ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- PiTensorProduct.mapL_add
protected theorem mapL_add [DecidableEq ι] (i : ι) (u v : E i →L[𝕜] E' i) :
    mapL (update f i (u + v)) = mapL (update f i u) + mapL (update f i v) 
Arguments: {ι : Type uι} [Fintype ι] {𝕜 : Type u𝕜} [NontriviallyNormedField 𝕜] {E : ι → Type uE} [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] {E' : ι → Type u_1} [(i : ι) → SeminormedAddCommGroup (E' i)] [(i : ι) → NormedSpace 𝕜 (E' i)] (f : (i : ι) → E i →L[𝕜] E' i) [DecidableEq ι] (i : ι) (u v : E i →L[𝕜] E' i)
Goal: PiTensorProduct.mapL (Function.update f i (u + v)) =
  PiTensorProduct.mapL (Function.update f i u) + PiTensorProduct.mapL (Function.update f i v)
-- PiTensorProduct.mapL_smul
protected theorem mapL_smul [DecidableEq ι] (i : ι) (c : 𝕜) (u : E i →L[𝕜] E' i) :
    mapL (update f i (c • u)) = c • mapL (update f i u) 
Arguments: {ι : Type uι} [Fintype ι] {𝕜 : Type u𝕜} [NontriviallyNormedField 𝕜] {E : ι → Type uE} [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] {E' : ι → Type u_1} [(i : ι) → SeminormedAddCommGroup (E' i)] [(i : ι) → NormedSpace 𝕜 (E' i)] (f : (i : ι) → E i →L[𝕜] E' i) [DecidableEq ι] (i : ι) (c : 𝕜) (u : E i →L[𝕜] E' i)
Goal: PiTensorProduct.mapL (Function.update f i (c • u)) = c • PiTensorProduct.mapL (Function.update f i u)
-- PiTensorProduct.mapL_opNorm
theorem mapL_opNorm : ‖mapL f‖ ≤ ∏ i, ‖f i‖ 
Arguments: {ι : Type uι} [Fintype ι] {𝕜 : Type u𝕜} [NontriviallyNormedField 𝕜] {E : ι → Type uE} [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] {E' : ι → Type u_1} [(i : ι) → SeminormedAddCommGroup (E' i)] [(i : ι) → NormedSpace 𝕜 (E' i)] (f : (i : ι) → E i →L[𝕜] E' i)
Goal: ‖PiTensorProduct.mapL f‖ ≤ ∏ i : ι, ‖f i‖
-- PiTensorProduct.mapLMultilinear_toFun_toFun
@[simps!]
noncomputable def mapLMultilinear : ContinuousMultilinearMap 𝕜 (fun (i : ι) ↦ E i →L[𝕜] E' i)
    ((⨂[𝕜] i, E i) →L[𝕜] ⨂[𝕜] i, E' i) 
Arguments: {ι : Type uι} [Fintype ι] (𝕜 : Type u𝕜) [NontriviallyNormedField 𝕜] (E : ι → Type uE) [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] (E' : ι → Type u_1) [(i : ι) → SeminormedAddCommGroup (E' i)] [(i : ι) → NormedSpace 𝕜 (E' i)] (f : (i : ι) → E i →L[𝕜] E' i)
Goal: ∀ (a : ⨂[𝕜] (i : ι), E i),
  ((PiTensorProduct.mapLMultilinear 𝕜 E E') f) a =
    (PiTensorProduct.liftAux ((PiTensorProduct.tprodL 𝕜).compContinuousLinearMap f).toMultilinearMap) a
-- PiTensorProduct.mapLMultilinear_apply_apply
@[simps!]
noncomputable def mapLMultilinear : ContinuousMultilinearMap 𝕜 (fun (i : ι) ↦ E i →L[𝕜] E' i)
    ((⨂[𝕜] i, E i) →L[𝕜] ⨂[𝕜] i, E' i) 
Arguments: {ι : Type uι} [Fintype ι] (𝕜 : Type u𝕜) [NontriviallyNormedField 𝕜] (E : ι → Type uE) [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] (E' : ι → Type u_1) [(i : ι) → SeminormedAddCommGroup (E' i)] [(i : ι) → NormedSpace 𝕜 (E' i)] (f : (i : ι) → E i →L[𝕜] E' i)
Goal: ∀ (a : ⨂[𝕜] (i : ι), E i),
  ((PiTensorProduct.mapLMultilinear 𝕜 E E') f) a =
    (PiTensorProduct.liftAux ((PiTensorProduct.tprodL 𝕜).compContinuousLinearMap f).toMultilinearMap) a
-- PiTensorProduct.mapLMultilinear_apply_toFun
@[simps!]
noncomputable def mapLMultilinear : ContinuousMultilinearMap 𝕜 (fun (i : ι) ↦ E i →L[𝕜] E' i)
    ((⨂[𝕜] i, E i) →L[𝕜] ⨂[𝕜] i, E' i) 
Arguments: {ι : Type uι} [Fintype ι] (𝕜 : Type u𝕜) [NontriviallyNormedField 𝕜] (E : ι → Type uE) [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] (E' : ι → Type u_1) [(i : ι) → SeminormedAddCommGroup (E' i)] [(i : ι) → NormedSpace 𝕜 (E' i)] (f : (i : ι) → E i →L[𝕜] E' i)
Goal: ∀ (a : ⨂[𝕜] (i : ι), E i),
  ((PiTensorProduct.mapLMultilinear 𝕜 E E') f) a =
    (PiTensorProduct.liftAux ((PiTensorProduct.tprodL 𝕜).compContinuousLinearMap f).toMultilinearMap) a
-- PiTensorProduct.mapLMultilinear_toFun_apply
@[simps!]
noncomputable def mapLMultilinear : ContinuousMultilinearMap 𝕜 (fun (i : ι) ↦ E i →L[𝕜] E' i)
    ((⨂[𝕜] i, E i) →L[𝕜] ⨂[𝕜] i, E' i) 
Arguments: {ι : Type uι} [Fintype ι] (𝕜 : Type u𝕜) [NontriviallyNormedField 𝕜] (E : ι → Type uE) [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] (E' : ι → Type u_1) [(i : ι) → SeminormedAddCommGroup (E' i)] [(i : ι) → NormedSpace 𝕜 (E' i)] (f : (i : ι) → E i →L[𝕜] E' i)
Goal: ∀ (a : ⨂[𝕜] (i : ι), E i),
  ((PiTensorProduct.mapLMultilinear 𝕜 E E') f) a =
    (PiTensorProduct.liftAux ((PiTensorProduct.tprodL 𝕜).compContinuousLinearMap f).toMultilinearMap) a
-- PiTensorProduct.mapLMultilinear_opNorm
theorem mapLMultilinear_opNorm : ‖mapLMultilinear 𝕜 E E'‖ ≤ 1 
Arguments: {ι : Type uι} [Fintype ι] {𝕜 : Type u𝕜} [NontriviallyNormedField 𝕜] {E : ι → Type uE} [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] {E' : ι → Type u_1} [(i : ι) → SeminormedAddCommGroup (E' i)] [(i : ι) → NormedSpace 𝕜 (E' i)]
Goal: ‖PiTensorProduct.mapLMultilinear 𝕜 E E'‖ ≤ 1
-- IsBoundedLinearMap.bound
structure IsBoundedLinearMap (𝕜 : Type*) [NormedField 𝕜] {E : Type*} [NormedAddCommGroup E]
  [NormedSpace 𝕜 E] {F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (f : E → F) extends
  IsLinearMap 𝕜 f : Prop where
  bound : ∃ M, 0 < M ∧ ∀ x : E, ‖f x‖ ≤ M * ‖x‖
Arguments: {𝕜 : Type u_5} [NormedField 𝕜] {E : Type u_6} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_7} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} (self : IsBoundedLinearMap 𝕜 f)
Goal: ∃ M, 0 < M ∧ ∀ (x : E), ‖f x‖ ≤ M * ‖x‖
-- IsLinearMap.with_bound
theorem IsLinearMap.with_bound {f : E → F} (hf : IsLinearMap 𝕜 f) (M : ℝ)
    (h : ∀ x : E, ‖f x‖ ≤ M * ‖x‖) : IsBoundedLinearMap 𝕜 f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} (hf : IsLinearMap 𝕜 f) (M : ℝ) (h : ∀ (x : E), ‖f x‖ ≤ M * ‖x‖)
Goal: IsBoundedLinearMap 𝕜 f
-- ContinuousLinearMap.isBoundedLinearMap
theorem ContinuousLinearMap.isBoundedLinearMap (f : E →L[𝕜] F) : IsBoundedLinearMap 𝕜 f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (f : E →L[𝕜] F)
Goal: IsBoundedLinearMap 𝕜 ⇑f
-- IsBoundedLinearMap.zero
theorem zero : IsBoundedLinearMap 𝕜 fun _ : E => (0 : F) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
Goal: IsBoundedLinearMap 𝕜 fun x => 0
-- IsBoundedLinearMap.id
theorem id : IsBoundedLinearMap 𝕜 fun x : E => x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
Goal: IsBoundedLinearMap 𝕜 fun x => x
-- IsBoundedLinearMap.fst
theorem fst : IsBoundedLinearMap 𝕜 fun x : E × F => x.1 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
Goal: IsBoundedLinearMap 𝕜 fun x => x.1
-- IsBoundedLinearMap.snd
theorem snd : IsBoundedLinearMap 𝕜 fun x : E × F => x.2 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
Goal: IsBoundedLinearMap 𝕜 fun x => x.2
-- IsBoundedLinearMap.smul
theorem smul (c : 𝕜) (hf : IsBoundedLinearMap 𝕜 f) : IsBoundedLinearMap 𝕜 (c • f) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} (c : 𝕜) (hf : IsBoundedLinearMap 𝕜 f)
Goal: IsBoundedLinearMap 𝕜 (c • f)
-- IsBoundedLinearMap.neg
theorem neg (hf : IsBoundedLinearMap 𝕜 f) : IsBoundedLinearMap 𝕜 fun e => -f e 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} (hf : IsBoundedLinearMap 𝕜 f)
Goal: IsBoundedLinearMap 𝕜 fun e => -f e
-- IsBoundedLinearMap.add
theorem add (hf : IsBoundedLinearMap 𝕜 f) (hg : IsBoundedLinearMap 𝕜 g) :
    IsBoundedLinearMap 𝕜 fun e => f e + g e 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f g : E → F} (hf : IsBoundedLinearMap 𝕜 f) (hg : IsBoundedLinearMap 𝕜 g)
Goal: IsBoundedLinearMap 𝕜 fun e => f e + g e
-- IsBoundedLinearMap.sub
theorem sub (hf : IsBoundedLinearMap 𝕜 f) (hg : IsBoundedLinearMap 𝕜 g) :
    IsBoundedLinearMap 𝕜 fun e => f e - g e 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f g : E → F} (hf : IsBoundedLinearMap 𝕜 f) (hg : IsBoundedLinearMap 𝕜 g)
Goal: IsBoundedLinearMap 𝕜 fun e => f e - g e
-- IsBoundedLinearMap.comp
theorem comp {g : F → G} (hg : IsBoundedLinearMap 𝕜 g) (hf : IsBoundedLinearMap 𝕜 f) :
    IsBoundedLinearMap 𝕜 (g ∘ f) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E → F} {g : F → G} (hg : IsBoundedLinearMap 𝕜 g) (hf : IsBoundedLinearMap 𝕜 f)
Goal: IsBoundedLinearMap 𝕜 (g ∘ f)
-- IsBoundedLinearMap.tendsto
protected theorem tendsto (x : E) (hf : IsBoundedLinearMap 𝕜 f) : Tendsto f (𝓝 x) (𝓝 (f x)) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} (x : E) (hf : IsBoundedLinearMap 𝕜 f)
Goal: Filter.Tendsto f (𝓝 x) (𝓝 (f x))
-- IsBoundedLinearMap.continuous
theorem continuous (hf : IsBoundedLinearMap 𝕜 f) : Continuous f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} (hf : IsBoundedLinearMap 𝕜 f)
Goal: Continuous f
-- IsBoundedLinearMap.lim_zero_bounded_linear_map
theorem lim_zero_bounded_linear_map (hf : IsBoundedLinearMap 𝕜 f) : Tendsto f (𝓝 0) (𝓝 0) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} (hf : IsBoundedLinearMap 𝕜 f)
Goal: Filter.Tendsto f (𝓝 0) (𝓝 0)
-- IsBoundedLinearMap.isBigO_id
theorem isBigO_id {f : E → F} (h : IsBoundedLinearMap 𝕜 f) (l : Filter E) : f =O[l] fun x => x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} (h : IsBoundedLinearMap 𝕜 f) (l : Filter E)
Goal: f =O[l] fun x => x
-- IsBoundedLinearMap.isBigO_comp
theorem isBigO_comp {E : Type*} {g : F → G} (hg : IsBoundedLinearMap 𝕜 g) {f : E → F}
    (l : Filter E) : (fun x' => g (f x')) =O[l] f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {E : Type u_5} {g : F → G} (hg : IsBoundedLinearMap 𝕜 g) {f : E → F} (l : Filter E)
Goal: (fun x' => g (f x')) =O[l] f
-- IsBoundedLinearMap.isBigO_sub
theorem isBigO_sub {f : E → F} (h : IsBoundedLinearMap 𝕜 f) (l : Filter E) (x : E) :
    (fun x' => f (x' - x)) =O[l] fun x' => x' - x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} (h : IsBoundedLinearMap 𝕜 f) (l : Filter E) (x : E)
Goal: (fun x' => f (x' - x)) =O[l] fun x' => x' - x
-- isBoundedLinearMap_prod_multilinear
theorem isBoundedLinearMap_prod_multilinear {E : ι → Type*} [∀ i, NormedAddCommGroup (E i)]
    [∀ i, NormedSpace 𝕜 (E i)] :
    IsBoundedLinearMap 𝕜 fun p : ContinuousMultilinearMap 𝕜 E F × ContinuousMultilinearMap 𝕜 E G =>
      p.1.prod p.2 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {ι : Type u_5} [Fintype ι] {E : ι → Type u_6} [(i : ι) → NormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)]
Goal: IsBoundedLinearMap 𝕜 fun p => p.1.prod p.2
-- isBoundedLinearMap_continuousMultilinearMap_comp_linear
theorem isBoundedLinearMap_continuousMultilinearMap_comp_linear (g : G →L[𝕜] E) :
    IsBoundedLinearMap 𝕜 fun f : ContinuousMultilinearMap 𝕜 (fun _ : ι => E) F =>
      f.compContinuousLinearMap fun _ => g 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {ι : Type u_5} [Fintype ι] (g : G →L[𝕜] E)
Goal: IsBoundedLinearMap 𝕜 fun f => f.compContinuousLinearMap fun x => g
-- ContinuousLinearMap.map_add₂
theorem map_add₂ (f : M →SL[ρ₁₂] F →SL[σ₁₂] G') (x x' : M) (y : F) :
    f (x + x') y = f x y + f x' y 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {R : Type u_5} {𝕜₂ : Type u_6} {𝕜' : Type u_7} [NontriviallyNormedField 𝕜'] [NontriviallyNormedField 𝕜₂] {M : Type u_8} [TopologicalSpace M] {σ₁₂ : 𝕜 →+* 𝕜₂} {G' : Type u_9} [NormedAddCommGroup G'] [NormedSpace 𝕜₂ G'] [NormedSpace 𝕜' G'] [SMulCommClass 𝕜₂ 𝕜' G'] [Semiring R] [AddCommMonoid M] [Module R M] {ρ₁₂ : R →+* 𝕜'} (f : M →SL[ρ₁₂] F →SL[σ₁₂] G') (x x' : M) (y : F)
Goal: (f (x + x')) y = (f x) y + (f x') y
-- ContinuousLinearMap.map_zero₂
theorem map_zero₂ (f : M →SL[ρ₁₂] F →SL[σ₁₂] G') (y : F) : f 0 y = 0 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {R : Type u_5} {𝕜₂ : Type u_6} {𝕜' : Type u_7} [NontriviallyNormedField 𝕜'] [NontriviallyNormedField 𝕜₂] {M : Type u_8} [TopologicalSpace M] {σ₁₂ : 𝕜 →+* 𝕜₂} {G' : Type u_9} [NormedAddCommGroup G'] [NormedSpace 𝕜₂ G'] [NormedSpace 𝕜' G'] [SMulCommClass 𝕜₂ 𝕜' G'] [Semiring R] [AddCommMonoid M] [Module R M] {ρ₁₂ : R →+* 𝕜'} (f : M →SL[ρ₁₂] F →SL[σ₁₂] G') (y : F)
Goal: (f 0) y = 0
-- ContinuousLinearMap.map_smulₛₗ₂
theorem map_smulₛₗ₂ (f : M →SL[ρ₁₂] F →SL[σ₁₂] G') (c : R) (x : M) (y : F) :
    f (c • x) y = ρ₁₂ c • f x y 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {R : Type u_5} {𝕜₂ : Type u_6} {𝕜' : Type u_7} [NontriviallyNormedField 𝕜'] [NontriviallyNormedField 𝕜₂] {M : Type u_8} [TopologicalSpace M] {σ₁₂ : 𝕜 →+* 𝕜₂} {G' : Type u_9} [NormedAddCommGroup G'] [NormedSpace 𝕜₂ G'] [NormedSpace 𝕜' G'] [SMulCommClass 𝕜₂ 𝕜' G'] [Semiring R] [AddCommMonoid M] [Module R M] {ρ₁₂ : R →+* 𝕜'} (f : M →SL[ρ₁₂] F →SL[σ₁₂] G') (c : R) (x : M) (y : F)
Goal: (f (c • x)) y = ρ₁₂ c • (f x) y
-- ContinuousLinearMap.map_sub₂
theorem map_sub₂ (f : M →SL[ρ₁₂] F →SL[σ₁₂] G') (x x' : M) (y : F) :
    f (x - x') y = f x y - f x' y 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {R : Type u_5} {𝕜₂ : Type u_6} {𝕜' : Type u_7} [NontriviallyNormedField 𝕜'] [NontriviallyNormedField 𝕜₂] {M : Type u_8} [TopologicalSpace M] {σ₁₂ : 𝕜 →+* 𝕜₂} {G' : Type u_9} [NormedAddCommGroup G'] [NormedSpace 𝕜₂ G'] [NormedSpace 𝕜' G'] [SMulCommClass 𝕜₂ 𝕜' G'] [Ring R] [AddCommGroup M] [Module R M] {ρ₁₂ : R →+* 𝕜'} (f : M →SL[ρ₁₂] F →SL[σ₁₂] G') (x x' : M) (y : F)
Goal: (f (x - x')) y = (f x) y - (f x') y
-- ContinuousLinearMap.map_neg₂
theorem map_neg₂ (f : M →SL[ρ₁₂] F →SL[σ₁₂] G') (x : M) (y : F) : f (-x) y = -f x y 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {R : Type u_5} {𝕜₂ : Type u_6} {𝕜' : Type u_7} [NontriviallyNormedField 𝕜'] [NontriviallyNormedField 𝕜₂] {M : Type u_8} [TopologicalSpace M] {σ₁₂ : 𝕜 →+* 𝕜₂} {G' : Type u_9} [NormedAddCommGroup G'] [NormedSpace 𝕜₂ G'] [NormedSpace 𝕜' G'] [SMulCommClass 𝕜₂ 𝕜' G'] [Ring R] [AddCommGroup M] [Module R M] {ρ₁₂ : R →+* 𝕜'} (f : M →SL[ρ₁₂] F →SL[σ₁₂] G') (x : M) (y : F)
Goal: (f (-x)) y = -(f x) y
-- ContinuousLinearMap.map_smul₂
theorem map_smul₂ (f : E →L[𝕜] F →L[𝕜] G) (c : 𝕜) (x : E) (y : F) : f (c • x) y = c • f x y 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] (f : E →L[𝕜] F →L[𝕜] G) (c : 𝕜) (x : E) (y : F)
Goal: (f (c • x)) y = c • (f x) y
-- IsBoundedBilinearMap.add_right
structure IsBoundedBilinearMap (f : E × F → G) : Prop where
  add_left : ∀ (x₁ x₂ : E) (y : F), f (x₁ + x₂, y) = f (x₁, y) + f (x₂, y)
  smul_left : ∀ (c : 𝕜) (x : E) (y : F), f (c • x, y) = c • f (x, y)
  add_right : ∀ (x : E) (y₁ y₂ : F), f (x, y₁ + y₂) = f (x, y₁) + f (x, y₂)
  smul_right : ∀ (c : 𝕜) (x : E) (y : F), f (x, c • y) = c • f (x, y)
  bound : ∃ C > 0, ∀ (x : E) (y : F), ‖f (x, y)‖ ≤ C * ‖x‖ * ‖y‖
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E × F → G} (self : IsBoundedBilinearMap 𝕜 f) (x : E) (y₁ y₂ : F)
Goal: f (x, y₁ + y₂) = f (x, y₁) + f (x, y₂)
-- IsBoundedBilinearMap.smul_left
structure IsBoundedBilinearMap (f : E × F → G) : Prop where
  add_left : ∀ (x₁ x₂ : E) (y : F), f (x₁ + x₂, y) = f (x₁, y) + f (x₂, y)
  smul_left : ∀ (c : 𝕜) (x : E) (y : F), f (c • x, y) = c • f (x, y)
  add_right : ∀ (x : E) (y₁ y₂ : F), f (x, y₁ + y₂) = f (x, y₁) + f (x, y₂)
  smul_right : ∀ (c : 𝕜) (x : E) (y : F), f (x, c • y) = c • f (x, y)
  bound : ∃ C > 0, ∀ (x : E) (y : F), ‖f (x, y)‖ ≤ C * ‖x‖ * ‖y‖
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E × F → G} (self : IsBoundedBilinearMap 𝕜 f) (c : 𝕜) (x : E) (y : F)
Goal: f (c • x, y) = c • f (x, y)
-- IsBoundedBilinearMap.smul_right
structure IsBoundedBilinearMap (f : E × F → G) : Prop where
  add_left : ∀ (x₁ x₂ : E) (y : F), f (x₁ + x₂, y) = f (x₁, y) + f (x₂, y)
  smul_left : ∀ (c : 𝕜) (x : E) (y : F), f (c • x, y) = c • f (x, y)
  add_right : ∀ (x : E) (y₁ y₂ : F), f (x, y₁ + y₂) = f (x, y₁) + f (x, y₂)
  smul_right : ∀ (c : 𝕜) (x : E) (y : F), f (x, c • y) = c • f (x, y)
  bound : ∃ C > 0, ∀ (x : E) (y : F), ‖f (x, y)‖ ≤ C * ‖x‖ * ‖y‖
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E × F → G} (self : IsBoundedBilinearMap 𝕜 f) (c : 𝕜) (x : E) (y : F)
Goal: f (x, c • y) = c • f (x, y)
-- IsBoundedBilinearMap.bound
structure IsBoundedBilinearMap (f : E × F → G) : Prop where
  add_left : ∀ (x₁ x₂ : E) (y : F), f (x₁ + x₂, y) = f (x₁, y) + f (x₂, y)
  smul_left : ∀ (c : 𝕜) (x : E) (y : F), f (c • x, y) = c • f (x, y)
  add_right : ∀ (x : E) (y₁ y₂ : F), f (x, y₁ + y₂) = f (x, y₁) + f (x, y₂)
  smul_right : ∀ (c : 𝕜) (x : E) (y : F), f (x, c • y) = c • f (x, y)
  bound : ∃ C > 0, ∀ (x : E) (y : F), ‖f (x, y)‖ ≤ C * ‖x‖ * ‖y‖
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E × F → G} (self : IsBoundedBilinearMap 𝕜 f)
Goal: ∃ C > 0, ∀ (x : E) (y : F), ‖f (x, y)‖ ≤ C * ‖x‖ * ‖y‖
-- IsBoundedBilinearMap.add_left
structure IsBoundedBilinearMap (f : E × F → G) : Prop where
  add_left : ∀ (x₁ x₂ : E) (y : F), f (x₁ + x₂, y) = f (x₁, y) + f (x₂, y)
  smul_left : ∀ (c : 𝕜) (x : E) (y : F), f (c • x, y) = c • f (x, y)
  add_right : ∀ (x : E) (y₁ y₂ : F), f (x, y₁ + y₂) = f (x, y₁) + f (x, y₂)
  smul_right : ∀ (c : 𝕜) (x : E) (y : F), f (x, c • y) = c • f (x, y)
  bound : ∃ C > 0, ∀ (x : E) (y : F), ‖f (x, y)‖ ≤ C * ‖x‖ * ‖y‖
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E × F → G} (self : IsBoundedBilinearMap 𝕜 f) (x₁ x₂ : E) (y : F)
Goal: f (x₁ + x₂, y) = f (x₁, y) + f (x₂, y)
-- ContinuousLinearMap.isBoundedBilinearMap
theorem ContinuousLinearMap.isBoundedBilinearMap (f : E →L[𝕜] F →L[𝕜] G) :
    IsBoundedBilinearMap 𝕜 fun x : E × F => f x.1 x.2 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] (f : E →L[𝕜] F →L[𝕜] G)
Goal: IsBoundedBilinearMap 𝕜 fun x => (f x.1) x.2
-- IsBoundedBilinearMap.isBigO
protected theorem IsBoundedBilinearMap.isBigO (h : IsBoundedBilinearMap 𝕜 f) :
    f =O[⊤] fun p : E × F => ‖p.1‖ * ‖p.2‖ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E × F → G} (h : IsBoundedBilinearMap 𝕜 f)
Goal: f =O[⊤] fun p => ‖p.1‖ * ‖p.2‖
-- IsBoundedBilinearMap.isBigO_comp
theorem IsBoundedBilinearMap.isBigO_comp {α : Type*} (H : IsBoundedBilinearMap 𝕜 f) {g : α → E}
    {h : α → F} {l : Filter α} : (fun x => f (g x, h x)) =O[l] fun x => ‖g x‖ * ‖h x‖ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E × F → G} {α : Type u_5} (H : IsBoundedBilinearMap 𝕜 f) {g : α → E} {h : α → F} {l : Filter α}
Goal: (fun x => f (g x, h x)) =O[l] fun x => ‖g x‖ * ‖h x‖
-- IsBoundedBilinearMap.isBigO'
protected theorem IsBoundedBilinearMap.isBigO' (h : IsBoundedBilinearMap 𝕜 f) :
    f =O[⊤] fun p : E × F => ‖p‖ * ‖p‖ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E × F → G} (h : IsBoundedBilinearMap 𝕜 f)
Goal: f =O[⊤] fun p => ‖p‖ * ‖p‖
-- IsBoundedBilinearMap.map_sub_left
theorem IsBoundedBilinearMap.map_sub_left (h : IsBoundedBilinearMap 𝕜 f) {x y : E} {z : F} :
    f (x - y, z) = f (x, z) - f (y, z) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E × F → G} (h : IsBoundedBilinearMap 𝕜 f) {x y : E} {z : F}
Goal: f (x - y, z) = f (x, z) - f (y, z)
-- IsBoundedBilinearMap.map_sub_right
theorem IsBoundedBilinearMap.map_sub_right (h : IsBoundedBilinearMap 𝕜 f) {x : E} {y z : F} :
    f (x, y - z) = f (x, y) - f (x, z) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E × F → G} (h : IsBoundedBilinearMap 𝕜 f) {x : E} {y z : F}
Goal: f (x, y - z) = f (x, y) - f (x, z)
-- IsBoundedBilinearMap.continuous
theorem IsBoundedBilinearMap.continuous (h : IsBoundedBilinearMap 𝕜 f) : Continuous f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E × F → G} (h : IsBoundedBilinearMap 𝕜 f)
Goal: Continuous f
-- IsBoundedBilinearMap.continuous_left
theorem IsBoundedBilinearMap.continuous_left (h : IsBoundedBilinearMap 𝕜 f) {e₂ : F} :
    Continuous fun e₁ => f (e₁, e₂) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E × F → G} (h : IsBoundedBilinearMap 𝕜 f) {e₂ : F}
Goal: Continuous fun e₁ => f (e₁, e₂)