ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Valued.cauchy_iff
theorem cauchy_iff {F : Filter R} : Cauchy F ↔
    F.NeBot ∧ ∀ γ : Γ₀ˣ, ∃ M ∈ F, ∀ᵉ (x ∈ M) (y ∈ M), (v (y - x) : Γ₀) < γ 
Arguments: {R : Type u} [Ring R] {Γ₀ : Type v} [LinearOrderedCommGroupWithZero Γ₀] [_i : Valued R Γ₀] {F : Filter R}
Goal: Cauchy F ↔ F.NeBot ∧ ∀ (γ : Γ₀ˣ), ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, Valued.v (y - x) < ↑γ
-- Valued.integer_isOpen
theorem integer_isOpen : IsOpen (_i.v.integer : Set R) 
Arguments: (R : Type u) [Ring R] {Γ₀ : Type v} [LinearOrderedCommGroupWithZero Γ₀] [_i : Valued R Γ₀]
Goal: IsOpen ↑Valued.v.integer
-- Valued.valuationSubring_isOpen
theorem valuationSubring_isOpen (K : Type u) [Field K] [hv : Valued K Γ₀] :
    IsOpen (hv.v.valuationSubring : Set K) 
Arguments: {Γ₀ : Type v} [LinearOrderedCommGroupWithZero Γ₀] (K : Type u) [Field K] [hv : Valued K Γ₀]
Goal: IsOpen ↑Valued.v.valuationSubring
-- TopologicalSpace.exists_inducing_l_infty
theorem exists_inducing_l_infty : ∃ f : X → ℕ →ᵇ ℝ, Inducing f 
Arguments: (X : Type u_1) [TopologicalSpace X] [RegularSpace X] [SecondCountableTopology X]
Goal: ∃ f, Inducing f
-- TopologicalSpace.exists_embedding_l_infty
theorem exists_embedding_l_infty : ∃ f : X → ℕ →ᵇ ℝ, Embedding f 
Arguments: (X : Type u_1) [TopologicalSpace X] [T3Space X] [SecondCountableTopology X]
Goal: ∃ f, Embedding f
-- IsLocallyConstant.tfae
protected theorem tfae (f : X → Y) :
    TFAE [IsLocallyConstant f,
      ∀ x, ∀ᶠ x' in 𝓝 x, f x' = f x,
      ∀ x, IsOpen { x' | f x' = f x },
      ∀ y, IsOpen (f ⁻¹' {y}),
      ∀ x, ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ ∀ x' ∈ U, f x' = f x] 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] (f : X → Y)
Goal: [IsLocallyConstant f, ∀ (x : X), ∀ᶠ (x' : X) in 𝓝 x, f x' = f x, ∀ (x : X), IsOpen {x' | f x' = f x},
    ∀ (y : Y), IsOpen (f ⁻¹' {y}), ∀ (x : X), ∃ U, IsOpen U ∧ x ∈ U ∧ ∀ x' ∈ U, f x' = f x].TFAE
-- IsLocallyConstant.of_discrete
@[nontriviality]
theorem of_discrete [DiscreteTopology X] (f : X → Y) : IsLocallyConstant f 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [DiscreteTopology X] (f : X → Y)
Goal: IsLocallyConstant f
-- IsLocallyConstant.isOpen_fiber
theorem isOpen_fiber {f : X → Y} (hf : IsLocallyConstant f) (y : Y) : IsOpen { x | f x = y } 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {f : X → Y} (hf : IsLocallyConstant f) (y : Y)
Goal: IsOpen {x | f x = y}
-- IsLocallyConstant.isClosed_fiber
theorem isClosed_fiber {f : X → Y} (hf : IsLocallyConstant f) (y : Y) : IsClosed { x | f x = y } 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {f : X → Y} (hf : IsLocallyConstant f) (y : Y)
Goal: IsClosed {x | f x = y}
-- IsLocallyConstant.isClopen_fiber
theorem isClopen_fiber {f : X → Y} (hf : IsLocallyConstant f) (y : Y) : IsClopen { x | f x = y } 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {f : X → Y} (hf : IsLocallyConstant f) (y : Y)
Goal: IsClopen {x | f x = y}
-- IsLocallyConstant.iff_exists_open
theorem iff_exists_open (f : X → Y) :
    IsLocallyConstant f ↔ ∀ x, ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ ∀ x' ∈ U, f x' = f x 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] (f : X → Y)
Goal: IsLocallyConstant f ↔ ∀ (x : X), ∃ U, IsOpen U ∧ x ∈ U ∧ ∀ x' ∈ U, f x' = f x
-- IsLocallyConstant.iff_eventually_eq
theorem iff_eventually_eq (f : X → Y) : IsLocallyConstant f ↔ ∀ x, ∀ᶠ y in 𝓝 x, f y = f x 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] (f : X → Y)
Goal: IsLocallyConstant f ↔ ∀ (x : X), ∀ᶠ (y : X) in 𝓝 x, f y = f x
-- IsLocallyConstant.exists_open
theorem exists_open {f : X → Y} (hf : IsLocallyConstant f) (x : X) :
    ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ ∀ x' ∈ U, f x' = f x 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {f : X → Y} (hf : IsLocallyConstant f) (x : X)
Goal: ∃ U, IsOpen U ∧ x ∈ U ∧ ∀ x' ∈ U, f x' = f x
-- IsLocallyConstant.eventually_eq
protected theorem eventually_eq {f : X → Y} (hf : IsLocallyConstant f) (x : X) :
    ∀ᶠ y in 𝓝 x, f y = f x 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {f : X → Y} (hf : IsLocallyConstant f) (x : X)
Goal: ∀ᶠ (y : X) in 𝓝 x, f y = f x
-- IsLocallyConstant.iff_isOpen_fiber_apply
theorem iff_isOpen_fiber_apply {f : X → Y} : IsLocallyConstant f ↔ ∀ x, IsOpen (f ⁻¹' {f x}) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {f : X → Y}
Goal: IsLocallyConstant f ↔ ∀ (x : X), IsOpen (f ⁻¹' {f x})
-- IsLocallyConstant.iff_isOpen_fiber
theorem iff_isOpen_fiber {f : X → Y} : IsLocallyConstant f ↔ ∀ y, IsOpen (f ⁻¹' {y}) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {f : X → Y}
Goal: IsLocallyConstant f ↔ ∀ (y : Y), IsOpen (f ⁻¹' {y})
-- IsLocallyConstant.continuous
protected theorem continuous [TopologicalSpace Y] {f : X → Y} (hf : IsLocallyConstant f) :
    Continuous f 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : IsLocallyConstant f)
Goal: Continuous f
-- IsLocallyConstant.iff_continuous
theorem iff_continuous {_ : TopologicalSpace Y} [DiscreteTopology Y] (f : X → Y) :
    IsLocallyConstant f ↔ Continuous f 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X]
Goal: ∀ {x : TopologicalSpace Y} [inst : DiscreteTopology Y] (f : X → Y), IsLocallyConstant f ↔ Continuous f
-- IsLocallyConstant.of_constant
theorem of_constant (f : X → Y) (h : ∀ x y, f x = f y) : IsLocallyConstant f 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] (f : X → Y) (h : ∀ (x y : X), f x = f y)
Goal: IsLocallyConstant f
-- IsLocallyConstant.const
protected theorem const (y : Y) : IsLocallyConstant (Function.const X y) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] (y : Y)
Goal: IsLocallyConstant (Function.const X y)
-- IsLocallyConstant.comp
protected theorem comp {f : X → Y} (hf : IsLocallyConstant f) (g : Y → Z) :
    IsLocallyConstant (g ∘ f) 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] {f : X → Y} (hf : IsLocallyConstant f) (g : Y → Z)
Goal: IsLocallyConstant (g ∘ f)
-- IsLocallyConstant.prod_mk
theorem prod_mk {Y'} {f : X → Y} {f' : X → Y'} (hf : IsLocallyConstant f)
    (hf' : IsLocallyConstant f') : IsLocallyConstant fun x => (f x, f' x) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {Y' : Type u_5} {f : X → Y} {f' : X → Y'} (hf : IsLocallyConstant f) (hf' : IsLocallyConstant f')
Goal: IsLocallyConstant fun x => (f x, f' x)
-- IsLocallyConstant.comp₂
theorem comp₂ {Y₁ Y₂ Z : Type*} {f : X → Y₁} {g : X → Y₂} (hf : IsLocallyConstant f)
    (hg : IsLocallyConstant g) (h : Y₁ → Y₂ → Z) : IsLocallyConstant fun x => h (f x) (g x) 
Arguments: {X : Type u_1} [TopologicalSpace X] {Y₁ : Type u_5} {Y₂ : Type u_6} {Z : Type u_7} {f : X → Y₁} {g : X → Y₂} (hf : IsLocallyConstant f) (hg : IsLocallyConstant g) (h : Y₁ → Y₂ → Z)
Goal: IsLocallyConstant fun x => h (f x) (g x)
-- IsLocallyConstant.comp_continuous
theorem comp_continuous [TopologicalSpace Y] {g : Y → Z} {f : X → Y} (hg : IsLocallyConstant g)
    (hf : Continuous f) : IsLocallyConstant (g ∘ f) 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] {g : Y → Z} {f : X → Y} (hg : IsLocallyConstant g) (hf : Continuous f)
Goal: IsLocallyConstant (g ∘ f)
-- IsLocallyConstant.apply_eq_of_isPreconnected
theorem apply_eq_of_isPreconnected {f : X → Y} (hf : IsLocallyConstant f) {s : Set X}
    (hs : IsPreconnected s) {x y : X} (hx : x ∈ s) (hy : y ∈ s) : f x = f y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {f : X → Y} (hf : IsLocallyConstant f) {s : Set X} (hs : IsPreconnected s) {x y : X} (hx : x ∈ s) (hy : y ∈ s)
Goal: f x = f y
-- IsLocallyConstant.apply_eq_of_preconnectedSpace
theorem apply_eq_of_preconnectedSpace [PreconnectedSpace X] {f : X → Y} (hf : IsLocallyConstant f)
    (x y : X) : f x = f y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [PreconnectedSpace X] {f : X → Y} (hf : IsLocallyConstant f) (x y : X)
Goal: f x = f y
-- IsLocallyConstant.eq_const
theorem eq_const [PreconnectedSpace X] {f : X → Y} (hf : IsLocallyConstant f) (x : X) :
    f = Function.const X (f x) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [PreconnectedSpace X] {f : X → Y} (hf : IsLocallyConstant f) (x : X)
Goal: f = Function.const X (f x)
-- IsLocallyConstant.exists_eq_const
theorem exists_eq_const [PreconnectedSpace X] [Nonempty Y] {f : X → Y} (hf : IsLocallyConstant f) :
    ∃ y, f = Function.const X y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [PreconnectedSpace X] [Nonempty Y] {f : X → Y} (hf : IsLocallyConstant f)
Goal: ∃ y, f = Function.const X y
-- IsLocallyConstant.iff_is_const
theorem iff_is_const [PreconnectedSpace X] {f : X → Y} : IsLocallyConstant f ↔ ∀ x y, f x = f y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [PreconnectedSpace X] {f : X → Y}
Goal: IsLocallyConstant f ↔ ∀ (x y : X), f x = f y
-- IsLocallyConstant.range_finite
theorem range_finite [CompactSpace X] {f : X → Y} (hf : IsLocallyConstant f) :
    (Set.range f).Finite 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [CompactSpace X] {f : X → Y} (hf : IsLocallyConstant f)
Goal: (Set.range f).Finite
-- IsLocallyConstant.zero
@[to_additive]
theorem one [One Y] : IsLocallyConstant (1 : X → Y) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Zero Y]
Goal: IsLocallyConstant 0
-- IsLocallyConstant.one
@[to_additive]
theorem one [One Y] : IsLocallyConstant (1 : X → Y) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [One Y]
Goal: IsLocallyConstant 1
-- IsLocallyConstant.neg
@[to_additive]
theorem inv [Inv Y] ⦃f : X → Y⦄ (hf : IsLocallyConstant f) : IsLocallyConstant f⁻¹ 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Neg Y] ⦃f : X → Y⦄ (hf : IsLocallyConstant f)
Goal: IsLocallyConstant (-f)
-- IsLocallyConstant.inv
@[to_additive]
theorem inv [Inv Y] ⦃f : X → Y⦄ (hf : IsLocallyConstant f) : IsLocallyConstant f⁻¹ 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Inv Y] ⦃f : X → Y⦄ (hf : IsLocallyConstant f)
Goal: IsLocallyConstant f⁻¹
-- IsLocallyConstant.add
@[to_additive]
theorem mul [Mul Y] ⦃f g : X → Y⦄ (hf : IsLocallyConstant f) (hg : IsLocallyConstant g) :
    IsLocallyConstant (f * g) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Add Y] ⦃f g : X → Y⦄ (hf : IsLocallyConstant f) (hg : IsLocallyConstant g)
Goal: IsLocallyConstant (f + g)
-- IsLocallyConstant.mul
@[to_additive]
theorem mul [Mul Y] ⦃f g : X → Y⦄ (hf : IsLocallyConstant f) (hg : IsLocallyConstant g) :
    IsLocallyConstant (f * g) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Mul Y] ⦃f g : X → Y⦄ (hf : IsLocallyConstant f) (hg : IsLocallyConstant g)
Goal: IsLocallyConstant (f * g)
-- IsLocallyConstant.sub
@[to_additive]
theorem div [Div Y] ⦃f g : X → Y⦄ (hf : IsLocallyConstant f) (hg : IsLocallyConstant g) :
    IsLocallyConstant (f / g) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Sub Y] ⦃f g : X → Y⦄ (hf : IsLocallyConstant f) (hg : IsLocallyConstant g)
Goal: IsLocallyConstant (f - g)
-- IsLocallyConstant.div
@[to_additive]
theorem div [Div Y] ⦃f g : X → Y⦄ (hf : IsLocallyConstant f) (hg : IsLocallyConstant g) :
    IsLocallyConstant (f / g) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [Div Y] ⦃f g : X → Y⦄ (hf : IsLocallyConstant f) (hg : IsLocallyConstant g)
Goal: IsLocallyConstant (f / g)
-- IsLocallyConstant.desc
theorem desc {α β : Type*} (f : X → α) (g : α → β) (h : IsLocallyConstant (g ∘ f))
    (inj : Function.Injective g) : IsLocallyConstant f 
Arguments: {X : Type u_1} [TopologicalSpace X] {α : Type u_5} {β : Type u_6} (f : X → α) (g : α → β) (h : IsLocallyConstant (g ∘ f)) (inj : Function.Injective g)
Goal: IsLocallyConstant f
-- IsLocallyConstant.of_constant_on_connected_components
theorem of_constant_on_connected_components [LocallyConnectedSpace X] {f : X → Y}
    (h : ∀ x, ∀ y ∈ connectedComponent x, f y = f x) : IsLocallyConstant f 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [LocallyConnectedSpace X] {f : X → Y} (h : ∀ (x y : X), y ∈ connectedComponent x → f y = f x)
Goal: IsLocallyConstant f
-- IsLocallyConstant.of_constant_on_connected_clopens
theorem of_constant_on_connected_clopens [LocallyConnectedSpace X] {f : X → Y}
    (h : ∀ U : Set X, IsConnected U → IsClopen U → ∀ x ∈ U, ∀ y ∈ U, f y = f x) :
    IsLocallyConstant f 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [LocallyConnectedSpace X] {f : X → Y} (h : ∀ (U : Set X), IsConnected U → IsClopen U → ∀ x ∈ U, ∀ y ∈ U, f y = f x)
Goal: IsLocallyConstant f
-- IsLocallyConstant.of_constant_on_preconnected_clopens
theorem of_constant_on_preconnected_clopens [LocallyConnectedSpace X] {f : X → Y}
    (h : ∀ U : Set X, IsPreconnected U → IsClopen U → ∀ x ∈ U, ∀ y ∈ U, f y = f x) :
    IsLocallyConstant f 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [LocallyConnectedSpace X] {f : X → Y} (h : ∀ (U : Set X), IsPreconnected U → IsClopen U → ∀ x ∈ U, ∀ y ∈ U, f y = f x)
Goal: IsLocallyConstant f
-- LocallyConstant.isLocallyConstant
structure LocallyConstant (X Y : Type*) [TopologicalSpace X] where
    protected toFun : X → Y
    protected isLocallyConstant : IsLocallyConstant toFun
Arguments: {X : Type u_5} {Y : Type u_6} [TopologicalSpace X] (self : LocallyConstant X Y)
Goal: IsLocallyConstant self.toFun
-- LocallyConstant.toFun_eq_coe
@[simp]
theorem toFun_eq_coe (f : LocallyConstant X Y) : f.toFun = f 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] (f : LocallyConstant X Y)
Goal: f.toFun = ⇑f
-- LocallyConstant.coe_mk
@[simp]
theorem coe_mk (f : X → Y) (h) : ⇑(⟨f, h⟩ : LocallyConstant X Y) = f 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] (f : X → Y) (h : IsLocallyConstant f)
Goal: ⇑{ toFun := f, isLocallyConstant := h } = f
-- LocallyConstant.congr_fun
protected theorem congr_fun {f g : LocallyConstant X Y} (h : f = g) (x : X) : f x = g x 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {f g : LocallyConstant X Y} (h : f = g) (x : X)
Goal: f x = g x
-- LocallyConstant.congr_arg
protected theorem congr_arg (f : LocallyConstant X Y) {x y : X} (h : x = y) : f x = f y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] (f : LocallyConstant X Y) {x y : X} (h : x = y)
Goal: f x = f y
-- LocallyConstant.coe_injective
theorem coe_injective : @Function.Injective (LocallyConstant X Y) (X → Y) (↑) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X]
Goal: Function.Injective DFunLike.coe
-- LocallyConstant.coe_inj
@[norm_cast]
theorem coe_inj {f g : LocallyConstant X Y} : (f : X → Y) = g ↔ f = g 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {f g : LocallyConstant X Y}
Goal: ⇑f = ⇑g ↔ f = g
-- LocallyConstant.ext
@[ext]
theorem ext ⦃f g : LocallyConstant X Y⦄ (h : ∀ x, f x = g x) : f = g 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] ⦃f g : LocallyConstant X Y⦄ (h : ∀ (x : X), f x = g x)
Goal: f = g
-- LocallyConstant.ext_iff
theorem ext_iff {f g : LocallyConstant X Y} : f = g ↔ ∀ x, f x = g x 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {f g : LocallyConstant X Y}
Goal: f = g ↔ ∀ (x : X), f x = g x
-- LocallyConstant.continuous
protected theorem continuous : Continuous f 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (f : LocallyConstant X Y)
Goal: Continuous ⇑f
-- LocallyConstant.coe_continuousMap
@[simp] theorem coe_continuousMap : ((f : C(X, Y)) : X → Y) = (f : X → Y) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (f : LocallyConstant X Y)
Goal: ⇑↑f = ⇑f
-- LocallyConstant.toContinuousMap_injective
theorem toContinuousMap_injective :
    Function.Injective (toContinuousMap : LocallyConstant X Y → C(X, Y)) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y]
Goal: Function.Injective LocallyConstant.toContinuousMap
-- LocallyConstant.coe_const
@[simp]
theorem coe_const (y : Y) : (const X y : X → Y) = Function.const X y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] (y : Y)
Goal: ⇑(LocallyConstant.const X y) = Function.const X y
-- LocallyConstant.ofIsClopen_fiber_zero
@[simp]
theorem ofIsClopen_fiber_zero {X : Type*} [TopologicalSpace X] {U : Set X} [∀ x, Decidable (x ∈ U)]
    (hU : IsClopen U) : ofIsClopen hU ⁻¹' ({0} : Set (Fin 2)) = U 
Arguments: {X : Type u_5} [TopologicalSpace X] {U : Set X} [(x : X) → Decidable (x ∈ U)] (hU : IsClopen U)
Goal: ⇑(LocallyConstant.ofIsClopen hU) ⁻¹' {0} = U
-- LocallyConstant.ofIsClopen_fiber_one
@[simp]
theorem ofIsClopen_fiber_one {X : Type*} [TopologicalSpace X] {U : Set X} [∀ x, Decidable (x ∈ U)]
    (hU : IsClopen U) : ofIsClopen hU ⁻¹' ({1} : Set (Fin 2)) = Uᶜ 
Arguments: {X : Type u_5} [TopologicalSpace X] {U : Set X} [(x : X) → Decidable (x ∈ U)] (hU : IsClopen U)
Goal: ⇑(LocallyConstant.ofIsClopen hU) ⁻¹' {1} = Uᶜ
-- LocallyConstant.locallyConstant_eq_of_fiber_zero_eq
theorem locallyConstant_eq_of_fiber_zero_eq {X : Type*} [TopologicalSpace X]
    (f g : LocallyConstant X (Fin 2)) (h : f ⁻¹' ({0} : Set (Fin 2)) = g ⁻¹' {0}) : f = g 
Arguments: {X : Type u_5} [TopologicalSpace X] (f g : LocallyConstant X (Fin 2)) (h : ⇑f ⁻¹' {0} = ⇑g ⁻¹' {0})
Goal: f = g
-- LocallyConstant.range_finite
theorem range_finite [CompactSpace X] (f : LocallyConstant X Y) : (Set.range f).Finite 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [CompactSpace X] (f : LocallyConstant X Y)
Goal: (Set.range ⇑f).Finite
-- LocallyConstant.apply_eq_of_isPreconnected
theorem apply_eq_of_isPreconnected (f : LocallyConstant X Y) {s : Set X} (hs : IsPreconnected s)
    {x y : X} (hx : x ∈ s) (hy : y ∈ s) : f x = f y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] (f : LocallyConstant X Y) {s : Set X} (hs : IsPreconnected s) {x y : X} (hx : x ∈ s) (hy : y ∈ s)
Goal: f x = f y
-- LocallyConstant.apply_eq_of_preconnectedSpace
theorem apply_eq_of_preconnectedSpace [PreconnectedSpace X] (f : LocallyConstant X Y) (x y : X) :
    f x = f y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [PreconnectedSpace X] (f : LocallyConstant X Y) (x y : X)
Goal: f x = f y
-- LocallyConstant.eq_const
theorem eq_const [PreconnectedSpace X] (f : LocallyConstant X Y) (x : X) : f = const X (f x) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [PreconnectedSpace X] (f : LocallyConstant X Y) (x : X)
Goal: f = LocallyConstant.const X (f x)
-- LocallyConstant.exists_eq_const
theorem exists_eq_const [PreconnectedSpace X] [Nonempty Y] (f : LocallyConstant X Y) :
    ∃ y, f = const X y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [PreconnectedSpace X] [Nonempty Y] (f : LocallyConstant X Y)
Goal: ∃ y, f = LocallyConstant.const X y
-- LocallyConstant.map_apply
@[simp]
theorem map_apply (f : Y → Z) (g : LocallyConstant X Y) : ⇑(map f g) = f ∘ g 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] (f : Y → Z) (g : LocallyConstant X Y)
Goal: ⇑(LocallyConstant.map f g) = f ∘ ⇑g
-- LocallyConstant.map_id
@[simp]
theorem map_id : @map X Y Y _ id = id 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X]
Goal: LocallyConstant.map id = id
-- LocallyConstant.map_comp
@[simp]
theorem map_comp {Y₁ Y₂ Y₃ : Type*} (g : Y₂ → Y₃) (f : Y₁ → Y₂) :
    @map X _ _ _ g ∘ map f = map (g ∘ f) 
Arguments: {X : Type u_1} [TopologicalSpace X] {Y₁ : Type u_5} {Y₂ : Type u_6} {Y₃ : Type u_7} (g : Y₂ → Y₃) (f : Y₁ → Y₂)
Goal: LocallyConstant.map g ∘ LocallyConstant.map f = LocallyConstant.map (g ∘ f)
-- LocallyConstant.unflip_flip
@[simp]
theorem unflip_flip {X α β : Type*} [Finite α] [TopologicalSpace X]
    (f : LocallyConstant X (α → β)) : unflip f.flip = f 
Arguments: {X : Type u_5} {α : Type u_6} {β : Type u_7} [Finite α] [TopologicalSpace X] (f : LocallyConstant X (α → β))
Goal: LocallyConstant.unflip f.flip = f
-- LocallyConstant.flip_unflip
@[simp]
theorem flip_unflip {X α β : Type*} [Finite α] [TopologicalSpace X]
    (f : α → LocallyConstant X β) : (unflip f).flip = f 
Arguments: {X : Type u_5} {α : Type u_6} {β : Type u_7} [Finite α] [TopologicalSpace X] (f : α → LocallyConstant X β)
Goal: (LocallyConstant.unflip f).flip = f
-- LocallyConstant.coe_comap
@[simp]
theorem coe_comap (f : C(X, Y)) (g : LocallyConstant Y Z) :
    (comap f g) = g ∘ f 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (f : C(X, Y)) (g : LocallyConstant Y Z)
Goal: ⇑(LocallyConstant.comap f g) = ⇑g ∘ ⇑f
-- LocallyConstant.coe_comap_apply
theorem coe_comap_apply (f : C(X, Y)) (g : LocallyConstant Y Z) (x : X) :
    comap f g x = g (f x) 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (f : C(X, Y)) (g : LocallyConstant Y Z) (x : X)
Goal: (LocallyConstant.comap f g) x = g (f x)
-- LocallyConstant.comap_id
@[simp]
theorem comap_id : comap (@ContinuousMap.id X _) = @id (LocallyConstant X Z) 
Arguments: {X : Type u_1} {Z : Type u_3} [TopologicalSpace X]
Goal: LocallyConstant.comap (ContinuousMap.id X) = id
-- LocallyConstant.comap_comp
theorem comap_comp {W : Type*} [TopologicalSpace W] (f : C(W, X)) (g : C(X, Y)) :
    comap (Z := Z) (g.comp f) = comap f ∘ comap g 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] {W : Type u_5} [TopologicalSpace W] (f : C(W, X)) (g : C(X, Y))
Goal: LocallyConstant.comap (g.comp f) = LocallyConstant.comap f ∘ LocallyConstant.comap g
-- LocallyConstant.comap_comap
theorem comap_comap {W : Type*} [TopologicalSpace W] (f : C(W, X)) (g : C(X, Y))
    (x : LocallyConstant Y Z) : comap f (comap g x) = comap (g.comp f) x 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] {W : Type u_5} [TopologicalSpace W] (f : C(W, X)) (g : C(X, Y)) (x : LocallyConstant Y Z)
Goal: LocallyConstant.comap f (LocallyConstant.comap g x) = LocallyConstant.comap (g.comp f) x
-- LocallyConstant.comap_const
theorem comap_const (f : C(X, Y)) (y : Y) (h : ∀ x, f x = y) :
    (comap f : LocallyConstant Y Z → LocallyConstant X Z) = fun g => const X (g y) 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (f : C(X, Y)) (y : Y) (h : ∀ (x : X), f x = y)
Goal: LocallyConstant.comap f = fun g => LocallyConstant.const X (g y)
-- LocallyConstant.comap_injective
lemma comap_injective (f : C(X, Y)) (hfs : f.1.Surjective) :
    (comap (Z := Z) f).Injective 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (f : C(X, Y)) (hfs : Function.Surjective f.toFun)
Goal: Function.Injective (LocallyConstant.comap f)
-- LocallyConstant.coe_desc
@[simp]
theorem coe_desc {X α β : Type*} [TopologicalSpace X] (f : X → α) (g : α → β)
    (h : LocallyConstant X β) (cond : g ∘ f = h) (inj : Function.Injective g) :
    ⇑(desc f h cond inj) = f 
Arguments: {X : Type u_5} {α : Type u_6} {β : Type u_7} [TopologicalSpace X] (f : X → α) (g : α → β) (h : LocallyConstant X β) (cond : g ∘ f = ⇑h) (inj : Function.Injective g)
Goal: ⇑(LocallyConstant.desc f h cond inj) = f
-- LocallyConstant.mulIndicator_apply
@[to_additive (attr := simps) "Given a clopen set `U` and a locally constant function `f`,
  `LocallyConstant.indicator` returns the locally constant function that is `f` on `U` and `0`
  otherwise. "]
noncomputable def mulIndicator (hU : IsClopen U) : LocallyConstant X R 
Arguments: {X : Type u_1} [TopologicalSpace X] {R : Type u_5} [One R] {U : Set X} (f : LocallyConstant X R) (hU : IsClopen U) (x : X)
Goal: (f.mulIndicator hU) x = U.mulIndicator (⇑f) x
-- LocallyConstant.indicator_apply
@[to_additive (attr := simps) "Given a clopen set `U` and a locally constant function `f`,
  `LocallyConstant.indicator` returns the locally constant function that is `f` on `U` and `0`
  otherwise. "]
noncomputable def mulIndicator (hU : IsClopen U) : LocallyConstant X R 
Arguments: {X : Type u_1} [TopologicalSpace X] {R : Type u_5} [Zero R] {U : Set X} (f : LocallyConstant X R) (hU : IsClopen U) (x : X)
Goal: (f.indicator hU) x = U.indicator (⇑f) x
-- LocallyConstant.mulIndicator_apply_eq_if
@[to_additive]
theorem mulIndicator_apply_eq_if (hU : IsClopen U) :
    mulIndicator f hU a = if a ∈ U then f a else 1 
Arguments: {X : Type u_1} [TopologicalSpace X] {R : Type u_5} [One R] {U : Set X} (f : LocallyConstant X R) (a : X) (hU : IsClopen U)
Goal: (f.mulIndicator hU) a = if a ∈ U then f a else 1