ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Complex.arg_neg_I
@[simp]
theorem arg_neg_I : arg (-I) = -(π / 2) 
Goal: (-Complex.I).arg = -(π / 2)
-- Complex.tan_arg
@[simp]
theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re 
Arguments: (x : ℂ)
Goal: Real.tan x.arg = x.im / x.re
-- Complex.arg_ofReal_of_nonneg
theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 
Arguments: {x : ℝ} (hx : 0 ≤ x)
Goal: (↑x).arg = 0
-- Complex.natCast_arg
@[simp, norm_cast]
lemma natCast_arg {n : ℕ} : arg n = 0 
Arguments: {n : ℕ}
Goal: (↑n).arg = 0
-- Complex.ofNat_arg
@[simp]
lemma ofNat_arg {n : ℕ} [n.AtLeastTwo] : arg (no_index (OfNat.ofNat n)) = 0 
Arguments: {n : ℕ} [n.AtLeastTwo]
Goal: (OfNat.ofNat n).arg = 0
-- Complex.arg_eq_zero_iff
theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 
Arguments: {z : ℂ}
Goal: z.arg = 0 ↔ 0 ≤ z.re ∧ z.im = 0
-- Complex.arg_eq_zero_iff_zero_le
lemma arg_eq_zero_iff_zero_le {z : ℂ} : arg z = 0 ↔ 0 ≤ z 
Arguments: {z : ℂ}
Goal: z.arg = 0 ↔ 0 ≤ z
-- Complex.arg_eq_pi_iff
theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 
Arguments: {z : ℂ}
Goal: z.arg = π ↔ z.re < 0 ∧ z.im = 0
-- Complex.arg_eq_pi_iff_lt_zero
lemma arg_eq_pi_iff_lt_zero {z : ℂ} : arg z = π ↔ z < 0 
Arguments: {z : ℂ}
Goal: z.arg = π ↔ z < 0
-- Complex.arg_lt_pi_iff
theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 
Arguments: {z : ℂ}
Goal: z.arg < π ↔ 0 ≤ z.re ∨ z.im ≠ 0
-- Complex.arg_ofReal_of_neg
theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π 
Arguments: {x : ℝ} (hx : x < 0)
Goal: (↑x).arg = π
-- Complex.arg_eq_pi_div_two_iff
theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im 
Arguments: {z : ℂ}
Goal: z.arg = π / 2 ↔ z.re = 0 ∧ 0 < z.im
-- Complex.arg_eq_neg_pi_div_two_iff
theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 
Arguments: {z : ℂ}
Goal: z.arg = -(π / 2) ↔ z.re = 0 ∧ z.im < 0
-- Complex.arg_of_re_nonneg
theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) 
Arguments: {x : ℂ} (hx : 0 ≤ x.re)
Goal: x.arg = Real.arcsin (x.im / Complex.abs x)
-- Complex.arg_of_re_neg_of_im_nonneg
theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) :
    arg x = Real.arcsin ((-x).im / abs x) + π 
Arguments: {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im)
Goal: x.arg = Real.arcsin ((-x).im / Complex.abs x) + π
-- Complex.arg_of_re_neg_of_im_neg
theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) :
    arg x = Real.arcsin ((-x).im / abs x) - π 
Arguments: {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0)
Goal: x.arg = Real.arcsin ((-x).im / Complex.abs x) - π
-- Complex.arg_of_im_nonneg_of_ne_zero
theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) :
    arg z = Real.arccos (z.re / abs z) 
Arguments: {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0)
Goal: z.arg = Real.arccos (z.re / Complex.abs z)
-- Complex.arg_of_im_pos
theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) 
Arguments: {z : ℂ} (hz : 0 < z.im)
Goal: z.arg = Real.arccos (z.re / Complex.abs z)
-- Complex.arg_of_im_neg
theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) 
Arguments: {z : ℂ} (hz : z.im < 0)
Goal: z.arg = -Real.arccos (z.re / Complex.abs z)
-- Complex.arg_conj
theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x 
Arguments: (x : ℂ)
Goal: ((starRingEnd ℂ) x).arg = if x.arg = π then π else -x.arg
-- Complex.arg_inv
theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x 
Arguments: (x : ℂ)
Goal: x⁻¹.arg = if x.arg = π then π else -x.arg
-- Complex.abs_arg_inv
@[simp] lemma abs_arg_inv (x : ℂ) : |x⁻¹.arg| = |x.arg| 
Arguments: (x : ℂ)
Goal: |x⁻¹.arg| = |x.arg|
-- Complex.abs_eq_one_iff'
lemma abs_eq_one_iff' : abs x = 1 ↔ ∃ θ ∈ Set.Ioc (-π) π, exp (θ * I) = x 
Arguments: {x : ℂ}
Goal: Complex.abs x = 1 ↔ ∃ θ ∈ Set.Ioc (-π) π, Complex.exp (↑θ * Complex.I) = x
-- Complex.image_exp_Ioc_eq_sphere
lemma image_exp_Ioc_eq_sphere : (fun θ : ℝ ↦ exp (θ * I)) '' Set.Ioc (-π) π = sphere 0 1 
Goal: (fun θ => Complex.exp (↑θ * Complex.I)) '' Set.Ioc (-π) π = Metric.sphere 0 1
-- Complex.arg_le_pi_div_two_iff
theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 
Arguments: {z : ℂ}
Goal: z.arg ≤ π / 2 ↔ 0 ≤ z.re ∨ z.im < 0
-- Complex.neg_pi_div_two_le_arg_iff
theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z 
Arguments: {z : ℂ}
Goal: -(π / 2) ≤ z.arg ↔ 0 ≤ z.re ∨ 0 ≤ z.im
-- Complex.neg_pi_div_two_lt_arg_iff
lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z 
Arguments: {z : ℂ}
Goal: -(π / 2) < z.arg ↔ 0 < z.re ∨ 0 ≤ z.im
-- Complex.arg_lt_pi_div_two_iff
lemma arg_lt_pi_div_two_iff {z : ℂ} : arg z < π / 2 ↔ 0 < re z ∨ im z < 0 ∨ z = 0 
Arguments: {z : ℂ}
Goal: z.arg < π / 2 ↔ 0 < z.re ∨ z.im < 0 ∨ z = 0
-- Complex.abs_arg_le_pi_div_two_iff
@[simp]
theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z 
Arguments: {z : ℂ}
Goal: |z.arg| ≤ π / 2 ↔ 0 ≤ z.re
-- Complex.abs_arg_lt_pi_div_two_iff
@[simp]
theorem abs_arg_lt_pi_div_two_iff {z : ℂ} : |arg z| < π / 2 ↔ 0 < re z ∨ z = 0 
Arguments: {z : ℂ}
Goal: |z.arg| < π / 2 ↔ 0 < z.re ∨ z = 0
-- Complex.arg_conj_coe_angle
@[simp]
theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x 
Arguments: (x : ℂ)
Goal: ↑((starRingEnd ℂ) x).arg = -↑x.arg
-- Complex.arg_inv_coe_angle
@[simp]
theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x 
Arguments: (x : ℂ)
Goal: ↑x⁻¹.arg = -↑x.arg
-- Complex.arg_neg_eq_arg_sub_pi_of_im_pos
theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π 
Arguments: {x : ℂ} (hi : 0 < x.im)
Goal: (-x).arg = x.arg - π
-- Complex.arg_neg_eq_arg_add_pi_of_im_neg
theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π 
Arguments: {x : ℂ} (hi : x.im < 0)
Goal: (-x).arg = x.arg + π
-- Complex.arg_neg_eq_arg_sub_pi_iff
theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} :
    arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 
Arguments: {x : ℂ}
Goal: (-x).arg = x.arg - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0
-- Complex.arg_neg_eq_arg_add_pi_iff
theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} :
    arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re 
Arguments: {x : ℂ}
Goal: (-x).arg = x.arg + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re
-- Complex.arg_neg_coe_angle
theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π 
Arguments: {x : ℂ} (hx : x ≠ 0)
Goal: ↑(-x).arg = ↑x.arg + ↑π
-- Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod
theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) :
    arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ 
Arguments: {r : ℝ} (hr : 0 < r) (θ : ℝ)
Goal: (↑r * (Complex.cos ↑θ + Complex.sin ↑θ * Complex.I)).arg = toIocMod Real.two_pi_pos (-π) θ
-- Complex.arg_cos_add_sin_mul_I_eq_toIocMod
theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) :
    arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ 
Arguments: (θ : ℝ)
Goal: (Complex.cos ↑θ + Complex.sin ↑θ * Complex.I).arg = toIocMod Real.two_pi_pos (-π) θ
-- Complex.arg_mul_cos_add_sin_mul_I_sub
theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) :
    arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ 
Arguments: {r : ℝ} (hr : 0 < r) (θ : ℝ)
Goal: (↑r * (Complex.cos ↑θ + Complex.sin ↑θ * Complex.I)).arg - θ = 2 * π * ↑⌊(π - θ) / (2 * π)⌋
-- Complex.arg_cos_add_sin_mul_I_sub
theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) :
    arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ 
Arguments: (θ : ℝ)
Goal: (Complex.cos ↑θ + Complex.sin ↑θ * Complex.I).arg - θ = 2 * π * ↑⌊(π - θ) / (2 * π)⌋
-- Complex.arg_mul_cos_add_sin_mul_I_coe_angle
theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) :
    (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ 
Arguments: {r : ℝ} (hr : 0 < r) (θ : Real.Angle)
Goal: ↑(↑r * (↑θ.cos + ↑θ.sin * Complex.I)).arg = θ
-- Complex.arg_cos_add_sin_mul_I_coe_angle
theorem arg_cos_add_sin_mul_I_coe_angle (θ : Real.Angle) :
    (arg (Real.Angle.cos θ + Real.Angle.sin θ * I) : Real.Angle) = θ 
Arguments: (θ : Real.Angle)
Goal: ↑(↑θ.cos + ↑θ.sin * Complex.I).arg = θ
-- Complex.arg_mul_coe_angle
theorem arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) :
    (arg (x * y) : Real.Angle) = arg x + arg y 
Arguments: {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0)
Goal: ↑(x * y).arg = ↑x.arg + ↑y.arg
-- Complex.arg_div_coe_angle
theorem arg_div_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) :
    (arg (x / y) : Real.Angle) = arg x - arg y 
Arguments: {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0)
Goal: ↑(x / y).arg = ↑x.arg - ↑y.arg
-- Complex.arg_coe_angle_toReal_eq_arg
@[simp]
theorem arg_coe_angle_toReal_eq_arg (z : ℂ) : (arg z : Real.Angle).toReal = arg z 
Arguments: (z : ℂ)
Goal: (↑z.arg).toReal = z.arg
-- Complex.arg_coe_angle_eq_iff_eq_toReal
theorem arg_coe_angle_eq_iff_eq_toReal {z : ℂ} {θ : Real.Angle} :
    (arg z : Real.Angle) = θ ↔ arg z = θ.toReal 
Arguments: {z : ℂ} {θ : Real.Angle}
Goal: ↑z.arg = θ ↔ z.arg = θ.toReal
-- Complex.arg_coe_angle_eq_iff
@[simp]
theorem arg_coe_angle_eq_iff {x y : ℂ} : (arg x : Real.Angle) = arg y ↔ arg x = arg y 
Arguments: {x y : ℂ}
Goal: ↑x.arg = ↑y.arg ↔ x.arg = y.arg
-- Complex.arg_mul_eq_add_arg_iff
lemma arg_mul_eq_add_arg_iff {x y : ℂ} (hx₀ : x ≠ 0) (hy₀ : y ≠ 0) :
    (x * y).arg = x.arg + y.arg ↔ arg x + arg y ∈ Set.Ioc (-π) π 
Arguments: {x y : ℂ} (hx₀ : x ≠ 0) (hy₀ : y ≠ 0)
Goal: (x * y).arg = x.arg + y.arg ↔ x.arg + y.arg ∈ Set.Ioc (-π) π
-- Complex.mem_slitPlane_iff_arg
lemma mem_slitPlane_iff_arg {z : ℂ} : z ∈ slitPlane ↔ z.arg ≠ π ∧ z ≠ 0 
Arguments: {z : ℂ}
Goal: z ∈ Complex.slitPlane ↔ z.arg ≠ π ∧ z ≠ 0
-- Complex.slitPlane_arg_ne_pi
lemma slitPlane_arg_ne_pi {z : ℂ} (hz : z ∈ slitPlane) : z.arg ≠ Real.pi 
Arguments: {z : ℂ} (hz : z ∈ Complex.slitPlane)
Goal: z.arg ≠ π
-- Complex.arg_eq_nhds_of_re_pos
theorem arg_eq_nhds_of_re_pos (hx : 0 < x.re) : arg =ᶠ[𝓝 x] fun x => Real.arcsin (x.im / abs x) 
Arguments: {x : ℂ} (hx : 0 < x.re)
Goal: Complex.arg =ᶠ[𝓝 x] fun x => Real.arcsin (x.im / Complex.abs x)
-- Complex.arg_eq_nhds_of_re_neg_of_im_pos
theorem arg_eq_nhds_of_re_neg_of_im_pos (hx_re : x.re < 0) (hx_im : 0 < x.im) :
    arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) + π 
Arguments: {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 < x.im)
Goal: Complex.arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / Complex.abs x) + π
-- Complex.arg_eq_nhds_of_re_neg_of_im_neg
theorem arg_eq_nhds_of_re_neg_of_im_neg (hx_re : x.re < 0) (hx_im : x.im < 0) :
    arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) - π 
Arguments: {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0)
Goal: Complex.arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / Complex.abs x) - π
-- Complex.arg_eq_nhds_of_im_pos
theorem arg_eq_nhds_of_im_pos (hz : 0 < im z) : arg =ᶠ[𝓝 z] fun x => Real.arccos (x.re / abs x) 
Arguments: {z : ℂ} (hz : 0 < z.im)
Goal: Complex.arg =ᶠ[𝓝 z] fun x => Real.arccos (x.re / Complex.abs x)
-- Complex.arg_eq_nhds_of_im_neg
theorem arg_eq_nhds_of_im_neg (hz : im z < 0) : arg =ᶠ[𝓝 z] fun x => -Real.arccos (x.re / abs x) 
Arguments: {z : ℂ} (hz : z.im < 0)
Goal: Complex.arg =ᶠ[𝓝 z] fun x => -Real.arccos (x.re / Complex.abs x)
-- Complex.continuousAt_arg
theorem continuousAt_arg (h : x ∈ slitPlane) : ContinuousAt arg x 
Arguments: {x : ℂ} (h : x ∈ Complex.slitPlane)
Goal: ContinuousAt Complex.arg x
-- Complex.tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero
theorem tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0)
    (him : z.im = 0) : Tendsto arg (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) 
Arguments: {z : ℂ} (hre : z.re < 0) (him : z.im = 0)
Goal: Filter.Tendsto Complex.arg (𝓝[{z | z.im < 0}] z) (𝓝 (-π))
-- Complex.continuousWithinAt_arg_of_re_neg_of_im_zero
theorem continuousWithinAt_arg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) :
    ContinuousWithinAt arg { z : ℂ | 0 ≤ z.im } z 
Arguments: {z : ℂ} (hre : z.re < 0) (him : z.im = 0)
Goal: ContinuousWithinAt Complex.arg {z | 0 ≤ z.im} z
-- Complex.tendsto_arg_nhdsWithin_im_nonneg_of_re_neg_of_im_zero
theorem tendsto_arg_nhdsWithin_im_nonneg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0)
    (him : z.im = 0) : Tendsto arg (𝓝[{ z : ℂ | 0 ≤ z.im }] z) (𝓝 π) 
Arguments: {z : ℂ} (hre : z.re < 0) (him : z.im = 0)
Goal: Filter.Tendsto Complex.arg (𝓝[{z | 0 ≤ z.im}] z) (𝓝 π)
-- Complex.continuousAt_arg_coe_angle
theorem continuousAt_arg_coe_angle (h : x ≠ 0) : ContinuousAt ((↑) ∘ arg : ℂ → Real.Angle) x 
Arguments: {x : ℂ} (h : x ≠ 0)
Goal: ContinuousAt (Real.Angle.coe ∘ Complex.arg) x
-- IsLprojection.proj
structure IsLprojection (P : M) : Prop where
  proj : IsIdempotentElem P
  Lnorm : ∀ x : X, ‖x‖ = ‖P • x‖ + ‖(1 - P) • x‖
Arguments: {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] {P : M} (self : IsLprojection X P)
Goal: IsIdempotentElem P
-- IsLprojection.Lnorm
structure IsLprojection (P : M) : Prop where
  proj : IsIdempotentElem P
  Lnorm : ∀ x : X, ‖x‖ = ‖P • x‖ + ‖(1 - P) • x‖
Arguments: {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] {P : M} (self : IsLprojection X P) (x : X)
Goal: ‖x‖ = ‖P • x‖ + ‖(1 - P) • x‖
-- IsMprojection.proj
structure IsMprojection (P : M) : Prop where
  proj : IsIdempotentElem P
  Mnorm : ∀ x : X, ‖x‖ = max ‖P • x‖ ‖(1 - P) • x‖
Arguments: {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] {P : M} (self : IsMprojection X P)
Goal: IsIdempotentElem P
-- IsMprojection.Mnorm
structure IsMprojection (P : M) : Prop where
  proj : IsIdempotentElem P
  Mnorm : ∀ x : X, ‖x‖ = max ‖P • x‖ ‖(1 - P) • x‖
Arguments: {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] {P : M} (self : IsMprojection X P) (x : X)
Goal: ‖x‖ = max ‖P • x‖ ‖(1 - P) • x‖
-- IsLprojection.Lcomplement
theorem Lcomplement {P : M} (h : IsLprojection X P) : IsLprojection X (1 - P) 
Arguments: {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] {P : M} (h : IsLprojection X P)
Goal: IsLprojection X (1 - P)
-- IsLprojection.Lcomplement_iff
theorem Lcomplement_iff (P : M) : IsLprojection X P ↔ IsLprojection X (1 - P) 
Arguments: {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] (P : M)
Goal: IsLprojection X P ↔ IsLprojection X (1 - P)
-- IsLprojection.commute
theorem commute [FaithfulSMul M X] {P Q : M} (h₁ : IsLprojection X P) (h₂ : IsLprojection X Q) :
    Commute P Q 
Arguments: {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] [FaithfulSMul M X] {P Q : M} (h₁ : IsLprojection X P) (h₂ : IsLprojection X Q)
Goal: Commute P Q
-- IsLprojection.mul
theorem mul [FaithfulSMul M X] {P Q : M} (h₁ : IsLprojection X P) (h₂ : IsLprojection X Q) :
    IsLprojection X (P * Q) 
Arguments: {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] [FaithfulSMul M X] {P Q : M} (h₁ : IsLprojection X P) (h₂ : IsLprojection X Q)
Goal: IsLprojection X (P * Q)
-- IsLprojection.join
theorem join [FaithfulSMul M X] {P Q : M} (h₁ : IsLprojection X P) (h₂ : IsLprojection X Q) :
    IsLprojection X (P + Q - P * Q) 
Arguments: {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] [FaithfulSMul M X] {P Q : M} (h₁ : IsLprojection X P) (h₂ : IsLprojection X Q)
Goal: IsLprojection X (P + Q - P * Q)
-- IsLprojection.coe_compl
@[simp]
theorem coe_compl (P : { P : M // IsLprojection X P }) : ↑Pᶜ = (1 : M) - ↑P 
Arguments: {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] (P : { P // IsLprojection X P })
Goal: ↑Pᶜ = 1 - ↑P
-- IsLprojection.coe_inf
@[simp]
theorem coe_inf [FaithfulSMul M X] (P Q : { P : M // IsLprojection X P }) :
    ↑(P ⊓ Q) = (↑P : M) * ↑Q 
Arguments: {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] [FaithfulSMul M X] (P Q : { P // IsLprojection X P })
Goal: ↑(P ⊓ Q) = ↑P * ↑Q
-- IsLprojection.coe_sup
@[simp]
theorem coe_sup [FaithfulSMul M X] (P Q : { P : M // IsLprojection X P }) :
    ↑(P ⊔ Q) = (↑P : M) + ↑Q - ↑P * ↑Q 
Arguments: {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] [FaithfulSMul M X] (P Q : { P // IsLprojection X P })
Goal: ↑(P ⊔ Q) = ↑P + ↑Q - ↑P * ↑Q
-- IsLprojection.coe_sdiff
@[simp]
theorem coe_sdiff [FaithfulSMul M X] (P Q : { P : M // IsLprojection X P }) :
    ↑(P \ Q) = (↑P : M) * (1 - ↑Q) 
Arguments: {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] [FaithfulSMul M X] (P Q : { P // IsLprojection X P })
Goal: ↑(P \ Q) = ↑P * (1 - ↑Q)
-- IsLprojection.le_def
theorem le_def [FaithfulSMul M X] (P Q : { P : M // IsLprojection X P }) :
    P ≤ Q ↔ (P : M) = ↑(P ⊓ Q) 
Arguments: {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] [FaithfulSMul M X] (P Q : { P // IsLprojection X P })
Goal: P ≤ Q ↔ ↑P = ↑(P ⊓ Q)
-- IsLprojection.coe_zero
@[simp]
theorem coe_zero : ↑(0 : { P : M // IsLprojection X P }) = (0 : M) 
Arguments: {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X]
Goal: ↑0 = 0
-- IsLprojection.coe_one
@[simp]
theorem coe_one : ↑(1 : { P : M // IsLprojection X P }) = (1 : M) 
Arguments: {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X]
Goal: ↑1 = 1
-- IsLprojection.coe_bot
@[simp]
theorem coe_bot [FaithfulSMul M X] :
    -- Porting note: Manual correction of name required here
    ↑(BoundedOrder.toOrderBot.toBot.bot : { P : M // IsLprojection X P }) = (0 : M) 
Arguments: {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] [FaithfulSMul M X]
Goal: ↑⊥ = 0
-- IsLprojection.coe_top
@[simp]
theorem coe_top [FaithfulSMul M X] :
    -- Porting note: Manual correction of name required here
    ↑(BoundedOrder.toOrderTop.toTop.top : { P : M // IsLprojection X P }) = (1 : M) 
Arguments: {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] [FaithfulSMul M X]
Goal: ↑⊤ = 1
-- IsLprojection.compl_mul
theorem compl_mul {P : { P : M // IsLprojection X P }} {Q : M} : ↑Pᶜ * Q = Q - ↑P * Q 
Arguments: {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] {P : { P // IsLprojection X P }} {Q : M}
Goal: ↑Pᶜ * Q = Q - ↑P * Q
-- IsLprojection.mul_compl_self
theorem mul_compl_self {P : { P : M // IsLprojection X P }} : (↑P : M) * ↑Pᶜ = 0 
Arguments: {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] {P : { P // IsLprojection X P }}
Goal: ↑P * ↑Pᶜ = 0
-- IsLprojection.distrib_lattice_lemma
theorem distrib_lattice_lemma [FaithfulSMul M X] {P Q R : { P : M // IsLprojection X P }} :
    ((↑P : M) + ↑Pᶜ * R) * (↑P + ↑Q * ↑R * ↑Pᶜ) = ↑P + ↑Q * ↑R * ↑Pᶜ 
Arguments: {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] [FaithfulSMul M X] {P Q R : { P // IsLprojection X P }}
Goal: (↑P + ↑Pᶜ * ↑R) * (↑P + ↑Q * ↑R * ↑Pᶜ) = ↑P + ↑Q * ↑R * ↑Pᶜ
-- IsometryEquiv.midpoint_fixed
theorem midpoint_fixed {x y : PE} :
    ∀ e : PE ≃ᵢ PE, e x = x → e y = y → e (midpoint ℝ x y) = midpoint ℝ x y 
Arguments: {E : Type u_1} {PE : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [MetricSpace PE] [NormedAddTorsor E PE] {x y : PE} (e : PE ≃ᵢ PE)
Goal: e x = x → e y = y → e (midpoint ℝ x y) = midpoint ℝ x y
-- IsometryEquiv.map_midpoint
theorem map_midpoint (f : PE ≃ᵢ PF) (x y : PE) : f (midpoint ℝ x y) = midpoint ℝ (f x) (f y) 
Arguments: {E : Type u_1} {PE : Type u_2} {F : Type u_3} {PF : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E] [MetricSpace PE] [NormedAddTorsor E PE] [NormedAddCommGroup F] [NormedSpace ℝ F] [MetricSpace PF] [NormedAddTorsor F PF] (f : PE ≃ᵢ PF) (x y : PE)
Goal: f (midpoint ℝ x y) = midpoint ℝ (f x) (f y)
-- IsometryEquiv.coe_toRealLinearIsometryEquivOfMapZero
@[simp]
theorem coe_toRealLinearIsometryEquivOfMapZero (f : E ≃ᵢ F) (h0 : f 0 = 0) :
    ⇑(f.toRealLinearIsometryEquivOfMapZero h0) = f 
Arguments: {E : Type u_1} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] (f : E ≃ᵢ F) (h0 : f 0 = 0)
Goal: ⇑(f.toRealLinearIsometryEquivOfMapZero h0) = ⇑f
-- IsometryEquiv.coe_toRealLinearIsometryEquivOfMapZero_symm
@[simp]
theorem coe_toRealLinearIsometryEquivOfMapZero_symm (f : E ≃ᵢ F) (h0 : f 0 = 0) :
    ⇑(f.toRealLinearIsometryEquivOfMapZero h0).symm = f.symm 
Arguments: {E : Type u_1} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] (f : E ≃ᵢ F) (h0 : f 0 = 0)
Goal: ⇑(f.toRealLinearIsometryEquivOfMapZero h0).symm = ⇑f.symm
-- IsometryEquiv.toRealLinearIsometryEquiv_apply
@[simp]
theorem toRealLinearIsometryEquiv_apply (f : E ≃ᵢ F) (x : E) :
    (f.toRealLinearIsometryEquiv : E → F) x = f x - f 0 
Arguments: {E : Type u_1} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] (f : E ≃ᵢ F) (x : E)
Goal: f.toRealLinearIsometryEquiv x = f x - f 0
-- IsometryEquiv.toRealLinearIsometryEquiv_symm_apply
@[simp]
theorem toRealLinearIsometryEquiv_symm_apply (f : E ≃ᵢ F) (y : F) :
    (f.toRealLinearIsometryEquiv.symm : F → E) y = f.symm (y + f 0) 
Arguments: {E : Type u_1} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] (f : E ≃ᵢ F) (y : F)
Goal: f.toRealLinearIsometryEquiv.symm y = f.symm (y + f 0)
-- IsometryEquiv.coeFn_toRealAffineIsometryEquiv
@[simp]
theorem coeFn_toRealAffineIsometryEquiv (f : PE ≃ᵢ PF) : ⇑f.toRealAffineIsometryEquiv = f 
Arguments: {E : Type u_1} {PE : Type u_2} {F : Type u_3} {PF : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E] [MetricSpace PE] [NormedAddTorsor E PE] [NormedAddCommGroup F] [NormedSpace ℝ F] [MetricSpace PF] [NormedAddTorsor F PF] (f : PE ≃ᵢ PF)
Goal: ⇑f.toRealAffineIsometryEquiv = ⇑f
-- IsometryEquiv.coe_toRealAffineIsometryEquiv
@[simp]
theorem coe_toRealAffineIsometryEquiv (f : PE ≃ᵢ PF) :
    f.toRealAffineIsometryEquiv.toIsometryEquiv = f 
Arguments: {E : Type u_1} {PE : Type u_2} {F : Type u_3} {PF : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E] [MetricSpace PE] [NormedAddTorsor E PE] [NormedAddCommGroup F] [NormedSpace ℝ F] [MetricSpace PF] [NormedAddTorsor F PF] (f : PE ≃ᵢ PF)
Goal: f.toRealAffineIsometryEquiv.toIsometryEquiv = f
-- Complex.hasStrictFDerivAt_cpow
theorem hasStrictFDerivAt_cpow {p : ℂ × ℂ} (hp : p.1 ∈ slitPlane) :
    HasStrictFDerivAt (fun x : ℂ × ℂ => x.1 ^ x.2)
      ((p.2 * p.1 ^ (p.2 - 1)) • ContinuousLinearMap.fst ℂ ℂ ℂ +
        (p.1 ^ p.2 * log p.1) • ContinuousLinearMap.snd ℂ ℂ ℂ) p 
Arguments: {p : ℂ × ℂ} (hp : p.1 ∈ Complex.slitPlane)
Goal: HasStrictFDerivAt (fun x => x.1 ^ x.2)
  ((p.2 * p.1 ^ (p.2 - 1)) • ContinuousLinearMap.fst ℂ ℂ ℂ +
    (p.1 ^ p.2 * Complex.log p.1) • ContinuousLinearMap.snd ℂ ℂ ℂ)
  p
-- Complex.hasStrictFDerivAt_cpow'
theorem hasStrictFDerivAt_cpow' {x y : ℂ} (hp : x ∈ slitPlane) :
    HasStrictFDerivAt (fun x : ℂ × ℂ => x.1 ^ x.2)
      ((y * x ^ (y - 1)) • ContinuousLinearMap.fst ℂ ℂ ℂ +
        (x ^ y * log x) • ContinuousLinearMap.snd ℂ ℂ ℂ) (x, y) 
Arguments: {x y : ℂ} (hp : x ∈ Complex.slitPlane)
Goal: HasStrictFDerivAt (fun x => x.1 ^ x.2)
  ((y * x ^ (y - 1)) • ContinuousLinearMap.fst ℂ ℂ ℂ + (x ^ y * Complex.log x) • ContinuousLinearMap.snd ℂ ℂ ℂ) (x, y)
-- Complex.hasStrictDerivAt_const_cpow
theorem hasStrictDerivAt_const_cpow {x y : ℂ} (h : x ≠ 0 ∨ y ≠ 0) :
    HasStrictDerivAt (fun y => x ^ y) (x ^ y * log x) y 
Arguments: {x y : ℂ} (h : x ≠ 0 ∨ y ≠ 0)
Goal: HasStrictDerivAt (fun y => x ^ y) (x ^ y * Complex.log x) y
-- Complex.hasFDerivAt_cpow
theorem hasFDerivAt_cpow {p : ℂ × ℂ} (hp : p.1 ∈ slitPlane) :
    HasFDerivAt (fun x : ℂ × ℂ => x.1 ^ x.2)
      ((p.2 * p.1 ^ (p.2 - 1)) • ContinuousLinearMap.fst ℂ ℂ ℂ +
        (p.1 ^ p.2 * log p.1) • ContinuousLinearMap.snd ℂ ℂ ℂ) p 
Arguments: {p : ℂ × ℂ} (hp : p.1 ∈ Complex.slitPlane)
Goal: HasFDerivAt (fun x => x.1 ^ x.2)
  ((p.2 * p.1 ^ (p.2 - 1)) • ContinuousLinearMap.fst ℂ ℂ ℂ +
    (p.1 ^ p.2 * Complex.log p.1) • ContinuousLinearMap.snd ℂ ℂ ℂ)
  p