ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContDiffAt.rpow
theorem ContDiffAt.rpow (hf : ContDiffAt ℝ n f x) (hg : ContDiffAt ℝ n g x) (h : f x ≠ 0) :
    ContDiffAt ℝ n (fun x => f x ^ g x) x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f g : E → ℝ} {x : E} {n : ℕ∞} (hf : ContDiffAt ℝ n f x) (hg : ContDiffAt ℝ n g x) (h : f x ≠ 0)
Goal: ContDiffAt ℝ n (fun x => f x ^ g x) x
-- ContDiffOn.rpow
theorem ContDiffOn.rpow (hf : ContDiffOn ℝ n f s) (hg : ContDiffOn ℝ n g s) (h : ∀ x ∈ s, f x ≠ 0) :
    ContDiffOn ℝ n (fun x => f x ^ g x) s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f g : E → ℝ} {s : Set E} {n : ℕ∞} (hf : ContDiffOn ℝ n f s) (hg : ContDiffOn ℝ n g s) (h : ∀ x ∈ s, f x ≠ 0)
Goal: ContDiffOn ℝ n (fun x => f x ^ g x) s
-- ContDiff.rpow
theorem ContDiff.rpow (hf : ContDiff ℝ n f) (hg : ContDiff ℝ n g) (h : ∀ x, f x ≠ 0) :
    ContDiff ℝ n fun x => f x ^ g x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f g : E → ℝ} {n : ℕ∞} (hf : ContDiff ℝ n f) (hg : ContDiff ℝ n g) (h : ∀ (x : E), f x ≠ 0)
Goal: ContDiff ℝ n fun x => f x ^ g x
-- ContDiffWithinAt.rpow_const_of_ne
theorem ContDiffWithinAt.rpow_const_of_ne (hf : ContDiffWithinAt ℝ n f s x) (h : f x ≠ 0) :
    ContDiffWithinAt ℝ n (fun x => f x ^ p) s x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {s : Set E} {p : ℝ} {n : ℕ∞} (hf : ContDiffWithinAt ℝ n f s x) (h : f x ≠ 0)
Goal: ContDiffWithinAt ℝ n (fun x => f x ^ p) s x
-- ContDiffAt.rpow_const_of_ne
theorem ContDiffAt.rpow_const_of_ne (hf : ContDiffAt ℝ n f x) (h : f x ≠ 0) :
    ContDiffAt ℝ n (fun x => f x ^ p) x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {p : ℝ} {n : ℕ∞} (hf : ContDiffAt ℝ n f x) (h : f x ≠ 0)
Goal: ContDiffAt ℝ n (fun x => f x ^ p) x
-- ContDiffOn.rpow_const_of_ne
theorem ContDiffOn.rpow_const_of_ne (hf : ContDiffOn ℝ n f s) (h : ∀ x ∈ s, f x ≠ 0) :
    ContDiffOn ℝ n (fun x => f x ^ p) s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {s : Set E} {p : ℝ} {n : ℕ∞} (hf : ContDiffOn ℝ n f s) (h : ∀ x ∈ s, f x ≠ 0)
Goal: ContDiffOn ℝ n (fun x => f x ^ p) s
-- ContDiff.rpow_const_of_ne
theorem ContDiff.rpow_const_of_ne (hf : ContDiff ℝ n f) (h : ∀ x, f x ≠ 0) :
    ContDiff ℝ n fun x => f x ^ p 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {p : ℝ} {n : ℕ∞} (hf : ContDiff ℝ n f) (h : ∀ (x : E), f x ≠ 0)
Goal: ContDiff ℝ n fun x => f x ^ p
-- ContDiffWithinAt.rpow_const_of_le
theorem ContDiffWithinAt.rpow_const_of_le (hf : ContDiffWithinAt ℝ m f s x) (h : ↑m ≤ p) :
    ContDiffWithinAt ℝ m (fun x => f x ^ p) s x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {s : Set E} {p : ℝ} {m : ℕ} (hf : ContDiffWithinAt ℝ (↑m) f s x) (h : ↑m ≤ p)
Goal: ContDiffWithinAt ℝ (↑m) (fun x => f x ^ p) s x
-- ContDiffAt.rpow_const_of_le
theorem ContDiffAt.rpow_const_of_le (hf : ContDiffAt ℝ m f x) (h : ↑m ≤ p) :
    ContDiffAt ℝ m (fun x => f x ^ p) x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {p : ℝ} {m : ℕ} (hf : ContDiffAt ℝ (↑m) f x) (h : ↑m ≤ p)
Goal: ContDiffAt ℝ (↑m) (fun x => f x ^ p) x
-- ContDiffOn.rpow_const_of_le
theorem ContDiffOn.rpow_const_of_le (hf : ContDiffOn ℝ m f s) (h : ↑m ≤ p) :
    ContDiffOn ℝ m (fun x => f x ^ p) s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {s : Set E} {p : ℝ} {m : ℕ} (hf : ContDiffOn ℝ (↑m) f s) (h : ↑m ≤ p)
Goal: ContDiffOn ℝ (↑m) (fun x => f x ^ p) s
-- ContDiff.rpow_const_of_le
theorem ContDiff.rpow_const_of_le (hf : ContDiff ℝ m f) (h : ↑m ≤ p) :
    ContDiff ℝ m fun x => f x ^ p 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {p : ℝ} {m : ℕ} (hf : ContDiff ℝ (↑m) f) (h : ↑m ≤ p)
Goal: ContDiff ℝ ↑m fun x => f x ^ p
-- HasDerivWithinAt.rpow
theorem HasDerivWithinAt.rpow (hf : HasDerivWithinAt f f' s x) (hg : HasDerivWithinAt g g' s x)
    (h : 0 < f x) : HasDerivWithinAt (fun x => f x ^ g x)
      (f' * g x * f x ^ (g x - 1) + g' * f x ^ g x * Real.log (f x)) s x 
Arguments: {f g : ℝ → ℝ} {f' g' x : ℝ} {s : Set ℝ} (hf : HasDerivWithinAt f f' s x) (hg : HasDerivWithinAt g g' s x) (h : 0 < f x)
Goal: HasDerivWithinAt (fun x => f x ^ g x) (f' * g x * f x ^ (g x - 1) + g' * f x ^ g x * Real.log (f x)) s x
-- HasDerivAt.rpow
theorem HasDerivAt.rpow (hf : HasDerivAt f f' x) (hg : HasDerivAt g g' x) (h : 0 < f x) :
    HasDerivAt (fun x => f x ^ g x)
      (f' * g x * f x ^ (g x - 1) + g' * f x ^ g x * Real.log (f x)) x 
Arguments: {f g : ℝ → ℝ} {f' g' x : ℝ} (hf : HasDerivAt f f' x) (hg : HasDerivAt g g' x) (h : 0 < f x)
Goal: HasDerivAt (fun x => f x ^ g x) (f' * g x * f x ^ (g x - 1) + g' * f x ^ g x * Real.log (f x)) x
-- HasDerivWithinAt.rpow_const
theorem HasDerivWithinAt.rpow_const (hf : HasDerivWithinAt f f' s x) (hx : f x ≠ 0 ∨ 1 ≤ p) :
    HasDerivWithinAt (fun y => f y ^ p) (f' * p * f x ^ (p - 1)) s x 
Arguments: {f : ℝ → ℝ} {f' x p : ℝ} {s : Set ℝ} (hf : HasDerivWithinAt f f' s x) (hx : f x ≠ 0 ∨ 1 ≤ p)
Goal: HasDerivWithinAt (fun y => f y ^ p) (f' * p * f x ^ (p - 1)) s x
-- HasDerivAt.rpow_const
theorem HasDerivAt.rpow_const (hf : HasDerivAt f f' x) (hx : f x ≠ 0 ∨ 1 ≤ p) :
    HasDerivAt (fun y => f y ^ p) (f' * p * f x ^ (p - 1)) x 
Arguments: {f : ℝ → ℝ} {f' x p : ℝ} (hf : HasDerivAt f f' x) (hx : f x ≠ 0 ∨ 1 ≤ p)
Goal: HasDerivAt (fun y => f y ^ p) (f' * p * f x ^ (p - 1)) x
-- derivWithin_rpow_const
theorem derivWithin_rpow_const (hf : DifferentiableWithinAt ℝ f s x) (hx : f x ≠ 0 ∨ 1 ≤ p)
    (hxs : UniqueDiffWithinAt ℝ s x) :
    derivWithin (fun x => f x ^ p) s x = derivWithin f s x * p * f x ^ (p - 1) 
Arguments: {f : ℝ → ℝ} {x p : ℝ} {s : Set ℝ} (hf : DifferentiableWithinAt ℝ f s x) (hx : f x ≠ 0 ∨ 1 ≤ p) (hxs : UniqueDiffWithinAt ℝ s x)
Goal: derivWithin (fun x => f x ^ p) s x = derivWithin f s x * p * f x ^ (p - 1)
-- deriv_rpow_const
@[simp]
theorem deriv_rpow_const (hf : DifferentiableAt ℝ f x) (hx : f x ≠ 0 ∨ 1 ≤ p) :
    deriv (fun x => f x ^ p) x = deriv f x * p * f x ^ (p - 1) 
Arguments: {f : ℝ → ℝ} {x p : ℝ} (hf : DifferentiableAt ℝ f x) (hx : f x ≠ 0 ∨ 1 ≤ p)
Goal: deriv (fun x => f x ^ p) x = deriv f x * p * f x ^ (p - 1)
-- isTheta_deriv_rpow_const_atTop
lemma isTheta_deriv_rpow_const_atTop {p : ℝ} (hp : p ≠ 0) :
    deriv (fun (x : ℝ) => x ^ p) =Θ[atTop] fun x => x ^ (p-1) 
Arguments: {p : ℝ} (hp : p ≠ 0)
Goal: (deriv fun x => x ^ p) =Θ[Filter.atTop] fun x => x ^ (p - 1)
-- isBigO_deriv_rpow_const_atTop
lemma isBigO_deriv_rpow_const_atTop (p : ℝ) :
    deriv (fun (x : ℝ) => x ^ p) =O[atTop] fun x => x ^ (p-1) 
Arguments: (p : ℝ)
Goal: (deriv fun x => x ^ p) =O[Filter.atTop] fun x => x ^ (p - 1)
-- tendsto_one_plus_div_rpow_exp
theorem tendsto_one_plus_div_rpow_exp (t : ℝ) :
    Tendsto (fun x : ℝ => (1 + t / x) ^ x) atTop (𝓝 (exp t)) 
Arguments: (t : ℝ)
Goal: Filter.Tendsto (fun x => (1 + t / x) ^ x) Filter.atTop (𝓝 (rexp t))
-- tendsto_one_plus_div_pow_exp
theorem tendsto_one_plus_div_pow_exp (t : ℝ) :
    Tendsto (fun x : ℕ => (1 + t / (x : ℝ)) ^ x) atTop (𝓝 (Real.exp t)) 
Arguments: (t : ℝ)
Goal: Filter.Tendsto (fun x => (1 + t / ↑x) ^ x) Filter.atTop (𝓝 (rexp t))
-- ContinuousLinearMap.mul_apply'
@[simp]
theorem mul_apply' (x y : 𝕜') : mul 𝕜 𝕜' x y = x * y 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] (𝕜' : Type u_3) [NonUnitalSeminormedRing 𝕜'] [NormedSpace 𝕜 𝕜'] [IsScalarTower 𝕜 𝕜' 𝕜'] [SMulCommClass 𝕜 𝕜' 𝕜'] (x y : 𝕜')
Goal: ((ContinuousLinearMap.mul 𝕜 𝕜') x) y = x * y
-- ContinuousLinearMap.opNorm_mul_apply_le
@[simp]
theorem opNorm_mul_apply_le (x : 𝕜') : ‖mul 𝕜 𝕜' x‖ ≤ ‖x‖ 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] (𝕜' : Type u_3) [NonUnitalSeminormedRing 𝕜'] [NormedSpace 𝕜 𝕜'] [IsScalarTower 𝕜 𝕜' 𝕜'] [SMulCommClass 𝕜 𝕜' 𝕜'] (x : 𝕜')
Goal: ‖(ContinuousLinearMap.mul 𝕜 𝕜') x‖ ≤ ‖x‖
-- ContinuousLinearMap.opNorm_mul_le
theorem opNorm_mul_le : ‖mul 𝕜 𝕜'‖ ≤ 1 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] (𝕜' : Type u_3) [NonUnitalSeminormedRing 𝕜'] [NormedSpace 𝕜 𝕜'] [IsScalarTower 𝕜 𝕜' 𝕜'] [SMulCommClass 𝕜 𝕜' 𝕜']
Goal: ‖ContinuousLinearMap.mul 𝕜 𝕜'‖ ≤ 1
-- NonUnitalAlgHom.coe_Lmul
@[simp]
theorem _root_.NonUnitalAlgHom.coe_Lmul : ⇑(NonUnitalAlgHom.Lmul 𝕜 𝕜') = mul 𝕜 𝕜' 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {𝕜' : Type u_3} [NonUnitalSeminormedRing 𝕜'] [NormedSpace 𝕜 𝕜'] [IsScalarTower 𝕜 𝕜' 𝕜'] [SMulCommClass 𝕜 𝕜' 𝕜']
Goal: ⇑(NonUnitalAlgHom.Lmul 𝕜 𝕜') = ⇑(ContinuousLinearMap.mul 𝕜 𝕜')
-- ContinuousLinearMap.mulLeftRight_apply
@[simp]
theorem mulLeftRight_apply (x y z : 𝕜') : mulLeftRight 𝕜 𝕜' x y z = x * z * y 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] (𝕜' : Type u_3) [NonUnitalSeminormedRing 𝕜'] [NormedSpace 𝕜 𝕜'] [IsScalarTower 𝕜 𝕜' 𝕜'] [SMulCommClass 𝕜 𝕜' 𝕜'] (x y z : 𝕜')
Goal: (((ContinuousLinearMap.mulLeftRight 𝕜 𝕜') x) y) z = x * z * y
-- ContinuousLinearMap.opNorm_mulLeftRight_apply_apply_le
theorem opNorm_mulLeftRight_apply_apply_le (x y : 𝕜') : ‖mulLeftRight 𝕜 𝕜' x y‖ ≤ ‖x‖ * ‖y‖ 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] (𝕜' : Type u_3) [NonUnitalSeminormedRing 𝕜'] [NormedSpace 𝕜 𝕜'] [IsScalarTower 𝕜 𝕜' 𝕜'] [SMulCommClass 𝕜 𝕜' 𝕜'] (x y : 𝕜')
Goal: ‖((ContinuousLinearMap.mulLeftRight 𝕜 𝕜') x) y‖ ≤ ‖x‖ * ‖y‖
-- ContinuousLinearMap.opNorm_mulLeftRight_apply_le
theorem opNorm_mulLeftRight_apply_le (x : 𝕜') : ‖mulLeftRight 𝕜 𝕜' x‖ ≤ ‖x‖ 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] (𝕜' : Type u_3) [NonUnitalSeminormedRing 𝕜'] [NormedSpace 𝕜 𝕜'] [IsScalarTower 𝕜 𝕜' 𝕜'] [SMulCommClass 𝕜 𝕜' 𝕜'] (x : 𝕜')
Goal: ‖(ContinuousLinearMap.mulLeftRight 𝕜 𝕜') x‖ ≤ ‖x‖
-- ContinuousLinearMap.opNorm_mulLeftRight_le
theorem opNorm_mulLeftRight_le :
    ‖mulLeftRight 𝕜 𝕜'‖ ≤ 1 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] (𝕜' : Type u_3) [NonUnitalSeminormedRing 𝕜'] [NormedSpace 𝕜 𝕜'] [IsScalarTower 𝕜 𝕜' 𝕜'] [SMulCommClass 𝕜 𝕜' 𝕜']
Goal: ‖ContinuousLinearMap.mulLeftRight 𝕜 𝕜'‖ ≤ 1
-- RegularNormedAlgebra.isometry_mul'
class _root_.RegularNormedAlgebra : Prop :=
    isometry_mul' : Isometry (mul 𝕜 𝕜')
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {𝕜' : Type u_3} [NonUnitalSeminormedRing 𝕜'] [NormedSpace 𝕜 𝕜'] [IsScalarTower 𝕜 𝕜' 𝕜'] [SMulCommClass 𝕜 𝕜' 𝕜'] [self : RegularNormedAlgebra 𝕜 𝕜']
Goal: Isometry ⇑(ContinuousLinearMap.mul 𝕜 𝕜')
-- ContinuousLinearMap.isometry_mul
lemma isometry_mul : Isometry (mul 𝕜 𝕜') 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] (𝕜' : Type u_3) [NonUnitalSeminormedRing 𝕜'] [NormedSpace 𝕜 𝕜'] [IsScalarTower 𝕜 𝕜' 𝕜'] [SMulCommClass 𝕜 𝕜' 𝕜'] [RegularNormedAlgebra 𝕜 𝕜']
Goal: Isometry ⇑(ContinuousLinearMap.mul 𝕜 𝕜')
-- ContinuousLinearMap.opNorm_mul_apply
@[simp]
lemma opNorm_mul_apply (x : 𝕜') : ‖mul 𝕜 𝕜' x‖ = ‖x‖ 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] (𝕜' : Type u_3) [NonUnitalSeminormedRing 𝕜'] [NormedSpace 𝕜 𝕜'] [IsScalarTower 𝕜 𝕜' 𝕜'] [SMulCommClass 𝕜 𝕜' 𝕜'] [RegularNormedAlgebra 𝕜 𝕜'] (x : 𝕜')
Goal: ‖(ContinuousLinearMap.mul 𝕜 𝕜') x‖ = ‖x‖
-- ContinuousLinearMap.opNNNorm_mul_apply
@[simp]
lemma opNNNorm_mul_apply (x : 𝕜') : ‖mul 𝕜 𝕜' x‖₊ = ‖x‖₊ 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] (𝕜' : Type u_3) [NonUnitalSeminormedRing 𝕜'] [NormedSpace 𝕜 𝕜'] [IsScalarTower 𝕜 𝕜' 𝕜'] [SMulCommClass 𝕜 𝕜' 𝕜'] [RegularNormedAlgebra 𝕜 𝕜'] (x : 𝕜')
Goal: ‖(ContinuousLinearMap.mul 𝕜 𝕜') x‖₊ = ‖x‖₊
-- ContinuousLinearMap.coe_mulₗᵢ
@[simp]
theorem coe_mulₗᵢ : ⇑(mulₗᵢ 𝕜 𝕜') = mul 𝕜 𝕜' 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] (𝕜' : Type u_3) [NonUnitalSeminormedRing 𝕜'] [NormedSpace 𝕜 𝕜'] [IsScalarTower 𝕜 𝕜' 𝕜'] [SMulCommClass 𝕜 𝕜' 𝕜'] [RegularNormedAlgebra 𝕜 𝕜']
Goal: ⇑(ContinuousLinearMap.mulₗᵢ 𝕜 𝕜') = ⇑(ContinuousLinearMap.mul 𝕜 𝕜')
-- ContinuousLinearMap.lsmul_apply
@[simp]
theorem lsmul_apply (c : 𝕜') (x : E) : lsmul 𝕜 𝕜' c x = c • x 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] (𝕜' : Type u_3) [NormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' E] [IsScalarTower 𝕜 𝕜' E] (c : 𝕜') (x : E)
Goal: ((ContinuousLinearMap.lsmul 𝕜 𝕜') c) x = c • x
-- ContinuousLinearMap.norm_toSpanSingleton
theorem norm_toSpanSingleton (x : E) : ‖toSpanSingleton 𝕜 x‖ = ‖x‖ 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] (x : E)
Goal: ‖ContinuousLinearMap.toSpanSingleton 𝕜 x‖ = ‖x‖
-- ContinuousLinearMap.opNorm_lsmul_apply_le
theorem opNorm_lsmul_apply_le (x : 𝕜') : ‖(lsmul 𝕜 𝕜' x : E →L[𝕜] E)‖ ≤ ‖x‖ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] {𝕜' : Type u_3} [NormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' E] [IsScalarTower 𝕜 𝕜' E] (x : 𝕜')
Goal: ‖(ContinuousLinearMap.lsmul 𝕜 𝕜') x‖ ≤ ‖x‖
-- ContinuousLinearMap.opNorm_lsmul_le
theorem opNorm_lsmul_le : ‖(lsmul 𝕜 𝕜' : 𝕜' →L[𝕜] E →L[𝕜] E)‖ ≤ 1 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] {𝕜' : Type u_3} [NormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' E] [IsScalarTower 𝕜 𝕜' E]
Goal: ‖ContinuousLinearMap.lsmul 𝕜 𝕜'‖ ≤ 1
-- ContinuousLinearMap.opNorm_mul
@[simp]
theorem opNorm_mul : ‖mul 𝕜 𝕜'‖ = 1 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] (𝕜' : Type u_3) [NonUnitalNormedRing 𝕜'] [NormedSpace 𝕜 𝕜'] [IsScalarTower 𝕜 𝕜' 𝕜'] [SMulCommClass 𝕜 𝕜' 𝕜'] [RegularNormedAlgebra 𝕜 𝕜'] [Nontrivial 𝕜']
Goal: ‖ContinuousLinearMap.mul 𝕜 𝕜'‖ = 1
-- ContinuousLinearMap.opNNNorm_mul
@[simp]
theorem opNNNorm_mul : ‖mul 𝕜 𝕜'‖₊ = 1 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] (𝕜' : Type u_3) [NonUnitalNormedRing 𝕜'] [NormedSpace 𝕜 𝕜'] [IsScalarTower 𝕜 𝕜' 𝕜'] [SMulCommClass 𝕜 𝕜' 𝕜'] [RegularNormedAlgebra 𝕜 𝕜'] [Nontrivial 𝕜']
Goal: ‖ContinuousLinearMap.mul 𝕜 𝕜'‖₊ = 1
-- ContinuousLinearMap.opNorm_lsmul
@[simp]
theorem opNorm_lsmul [NormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' E]
    [IsScalarTower 𝕜 𝕜' E] [Nontrivial E] : ‖(lsmul 𝕜 𝕜' : 𝕜' →L[𝕜] E →L[𝕜] E)‖ = 1 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] (𝕜' : Type u_3) [NormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' E] [IsScalarTower 𝕜 𝕜' E] [Nontrivial E]
Goal: ‖ContinuousLinearMap.lsmul 𝕜 𝕜'‖ = 1
-- ContinuousLinearMap.norm_fst_le
lemma norm_fst_le : ‖fst 𝕜 E F‖ ≤ 1 
Arguments: (𝕜 : Type u_1) (E : Type u_2) (F : Type u_3) [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F]
Goal: ‖ContinuousLinearMap.fst 𝕜 E F‖ ≤ 1
-- ContinuousLinearMap.norm_snd_le
lemma norm_snd_le : ‖snd 𝕜 E F‖ ≤ 1 
Arguments: (𝕜 : Type u_1) (E : Type u_2) (F : Type u_3) [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F]
Goal: ‖ContinuousLinearMap.snd 𝕜 E F‖ ≤ 1
-- ContinuousLinearMap.opNorm_prod
@[simp]
theorem opNorm_prod (f : E →L[𝕜] F) (g : E →L[𝕜] G) : ‖f.prod g‖ = ‖(f, g)‖ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] [NormedSpace 𝕜 G] (f : E →L[𝕜] F) (g : E →L[𝕜] G)
Goal: ‖f.prod g‖ = ‖(f, g)‖
-- ContinuousLinearMap.opNNNorm_prod
@[simp]
theorem opNNNorm_prod (f : E →L[𝕜] F) (g : E →L[𝕜] G) : ‖f.prod g‖₊ = ‖(f, g)‖₊ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] [NormedSpace 𝕜 G] (f : E →L[𝕜] F) (g : E →L[𝕜] G)
Goal: ‖f.prod g‖₊ = ‖(f, g)‖₊
-- ContinuousLinearMap.prodMapL_apply
@[simp]
theorem prodMapL_apply (p : (M₁ →L[𝕜] M₂) × (M₃ →L[𝕜] M₄)) :
    ContinuousLinearMap.prodMapL 𝕜 M₁ M₂ M₃ M₄ p = p.1.prodMap p.2 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {M₁ : Type u_5} {M₂ : Type u_6} {M₃ : Type u_7} {M₄ : Type u_8} [SeminormedAddCommGroup M₁] [NormedSpace 𝕜 M₁] [SeminormedAddCommGroup M₂] [NormedSpace 𝕜 M₂] [SeminormedAddCommGroup M₃] [NormedSpace 𝕜 M₃] [SeminormedAddCommGroup M₄] [NormedSpace 𝕜 M₄] (p : (M₁ →L[𝕜] M₂) × (M₃ →L[𝕜] M₄))
Goal: (ContinuousLinearMap.prodMapL 𝕜 M₁ M₂ M₃ M₄) p = p.1.prodMap p.2
-- Continuous.prod_mapL
theorem _root_.Continuous.prod_mapL {f : X → M₁ →L[𝕜] M₂} {g : X → M₃ →L[𝕜] M₄} (hf : Continuous f)
    (hg : Continuous g) : Continuous fun x => (f x).prodMap (g x) 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {M₁ : Type u_5} {M₂ : Type u_6} {M₃ : Type u_7} {M₄ : Type u_8} [SeminormedAddCommGroup M₁] [NormedSpace 𝕜 M₁] [SeminormedAddCommGroup M₂] [NormedSpace 𝕜 M₂] [SeminormedAddCommGroup M₃] [NormedSpace 𝕜 M₃] [SeminormedAddCommGroup M₄] [NormedSpace 𝕜 M₄] {X : Type u_9} [TopologicalSpace X] {f : X → M₁ →L[𝕜] M₂} {g : X → M₃ →L[𝕜] M₄} (hf : Continuous f) (hg : Continuous g)
Goal: Continuous fun x => (f x).prodMap (g x)
-- Continuous.prod_map_equivL
theorem _root_.Continuous.prod_map_equivL {f : X → M₁ ≃L[𝕜] M₂} {g : X → M₃ ≃L[𝕜] M₄}
    (hf : Continuous fun x => (f x : M₁ →L[𝕜] M₂)) (hg : Continuous fun x => (g x : M₃ →L[𝕜] M₄)) :
    Continuous fun x => ((f x).prod (g x) : M₁ × M₃ →L[𝕜] M₂ × M₄) 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {M₁ : Type u_5} {M₂ : Type u_6} {M₃ : Type u_7} {M₄ : Type u_8} [SeminormedAddCommGroup M₁] [NormedSpace 𝕜 M₁] [SeminormedAddCommGroup M₂] [NormedSpace 𝕜 M₂] [SeminormedAddCommGroup M₃] [NormedSpace 𝕜 M₃] [SeminormedAddCommGroup M₄] [NormedSpace 𝕜 M₄] {X : Type u_9} [TopologicalSpace X] {f : X → M₁ ≃L[𝕜] M₂} {g : X → M₃ ≃L[𝕜] M₄} (hf : Continuous fun x => ↑(f x)) (hg : Continuous fun x => ↑(g x))
Goal: Continuous fun x => ↑((f x).prod (g x))
-- ContinuousOn.prod_mapL
theorem _root_.ContinuousOn.prod_mapL {f : X → M₁ →L[𝕜] M₂} {g : X → M₃ →L[𝕜] M₄} {s : Set X}
    (hf : ContinuousOn f s) (hg : ContinuousOn g s) :
    ContinuousOn (fun x => (f x).prodMap (g x)) s 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {M₁ : Type u_5} {M₂ : Type u_6} {M₃ : Type u_7} {M₄ : Type u_8} [SeminormedAddCommGroup M₁] [NormedSpace 𝕜 M₁] [SeminormedAddCommGroup M₂] [NormedSpace 𝕜 M₂] [SeminormedAddCommGroup M₃] [NormedSpace 𝕜 M₃] [SeminormedAddCommGroup M₄] [NormedSpace 𝕜 M₄] {X : Type u_9} [TopologicalSpace X] {f : X → M₁ →L[𝕜] M₂} {g : X → M₃ →L[𝕜] M₄} {s : Set X} (hf : ContinuousOn f s) (hg : ContinuousOn g s)
Goal: ContinuousOn (fun x => (f x).prodMap (g x)) s
-- ContinuousOn.prod_map_equivL
theorem _root_.ContinuousOn.prod_map_equivL {f : X → M₁ ≃L[𝕜] M₂} {g : X → M₃ ≃L[𝕜] M₄} {s : Set X}
    (hf : ContinuousOn (fun x => (f x : M₁ →L[𝕜] M₂)) s)
    (hg : ContinuousOn (fun x => (g x : M₃ →L[𝕜] M₄)) s) :
    ContinuousOn (fun x => ((f x).prod (g x) : M₁ × M₃ →L[𝕜] M₂ × M₄)) s 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {M₁ : Type u_5} {M₂ : Type u_6} {M₃ : Type u_7} {M₄ : Type u_8} [SeminormedAddCommGroup M₁] [NormedSpace 𝕜 M₁] [SeminormedAddCommGroup M₂] [NormedSpace 𝕜 M₂] [SeminormedAddCommGroup M₃] [NormedSpace 𝕜 M₃] [SeminormedAddCommGroup M₄] [NormedSpace 𝕜 M₄] {X : Type u_9} [TopologicalSpace X] {f : X → M₁ ≃L[𝕜] M₂} {g : X → M₃ ≃L[𝕜] M₄} {s : Set X} (hf : ContinuousOn (fun x => ↑(f x)) s) (hg : ContinuousOn (fun x => ↑(g x)) s)
Goal: ContinuousOn (fun x => ↑((f x).prod (g x))) s
-- ContinuousLinearMap.norm_fst
@[simp] lemma norm_fst [NormedAddCommGroup E] [NormedSpace 𝕜 E]
    [SeminormedAddCommGroup F] [NormedSpace 𝕜 F] [Nontrivial E] :
    ‖fst 𝕜 E F‖ = 1 
Arguments: (𝕜 : Type u_1) (E : Type u_2) (F : Type u_3) [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [SeminormedAddCommGroup F] [NormedSpace 𝕜 F] [Nontrivial E]
Goal: ‖ContinuousLinearMap.fst 𝕜 E F‖ = 1
-- ContinuousLinearMap.norm_snd
@[simp] lemma norm_snd [SeminormedAddCommGroup E] [NormedSpace 𝕜 E]
    [NormedAddCommGroup F] [NormedSpace 𝕜 F] [Nontrivial F]  :
    ‖snd 𝕜 E F‖ = 1 
Arguments: (𝕜 : Type u_1) (E : Type u_2) (F : Type u_3) [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [Nontrivial F]
Goal: ‖ContinuousLinearMap.snd 𝕜 E F‖ = 1
-- separate_convex_open_set
theorem separate_convex_open_set [TopologicalSpace E] [AddCommGroup E] [TopologicalAddGroup E]
    [Module ℝ E] [ContinuousSMul ℝ E] {s : Set E} (hs₀ : (0 : E) ∈ s) (hs₁ : Convex ℝ s)
    (hs₂ : IsOpen s) {x₀ : E} (hx₀ : x₀ ∉ s) : ∃ f : E →L[ℝ] ℝ, f x₀ = 1 ∧ ∀ x ∈ s, f x < 1 
Arguments: {E : Type u_2} [TopologicalSpace E] [AddCommGroup E] [TopologicalAddGroup E] [Module ℝ E] [ContinuousSMul ℝ E] {s : Set E} (hs₀ : 0 ∈ s) (hs₁ : Convex ℝ s) (hs₂ : IsOpen s) {x₀ : E} (hx₀ : x₀ ∉ s)
Goal: ∃ f, f x₀ = 1 ∧ ∀ x ∈ s, f x < 1
-- geometric_hahn_banach_open
theorem geometric_hahn_banach_open (hs₁ : Convex ℝ s) (hs₂ : IsOpen s) (ht : Convex ℝ t)
    (disj : Disjoint s t) : ∃ (f : E →L[ℝ] ℝ) (u : ℝ), (∀ a ∈ s, f a < u) ∧ ∀ b ∈ t, u ≤ f b 
Arguments: {E : Type u_2} [TopologicalSpace E] [AddCommGroup E] [TopologicalAddGroup E] [Module ℝ E] [ContinuousSMul ℝ E] {s t : Set E} (hs₁ : Convex ℝ s) (hs₂ : IsOpen s) (ht : Convex ℝ t) (disj : Disjoint s t)
Goal: ∃ f u, (∀ a ∈ s, f a < u) ∧ ∀ b ∈ t, u ≤ f b
-- geometric_hahn_banach_open_point
theorem geometric_hahn_banach_open_point (hs₁ : Convex ℝ s) (hs₂ : IsOpen s) (disj : x ∉ s) :
    ∃ f : E →L[ℝ] ℝ, ∀ a ∈ s, f a < f x 
Arguments: {E : Type u_2} [TopologicalSpace E] [AddCommGroup E] [TopologicalAddGroup E] [Module ℝ E] [ContinuousSMul ℝ E] {s : Set E} {x : E} (hs₁ : Convex ℝ s) (hs₂ : IsOpen s) (disj : x ∉ s)
Goal: ∃ f, ∀ a ∈ s, f a < f x
-- geometric_hahn_banach_point_open
theorem geometric_hahn_banach_point_open (ht₁ : Convex ℝ t) (ht₂ : IsOpen t) (disj : x ∉ t) :
    ∃ f : E →L[ℝ] ℝ, ∀ b ∈ t, f x < f b 
Arguments: {E : Type u_2} [TopologicalSpace E] [AddCommGroup E] [TopologicalAddGroup E] [Module ℝ E] [ContinuousSMul ℝ E] {t : Set E} {x : E} (ht₁ : Convex ℝ t) (ht₂ : IsOpen t) (disj : x ∉ t)
Goal: ∃ f, ∀ b ∈ t, f x < f b
-- geometric_hahn_banach_open_open
theorem geometric_hahn_banach_open_open (hs₁ : Convex ℝ s) (hs₂ : IsOpen s) (ht₁ : Convex ℝ t)
    (ht₃ : IsOpen t) (disj : Disjoint s t) :
    ∃ (f : E →L[ℝ] ℝ) (u : ℝ), (∀ a ∈ s, f a < u) ∧ ∀ b ∈ t, u < f b 
Arguments: {E : Type u_2} [TopologicalSpace E] [AddCommGroup E] [TopologicalAddGroup E] [Module ℝ E] [ContinuousSMul ℝ E] {s t : Set E} (hs₁ : Convex ℝ s) (hs₂ : IsOpen s) (ht₁ : Convex ℝ t) (ht₃ : IsOpen t) (disj : Disjoint s t)
Goal: ∃ f u, (∀ a ∈ s, f a < u) ∧ ∀ b ∈ t, u < f b
-- geometric_hahn_banach_compact_closed
theorem geometric_hahn_banach_compact_closed (hs₁ : Convex ℝ s) (hs₂ : IsCompact s)
    (ht₁ : Convex ℝ t) (ht₂ : IsClosed t) (disj : Disjoint s t) :
    ∃ (f : E →L[ℝ] ℝ) (u v : ℝ), (∀ a ∈ s, f a < u) ∧ u < v ∧ ∀ b ∈ t, v < f b 
Arguments: {E : Type u_2} [TopologicalSpace E] [AddCommGroup E] [TopologicalAddGroup E] [Module ℝ E] [ContinuousSMul ℝ E] {s t : Set E} [LocallyConvexSpace ℝ E] (hs₁ : Convex ℝ s) (hs₂ : IsCompact s) (ht₁ : Convex ℝ t) (ht₂ : IsClosed t) (disj : Disjoint s t)
Goal: ∃ f u v, (∀ a ∈ s, f a < u) ∧ u < v ∧ ∀ b ∈ t, v < f b
-- geometric_hahn_banach_closed_compact
theorem geometric_hahn_banach_closed_compact (hs₁ : Convex ℝ s) (hs₂ : IsClosed s)
    (ht₁ : Convex ℝ t) (ht₂ : IsCompact t) (disj : Disjoint s t) :
    ∃ (f : E →L[ℝ] ℝ) (u v : ℝ), (∀ a ∈ s, f a < u) ∧ u < v ∧ ∀ b ∈ t, v < f b 
Arguments: {E : Type u_2} [TopologicalSpace E] [AddCommGroup E] [TopologicalAddGroup E] [Module ℝ E] [ContinuousSMul ℝ E] {s t : Set E} [LocallyConvexSpace ℝ E] (hs₁ : Convex ℝ s) (hs₂ : IsClosed s) (ht₁ : Convex ℝ t) (ht₂ : IsCompact t) (disj : Disjoint s t)
Goal: ∃ f u v, (∀ a ∈ s, f a < u) ∧ u < v ∧ ∀ b ∈ t, v < f b
-- geometric_hahn_banach_point_closed
theorem geometric_hahn_banach_point_closed (ht₁ : Convex ℝ t) (ht₂ : IsClosed t) (disj : x ∉ t) :
    ∃ (f : E →L[ℝ] ℝ) (u : ℝ), f x < u ∧ ∀ b ∈ t, u < f b 
Arguments: {E : Type u_2} [TopologicalSpace E] [AddCommGroup E] [TopologicalAddGroup E] [Module ℝ E] [ContinuousSMul ℝ E] {t : Set E} {x : E} [LocallyConvexSpace ℝ E] (ht₁ : Convex ℝ t) (ht₂ : IsClosed t) (disj : x ∉ t)
Goal: ∃ f u, f x < u ∧ ∀ b ∈ t, u < f b