ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- LinearMap.mkContinuous₂_apply
@[simp]
theorem mkContinuous₂_apply (f : E →ₛₗ[σ₁₃] F →ₛₗ[σ₂₃] G) {C : ℝ}
    (hC : ∀ x y, ‖f x y‖ ≤ C * ‖x‖ * ‖y‖) (x : E) (y : F) : f.mkContinuous₂ C hC x y = f x y 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomIsometric σ₂₃] (f : E →ₛₗ[σ₁₃] F →ₛₗ[σ₂₃] G) {C : ℝ} (hC : ∀ (x : E) (y : F), ‖(f x) y‖ ≤ C * ‖x‖ * ‖y‖) (x : E) (y : F)
Goal: ((f.mkContinuous₂ C hC) x) y = (f x) y
-- LinearMap.mkContinuous₂_norm_le'
theorem mkContinuous₂_norm_le' (f : E →ₛₗ[σ₁₃] F →ₛₗ[σ₂₃] G) {C : ℝ}
    (hC : ∀ x y, ‖f x y‖ ≤ C * ‖x‖ * ‖y‖) : ‖f.mkContinuous₂ C hC‖ ≤ max C 0 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomIsometric σ₂₃] (f : E →ₛₗ[σ₁₃] F →ₛₗ[σ₂₃] G) {C : ℝ} (hC : ∀ (x : E) (y : F), ‖(f x) y‖ ≤ C * ‖x‖ * ‖y‖)
Goal: ‖f.mkContinuous₂ C hC‖ ≤ max C 0
-- LinearMap.mkContinuous₂_norm_le
theorem mkContinuous₂_norm_le (f : E →ₛₗ[σ₁₃] F →ₛₗ[σ₂₃] G) {C : ℝ} (h0 : 0 ≤ C)
    (hC : ∀ x y, ‖f x y‖ ≤ C * ‖x‖ * ‖y‖) : ‖f.mkContinuous₂ C hC‖ ≤ C 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomIsometric σ₂₃] (f : E →ₛₗ[σ₁₃] F →ₛₗ[σ₂₃] G) {C : ℝ} (h0 : 0 ≤ C) (hC : ∀ (x : E) (y : F), ‖(f x) y‖ ≤ C * ‖x‖ * ‖y‖)
Goal: ‖f.mkContinuous₂ C hC‖ ≤ C
-- _private.Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear.0.ContinuousLinearMap.le_norm_flip
private theorem le_norm_flip (f : E →SL[σ₁₃] F →SL[σ₂₃] G) : ‖f‖ ≤ ‖flip f‖ 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomIsometric σ₂₃] [RingHomIsometric σ₁₃] (f : E →SL[σ₁₃] F →SL[σ₂₃] G)
Goal: ‖f‖ ≤ ‖f.flip‖
-- ContinuousLinearMap.flip_apply
@[simp]
theorem flip_apply (f : E →SL[σ₁₃] F →SL[σ₂₃] G) (x : E) (y : F) : f.flip y x = f x y 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomIsometric σ₂₃] [RingHomIsometric σ₁₃] (f : E →SL[σ₁₃] F →SL[σ₂₃] G) (x : E) (y : F)
Goal: (f.flip y) x = (f x) y
-- ContinuousLinearMap.flip_flip
@[simp]
theorem flip_flip (f : E →SL[σ₁₃] F →SL[σ₂₃] G) : f.flip.flip = f 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomIsometric σ₂₃] [RingHomIsometric σ₁₃] (f : E →SL[σ₁₃] F →SL[σ₂₃] G)
Goal: f.flip.flip = f
-- ContinuousLinearMap.opNorm_flip
@[simp]
theorem opNorm_flip (f : E →SL[σ₁₃] F →SL[σ₂₃] G) : ‖f.flip‖ = ‖f‖ 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomIsometric σ₂₃] [RingHomIsometric σ₁₃] (f : E →SL[σ₁₃] F →SL[σ₂₃] G)
Goal: ‖f.flip‖ = ‖f‖
-- ContinuousLinearMap.flip_add
@[simp]
theorem flip_add (f g : E →SL[σ₁₃] F →SL[σ₂₃] G) : (f + g).flip = f.flip + g.flip 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomIsometric σ₂₃] [RingHomIsometric σ₁₃] (f g : E →SL[σ₁₃] F →SL[σ₂₃] G)
Goal: (f + g).flip = f.flip + g.flip
-- ContinuousLinearMap.flip_smul
@[simp]
theorem flip_smul (c : 𝕜₃) (f : E →SL[σ₁₃] F →SL[σ₂₃] G) : (c • f).flip = c • f.flip 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomIsometric σ₂₃] [RingHomIsometric σ₁₃] (c : 𝕜₃) (f : E →SL[σ₁₃] F →SL[σ₂₃] G)
Goal: (c • f).flip = c • f.flip
-- ContinuousLinearMap.flipₗᵢ'_symm
@[simp]
theorem flipₗᵢ'_symm : (flipₗᵢ' E F G σ₂₃ σ₁₃).symm = flipₗᵢ' F E G σ₁₃ σ₂₃ 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomIsometric σ₂₃] [RingHomIsometric σ₁₃]
Goal: (ContinuousLinearMap.flipₗᵢ' E F G σ₂₃ σ₁₃).symm = ContinuousLinearMap.flipₗᵢ' F E G σ₁₃ σ₂₃
-- ContinuousLinearMap.coe_flipₗᵢ'
@[simp]
theorem coe_flipₗᵢ' : ⇑(flipₗᵢ' E F G σ₂₃ σ₁₃) = flip 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomIsometric σ₂₃] [RingHomIsometric σ₁₃]
Goal: ⇑(ContinuousLinearMap.flipₗᵢ' E F G σ₂₃ σ₁₃) = ContinuousLinearMap.flip
-- ContinuousLinearMap.flipₗᵢ_symm
@[simp]
theorem flipₗᵢ_symm : (flipₗᵢ 𝕜 E Fₗ Gₗ).symm = flipₗᵢ 𝕜 Fₗ E Gₗ 
Arguments: {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} {Gₗ : Type u_9} [SeminormedAddCommGroup E] [SeminormedAddCommGroup Fₗ] [SeminormedAddCommGroup Gₗ] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Fₗ] [NormedSpace 𝕜 Gₗ]
Goal: (ContinuousLinearMap.flipₗᵢ 𝕜 E Fₗ Gₗ).symm = ContinuousLinearMap.flipₗᵢ 𝕜 Fₗ E Gₗ
-- ContinuousLinearMap.coe_flipₗᵢ
@[simp]
theorem coe_flipₗᵢ : ⇑(flipₗᵢ 𝕜 E Fₗ Gₗ) = flip 
Arguments: {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} {Gₗ : Type u_9} [SeminormedAddCommGroup E] [SeminormedAddCommGroup Fₗ] [SeminormedAddCommGroup Gₗ] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Fₗ] [NormedSpace 𝕜 Gₗ]
Goal: ⇑(ContinuousLinearMap.flipₗᵢ 𝕜 E Fₗ Gₗ) = ContinuousLinearMap.flip
-- ContinuousLinearMap.apply_apply'
@[simp]
theorem apply_apply' (v : E) (f : E →SL[σ₁₂] F) : apply' F σ₁₂ v f = f v 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] (v : E) (f : E →SL[σ₁₂] F)
Goal: ((ContinuousLinearMap.apply' F σ₁₂) v) f = f v
-- ContinuousLinearMap.apply_apply
@[simp]
theorem apply_apply (v : E) (f : E →L[𝕜] Fₗ) : apply 𝕜 Fₗ v f = f v 
Arguments: {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} [SeminormedAddCommGroup E] [SeminormedAddCommGroup Fₗ] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Fₗ] (v : E) (f : E →L[𝕜] Fₗ)
Goal: ((ContinuousLinearMap.apply 𝕜 Fₗ) v) f = f v
-- ContinuousLinearMap.norm_compSL_le
theorem norm_compSL_le : ‖compSL E F G σ₁₂ σ₂₃‖ ≤ 1 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} (E : Type u_4) (F : Type u_6) (G : Type u_8) [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] (σ₁₂ : 𝕜 →+* 𝕜₂) (σ₂₃ : 𝕜₂ →+* 𝕜₃) {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [RingHomIsometric σ₂₃] [RingHomIsometric σ₁₃] [RingHomIsometric σ₁₂]
Goal: ‖ContinuousLinearMap.compSL E F G σ₁₂ σ₂₃‖ ≤ 1
-- ContinuousLinearMap.compSL_apply
@[simp]
theorem compSL_apply (f : F →SL[σ₂₃] G) (g : E →SL[σ₁₂] F) : compSL E F G σ₁₂ σ₂₃ f g = f.comp g 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [RingHomIsometric σ₂₃] [RingHomIsometric σ₁₃] [RingHomIsometric σ₁₂] (f : F →SL[σ₂₃] G) (g : E →SL[σ₁₂] F)
Goal: ((ContinuousLinearMap.compSL E F G σ₁₂ σ₂₃) f) g = f.comp g
-- Continuous.const_clm_comp
theorem _root_.Continuous.const_clm_comp {X} [TopologicalSpace X] {f : X → E →SL[σ₁₂] F}
    (hf : Continuous f) (g : F →SL[σ₂₃] G) :
    Continuous (fun x => g.comp (f x) : X → E →SL[σ₁₃] G) 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [RingHomIsometric σ₂₃] [RingHomIsometric σ₁₃] [RingHomIsometric σ₁₂] {X : Type u_11} [TopologicalSpace X] {f : X → E →SL[σ₁₂] F} (hf : Continuous f) (g : F →SL[σ₂₃] G)
Goal: Continuous fun x => g.comp (f x)
-- Continuous.clm_comp_const
theorem _root_.Continuous.clm_comp_const {X} [TopologicalSpace X] {g : X → F →SL[σ₂₃] G}
    (hg : Continuous g) (f : E →SL[σ₁₂] F) :
    Continuous (fun x => (g x).comp f : X → E →SL[σ₁₃] G) 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [RingHomIsometric σ₂₃] [RingHomIsometric σ₁₃] [RingHomIsometric σ₁₂] {X : Type u_11} [TopologicalSpace X] {g : X → F →SL[σ₂₃] G} (hg : Continuous g) (f : E →SL[σ₁₂] F)
Goal: Continuous fun x => (g x).comp f
-- ContinuousLinearMap.norm_compL_le
theorem norm_compL_le : ‖compL 𝕜 E Fₗ Gₗ‖ ≤ 1 
Arguments: (𝕜 : Type u_1) (E : Type u_4) (Fₗ : Type u_7) (Gₗ : Type u_9) [SeminormedAddCommGroup E] [SeminormedAddCommGroup Fₗ] [SeminormedAddCommGroup Gₗ] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Fₗ] [NormedSpace 𝕜 Gₗ]
Goal: ‖ContinuousLinearMap.compL 𝕜 E Fₗ Gₗ‖ ≤ 1
-- ContinuousLinearMap.compL_apply
@[simp]
theorem compL_apply (f : Fₗ →L[𝕜] Gₗ) (g : E →L[𝕜] Fₗ) : compL 𝕜 E Fₗ Gₗ f g = f.comp g 
Arguments: (𝕜 : Type u_1) (E : Type u_4) (Fₗ : Type u_7) (Gₗ : Type u_9) [SeminormedAddCommGroup E] [SeminormedAddCommGroup Fₗ] [SeminormedAddCommGroup Gₗ] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Fₗ] [NormedSpace 𝕜 Gₗ] (f : Fₗ →L[𝕜] Gₗ) (g : E →L[𝕜] Fₗ)
Goal: ((ContinuousLinearMap.compL 𝕜 E Fₗ Gₗ) f) g = f.comp g
-- ContinuousLinearMap.precompR_apply
@[simps! apply]
def precompR (L : E →L[𝕜] Fₗ →L[𝕜] Gₗ) : E →L[𝕜] (Eₗ →L[𝕜] Fₗ) →L[𝕜] Eₗ →L[𝕜] Gₗ 
Arguments: {𝕜 : Type u_1} {E : Type u_4} (Eₗ : Type u_5) {Fₗ : Type u_7} {Gₗ : Type u_9} [SeminormedAddCommGroup E] [SeminormedAddCommGroup Eₗ] [SeminormedAddCommGroup Fₗ] [SeminormedAddCommGroup Gₗ] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Eₗ] [NormedSpace 𝕜 Fₗ] [NormedSpace 𝕜 Gₗ] (L : E →L[𝕜] Fₗ →L[𝕜] Gₗ)
Goal: ∀ (a : E), (ContinuousLinearMap.precompR Eₗ L) a = (ContinuousLinearMap.compL 𝕜 Eₗ Fₗ Gₗ) (L a)
-- ContinuousLinearMap.precompL_apply
@[simp] lemma precompL_apply (L : E →L[𝕜] Fₗ →L[𝕜] Gₗ) (u : Eₗ →L[𝕜] E) (f : Fₗ) (g : Eₗ) :
    precompL Eₗ L u f g = L (u g) f 
Arguments: {𝕜 : Type u_1} {E : Type u_4} (Eₗ : Type u_5) {Fₗ : Type u_7} {Gₗ : Type u_9} [SeminormedAddCommGroup E] [SeminormedAddCommGroup Eₗ] [SeminormedAddCommGroup Fₗ] [SeminormedAddCommGroup Gₗ] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Eₗ] [NormedSpace 𝕜 Fₗ] [NormedSpace 𝕜 Gₗ] (L : E →L[𝕜] Fₗ →L[𝕜] Gₗ) (u : Eₗ →L[𝕜] E) (f : Fₗ) (g : Eₗ)
Goal: (((ContinuousLinearMap.precompL Eₗ L) u) f) g = (L (u g)) f
-- ContinuousLinearMap.norm_precompR_le
theorem norm_precompR_le (L : E →L[𝕜] Fₗ →L[𝕜] Gₗ) : ‖precompR Eₗ L‖ ≤ ‖L‖ 
Arguments: {𝕜 : Type u_1} {E : Type u_4} (Eₗ : Type u_5) {Fₗ : Type u_7} {Gₗ : Type u_9} [SeminormedAddCommGroup E] [SeminormedAddCommGroup Eₗ] [SeminormedAddCommGroup Fₗ] [SeminormedAddCommGroup Gₗ] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Eₗ] [NormedSpace 𝕜 Fₗ] [NormedSpace 𝕜 Gₗ] (L : E →L[𝕜] Fₗ →L[𝕜] Gₗ)
Goal: ‖ContinuousLinearMap.precompR Eₗ L‖ ≤ ‖L‖
-- ContinuousLinearMap.norm_precompL_le
theorem norm_precompL_le (L : E →L[𝕜] Fₗ →L[𝕜] Gₗ) : ‖precompL Eₗ L‖ ≤ ‖L‖ 
Arguments: {𝕜 : Type u_1} {E : Type u_4} (Eₗ : Type u_5) {Fₗ : Type u_7} {Gₗ : Type u_9} [SeminormedAddCommGroup E] [SeminormedAddCommGroup Eₗ] [SeminormedAddCommGroup Fₗ] [SeminormedAddCommGroup Gₗ] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Eₗ] [NormedSpace 𝕜 Fₗ] [NormedSpace 𝕜 Gₗ] (L : E →L[𝕜] Fₗ →L[𝕜] Gₗ)
Goal: ‖ContinuousLinearMap.precompL Eₗ L‖ ≤ ‖L‖
-- ContinuousLinearMap.bilinearComp_apply
@[simp]
theorem bilinearComp_apply (f : E →SL[σ₁₃] F →SL[σ₂₃] G) (gE : E' →SL[σ₁'] E) (gF : F' →SL[σ₂'] F)
    (x : E') (y : F') : f.bilinearComp gE gF x y = f (gE x) (gF y) 
Arguments: {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} {E' : Type u_11} {F' : Type u_12} [SeminormedAddCommGroup E'] [SeminormedAddCommGroup F'] {𝕜₁' : Type u_13} {𝕜₂' : Type u_14} [NontriviallyNormedField 𝕜₁'] [NontriviallyNormedField 𝕜₂'] [NormedSpace 𝕜₁' E'] [NormedSpace 𝕜₂' F'] {σ₁' : 𝕜₁' →+* 𝕜} {σ₁₃' : 𝕜₁' →+* 𝕜₃} {σ₂' : 𝕜₂' →+* 𝕜₂} {σ₂₃' : 𝕜₂' →+* 𝕜₃} [RingHomCompTriple σ₁' σ₁₃ σ₁₃'] [RingHomCompTriple σ₂' σ₂₃ σ₂₃'] [RingHomIsometric σ₂₃] [RingHomIsometric σ₁₃'] [RingHomIsometric σ₂₃'] (f : E →SL[σ₁₃] F →SL[σ₂₃] G) (gE : E' →SL[σ₁'] E) (gF : F' →SL[σ₂'] F) (x : E') (y : F')
Goal: ((f.bilinearComp gE gF) x) y = (f (gE x)) (gF y)
-- ContinuousLinearMap.coe_deriv₂
@[simp]
theorem coe_deriv₂ (f : E →L[𝕜] Fₗ →L[𝕜] Gₗ) (p : E × Fₗ) :
    ⇑(f.deriv₂ p) = fun q : E × Fₗ => f p.1 q.2 + f q.1 p.2 
Arguments: {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} {Gₗ : Type u_9} [SeminormedAddCommGroup E] [SeminormedAddCommGroup Fₗ] [SeminormedAddCommGroup Gₗ] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Fₗ] [NormedSpace 𝕜 Gₗ] (f : E →L[𝕜] Fₗ →L[𝕜] Gₗ) (p : E × Fₗ)
Goal: ⇑(f.deriv₂ p) = fun q => (f p.1) q.2 + (f q.1) p.2
-- ContinuousLinearMap.map_add_add
theorem map_add_add (f : E →L[𝕜] Fₗ →L[𝕜] Gₗ) (x x' : E) (y y' : Fₗ) :
    f (x + x') (y + y') = f x y + f.deriv₂ (x, y) (x', y') + f x' y' 
Arguments: {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} {Gₗ : Type u_9} [SeminormedAddCommGroup E] [SeminormedAddCommGroup Fₗ] [SeminormedAddCommGroup Gₗ] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Fₗ] [NormedSpace 𝕜 Gₗ] (f : E →L[𝕜] Fₗ →L[𝕜] Gₗ) (x x' : E) (y y' : Fₗ)
Goal: (f (x + x')) (y + y') = (f x) y + (f.deriv₂ (x, y)) (x', y') + (f x') y'
-- HasSolidNorm.solid
class HasSolidNorm (α : Type*) [NormedAddCommGroup α] [Lattice α] : Prop where
  solid : ∀ ⦃x y : α⦄, |x| ≤ |y| → ‖x‖ ≤ ‖y‖
Arguments: {α : Type u_1} [NormedAddCommGroup α] [Lattice α] [self : HasSolidNorm α] ⦃x y : α⦄
Goal: |x| ≤ |y| → ‖x‖ ≤ ‖y‖
-- norm_le_norm_of_abs_le_abs
theorem norm_le_norm_of_abs_le_abs {a b : α} (h : |a| ≤ |b|) : ‖a‖ ≤ ‖b‖ 
Arguments: {α : Type u_1} [NormedAddCommGroup α] [Lattice α] [HasSolidNorm α] {a b : α} (h : |a| ≤ |b|)
Goal: ‖a‖ ≤ ‖b‖
-- LatticeOrderedAddCommGroup.isSolid_ball
theorem LatticeOrderedAddCommGroup.isSolid_ball (r : ℝ) :
    LatticeOrderedAddCommGroup.IsSolid (Metric.ball (0 : α) r) 
Arguments: {α : Type u_1} [NormedAddCommGroup α] [Lattice α] [HasSolidNorm α] (r : ℝ)
Goal: LatticeOrderedAddCommGroup.IsSolid (Metric.ball 0 r)
-- NormedLatticeAddCommGroup.add_le_add_left
class NormedLatticeAddCommGroup (α : Type*) extends
    NormedAddCommGroup α, Lattice α, HasSolidNorm α where
  add_le_add_left : ∀ a b : α, a ≤ b → ∀ c : α, c + a ≤ c + b
Arguments: {α : Type u_1} [self : NormedLatticeAddCommGroup α] (a b : α)
Goal: a ≤ b → ∀ (c : α), c + a ≤ c + b
-- dual_solid
theorem dual_solid (a b : α) (h : b ⊓ -b ≤ a ⊓ -a) : ‖a‖ ≤ ‖b‖ 
Arguments: {α : Type u_1} [NormedLatticeAddCommGroup α] (a b : α) (h : b ⊓ -b ≤ a ⊓ -a)
Goal: ‖a‖ ≤ ‖b‖
-- norm_abs_eq_norm
theorem norm_abs_eq_norm (a : α) : ‖|a|‖ = ‖a‖ 
Arguments: {α : Type u_1} [NormedLatticeAddCommGroup α] (a : α)
Goal: ‖|a|‖ = ‖a‖
-- norm_inf_sub_inf_le_add_norm
theorem norm_inf_sub_inf_le_add_norm (a b c d : α) : ‖a ⊓ b - c ⊓ d‖ ≤ ‖a - c‖ + ‖b - d‖ 
Arguments: {α : Type u_1} [NormedLatticeAddCommGroup α] (a b c d : α)
Goal: ‖a ⊓ b - c ⊓ d‖ ≤ ‖a - c‖ + ‖b - d‖
-- norm_sup_sub_sup_le_add_norm
theorem norm_sup_sub_sup_le_add_norm (a b c d : α) : ‖a ⊔ b - c ⊔ d‖ ≤ ‖a - c‖ + ‖b - d‖ 
Arguments: {α : Type u_1} [NormedLatticeAddCommGroup α] (a b c d : α)
Goal: ‖a ⊔ b - c ⊔ d‖ ≤ ‖a - c‖ + ‖b - d‖
-- norm_inf_le_add
theorem norm_inf_le_add (x y : α) : ‖x ⊓ y‖ ≤ ‖x‖ + ‖y‖ 
Arguments: {α : Type u_1} [NormedLatticeAddCommGroup α] (x y : α)
Goal: ‖x ⊓ y‖ ≤ ‖x‖ + ‖y‖
-- norm_sup_le_add
theorem norm_sup_le_add (x y : α) : ‖x ⊔ y‖ ≤ ‖x‖ + ‖y‖ 
Arguments: {α : Type u_1} [NormedLatticeAddCommGroup α] (x y : α)
Goal: ‖x ⊔ y‖ ≤ ‖x‖ + ‖y‖
-- norm_abs_sub_abs
theorem norm_abs_sub_abs (a b : α) : ‖|a| - |b|‖ ≤ ‖a - b‖ 
Arguments: {α : Type u_1} [NormedLatticeAddCommGroup α] (a b : α)
Goal: ‖|a| - |b|‖ ≤ ‖a - b‖
-- norm_sup_sub_sup_le_norm
theorem norm_sup_sub_sup_le_norm (x y z : α) : ‖x ⊔ z - y ⊔ z‖ ≤ ‖x - y‖ 
Arguments: {α : Type u_1} [NormedLatticeAddCommGroup α] (x y z : α)
Goal: ‖x ⊔ z - y ⊔ z‖ ≤ ‖x - y‖
-- norm_inf_sub_inf_le_norm
theorem norm_inf_sub_inf_le_norm (x y z : α) : ‖x ⊓ z - y ⊓ z‖ ≤ ‖x - y‖ 
Arguments: {α : Type u_1} [NormedLatticeAddCommGroup α] (x y z : α)
Goal: ‖x ⊓ z - y ⊓ z‖ ≤ ‖x - y‖
-- lipschitzWith_sup_right
theorem lipschitzWith_sup_right (z : α) : LipschitzWith 1 fun x => x ⊔ z 
Arguments: {α : Type u_1} [NormedLatticeAddCommGroup α] (z : α)
Goal: LipschitzWith 1 fun x => x ⊔ z
-- lipschitzWith_posPart
lemma lipschitzWith_posPart : LipschitzWith 1 (posPart : α → α) 
Arguments: {α : Type u_1} [NormedLatticeAddCommGroup α]
Goal: LipschitzWith 1 posPart
-- lipschitzWith_negPart
lemma lipschitzWith_negPart : LipschitzWith 1 (negPart : α → α) 
Arguments: {α : Type u_1} [NormedLatticeAddCommGroup α]
Goal: LipschitzWith 1 negPart
-- continuous_posPart
lemma continuous_posPart : Continuous (posPart : α → α) 
Arguments: {α : Type u_1} [NormedLatticeAddCommGroup α]
Goal: Continuous posPart
-- continuous_negPart
lemma continuous_negPart : Continuous (negPart : α → α) 
Arguments: {α : Type u_1} [NormedLatticeAddCommGroup α]
Goal: Continuous negPart
-- isClosed_nonneg
lemma isClosed_nonneg : IsClosed {x : α | 0 ≤ x} 
Arguments: {α : Type u_1} [NormedLatticeAddCommGroup α]
Goal: IsClosed {x | 0 ≤ x}
-- isClosed_le_of_isClosed_nonneg
theorem isClosed_le_of_isClosed_nonneg {G} [OrderedAddCommGroup G] [TopologicalSpace G]
    [ContinuousSub G] (h : IsClosed { x : G | 0 ≤ x }) :
    IsClosed { p : G × G | p.fst ≤ p.snd } 
Arguments: {G : Type u_2} [OrderedAddCommGroup G] [TopologicalSpace G] [ContinuousSub G] (h : IsClosed {x | 0 ≤ x})
Goal: IsClosed {p | p.1 ≤ p.2}
-- Asymptotics.IsEquivalent.isLittleO
theorem IsEquivalent.isLittleO (h : u ~[l] v) : (u - v) =o[l] v 
Arguments: {α : Type u_1} {β : Type u_2} [NormedAddCommGroup β] {u v : α → β} {l : Filter α} (h : u ~[l] v)
Goal: (u - v) =o[l] v
-- Asymptotics.IsEquivalent.isBigO
nonrec theorem IsEquivalent.isBigO (h : u ~[l] v) : u =O[l] v 
Arguments: {α : Type u_1} {β : Type u_2} [NormedAddCommGroup β] {u v : α → β} {l : Filter α} (h : u ~[l] v)
Goal: u =O[l] v
-- Asymptotics.IsEquivalent.isBigO_symm
theorem IsEquivalent.isBigO_symm (h : u ~[l] v) : v =O[l] u 
Arguments: {α : Type u_1} {β : Type u_2} [NormedAddCommGroup β] {u v : α → β} {l : Filter α} (h : u ~[l] v)
Goal: v =O[l] u
-- Asymptotics.IsEquivalent.isTheta
theorem IsEquivalent.isTheta (h : u ~[l] v) : u =Θ[l] v 
Arguments: {α : Type u_1} {β : Type u_2} [NormedAddCommGroup β] {u v : α → β} {l : Filter α} (h : u ~[l] v)
Goal: u =Θ[l] v
-- Asymptotics.IsEquivalent.isTheta_symm
theorem IsEquivalent.isTheta_symm (h : u ~[l] v) : v =Θ[l] u 
Arguments: {α : Type u_1} {β : Type u_2} [NormedAddCommGroup β] {u v : α → β} {l : Filter α} (h : u ~[l] v)
Goal: v =Θ[l] u
-- Asymptotics.IsEquivalent.refl
@[refl]
theorem IsEquivalent.refl : u ~[l] u 
Arguments: {α : Type u_1} {β : Type u_2} [NormedAddCommGroup β] {u : α → β} {l : Filter α}
Goal: u ~[l] u
-- Asymptotics.IsEquivalent.symm
@[symm]
theorem IsEquivalent.symm (h : u ~[l] v) : v ~[l] u 
Arguments: {α : Type u_1} {β : Type u_2} [NormedAddCommGroup β] {u v : α → β} {l : Filter α} (h : u ~[l] v)
Goal: v ~[l] u
-- Asymptotics.IsEquivalent.trans
@[trans]
theorem IsEquivalent.trans {l : Filter α} {u v w : α → β} (huv : u ~[l] v) (hvw : v ~[l] w) :
    u ~[l] w 
Arguments: {α : Type u_1} {β : Type u_2} [NormedAddCommGroup β] {l : Filter α} {u v w : α → β} (huv : u ~[l] v) (hvw : v ~[l] w)
Goal: u ~[l] w
-- Asymptotics.IsEquivalent.congr_left
theorem IsEquivalent.congr_left {u v w : α → β} {l : Filter α} (huv : u ~[l] v) (huw : u =ᶠ[l] w) :
    w ~[l] v 
Arguments: {α : Type u_1} {β : Type u_2} [NormedAddCommGroup β] {u v w : α → β} {l : Filter α} (huv : u ~[l] v) (huw : u =ᶠ[l] w)
Goal: w ~[l] v
-- Asymptotics.IsEquivalent.congr_right
theorem IsEquivalent.congr_right {u v w : α → β} {l : Filter α} (huv : u ~[l] v) (hvw : v =ᶠ[l] w) :
    u ~[l] w 
Arguments: {α : Type u_1} {β : Type u_2} [NormedAddCommGroup β] {u v w : α → β} {l : Filter α} (huv : u ~[l] v) (hvw : v =ᶠ[l] w)
Goal: u ~[l] w
-- Asymptotics.isEquivalent_zero_iff_eventually_zero
theorem isEquivalent_zero_iff_eventually_zero : u ~[l] 0 ↔ u =ᶠ[l] 0 
Arguments: {α : Type u_1} {β : Type u_2} [NormedAddCommGroup β] {u : α → β} {l : Filter α}
Goal: u ~[l] 0 ↔ u =ᶠ[l] 0
-- Asymptotics.isEquivalent_zero_iff_isBigO_zero
theorem isEquivalent_zero_iff_isBigO_zero : u ~[l] 0 ↔ u =O[l] (0 : α → β) 
Arguments: {α : Type u_1} {β : Type u_2} [NormedAddCommGroup β] {u : α → β} {l : Filter α}
Goal: u ~[l] 0 ↔ u =O[l] 0
-- Asymptotics.isEquivalent_const_iff_tendsto
theorem isEquivalent_const_iff_tendsto {c : β} (h : c ≠ 0) :
    u ~[l] const _ c ↔ Tendsto u l (𝓝 c) 
Arguments: {α : Type u_1} {β : Type u_2} [NormedAddCommGroup β] {u : α → β} {l : Filter α} {c : β} (h : c ≠ 0)
Goal: u ~[l] Function.const α c ↔ Filter.Tendsto u l (nhds c)