ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- RCLike.ofReal_finsupp_sum
@[simp, rclike_simps, norm_cast]
theorem ofReal_finsupp_sum {α M : Type*} [Zero M] (f : α →₀ M) (g : α → M → ℝ) :
    ((f.sum fun a b => g a b : ℝ) : K) = f.sum fun a b => (g a b : K) 
Arguments: {K : Type u_1} [RCLike K] {α : Type u_3} {M : Type u_4} [Zero M] (f : α →₀ M) (g : α → M → ℝ)
Goal: ↑(f.sum fun a b => g a b) = f.sum fun a b => ↑(g a b)
-- RCLike.ofReal_mul
@[rclike_simps, norm_cast]
theorem ofReal_mul (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_pow
@[rclike_simps, norm_cast]
theorem ofReal_pow (r : ℝ) (n : ℕ) : ((r ^ n : ℝ) : K) = (r : K) ^ n 
Arguments: {K : Type u_1} [RCLike K] (r : ℝ) (n : ℕ)
Goal: ↑(r ^ n) = ↑r ^ n
-- RCLike.ofReal_prod
@[rclike_simps, norm_cast]
theorem ofReal_prod {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)
-- RCLike.ofReal_finsupp_prod
@[simp, rclike_simps, norm_cast]
theorem ofReal_finsupp_prod {α M : Type*} [Zero M] (f : α →₀ M) (g : α → M → ℝ) :
    ((f.prod fun a b => g a b : ℝ) : K) = f.prod fun a b => (g a b : K) 
Arguments: {K : Type u_1} [RCLike K] {α : Type u_3} {M : Type u_4} [Zero M] (f : α →₀ M) (g : α → M → ℝ)
Goal: ↑(f.prod fun a b => g a b) = f.prod fun a b => ↑(g a b)
-- RCLike.real_smul_ofReal
@[simp, norm_cast, rclike_simps]
theorem real_smul_ofReal (r x : ℝ) : r • (x : K) = (r : K) * (x : K) 
Arguments: {K : Type u_1} [RCLike K] (r x : ℝ)
Goal: r • ↑x = ↑r * ↑x
-- RCLike.re_ofReal_mul
@[rclike_simps]
theorem re_ofReal_mul (r : ℝ) (z : K) : re (↑r * z) = r * re z 
Arguments: {K : Type u_1} [RCLike K] (r : ℝ) (z : K)
Goal: RCLike.re (↑r * z) = r * RCLike.re z
-- RCLike.im_ofReal_mul
@[rclike_simps]
theorem im_ofReal_mul (r : ℝ) (z : K) : im (↑r * z) = r * im z 
Arguments: {K : Type u_1} [RCLike K] (r : ℝ) (z : K)
Goal: RCLike.im (↑r * z) = r * RCLike.im z
-- RCLike.smul_re
@[rclike_simps]
theorem smul_re (r : ℝ) (z : K) : re (r • z) = r * re z 
Arguments: {K : Type u_1} [RCLike K] (r : ℝ) (z : K)
Goal: RCLike.re (r • z) = r * RCLike.re z
-- RCLike.smul_im
@[rclike_simps]
theorem smul_im (r : ℝ) (z : K) : im (r • z) = r * im z 
Arguments: {K : Type u_1} [RCLike K] (r : ℝ) (z : K)
Goal: RCLike.im (r • z) = r * RCLike.im z
-- RCLike.norm_ofReal
@[rclike_simps, norm_cast]
theorem norm_ofReal (r : ℝ) : ‖(r : K)‖ = |r| 
Arguments: {K : Type u_1} [RCLike K] (r : ℝ)
Goal: ‖↑r‖ = |r|
-- RCLike.I_re
@[simp, rclike_simps]
theorem I_re : re (I : K) = 0 
Arguments: {K : Type u_1} [RCLike K]
Goal: RCLike.re RCLike.I = 0
-- RCLike.I_im
@[simp, rclike_simps]
theorem I_im (z : K) : im z * im (I : K) = im z 
Arguments: {K : Type u_1} [RCLike K] (z : K)
Goal: RCLike.im z * RCLike.im RCLike.I = RCLike.im z
-- RCLike.I_im'
@[simp, rclike_simps]
theorem I_im' (z : K) : im (I : K) * im z = im z 
Arguments: {K : Type u_1} [RCLike K] (z : K)
Goal: RCLike.im RCLike.I * RCLike.im z = RCLike.im z
-- RCLike.I_mul_re
@[rclike_simps] -- porting note (#10618): was `simp`
theorem I_mul_re (z : K) : re (I * z) = -im z 
Arguments: {K : Type u_1} [RCLike K] (z : K)
Goal: RCLike.re (RCLike.I * z) = -RCLike.im z
-- RCLike.I_mul_I
theorem I_mul_I : (I : K) = 0 ∨ (I : K) * I = -1 
Arguments: {K : Type u_1} [RCLike K]
Goal: RCLike.I = 0 ∨ RCLike.I * RCLike.I = -1
-- RCLike.I_eq_zero_or_im_I_eq_one
lemma I_eq_zero_or_im_I_eq_one : (I : K) = 0 ∨ im (I : K) = 1 
Arguments: {K : Type u_1} [RCLike K]
Goal: RCLike.I = 0 ∨ RCLike.im RCLike.I = 1
-- RCLike.conj_re
@[simp, rclike_simps]
theorem conj_re (z : K) : re (conj z) = re z 
Arguments: {K : Type u_1} [RCLike K] (z : K)
Goal: RCLike.re ((starRingEnd K) z) = RCLike.re z
-- RCLike.conj_im
@[simp, rclike_simps]
theorem conj_im (z : K) : im (conj z) = -im z 
Arguments: {K : Type u_1} [RCLike K] (z : K)
Goal: RCLike.im ((starRingEnd K) z) = -RCLike.im z
-- RCLike.conj_I
@[simp, rclike_simps]
theorem conj_I : conj (I : K) = -I 
Arguments: {K : Type u_1} [RCLike K]
Goal: (starRingEnd K) RCLike.I = -RCLike.I
-- RCLike.conj_ofReal
@[simp, rclike_simps]
theorem conj_ofReal (r : ℝ) : conj (r : K) = (r : K) 
Arguments: {K : Type u_1} [RCLike K] (r : ℝ)
Goal: (starRingEnd K) ↑r = ↑r
-- RCLike.conj_nat_cast
theorem conj_nat_cast (n : ℕ) : conj (n : K) = n 
Arguments: {K : Type u_1} [RCLike K] (n : ℕ)
Goal: (starRingEnd K) ↑n = ↑n
-- RCLike.conj_ofNat
theorem conj_ofNat (n : ℕ) [n.AtLeastTwo] : conj (no_index (OfNat.ofNat n : K)) = OfNat.ofNat n 
Arguments: {K : Type u_1} [RCLike K] (n : ℕ) [n.AtLeastTwo]
Goal: (starRingEnd K) (OfNat.ofNat n) = OfNat.ofNat n
-- RCLike.conj_neg_I
@[rclike_simps] -- Porting note (#10618): was a `simp` but `simp` can prove it
theorem conj_neg_I : conj (-I) = (I : K) 
Arguments: {K : Type u_1} [RCLike K]
Goal: (starRingEnd K) (-RCLike.I) = RCLike.I
-- RCLike.conj_eq_re_sub_im
theorem conj_eq_re_sub_im (z : K) : conj z = re z - im z * I 
Arguments: {K : Type u_1} [RCLike K] (z : K)
Goal: (starRingEnd K) z = ↑(RCLike.re z) - ↑(RCLike.im z) * RCLike.I
-- RCLike.sub_conj
theorem sub_conj (z : K) : z - conj z = 2 * im z * I 
Arguments: {K : Type u_1} [RCLike K] (z : K)
Goal: z - (starRingEnd K) z = 2 * ↑(RCLike.im z) * RCLike.I
-- RCLike.conj_smul
@[rclike_simps]
theorem conj_smul (r : ℝ) (z : K) : conj (r • z) = r • conj z 
Arguments: {K : Type u_1} [RCLike K] (r : ℝ) (z : K)
Goal: (starRingEnd K) (r • z) = r • (starRingEnd K) z
-- RCLike.add_conj
theorem add_conj (z : K) : z + conj z = 2 * re z 
Arguments: {K : Type u_1} [RCLike K] (z : K)
Goal: z + (starRingEnd K) z = 2 * ↑(RCLike.re z)
-- RCLike.re_eq_add_conj
theorem re_eq_add_conj (z : K) : ↑(re z) = (z + conj z) / 2 
Arguments: {K : Type u_1} [RCLike K] (z : K)
Goal: ↑(RCLike.re z) = (z + (starRingEnd K) z) / 2
-- RCLike.im_eq_conj_sub
theorem im_eq_conj_sub (z : K) : ↑(im z) = I * (conj z - z) / 2 
Arguments: {K : Type u_1} [RCLike K] (z : K)
Goal: ↑(RCLike.im z) = RCLike.I * ((starRingEnd K) z - z) / 2
-- RCLike.is_real_TFAE
theorem is_real_TFAE (z : K) : TFAE [conj z = z, ∃ r : ℝ, (r : K) = z, ↑(re z) = z, im z = 0] 
Arguments: {K : Type u_1} [RCLike K] (z : K)
Goal: [(starRingEnd K) z = z, ∃ r, ↑r = z, ↑(RCLike.re z) = z, RCLike.im z = 0].TFAE
-- RCLike.conj_eq_iff_real
theorem conj_eq_iff_real {z : K} : conj z = z ↔ ∃ r : ℝ, z = (r : K) 
Arguments: {K : Type u_1} [RCLike K] {z : K}
Goal: (starRingEnd K) z = z ↔ ∃ r, z = ↑r
-- RCLike.conj_eq_iff_re
theorem conj_eq_iff_re {z : K} : conj z = z ↔ (re z : K) = z 
Arguments: {K : Type u_1} [RCLike K] {z : K}
Goal: (starRingEnd K) z = z ↔ ↑(RCLike.re z) = z
-- RCLike.conj_eq_iff_im
theorem conj_eq_iff_im {z : K} : conj z = z ↔ im z = 0 
Arguments: {K : Type u_1} [RCLike K] {z : K}
Goal: (starRingEnd K) z = z ↔ RCLike.im z = 0
-- RCLike.star_def
@[simp]
theorem star_def : (Star.star : K → K) = conj 
Arguments: {K : Type u_1} [RCLike K]
Goal: star = ⇑(starRingEnd K)
-- RCLike.normSq_apply
theorem normSq_apply (z : K) : normSq z = re z * re z + im z * im z 
Arguments: {K : Type u_1} [RCLike K] (z : K)
Goal: RCLike.normSq z = RCLike.re z * RCLike.re z + RCLike.im z * RCLike.im z
-- RCLike.norm_sq_eq_def
theorem norm_sq_eq_def {z : K} : ‖z‖ ^ 2 = re z * re z + im z * im z 
Arguments: {K : Type u_1} [RCLike K] {z : K}
Goal: ‖z‖ ^ 2 = RCLike.re z * RCLike.re z + RCLike.im z * RCLike.im z
-- RCLike.normSq_eq_def'
theorem normSq_eq_def' (z : K) : normSq z = ‖z‖ ^ 2 
Arguments: {K : Type u_1} [RCLike K] (z : K)
Goal: RCLike.normSq z = ‖z‖ ^ 2
-- RCLike.normSq_zero
@[rclike_simps]
theorem normSq_zero : normSq (0 : K) = 0 
Arguments: {K : Type u_1} [RCLike K]
Goal: RCLike.normSq 0 = 0
-- RCLike.normSq_one
@[rclike_simps]
theorem normSq_one : normSq (1 : K) = 1 
Arguments: {K : Type u_1} [RCLike K]
Goal: RCLike.normSq 1 = 1
-- RCLike.normSq_nonneg
theorem normSq_nonneg (z : K) : 0 ≤ normSq z 
Arguments: {K : Type u_1} [RCLike K] (z : K)
Goal: 0 ≤ RCLike.normSq z
-- RCLike.normSq_eq_zero
@[rclike_simps] -- porting note (#10618): was `simp`
theorem normSq_eq_zero {z : K} : normSq z = 0 ↔ z = 0 
Arguments: {K : Type u_1} [RCLike K] {z : K}
Goal: RCLike.normSq z = 0 ↔ z = 0
-- RCLike.normSq_pos
@[simp, rclike_simps]
theorem normSq_pos {z : K} : 0 < normSq z ↔ z ≠ 0 
Arguments: {K : Type u_1} [RCLike K] {z : K}
Goal: 0 < RCLike.normSq z ↔ z ≠ 0
-- RCLike.normSq_neg
@[simp, rclike_simps]
theorem normSq_neg (z : K) : normSq (-z) = normSq z 
Arguments: {K : Type u_1} [RCLike K] (z : K)
Goal: RCLike.normSq (-z) = RCLike.normSq z
-- RCLike.normSq_conj
@[simp, rclike_simps]
theorem normSq_conj (z : K) : normSq (conj z) = normSq z 
Arguments: {K : Type u_1} [RCLike K] (z : K)
Goal: RCLike.normSq ((starRingEnd K) z) = RCLike.normSq z
-- RCLike.normSq_mul
@[rclike_simps] -- porting note (#10618): was `simp`
theorem normSq_mul (z w : K) : normSq (z * w) = normSq z * normSq w 
Arguments: {K : Type u_1} [RCLike K] (z w : K)
Goal: RCLike.normSq (z * w) = RCLike.normSq z * RCLike.normSq w
-- RCLike.normSq_add
theorem normSq_add (z w : K) : normSq (z + w) = normSq z + normSq w + 2 * re (z * conj w) 
Arguments: {K : Type u_1} [RCLike K] (z w : K)
Goal: RCLike.normSq (z + w) = RCLike.normSq z + RCLike.normSq w + 2 * RCLike.re (z * (starRingEnd K) w)
-- RCLike.re_sq_le_normSq
theorem re_sq_le_normSq (z : K) : re z * re z ≤ normSq z 
Arguments: {K : Type u_1} [RCLike K] (z : K)
Goal: RCLike.re z * RCLike.re z ≤ RCLike.normSq z
-- RCLike.im_sq_le_normSq
theorem im_sq_le_normSq (z : K) : im z * im z ≤ normSq z 
Arguments: {K : Type u_1} [RCLike K] (z : K)
Goal: RCLike.im z * RCLike.im z ≤ RCLike.normSq z
-- RCLike.mul_conj
theorem mul_conj (z : K) : z * conj z = ‖z‖ ^ 2 
Arguments: {K : Type u_1} [RCLike K] (z : K)
Goal: z * (starRingEnd K) z = ↑‖z‖ ^ 2
-- RCLike.conj_mul
theorem conj_mul (z : K) : conj z * z = ‖z‖ ^ 2 
Arguments: {K : Type u_1} [RCLike K] (z : K)
Goal: (starRingEnd K) z * z = ↑‖z‖ ^ 2
-- RCLike.inv_eq_conj
lemma inv_eq_conj (hz : ‖z‖ = 1) : z⁻¹ = conj z 
Arguments: {K : Type u_1} [RCLike K] {z : K} (hz : ‖z‖ = 1)
Goal: z⁻¹ = (starRingEnd K) z
-- RCLike.normSq_sub
theorem normSq_sub (z w : K) : normSq (z - w) = normSq z + normSq w - 2 * re (z * conj w) 
Arguments: {K : Type u_1} [RCLike K] (z w : K)
Goal: RCLike.normSq (z - w) = RCLike.normSq z + RCLike.normSq w - 2 * RCLike.re (z * (starRingEnd K) w)
-- RCLike.sqrt_normSq_eq_norm
theorem sqrt_normSq_eq_norm {z : K} : √(normSq z) = ‖z‖ 
Arguments: {K : Type u_1} [RCLike K] {z : K}
Goal: √(RCLike.normSq z) = ‖z‖
-- RCLike.ofReal_inv
@[rclike_simps, norm_cast]
theorem ofReal_inv (r : ℝ) : ((r⁻¹ : ℝ) : K) = (r : K)⁻¹ 
Arguments: {K : Type u_1} [RCLike K] (r : ℝ)
Goal: ↑r⁻¹ = (↑r)⁻¹
-- RCLike.inv_def
theorem inv_def (z : K) : z⁻¹ = conj z * ((‖z‖ ^ 2)⁻¹ : ℝ) 
Arguments: {K : Type u_1} [RCLike K] (z : K)
Goal: z⁻¹ = (starRingEnd K) z * ↑(‖z‖ ^ 2)⁻¹
-- RCLike.inv_re
@[simp, rclike_simps]
theorem inv_re (z : K) : re z⁻¹ = re z / normSq z 
Arguments: {K : Type u_1} [RCLike K] (z : K)
Goal: RCLike.re z⁻¹ = RCLike.re z / RCLike.normSq z
-- RCLike.inv_im
@[simp, rclike_simps]
theorem inv_im (z : K) : im z⁻¹ = -im z / normSq z 
Arguments: {K : Type u_1} [RCLike K] (z : K)
Goal: RCLike.im z⁻¹ = -RCLike.im z / RCLike.normSq z
-- RCLike.div_re
theorem div_re (z w : K) : re (z / w) = re z * re w / normSq w + im z * im w / normSq w 
Arguments: {K : Type u_1} [RCLike K] (z w : K)
Goal: RCLike.re (z / w) = RCLike.re z * RCLike.re w / RCLike.normSq w + RCLike.im z * RCLike.im w / RCLike.normSq w
-- RCLike.div_im
theorem div_im (z w : K) : im (z / w) = im z * re w / normSq w - re z * im w / normSq w 
Arguments: {K : Type u_1} [RCLike K] (z w : K)
Goal: RCLike.im (z / w) = RCLike.im z * RCLike.re w / RCLike.normSq w - RCLike.re z * RCLike.im w / RCLike.normSq w
-- RCLike.conj_inv
@[rclike_simps] -- porting note (#10618): was `simp`
theorem conj_inv (x : K) : conj x⁻¹ = (conj x)⁻¹ 
Arguments: {K : Type u_1} [RCLike K] (x : K)
Goal: (starRingEnd K) x⁻¹ = ((starRingEnd K) x)⁻¹
-- RCLike.conj_div
lemma conj_div (x y : K) : conj (x / y) = conj x / conj y 
Arguments: {K : Type u_1} [RCLike K] (x y : K)
Goal: (starRingEnd K) (x / y) = (starRingEnd K) x / (starRingEnd K) y
-- RCLike.exists_norm_eq_mul_self
lemma exists_norm_eq_mul_self (x : K) : ∃ c, ‖c‖ = 1 ∧ ↑‖x‖ = c * x 
Arguments: {K : Type u_1} [RCLike K] (x : K)
Goal: ∃ c, ‖c‖ = 1 ∧ ↑‖x‖ = c * x
-- RCLike.exists_norm_mul_eq_self
lemma exists_norm_mul_eq_self (x : K) : ∃ c, ‖c‖ = 1 ∧ c * ‖x‖ = x 
Arguments: {K : Type u_1} [RCLike K] (x : K)
Goal: ∃ c, ‖c‖ = 1 ∧ c * ↑‖x‖ = x
-- RCLike.ofReal_div
@[rclike_simps, norm_cast]
theorem ofReal_div (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.div_re_ofReal
theorem div_re_ofReal {z : K} {r : ℝ} : re (z / r) = re z / r 
Arguments: {K : Type u_1} [RCLike K] {z : K} {r : ℝ}
Goal: RCLike.re (z / ↑r) = RCLike.re z / r
-- RCLike.ofReal_zpow
@[rclike_simps, norm_cast]
theorem ofReal_zpow (r : ℝ) (n : ℤ) : ((r ^ n : ℝ) : K) = (r : K) ^ n 
Arguments: {K : Type u_1} [RCLike K] (r : ℝ) (n : ℤ)
Goal: ↑(r ^ n) = ↑r ^ n
-- RCLike.I_mul_I_of_nonzero
theorem I_mul_I_of_nonzero : (I : K) ≠ 0 → (I : K) * I = -1 
Arguments: {K : Type u_1} [RCLike K]
Goal: RCLike.I ≠ 0 → RCLike.I * RCLike.I = -1
-- RCLike.inv_I
@[simp, rclike_simps]
theorem inv_I : (I : K)⁻¹ = -I 
Arguments: {K : Type u_1} [RCLike K]
Goal: RCLike.I⁻¹ = -RCLike.I
-- RCLike.div_I
@[simp, rclike_simps]
theorem div_I (z : K) : z / I = -(z * I) 
Arguments: {K : Type u_1} [RCLike K] (z : K)
Goal: z / RCLike.I = -(z * RCLike.I)
-- RCLike.normSq_inv
@[rclike_simps] -- porting note (#10618): was `simp`
theorem normSq_inv (z : K) : normSq z⁻¹ = (normSq z)⁻¹ 
Arguments: {K : Type u_1} [RCLike K] (z : K)
Goal: RCLike.normSq z⁻¹ = (RCLike.normSq z)⁻¹
-- RCLike.normSq_div
@[rclike_simps] -- porting note (#10618): was `simp`
theorem normSq_div (z w : K) : normSq (z / w) = normSq z / normSq w 
Arguments: {K : Type u_1} [RCLike K] (z w : K)
Goal: RCLike.normSq (z / w) = RCLike.normSq z / RCLike.normSq w
-- RCLike.norm_conj
@[rclike_simps] -- porting note (#10618): was `simp`
theorem norm_conj {z : K} : ‖conj z‖ = ‖z‖ 
Arguments: {K : Type u_1} [RCLike K] {z : K}
Goal: ‖(starRingEnd K) z‖ = ‖z‖
-- RCLike.ofReal_natCast
@[rclike_simps, norm_cast]
theorem ofReal_natCast (n : ℕ) : ((n : ℝ) : K) = n 
Arguments: {K : Type u_1} [RCLike K] (n : ℕ)
Goal: ↑↑n = ↑n
-- RCLike.natCast_re
@[simp, rclike_simps] -- Porting note: removed `norm_cast`
theorem natCast_re (n : ℕ) : re (n : K) = n 
Arguments: {K : Type u_1} [RCLike K] (n : ℕ)
Goal: RCLike.re ↑n = ↑n
-- RCLike.natCast_im
@[simp, rclike_simps, norm_cast]
theorem natCast_im (n : ℕ) : im (n : K) = 0 
Arguments: {K : Type u_1} [RCLike K] (n : ℕ)
Goal: RCLike.im ↑n = 0
-- RCLike.ofNat_re
@[simp, rclike_simps]
theorem ofNat_re (n : ℕ) [n.AtLeastTwo] : re (no_index (OfNat.ofNat n) : K) = OfNat.ofNat n 
Arguments: {K : Type u_1} [RCLike K] (n : ℕ) [n.AtLeastTwo]
Goal: RCLike.re (OfNat.ofNat n) = OfNat.ofNat n
-- RCLike.ofNat_im
@[simp, rclike_simps]
theorem ofNat_im (n : ℕ) [n.AtLeastTwo] : im (no_index (OfNat.ofNat n) : K) = 0 
Arguments: {K : Type u_1} [RCLike K] (n : ℕ) [n.AtLeastTwo]
Goal: RCLike.im (OfNat.ofNat n) = 0
-- RCLike.ofReal_ofNat
@[rclike_simps, norm_cast]
theorem ofReal_ofNat (n : ℕ) [n.AtLeastTwo] :
    ((no_index (OfNat.ofNat n) : ℝ) : K) = OfNat.ofNat n 
Arguments: {K : Type u_1} [RCLike K] (n : ℕ) [n.AtLeastTwo]
Goal: ↑(OfNat.ofNat n) = OfNat.ofNat n
-- RCLike.ofNat_mul_re
theorem ofNat_mul_re (n : ℕ) [n.AtLeastTwo] (z : K) :
    re (OfNat.ofNat n * z) = OfNat.ofNat n * re z 
Arguments: {K : Type u_1} [RCLike K] (n : ℕ) [n.AtLeastTwo] (z : K)
Goal: RCLike.re (OfNat.ofNat n * z) = OfNat.ofNat n * RCLike.re z
-- RCLike.ofNat_mul_im
theorem ofNat_mul_im (n : ℕ) [n.AtLeastTwo] (z : K) :
    im (OfNat.ofNat n * z) = OfNat.ofNat n * im z 
Arguments: {K : Type u_1} [RCLike K] (n : ℕ) [n.AtLeastTwo] (z : K)
Goal: RCLike.im (OfNat.ofNat n * z) = OfNat.ofNat n * RCLike.im z
-- RCLike.ofReal_intCast
@[rclike_simps, norm_cast]
theorem ofReal_intCast (n : ℤ) : ((n : ℝ) : K) = n 
Arguments: {K : Type u_1} [RCLike K] (n : ℤ)
Goal: ↑↑n = ↑n
-- RCLike.intCast_re
@[simp, rclike_simps] -- Porting note: removed `norm_cast`
theorem intCast_re (n : ℤ) : re (n : K) = n 
Arguments: {K : Type u_1} [RCLike K] (n : ℤ)
Goal: RCLike.re ↑n = ↑n
-- RCLike.intCast_im
@[simp, rclike_simps, norm_cast]
theorem intCast_im (n : ℤ) : im (n : K) = 0 
Arguments: {K : Type u_1} [RCLike K] (n : ℤ)
Goal: RCLike.im ↑n = 0
-- RCLike.ofReal_ratCast
@[rclike_simps, norm_cast]
theorem ofReal_ratCast (n : ℚ) : ((n : ℝ) : K) = n 
Arguments: {K : Type u_1} [RCLike K] (n : ℚ)
Goal: ↑↑n = ↑n
-- RCLike.ratCast_re
@[simp, rclike_simps] -- Porting note: removed `norm_cast`
theorem ratCast_re (q : ℚ) : re (q : K) = q 
Arguments: {K : Type u_1} [RCLike K] (q : ℚ)
Goal: RCLike.re ↑q = ↑q
-- RCLike.ratCast_im
@[simp, rclike_simps, norm_cast]
theorem ratCast_im (q : ℚ) : im (q : K) = 0 
Arguments: {K : Type u_1} [RCLike K] (q : ℚ)
Goal: RCLike.im ↑q = 0
-- RCLike.norm_of_nonneg
theorem norm_of_nonneg {r : ℝ} (h : 0 ≤ r) : ‖(r : K)‖ = r 
Arguments: {K : Type u_1} [RCLike K] {r : ℝ} (h : 0 ≤ r)
Goal: ‖↑r‖ = r
-- RCLike.norm_natCast
@[simp, rclike_simps, norm_cast]
theorem norm_natCast (n : ℕ) : ‖(n : K)‖ = n 
Arguments: {K : Type u_1} [RCLike K] (n : ℕ)
Goal: ‖↑n‖ = ↑n
-- RCLike.norm_ofNat
@[simp, rclike_simps]
theorem norm_ofNat (n : ℕ) [n.AtLeastTwo] : ‖(no_index (OfNat.ofNat n) : K)‖ = OfNat.ofNat n 
Arguments: {K : Type u_1} [RCLike K] (n : ℕ) [n.AtLeastTwo]
Goal: ‖OfNat.ofNat n‖ = OfNat.ofNat n
-- RCLike.norm_nsmul
lemma norm_nsmul [NormedAddCommGroup E] [NormedSpace K E] (n : ℕ) (x : E) : ‖n • x‖ = n • ‖x‖ 
Arguments: (K : Type u_1) {E : Type u_2} [RCLike K] [NormedAddCommGroup E] [NormedSpace K E] (n : ℕ) (x : E)
Goal: ‖n • x‖ = n • ‖x‖
-- RCLike.mul_self_norm
theorem mul_self_norm (z : K) : ‖z‖ * ‖z‖ = normSq z 
Arguments: {K : Type u_1} [RCLike K] (z : K)
Goal: ‖z‖ * ‖z‖ = RCLike.normSq z
-- RCLike.norm_two
theorem norm_two : ‖(2 : K)‖ = 2 
Arguments: {K : Type u_1} [RCLike K]
Goal: ‖2‖ = 2
-- RCLike.abs_re_le_norm
theorem abs_re_le_norm (z : K) : |re z| ≤ ‖z‖ 
Arguments: {K : Type u_1} [RCLike K] (z : K)
Goal: |RCLike.re z| ≤ ‖z‖
-- RCLike.abs_im_le_norm
theorem abs_im_le_norm (z : K) : |im z| ≤ ‖z‖ 
Arguments: {K : Type u_1} [RCLike K] (z : K)
Goal: |RCLike.im z| ≤ ‖z‖
-- RCLike.norm_re_le_norm
theorem norm_re_le_norm (z : K) : ‖re z‖ ≤ ‖z‖ 
Arguments: {K : Type u_1} [RCLike K] (z : K)
Goal: ‖RCLike.re z‖ ≤ ‖z‖
-- RCLike.norm_im_le_norm
theorem norm_im_le_norm (z : K) : ‖im z‖ ≤ ‖z‖ 
Arguments: {K : Type u_1} [RCLike K] (z : K)
Goal: ‖RCLike.im z‖ ≤ ‖z‖
-- RCLike.re_le_norm
theorem re_le_norm (z : K) : re z ≤ ‖z‖ 
Arguments: {K : Type u_1} [RCLike K] (z : K)
Goal: RCLike.re z ≤ ‖z‖
-- RCLike.im_le_norm
theorem im_le_norm (z : K) : im z ≤ ‖z‖ 
Arguments: {K : Type u_1} [RCLike K] (z : K)
Goal: RCLike.im z ≤ ‖z‖
-- RCLike.im_eq_zero_of_le
theorem im_eq_zero_of_le {a : K} (h : ‖a‖ ≤ re a) : im a = 0 
Arguments: {K : Type u_1} [RCLike K] {a : K} (h : ‖a‖ ≤ RCLike.re a)
Goal: RCLike.im a = 0
-- RCLike.re_eq_self_of_le
theorem re_eq_self_of_le {a : K} (h : ‖a‖ ≤ re a) : (re a : K) = a 
Arguments: {K : Type u_1} [RCLike K] {a : K} (h : ‖a‖ ≤ RCLike.re a)
Goal: ↑(RCLike.re a) = a
-- RCLike.abs_re_div_norm_le_one
theorem abs_re_div_norm_le_one (z : K) : |re z / ‖z‖| ≤ 1 
Arguments: {K : Type u_1} [RCLike K] (z : K)
Goal: |RCLike.re z / ‖z‖| ≤ 1
-- RCLike.abs_im_div_norm_le_one
theorem abs_im_div_norm_le_one (z : K) : |im z / ‖z‖| ≤ 1 
Arguments: {K : Type u_1} [RCLike K] (z : K)
Goal: |RCLike.im z / ‖z‖| ≤ 1
-- RCLike.norm_I_of_ne_zero
theorem norm_I_of_ne_zero (hI : (I : K) ≠ 0) : ‖(I : K)‖ = 1 
Arguments: {K : Type u_1} [RCLike K] (hI : RCLike.I ≠ 0)
Goal: ‖RCLike.I‖ = 1
-- RCLike.re_eq_norm_of_mul_conj
theorem re_eq_norm_of_mul_conj (x : K) : re (x * conj x) = ‖x * conj x‖ 
Arguments: {K : Type u_1} [RCLike K] (x : K)
Goal: RCLike.re (x * (starRingEnd K) x) = ‖x * (starRingEnd K) x‖
-- RCLike.norm_sq_re_add_conj
theorem norm_sq_re_add_conj (x : K) : ‖x + conj x‖ ^ 2 = re (x + conj x) ^ 2 
Arguments: {K : Type u_1} [RCLike K] (x : K)
Goal: ‖x + (starRingEnd K) x‖ ^ 2 = RCLike.re (x + (starRingEnd K) x) ^ 2
-- RCLike.norm_sq_re_conj_add
theorem norm_sq_re_conj_add (x : K) : ‖conj x + x‖ ^ 2 = re (conj x + x) ^ 2 
Arguments: {K : Type u_1} [RCLike K] (x : K)
Goal: ‖(starRingEnd K) x + x‖ ^ 2 = RCLike.re ((starRingEnd K) x + x) ^ 2
-- RCLike.isCauSeq_re
theorem isCauSeq_re (f : CauSeq K norm) : IsCauSeq abs fun n => re (f n) 
Arguments: {K : Type u_1} [RCLike K] (f : CauSeq K norm)
Goal: IsCauSeq abs fun n => RCLike.re (↑f n)
-- RCLike.isCauSeq_im
theorem isCauSeq_im (f : CauSeq K norm) : IsCauSeq abs fun n => im (f n) 
Arguments: {K : Type u_1} [RCLike K] (f : CauSeq K norm)
Goal: IsCauSeq abs fun n => RCLike.im (↑f n)
-- RCLike.isCauSeq_norm
theorem isCauSeq_norm {f : ℕ → K} (hf : IsCauSeq norm f) : IsCauSeq abs (norm ∘ f) 
Arguments: {K : Type u_1} [RCLike K] {f : ℕ → K} (hf : IsCauSeq norm f)
Goal: IsCauSeq abs (norm ∘ f)
-- RCLike.lt_iff_re_im
theorem lt_iff_re_im : 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.nonneg_iff
theorem nonneg_iff : 0 ≤ z ↔ 0 ≤ re z ∧ im z = 0 
Arguments: {K : Type u_1} [RCLike K] {z : K}
Goal: 0 ≤ z ↔ 0 ≤ RCLike.re z ∧ RCLike.im z = 0
-- RCLike.pos_iff
theorem pos_iff : 0 < z ↔ 0 < re z ∧ im z = 0 
Arguments: {K : Type u_1} [RCLike K] {z : K}
Goal: 0 < z ↔ 0 < RCLike.re z ∧ RCLike.im z = 0
-- RCLike.nonpos_iff
theorem nonpos_iff : z ≤ 0 ↔ re z ≤ 0 ∧ im z = 0 
Arguments: {K : Type u_1} [RCLike K] {z : K}
Goal: z ≤ 0 ↔ RCLike.re z ≤ 0 ∧ RCLike.im z = 0
-- RCLike.neg_iff
theorem neg_iff : z < 0 ↔ re z < 0 ∧ im z = 0 
Arguments: {K : Type u_1} [RCLike K] {z : K}
Goal: z < 0 ↔ RCLike.re z < 0 ∧ RCLike.im z = 0
-- RCLike.nonneg_iff_exists_ofReal
lemma nonneg_iff_exists_ofReal : 0 ≤ z ↔ ∃ x ≥ (0 : ℝ), x = z 
Arguments: {K : Type u_1} [RCLike K] {z : K}
Goal: 0 ≤ z ↔ ∃ x ≥ 0, ↑x = z
-- RCLike.pos_iff_exists_ofReal
lemma pos_iff_exists_ofReal : 0 < z ↔ ∃ x > (0 : ℝ), x = z 
Arguments: {K : Type u_1} [RCLike K] {z : K}
Goal: 0 < z ↔ ∃ x > 0, ↑x = z
-- RCLike.nonpos_iff_exists_ofReal
lemma nonpos_iff_exists_ofReal : z ≤ 0 ↔ ∃ x ≤ (0 : ℝ), x = z 
Arguments: {K : Type u_1} [RCLike K] {z : K}
Goal: z ≤ 0 ↔ ∃ x ≤ 0, ↑x = z
-- RCLike.neg_iff_exists_ofReal
lemma neg_iff_exists_ofReal : z < 0 ↔ ∃ x < (0 : ℝ), x = z 
Arguments: {K : Type u_1} [RCLike K] {z : K}
Goal: z < 0 ↔ ∃ x < 0, ↑x = z
-- RCLike.ofReal_le_ofReal
@[simp, norm_cast]
lemma ofReal_le_ofReal {x y : ℝ} : (x : K) ≤ (y : K) ↔ x ≤ y 
Arguments: {K : Type u_1} [RCLike K] {x y : ℝ}
Goal: ↑x ≤ ↑y ↔ x ≤ y
-- RCLike.ofReal_lt_ofReal
@[simp, norm_cast]
lemma ofReal_lt_ofReal {x y : ℝ} : (x : K) < (y : K) ↔ x < y 
Arguments: {K : Type u_1} [RCLike K] {x y : ℝ}
Goal: ↑x < ↑y ↔ x < y
-- RCLike.ofReal_nonneg
@[simp, norm_cast]
lemma ofReal_nonneg {x : ℝ} : 0 ≤ (x : K) ↔ 0 ≤ x 
Arguments: {K : Type u_1} [RCLike K] {x : ℝ}
Goal: 0 ≤ ↑x ↔ 0 ≤ x
-- RCLike.ofReal_nonpos
@[simp, norm_cast]
lemma ofReal_nonpos {x : ℝ} : (x : K) ≤ 0 ↔ x ≤ 0 
Arguments: {K : Type u_1} [RCLike K] {x : ℝ}
Goal: ↑x ≤ 0 ↔ x ≤ 0
-- RCLike.ofReal_pos
@[simp, norm_cast]
lemma ofReal_pos {x : ℝ} : 0 < (x : K) ↔ 0 < x 
Arguments: {K : Type u_1} [RCLike K] {x : ℝ}
Goal: 0 < ↑x ↔ 0 < x
-- RCLike.ofReal_lt_zero
@[simp, norm_cast]
lemma ofReal_lt_zero {x : ℝ} : (x : K) < 0 ↔ x < 0 
Arguments: {K : Type u_1} [RCLike K] {x : ℝ}
Goal: ↑x < 0 ↔ x < 0
-- RCLike.inv_pos_of_pos
protected lemma inv_pos_of_pos (hz : 0 < z) : 0 < z⁻¹ 
Arguments: {K : Type u_1} [RCLike K] {z : K} (hz : 0 < z)
Goal: 0 < z⁻¹
-- RCLike.inv_pos
protected lemma inv_pos : 0 < z⁻¹ ↔ 0 < z 
Arguments: {K : Type u_1} [RCLike K] {z : K}
Goal: 0 < z⁻¹ ↔ 0 < z
-- RCLike.toStarOrderedRing
lemma toStarOrderedRing : StarOrderedRing K 
Arguments: {K : Type u_1} [RCLike K]
Goal: StarOrderedRing K
-- RCLike.toOrderedSMul
theorem toOrderedSMul : OrderedSMul ℝ K 
Arguments: {K : Type u_1} [RCLike K]
Goal: OrderedSMul ℝ K