ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- affineHomeomorph_apply
@[simps]
def affineHomeomorph (a b : 𝕜) (h : a ≠ 0) : 𝕜 ≃ₜ 𝕜 
Arguments: {𝕜 : Type u_2} [Field 𝕜] [TopologicalSpace 𝕜] [TopologicalRing 𝕜] (a b : 𝕜) (h : a ≠ 0) (x : 𝕜)
Goal: (affineHomeomorph a b h) x = a * x + b
-- IsLocalMin.inv
theorem IsLocalMin.inv {f : α → β} {a : α} (h1 : IsLocalMin f a) (h2 : ∀ᶠ z in 𝓝 a, 0 < f z) :
    IsLocalMax f⁻¹ a 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [LinearOrderedSemifield β] {f : α → β} {a : α} (h1 : IsLocalMin f a) (h2 : ∀ᶠ (z : α) in nhds a, 0 < f z)
Goal: IsLocalMax f⁻¹ a
-- IsPreconnected.eq_one_or_eq_neg_one_of_sq_eq
theorem IsPreconnected.eq_one_or_eq_neg_one_of_sq_eq [Ring 𝕜] [NoZeroDivisors 𝕜]
    (hS : IsPreconnected S) (hf : ContinuousOn f S) (hsq : EqOn (f ^ 2) 1 S) :
    EqOn f 1 S ∨ EqOn f (-1) S 
Arguments: {α : Type u_2} {𝕜 : Type u_3} {f : α → 𝕜} {S : Set α} [TopologicalSpace α] [TopologicalSpace 𝕜] [T1Space 𝕜] [Ring 𝕜] [NoZeroDivisors 𝕜] (hS : IsPreconnected S) (hf : ContinuousOn f S) (hsq : Set.EqOn (f ^ 2) 1 S)
Goal: Set.EqOn f 1 S ∨ Set.EqOn f (-1) S
-- IsPreconnected.eq_or_eq_neg_of_sq_eq
theorem IsPreconnected.eq_or_eq_neg_of_sq_eq [Field 𝕜] [HasContinuousInv₀ 𝕜] [ContinuousMul 𝕜]
    (hS : IsPreconnected S) (hf : ContinuousOn f S) (hg : ContinuousOn g S)
    (hsq : EqOn (f ^ 2) (g ^ 2) S) (hg_ne : ∀ {x : α}, x ∈ S → g x ≠ 0) :
    EqOn f g S ∨ EqOn f (-g) S 
Arguments: {α : Type u_2} {𝕜 : Type u_3} {f g : α → 𝕜} {S : Set α} [TopologicalSpace α] [TopologicalSpace 𝕜] [T1Space 𝕜] [Field 𝕜] [HasContinuousInv₀ 𝕜] [ContinuousMul 𝕜] (hS : IsPreconnected S) (hf : ContinuousOn f S) (hg : ContinuousOn g S) (hsq : Set.EqOn (f ^ 2) (g ^ 2) S) (hg_ne : ∀ {x : α}, x ∈ S → g x ≠ 0)
Goal: Set.EqOn f g S ∨ Set.EqOn f (-g) S
-- IsPreconnected.eq_of_sq_eq
theorem IsPreconnected.eq_of_sq_eq [Field 𝕜] [HasContinuousInv₀ 𝕜] [ContinuousMul 𝕜]
    (hS : IsPreconnected S) (hf : ContinuousOn f S) (hg : ContinuousOn g S)
    (hsq : EqOn (f ^ 2) (g ^ 2) S) (hg_ne : ∀ {x : α}, x ∈ S → g x ≠ 0) {y : α} (hy : y ∈ S)
    (hy' : f y = g y) : EqOn f g S 
Arguments: {α : Type u_2} {𝕜 : Type u_3} {f g : α → 𝕜} {S : Set α} [TopologicalSpace α] [TopologicalSpace 𝕜] [T1Space 𝕜] [Field 𝕜] [HasContinuousInv₀ 𝕜] [ContinuousMul 𝕜] (hS : IsPreconnected S) (hf : ContinuousOn f S) (hg : ContinuousOn g S) (hsq : Set.EqOn (f ^ 2) (g ^ 2) S) (hg_ne : ∀ {x : α}, x ∈ S → g x ≠ 0) {y : α} (hy : y ∈ S) (hy' : f y = g y)
Goal: Set.EqOn f g S
-- HasSum.const_smul
theorem HasSum.const_smul {a : α} (b : γ) (hf : HasSum f a) : HasSum (fun i ↦ b • f i) (b • a) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [Monoid γ] [TopologicalSpace α] [AddCommMonoid α] [DistribMulAction γ α] [ContinuousConstSMul γ α] {f : β → α} {a : α} (b : γ) (hf : HasSum f a)
Goal: HasSum (fun i => b • f i) (b • a)
-- Summable.const_smul
theorem Summable.const_smul (b : γ) (hf : Summable f) : Summable fun i ↦ b • f i 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [Monoid γ] [TopologicalSpace α] [AddCommMonoid α] [DistribMulAction γ α] [ContinuousConstSMul γ α] {f : β → α} (b : γ) (hf : Summable f)
Goal: Summable fun i => b • f i
-- tsum_const_smul
theorem tsum_const_smul [T2Space α] (b : γ) (hf : Summable f) : ∑' i, b • f i = b • ∑' i, f i 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [Monoid γ] [TopologicalSpace α] [AddCommMonoid α] [DistribMulAction γ α] [ContinuousConstSMul γ α] {f : β → α} [T2Space α] (b : γ) (hf : Summable f)
Goal: ∑' (i : β), b • f i = b • ∑' (i : β), f i
-- tsum_const_smul'
lemma tsum_const_smul' {γ : Type*} [Group γ] [DistribMulAction γ α] [ContinuousConstSMul γ α]
    [T2Space α] (g : γ) : ∑' (i : β), g • f i = g • ∑' (i : β), f i 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [AddCommMonoid α] {f : β → α} {γ : Type u_5} [Group γ] [DistribMulAction γ α] [ContinuousConstSMul γ α] [T2Space α] (g : γ)
Goal: ∑' (i : β), g • f i = g • ∑' (i : β), f i
-- tsum_const_smul''
lemma tsum_const_smul'' {γ : Type*} [DivisionRing γ] [Module γ α] [ContinuousConstSMul γ α]
    [T2Space α] (g : γ) : ∑' (i : β), g • f i = g • ∑' (i : β), f i 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [AddCommMonoid α] {f : β → α} {γ : Type u_5} [DivisionRing γ] [Module γ α] [ContinuousConstSMul γ α] [T2Space α] (g : γ)
Goal: ∑' (i : β), g • f i = g • ∑' (i : β), f i
-- HasSum.smul_const
theorem HasSum.smul_const {r : R} (hf : HasSum f r) (a : M) : HasSum (fun z ↦ f z • a) (r • a) 
Arguments: {ι : Type u_5} {R : Type u_7} {M : Type u_9} [Semiring R] [TopologicalSpace R] [TopologicalSpace M] [AddCommMonoid M] [Module R M] [ContinuousSMul R M] {f : ι → R} {r : R} (hf : HasSum f r) (a : M)
Goal: HasSum (fun z => f z • a) (r • a)
-- Summable.smul_const
theorem Summable.smul_const (hf : Summable f) (a : M) : Summable fun z ↦ f z • a 
Arguments: {ι : Type u_5} {R : Type u_7} {M : Type u_9} [Semiring R] [TopologicalSpace R] [TopologicalSpace M] [AddCommMonoid M] [Module R M] [ContinuousSMul R M] {f : ι → R} (hf : Summable f) (a : M)
Goal: Summable fun z => f z • a
-- tsum_smul_const
theorem tsum_smul_const [T2Space M] (hf : Summable f) (a : M) : ∑' z, f z • a = (∑' z, f z) • a 
Arguments: {ι : Type u_5} {R : Type u_7} {M : Type u_9} [Semiring R] [TopologicalSpace R] [TopologicalSpace M] [AddCommMonoid M] [Module R M] [ContinuousSMul R M] {f : ι → R} [T2Space M] (hf : Summable f) (a : M)
Goal: ∑' (z : ι), f z • a = (∑' (z : ι), f z) • a
-- HasSum.smul_eq
theorem HasSum.smul_eq (hf : HasSum f s) (hg : HasSum g t)
    (hfg : HasSum (fun x : ι × κ ↦ f x.1 • g x.2) u) : s • t = u 
Arguments: {ι : Type u_5} {κ : Type u_6} {R : Type u_7} {M : Type u_9} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace R] [TopologicalSpace M] [T3Space M] [ContinuousAdd M] [ContinuousSMul R M] {f : ι → R} {g : κ → M} {s : R} {t u : M} (hf : HasSum f s) (hg : HasSum g t) (hfg : HasSum (fun x => f x.1 • g x.2) u)
Goal: s • t = u
-- HasSum.smul
theorem HasSum.smul (hf : HasSum f s) (hg : HasSum g t)
    (hfg : Summable fun x : ι × κ ↦ f x.1 • g x.2) :
    HasSum (fun x : ι × κ ↦ f x.1 • g x.2) (s • t) 
Arguments: {ι : Type u_5} {κ : Type u_6} {R : Type u_7} {M : Type u_9} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace R] [TopologicalSpace M] [T3Space M] [ContinuousAdd M] [ContinuousSMul R M] {f : ι → R} {g : κ → M} {s : R} {t : M} (hf : HasSum f s) (hg : HasSum g t) (hfg : Summable fun x => f x.1 • g x.2)
Goal: HasSum (fun x => f x.1 • g x.2) (s • t)
-- tsum_smul_tsum
theorem tsum_smul_tsum (hf : Summable f) (hg : Summable g)
    (hfg : Summable fun x : ι × κ ↦ f x.1 • g x.2) :
    ((∑' x, f x) • ∑' y, g y) = ∑' z : ι × κ, f z.1 • g z.2 
Arguments: {ι : Type u_5} {κ : Type u_6} {R : Type u_7} {M : Type u_9} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace R] [TopologicalSpace M] [T3Space M] [ContinuousAdd M] [ContinuousSMul R M] {f : ι → R} {g : κ → M} (hf : Summable f) (hg : Summable g) (hfg : Summable fun x => f x.1 • g x.2)
Goal: (∑' (x : ι), f x) • ∑' (y : κ), g y = ∑' (z : ι × κ), f z.1 • g z.2
-- ContinuousLinearMap.hasSum
protected theorem ContinuousLinearMap.hasSum {f : ι → M} (φ : M →SL[σ] M₂) {x : M}
    (hf : HasSum f x) : HasSum (fun b : ι ↦ φ (f b)) (φ x) 
Arguments: {ι : Type u_5} {R : Type u_7} {R₂ : Type u_8} {M : Type u_9} {M₂ : Type u_10} [Semiring R] [Semiring R₂] [AddCommMonoid M] [Module R M] [AddCommMonoid M₂] [Module R₂ M₂] [TopologicalSpace M] [TopologicalSpace M₂] {σ : R →+* R₂} {f : ι → M} (φ : M →SL[σ] M₂) {x : M} (hf : HasSum f x)
Goal: HasSum (fun b => φ (f b)) (φ x)
-- ContinuousLinearMap.summable
protected theorem ContinuousLinearMap.summable {f : ι → M} (φ : M →SL[σ] M₂) (hf : Summable f) :
    Summable fun b : ι ↦ φ (f b) 
Arguments: {ι : Type u_5} {R : Type u_7} {R₂ : Type u_8} {M : Type u_9} {M₂ : Type u_10} [Semiring R] [Semiring R₂] [AddCommMonoid M] [Module R M] [AddCommMonoid M₂] [Module R₂ M₂] [TopologicalSpace M] [TopologicalSpace M₂] {σ : R →+* R₂} {f : ι → M} (φ : M →SL[σ] M₂) (hf : Summable f)
Goal: Summable fun b => φ (f b)
-- ContinuousLinearMap.map_tsum
protected theorem ContinuousLinearMap.map_tsum [T2Space M₂] {f : ι → M} (φ : M →SL[σ] M₂)
    (hf : Summable f) : φ (∑' z, f z) = ∑' z, φ (f z) 
Arguments: {ι : Type u_5} {R : Type u_7} {R₂ : Type u_8} {M : Type u_9} {M₂ : Type u_10} [Semiring R] [Semiring R₂] [AddCommMonoid M] [Module R M] [AddCommMonoid M₂] [Module R₂ M₂] [TopologicalSpace M] [TopologicalSpace M₂] {σ : R →+* R₂} [T2Space M₂] {f : ι → M} (φ : M →SL[σ] M₂) (hf : Summable f)
Goal: φ (∑' (z : ι), f z) = ∑' (z : ι), φ (f z)
-- ContinuousLinearEquiv.hasSum
protected theorem ContinuousLinearEquiv.hasSum {f : ι → M} (e : M ≃SL[σ] M₂) {y : M₂} :
    HasSum (fun b : ι ↦ e (f b)) y ↔ HasSum f (e.symm y) 
Arguments: {ι : Type u_5} {R : Type u_7} {R₂ : Type u_8} {M : Type u_9} {M₂ : Type u_10} [Semiring R] [Semiring R₂] [AddCommMonoid M] [Module R M] [AddCommMonoid M₂] [Module R₂ M₂] [TopologicalSpace M] [TopologicalSpace M₂] {σ : R →+* R₂} {σ' : R₂ →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] {f : ι → M} (e : M ≃SL[σ] M₂) {y : M₂}
Goal: HasSum (fun b => e (f b)) y ↔ HasSum f (e.symm y)
-- ContinuousLinearEquiv.hasSum'
protected theorem ContinuousLinearEquiv.hasSum' {f : ι → M} (e : M ≃SL[σ] M₂) {x : M} :
    HasSum (fun b : ι ↦ e (f b)) (e x) ↔ HasSum f x 
Arguments: {ι : Type u_5} {R : Type u_7} {R₂ : Type u_8} {M : Type u_9} {M₂ : Type u_10} [Semiring R] [Semiring R₂] [AddCommMonoid M] [Module R M] [AddCommMonoid M₂] [Module R₂ M₂] [TopologicalSpace M] [TopologicalSpace M₂] {σ : R →+* R₂} {σ' : R₂ →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] {f : ι → M} (e : M ≃SL[σ] M₂) {x : M}
Goal: HasSum (fun b => e (f b)) (e x) ↔ HasSum f x
-- ContinuousLinearEquiv.summable
protected theorem ContinuousLinearEquiv.summable {f : ι → M} (e : M ≃SL[σ] M₂) :
    (Summable fun b : ι ↦ e (f b)) ↔ Summable f 
Arguments: {ι : Type u_5} {R : Type u_7} {R₂ : Type u_8} {M : Type u_9} {M₂ : Type u_10} [Semiring R] [Semiring R₂] [AddCommMonoid M] [Module R M] [AddCommMonoid M₂] [Module R₂ M₂] [TopologicalSpace M] [TopologicalSpace M₂] {σ : R →+* R₂} {σ' : R₂ →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] {f : ι → M} (e : M ≃SL[σ] M₂)
Goal: (Summable fun b => e (f b)) ↔ Summable f
-- ContinuousLinearEquiv.tsum_eq_iff
theorem ContinuousLinearEquiv.tsum_eq_iff [T2Space M] [T2Space M₂] {f : ι → M} (e : M ≃SL[σ] M₂)
    {y : M₂} : (∑' z, e (f z)) = y ↔ ∑' z, f z = e.symm y 
Arguments: {ι : Type u_5} {R : Type u_7} {R₂ : Type u_8} {M : Type u_9} {M₂ : Type u_10} [Semiring R] [Semiring R₂] [AddCommMonoid M] [Module R M] [AddCommMonoid M₂] [Module R₂ M₂] [TopologicalSpace M] [TopologicalSpace M₂] {σ : R →+* R₂} {σ' : R₂ →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] [T2Space M] [T2Space M₂] {f : ι → M} (e : M ≃SL[σ] M₂) {y : M₂}
Goal: ∑' (z : ι), e (f z) = y ↔ ∑' (z : ι), f z = e.symm y
-- ContinuousLinearEquiv.map_tsum
protected theorem ContinuousLinearEquiv.map_tsum [T2Space M] [T2Space M₂] {f : ι → M}
    (e : M ≃SL[σ] M₂) : e (∑' z, f z) = ∑' z, e (f z) 
Arguments: {ι : Type u_5} {R : Type u_7} {R₂ : Type u_8} {M : Type u_9} {M₂ : Type u_10} [Semiring R] [Semiring R₂] [AddCommMonoid M] [Module R M] [AddCommMonoid M₂] [Module R₂ M₂] [TopologicalSpace M] [TopologicalSpace M₂] {σ : R →+* R₂} {σ' : R₂ →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] [T2Space M] [T2Space M₂] {f : ι → M} (e : M ≃SL[σ] M₂)
Goal: e (∑' (z : ι), f z) = ∑' (z : ι), e (f z)
-- MulAction.automorphize_smul_left
lemma MulAction.automorphize_smul_left [Group α] [MulAction α β] (f : β → M)
    (g : Quotient (MulAction.orbitRel α β) → R) :
    MulAction.automorphize ((g ∘ (@Quotient.mk' _ (_))) • f)
      = g • (MulAction.automorphize f : Quotient (MulAction.orbitRel α β) → M) 
Arguments: {α : Type u_1} {β : Type u_2} {M : Type u_11} [TopologicalSpace M] [AddCommMonoid M] [T2Space M] {R : Type u_12} [DivisionRing R] [Module R M] [ContinuousConstSMul R M] [Group α] [MulAction α β] (f : β → M) (g : Quotient (MulAction.orbitRel α β) → R)
Goal: MulAction.automorphize (g ∘ Quotient.mk' • f) = g • MulAction.automorphize f
-- AddAction.automorphize_smul_left
lemma AddAction.automorphize_smul_left [AddGroup α] [AddAction α β]  (f : β → M)
    (g : Quotient (AddAction.orbitRel α β) → R) :
    AddAction.automorphize ((g ∘ (@Quotient.mk' _ (_))) • f)
      = g • (AddAction.automorphize f : Quotient (AddAction.orbitRel α β) → M) 
Arguments: {α : Type u_1} {β : Type u_2} {M : Type u_11} [TopologicalSpace M] [AddCommMonoid M] [T2Space M] {R : Type u_12} [DivisionRing R] [Module R M] [ContinuousConstSMul R M] [AddGroup α] [AddAction α β] (f : β → M) (g : Quotient (AddAction.orbitRel α β) → R)
Goal: AddAction.automorphize (g ∘ Quotient.mk' • f) = g • AddAction.automorphize f
-- QuotientGroup.automorphize_smul_left
lemma QuotientGroup.automorphize_smul_left (f : G → M) (g : G ⧸ Γ → R) :
    (QuotientGroup.automorphize ((g ∘ (@Quotient.mk' _ (_)) : G → R) • f) : G ⧸ Γ → M)
      = g • (QuotientGroup.automorphize f : G ⧸ Γ → M) 
Arguments: {M : Type u_11} [TopologicalSpace M] [AddCommMonoid M] [T2Space M] {R : Type u_12} [DivisionRing R] [Module R M] [ContinuousConstSMul R M] {G : Type u_13} [Group G] {Γ : Subgroup G} (f : G → M) (g : G ⧸ Γ → R)
Goal: QuotientGroup.automorphize (g ∘ Quotient.mk' • f) = g • QuotientGroup.automorphize f
-- QuotientAddGroup.automorphize_smul_left
lemma QuotientAddGroup.automorphize_smul_left (f : G → M) (g : G ⧸ Γ → R) :
    QuotientAddGroup.automorphize ((g ∘ (@Quotient.mk' _ (_))) • f)
      = g • (QuotientAddGroup.automorphize f : G ⧸ Γ → M) 
Arguments: {M : Type u_11} [TopologicalSpace M] [AddCommMonoid M] [T2Space M] {R : Type u_12} [DivisionRing R] [Module R M] [ContinuousConstSMul R M] {G : Type u_13} [AddGroup G] {Γ : AddSubgroup G} (f : G → M) (g : G ⧸ Γ → R)
Goal: QuotientAddGroup.automorphize (g ∘ Quotient.mk' • f) = g • QuotientAddGroup.automorphize f
-- BoundedContinuousFunction.map_bounded'
structure BoundedContinuousFunction (α : Type u) (β : Type v) [TopologicalSpace α]
    [PseudoMetricSpace β] extends ContinuousMap α β : Type max u v where
  map_bounded' : ∃ C, ∀ x y, dist (toFun x) (toFun y) ≤ C
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] (self : BoundedContinuousFunction α β)
Goal: ∃ C, ∀ (x y : α), dist (self.toFun x) (self.toFun y) ≤ C
-- BoundedContinuousMapClass.map_bounded
class BoundedContinuousMapClass (F : Type*) (α β : outParam Type*) [TopologicalSpace α]
    [PseudoMetricSpace β] [FunLike F α β] extends ContinuousMapClass F α β : Prop where
  map_bounded (f : F) : ∃ C, ∀ x y, dist (f x) (f y) ≤ C
Arguments: {F : Type u_2} {α : outParam (Type u_3)} {β : outParam (Type u_4)} [TopologicalSpace α] [PseudoMetricSpace β] [FunLike F α β] [self : BoundedContinuousMapClass F α β] (f : F)
Goal: ∃ C, ∀ (x y : α), dist (f x) (f y) ≤ C
-- BoundedContinuousFunction.coe_to_continuous_fun
@[simp]
theorem coe_to_continuous_fun (f : α →ᵇ β) : (f.toContinuousMap : α → β) = f 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] (f : BoundedContinuousFunction α β)
Goal: ⇑f.toContinuousMap = ⇑f
-- BoundedContinuousFunction.bounded
protected theorem bounded (f : α →ᵇ β) : ∃ C, ∀ x y : α, dist (f x) (f y) ≤ C 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] (f : BoundedContinuousFunction α β)
Goal: ∃ C, ∀ (x y : α), dist (f x) (f y) ≤ C
-- BoundedContinuousFunction.continuous
protected theorem continuous (f : α →ᵇ β) : Continuous f 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] (f : BoundedContinuousFunction α β)
Goal: Continuous ⇑f
-- BoundedContinuousFunction.ext
@[ext]
theorem ext (h : ∀ x, f x = g x) : f = g 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {f g : BoundedContinuousFunction α β} (h : ∀ (x : α), f x = g x)
Goal: f = g
-- BoundedContinuousFunction.isBounded_range
theorem isBounded_range (f : α →ᵇ β) : IsBounded (range f) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] (f : BoundedContinuousFunction α β)
Goal: Bornology.IsBounded (Set.range ⇑f)
-- BoundedContinuousFunction.isBounded_image
theorem isBounded_image (f : α →ᵇ β) (s : Set α) : IsBounded (f '' s) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] (f : BoundedContinuousFunction α β) (s : Set α)
Goal: Bornology.IsBounded (⇑f '' s)
-- BoundedContinuousFunction.eq_of_empty
theorem eq_of_empty [h : IsEmpty α] (f g : α →ᵇ β) : f = g 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [h : IsEmpty α] (f g : BoundedContinuousFunction α β)
Goal: f = g
-- BoundedContinuousFunction.mkOfBound_coe
@[simp]
theorem mkOfBound_coe {f} {C} {h} : (mkOfBound f C h : α → β) = (f : α → β) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {f : C(α, β)} {C : ℝ} {h : ∀ (x y : α), dist (f x) (f y) ≤ C}
Goal: ⇑(BoundedContinuousFunction.mkOfBound f C h) = ⇑f
-- BoundedContinuousFunction.mkOfCompact_apply
@[simp]
theorem mkOfCompact_apply [CompactSpace α] (f : C(α, β)) (a : α) : mkOfCompact f a = f a 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [CompactSpace α] (f : C(α, β)) (a : α)
Goal: (BoundedContinuousFunction.mkOfCompact f) a = f a
-- BoundedContinuousFunction.mkOfDiscrete_toFun
@[simps]
def mkOfDiscrete [DiscreteTopology α] (f : α → β) (C : ℝ) (h : ∀ x y : α, dist (f x) (f y) ≤ C) :
    α →ᵇ β 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [DiscreteTopology α] (f : α → β) (C : ℝ) (h : ∀ (x y : α), dist (f x) (f y) ≤ C)
Goal: ∀ (a : α), (BoundedContinuousFunction.mkOfDiscrete f C h) a = f a
-- BoundedContinuousFunction.mkOfDiscrete_apply
@[simps]
def mkOfDiscrete [DiscreteTopology α] (f : α → β) (C : ℝ) (h : ∀ x y : α, dist (f x) (f y) ≤ C) :
    α →ᵇ β 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [DiscreteTopology α] (f : α → β) (C : ℝ) (h : ∀ (x y : α), dist (f x) (f y) ≤ C)
Goal: ∀ (a : α), (BoundedContinuousFunction.mkOfDiscrete f C h) a = f a
-- BoundedContinuousFunction.dist_eq
theorem dist_eq : dist f g = sInf { C | 0 ≤ C ∧ ∀ x : α, dist (f x) (g x) ≤ C } 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {f g : BoundedContinuousFunction α β}
Goal: dist f g = sInf {C | 0 ≤ C ∧ ∀ (x : α), dist (f x) (g x) ≤ C}
-- BoundedContinuousFunction.dist_set_exists
theorem dist_set_exists : ∃ C, 0 ≤ C ∧ ∀ x : α, dist (f x) (g x) ≤ C 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {f g : BoundedContinuousFunction α β}
Goal: ∃ C, 0 ≤ C ∧ ∀ (x : α), dist (f x) (g x) ≤ C
-- BoundedContinuousFunction.dist_coe_le_dist
theorem dist_coe_le_dist (x : α) : dist (f x) (g x) ≤ dist f g 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {f g : BoundedContinuousFunction α β} (x : α)
Goal: dist (f x) (g x) ≤ dist f g
-- _private.Mathlib.Topology.ContinuousFunction.Bounded.0.BoundedContinuousFunction.dist_nonneg'
private theorem dist_nonneg' : 0 ≤ dist f g 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {f g : BoundedContinuousFunction α β}
Goal: 0 ≤ dist f g
-- BoundedContinuousFunction.dist_le
theorem dist_le (C0 : (0 : ℝ) ≤ C) : dist f g ≤ C ↔ ∀ x : α, dist (f x) (g x) ≤ C 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {f g : BoundedContinuousFunction α β} {C : ℝ} (C0 : 0 ≤ C)
Goal: dist f g ≤ C ↔ ∀ (x : α), dist (f x) (g x) ≤ C
-- BoundedContinuousFunction.dist_le_iff_of_nonempty
theorem dist_le_iff_of_nonempty [Nonempty α] : dist f g ≤ C ↔ ∀ x, dist (f x) (g x) ≤ C 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {f g : BoundedContinuousFunction α β} {C : ℝ} [Nonempty α]
Goal: dist f g ≤ C ↔ ∀ (x : α), dist (f x) (g x) ≤ C
-- BoundedContinuousFunction.dist_lt_of_nonempty_compact
theorem dist_lt_of_nonempty_compact [Nonempty α] [CompactSpace α]
    (w : ∀ x : α, dist (f x) (g x) < C) : dist f g < C 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {f g : BoundedContinuousFunction α β} {C : ℝ} [Nonempty α] [CompactSpace α] (w : ∀ (x : α), dist (f x) (g x) < C)
Goal: dist f g < C
-- BoundedContinuousFunction.dist_lt_iff_of_compact
theorem dist_lt_iff_of_compact [CompactSpace α] (C0 : (0 : ℝ) < C) :
    dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {f g : BoundedContinuousFunction α β} {C : ℝ} [CompactSpace α] (C0 : 0 < C)
Goal: dist f g < C ↔ ∀ (x : α), dist (f x) (g x) < C
-- BoundedContinuousFunction.dist_lt_iff_of_nonempty_compact
theorem dist_lt_iff_of_nonempty_compact [Nonempty α] [CompactSpace α] :
    dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {f g : BoundedContinuousFunction α β} {C : ℝ} [Nonempty α] [CompactSpace α]
Goal: dist f g < C ↔ ∀ (x : α), dist (f x) (g x) < C
-- BoundedContinuousFunction.nndist_eq
theorem nndist_eq : nndist f g = sInf { C | ∀ x : α, nndist (f x) (g x) ≤ C } 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {f g : BoundedContinuousFunction α β}
Goal: nndist f g = sInf {C | ∀ (x : α), nndist (f x) (g x) ≤ C}
-- BoundedContinuousFunction.nndist_set_exists
theorem nndist_set_exists : ∃ C, ∀ x : α, nndist (f x) (g x) ≤ C 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {f g : BoundedContinuousFunction α β}
Goal: ∃ C, ∀ (x : α), nndist (f x) (g x) ≤ C
-- BoundedContinuousFunction.nndist_coe_le_nndist
theorem nndist_coe_le_nndist (x : α) : nndist (f x) (g x) ≤ nndist f g 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {f g : BoundedContinuousFunction α β} (x : α)
Goal: nndist (f x) (g x) ≤ nndist f g
-- BoundedContinuousFunction.dist_zero_of_empty
theorem dist_zero_of_empty [IsEmpty α] : dist f g = 0 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {f g : BoundedContinuousFunction α β} [IsEmpty α]
Goal: dist f g = 0
-- BoundedContinuousFunction.dist_eq_iSup
theorem dist_eq_iSup : dist f g = ⨆ x : α, dist (f x) (g x) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {f g : BoundedContinuousFunction α β}
Goal: dist f g = ⨆ x, dist (f x) (g x)
-- BoundedContinuousFunction.nndist_eq_iSup
theorem nndist_eq_iSup : nndist f g = ⨆ x : α, nndist (f x) (g x) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {f g : BoundedContinuousFunction α β}
Goal: nndist f g = ⨆ x, nndist (f x) (g x)
-- BoundedContinuousFunction.tendsto_iff_tendstoUniformly
theorem tendsto_iff_tendstoUniformly {ι : Type*} {F : ι → α →ᵇ β} {f : α →ᵇ β} {l : Filter ι} :
    Tendsto F l (𝓝 f) ↔ TendstoUniformly (fun i => F i) f l 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] {ι : Type u_2} {F : ι → BoundedContinuousFunction α β} {f : BoundedContinuousFunction α β} {l : Filter ι}
Goal: Filter.Tendsto F l (𝓝 f) ↔ TendstoUniformly (fun i => ⇑(F i)) (⇑f) l
-- BoundedContinuousFunction.inducing_coeFn
theorem inducing_coeFn : Inducing (UniformFun.ofFun ∘ (⇑) : (α →ᵇ β) → α →ᵤ β) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β]
Goal: Inducing (⇑UniformFun.ofFun ∘ DFunLike.coe)
-- BoundedContinuousFunction.embedding_coeFn
theorem embedding_coeFn : Embedding (UniformFun.ofFun ∘ (⇑) : (α →ᵇ β) → α →ᵤ β) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β]
Goal: Embedding (⇑UniformFun.ofFun ∘ DFunLike.coe)
-- BoundedContinuousFunction.const_apply
@[simps! (config := .asFn)] -- Porting note: Changed `simps` to `simps!`
def const (b : β) : α →ᵇ β 
Arguments: (α : Type u) {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] (b : β)
Goal: ⇑(BoundedContinuousFunction.const α b) = fun x => b
-- BoundedContinuousFunction.const_toFun
@[simps! (config := .asFn)] -- Porting note: Changed `simps` to `simps!`
def const (b : β) : α →ᵇ β 
Arguments: (α : Type u) {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] (b : β)
Goal: ⇑(BoundedContinuousFunction.const α b) = fun x => b
-- BoundedContinuousFunction.const_apply'
theorem const_apply' (a : α) (b : β) : (const α b : α → β) a = b 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] (a : α) (b : β)
Goal: (BoundedContinuousFunction.const α b) a = b