ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasurableEquiv.measurable
@[measurability]
protected theorem measurable (e : α ≃ᵐ β) : Measurable (e : α → β) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β)
Goal: Measurable ⇑e
-- MeasurableEquiv.coe_mk
@[simp]
theorem coe_mk (e : α ≃ β) (h1 : Measurable e) (h2 : Measurable e.symm) :
    ((⟨e, h1, h2⟩ : α ≃ᵐ β) : α → β) = e 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ β) (h1 : Measurable ⇑e) (h2 : Measurable ⇑e.symm)
Goal: ⇑{ toEquiv := e, measurable_toFun := h1, measurable_invFun := h2 } = ⇑e
-- MeasurableEquiv.coe_trans
theorem coe_trans (ab : α ≃ᵐ β) (bc : β ≃ᵐ γ) : ⇑(ab.trans bc) = bc ∘ ab 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] (ab : α ≃ᵐ β) (bc : β ≃ᵐ γ)
Goal: ⇑(ab.trans bc) = ⇑bc ∘ ⇑ab
-- MeasurableEquiv.coe_toEquiv_symm
@[simp]
theorem coe_toEquiv_symm (e : α ≃ᵐ β) : (e.toEquiv.symm : β → α) = e.symm 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β)
Goal: ⇑e.symm = ⇑e.symm
-- MeasurableEquiv.ext
@[ext] theorem ext {e₁ e₂ : α ≃ᵐ β} (h : (e₁ : α → β) = e₂) : e₁ = e₂ 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {e₁ e₂ : α ≃ᵐ β} (h : ⇑e₁ = ⇑e₂)
Goal: e₁ = e₂
-- MeasurableEquiv.symm_mk
@[simp]
theorem symm_mk (e : α ≃ β) (h1 : Measurable e) (h2 : Measurable e.symm) :
    (⟨e, h1, h2⟩ : α ≃ᵐ β).symm = ⟨e.symm, h2, h1⟩ 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ β) (h1 : Measurable ⇑e) (h2 : Measurable ⇑e.symm)
Goal: { toEquiv := e, measurable_toFun := h1, measurable_invFun := h2 }.symm =
  { toEquiv := e.symm, measurable_toFun := h2, measurable_invFun := h1 }
-- MeasurableEquiv.symm_symm
@[simp]
theorem symm_symm (e : α ≃ᵐ β) : e.symm.symm = e 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β)
Goal: e.symm.symm = e
-- MeasurableEquiv.symm_bijective
theorem symm_bijective :
    Function.Bijective (MeasurableEquiv.symm : (α ≃ᵐ β) → β ≃ᵐ α) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β]
Goal: Function.Bijective MeasurableEquiv.symm
-- MeasurableEquiv.symm_refl
@[simp]
theorem symm_refl (α : Type*) [MeasurableSpace α] : (refl α).symm = refl α 
Arguments: (α : Type u_6) [MeasurableSpace α]
Goal: (MeasurableEquiv.refl α).symm = MeasurableEquiv.refl α
-- MeasurableEquiv.symm_comp_self
@[simp]
theorem symm_comp_self (e : α ≃ᵐ β) : e.symm ∘ e = id 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β)
Goal: ⇑e.symm ∘ ⇑e = id
-- MeasurableEquiv.self_comp_symm
@[simp]
theorem self_comp_symm (e : α ≃ᵐ β) : e ∘ e.symm = id 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β)
Goal: ⇑e ∘ ⇑e.symm = id
-- MeasurableEquiv.apply_symm_apply
@[simp]
theorem apply_symm_apply (e : α ≃ᵐ β) (y : β) : e (e.symm y) = y 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β) (y : β)
Goal: e (e.symm y) = y
-- MeasurableEquiv.symm_apply_apply
@[simp]
theorem symm_apply_apply (e : α ≃ᵐ β) (x : α) : e.symm (e x) = x 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β) (x : α)
Goal: e.symm (e x) = x
-- MeasurableEquiv.symm_trans_self
@[simp]
theorem symm_trans_self (e : α ≃ᵐ β) : e.symm.trans e = refl β 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β)
Goal: e.symm.trans e = MeasurableEquiv.refl β
-- MeasurableEquiv.self_trans_symm
@[simp]
theorem self_trans_symm (e : α ≃ᵐ β) : e.trans e.symm = refl α 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β)
Goal: e.trans e.symm = MeasurableEquiv.refl α
-- MeasurableEquiv.surjective
protected theorem surjective (e : α ≃ᵐ β) : Surjective e 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β)
Goal: Function.Surjective ⇑e
-- MeasurableEquiv.bijective
protected theorem bijective (e : α ≃ᵐ β) : Bijective e 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β)
Goal: Function.Bijective ⇑e
-- MeasurableEquiv.injective
protected theorem injective (e : α ≃ᵐ β) : Injective e 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β)
Goal: Function.Injective ⇑e
-- MeasurableEquiv.symm_preimage_preimage
@[simp]
theorem symm_preimage_preimage (e : α ≃ᵐ β) (s : Set β) : e.symm ⁻¹' (e ⁻¹' s) = s 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β) (s : Set β)
Goal: ⇑e.symm ⁻¹' (⇑e ⁻¹' s) = s
-- MeasurableEquiv.image_eq_preimage
theorem image_eq_preimage (e : α ≃ᵐ β) (s : Set α) : e '' s = e.symm ⁻¹' s 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β) (s : Set α)
Goal: ⇑e '' s = ⇑e.symm ⁻¹' s
-- MeasurableEquiv.preimage_symm
lemma preimage_symm (e : α ≃ᵐ β) (s : Set α) : e.symm ⁻¹' s = e '' s 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β) (s : Set α)
Goal: ⇑e.symm ⁻¹' s = ⇑e '' s
-- MeasurableEquiv.image_symm
lemma image_symm (e : α ≃ᵐ β) (s : Set β) : e.symm '' s = e ⁻¹' s 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β) (s : Set β)
Goal: ⇑e.symm '' s = ⇑e ⁻¹' s
-- MeasurableEquiv.eq_image_iff_symm_image_eq
lemma eq_image_iff_symm_image_eq (e : α ≃ᵐ β) (s : Set β) (t : Set α) :
    s = e '' t ↔ e.symm '' s = t 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β) (s : Set β) (t : Set α)
Goal: s = ⇑e '' t ↔ ⇑e.symm '' s = t
-- MeasurableEquiv.image_preimage
@[simp]
lemma image_preimage (e : α ≃ᵐ β) (s : Set β) : e '' (e ⁻¹' s) = s 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β) (s : Set β)
Goal: ⇑e '' (⇑e ⁻¹' s) = s
-- MeasurableEquiv.preimage_image
@[simp]
lemma preimage_image (e : α ≃ᵐ β) (s : Set α) : e ⁻¹' (e '' s) = s 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β) (s : Set α)
Goal: ⇑e ⁻¹' (⇑e '' s) = s
-- MeasurableEquiv.measurableSet_preimage
@[simp]
theorem measurableSet_preimage (e : α ≃ᵐ β) {s : Set β} :
    MeasurableSet (e ⁻¹' s) ↔ MeasurableSet s 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β) {s : Set β}
Goal: MeasurableSet (⇑e ⁻¹' s) ↔ MeasurableSet s
-- MeasurableEquiv.measurableSet_image
@[simp]
theorem measurableSet_image (e : α ≃ᵐ β) :
    MeasurableSet (e '' s) ↔ MeasurableSet s 
Arguments: {α : Type u_1} {β : Type u_2} {s : Set α} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β)
Goal: MeasurableSet (⇑e '' s) ↔ MeasurableSet s
-- MeasurableEquiv.map_eq
@[simp] theorem map_eq (e : α ≃ᵐ β) : MeasurableSpace.map e ‹_› = ‹_› 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β)
Goal: MeasurableSpace.map (⇑e) inst✝ = inst✝¹
-- MeasurableEquiv.measurableEmbedding
protected theorem measurableEmbedding (e : α ≃ᵐ β) : MeasurableEmbedding e 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β)
Goal: MeasurableEmbedding ⇑e
-- MeasurableEquiv.measurable_comp_iff
protected theorem measurable_comp_iff {f : β → γ} (e : α ≃ᵐ β) :
    Measurable (f ∘ e) ↔ Measurable f 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {f : β → γ} (e : α ≃ᵐ β)
Goal: Measurable (f ∘ ⇑e) ↔ Measurable f
-- MeasurableEquiv.coe_piCongrLeft
theorem coe_piCongrLeft (f : δ ≃ δ') :
    ⇑(MeasurableEquiv.piCongrLeft π f) = f.piCongrLeft π 
Arguments: {δ : Type u_4} {δ' : Type u_5} {π : δ' → Type u_6} [(x : δ') → MeasurableSpace (π x)] (f : δ ≃ δ')
Goal: ⇑(MeasurableEquiv.piCongrLeft π f) = ⇑(Equiv.piCongrLeft π f)
-- MeasurableEquiv.piCongrLeft_apply_apply
lemma piCongrLeft_apply_apply {ι ι' : Type*} (e : ι ≃ ι') {β : ι' → Type*}
    [∀ i', MeasurableSpace (β i')] (x : (i : ι) → β (e i)) (i : ι) :
    piCongrLeft (fun i' ↦ β i') e x (e i) = x i 
Arguments: {ι : Type u_8} {ι' : Type u_9} (e : ι ≃ ι') {β : ι' → Type u_10} [(i' : ι') → MeasurableSpace (β i')] (x : (i : ι) → β (e i)) (i : ι)
Goal: (MeasurableEquiv.piCongrLeft (fun i' => β i') e) x (e i) = x i
-- MeasurableEquiv.piMeasurableEquivTProd_apply
@[simps! (config := .asFn)]
def piMeasurableEquivTProd [DecidableEq δ'] {l : List δ'} (hnd : l.Nodup) (h : ∀ i, i ∈ l) :
    (∀ i, π i) ≃ᵐ List.TProd π l 
Arguments: {δ' : Type u_5} {π : δ' → Type u_6} [(x : δ') → MeasurableSpace (π x)] [DecidableEq δ'] {l : List δ'} (hnd : l.Nodup) (h : ∀ (i : δ'), i ∈ l)
Goal: ⇑(MeasurableEquiv.piMeasurableEquivTProd hnd h) = List.TProd.mk l
-- MeasurableEquiv.piMeasurableEquivTProd_symm_apply
@[simps! (config := .asFn)]
def piMeasurableEquivTProd [DecidableEq δ'] {l : List δ'} (hnd : l.Nodup) (h : ∀ i, i ∈ l) :
    (∀ i, π i) ≃ᵐ List.TProd π l 
Arguments: {δ' : Type u_5} {π : δ' → Type u_6} [(x : δ') → MeasurableSpace (π x)] [DecidableEq δ'] {l : List δ'} (hnd : l.Nodup) (h : ∀ (i : δ'), i ∈ l)
Goal: ⇑(MeasurableEquiv.piMeasurableEquivTProd hnd h).symm = List.TProd.elim' h
-- MeasurableEquiv.piUnique_symm_apply
@[simps! (config := .asFn)]
def piUnique [Unique δ'] : (∀ i, π i) ≃ᵐ π default 
Arguments: {δ' : Type u_5} (π : δ' → Type u_6) [(x : δ') → MeasurableSpace (π x)] [Unique δ']
Goal: ⇑(MeasurableEquiv.piUnique π).symm = uniqueElim
-- MeasurableEquiv.piUnique_apply
@[simps! (config := .asFn)]
def piUnique [Unique δ'] : (∀ i, π i) ≃ᵐ π default 
Arguments: {δ' : Type u_5} (π : δ' → Type u_6) [(x : δ') → MeasurableSpace (π x)] [Unique δ']
Goal: ⇑(MeasurableEquiv.piUnique π) = fun f => f default
-- MeasurableEquiv.funUnique_symm_apply
@[simps! (config := .asFn)]
def funUnique (α β : Type*) [Unique α] [MeasurableSpace β] : (α → β) ≃ᵐ β 
Arguments: (α : Type u_8) (β : Type u_9) [Unique α] [MeasurableSpace β]
Goal: ⇑(MeasurableEquiv.funUnique α β).symm = uniqueElim
-- MeasurableEquiv.funUnique_apply
@[simps! (config := .asFn)]
def funUnique (α β : Type*) [Unique α] [MeasurableSpace β] : (α → β) ≃ᵐ β 
Arguments: (α : Type u_8) (β : Type u_9) [Unique α] [MeasurableSpace β]
Goal: ⇑(MeasurableEquiv.funUnique α β) = fun f => f default
-- MeasurableEquiv.piFinTwo_symm_apply
@[simps! (config := .asFn)]
def piFinTwo (α : Fin 2 → Type*) [∀ i, MeasurableSpace (α i)] : (∀ i, α i) ≃ᵐ α 0 × α 1 
Arguments: (α : Fin 2 → Type u_8) [(i : Fin 2) → MeasurableSpace (α i)]
Goal: ⇑(MeasurableEquiv.piFinTwo α).symm = fun p => Fin.cons p.1 (Fin.cons p.2 finZeroElim)
-- MeasurableEquiv.piFinTwo_apply
@[simps! (config := .asFn)]
def piFinTwo (α : Fin 2 → Type*) [∀ i, MeasurableSpace (α i)] : (∀ i, α i) ≃ᵐ α 0 × α 1 
Arguments: (α : Fin 2 → Type u_8) [(i : Fin 2) → MeasurableSpace (α i)]
Goal: ⇑(MeasurableEquiv.piFinTwo α) = fun f => (f 0, f 1)
-- MeasurableEquiv.finTwoArrow_apply
@[simps! (config := .asFn)]
def finTwoArrow : (Fin 2 → α) ≃ᵐ α × α 
Arguments: {α : Type u_1} [MeasurableSpace α]
Goal: ⇑MeasurableEquiv.finTwoArrow = fun f => (f 0, f 1)
-- MeasurableEquiv.finTwoArrow_symm_apply
@[simps! (config := .asFn)]
def finTwoArrow : (Fin 2 → α) ≃ᵐ α × α 
Arguments: {α : Type u_1} [MeasurableSpace α]
Goal: ⇑MeasurableEquiv.finTwoArrow.symm = fun p => Fin.cons p.1 (Fin.cons p.2 finZeroElim)
-- MeasurableEquiv.piFinSuccAbove_symm_apply
@[simps! (config := .asFn)]
def piFinSuccAbove {n : ℕ} (α : Fin (n + 1) → Type*) [∀ i, MeasurableSpace (α i)]
    (i : Fin (n + 1)) : (∀ j, α j) ≃ᵐ α i × ∀ j, α (i.succAbove j) 
Arguments: {n : ℕ} (α : Fin (n + 1) → Type u_8) [(i : Fin (n + 1)) → MeasurableSpace (α i)] (i : Fin (n + 1))
Goal: ⇑(MeasurableEquiv.piFinSuccAbove α i).symm = fun f => i.insertNth f.1 f.2
-- MeasurableEquiv.piFinSuccAbove_apply
@[simps! (config := .asFn)]
def piFinSuccAbove {n : ℕ} (α : Fin (n + 1) → Type*) [∀ i, MeasurableSpace (α i)]
    (i : Fin (n + 1)) : (∀ j, α j) ≃ᵐ α i × ∀ j, α (i.succAbove j) 
Arguments: {n : ℕ} (α : Fin (n + 1) → Type u_8) [(i : Fin (n + 1)) → MeasurableSpace (α i)] (i : Fin (n + 1))
Goal: ⇑(MeasurableEquiv.piFinSuccAbove α i) = fun f => (f i, i.removeNth f)
-- MeasurableEquiv.piEquivPiSubtypeProd_apply
@[simps! (config := .asFn)]
def piEquivPiSubtypeProd (p : δ' → Prop) [DecidablePred p] :
    (∀ i, π i) ≃ᵐ (∀ i : Subtype p, π i) × ∀ i : { i // ¬p i }, π i 
Arguments: {δ' : Type u_5} (π : δ' → Type u_6) [(x : δ') → MeasurableSpace (π x)] (p : δ' → Prop) [DecidablePred p]
Goal: ⇑(MeasurableEquiv.piEquivPiSubtypeProd π p) = fun f => (fun x => f ↑x, fun x => f ↑x)
-- MeasurableEquiv.piEquivPiSubtypeProd_symm_apply
@[simps! (config := .asFn)]
def piEquivPiSubtypeProd (p : δ' → Prop) [DecidablePred p] :
    (∀ i, π i) ≃ᵐ (∀ i : Subtype p, π i) × ∀ i : { i // ¬p i }, π i 
Arguments: {δ' : Type u_5} (π : δ' → Type u_6) [(x : δ') → MeasurableSpace (π x)] (p : δ' → Prop) [DecidablePred p]
Goal: ⇑(MeasurableEquiv.piEquivPiSubtypeProd π p).symm = fun f x => if h : p x then f.1 ⟨x, h⟩ else f.2 ⟨x, h⟩
-- MeasurableEquiv.coe_sumPiEquivProdPi
theorem coe_sumPiEquivProdPi (α : δ ⊕ δ' → Type*) [∀ i, MeasurableSpace (α i)] :
    ⇑(MeasurableEquiv.sumPiEquivProdPi α) = Equiv.sumPiEquivProdPi α 
Arguments: {δ : Type u_4} {δ' : Type u_5} (α : δ ⊕ δ' → Type u_8) [(i : δ ⊕ δ') → MeasurableSpace (α i)]
Goal: ⇑(MeasurableEquiv.sumPiEquivProdPi α) = ⇑(Equiv.sumPiEquivProdPi α)
-- MeasurableEquiv.coe_sumPiEquivProdPi_symm
theorem coe_sumPiEquivProdPi_symm (α : δ ⊕ δ' → Type*) [∀ i, MeasurableSpace (α i)] :
    ⇑(MeasurableEquiv.sumPiEquivProdPi α).symm = (Equiv.sumPiEquivProdPi α).symm 
Arguments: {δ : Type u_4} {δ' : Type u_5} (α : δ ⊕ δ' → Type u_8) [(i : δ ⊕ δ') → MeasurableSpace (α i)]
Goal: ⇑(MeasurableEquiv.sumPiEquivProdPi α).symm = ⇑(Equiv.sumPiEquivProdPi α).symm
-- MeasurableEquiv.ofInvolutive_toEquiv
@[simps toEquiv]
def ofInvolutive (f : α → α) (hf : Involutive f) (hf' : Measurable f) : α ≃ᵐ α 
Arguments: {α : Type u_1} [MeasurableSpace α] (f : α → α) (hf : Function.Involutive f) (hf' : Measurable f)
Goal: (MeasurableEquiv.ofInvolutive f hf hf').toEquiv = Function.Involutive.toPerm f hf
-- MeasurableEquiv.ofInvolutive_apply
@[simp] theorem ofInvolutive_apply (f : α → α) (hf : Involutive f) (hf' : Measurable f) (a : α) :
    ofInvolutive f hf hf' a = f a 
Arguments: {α : Type u_1} [MeasurableSpace α] (f : α → α) (hf : Function.Involutive f) (hf' : Measurable f) (a : α)
Goal: (MeasurableEquiv.ofInvolutive f hf hf') a = f a
-- MeasurableEquiv.ofInvolutive_symm
@[simp] theorem ofInvolutive_symm (f : α → α) (hf : Involutive f) (hf' : Measurable f) :
    (ofInvolutive f hf hf').symm = ofInvolutive f hf hf' 
Arguments: {α : Type u_1} [MeasurableSpace α] (f : α → α) (hf : Function.Involutive f) (hf' : Measurable f)
Goal: (MeasurableEquiv.ofInvolutive f hf hf').symm = MeasurableEquiv.ofInvolutive f hf hf'
-- MeasurableEmbedding.comap_eq
@[simp] theorem comap_eq (hf : MeasurableEmbedding f) : MeasurableSpace.comap f ‹_› = ‹_› 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {f : α → β} (hf : MeasurableEmbedding f)
Goal: MeasurableSpace.comap f inst✝ = inst✝¹
-- MeasurableEmbedding.iff_comap_eq
theorem iff_comap_eq :
    MeasurableEmbedding f ↔
      Injective f ∧ MeasurableSpace.comap f ‹_› = ‹_› ∧ MeasurableSet (range f) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {f : α → β}
Goal: MeasurableEmbedding f ↔ Function.Injective f ∧ MeasurableSpace.comap f inst✝ = inst✝¹ ∧ MeasurableSet (Set.range f)
-- MeasurableEmbedding.of_measurable_inverse_on_range
theorem of_measurable_inverse_on_range {g : range f → α} (hf₁ : Measurable f)
    (hf₂ : MeasurableSet (range f)) (hg : Measurable g) (H : LeftInverse g (rangeFactorization f)) :
    MeasurableEmbedding f 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {f : α → β} {g : ↑(Set.range f) → α} (hf₁ : Measurable f) (hf₂ : MeasurableSet (Set.range f)) (hg : Measurable g) (H : Function.LeftInverse g (Set.rangeFactorization f))
Goal: MeasurableEmbedding f
-- MeasurableEmbedding.of_measurable_inverse
theorem of_measurable_inverse (hf₁ : Measurable f) (hf₂ : MeasurableSet (range f))
    (hg : Measurable g) (H : LeftInverse g f) : MeasurableEmbedding f 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {f : α → β} {g : β → α} (hf₁ : Measurable f) (hf₂ : MeasurableSet (Set.range f)) (hg : Measurable g) (H : Function.LeftInverse g f)
Goal: MeasurableEmbedding f
-- MeasurableSpace.comap_compl
theorem MeasurableSpace.comap_compl {m' : MeasurableSpace β} [BooleanAlgebra β]
    (h : Measurable (compl : β → β)) (f : α → β) :
    MeasurableSpace.comap (fun a => (f a)ᶜ) inferInstance =
      MeasurableSpace.comap f inferInstance 
Arguments: {α : Type u_1} {β : Type u_2} {m' : MeasurableSpace β} [BooleanAlgebra β] (h : Measurable compl) (f : α → β)
Goal: MeasurableSpace.comap (fun a => (f a)ᶜ) inferInstance = MeasurableSpace.comap f inferInstance
-- MeasurableSpace.comap_not
@[simp] theorem MeasurableSpace.comap_not (p : α → Prop) :
    MeasurableSpace.comap (fun a ↦ ¬p a) inferInstance = MeasurableSpace.comap p inferInstance 
Arguments: {α : Type u_1} (p : α → Prop)
Goal: MeasurableSpace.comap (fun a => ¬p a) inferInstance = MeasurableSpace.comap p inferInstance
-- MeasureTheory.setIntegral_congr_ae₀
theorem setIntegral_congr_ae₀ (hs : NullMeasurableSet s μ) (h : ∀ᵐ x ∂μ, x ∈ s → f x = g x) :
    ∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f g : X → E} {s : Set X} {μ : MeasureTheory.Measure X} (hs : MeasureTheory.NullMeasurableSet s μ) (h : ∀ᵐ (x : X) ∂μ, x ∈ s → f x = g x)
Goal: ∫ (x : X) in s, f x ∂μ = ∫ (x : X) in s, g x ∂μ
-- MeasureTheory.setIntegral_congr_ae
theorem setIntegral_congr_ae (hs : MeasurableSet s) (h : ∀ᵐ x ∂μ, x ∈ s → f x = g x) :
    ∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f g : X → E} {s : Set X} {μ : MeasureTheory.Measure X} (hs : MeasurableSet s) (h : ∀ᵐ (x : X) ∂μ, x ∈ s → f x = g x)
Goal: ∫ (x : X) in s, f x ∂μ = ∫ (x : X) in s, g x ∂μ
-- MeasureTheory.setIntegral_congr₀
theorem setIntegral_congr₀ (hs : NullMeasurableSet s μ) (h : EqOn f g s) :
    ∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f g : X → E} {s : Set X} {μ : MeasureTheory.Measure X} (hs : MeasureTheory.NullMeasurableSet s μ) (h : Set.EqOn f g s)
Goal: ∫ (x : X) in s, f x ∂μ = ∫ (x : X) in s, g x ∂μ
-- MeasureTheory.setIntegral_congr
theorem setIntegral_congr (hs : MeasurableSet s) (h : EqOn f g s) :
    ∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f g : X → E} {s : Set X} {μ : MeasureTheory.Measure X} (hs : MeasurableSet s) (h : Set.EqOn f g s)
Goal: ∫ (x : X) in s, f x ∂μ = ∫ (x : X) in s, g x ∂μ
-- MeasureTheory.setIntegral_congr_set_ae
theorem setIntegral_congr_set_ae (hst : s =ᵐ[μ] t) : ∫ x in s, f x ∂μ = ∫ x in t, f x ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {s t : Set X} {μ : MeasureTheory.Measure X} (hst : s =ᵐ[μ] t)
Goal: ∫ (x : X) in s, f x ∂μ = ∫ (x : X) in t, f x ∂μ
-- MeasureTheory.integral_union_ae
theorem integral_union_ae (hst : AEDisjoint μ s t) (ht : NullMeasurableSet t μ)
    (hfs : IntegrableOn f s μ) (hft : IntegrableOn f t μ) :
    ∫ x in s ∪ t, f x ∂μ = ∫ x in s, f x ∂μ + ∫ x in t, f x ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {s t : Set X} {μ : MeasureTheory.Measure X} (hst : MeasureTheory.AEDisjoint μ s t) (ht : MeasureTheory.NullMeasurableSet t μ) (hfs : MeasureTheory.IntegrableOn f s μ) (hft : MeasureTheory.IntegrableOn f t μ)
Goal: ∫ (x : X) in s ∪ t, f x ∂μ = ∫ (x : X) in s, f x ∂μ + ∫ (x : X) in t, f x ∂μ
-- MeasureTheory.integral_union
theorem integral_union (hst : Disjoint s t) (ht : MeasurableSet t) (hfs : IntegrableOn f s μ)
    (hft : IntegrableOn f t μ) : ∫ x in s ∪ t, f x ∂μ = ∫ x in s, f x ∂μ + ∫ x in t, f x ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {s t : Set X} {μ : MeasureTheory.Measure X} (hst : Disjoint s t) (ht : MeasurableSet t) (hfs : MeasureTheory.IntegrableOn f s μ) (hft : MeasureTheory.IntegrableOn f t μ)
Goal: ∫ (x : X) in s ∪ t, f x ∂μ = ∫ (x : X) in s, f x ∂μ + ∫ (x : X) in t, f x ∂μ
-- MeasureTheory.integral_diff
theorem integral_diff (ht : MeasurableSet t) (hfs : IntegrableOn f s μ) (hts : t ⊆ s) :
    ∫ x in s \ t, f x ∂μ = ∫ x in s, f x ∂μ - ∫ x in t, f x ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {s t : Set X} {μ : MeasureTheory.Measure X} (ht : MeasurableSet t) (hfs : MeasureTheory.IntegrableOn f s μ) (hts : t ⊆ s)
Goal: ∫ (x : X) in s \ t, f x ∂μ = ∫ (x : X) in s, f x ∂μ - ∫ (x : X) in t, f x ∂μ
-- MeasureTheory.integral_inter_add_diff₀
theorem integral_inter_add_diff₀ (ht : NullMeasurableSet t μ) (hfs : IntegrableOn f s μ) :
    ∫ x in s ∩ t, f x ∂μ + ∫ x in s \ t, f x ∂μ = ∫ x in s, f x ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {s t : Set X} {μ : MeasureTheory.Measure X} (ht : MeasureTheory.NullMeasurableSet t μ) (hfs : MeasureTheory.IntegrableOn f s μ)
Goal: ∫ (x : X) in s ∩ t, f x ∂μ + ∫ (x : X) in s \ t, f x ∂μ = ∫ (x : X) in s, f x ∂μ
-- MeasureTheory.integral_inter_add_diff
theorem integral_inter_add_diff (ht : MeasurableSet t) (hfs : IntegrableOn f s μ) :
    ∫ x in s ∩ t, f x ∂μ + ∫ x in s \ t, f x ∂μ = ∫ x in s, f x ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {s t : Set X} {μ : MeasureTheory.Measure X} (ht : MeasurableSet t) (hfs : MeasureTheory.IntegrableOn f s μ)
Goal: ∫ (x : X) in s ∩ t, f x ∂μ + ∫ (x : X) in s \ t, f x ∂μ = ∫ (x : X) in s, f x ∂μ
-- MeasureTheory.integral_finset_biUnion
theorem integral_finset_biUnion {ι : Type*} (t : Finset ι) {s : ι → Set X}
    (hs : ∀ i ∈ t, MeasurableSet (s i)) (h's : Set.Pairwise (↑t) (Disjoint on s))
    (hf : ∀ i ∈ t, IntegrableOn f (s i) μ) :
    ∫ x in ⋃ i ∈ t, s i, f x ∂μ = ∑ i ∈ t, ∫ x in s i, f x ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {μ : MeasureTheory.Measure X} {ι : Type u_5} (t : Finset ι) {s : ι → Set X} (hs : ∀ i ∈ t, MeasurableSet (s i)) (h's : (↑t).Pairwise (Disjoint on s)) (hf : ∀ i ∈ t, MeasureTheory.IntegrableOn f (s i) μ)
Goal: ∫ (x : X) in ⋃ i ∈ t, s i, f x ∂μ = ∑ i ∈ t, ∫ (x : X) in s i, f x ∂μ
-- MeasureTheory.integral_fintype_iUnion
theorem integral_fintype_iUnion {ι : Type*} [Fintype ι] {s : ι → Set X}
    (hs : ∀ i, MeasurableSet (s i)) (h's : Pairwise (Disjoint on s))
    (hf : ∀ i, IntegrableOn f (s i) μ) : ∫ x in ⋃ i, s i, f x ∂μ = ∑ i, ∫ x in s i, f x ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {μ : MeasureTheory.Measure X} {ι : Type u_5} [Fintype ι] {s : ι → Set X} (hs : ∀ (i : ι), MeasurableSet (s i)) (h's : Pairwise (Disjoint on s)) (hf : ∀ (i : ι), MeasureTheory.IntegrableOn f (s i) μ)
Goal: ∫ (x : X) in ⋃ i, s i, f x ∂μ = ∑ i : ι, ∫ (x : X) in s i, f x ∂μ
-- MeasureTheory.integral_empty
theorem integral_empty : ∫ x in ∅, f x ∂μ = 0 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {μ : MeasureTheory.Measure X}
Goal: ∫ (x : X) in ∅, f x ∂μ = 0
-- MeasureTheory.integral_univ
theorem integral_univ : ∫ x in univ, f x ∂μ = ∫ x, f x ∂μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {μ : MeasureTheory.Measure X}
Goal: ∫ (x : X) in Set.univ, f x ∂μ = ∫ (x : X), f x ∂μ