ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.SimpleFunc.preimage_eq_empty_iff
theorem preimage_eq_empty_iff (f : α →ₛ β) (b : β) : f ⁻¹' {b} = ∅ ↔ b ∉ f.range 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] (f : MeasureTheory.SimpleFunc α β) (b : β)
Goal: ↑f ⁻¹' {b} = ∅ ↔ b ∉ f.range
-- MeasureTheory.SimpleFunc.exists_forall_le
theorem exists_forall_le [Nonempty β] [Preorder β] [IsDirected β (· ≤ ·)] (f : α →ₛ β) :
    ∃ C, ∀ x, f x ≤ C 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [Nonempty β] [Preorder β] [IsDirected β fun x x_1 => x ≤ x_1] (f : MeasureTheory.SimpleFunc α β)
Goal: ∃ C, ∀ (x : α), ↑f x ≤ C
-- MeasureTheory.SimpleFunc.const_apply
theorem const_apply (a : α) (b : β) : (const α b) a = b 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] (a : α) (b : β)
Goal: ↑(MeasureTheory.SimpleFunc.const α b) a = b
-- MeasureTheory.SimpleFunc.coe_const
@[simp]
theorem coe_const (b : β) : ⇑(const α b) = Function.const α b 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] (b : β)
Goal: ↑(MeasureTheory.SimpleFunc.const α b) = Function.const α b
-- MeasureTheory.SimpleFunc.range_const
@[simp]
theorem range_const (α) [MeasurableSpace α] [Nonempty α] (b : β) : (const α b).range = {b} 
Arguments: {β : Type u_2} (α : Type u_5) [MeasurableSpace α] [Nonempty α] (b : β)
Goal: (MeasureTheory.SimpleFunc.const α b).range = {b}
-- MeasureTheory.SimpleFunc.range_const_subset
theorem range_const_subset (α) [MeasurableSpace α] (b : β) : (const α b).range ⊆ {b} 
Arguments: {β : Type u_2} (α : Type u_5) [MeasurableSpace α] (b : β)
Goal: (MeasureTheory.SimpleFunc.const α b).range ⊆ {b}
-- MeasureTheory.SimpleFunc.simpleFunc_bot
theorem simpleFunc_bot {α} (f : @SimpleFunc α ⊥ β) [Nonempty β] : ∃ c, ∀ x, f x = c 
Arguments: {β : Type u_2} {α : Type u_5} (f : MeasureTheory.SimpleFunc α β) [Nonempty β]
Goal: ∃ c, ∀ (x : α), ↑f x = c
-- MeasureTheory.SimpleFunc.simpleFunc_bot'
theorem simpleFunc_bot' {α} [Nonempty β] (f : @SimpleFunc α ⊥ β) :
    ∃ c, f = @SimpleFunc.const α _ ⊥ c 
Arguments: {β : Type u_2} {α : Type u_5} [Nonempty β] (f : MeasureTheory.SimpleFunc α β)
Goal: ∃ c, f = MeasureTheory.SimpleFunc.const α c
-- MeasureTheory.SimpleFunc.measurableSet_cut
theorem measurableSet_cut (r : α → β → Prop) (f : α →ₛ β) (h : ∀ b, MeasurableSet { a | r a b }) :
    MeasurableSet { a | r a (f a) } 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] (r : α → β → Prop) (f : MeasureTheory.SimpleFunc α β) (h : ∀ (b : β), MeasurableSet {a | r a b})
Goal: MeasurableSet {a | r a (↑f a)}
-- MeasureTheory.SimpleFunc.measurableSet_preimage
@[measurability]
theorem measurableSet_preimage (f : α →ₛ β) (s) : MeasurableSet (f ⁻¹' s) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] (f : MeasureTheory.SimpleFunc α β) (s : Set β)
Goal: MeasurableSet (↑f ⁻¹' s)
-- MeasureTheory.SimpleFunc.measurable
@[measurability]
protected theorem measurable [MeasurableSpace β] (f : α →ₛ β) : Measurable f 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (f : MeasureTheory.SimpleFunc α β)
Goal: Measurable ↑f
-- MeasureTheory.SimpleFunc.aemeasurable
@[measurability]
protected theorem aemeasurable [MeasurableSpace β] {μ : Measure α} (f : α →ₛ β) :
    AEMeasurable f μ 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} (f : MeasureTheory.SimpleFunc α β)
Goal: AEMeasurable (↑f) μ
-- MeasureTheory.SimpleFunc.sum_measure_preimage_singleton
protected theorem sum_measure_preimage_singleton (f : α →ₛ β) {μ : Measure α} (s : Finset β) :
    (∑ y ∈ s, μ (f ⁻¹' {y})) = μ (f ⁻¹' ↑s) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] (f : MeasureTheory.SimpleFunc α β) {μ : MeasureTheory.Measure α} (s : Finset β)
Goal: ∑ y ∈ s, μ (↑f ⁻¹' {y}) = μ (↑f ⁻¹' ↑s)
-- MeasureTheory.SimpleFunc.sum_range_measure_preimage_singleton
theorem sum_range_measure_preimage_singleton (f : α →ₛ β) (μ : Measure α) :
    (∑ y ∈ f.range, μ (f ⁻¹' {y})) = μ univ 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] (f : MeasureTheory.SimpleFunc α β) (μ : MeasureTheory.Measure α)
Goal: ∑ y ∈ f.range, μ (↑f ⁻¹' {y}) = μ Set.univ
-- MeasureTheory.SimpleFunc.coe_piecewise
@[simp]
theorem coe_piecewise {s : Set α} (hs : MeasurableSet s) (f g : α →ₛ β) :
    ⇑(piecewise s hs f g) = s.piecewise f g 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {s : Set α} (hs : MeasurableSet s) (f g : MeasureTheory.SimpleFunc α β)
Goal: ↑(MeasureTheory.SimpleFunc.piecewise s hs f g) = s.piecewise ↑f ↑g
-- MeasureTheory.SimpleFunc.piecewise_apply
theorem piecewise_apply {s : Set α} (hs : MeasurableSet s) (f g : α →ₛ β) (a) :
    piecewise s hs f g a = if a ∈ s then f a else g a 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {s : Set α} (hs : MeasurableSet s) (f g : MeasureTheory.SimpleFunc α β) (a : α)
Goal: ↑(MeasureTheory.SimpleFunc.piecewise s hs f g) a = if a ∈ s then ↑f a else ↑g a
-- MeasureTheory.SimpleFunc.piecewise_compl
@[simp]
theorem piecewise_compl {s : Set α} (hs : MeasurableSet sᶜ) (f g : α →ₛ β) :
    piecewise sᶜ hs f g = piecewise s hs.of_compl g f 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {s : Set α} (hs : MeasurableSet sᶜ) (f g : MeasureTheory.SimpleFunc α β)
Goal: MeasureTheory.SimpleFunc.piecewise sᶜ hs f g = MeasureTheory.SimpleFunc.piecewise s ⋯ g f
-- MeasureTheory.SimpleFunc.piecewise_univ
@[simp]
theorem piecewise_univ (f g : α →ₛ β) : piecewise univ MeasurableSet.univ f g = f 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] (f g : MeasureTheory.SimpleFunc α β)
Goal: MeasureTheory.SimpleFunc.piecewise Set.univ ⋯ f g = f
-- MeasureTheory.SimpleFunc.piecewise_empty
@[simp]
theorem piecewise_empty (f g : α →ₛ β) : piecewise ∅ MeasurableSet.empty f g = g 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] (f g : MeasureTheory.SimpleFunc α β)
Goal: MeasureTheory.SimpleFunc.piecewise ∅ ⋯ f g = g
-- MeasureTheory.SimpleFunc.piecewise_same
@[simp]
theorem piecewise_same (f : α →ₛ β) {s : Set α} (hs : MeasurableSet s) :
    piecewise s hs f f = f 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] (f : MeasureTheory.SimpleFunc α β) {s : Set α} (hs : MeasurableSet s)
Goal: MeasureTheory.SimpleFunc.piecewise s hs f f = f
-- MeasureTheory.SimpleFunc.support_indicator
theorem support_indicator [Zero β] {s : Set α} (hs : MeasurableSet s) (f : α →ₛ β) :
    Function.support (f.piecewise s hs (SimpleFunc.const α 0)) = s ∩ Function.support f 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [Zero β] {s : Set α} (hs : MeasurableSet s) (f : MeasureTheory.SimpleFunc α β)
Goal: Function.support ↑(MeasureTheory.SimpleFunc.piecewise s hs f (MeasureTheory.SimpleFunc.const α 0)) =
  s ∩ Function.support ↑f
-- MeasureTheory.SimpleFunc.range_indicator
theorem range_indicator {s : Set α} (hs : MeasurableSet s) (hs_nonempty : s.Nonempty)
    (hs_ne_univ : s ≠ univ) (x y : β) :
    (piecewise s hs (const α x) (const α y)).range = {x, y} 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {s : Set α} (hs : MeasurableSet s) (hs_nonempty : s.Nonempty) (hs_ne_univ : s ≠ Set.univ) (x y : β)
Goal: (MeasureTheory.SimpleFunc.piecewise s hs (MeasureTheory.SimpleFunc.const α x)
      (MeasureTheory.SimpleFunc.const α y)).range =
  {x, y}
-- MeasureTheory.SimpleFunc.measurable_bind
theorem measurable_bind [MeasurableSpace γ] (f : α →ₛ β) (g : β → α → γ)
    (hg : ∀ b, Measurable (g b)) : Measurable fun a => g (f a) a 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace γ] (f : MeasureTheory.SimpleFunc α β) (g : β → α → γ) (hg : ∀ (b : β), Measurable (g b))
Goal: Measurable fun a => g (↑f a) a
-- MeasureTheory.SimpleFunc.bind_apply
@[simp]
theorem bind_apply (f : α →ₛ β) (g : β → α →ₛ γ) (a) : f.bind g a = g (f a) a 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] (f : MeasureTheory.SimpleFunc α β) (g : β → MeasureTheory.SimpleFunc α γ) (a : α)
Goal: ↑(f.bind g) a = ↑(g (↑f a)) a
-- MeasureTheory.SimpleFunc.map_apply
theorem map_apply (g : β → γ) (f : α →ₛ β) (a) : f.map g a = g (f a) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] (g : β → γ) (f : MeasureTheory.SimpleFunc α β) (a : α)
Goal: ↑(MeasureTheory.SimpleFunc.map g f) a = g (↑f a)
-- MeasureTheory.SimpleFunc.map_map
theorem map_map (g : β → γ) (h : γ → δ) (f : α →ₛ β) : (f.map g).map h = f.map (h ∘ g) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [MeasurableSpace α] (g : β → γ) (h : γ → δ) (f : MeasureTheory.SimpleFunc α β)
Goal: MeasureTheory.SimpleFunc.map h (MeasureTheory.SimpleFunc.map g f) = MeasureTheory.SimpleFunc.map (h ∘ g) f
-- MeasureTheory.SimpleFunc.coe_map
@[simp]
theorem coe_map (g : β → γ) (f : α →ₛ β) : (f.map g : α → γ) = g ∘ f 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] (g : β → γ) (f : MeasureTheory.SimpleFunc α β)
Goal: ↑(MeasureTheory.SimpleFunc.map g f) = g ∘ ↑f
-- MeasureTheory.SimpleFunc.range_map
@[simp]
theorem range_map [DecidableEq γ] (g : β → γ) (f : α →ₛ β) : (f.map g).range = f.range.image g 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [DecidableEq γ] (g : β → γ) (f : MeasureTheory.SimpleFunc α β)
Goal: (MeasureTheory.SimpleFunc.map g f).range = Finset.image g f.range
-- MeasureTheory.SimpleFunc.map_const
@[simp]
theorem map_const (g : β → γ) (b : β) : (const α b).map g = const α (g b) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] (g : β → γ) (b : β)
Goal: MeasureTheory.SimpleFunc.map g (MeasureTheory.SimpleFunc.const α b) = MeasureTheory.SimpleFunc.const α (g b)
-- MeasureTheory.SimpleFunc.map_preimage
theorem map_preimage (f : α →ₛ β) (g : β → γ) (s : Set γ) :
    f.map g ⁻¹' s = f ⁻¹' ↑(f.range.filter fun b => g b ∈ s) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] (f : MeasureTheory.SimpleFunc α β) (g : β → γ) (s : Set γ)
Goal: ↑(MeasureTheory.SimpleFunc.map g f) ⁻¹' s = ↑f ⁻¹' ↑(Finset.filter (fun b => g b ∈ s) f.range)
-- MeasureTheory.SimpleFunc.map_preimage_singleton
theorem map_preimage_singleton (f : α →ₛ β) (g : β → γ) (c : γ) :
    f.map g ⁻¹' {c} = f ⁻¹' ↑(f.range.filter fun b => g b = c) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] (f : MeasureTheory.SimpleFunc α β) (g : β → γ) (c : γ)
Goal: ↑(MeasureTheory.SimpleFunc.map g f) ⁻¹' {c} = ↑f ⁻¹' ↑(Finset.filter (fun b => g b = c) f.range)
-- MeasureTheory.SimpleFunc.coe_comp
@[simp]
theorem coe_comp [MeasurableSpace β] (f : β →ₛ γ) {g : α → β} (hgm : Measurable g) :
    ⇑(f.comp g hgm) = f ∘ g 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] (f : MeasureTheory.SimpleFunc β γ) {g : α → β} (hgm : Measurable g)
Goal: ↑(f.comp g hgm) = ↑f ∘ g
-- MeasureTheory.SimpleFunc.range_comp_subset_range
theorem range_comp_subset_range [MeasurableSpace β] (f : β →ₛ γ) {g : α → β} (hgm : Measurable g) :
    (f.comp g hgm).range ⊆ f.range 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] (f : MeasureTheory.SimpleFunc β γ) {g : α → β} (hgm : Measurable g)
Goal: (f.comp g hgm).range ⊆ f.range
-- MeasureTheory.SimpleFunc.extend_apply
@[simp]
theorem extend_apply [MeasurableSpace β] (f₁ : α →ₛ γ) {g : α → β} (hg : MeasurableEmbedding g)
    (f₂ : β →ₛ γ) (x : α) : (f₁.extend g hg f₂) (g x) = f₁ x 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] (f₁ : MeasureTheory.SimpleFunc α γ) {g : α → β} (hg : MeasurableEmbedding g) (f₂ : MeasureTheory.SimpleFunc β γ) (x : α)
Goal: ↑(f₁.extend g hg f₂) (g x) = ↑f₁ x
-- MeasureTheory.SimpleFunc.extend_apply'
@[simp]
theorem extend_apply' [MeasurableSpace β] (f₁ : α →ₛ γ) {g : α → β} (hg : MeasurableEmbedding g)
    (f₂ : β →ₛ γ) {y : β} (h : ¬∃ x, g x = y) : (f₁.extend g hg f₂) y = f₂ y 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] (f₁ : MeasureTheory.SimpleFunc α γ) {g : α → β} (hg : MeasurableEmbedding g) (f₂ : MeasureTheory.SimpleFunc β γ) {y : β} (h : ¬∃ x, g x = y)
Goal: ↑(f₁.extend g hg f₂) y = ↑f₂ y
-- MeasureTheory.SimpleFunc.extend_comp_eq'
@[simp]
theorem extend_comp_eq' [MeasurableSpace β] (f₁ : α →ₛ γ) {g : α → β} (hg : MeasurableEmbedding g)
    (f₂ : β →ₛ γ) : f₁.extend g hg f₂ ∘ g = f₁ 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] (f₁ : MeasureTheory.SimpleFunc α γ) {g : α → β} (hg : MeasurableEmbedding g) (f₂ : MeasureTheory.SimpleFunc β γ)
Goal: ↑(f₁.extend g hg f₂) ∘ g = ↑f₁
-- MeasureTheory.SimpleFunc.extend_comp_eq
@[simp]
theorem extend_comp_eq [MeasurableSpace β] (f₁ : α →ₛ γ) {g : α → β} (hg : MeasurableEmbedding g)
    (f₂ : β →ₛ γ) : (f₁.extend g hg f₂).comp g hg.measurable = f₁ 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] (f₁ : MeasureTheory.SimpleFunc α γ) {g : α → β} (hg : MeasurableEmbedding g) (f₂ : MeasureTheory.SimpleFunc β γ)
Goal: (f₁.extend g hg f₂).comp g ⋯ = f₁
-- MeasureTheory.SimpleFunc.seq_apply
@[simp]
theorem seq_apply (f : α →ₛ β → γ) (g : α →ₛ β) (a : α) : f.seq g a = f a (g a) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] (f : MeasureTheory.SimpleFunc α (β → γ)) (g : MeasureTheory.SimpleFunc α β) (a : α)
Goal: ↑(f.seq g) a = ↑f a (↑g a)
-- MeasureTheory.SimpleFunc.pair_apply
@[simp]
theorem pair_apply (f : α →ₛ β) (g : α →ₛ γ) (a) : pair f g a = (f a, g a) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] (f : MeasureTheory.SimpleFunc α β) (g : MeasureTheory.SimpleFunc α γ) (a : α)
Goal: ↑(f.pair g) a = (↑f a, ↑g a)
-- MeasureTheory.SimpleFunc.pair_preimage
theorem pair_preimage (f : α →ₛ β) (g : α →ₛ γ) (s : Set β) (t : Set γ) :
    pair f g ⁻¹' s ×ˢ t = f ⁻¹' s ∩ g ⁻¹' t 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] (f : MeasureTheory.SimpleFunc α β) (g : MeasureTheory.SimpleFunc α γ) (s : Set β) (t : Set γ)
Goal: ↑(f.pair g) ⁻¹' s ×ˢ t = ↑f ⁻¹' s ∩ ↑g ⁻¹' t
-- MeasureTheory.SimpleFunc.pair_preimage_singleton
theorem pair_preimage_singleton (f : α →ₛ β) (g : α →ₛ γ) (b : β) (c : γ) :
    pair f g ⁻¹' {(b, c)} = f ⁻¹' {b} ∩ g ⁻¹' {c} 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] (f : MeasureTheory.SimpleFunc α β) (g : MeasureTheory.SimpleFunc α γ) (b : β) (c : γ)
Goal: ↑(f.pair g) ⁻¹' {(b, c)} = ↑f ⁻¹' {b} ∩ ↑g ⁻¹' {c}
-- MeasureTheory.SimpleFunc.bind_const
theorem bind_const (f : α →ₛ β) : f.bind (const α) = f 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] (f : MeasureTheory.SimpleFunc α β)
Goal: f.bind (MeasureTheory.SimpleFunc.const α) = f
-- MeasureTheory.SimpleFunc.const_zero
@[to_additive (attr := simp)]
theorem const_one [One β] : const α (1 : β) = 1 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [Zero β]
Goal: MeasureTheory.SimpleFunc.const α 0 = 0
-- MeasureTheory.SimpleFunc.const_one
@[to_additive (attr := simp)]
theorem const_one [One β] : const α (1 : β) = 1 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [One β]
Goal: MeasureTheory.SimpleFunc.const α 1 = 1
-- MeasureTheory.SimpleFunc.coe_one
@[to_additive (attr := simp, norm_cast)]
theorem coe_one [One β] : ⇑(1 : α →ₛ β) = 1 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [One β]
Goal: ↑1 = 1
-- MeasureTheory.SimpleFunc.coe_zero
@[to_additive (attr := simp, norm_cast)]
theorem coe_one [One β] : ⇑(1 : α →ₛ β) = 1 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [Zero β]
Goal: ↑0 = 0
-- MeasureTheory.SimpleFunc.coe_mul
@[to_additive (attr := simp, norm_cast)]
theorem coe_mul [Mul β] (f g : α →ₛ β) : ⇑(f * g) = ⇑f * ⇑g 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [Mul β] (f g : MeasureTheory.SimpleFunc α β)
Goal: ↑(f * g) = ↑f * ↑g
-- MeasureTheory.SimpleFunc.coe_add
@[to_additive (attr := simp, norm_cast)]
theorem coe_mul [Mul β] (f g : α →ₛ β) : ⇑(f * g) = ⇑f * ⇑g 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [Add β] (f g : MeasureTheory.SimpleFunc α β)
Goal: ↑(f + g) = ↑f + ↑g
-- MeasureTheory.SimpleFunc.coe_neg
@[to_additive (attr := simp, norm_cast)]
theorem coe_inv [Inv β] (f : α →ₛ β) : ⇑(f⁻¹) = (⇑f)⁻¹ 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [Neg β] (f : MeasureTheory.SimpleFunc α β)
Goal: ↑(-f) = -↑f
-- MeasureTheory.SimpleFunc.coe_inv
@[to_additive (attr := simp, norm_cast)]
theorem coe_inv [Inv β] (f : α →ₛ β) : ⇑(f⁻¹) = (⇑f)⁻¹ 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [Inv β] (f : MeasureTheory.SimpleFunc α β)
Goal: ↑f⁻¹ = (↑f)⁻¹
-- MeasureTheory.SimpleFunc.coe_div
@[to_additive (attr := simp, norm_cast)]
theorem coe_div [Div β] (f g : α →ₛ β) : ⇑(f / g) = ⇑f / ⇑g 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [Div β] (f g : MeasureTheory.SimpleFunc α β)
Goal: ↑(f / g) = ↑f / ↑g
-- MeasureTheory.SimpleFunc.coe_sub
@[to_additive (attr := simp, norm_cast)]
theorem coe_div [Div β] (f g : α →ₛ β) : ⇑(f / g) = ⇑f / ⇑g 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [Sub β] (f g : MeasureTheory.SimpleFunc α β)
Goal: ↑(f - g) = ↑f - ↑g
-- MeasureTheory.SimpleFunc.coe_le
@[simp, norm_cast]
theorem coe_le [Preorder β] {f g : α →ₛ β} : (f : α → β) ≤ g ↔ f ≤ g 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [Preorder β] {f g : MeasureTheory.SimpleFunc α β}
Goal: ↑f ≤ ↑g ↔ f ≤ g
-- MeasureTheory.SimpleFunc.coe_sup
@[simp, norm_cast]
theorem coe_sup [Sup β] (f g : α →ₛ β) : ⇑(f ⊔ g) = ⇑f ⊔ ⇑g 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [Sup β] (f g : MeasureTheory.SimpleFunc α β)
Goal: ↑(f ⊔ g) = ↑f ⊔ ↑g
-- MeasureTheory.SimpleFunc.coe_inf
@[simp, norm_cast]
theorem coe_inf [Inf β] (f g : α →ₛ β) : ⇑(f ⊓ g) = ⇑f ⊓ ⇑g 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [Inf β] (f g : MeasureTheory.SimpleFunc α β)
Goal: ↑(f ⊓ g) = ↑f ⊓ ↑g
-- MeasureTheory.SimpleFunc.add_apply
@[to_additive]
theorem mul_apply [Mul β] (f g : α →ₛ β) (a : α) : (f * g) a = f a * g a 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [Add β] (f g : MeasureTheory.SimpleFunc α β) (a : α)
Goal: ↑(f + g) a = ↑f a + ↑g a
-- MeasureTheory.SimpleFunc.mul_apply
@[to_additive]
theorem mul_apply [Mul β] (f g : α →ₛ β) (a : α) : (f * g) a = f a * g a 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [Mul β] (f g : MeasureTheory.SimpleFunc α β) (a : α)
Goal: ↑(f * g) a = ↑f a * ↑g a
-- MeasureTheory.SimpleFunc.sub_apply
@[to_additive]
theorem div_apply [Div β] (f g : α →ₛ β) (x : α) : (f / g) x = f x / g x 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [Sub β] (f g : MeasureTheory.SimpleFunc α β) (x : α)
Goal: ↑(f - g) x = ↑f x - ↑g x
-- MeasureTheory.SimpleFunc.div_apply
@[to_additive]
theorem div_apply [Div β] (f g : α →ₛ β) (x : α) : (f / g) x = f x / g x 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [Div β] (f g : MeasureTheory.SimpleFunc α β) (x : α)
Goal: ↑(f / g) x = ↑f x / ↑g x
-- MeasureTheory.SimpleFunc.inv_apply
@[to_additive]
theorem inv_apply [Inv β] (f : α →ₛ β) (x : α) : f⁻¹ x = (f x)⁻¹ 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [Inv β] (f : MeasureTheory.SimpleFunc α β) (x : α)
Goal: ↑f⁻¹ x = (↑f x)⁻¹
-- MeasureTheory.SimpleFunc.neg_apply
@[to_additive]
theorem inv_apply [Inv β] (f : α →ₛ β) (x : α) : f⁻¹ x = (f x)⁻¹ 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [Neg β] (f : MeasureTheory.SimpleFunc α β) (x : α)
Goal: ↑(-f) x = -↑f x
-- MeasureTheory.SimpleFunc.sup_apply
theorem sup_apply [Sup β] (f g : α →ₛ β) (a : α) : (f ⊔ g) a = f a ⊔ g a 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [Sup β] (f g : MeasureTheory.SimpleFunc α β) (a : α)
Goal: ↑(f ⊔ g) a = ↑f a ⊔ ↑g a
-- MeasureTheory.SimpleFunc.inf_apply
theorem inf_apply [Inf β] (f g : α →ₛ β) (a : α) : (f ⊓ g) a = f a ⊓ g a 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [Inf β] (f g : MeasureTheory.SimpleFunc α β) (a : α)
Goal: ↑(f ⊓ g) a = ↑f a ⊓ ↑g a
-- MeasureTheory.SimpleFunc.range_zero
@[to_additive (attr := simp)]
theorem range_one [Nonempty α] [One β] : (1 : α →ₛ β).range = {1} 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [Nonempty α] [Zero β]
Goal: MeasureTheory.SimpleFunc.range 0 = {0}
-- MeasureTheory.SimpleFunc.range_one
@[to_additive (attr := simp)]
theorem range_one [Nonempty α] [One β] : (1 : α →ₛ β).range = {1} 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [Nonempty α] [One β]
Goal: MeasureTheory.SimpleFunc.range 1 = {1}
-- MeasureTheory.SimpleFunc.range_eq_empty_of_isEmpty
@[simp]
theorem range_eq_empty_of_isEmpty {β} [hα : IsEmpty α] (f : α →ₛ β) : f.range = ∅ 
Arguments: {α : Type u_1} [MeasurableSpace α] {β : Type u_5} [hα : IsEmpty α] (f : MeasureTheory.SimpleFunc α β)
Goal: f.range = ∅
-- MeasureTheory.SimpleFunc.eq_zero_of_mem_range_zero
theorem eq_zero_of_mem_range_zero [Zero β] : ∀ {y : β}, y ∈ (0 : α →ₛ β).range → y = 0 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [Zero β] {y : β}
Goal: y ∈ MeasureTheory.SimpleFunc.range 0 → y = 0
-- MeasureTheory.SimpleFunc.add_eq_map₂
@[to_additive]
theorem mul_eq_map₂ [Mul β] (f g : α →ₛ β) : f * g = (pair f g).map fun p : β × β => p.1 * p.2 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [Add β] (f g : MeasureTheory.SimpleFunc α β)
Goal: f + g = MeasureTheory.SimpleFunc.map (fun p => p.1 + p.2) (f.pair g)
-- MeasureTheory.SimpleFunc.mul_eq_map₂
@[to_additive]
theorem mul_eq_map₂ [Mul β] (f g : α →ₛ β) : f * g = (pair f g).map fun p : β × β => p.1 * p.2 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [Mul β] (f g : MeasureTheory.SimpleFunc α β)
Goal: f * g = MeasureTheory.SimpleFunc.map (fun p => p.1 * p.2) (f.pair g)
-- MeasureTheory.SimpleFunc.sup_eq_map₂
theorem sup_eq_map₂ [Sup β] (f g : α →ₛ β) : f ⊔ g = (pair f g).map fun p : β × β => p.1 ⊔ p.2 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [Sup β] (f g : MeasureTheory.SimpleFunc α β)
Goal: f ⊔ g = MeasureTheory.SimpleFunc.map (fun p => p.1 ⊔ p.2) (f.pair g)
-- MeasureTheory.SimpleFunc.const_add_eq_map
@[to_additive]
theorem const_mul_eq_map [Mul β] (f : α →ₛ β) (b : β) : const α b * f = f.map fun a => b * a 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [Add β] (f : MeasureTheory.SimpleFunc α β) (b : β)
Goal: MeasureTheory.SimpleFunc.const α b + f = MeasureTheory.SimpleFunc.map (fun a => b + a) f
-- MeasureTheory.SimpleFunc.const_mul_eq_map
@[to_additive]
theorem const_mul_eq_map [Mul β] (f : α →ₛ β) (b : β) : const α b * f = f.map fun a => b * a 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [Mul β] (f : MeasureTheory.SimpleFunc α β) (b : β)
Goal: MeasureTheory.SimpleFunc.const α b * f = MeasureTheory.SimpleFunc.map (fun a => b * a) f
-- MeasureTheory.SimpleFunc.map_add
@[to_additive]
theorem map_mul [Mul β] [Mul γ] {g : β → γ} (hg : ∀ x y, g (x * y) = g x * g y) (f₁ f₂ : α →ₛ β) :
    (f₁ * f₂).map g = f₁.map g * f₂.map g 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [Add β] [Add γ] {g : β → γ} (hg : ∀ (x y : β), g (x + y) = g x + g y) (f₁ f₂ : MeasureTheory.SimpleFunc α β)
Goal: MeasureTheory.SimpleFunc.map g (f₁ + f₂) = MeasureTheory.SimpleFunc.map g f₁ + MeasureTheory.SimpleFunc.map g f₂
-- MeasureTheory.SimpleFunc.map_mul
@[to_additive]
theorem map_mul [Mul β] [Mul γ] {g : β → γ} (hg : ∀ x y, g (x * y) = g x * g y) (f₁ f₂ : α →ₛ β) :
    (f₁ * f₂).map g = f₁.map g * f₂.map g 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [Mul β] [Mul γ] {g : β → γ} (hg : ∀ (x y : β), g (x * y) = g x * g y) (f₁ f₂ : MeasureTheory.SimpleFunc α β)
Goal: MeasureTheory.SimpleFunc.map g (f₁ * f₂) = MeasureTheory.SimpleFunc.map g f₁ * MeasureTheory.SimpleFunc.map g f₂
-- MeasureTheory.SimpleFunc.coe_smul
@[to_additive (attr := simp)]
theorem coe_smul [SMul K β] (c : K) (f : α →ₛ β) : ⇑(c • f) = c • ⇑f 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {K : Type u_5} [SMul K β] (c : K) (f : MeasureTheory.SimpleFunc α β)
Goal: ↑(c • f) = c • ↑f
-- MeasureTheory.SimpleFunc.coe_vadd
@[to_additive (attr := simp)]
theorem coe_smul [SMul K β] (c : K) (f : α →ₛ β) : ⇑(c • f) = c • ⇑f 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {K : Type u_5} [VAdd K β] (c : K) (f : MeasureTheory.SimpleFunc α β)
Goal: ↑(c +ᵥ f) = c +ᵥ ↑f