ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- BoundedContinuousFunction.lipschitz_evalx
theorem lipschitz_evalx (x : α) : LipschitzWith 1 fun f : α →ᵇ β => f x 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] (x : α)
Goal: LipschitzWith 1 fun f => f x
-- BoundedContinuousFunction.uniformContinuous_coe
theorem uniformContinuous_coe : @UniformContinuous (α →ᵇ β) (α → β) _ _ (⇑) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β]
Goal: UniformContinuous DFunLike.coe
-- BoundedContinuousFunction.continuous_coe
theorem continuous_coe : Continuous fun (f : α →ᵇ β) x => f x 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β]
Goal: Continuous fun f x => f x
-- BoundedContinuousFunction.continuous_eval_const
@[continuity]
theorem continuous_eval_const {x : α} : Continuous fun f : α →ᵇ β => f x 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {x : α}
Goal: Continuous fun f => f x
-- BoundedContinuousFunction.continuous_eval
@[continuity]
theorem continuous_eval : Continuous fun p : (α →ᵇ β) × α => p.1 p.2 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β]
Goal: Continuous fun p => p.1 p.2
-- BoundedContinuousFunction.coe_compContinuous
@[simp]
theorem coe_compContinuous {δ : Type*} [TopologicalSpace δ] (f : α →ᵇ β) (g : C(δ, α)) :
    ⇑(f.compContinuous g) = f ∘ g 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {δ : Type u_2} [TopologicalSpace δ] (f : BoundedContinuousFunction α β) (g : C(δ, α))
Goal: ⇑(f.compContinuous g) = ⇑f ∘ ⇑g
-- BoundedContinuousFunction.compContinuous_apply
@[simp]
theorem compContinuous_apply {δ : Type*} [TopologicalSpace δ] (f : α →ᵇ β) (g : C(δ, α)) (x : δ) :
    f.compContinuous g x = f (g x) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {δ : Type u_2} [TopologicalSpace δ] (f : BoundedContinuousFunction α β) (g : C(δ, α)) (x : δ)
Goal: (f.compContinuous g) x = f (g x)
-- BoundedContinuousFunction.lipschitz_compContinuous
theorem lipschitz_compContinuous {δ : Type*} [TopologicalSpace δ] (g : C(δ, α)) :
    LipschitzWith 1 fun f : α →ᵇ β => f.compContinuous g 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {δ : Type u_2} [TopologicalSpace δ] (g : C(δ, α))
Goal: LipschitzWith 1 fun f => f.compContinuous g
-- BoundedContinuousFunction.continuous_compContinuous
theorem continuous_compContinuous {δ : Type*} [TopologicalSpace δ] (g : C(δ, α)) :
    Continuous fun f : α →ᵇ β => f.compContinuous g 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {δ : Type u_2} [TopologicalSpace δ] (g : C(δ, α))
Goal: Continuous fun f => f.compContinuous g
-- BoundedContinuousFunction.coe_restrict
@[simp]
theorem coe_restrict (f : α →ᵇ β) (s : Set α) : ⇑(f.restrict s) = f ∘ (↑) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] (f : BoundedContinuousFunction α β) (s : Set α)
Goal: ⇑(f.restrict s) = ⇑f ∘ Subtype.val
-- BoundedContinuousFunction.restrict_apply
@[simp]
theorem restrict_apply (f : α →ᵇ β) (s : Set α) (x : s) : f.restrict s x = f x 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] (f : BoundedContinuousFunction α β) (s : Set α) (x : ↑s)
Goal: (f.restrict s) x = f ↑x
-- BoundedContinuousFunction.lipschitz_comp
theorem lipschitz_comp {G : β → γ} {C : ℝ≥0} (H : LipschitzWith C G) :
    LipschitzWith C (comp G H : (α →ᵇ β) → α →ᵇ γ) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [TopologicalSpace α] [PseudoMetricSpace β] [PseudoMetricSpace γ] {G : β → γ} {C : ℝ≥0} (H : LipschitzWith C G)
Goal: LipschitzWith C (BoundedContinuousFunction.comp G H)
-- BoundedContinuousFunction.uniformContinuous_comp
theorem uniformContinuous_comp {G : β → γ} {C : ℝ≥0} (H : LipschitzWith C G) :
    UniformContinuous (comp G H : (α →ᵇ β) → α →ᵇ γ) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [TopologicalSpace α] [PseudoMetricSpace β] [PseudoMetricSpace γ] {G : β → γ} {C : ℝ≥0} (H : LipschitzWith C G)
Goal: UniformContinuous (BoundedContinuousFunction.comp G H)
-- BoundedContinuousFunction.continuous_comp
theorem continuous_comp {G : β → γ} {C : ℝ≥0} (H : LipschitzWith C G) :
    Continuous (comp G H : (α →ᵇ β) → α →ᵇ γ) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [TopologicalSpace α] [PseudoMetricSpace β] [PseudoMetricSpace γ] {G : β → γ} {C : ℝ≥0} (H : LipschitzWith C G)
Goal: Continuous (BoundedContinuousFunction.comp G H)
-- BoundedContinuousFunction.extend_apply
@[simp]
theorem extend_apply (f : α ↪ δ) (g : α →ᵇ β) (h : δ →ᵇ β) (x : α) : extend f g h (f x) = g x 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {δ : Type u_2} [TopologicalSpace δ] [DiscreteTopology δ] (f : α ↪ δ) (g : BoundedContinuousFunction α β) (h : BoundedContinuousFunction δ β) (x : α)
Goal: (BoundedContinuousFunction.extend f g h) (f x) = g x
-- BoundedContinuousFunction.extend_comp
@[simp]
nonrec theorem extend_comp (f : α ↪ δ) (g : α →ᵇ β) (h : δ →ᵇ β) : extend f g h ∘ f = g 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {δ : Type u_2} [TopologicalSpace δ] [DiscreteTopology δ] (f : α ↪ δ) (g : BoundedContinuousFunction α β) (h : BoundedContinuousFunction δ β)
Goal: ⇑(BoundedContinuousFunction.extend f g h) ∘ ⇑f = ⇑g
-- BoundedContinuousFunction.extend_apply'
nonrec theorem extend_apply' {f : α ↪ δ} {x : δ} (hx : x ∉ range f) (g : α →ᵇ β) (h : δ →ᵇ β) :
    extend f g h x = h x 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {δ : Type u_2} [TopologicalSpace δ] [DiscreteTopology δ] {f : α ↪ δ} {x : δ} (hx : x ∉ Set.range ⇑f) (g : BoundedContinuousFunction α β) (h : BoundedContinuousFunction δ β)
Goal: (BoundedContinuousFunction.extend f g h) x = h x
-- BoundedContinuousFunction.extend_of_empty
theorem extend_of_empty [IsEmpty α] (f : α ↪ δ) (g : α →ᵇ β) (h : δ →ᵇ β) : extend f g h = h 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {δ : Type u_2} [TopologicalSpace δ] [DiscreteTopology δ] [IsEmpty α] (f : α ↪ δ) (g : BoundedContinuousFunction α β) (h : BoundedContinuousFunction δ β)
Goal: BoundedContinuousFunction.extend f g h = h
-- BoundedContinuousFunction.dist_extend_extend
@[simp]
theorem dist_extend_extend (f : α ↪ δ) (g₁ g₂ : α →ᵇ β) (h₁ h₂ : δ →ᵇ β) :
    dist (g₁.extend f h₁) (g₂.extend f h₂) =
      max (dist g₁ g₂) (dist (h₁.restrict (range f)ᶜ) (h₂.restrict (range f)ᶜ)) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {δ : Type u_2} [TopologicalSpace δ] [DiscreteTopology δ] (f : α ↪ δ) (g₁ g₂ : BoundedContinuousFunction α β) (h₁ h₂ : BoundedContinuousFunction δ β)
Goal: dist (BoundedContinuousFunction.extend f g₁ h₁) (BoundedContinuousFunction.extend f g₂ h₂) =
  max (dist g₁ g₂) (dist (h₁.restrict (Set.range ⇑f)ᶜ) (h₂.restrict (Set.range ⇑f)ᶜ))
-- BoundedContinuousFunction.isometry_extend
theorem isometry_extend (f : α ↪ δ) (h : δ →ᵇ β) : Isometry fun g : α →ᵇ β => extend f g h 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {δ : Type u_2} [TopologicalSpace δ] [DiscreteTopology δ] (f : α ↪ δ) (h : BoundedContinuousFunction δ β)
Goal: Isometry fun g => BoundedContinuousFunction.extend f g h
-- BoundedContinuousFunction.arzela_ascoli₁
theorem arzela_ascoli₁ [CompactSpace β] (A : Set (α →ᵇ β)) (closed : IsClosed A)
    (H : Equicontinuous ((↑) : A → α → β)) : IsCompact A 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [CompactSpace α] [PseudoMetricSpace β] [CompactSpace β] (A : Set (BoundedContinuousFunction α β)) (closed : IsClosed A) (H : Equicontinuous fun x => ⇑↑x)
Goal: IsCompact A
-- BoundedContinuousFunction.arzela_ascoli₂
theorem arzela_ascoli₂ (s : Set β) (hs : IsCompact s) (A : Set (α →ᵇ β)) (closed : IsClosed A)
    (in_s : ∀ (f : α →ᵇ β) (x : α), f ∈ A → f x ∈ s) (H : Equicontinuous ((↑) : A → α → β)) :
    IsCompact A 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [CompactSpace α] [PseudoMetricSpace β] (s : Set β) (hs : IsCompact s) (A : Set (BoundedContinuousFunction α β)) (closed : IsClosed A) (in_s : ∀ (f : BoundedContinuousFunction α β) (x : α), f ∈ A → f x ∈ s) (H : Equicontinuous fun x => ⇑↑x)
Goal: IsCompact A
-- BoundedContinuousFunction.arzela_ascoli
theorem arzela_ascoli [T2Space β] (s : Set β) (hs : IsCompact s) (A : Set (α →ᵇ β))
    (in_s : ∀ (f : α →ᵇ β) (x : α), f ∈ A → f x ∈ s) (H : Equicontinuous ((↑) : A → α → β)) :
    IsCompact (closure A) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [CompactSpace α] [PseudoMetricSpace β] [T2Space β] (s : Set β) (hs : IsCompact s) (A : Set (BoundedContinuousFunction α β)) (in_s : ∀ (f : BoundedContinuousFunction α β) (x : α), f ∈ A → f x ∈ s) (H : Equicontinuous fun x => ⇑↑x)
Goal: IsCompact (closure A)
-- BoundedContinuousFunction.coe_one
@[to_additive (attr := simp)]
theorem coe_one : ((1 : α →ᵇ β) : α → β) = 1 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [One β]
Goal: ⇑1 = 1
-- BoundedContinuousFunction.coe_zero
@[to_additive (attr := simp)]
theorem coe_one : ((1 : α →ᵇ β) : α → β) = 1 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [Zero β]
Goal: ⇑0 = 0
-- BoundedContinuousFunction.mkOfCompact_zero
@[to_additive (attr := simp)]
theorem mkOfCompact_one [CompactSpace α] : mkOfCompact (1 : C(α, β)) = 1 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [Zero β] [CompactSpace α]
Goal: BoundedContinuousFunction.mkOfCompact 0 = 0
-- BoundedContinuousFunction.mkOfCompact_one
@[to_additive (attr := simp)]
theorem mkOfCompact_one [CompactSpace α] : mkOfCompact (1 : C(α, β)) = 1 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [One β] [CompactSpace α]
Goal: BoundedContinuousFunction.mkOfCompact 1 = 1
-- BoundedContinuousFunction.forall_coe_zero_iff_zero
@[to_additive]
theorem forall_coe_one_iff_one (f : α →ᵇ β) : (∀ x, f x = 1) ↔ f = 1 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [Zero β] (f : BoundedContinuousFunction α β)
Goal: (∀ (x : α), f x = 0) ↔ f = 0
-- BoundedContinuousFunction.forall_coe_one_iff_one
@[to_additive]
theorem forall_coe_one_iff_one (f : α →ᵇ β) : (∀ x, f x = 1) ↔ f = 1 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [One β] (f : BoundedContinuousFunction α β)
Goal: (∀ (x : α), f x = 1) ↔ f = 1
-- BoundedContinuousFunction.zero_compContinuous
@[to_additive (attr := simp)]
theorem one_compContinuous [TopologicalSpace γ] (f : C(γ, α)) : (1 : α →ᵇ β).compContinuous f = 1 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [TopologicalSpace α] [PseudoMetricSpace β] [Zero β] [TopologicalSpace γ] (f : C(γ, α))
Goal: BoundedContinuousFunction.compContinuous 0 f = 0
-- BoundedContinuousFunction.one_compContinuous
@[to_additive (attr := simp)]
theorem one_compContinuous [TopologicalSpace γ] (f : C(γ, α)) : (1 : α →ᵇ β).compContinuous f = 1 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [TopologicalSpace α] [PseudoMetricSpace β] [One β] [TopologicalSpace γ] (f : C(γ, α))
Goal: BoundedContinuousFunction.compContinuous 1 f = 1
-- BoundedContinuousFunction.coe_add
@[simp]
theorem coe_add : ⇑(f + g) = f + g 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [AddMonoid β] [BoundedAdd β] [ContinuousAdd β] (f g : BoundedContinuousFunction α β)
Goal: ⇑(f + g) = ⇑f + ⇑g
-- BoundedContinuousFunction.add_apply
theorem add_apply : (f + g) x = f x + g x 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [AddMonoid β] [BoundedAdd β] [ContinuousAdd β] (f g : BoundedContinuousFunction α β) {x : α}
Goal: (f + g) x = f x + g x
-- BoundedContinuousFunction.mkOfCompact_add
@[simp]
theorem mkOfCompact_add [CompactSpace α] (f g : C(α, β)) :
    mkOfCompact (f + g) = mkOfCompact f + mkOfCompact g 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [AddMonoid β] [BoundedAdd β] [ContinuousAdd β] [CompactSpace α] (f g : C(α, β))
Goal: BoundedContinuousFunction.mkOfCompact (f + g) =
  BoundedContinuousFunction.mkOfCompact f + BoundedContinuousFunction.mkOfCompact g
-- BoundedContinuousFunction.add_compContinuous
theorem add_compContinuous [TopologicalSpace γ] (h : C(γ, α)) :
    (g + f).compContinuous h = g.compContinuous h + f.compContinuous h 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [TopologicalSpace α] [PseudoMetricSpace β] [AddMonoid β] [BoundedAdd β] [ContinuousAdd β] (f g : BoundedContinuousFunction α β) [TopologicalSpace γ] (h : C(γ, α))
Goal: (g + f).compContinuous h = g.compContinuous h + f.compContinuous h
-- BoundedContinuousFunction.coe_nsmulRec
@[simp]
theorem coe_nsmulRec : ∀ n, ⇑(nsmulRec n f) = n • ⇑f
  
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [AddMonoid β] [BoundedAdd β] [ContinuousAdd β] (f : BoundedContinuousFunction α β) (n : ℕ)
Goal: ⇑(nsmulRec n f) = n • ⇑f
-- BoundedContinuousFunction.coe_nsmul
@[simp]
theorem coe_nsmul (r : ℕ) (f : α →ᵇ β) : ⇑(r • f) = r • ⇑f 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [AddMonoid β] [BoundedAdd β] [ContinuousAdd β] (r : ℕ) (f : BoundedContinuousFunction α β)
Goal: ⇑(r • f) = r • ⇑f
-- BoundedContinuousFunction.nsmul_apply
@[simp]
theorem nsmul_apply (r : ℕ) (f : α →ᵇ β) (v : α) : (r • f) v = r • f v 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [AddMonoid β] [BoundedAdd β] [ContinuousAdd β] (r : ℕ) (f : BoundedContinuousFunction α β) (v : α)
Goal: (r • f) v = r • f v
-- BoundedContinuousFunction.coeFnAddHom_apply
@[simps]
def coeFnAddHom : (α →ᵇ β) →+ α → β 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [AddMonoid β] [BoundedAdd β] [ContinuousAdd β]
Goal: ∀ (a : BoundedContinuousFunction α β) (a_1 : α), BoundedContinuousFunction.coeFnAddHom a a_1 = a a_1
-- BoundedContinuousFunction.toContinuousMapAddHom_apply
@[simps]
def toContinuousMapAddHom : (α →ᵇ β) →+ C(α, β) 
Arguments: (α : Type u) (β : Type v) [TopologicalSpace α] [PseudoMetricSpace β] [AddMonoid β] [BoundedAdd β] [ContinuousAdd β] (self : BoundedContinuousFunction α β)
Goal: (BoundedContinuousFunction.toContinuousMapAddHom α β) self = self.toContinuousMap
-- BoundedContinuousFunction.coe_sum
@[simp]
theorem coe_sum {ι : Type*} (s : Finset ι) (f : ι → α →ᵇ β) :
    ⇑(∑ i ∈ s, f i) = ∑ i ∈ s, (f i : α → β) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [AddCommMonoid β] [BoundedAdd β] [ContinuousAdd β] {ι : Type u_2} (s : Finset ι) (f : ι → BoundedContinuousFunction α β)
Goal: ⇑(∑ i ∈ s, f i) = ∑ i ∈ s, ⇑(f i)
-- BoundedContinuousFunction.sum_apply
theorem sum_apply {ι : Type*} (s : Finset ι) (f : ι → α →ᵇ β) (a : α) :
    (∑ i ∈ s, f i) a = ∑ i ∈ s, f i a 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [AddCommMonoid β] [BoundedAdd β] [ContinuousAdd β] {ι : Type u_2} (s : Finset ι) (f : ι → BoundedContinuousFunction α β) (a : α)
Goal: (∑ i ∈ s, f i) a = ∑ i ∈ s, (f i) a
-- BoundedContinuousFunction.sub_apply
theorem sub_apply {x : α} : (f - g) x = f x - g x 
Arguments: {α : Type u} [TopologicalSpace α] {R : Type u_2} [PseudoMetricSpace R] [Sub R] [BoundedSub R] [ContinuousSub R] (f g : BoundedContinuousFunction α R) {x : α}
Goal: (f - g) x = f x - g x
-- BoundedContinuousFunction.coe_sub
@[simp]
theorem coe_sub : ⇑(f - g) = f - g 
Arguments: {α : Type u} [TopologicalSpace α] {R : Type u_2} [PseudoMetricSpace R] [Sub R] [BoundedSub R] [ContinuousSub R] (f g : BoundedContinuousFunction α R)
Goal: ⇑(f - g) = ⇑f - ⇑g
-- BoundedContinuousFunction.natCast_apply
@[simp]
theorem natCast_apply [NatCast β] (n : ℕ) (x : α) : (n : α →ᵇ β) x = n 
Arguments: {α : Type u} [TopologicalSpace α] {β : Type u_2} [PseudoMetricSpace β] [NatCast β] (n : ℕ) (x : α)
Goal: ↑n x = ↑n
-- BoundedContinuousFunction.intCast_apply
@[simp]
theorem intCast_apply [IntCast β] (m : ℤ) (x : α) : (m : α →ᵇ β) x = m 
Arguments: {α : Type u} [TopologicalSpace α] {β : Type u_2} [PseudoMetricSpace β] [IntCast β] (m : ℤ) (x : α)
Goal: ↑m x = ↑m
-- BoundedContinuousFunction.coe_mul
@[simp]
theorem coe_mul [Mul R] [BoundedMul R] [ContinuousMul R] (f g : α →ᵇ R) : ⇑(f * g) = f * g 
Arguments: {α : Type u} [TopologicalSpace α] {R : Type u_2} [PseudoMetricSpace R] [Mul R] [BoundedMul R] [ContinuousMul R] (f g : BoundedContinuousFunction α R)
Goal: ⇑(f * g) = ⇑f * ⇑g
-- BoundedContinuousFunction.mul_apply
theorem mul_apply [Mul R] [BoundedMul R] [ContinuousMul R] (f g : α →ᵇ R) (x : α) :
    (f * g) x = f x * g x 
Arguments: {α : Type u} [TopologicalSpace α] {R : Type u_2} [PseudoMetricSpace R] [Mul R] [BoundedMul R] [ContinuousMul R] (f g : BoundedContinuousFunction α R) (x : α)
Goal: (f * g) x = f x * g x
-- BoundedContinuousFunction.coe_pow
theorem coe_pow [Monoid R] [BoundedMul R] [ContinuousMul R] (n : ℕ) (f : α →ᵇ R) :
    ⇑(f ^ n) = (⇑f) ^ n 
Arguments: {α : Type u} [TopologicalSpace α] {R : Type u_2} [PseudoMetricSpace R] [Monoid R] [BoundedMul R] [ContinuousMul R] (n : ℕ) (f : BoundedContinuousFunction α R)
Goal: ⇑(f ^ n) = ⇑f ^ n
-- BoundedContinuousFunction.pow_apply
@[simp]
theorem pow_apply [Monoid R] [BoundedMul R] [ContinuousMul R] (n : ℕ) (f : α →ᵇ R) (x : α) :
    (f ^ n) x = f x ^ n 
Arguments: {α : Type u} [TopologicalSpace α] {R : Type u_2} [PseudoMetricSpace R] [Monoid R] [BoundedMul R] [ContinuousMul R] (n : ℕ) (f : BoundedContinuousFunction α R) (x : α)
Goal: (f ^ n) x = f x ^ n
-- BoundedContinuousFunction.norm_def
theorem norm_def : ‖f‖ = dist f 0 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] (f : BoundedContinuousFunction α β)
Goal: ‖f‖ = dist f 0
-- BoundedContinuousFunction.norm_eq
theorem norm_eq (f : α →ᵇ β) : ‖f‖ = sInf { C : ℝ | 0 ≤ C ∧ ∀ x : α, ‖f x‖ ≤ C } 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] (f : BoundedContinuousFunction α β)
Goal: ‖f‖ = sInf {C | 0 ≤ C ∧ ∀ (x : α), ‖f x‖ ≤ C}
-- BoundedContinuousFunction.norm_eq_of_nonempty
theorem norm_eq_of_nonempty [h : Nonempty α] : ‖f‖ = sInf { C : ℝ | ∀ x : α, ‖f x‖ ≤ C } 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] (f : BoundedContinuousFunction α β) [h : Nonempty α]
Goal: ‖f‖ = sInf {C | ∀ (x : α), ‖f x‖ ≤ C}
-- BoundedContinuousFunction.norm_eq_zero_of_empty
@[simp]
theorem norm_eq_zero_of_empty [IsEmpty α] : ‖f‖ = 0 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] (f : BoundedContinuousFunction α β) [IsEmpty α]
Goal: ‖f‖ = 0
-- BoundedContinuousFunction.norm_coe_le_norm
theorem norm_coe_le_norm (x : α) : ‖f x‖ ≤ ‖f‖ 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] (f : BoundedContinuousFunction α β) (x : α)
Goal: ‖f x‖ ≤ ‖f‖
-- BoundedContinuousFunction.neg_norm_le_apply
lemma neg_norm_le_apply (f : α →ᵇ ℝ) (x : α) :
    -‖f‖ ≤ f x 
Arguments: {α : Type u} [TopologicalSpace α] (f : BoundedContinuousFunction α ℝ) (x : α)
Goal: -‖f‖ ≤ f x
-- BoundedContinuousFunction.apply_le_norm
lemma apply_le_norm (f : α →ᵇ ℝ) (x : α) :
    f x ≤ ‖f‖ 
Arguments: {α : Type u} [TopologicalSpace α] (f : BoundedContinuousFunction α ℝ) (x : α)
Goal: f x ≤ ‖f‖
-- BoundedContinuousFunction.dist_le_two_norm'
theorem dist_le_two_norm' {f : γ → β} {C : ℝ} (hC : ∀ x, ‖f x‖ ≤ C) (x y : γ) :
    dist (f x) (f y) ≤ 2 * C 
Arguments: {β : Type v} {γ : Type w} [SeminormedAddCommGroup β] {f : γ → β} {C : ℝ} (hC : ∀ (x : γ), ‖f x‖ ≤ C) (x y : γ)
Goal: dist (f x) (f y) ≤ 2 * C
-- BoundedContinuousFunction.dist_le_two_norm
theorem dist_le_two_norm (x y : α) : dist (f x) (f y) ≤ 2 * ‖f‖ 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] (f : BoundedContinuousFunction α β) (x y : α)
Goal: dist (f x) (f y) ≤ 2 * ‖f‖
-- BoundedContinuousFunction.norm_le
theorem norm_le (C0 : (0 : ℝ) ≤ C) : ‖f‖ ≤ C ↔ ∀ x : α, ‖f x‖ ≤ C 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] {f : BoundedContinuousFunction α β} {C : ℝ} (C0 : 0 ≤ C)
Goal: ‖f‖ ≤ C ↔ ∀ (x : α), ‖f x‖ ≤ C
-- BoundedContinuousFunction.norm_le_of_nonempty
theorem norm_le_of_nonempty [Nonempty α] {f : α →ᵇ β} {M : ℝ} : ‖f‖ ≤ M ↔ ∀ x, ‖f x‖ ≤ M 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] [Nonempty α] {f : BoundedContinuousFunction α β} {M : ℝ}
Goal: ‖f‖ ≤ M ↔ ∀ (x : α), ‖f x‖ ≤ M
-- BoundedContinuousFunction.norm_lt_iff_of_compact
theorem norm_lt_iff_of_compact [CompactSpace α] {f : α →ᵇ β} {M : ℝ} (M0 : 0 < M) :
    ‖f‖ < M ↔ ∀ x, ‖f x‖ < M 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] [CompactSpace α] {f : BoundedContinuousFunction α β} {M : ℝ} (M0 : 0 < M)
Goal: ‖f‖ < M ↔ ∀ (x : α), ‖f x‖ < M
-- BoundedContinuousFunction.norm_lt_iff_of_nonempty_compact
theorem norm_lt_iff_of_nonempty_compact [Nonempty α] [CompactSpace α] {f : α →ᵇ β} {M : ℝ} :
    ‖f‖ < M ↔ ∀ x, ‖f x‖ < M 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] [Nonempty α] [CompactSpace α] {f : BoundedContinuousFunction α β} {M : ℝ}
Goal: ‖f‖ < M ↔ ∀ (x : α), ‖f x‖ < M
-- BoundedContinuousFunction.norm_const_le
theorem norm_const_le (b : β) : ‖const α b‖ ≤ ‖b‖ 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] (b : β)
Goal: ‖BoundedContinuousFunction.const α b‖ ≤ ‖b‖
-- BoundedContinuousFunction.norm_const_eq
@[simp]
theorem norm_const_eq [h : Nonempty α] (b : β) : ‖const α b‖ = ‖b‖ 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] [h : Nonempty α] (b : β)
Goal: ‖BoundedContinuousFunction.const α b‖ = ‖b‖
-- BoundedContinuousFunction.coe_ofNormedAddCommGroup
@[simp]
theorem coe_ofNormedAddCommGroup {α : Type u} {β : Type v} [TopologicalSpace α]
    [SeminormedAddCommGroup β] (f : α → β) (Hf : Continuous f) (C : ℝ) (H : ∀ x, ‖f x‖ ≤ C) :
    (ofNormedAddCommGroup f Hf C H : α → β) = f 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] (f : α → β) (Hf : Continuous f) (C : ℝ) (H : ∀ (x : α), ‖f x‖ ≤ C)
Goal: ⇑(BoundedContinuousFunction.ofNormedAddCommGroup f Hf C H) = f
-- BoundedContinuousFunction.norm_ofNormedAddCommGroup_le
theorem norm_ofNormedAddCommGroup_le {f : α → β} (hfc : Continuous f) {C : ℝ} (hC : 0 ≤ C)
    (hfC : ∀ x, ‖f x‖ ≤ C) : ‖ofNormedAddCommGroup f hfc C hfC‖ ≤ C 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] {f : α → β} (hfc : Continuous f) {C : ℝ} (hC : 0 ≤ C) (hfC : ∀ (x : α), ‖f x‖ ≤ C)
Goal: ‖BoundedContinuousFunction.ofNormedAddCommGroup f hfc C hfC‖ ≤ C
-- BoundedContinuousFunction.coe_ofNormedAddCommGroupDiscrete
@[simp]
theorem coe_ofNormedAddCommGroupDiscrete {α : Type u} {β : Type v} [TopologicalSpace α]
    [DiscreteTopology α] [SeminormedAddCommGroup β] (f : α → β) (C : ℝ) (H : ∀ x, ‖f x‖ ≤ C) :
    (ofNormedAddCommGroupDiscrete f C H : α → β) = f 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [DiscreteTopology α] [SeminormedAddCommGroup β] (f : α → β) (C : ℝ) (H : ∀ (x : α), ‖f x‖ ≤ C)
Goal: ⇑(BoundedContinuousFunction.ofNormedAddCommGroupDiscrete f C H) = f
-- BoundedContinuousFunction.coe_normComp
@[simp]
theorem coe_normComp : (f.normComp : α → ℝ) = norm ∘ f 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] (f : BoundedContinuousFunction α β)
Goal: ⇑f.normComp = norm ∘ ⇑f
-- BoundedContinuousFunction.norm_normComp
@[simp]
theorem norm_normComp : ‖f.normComp‖ = ‖f‖ 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] (f : BoundedContinuousFunction α β)
Goal: ‖f.normComp‖ = ‖f‖
-- BoundedContinuousFunction.bddAbove_range_norm_comp
theorem bddAbove_range_norm_comp : BddAbove <| Set.range <| norm ∘ f 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] (f : BoundedContinuousFunction α β)
Goal: BddAbove (Set.range (norm ∘ ⇑f))
-- BoundedContinuousFunction.norm_eq_iSup_norm
theorem norm_eq_iSup_norm : ‖f‖ = ⨆ x : α, ‖f x‖ 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] (f : BoundedContinuousFunction α β)
Goal: ‖f‖ = ⨆ x, ‖f x‖