ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.IsFundamentalDomain.measurePreserving_quotient_mk
@[to_additive IsAddFundamentalDomain.measurePreserving_add_quotient_mk]
theorem IsFundamentalDomain.measurePreserving_quotient_mk
    {𝓕 : Set α} (h𝓕 : IsFundamentalDomain G 𝓕 ν)
    (μ : Measure (Quotient α_mod_G)) [QuotientMeasureEqMeasurePreimage ν μ] :
    MeasurePreserving π (ν.restrict 𝓕) μ 
Arguments: {G : Type u_1} {α : Type u_3} [Group G] [MulAction G α] [MeasurableSpace α] {ν : MeasureTheory.Measure α} {𝓕 : Set α} (h𝓕 : MeasureTheory.IsFundamentalDomain G 𝓕 ν) (μ : MeasureTheory.Measure (Quotient (MulAction.orbitRel G α))) [MeasureTheory.QuotientMeasureEqMeasurePreimage ν μ]
Goal: MeasureTheory.MeasurePreserving (Quotient.mk (MulAction.orbitRel G α)) (ν.restrict 𝓕) μ
-- MeasureTheory.IsAddFundamentalDomain.addQuotientMeasureEqMeasurePreimage_of_zero
@[to_additive]
theorem IsFundamentalDomain.quotientMeasureEqMeasurePreimage_of_zero
    {s : Set α} (fund_dom_s : IsFundamentalDomain G s ν)
    (vol_s : ν s = 0) :
    QuotientMeasureEqMeasurePreimage ν (0 : Measure (Quotient α_mod_G)) 
Arguments: {G : Type u_1} {α : Type u_3} [AddGroup G] [AddAction G α] [MeasurableSpace α] {ν : MeasureTheory.Measure α} [Countable G] [MeasurableSpace G] [MeasureTheory.VAddInvariantMeasure G α ν] [MeasurableVAdd G α] {s : Set α} (fund_dom_s : MeasureTheory.IsAddFundamentalDomain G s ν) (vol_s : ν s = 0)
Goal: MeasureTheory.AddQuotientMeasureEqMeasurePreimage ν 0
-- MeasureTheory.IsFundamentalDomain.quotientMeasureEqMeasurePreimage_of_zero
@[to_additive]
theorem IsFundamentalDomain.quotientMeasureEqMeasurePreimage_of_zero
    {s : Set α} (fund_dom_s : IsFundamentalDomain G s ν)
    (vol_s : ν s = 0) :
    QuotientMeasureEqMeasurePreimage ν (0 : Measure (Quotient α_mod_G)) 
Arguments: {G : Type u_1} {α : Type u_3} [Group G] [MulAction G α] [MeasurableSpace α] {ν : MeasureTheory.Measure α} [Countable G] [MeasurableSpace G] [MeasureTheory.SMulInvariantMeasure G α ν] [MeasurableSMul G α] {s : Set α} (fund_dom_s : MeasureTheory.IsFundamentalDomain G s ν) (vol_s : ν s = 0)
Goal: MeasureTheory.QuotientMeasureEqMeasurePreimage ν 0
-- MeasureTheory.AddQuotientMeasureEqMeasurePreimage.sigmaFiniteQuotient
@[to_additive]
lemma QuotientMeasureEqMeasurePreimage.sigmaFiniteQuotient
    [i : SigmaFinite ν] [i' : HasFundamentalDomain G α ν]
    (μ : Measure (Quotient α_mod_G)) [QuotientMeasureEqMeasurePreimage ν μ] :
    SigmaFinite μ 
Arguments: {G : Type u_1} {α : Type u_3} [AddGroup G] [AddAction G α] [MeasurableSpace α] {ν : MeasureTheory.Measure α} [Countable G] [MeasurableSpace G] [MeasureTheory.VAddInvariantMeasure G α ν] [MeasurableVAdd G α] [i : MeasureTheory.SigmaFinite ν] [i' : MeasureTheory.HasAddFundamentalDomain G α ν] (μ : MeasureTheory.Measure (Quotient (AddAction.orbitRel G α))) [MeasureTheory.AddQuotientMeasureEqMeasurePreimage ν μ]
Goal: MeasureTheory.SigmaFinite μ
-- MeasureTheory.QuotientMeasureEqMeasurePreimage.sigmaFiniteQuotient
@[to_additive]
lemma QuotientMeasureEqMeasurePreimage.sigmaFiniteQuotient
    [i : SigmaFinite ν] [i' : HasFundamentalDomain G α ν]
    (μ : Measure (Quotient α_mod_G)) [QuotientMeasureEqMeasurePreimage ν μ] :
    SigmaFinite μ 
Arguments: {G : Type u_1} {α : Type u_3} [Group G] [MulAction G α] [MeasurableSpace α] {ν : MeasureTheory.Measure α} [Countable G] [MeasurableSpace G] [MeasureTheory.SMulInvariantMeasure G α ν] [MeasurableSMul G α] [i : MeasureTheory.SigmaFinite ν] [i' : MeasureTheory.HasFundamentalDomain G α ν] (μ : MeasureTheory.Measure (Quotient (MulAction.orbitRel G α))) [MeasureTheory.QuotientMeasureEqMeasurePreimage ν μ]
Goal: MeasureTheory.SigmaFinite μ
-- MeasureTheory.AddQuotientMeasureEqMeasurePreimage.isFiniteMeasure_quotient
@[to_additive]
theorem QuotientMeasureEqMeasurePreimage.isFiniteMeasure_quotient
    (μ : Measure (Quotient α_mod_G)) [QuotientMeasureEqMeasurePreimage ν μ]
    [hasFun : HasFundamentalDomain G α ν] (h : covolume G α ν ≠ ∞) :
    IsFiniteMeasure μ 
Arguments: {G : Type u_1} {α : Type u_3} [AddGroup G] [AddAction G α] [MeasurableSpace α] {ν : MeasureTheory.Measure α} [Countable G] [MeasurableSpace G] [MeasureTheory.VAddInvariantMeasure G α ν] [MeasurableVAdd G α] (μ : MeasureTheory.Measure (Quotient (AddAction.orbitRel G α))) [MeasureTheory.AddQuotientMeasureEqMeasurePreimage ν μ] [hasFun : MeasureTheory.HasAddFundamentalDomain G α ν] (h : MeasureTheory.addCovolume G α ν ≠ ⊤)
Goal: MeasureTheory.IsFiniteMeasure μ
-- MeasureTheory.QuotientMeasureEqMeasurePreimage.isFiniteMeasure_quotient
@[to_additive]
theorem QuotientMeasureEqMeasurePreimage.isFiniteMeasure_quotient
    (μ : Measure (Quotient α_mod_G)) [QuotientMeasureEqMeasurePreimage ν μ]
    [hasFun : HasFundamentalDomain G α ν] (h : covolume G α ν ≠ ∞) :
    IsFiniteMeasure μ 
Arguments: {G : Type u_1} {α : Type u_3} [Group G] [MulAction G α] [MeasurableSpace α] {ν : MeasureTheory.Measure α} [Countable G] [MeasurableSpace G] [MeasureTheory.SMulInvariantMeasure G α ν] [MeasurableSMul G α] (μ : MeasureTheory.Measure (Quotient (MulAction.orbitRel G α))) [MeasureTheory.QuotientMeasureEqMeasurePreimage ν μ] [hasFun : MeasureTheory.HasFundamentalDomain G α ν] (h : MeasureTheory.covolume G α ν ≠ ⊤)
Goal: MeasureTheory.IsFiniteMeasure μ
-- MeasureTheory.QuotientMeasureEqMeasurePreimage.covolume_ne_top
@[to_additive]
theorem QuotientMeasureEqMeasurePreimage.covolume_ne_top
    (μ : Measure (Quotient α_mod_G)) [QuotientMeasureEqMeasurePreimage ν μ] [IsFiniteMeasure μ] :
    covolume G α ν < ∞ 
Arguments: {G : Type u_1} {α : Type u_3} [Group G] [MulAction G α] [MeasurableSpace α] {ν : MeasureTheory.Measure α} [Countable G] [MeasurableSpace G] [MeasureTheory.SMulInvariantMeasure G α ν] [MeasurableSMul G α] (μ : MeasureTheory.Measure (Quotient (MulAction.orbitRel G α))) [MeasureTheory.QuotientMeasureEqMeasurePreimage ν μ] [MeasureTheory.IsFiniteMeasure μ]
Goal: MeasureTheory.covolume G α ν < ⊤
-- MeasureTheory.AddQuotientMeasureEqMeasurePreimage.covolume_ne_top
@[to_additive]
theorem QuotientMeasureEqMeasurePreimage.covolume_ne_top
    (μ : Measure (Quotient α_mod_G)) [QuotientMeasureEqMeasurePreimage ν μ] [IsFiniteMeasure μ] :
    covolume G α ν < ∞ 
Arguments: {G : Type u_1} {α : Type u_3} [AddGroup G] [AddAction G α] [MeasurableSpace α] {ν : MeasureTheory.Measure α} [Countable G] [MeasurableSpace G] [MeasureTheory.VAddInvariantMeasure G α ν] [MeasurableVAdd G α] (μ : MeasureTheory.Measure (Quotient (AddAction.orbitRel G α))) [MeasureTheory.AddQuotientMeasureEqMeasurePreimage ν μ] [MeasureTheory.IsFiniteMeasure μ]
Goal: MeasureTheory.addCovolume G α ν < ⊤
-- RCLike.I_re_ax
class RCLike (K : semiOutParam Type*) extends DenselyNormedField K, StarRing K,
    NormedAlgebra ℝ K, CompleteSpace K where
  re : K →+ ℝ
  im : K →+ ℝ
    I : K
  I_re_ax : re I = 0
  I_mul_I_ax : I = 0 ∨ I * I = -1
  re_add_im_ax : ∀ z : K, 𝓚 (re z) + 𝓚 (im z) * I = z
  ofReal_re_ax : ∀ r : ℝ, re (𝓚 r) = r
  ofReal_im_ax : ∀ r : ℝ, im (𝓚 r) = 0
  mul_re_ax : ∀ z w : K, re (z * w) = re z * re w - im z * im w
  mul_im_ax : ∀ z w : K, im (z * w) = re z * im w + im z * re w
  conj_re_ax : ∀ z : K, re (conj z) = re z
  conj_im_ax : ∀ z : K, im (conj z) = -im z
  conj_I_ax : conj I = -I
  norm_sq_eq_def_ax : ∀ z : K, ‖z‖ ^ 2 = re z * re z + im z * im z
  mul_im_I_ax : ∀ z : K, im z * im I = im z
    [toPartialOrder : PartialOrder K]
  le_iff_re_im {z w : K} : z ≤ w ↔ re z ≤ re w ∧ im z = im w
  -- note we cannot put this in the `extends` clause
  [toDecidableEq : DecidableEq K]
Arguments: {K : semiOutParam (Type u_1)} [self : RCLike K]
Goal: RCLike.re RCLike.I = 0
-- RCLike.ofReal_im_ax
class RCLike (K : semiOutParam Type*) extends DenselyNormedField K, StarRing K,
    NormedAlgebra ℝ K, CompleteSpace K where
  re : K →+ ℝ
  im : K →+ ℝ
    I : K
  I_re_ax : re I = 0
  I_mul_I_ax : I = 0 ∨ I * I = -1
  re_add_im_ax : ∀ z : K, 𝓚 (re z) + 𝓚 (im z) * I = z
  ofReal_re_ax : ∀ r : ℝ, re (𝓚 r) = r
  ofReal_im_ax : ∀ r : ℝ, im (𝓚 r) = 0
  mul_re_ax : ∀ z w : K, re (z * w) = re z * re w - im z * im w
  mul_im_ax : ∀ z w : K, im (z * w) = re z * im w + im z * re w
  conj_re_ax : ∀ z : K, re (conj z) = re z
  conj_im_ax : ∀ z : K, im (conj z) = -im z
  conj_I_ax : conj I = -I
  norm_sq_eq_def_ax : ∀ z : K, ‖z‖ ^ 2 = re z * re z + im z * im z
  mul_im_I_ax : ∀ z : K, im z * im I = im z
    [toPartialOrder : PartialOrder K]
  le_iff_re_im {z w : K} : z ≤ w ↔ re z ≤ re w ∧ im z = im w
  -- note we cannot put this in the `extends` clause
  [toDecidableEq : DecidableEq K]
Arguments: {K : semiOutParam (Type u_1)} [self : RCLike K] (r : ℝ)
Goal: RCLike.im ((algebraMap ℝ K) r) = 0
-- RCLike.conj_im_ax
class RCLike (K : semiOutParam Type*) extends DenselyNormedField K, StarRing K,
    NormedAlgebra ℝ K, CompleteSpace K where
  re : K →+ ℝ
  im : K →+ ℝ
    I : K
  I_re_ax : re I = 0
  I_mul_I_ax : I = 0 ∨ I * I = -1
  re_add_im_ax : ∀ z : K, 𝓚 (re z) + 𝓚 (im z) * I = z
  ofReal_re_ax : ∀ r : ℝ, re (𝓚 r) = r
  ofReal_im_ax : ∀ r : ℝ, im (𝓚 r) = 0
  mul_re_ax : ∀ z w : K, re (z * w) = re z * re w - im z * im w
  mul_im_ax : ∀ z w : K, im (z * w) = re z * im w + im z * re w
  conj_re_ax : ∀ z : K, re (conj z) = re z
  conj_im_ax : ∀ z : K, im (conj z) = -im z
  conj_I_ax : conj I = -I
  norm_sq_eq_def_ax : ∀ z : K, ‖z‖ ^ 2 = re z * re z + im z * im z
  mul_im_I_ax : ∀ z : K, im z * im I = im z
    [toPartialOrder : PartialOrder K]
  le_iff_re_im {z w : K} : z ≤ w ↔ re z ≤ re w ∧ im z = im w
  -- note we cannot put this in the `extends` clause
  [toDecidableEq : DecidableEq K]
Arguments: {K : semiOutParam (Type u_1)} [self : RCLike K] (z : K)
Goal: RCLike.im ((starRingEnd K) z) = -RCLike.im z
-- RCLike.ofReal_re_ax
class RCLike (K : semiOutParam Type*) extends DenselyNormedField K, StarRing K,
    NormedAlgebra ℝ K, CompleteSpace K where
  re : K →+ ℝ
  im : K →+ ℝ
    I : K
  I_re_ax : re I = 0
  I_mul_I_ax : I = 0 ∨ I * I = -1
  re_add_im_ax : ∀ z : K, 𝓚 (re z) + 𝓚 (im z) * I = z
  ofReal_re_ax : ∀ r : ℝ, re (𝓚 r) = r
  ofReal_im_ax : ∀ r : ℝ, im (𝓚 r) = 0
  mul_re_ax : ∀ z w : K, re (z * w) = re z * re w - im z * im w
  mul_im_ax : ∀ z w : K, im (z * w) = re z * im w + im z * re w
  conj_re_ax : ∀ z : K, re (conj z) = re z
  conj_im_ax : ∀ z : K, im (conj z) = -im z
  conj_I_ax : conj I = -I
  norm_sq_eq_def_ax : ∀ z : K, ‖z‖ ^ 2 = re z * re z + im z * im z
  mul_im_I_ax : ∀ z : K, im z * im I = im z
    [toPartialOrder : PartialOrder K]
  le_iff_re_im {z w : K} : z ≤ w ↔ re z ≤ re w ∧ im z = im w
  -- note we cannot put this in the `extends` clause
  [toDecidableEq : DecidableEq K]
Arguments: {K : semiOutParam (Type u_1)} [self : RCLike K] (r : ℝ)
Goal: RCLike.re ((algebraMap ℝ K) r) = r
-- RCLike.conj_I_ax
class RCLike (K : semiOutParam Type*) extends DenselyNormedField K, StarRing K,
    NormedAlgebra ℝ K, CompleteSpace K where
  re : K →+ ℝ
  im : K →+ ℝ
    I : K
  I_re_ax : re I = 0
  I_mul_I_ax : I = 0 ∨ I * I = -1
  re_add_im_ax : ∀ z : K, 𝓚 (re z) + 𝓚 (im z) * I = z
  ofReal_re_ax : ∀ r : ℝ, re (𝓚 r) = r
  ofReal_im_ax : ∀ r : ℝ, im (𝓚 r) = 0
  mul_re_ax : ∀ z w : K, re (z * w) = re z * re w - im z * im w
  mul_im_ax : ∀ z w : K, im (z * w) = re z * im w + im z * re w
  conj_re_ax : ∀ z : K, re (conj z) = re z
  conj_im_ax : ∀ z : K, im (conj z) = -im z
  conj_I_ax : conj I = -I
  norm_sq_eq_def_ax : ∀ z : K, ‖z‖ ^ 2 = re z * re z + im z * im z
  mul_im_I_ax : ∀ z : K, im z * im I = im z
    [toPartialOrder : PartialOrder K]
  le_iff_re_im {z w : K} : z ≤ w ↔ re z ≤ re w ∧ im z = im w
  -- note we cannot put this in the `extends` clause
  [toDecidableEq : DecidableEq K]
Arguments: {K : semiOutParam (Type u_1)} [self : RCLike K]
Goal: (starRingEnd K) RCLike.I = -RCLike.I
-- RCLike.norm_sq_eq_def_ax
class RCLike (K : semiOutParam Type*) extends DenselyNormedField K, StarRing K,
    NormedAlgebra ℝ K, CompleteSpace K where
  re : K →+ ℝ
  im : K →+ ℝ
    I : K
  I_re_ax : re I = 0
  I_mul_I_ax : I = 0 ∨ I * I = -1
  re_add_im_ax : ∀ z : K, 𝓚 (re z) + 𝓚 (im z) * I = z
  ofReal_re_ax : ∀ r : ℝ, re (𝓚 r) = r
  ofReal_im_ax : ∀ r : ℝ, im (𝓚 r) = 0
  mul_re_ax : ∀ z w : K, re (z * w) = re z * re w - im z * im w
  mul_im_ax : ∀ z w : K, im (z * w) = re z * im w + im z * re w
  conj_re_ax : ∀ z : K, re (conj z) = re z
  conj_im_ax : ∀ z : K, im (conj z) = -im z
  conj_I_ax : conj I = -I
  norm_sq_eq_def_ax : ∀ z : K, ‖z‖ ^ 2 = re z * re z + im z * im z
  mul_im_I_ax : ∀ z : K, im z * im I = im z
    [toPartialOrder : PartialOrder K]
  le_iff_re_im {z w : K} : z ≤ w ↔ re z ≤ re w ∧ im z = im w
  -- note we cannot put this in the `extends` clause
  [toDecidableEq : DecidableEq K]
Arguments: {K : semiOutParam (Type u_1)} [self : RCLike K] (z : K)
Goal: ‖z‖ ^ 2 = RCLike.re z * RCLike.re z + RCLike.im z * RCLike.im z
-- RCLike.re_add_im_ax
class RCLike (K : semiOutParam Type*) extends DenselyNormedField K, StarRing K,
    NormedAlgebra ℝ K, CompleteSpace K where
  re : K →+ ℝ
  im : K →+ ℝ
    I : K
  I_re_ax : re I = 0
  I_mul_I_ax : I = 0 ∨ I * I = -1
  re_add_im_ax : ∀ z : K, 𝓚 (re z) + 𝓚 (im z) * I = z
  ofReal_re_ax : ∀ r : ℝ, re (𝓚 r) = r
  ofReal_im_ax : ∀ r : ℝ, im (𝓚 r) = 0
  mul_re_ax : ∀ z w : K, re (z * w) = re z * re w - im z * im w
  mul_im_ax : ∀ z w : K, im (z * w) = re z * im w + im z * re w
  conj_re_ax : ∀ z : K, re (conj z) = re z
  conj_im_ax : ∀ z : K, im (conj z) = -im z
  conj_I_ax : conj I = -I
  norm_sq_eq_def_ax : ∀ z : K, ‖z‖ ^ 2 = re z * re z + im z * im z
  mul_im_I_ax : ∀ z : K, im z * im I = im z
    [toPartialOrder : PartialOrder K]
  le_iff_re_im {z w : K} : z ≤ w ↔ re z ≤ re w ∧ im z = im w
  -- note we cannot put this in the `extends` clause
  [toDecidableEq : DecidableEq K]
Arguments: {K : semiOutParam (Type u_1)} [self : RCLike K] (z : K)
Goal: (algebraMap ℝ K) (RCLike.re z) + (algebraMap ℝ K) (RCLike.im z) * RCLike.I = z
-- RCLike.mul_re_ax
class RCLike (K : semiOutParam Type*) extends DenselyNormedField K, StarRing K,
    NormedAlgebra ℝ K, CompleteSpace K where
  re : K →+ ℝ
  im : K →+ ℝ
    I : K
  I_re_ax : re I = 0
  I_mul_I_ax : I = 0 ∨ I * I = -1
  re_add_im_ax : ∀ z : K, 𝓚 (re z) + 𝓚 (im z) * I = z
  ofReal_re_ax : ∀ r : ℝ, re (𝓚 r) = r
  ofReal_im_ax : ∀ r : ℝ, im (𝓚 r) = 0
  mul_re_ax : ∀ z w : K, re (z * w) = re z * re w - im z * im w
  mul_im_ax : ∀ z w : K, im (z * w) = re z * im w + im z * re w
  conj_re_ax : ∀ z : K, re (conj z) = re z
  conj_im_ax : ∀ z : K, im (conj z) = -im z
  conj_I_ax : conj I = -I
  norm_sq_eq_def_ax : ∀ z : K, ‖z‖ ^ 2 = re z * re z + im z * im z
  mul_im_I_ax : ∀ z : K, im z * im I = im z
    [toPartialOrder : PartialOrder K]
  le_iff_re_im {z w : K} : z ≤ w ↔ re z ≤ re w ∧ im z = im w
  -- note we cannot put this in the `extends` clause
  [toDecidableEq : DecidableEq K]
Arguments: {K : semiOutParam (Type u_1)} [self : RCLike K] (z w : K)
Goal: RCLike.re (z * w) = RCLike.re z * RCLike.re w - RCLike.im z * RCLike.im w
-- RCLike.conj_re_ax
class RCLike (K : semiOutParam Type*) extends DenselyNormedField K, StarRing K,
    NormedAlgebra ℝ K, CompleteSpace K where
  re : K →+ ℝ
  im : K →+ ℝ
    I : K
  I_re_ax : re I = 0
  I_mul_I_ax : I = 0 ∨ I * I = -1
  re_add_im_ax : ∀ z : K, 𝓚 (re z) + 𝓚 (im z) * I = z
  ofReal_re_ax : ∀ r : ℝ, re (𝓚 r) = r
  ofReal_im_ax : ∀ r : ℝ, im (𝓚 r) = 0
  mul_re_ax : ∀ z w : K, re (z * w) = re z * re w - im z * im w
  mul_im_ax : ∀ z w : K, im (z * w) = re z * im w + im z * re w
  conj_re_ax : ∀ z : K, re (conj z) = re z
  conj_im_ax : ∀ z : K, im (conj z) = -im z
  conj_I_ax : conj I = -I
  norm_sq_eq_def_ax : ∀ z : K, ‖z‖ ^ 2 = re z * re z + im z * im z
  mul_im_I_ax : ∀ z : K, im z * im I = im z
    [toPartialOrder : PartialOrder K]
  le_iff_re_im {z w : K} : z ≤ w ↔ re z ≤ re w ∧ im z = im w
  -- note we cannot put this in the `extends` clause
  [toDecidableEq : DecidableEq K]
Arguments: {K : semiOutParam (Type u_1)} [self : RCLike K] (z : K)
Goal: RCLike.re ((starRingEnd K) z) = RCLike.re z
-- RCLike.mul_im_I_ax
class RCLike (K : semiOutParam Type*) extends DenselyNormedField K, StarRing K,
    NormedAlgebra ℝ K, CompleteSpace K where
  re : K →+ ℝ
  im : K →+ ℝ
    I : K
  I_re_ax : re I = 0
  I_mul_I_ax : I = 0 ∨ I * I = -1
  re_add_im_ax : ∀ z : K, 𝓚 (re z) + 𝓚 (im z) * I = z
  ofReal_re_ax : ∀ r : ℝ, re (𝓚 r) = r
  ofReal_im_ax : ∀ r : ℝ, im (𝓚 r) = 0
  mul_re_ax : ∀ z w : K, re (z * w) = re z * re w - im z * im w
  mul_im_ax : ∀ z w : K, im (z * w) = re z * im w + im z * re w
  conj_re_ax : ∀ z : K, re (conj z) = re z
  conj_im_ax : ∀ z : K, im (conj z) = -im z
  conj_I_ax : conj I = -I
  norm_sq_eq_def_ax : ∀ z : K, ‖z‖ ^ 2 = re z * re z + im z * im z
  mul_im_I_ax : ∀ z : K, im z * im I = im z
    [toPartialOrder : PartialOrder K]
  le_iff_re_im {z w : K} : z ≤ w ↔ re z ≤ re w ∧ im z = im w
  -- note we cannot put this in the `extends` clause
  [toDecidableEq : DecidableEq K]
Arguments: {K : semiOutParam (Type u_1)} [self : RCLike K] (z : K)
Goal: RCLike.im z * RCLike.im RCLike.I = RCLike.im z
-- RCLike.I_mul_I_ax
class RCLike (K : semiOutParam Type*) extends DenselyNormedField K, StarRing K,
    NormedAlgebra ℝ K, CompleteSpace K where
  re : K →+ ℝ
  im : K →+ ℝ
    I : K
  I_re_ax : re I = 0
  I_mul_I_ax : I = 0 ∨ I * I = -1
  re_add_im_ax : ∀ z : K, 𝓚 (re z) + 𝓚 (im z) * I = z
  ofReal_re_ax : ∀ r : ℝ, re (𝓚 r) = r
  ofReal_im_ax : ∀ r : ℝ, im (𝓚 r) = 0
  mul_re_ax : ∀ z w : K, re (z * w) = re z * re w - im z * im w
  mul_im_ax : ∀ z w : K, im (z * w) = re z * im w + im z * re w
  conj_re_ax : ∀ z : K, re (conj z) = re z
  conj_im_ax : ∀ z : K, im (conj z) = -im z
  conj_I_ax : conj I = -I
  norm_sq_eq_def_ax : ∀ z : K, ‖z‖ ^ 2 = re z * re z + im z * im z
  mul_im_I_ax : ∀ z : K, im z * im I = im z
    [toPartialOrder : PartialOrder K]
  le_iff_re_im {z w : K} : z ≤ w ↔ re z ≤ re w ∧ im z = im w
  -- note we cannot put this in the `extends` clause
  [toDecidableEq : DecidableEq K]
Arguments: {K : semiOutParam (Type u_1)} [self : RCLike K]
Goal: RCLike.I = 0 ∨ RCLike.I * RCLike.I = -1
-- RCLike.le_iff_re_im
class RCLike (K : semiOutParam Type*) extends DenselyNormedField K, StarRing K,
    NormedAlgebra ℝ K, CompleteSpace K where
  re : K →+ ℝ
  im : K →+ ℝ
    I : K
  I_re_ax : re I = 0
  I_mul_I_ax : I = 0 ∨ I * I = -1
  re_add_im_ax : ∀ z : K, 𝓚 (re z) + 𝓚 (im z) * I = z
  ofReal_re_ax : ∀ r : ℝ, re (𝓚 r) = r
  ofReal_im_ax : ∀ r : ℝ, im (𝓚 r) = 0
  mul_re_ax : ∀ z w : K, re (z * w) = re z * re w - im z * im w
  mul_im_ax : ∀ z w : K, im (z * w) = re z * im w + im z * re w
  conj_re_ax : ∀ z : K, re (conj z) = re z
  conj_im_ax : ∀ z : K, im (conj z) = -im z
  conj_I_ax : conj I = -I
  norm_sq_eq_def_ax : ∀ z : K, ‖z‖ ^ 2 = re z * re z + im z * im z
  mul_im_I_ax : ∀ z : K, im z * im I = im z
    [toPartialOrder : PartialOrder K]
  le_iff_re_im {z w : K} : z ≤ w ↔ re z ≤ re w ∧ im z = im w
  -- note we cannot put this in the `extends` clause
  [toDecidableEq : DecidableEq K]
Arguments: {K : semiOutParam (Type u_1)} [self : RCLike K] {z w : K}
Goal: z ≤ w ↔ RCLike.re z ≤ RCLike.re w ∧ RCLike.im z = RCLike.im w
-- RCLike.mul_im_ax
class RCLike (K : semiOutParam Type*) extends DenselyNormedField K, StarRing K,
    NormedAlgebra ℝ K, CompleteSpace K where
  re : K →+ ℝ
  im : K →+ ℝ
    I : K
  I_re_ax : re I = 0
  I_mul_I_ax : I = 0 ∨ I * I = -1
  re_add_im_ax : ∀ z : K, 𝓚 (re z) + 𝓚 (im z) * I = z
  ofReal_re_ax : ∀ r : ℝ, re (𝓚 r) = r
  ofReal_im_ax : ∀ r : ℝ, im (𝓚 r) = 0
  mul_re_ax : ∀ z w : K, re (z * w) = re z * re w - im z * im w
  mul_im_ax : ∀ z w : K, im (z * w) = re z * im w + im z * re w
  conj_re_ax : ∀ z : K, re (conj z) = re z
  conj_im_ax : ∀ z : K, im (conj z) = -im z
  conj_I_ax : conj I = -I
  norm_sq_eq_def_ax : ∀ z : K, ‖z‖ ^ 2 = re z * re z + im z * im z
  mul_im_I_ax : ∀ z : K, im z * im I = im z
    [toPartialOrder : PartialOrder K]
  le_iff_re_im {z w : K} : z ≤ w ↔ re z ≤ re w ∧ im z = im w
  -- note we cannot put this in the `extends` clause
  [toDecidableEq : DecidableEq K]
Arguments: {K : semiOutParam (Type u_1)} [self : RCLike K] (z w : K)
Goal: RCLike.im (z * w) = RCLike.re z * RCLike.im w + RCLike.im z * RCLike.re w
-- RCLike.ofReal_alg
theorem ofReal_alg (x : ℝ) : (x : K) = x • (1 : K) 
Arguments: {K : Type u_1} [RCLike K] (x : ℝ)
Goal: ↑x = x • 1
-- RCLike.real_smul_eq_coe_mul
theorem real_smul_eq_coe_mul (r : ℝ) (z : K) : r • z = (r : K) * z 
Arguments: {K : Type u_1} [RCLike K] (r : ℝ) (z : K)
Goal: r • z = ↑r * z
-- RCLike.real_smul_eq_coe_smul
theorem real_smul_eq_coe_smul [AddCommGroup E] [Module K E] [Module ℝ E] [IsScalarTower ℝ K E]
    (r : ℝ) (x : E) : r • x = (r : K) • x 
Arguments: {K : Type u_1} {E : Type u_2} [RCLike K] [AddCommGroup E] [Module K E] [Module ℝ E] [IsScalarTower ℝ K E] (r : ℝ) (x : E)
Goal: r • x = ↑r • x
-- RCLike.algebraMap_eq_ofReal
theorem algebraMap_eq_ofReal : ⇑(algebraMap ℝ K) = ofReal 
Arguments: {K : Type u_1} [RCLike K]
Goal: ⇑(algebraMap ℝ K) = RCLike.ofReal
-- RCLike.re_add_im
@[simp, rclike_simps]
theorem re_add_im (z : K) : (re z : K) + im z * I = z 
Arguments: {K : Type u_1} [RCLike K] (z : K)
Goal: ↑(RCLike.re z) + ↑(RCLike.im z) * RCLike.I = z
-- RCLike.ofReal_re
@[simp, norm_cast, rclike_simps]
theorem ofReal_re : ∀ r : ℝ, re (r : K) = r 
Arguments: {K : Type u_1} [RCLike K] (r : ℝ)
Goal: RCLike.re ↑r = r
-- RCLike.ofReal_im
@[simp, norm_cast, rclike_simps]
theorem ofReal_im : ∀ r : ℝ, im (r : K) = 0 
Arguments: {K : Type u_1} [RCLike K] (r : ℝ)
Goal: RCLike.im ↑r = 0
-- RCLike.mul_re
@[simp, rclike_simps]
theorem mul_re : ∀ z w : K, re (z * w) = re z * re w - im z * im w 
Arguments: {K : Type u_1} [RCLike K] (z w : K)
Goal: RCLike.re (z * w) = RCLike.re z * RCLike.re w - RCLike.im z * RCLike.im w
-- RCLike.mul_im
@[simp, rclike_simps]
theorem mul_im : ∀ z w : K, im (z * w) = re z * im w + im z * re w 
Arguments: {K : Type u_1} [RCLike K] (z w : K)
Goal: RCLike.im (z * w) = RCLike.re z * RCLike.im w + RCLike.im z * RCLike.re w
-- RCLike.ext_iff
theorem ext_iff {z w : K} : z = w ↔ re z = re w ∧ im z = im w 
Arguments: {K : Type u_1} [RCLike K] {z w : K}
Goal: z = w ↔ RCLike.re z = RCLike.re w ∧ RCLike.im z = RCLike.im w
-- RCLike.ext
theorem ext {z w : K} (hre : re z = re w) (him : im z = im w) : z = w 
Arguments: {K : Type u_1} [RCLike K] {z w : K} (hre : RCLike.re z = RCLike.re w) (him : RCLike.im z = RCLike.im w)
Goal: z = w
-- RCLike.ofReal_zero
@[norm_cast]
theorem ofReal_zero : ((0 : ℝ) : K) = 0 
Arguments: {K : Type u_1} [RCLike K]
Goal: ↑0 = 0
-- RCLike.zero_re'
@[rclike_simps]
theorem zero_re' : re (0 : K) = (0 : ℝ) 
Arguments: {K : Type u_1} [RCLike K]
Goal: RCLike.re 0 = 0
-- RCLike.ofReal_one
@[norm_cast]
theorem ofReal_one : ((1 : ℝ) : K) = 1 
Arguments: {K : Type u_1} [RCLike K]
Goal: ↑1 = 1
-- RCLike.one_re
@[simp, rclike_simps]
theorem one_re : re (1 : K) = 1 
Arguments: {K : Type u_1} [RCLike K]
Goal: RCLike.re 1 = 1
-- RCLike.one_im
@[simp, rclike_simps]
theorem one_im : im (1 : K) = 0 
Arguments: {K : Type u_1} [RCLike K]
Goal: RCLike.im 1 = 0
-- RCLike.ofReal_injective
theorem ofReal_injective : Function.Injective ((↑) : ℝ → K) 
Arguments: {K : Type u_1} [RCLike K]
Goal: Function.Injective RCLike.ofReal
-- RCLike.ofReal_inj
@[norm_cast]
theorem ofReal_inj {z w : ℝ} : (z : K) = (w : K) ↔ z = w 
Arguments: {K : Type u_1} [RCLike K] {z w : ℝ}
Goal: ↑z = ↑w ↔ z = w
-- RCLike.ofReal_eq_zero
theorem ofReal_eq_zero {x : ℝ} : (x : K) = 0 ↔ x = 0 
Arguments: {K : Type u_1} [RCLike K] {x : ℝ}
Goal: ↑x = 0 ↔ x = 0
-- RCLike.ofReal_ne_zero
theorem ofReal_ne_zero {x : ℝ} : (x : K) ≠ 0 ↔ x ≠ 0 
Arguments: {K : Type u_1} [RCLike K] {x : ℝ}
Goal: ↑x ≠ 0 ↔ x ≠ 0
-- RCLike.ofReal_add
@[rclike_simps, norm_cast]
theorem ofReal_add (r s : ℝ) : ((r + s : ℝ) : K) = r + s 
Arguments: {K : Type u_1} [RCLike K] (r s : ℝ)
Goal: ↑(r + s) = ↑r + ↑s
-- RCLike.ofReal_neg
@[rclike_simps, norm_cast]
theorem ofReal_neg (r : ℝ) : ((-r : ℝ) : K) = -r 
Arguments: {K : Type u_1} [RCLike K] (r : ℝ)
Goal: ↑(-r) = -↑r
-- RCLike.ofReal_sub
@[rclike_simps, norm_cast]
theorem ofReal_sub (r s : ℝ) : ((r - s : ℝ) : K) = r - s 
Arguments: {K : Type u_1} [RCLike K] (r s : ℝ)
Goal: ↑(r - s) = ↑r - ↑s
-- RCLike.ofReal_sum
@[rclike_simps, norm_cast]
theorem ofReal_sum {α : Type*} (s : Finset α) (f : α → ℝ) :
    ((∑ i ∈ s, f i : ℝ) : K) = ∑ i ∈ s, (f i : K) 
Arguments: {K : Type u_1} [RCLike K] {α : Type u_3} (s : Finset α) (f : α → ℝ)
Goal: ↑(∑ i ∈ s, f i) = ∑ i ∈ s, ↑(f i)