⁻¹' {r} 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [Zero β] (f : MeasureTheory.SimpleFunc α β) {s : Set α} (hs : MeasurableSet s) {r : β} (hr : r ≠ 0)
Goal: ↑(f.restrict s) ⁻¹' {r} = s ∩ ↑f ⁻¹' {r}
-- MeasureTheory.SimpleFunc.mem_restrict_range
theorem mem_restrict_range {r : β} {s : Set α} {f : α →ₛ β} (hs : MeasurableSet s) :
    r ∈ (restrict f s).range ↔ r = 0 ∧ s ≠ univ ∨ r ∈ f '' s 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [Zero β] {r : β} {s : Set α} {f : MeasureTheory.SimpleFunc α β} (hs : MeasurableSet s)
Goal: r ∈ (f.restrict s).range ↔ r = 0 ∧ s ≠ Set.univ ∨ r ∈ ↑f '' s
-- MeasureTheory.SimpleFunc.mem_image_of_mem_range_restrict
theorem mem_image_of_mem_range_restrict {r : β} {s : Set α} {f : α →ₛ β}
    (hr : r ∈ (restrict f s).range) (h0 : r ≠ 0) : r ∈ f '' s 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [Zero β] {r : β} {s : Set α} {f : MeasureTheory.SimpleFunc α β} (hr : r ∈ (f.restrict s).range) (h0 : r ≠ 0)
Goal: r ∈ ↑f '' s
-- MeasureTheory.SimpleFunc.restrict_mono
@[mono]
theorem restrict_mono [Preorder β] (s : Set α) {f g : α →ₛ β} (H : f ≤ g) :
    f.restrict s ≤ g.restrict s 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [Zero β] [Preorder β] (s : Set α) {f g : MeasureTheory.SimpleFunc α β} (H : f ≤ g)
Goal: f.restrict s ≤ g.restrict s
-- MeasureTheory.SimpleFunc.approx_apply
theorem approx_apply [TopologicalSpace β] [OrderClosedTopology β] [MeasurableSpace β]
    [OpensMeasurableSpace β] {i : ℕ → β} {f : α → β} {n : ℕ} (a : α) (hf : Measurable f) :
    (approx i f n : α →ₛ β) a = (Finset.range n).sup fun k => if i k ≤ f a then i k else 0 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [SemilatticeSup β] [OrderBot β] [Zero β] [TopologicalSpace β] [OrderClosedTopology β] [MeasurableSpace β] [OpensMeasurableSpace β] {i : ℕ → β} {f : α → β} {n : ℕ} (a : α) (hf : Measurable f)
Goal: ↑(MeasureTheory.SimpleFunc.approx i f n) a = (Finset.range n).sup fun k => if i k ≤ f a then i k else 0
-- MeasureTheory.SimpleFunc.monotone_approx
theorem monotone_approx (i : ℕ → β) (f : α → β) : Monotone (approx i f) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [SemilatticeSup β] [OrderBot β] [Zero β] (i : ℕ → β) (f : α → β)
Goal: Monotone (MeasureTheory.SimpleFunc.approx i f)
-- MeasureTheory.SimpleFunc.approx_comp
theorem approx_comp [TopologicalSpace β] [OrderClosedTopology β] [MeasurableSpace β]
    [OpensMeasurableSpace β] [MeasurableSpace γ] {i : ℕ → β} {f : γ → β} {g : α → γ} {n : ℕ} (a : α)
    (hf : Measurable f) (hg : Measurable g) :
    (approx i (f ∘ g) n : α →ₛ β) a = (approx i f n : γ →ₛ β) (g a) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [SemilatticeSup β] [OrderBot β] [Zero β] [TopologicalSpace β] [OrderClosedTopology β] [MeasurableSpace β] [OpensMeasurableSpace β] [MeasurableSpace γ] {i : ℕ → β} {f : γ → β} {g : α → γ} {n : ℕ} (a : α) (hf : Measurable f) (hg : Measurable g)
Goal: ↑(MeasureTheory.SimpleFunc.approx i (f ∘ g) n) a = ↑(MeasureTheory.SimpleFunc.approx i f n) (g a)
-- MeasureTheory.SimpleFunc.iSup_approx_apply
theorem iSup_approx_apply [TopologicalSpace β] [CompleteLattice β] [OrderClosedTopology β] [Zero β]
    [MeasurableSpace β] [OpensMeasurableSpace β] (i : ℕ → β) (f : α → β) (a : α) (hf : Measurable f)
    (h_zero : (0 : β) = ⊥) : ⨆ n, (approx i f n : α →ₛ β) a = ⨆ (k) (_ : i k ≤ f a), i k 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [TopologicalSpace β] [CompleteLattice β] [OrderClosedTopology β] [Zero β] [MeasurableSpace β] [OpensMeasurableSpace β] (i : ℕ → β) (f : α → β) (a : α) (hf : Measurable f) (h_zero : 0 = ⊥)
Goal: ⨆ n, ↑(MeasureTheory.SimpleFunc.approx i f n) a = ⨆ k, ⨆ (_ : i k ≤ f a), i k
-- MeasureTheory.SimpleFunc.ennrealRatEmbed_encode
theorem ennrealRatEmbed_encode (q : ℚ) :
    ennrealRatEmbed (Encodable.encode q) = Real.toNNReal q 
Arguments: (q : ℚ)
Goal: MeasureTheory.SimpleFunc.ennrealRatEmbed (Encodable.encode q) = ↑(↑q).toNNReal
-- MeasureTheory.SimpleFunc.eapprox_lt_top
theorem eapprox_lt_top (f : α → ℝ≥0∞) (n : ℕ) (a : α) : eapprox f n a < ∞ 
Arguments: {α : Type u_1} [MeasurableSpace α] (f : α → ℝ≥0∞) (n : ℕ) (a : α)
Goal: ↑(MeasureTheory.SimpleFunc.eapprox f n) a < ⊤
-- MeasureTheory.SimpleFunc.monotone_eapprox
@[mono]
theorem monotone_eapprox (f : α → ℝ≥0∞) : Monotone (eapprox f) 
Arguments: {α : Type u_1} [MeasurableSpace α] (f : α → ℝ≥0∞)
Goal: Monotone (MeasureTheory.SimpleFunc.eapprox f)
-- MeasureTheory.SimpleFunc.iSup_eapprox_apply
theorem iSup_eapprox_apply (f : α → ℝ≥0∞) (hf : Measurable f) (a : α) :
    ⨆ n, (eapprox f n : α →ₛ ℝ≥0∞) a = f a 
Arguments: {α : Type u_1} [MeasurableSpace α] (f : α → ℝ≥0∞) (hf : Measurable f) (a : α)
Goal: ⨆ n, ↑(MeasureTheory.SimpleFunc.eapprox f n) a = f a
-- MeasureTheory.SimpleFunc.eapprox_comp
theorem eapprox_comp [MeasurableSpace γ] {f : γ → ℝ≥0∞} {g : α → γ} {n : ℕ} (hf : Measurable f)
    (hg : Measurable g) : (eapprox (f ∘ g) n : α → ℝ≥0∞) = (eapprox f n : γ →ₛ ℝ≥0∞) ∘ g 
Arguments: {α : Type u_1} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace γ] {f : γ → ℝ≥0∞} {g : α → γ} {n : ℕ} (hf : Measurable f) (hg : Measurable g)
Goal: ↑(MeasureTheory.SimpleFunc.eapprox (f ∘ g) n) = ↑(MeasureTheory.SimpleFunc.eapprox f n) ∘ g
-- MeasureTheory.SimpleFunc.sum_eapproxDiff
theorem sum_eapproxDiff (f : α → ℝ≥0∞) (n : ℕ) (a : α) :
    (∑ k ∈ Finset.range (n + 1), (eapproxDiff f k a : ℝ≥0∞)) = eapprox f n a 
Arguments: {α : Type u_1} [MeasurableSpace α] (f : α → ℝ≥0∞) (n : ℕ) (a : α)
Goal: ∑ k ∈ Finset.range (n + 1), ↑(↑(MeasureTheory.SimpleFunc.eapproxDiff f k) a) = ↑(MeasureTheory.SimpleFunc.eapprox f n) a
-- MeasureTheory.SimpleFunc.tsum_eapproxDiff
theorem tsum_eapproxDiff (f : α → ℝ≥0∞) (hf : Measurable f) (a : α) :
    (∑' n, (eapproxDiff f n a : ℝ≥0∞)) = f a 
Arguments: {α : Type u_1} [MeasurableSpace α] (f : α → ℝ≥0∞) (hf : Measurable f) (a : α)
Goal: ∑' (n : ℕ), ↑(↑(MeasureTheory.SimpleFunc.eapproxDiff f n) a) = f a
-- MeasureTheory.SimpleFunc.lintegral_eq_of_subset
theorem lintegral_eq_of_subset (f : α →ₛ ℝ≥0∞) {s : Finset ℝ≥0∞}
    (hs : ∀ x, f x ≠ 0 → μ (f ⁻¹' {f x}) ≠ 0 → f x ∈ s) :
    f.lintegral μ = ∑ x ∈ s, x * μ (f ⁻¹' {x}) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : MeasureTheory.SimpleFunc α ℝ≥0∞) {s : Finset ℝ≥0∞} (hs : ∀ (x : α), ↑f x ≠ 0 → μ (↑f ⁻¹' {↑f x}) ≠ 0 → ↑f x ∈ s)
Goal: f.lintegral μ = ∑ x ∈ s, x * μ (↑f ⁻¹' {x})
-- MeasureTheory.SimpleFunc.lintegral_eq_of_subset'
theorem lintegral_eq_of_subset' (f : α →ₛ ℝ≥0∞) {s : Finset ℝ≥0∞} (hs : f.range \ {0} ⊆ s) :
    f.lintegral μ = ∑ x ∈ s, x * μ (f ⁻¹' {x}) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : MeasureTheory.SimpleFunc α ℝ≥0∞) {s : Finset ℝ≥0∞} (hs : f.range \ {0} ⊆ s)
Goal: f.lintegral μ = ∑ x ∈ s, x * μ (↑f ⁻¹' {x})
-- MeasureTheory.SimpleFunc.map_lintegral
theorem map_lintegral (g : β → ℝ≥0∞) (f : α →ₛ β) :
    (f.map g).lintegral μ = ∑ x ∈ f.range, g x * μ (f ⁻¹' {x}) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (g : β → ℝ≥0∞) (f : MeasureTheory.SimpleFunc α β)
Goal: (MeasureTheory.SimpleFunc.map g f).lintegral μ = ∑ x ∈ f.range, g x * μ (↑f ⁻¹' {x})
-- MeasureTheory.SimpleFunc.add_lintegral
theorem add_lintegral (f g : α →ₛ ℝ≥0∞) : (f + g).lintegral μ = f.lintegral μ + g.lintegral μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f g : MeasureTheory.SimpleFunc α ℝ≥0∞)
Goal: (f + g).lintegral μ = f.lintegral μ + g.lintegral μ
-- MeasureTheory.SimpleFunc.const_mul_lintegral
theorem const_mul_lintegral (f : α →ₛ ℝ≥0∞) (x : ℝ≥0∞) :
    (const α x * f).lintegral μ = x * f.lintegral μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : MeasureTheory.SimpleFunc α ℝ≥0∞) (x : ℝ≥0∞)
Goal: (MeasureTheory.SimpleFunc.const α x * f).lintegral μ = x * f.lintegral μ
-- MeasureTheory.SimpleFunc.zero_lintegral
@[simp]
theorem zero_lintegral : (0 : α →ₛ ℝ≥0∞).lintegral μ = 0 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α}
Goal: MeasureTheory.SimpleFunc.lintegral 0 μ = 0
-- MeasureTheory.SimpleFunc.lintegral_add
theorem lintegral_add {ν} (f : α →ₛ ℝ≥0∞) : f.lintegral (μ + ν) = f.lintegral μ + f.lintegral ν 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} (f : MeasureTheory.SimpleFunc α ℝ≥0∞)
Goal: f.lintegral (μ + ν) = f.lintegral μ + f.lintegral ν
-- MeasureTheory.SimpleFunc.lintegral_smul
theorem lintegral_smul (f : α →ₛ ℝ≥0∞) (c : ℝ≥0∞) : f.lintegral (c • μ) = c • f.lintegral μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : MeasureTheory.SimpleFunc α ℝ≥0∞) (c : ℝ≥0∞)
Goal: f.lintegral (c • μ) = c • f.lintegral μ
-- MeasureTheory.SimpleFunc.lintegral_zero
@[simp]
theorem lintegral_zero [MeasurableSpace α] (f : α →ₛ ℝ≥0∞) : f.lintegral 0 = 0 
Arguments: {α : Type u_1} [MeasurableSpace α] (f : MeasureTheory.SimpleFunc α ℝ≥0∞)
Goal: f.lintegral 0 = 0
-- MeasureTheory.SimpleFunc.lintegral_sum
theorem lintegral_sum {m : MeasurableSpace α} {ι} (f : α →ₛ ℝ≥0∞) (μ : ι → Measure α) :
    f.lintegral (Measure.sum μ) = ∑' i, f.lintegral (μ i) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {ι : Type u_5} (f : MeasureTheory.SimpleFunc α ℝ≥0∞) (μ : ι → MeasureTheory.Measure α)
Goal: f.lintegral (MeasureTheory.Measure.sum μ) = ∑' (i : ι), f.lintegral (μ i)
-- MeasureTheory.SimpleFunc.restrict_lintegral
theorem restrict_lintegral (f : α →ₛ ℝ≥0∞) {s : Set α} (hs : MeasurableSet s) :
    (restrict f s).lintegral μ = ∑ r ∈ f.range, r * μ (f ⁻¹' {r} ∩ s) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : MeasureTheory.SimpleFunc α ℝ≥0∞) {s : Set α} (hs : MeasurableSet s)
Goal: (f.restrict s).lintegral μ = ∑ r ∈ f.range, r * μ (↑f ⁻¹' {r} ∩ s)
-- MeasureTheory.SimpleFunc.lintegral_restrict
theorem lintegral_restrict {m : MeasurableSpace α} (f : α →ₛ ℝ≥0∞) (s : Set α) (μ : Measure α) :
    f.lintegral (μ.restrict s) = ∑ y ∈ f.range, y * μ (f ⁻¹' {y} ∩ s) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (f : MeasureTheory.SimpleFunc α ℝ≥0∞) (s : Set α) (μ : MeasureTheory.Measure α)
Goal: f.lintegral (μ.restrict s) = ∑ y ∈ f.range, y * μ (↑f ⁻¹' {y} ∩ s)
-- MeasureTheory.SimpleFunc.restrict_lintegral_eq_lintegral_restrict
theorem restrict_lintegral_eq_lintegral_restrict (f : α →ₛ ℝ≥0∞) {s : Set α}
    (hs : MeasurableSet s) : (restrict f s).lintegral μ = f.lintegral (μ.restrict s) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : MeasureTheory.SimpleFunc α ℝ≥0∞) {s : Set α} (hs : MeasurableSet s)
Goal: (f.restrict s).lintegral μ = f.lintegral (μ.restrict s)
-- MeasureTheory.SimpleFunc.lintegral_restrict_iUnion_of_directed
theorem lintegral_restrict_iUnion_of_directed {ι : Type*} [Countable ι]
    (f : α →ₛ ℝ≥0∞) {s : ι → Set α} (hd : Directed (· ⊆ ·) s) (μ : Measure α) :
    f.lintegral (μ.restrict (⋃ i, s i)) = ⨆ i, f.lintegral (μ.restrict (s i)) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {ι : Type u_5} [Countable ι] (f : MeasureTheory.SimpleFunc α ℝ≥0∞) {s : ι → Set α} (hd : Directed (fun x x_1 => x ⊆ x_1) s) (μ : MeasureTheory.Measure α)
Goal: f.lintegral (μ.restrict (⋃ i, s i)) = ⨆ i, f.lintegral (μ.restrict (s i))
-- MeasureTheory.SimpleFunc.const_lintegral
theorem const_lintegral (c : ℝ≥0∞) : (const α c).lintegral μ = c * μ univ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (c : ℝ≥0∞)
Goal: (MeasureTheory.SimpleFunc.const α c).lintegral μ = c * μ Set.univ
-- MeasureTheory.SimpleFunc.const_lintegral_restrict
theorem const_lintegral_restrict (c : ℝ≥0∞) (s : Set α) :
    (const α c).lintegral (μ.restrict s) = c * μ s 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (c : ℝ≥0∞) (s : Set α)
Goal: (MeasureTheory.SimpleFunc.const α c).lintegral (μ.restrict s) = c * μ s
-- MeasureTheory.SimpleFunc.restrict_const_lintegral
theorem restrict_const_lintegral (c : ℝ≥0∞) {s : Set α} (hs : MeasurableSet s) :
    ((const α c).restrict s).lintegral μ = c * μ s 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (c : ℝ≥0∞) {s : Set α} (hs : MeasurableSet s)
Goal: ((MeasureTheory.SimpleFunc.const α c).restrict s).lintegral μ = c * μ s
-- MeasureTheory.SimpleFunc.le_sup_lintegral
theorem le_sup_lintegral (f g : α →ₛ ℝ≥0∞) : f.lintegral μ ⊔ g.lintegral μ ≤ (f ⊔ g).lintegral μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f g : MeasureTheory.SimpleFunc α ℝ≥0∞)
Goal: f.lintegral μ ⊔ g.lintegral μ ≤ (f ⊔ g).lintegral μ
-- MeasureTheory.SimpleFunc.lintegral_mono
@[mono]
theorem lintegral_mono {f g : α →ₛ ℝ≥0∞} (hfg : f ≤ g) (hμν : μ ≤ ν) :
    f.lintegral μ ≤ g.lintegral ν 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} {f g : MeasureTheory.SimpleFunc α ℝ≥0∞} (hfg : f ≤ g) (hμν : μ ≤ ν)
Goal: f.lintegral μ ≤ g.lintegral ν
-- MeasureTheory.SimpleFunc.lintegral_eq_of_measure_preimage
theorem lintegral_eq_of_measure_preimage [MeasurableSpace β] {f : α →ₛ ℝ≥0∞} {g : β →ₛ ℝ≥0∞}
    {ν : Measure β} (H : ∀ y, μ (f ⁻¹' {y}) = ν (g ⁻¹' {y})) : f.lintegral μ = g.lintegral ν 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasurableSpace β] {f : MeasureTheory.SimpleFunc α ℝ≥0∞} {g : MeasureTheory.SimpleFunc β ℝ≥0∞} {ν : MeasureTheory.Measure β} (H : ∀ (y : ℝ≥0∞), μ (↑f ⁻¹' {y}) = ν (↑g ⁻¹' {y}))
Goal: f.lintegral μ = g.lintegral ν
-- MeasureTheory.SimpleFunc.lintegral_congr
theorem lintegral_congr {f g : α →ₛ ℝ≥0∞} (h : f =ᵐ[μ] g) : f.lintegral μ = g.lintegral μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : MeasureTheory.SimpleFunc α ℝ≥0∞} (h : ↑f =ᵐ[μ] ↑g)
Goal: f.lintegral μ = g.lintegral μ
-- MeasureTheory.SimpleFunc.lintegral_map'
theorem lintegral_map' {β} [MeasurableSpace β] {μ' : Measure β} (f : α →ₛ ℝ≥0∞) (g : β →ₛ ℝ≥0∞)
    (m' : α → β) (eq : ∀ a, f a = g (m' a)) (h : ∀ s, MeasurableSet s → μ' s = μ (m' ⁻¹' s)) :
    f.lintegral μ = g.lintegral μ' 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_5} [MeasurableSpace β] {μ' : MeasureTheory.Measure β} (f : MeasureTheory.SimpleFunc α ℝ≥0∞) (g : MeasureTheory.SimpleFunc β ℝ≥0∞) (m' : α → β) (eq : ∀ (a : α), ↑f a = ↑g (m' a)) (h : ∀ (s : Set β), MeasurableSet s → μ' s = μ (m' ⁻¹' s))
Goal: f.lintegral μ = g.lintegral μ'
-- MeasureTheory.SimpleFunc.lintegral_map
theorem lintegral_map {β} [MeasurableSpace β] (g : β →ₛ ℝ≥0∞) {f : α → β} (hf : Measurable f) :
    g.lintegral (Measure.map f μ) = (g.comp f hf).lintegral μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_5} [MeasurableSpace β] (g : MeasureTheory.SimpleFunc β ℝ≥0∞) {f : α → β} (hf : Measurable f)
Goal: g.lintegral (MeasureTheory.Measure.map f μ) = (g.comp f hf).lintegral μ
-- MeasureTheory.SimpleFunc.support_eq
theorem support_eq [MeasurableSpace α] [Zero β] (f : α →ₛ β) :
    support f = ⋃ y ∈ f.range.filter fun y => y ≠ 0, f ⁻¹' {y} 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [Zero β] (f : MeasureTheory.SimpleFunc α β)
Goal: Function.support ↑f = ⋃ y ∈ Finset.filter (fun y => y ≠ 0) f.range, ↑f ⁻¹' {y}
-- MeasureTheory.SimpleFunc.measurableSet_support
theorem measurableSet_support [MeasurableSpace α] (f : α →ₛ β) : MeasurableSet (support f) 
Arguments: {α : Type u_1} {β : Type u_2} [Zero β] [MeasurableSpace α] (f : MeasureTheory.SimpleFunc α β)
Goal: MeasurableSet (Function.support ↑f)
-- MeasureTheory.SimpleFunc.finMeasSupp_iff_support
theorem finMeasSupp_iff_support : f.FinMeasSupp μ ↔ μ (support f) < ∞ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} [Zero β] {μ : MeasureTheory.Measure α} {f : MeasureTheory.SimpleFunc α β}
Goal: f.FinMeasSupp μ ↔ μ (Function.support ↑f) < ⊤
-- MeasureTheory.SimpleFunc.finMeasSupp_iff
theorem finMeasSupp_iff : f.FinMeasSupp μ ↔ ∀ y, y ≠ 0 → μ (f ⁻¹' {y}) < ∞ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} [Zero β] {μ : MeasureTheory.Measure α} {f : MeasureTheory.SimpleFunc α β}
Goal: f.FinMeasSupp μ ↔ ∀ (y : β), y ≠ 0 → μ (↑f ⁻¹' {y}) < ⊤
-- MeasureTheory.SimpleFunc.FinMeasSupp.meas_preimage_singleton_ne_zero
theorem meas_preimage_singleton_ne_zero (h : f.FinMeasSupp μ) {y : β} (hy : y ≠ 0) :
    μ (f ⁻¹' {y}) < ∞ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} [Zero β] {μ : MeasureTheory.Measure α} {f : MeasureTheory.SimpleFunc α β} (h : f.FinMeasSupp μ) {y : β} (hy : y ≠ 0)
Goal: μ (↑f ⁻¹' {y}) < ⊤
-- MeasureTheory.SimpleFunc.FinMeasSupp.map
protected theorem map {g : β → γ} (hf : f.FinMeasSupp μ) (hg : g 0 = 0) : (f.map g).FinMeasSupp μ 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m : MeasurableSpace α} [Zero β] [Zero γ] {μ : MeasureTheory.Measure α} {f : MeasureTheory.SimpleFunc α β} {g : β → γ} (hf : f.FinMeasSupp μ) (hg : g 0 = 0)
Goal: (MeasureTheory.SimpleFunc.map g f).FinMeasSupp μ
-- MeasureTheory.SimpleFunc.FinMeasSupp.of_map
theorem of_map {g : β → γ} (h : (f.map g).FinMeasSupp μ) (hg : ∀ b, g b = 0 → b = 0) :
    f.FinMeasSupp μ 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m : MeasurableSpace α} [Zero β] [Zero γ] {μ : MeasureTheory.Measure α} {f : MeasureTheory.SimpleFunc α β} {g : β → γ} (h : (MeasureTheory.SimpleFunc.map g f).FinMeasSupp μ) (hg : ∀ (b : β), g b = 0 → b = 0)
Goal: f.FinMeasSupp μ
-- MeasureTheory.SimpleFunc.FinMeasSupp.map_iff
theorem map_iff {g : β → γ} (hg : ∀ {b}, g b = 0 ↔ b = 0) :
    (f.map g).FinMeasSupp μ ↔ f.FinMeasSupp μ 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m : MeasurableSpace α} [Zero β] [Zero γ] {μ : MeasureTheory.Measure α} {f : MeasureTheory.SimpleFunc α β} {g : β → γ} (hg : ∀ {b : β}, g b = 0 ↔ b = 0)
Goal: (MeasureTheory.SimpleFunc.map g f).FinMeasSupp μ ↔ f.FinMeasSupp μ
-- MeasureTheory.SimpleFunc.FinMeasSupp.pair
protected theorem pair {g : α →ₛ γ} (hf : f.FinMeasSupp μ) (hg : g.FinMeasSupp μ) :
    (pair f g).FinMeasSupp μ 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m : MeasurableSpace α} [Zero β] [Zero γ] {μ : MeasureTheory.Measure α} {f : MeasureTheory.SimpleFunc α β} {g : MeasureTheory.SimpleFunc α γ} (hf : f.FinMeasSupp μ) (hg : g.FinMeasSupp μ)
Goal: (f.pair g).FinMeasSupp μ