ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- GaussianInt.toComplex_div_re
theorem toComplex_div_re (x y : ℤ[i]) : ((x / y : ℤ[i]) : ℂ).re = round (x / y : ℂ).re 
Arguments: (x y : ℤ[i])
Goal: (GaussianInt.toComplex (x / y)).re = ↑(round (GaussianInt.toComplex x / GaussianInt.toComplex y).re)
-- GaussianInt.toComplex_div_im
theorem toComplex_div_im (x y : ℤ[i]) : ((x / y : ℤ[i]) : ℂ).im = round (x / y : ℂ).im 
Arguments: (x y : ℤ[i])
Goal: (GaussianInt.toComplex (x / y)).im = ↑(round (GaussianInt.toComplex x / GaussianInt.toComplex y).im)
-- GaussianInt.normSq_le_normSq_of_re_le_of_im_le
theorem normSq_le_normSq_of_re_le_of_im_le {x y : ℂ} (hre : |x.re| ≤ |y.re|)
    (him : |x.im| ≤ |y.im|) : Complex.normSq x ≤ Complex.normSq y 
Arguments: {x y : ℂ} (hre : |x.re| ≤ |y.re|) (him : |x.im| ≤ |y.im|)
Goal: Complex.normSq x ≤ Complex.normSq y
-- GaussianInt.normSq_div_sub_div_lt_one
theorem normSq_div_sub_div_lt_one (x y : ℤ[i]) :
    Complex.normSq ((x / y : ℂ) - ((x / y : ℤ[i]) : ℂ)) < 1 
Arguments: (x y : ℤ[i])
Goal: Complex.normSq (GaussianInt.toComplex x / GaussianInt.toComplex y - GaussianInt.toComplex (x / y)) < 1
-- GaussianInt.mod_def
theorem mod_def (x y : ℤ[i]) : x % y = x - y * (x / y) 
Arguments: (x y : ℤ[i])
Goal: x % y = x - y * (x / y)
-- GaussianInt.norm_mod_lt
theorem norm_mod_lt (x : ℤ[i]) {y : ℤ[i]} (hy : y ≠ 0) : (x % y).norm < y.norm 
Arguments: (x : ℤ[i]) {y : ℤ[i]} (hy : y ≠ 0)
Goal: Zsqrtd.norm (x % y) < Zsqrtd.norm y
-- GaussianInt.natAbs_norm_mod_lt
theorem natAbs_norm_mod_lt (x : ℤ[i]) {y : ℤ[i]} (hy : y ≠ 0) :
    (x % y).norm.natAbs < y.norm.natAbs 
Arguments: (x : ℤ[i]) {y : ℤ[i]} (hy : y ≠ 0)
Goal: (Zsqrtd.norm (x % y)).natAbs < (Zsqrtd.norm y).natAbs
-- GaussianInt.norm_le_norm_mul_left
theorem norm_le_norm_mul_left (x : ℤ[i]) {y : ℤ[i]} (hy : y ≠ 0) :
    (norm x).natAbs ≤ (norm (x * y)).natAbs 
Arguments: (x : ℤ[i]) {y : ℤ[i]} (hy : y ≠ 0)
Goal: (Zsqrtd.norm x).natAbs ≤ (Zsqrtd.norm (x * y)).natAbs
-- GaussianInt.sq_add_sq_of_nat_prime_of_not_irreducible
theorem sq_add_sq_of_nat_prime_of_not_irreducible (p : ℕ) [hp : Fact p.Prime]
    (hpi : ¬Irreducible (p : ℤ[i])) : ∃ a b, a ^ 2 + b ^ 2 = p 
Arguments: (p : ℕ) [hp : Fact (Nat.Prime p)] (hpi : ¬Irreducible ↑p)
Goal: ∃ a b, a ^ 2 + b ^ 2 = p
-- rootsOfUnity.integer_power_of_ringEquiv
theorem rootsOfUnity.integer_power_of_ringEquiv (g : L ≃+* L) :
    ∃ m : ℤ, ∀ t : rootsOfUnity n L, g (t : Lˣ) = (t ^ m : Lˣ) 
Arguments: {L : Type u} [CommRing L] [IsDomain L] (n : ℕ+) (g : L ≃+* L)
Goal: ∃ m, ∀ (t : ↥(rootsOfUnity n L)), g ↑↑t = ↑(↑t ^ m)
-- rootsOfUnity.integer_power_of_ringEquiv'
theorem rootsOfUnity.integer_power_of_ringEquiv' (g : L ≃+* L) :
    ∃ m : ℤ, ∀ t ∈ rootsOfUnity n L, g (t : Lˣ) = (t ^ m : Lˣ) 
Arguments: {L : Type u} [CommRing L] [IsDomain L] (n : ℕ+) (g : L ≃+* L)
Goal: ∃ m, ∀ t ∈ rootsOfUnity n L, g ↑t = ↑(t ^ m)
-- ModularCyclotomicCharacter_aux_spec
theorem ModularCyclotomicCharacter_aux_spec (g : L ≃+* L) (n : ℕ+) :
    ∀ t : rootsOfUnity n L, g (t : Lˣ) = (t ^ (ModularCyclotomicCharacter_aux g n) : Lˣ) 
Arguments: {L : Type u} [CommRing L] [IsDomain L] (g : L ≃+* L) (n : ℕ+) (t : ↥(rootsOfUnity n L))
Goal: g ↑↑t = ↑(↑t ^ ModularCyclotomicCharacter_aux g n)
-- ModularCyclotomicCharacter.toFun_spec
theorem toFun_spec (g : L ≃+* L) {n : ℕ+} (t : rootsOfUnity n L) :
    g (t : Lˣ) = (t ^ (χ₀ n g).val : Lˣ) 
Arguments: {L : Type u} [CommRing L] [IsDomain L] (g : L ≃+* L) {n : ℕ+} (t : ↥(rootsOfUnity n L))
Goal: g ↑↑t = ↑(↑t ^ (ModularCyclotomicCharacter.toFun n g).val)
-- ModularCyclotomicCharacter.toFun_spec'
theorem toFun_spec' (g : L ≃+* L) {n : ℕ+} {t : Lˣ} (ht : t ∈ rootsOfUnity n L) :
    g t = t ^ (χ₀ n g).val 
Arguments: {L : Type u} [CommRing L] [IsDomain L] (g : L ≃+* L) {n : ℕ+} {t : Lˣ} (ht : t ∈ rootsOfUnity n L)
Goal: g ↑t = ↑t ^ (ModularCyclotomicCharacter.toFun n g).val
-- ModularCyclotomicCharacter.toFun_spec''
theorem toFun_spec'' (g : L ≃+* L) {n : ℕ+} {t : L} (ht : IsPrimitiveRoot t n) :
    g t = t ^ (χ₀ n g).val 
Arguments: {L : Type u} [CommRing L] [IsDomain L] (g : L ≃+* L) {n : ℕ+} {t : L} (ht : IsPrimitiveRoot t ↑n)
Goal: g t = t ^ (ModularCyclotomicCharacter.toFun n g).val
-- ModularCyclotomicCharacter.toFun_unique
theorem toFun_unique (g : L ≃+* L) (c : ZMod (Fintype.card (rootsOfUnity n L)))
    (hc : ∀ t : rootsOfUnity n L, g (t : Lˣ) = (t ^ c.val : Lˣ)) : c = χ₀ n g 
Arguments: {L : Type u} [CommRing L] [IsDomain L] (n : ℕ+) (g : L ≃+* L) (c : ZMod (Fintype.card ↥(rootsOfUnity n L))) (hc : ∀ (t : ↥(rootsOfUnity n L)), g ↑↑t = ↑(↑t ^ c.val))
Goal: c = ModularCyclotomicCharacter.toFun n g
-- ModularCyclotomicCharacter.toFun_unique'
theorem toFun_unique' (g : L ≃+* L) (c : ZMod (Fintype.card (rootsOfUnity n L)))
    (hc : ∀ t ∈ rootsOfUnity n L, g t = t ^ c.val) : c = χ₀ n g 
Arguments: {L : Type u} [CommRing L] [IsDomain L] (n : ℕ+) (g : L ≃+* L) (c : ZMod (Fintype.card ↥(rootsOfUnity n L))) (hc : ∀ t ∈ rootsOfUnity n L, g ↑t = ↑t ^ c.val)
Goal: c = ModularCyclotomicCharacter.toFun n g
-- ModularCyclotomicCharacter.id
lemma id : χ₀ n (RingEquiv.refl L) = 1 
Arguments: {L : Type u} [CommRing L] [IsDomain L] (n : ℕ+)
Goal: ModularCyclotomicCharacter.toFun n (RingEquiv.refl L) = 1
-- ModularCyclotomicCharacter.comp
lemma comp (g h : L ≃+* L) : χ₀ n (g * h) =
    χ₀ n g * χ₀ n h 
Arguments: {L : Type u} [CommRing L] [IsDomain L] (n : ℕ+) (g h : L ≃+* L)
Goal: ModularCyclotomicCharacter.toFun n (g * h) = ModularCyclotomicCharacter.toFun n g * ModularCyclotomicCharacter.toFun n h
-- spec'
lemma spec' (g : L ≃+* L) {t : Lˣ} (ht : t ∈ rootsOfUnity n L) :
    g t = t ^ ((ModularCyclotomicCharacter' L n g) : ZMod
      (Fintype.card { x // x ∈ rootsOfUnity n L })).val 
Arguments: (L : Type u) [CommRing L] [IsDomain L] (n : ℕ+) (g : L ≃+* L) {t : Lˣ} (ht : t ∈ rootsOfUnity n L)
Goal: g ↑t = ↑t ^ (↑((ModularCyclotomicCharacter' L n) g)).val
-- unique'
lemma unique' (g : L ≃+* L) {c : ZMod (Fintype.card { x // x ∈ rootsOfUnity n L })}
    (hc : ∀ t ∈ rootsOfUnity n L, g t = t ^ c.val) :
    c = ModularCyclotomicCharacter' L n g 
Arguments: (L : Type u) [CommRing L] [IsDomain L] (n : ℕ+) (g : L ≃+* L) {c : ZMod (Fintype.card ↥(rootsOfUnity n L))} (hc : ∀ t ∈ rootsOfUnity n L, g ↑t = ↑t ^ c.val)
Goal: c = ↑((ModularCyclotomicCharacter' L n) g)
-- ModularCyclotomicCharacter.spec
lemma spec (g : L ≃+* L) {t : Lˣ} (ht : t ∈ rootsOfUnity n L) :
    g t = t ^ ((ModularCyclotomicCharacter L hn g) : ZMod n).val 
Arguments: (L : Type u) [CommRing L] [IsDomain L] {n : ℕ+} (hn : Fintype.card ↥(rootsOfUnity n L) = ↑n) (g : L ≃+* L) {t : Lˣ} (ht : t ∈ rootsOfUnity n L)
Goal: g ↑t = ↑t ^ (↑((ModularCyclotomicCharacter L hn) g)).val
-- ModularCyclotomicCharacter.unique
lemma unique (g : L ≃+* L) {c : ZMod n}  (hc : ∀ t ∈ rootsOfUnity n L, g t = t ^ c.val) :
    c = ModularCyclotomicCharacter L hn g 
Arguments: (L : Type u) [CommRing L] [IsDomain L] {n : ℕ+} (hn : Fintype.card ↥(rootsOfUnity n L) = ↑n) (g : L ≃+* L) {c : ZMod ↑n} (hc : ∀ t ∈ rootsOfUnity n L, g ↑t = ↑t ^ c.val)
Goal: c = ↑((ModularCyclotomicCharacter L hn) g)
-- IsPrimitiveRoot.autToPow_eq_ModularCyclotomicCharacter
lemma IsPrimitiveRoot.autToPow_eq_ModularCyclotomicCharacter (n : ℕ+)
    (R : Type*) [CommRing R] [Algebra R L] {μ : L} (hμ : IsPrimitiveRoot μ n) (g : L ≃ₐ[R] L) :
    hμ.autToPow R g = ModularCyclotomicCharacter L hμ.card_rootsOfUnity g 
Arguments: {L : Type u} [CommRing L] [IsDomain L] (n : ℕ+) (R : Type u_1) [CommRing R] [Algebra R L] {μ : L} (hμ : IsPrimitiveRoot μ ↑n) (g : L ≃ₐ[R] L)
Goal: (IsPrimitiveRoot.autToPow R hμ) g = (ModularCyclotomicCharacter L ⋯) ↑g
-- NumberField.Embeddings.card
theorem card : Fintype.card (K →+* A) = finrank ℚ K 
Arguments: (K : Type u_1) [Field K] [NumberField K] (A : Type u_2) [Field A] [CharZero A] [IsAlgClosed A]
Goal: Fintype.card (K →+* A) = FiniteDimensional.finrank ℚ K
-- NumberField.Embeddings.range_eval_eq_rootSet_minpoly
theorem range_eval_eq_rootSet_minpoly :
    (range fun φ : K →+* A => φ x) = (minpoly ℚ x).rootSet A 
Arguments: (K : Type u_1) (A : Type u_2) [Field K] [NumberField K] [Field A] [Algebra ℚ A] [IsAlgClosed A] (x : K)
Goal: (Set.range fun φ => φ x) = (minpoly ℚ x).rootSet A
-- NumberField.Embeddings.coeff_bdd_of_norm_le
theorem coeff_bdd_of_norm_le {B : ℝ} {x : K} (h : ∀ φ : K →+* A, ‖φ x‖ ≤ B) (i : ℕ) :
    ‖(minpoly ℚ x).coeff i‖ ≤ max B 1 ^ finrank ℚ K * (finrank ℚ K).choose (finrank ℚ K / 2) 
Arguments: {K : Type u_1} [Field K] [NumberField K] {A : Type u_2} [NormedField A] [IsAlgClosed A] [NormedAlgebra ℚ A] {B : ℝ} {x : K} (h : ∀ (φ : K →+* A), ‖φ x‖ ≤ B) (i : ℕ)
Goal: ‖(minpoly ℚ x).coeff i‖ ≤
  max B 1 ^ FiniteDimensional.finrank ℚ K *
    ↑((FiniteDimensional.finrank ℚ K).choose (FiniteDimensional.finrank ℚ K / 2))
-- NumberField.Embeddings.finite_of_norm_le
theorem finite_of_norm_le (B : ℝ) : {x : K | IsIntegral ℤ x ∧ ∀ φ : K →+* A, ‖φ x‖ ≤ B}.Finite 
Arguments: (K : Type u_1) [Field K] [NumberField K] (A : Type u_2) [NormedField A] [IsAlgClosed A] [NormedAlgebra ℚ A] (B : ℝ)
Goal: {x | IsIntegral ℤ x ∧ ∀ (φ : K →+* A), ‖φ x‖ ≤ B}.Finite
-- NumberField.Embeddings.pow_eq_one_of_norm_eq_one
theorem pow_eq_one_of_norm_eq_one {x : K} (hxi : IsIntegral ℤ x) (hx : ∀ φ : K →+* A, ‖φ x‖ = 1) :
    ∃ (n : ℕ) (_ : 0 < n), x ^ n = 1 
Arguments: (K : Type u_1) [Field K] [NumberField K] (A : Type u_2) [NormedField A] [IsAlgClosed A] [NormedAlgebra ℚ A] {x : K} (hxi : IsIntegral ℤ x) (hx : ∀ (φ : K →+* A), ‖φ x‖ = 1)
Goal: ∃ n, ∃ (_ : 0 < n), x ^ n = 1
-- NumberField.place_apply
@[simp]
theorem NumberField.place_apply (x : K) : (NumberField.place φ) x = norm (φ x) 
Arguments: {K : Type u_1} [Field K] {A : Type u_2} [NormedDivisionRing A] [Nontrivial A] (φ : K →+* A) (x : K)
Goal: (NumberField.place φ) x = ‖φ x‖
-- NumberField.ComplexEmbedding.conjugate_coe_eq
@[simp]
theorem conjugate_coe_eq (φ : K →+* ℂ) (x : K) : (conjugate φ) x = conj (φ x) 
Arguments: {K : Type u_1} [Field K] (φ : K →+* ℂ) (x : K)
Goal: (NumberField.ComplexEmbedding.conjugate φ) x = (starRingEnd ℂ) (φ x)
-- NumberField.ComplexEmbedding.place_conjugate
theorem place_conjugate (φ : K →+* ℂ) : place (conjugate φ) = place φ 
Arguments: {K : Type u_1} [Field K] (φ : K →+* ℂ)
Goal: NumberField.place (NumberField.ComplexEmbedding.conjugate φ) = NumberField.place φ
-- NumberField.ComplexEmbedding.isReal_iff
theorem isReal_iff {φ : K →+* ℂ} : IsReal φ ↔ conjugate φ = φ 
Arguments: {K : Type u_1} [Field K] {φ : K →+* ℂ}
Goal: NumberField.ComplexEmbedding.IsReal φ ↔ NumberField.ComplexEmbedding.conjugate φ = φ
-- NumberField.ComplexEmbedding.isReal_conjugate_iff
theorem isReal_conjugate_iff {φ : K →+* ℂ} : IsReal (conjugate φ) ↔ IsReal φ 
Arguments: {K : Type u_1} [Field K] {φ : K →+* ℂ}
Goal: NumberField.ComplexEmbedding.IsReal (NumberField.ComplexEmbedding.conjugate φ) ↔ NumberField.ComplexEmbedding.IsReal φ
-- NumberField.ComplexEmbedding.IsReal.coe_embedding_apply
@[simp]
theorem IsReal.coe_embedding_apply {φ : K →+* ℂ} (hφ : IsReal φ) (x : K) :
    (hφ.embedding x : ℂ) = φ x 
Arguments: {K : Type u_1} [Field K] {φ : K →+* ℂ} (hφ : NumberField.ComplexEmbedding.IsReal φ) (x : K)
Goal: ↑(hφ.embedding x) = φ x
-- NumberField.ComplexEmbedding.IsReal.comp
lemma IsReal.comp (f : k →+* K) {φ : K →+* ℂ} (hφ : IsReal φ) :
    IsReal (φ.comp f) 
Arguments: {K : Type u_1} [Field K] {k : Type u_2} [Field k] (f : k →+* K) {φ : K →+* ℂ} (hφ : NumberField.ComplexEmbedding.IsReal φ)
Goal: NumberField.ComplexEmbedding.IsReal (φ.comp f)
-- NumberField.ComplexEmbedding.isReal_comp_iff
lemma isReal_comp_iff {f : k ≃+* K} {φ : K →+* ℂ} :
    IsReal (φ.comp (f : k →+* K)) ↔ IsReal φ 
Arguments: {K : Type u_1} [Field K] {k : Type u_2} [Field k] {f : k ≃+* K} {φ : K →+* ℂ}
Goal: NumberField.ComplexEmbedding.IsReal (φ.comp ↑f) ↔ NumberField.ComplexEmbedding.IsReal φ
-- NumberField.ComplexEmbedding.exists_comp_symm_eq_of_comp_eq
lemma exists_comp_symm_eq_of_comp_eq [Algebra k K] [IsGalois k K] (φ ψ : K →+* ℂ)
    (h : φ.comp (algebraMap k K) = ψ.comp (algebraMap k K)) :
    ∃ σ : K ≃ₐ[k] K, φ.comp σ.symm = ψ 
Arguments: {K : Type u_1} [Field K] {k : Type u_2} [Field k] [Algebra k K] [IsGalois k K] (φ ψ : K →+* ℂ) (h : φ.comp (algebraMap k K) = ψ.comp (algebraMap k K))
Goal: ∃ σ, φ.comp ↑σ.symm = ψ
-- NumberField.ComplexEmbedding.IsConj.eq
lemma IsConj.eq (h : IsConj φ σ) (x) : φ (σ x) = star (φ x) 
Arguments: {K : Type u_1} [Field K] {k : Type u_2} [Field k] [Algebra k K] {φ : K →+* ℂ} {σ : K ≃ₐ[k] K} (h : NumberField.ComplexEmbedding.IsConj φ σ) (x : K)
Goal: φ (σ x) = star (φ x)
-- NumberField.ComplexEmbedding.IsConj.ext
lemma IsConj.ext {σ₁ σ₂ : K ≃ₐ[k] K} (h₁ : IsConj φ σ₁) (h₂ : IsConj φ σ₂) : σ₁ = σ₂ 
Arguments: {K : Type u_1} [Field K] {k : Type u_2} [Field k] [Algebra k K] {φ : K →+* ℂ} {σ₁ σ₂ : K ≃ₐ[k] K} (h₁ : NumberField.ComplexEmbedding.IsConj φ σ₁) (h₂ : NumberField.ComplexEmbedding.IsConj φ σ₂)
Goal: σ₁ = σ₂
-- NumberField.ComplexEmbedding.IsConj.ext_iff
lemma IsConj.ext_iff {σ₁ σ₂ : K ≃ₐ[k] K} (h₁ : IsConj φ σ₁) : σ₁ = σ₂ ↔ IsConj φ σ₂ 
Arguments: {K : Type u_1} [Field K] {k : Type u_2} [Field k] [Algebra k K] {φ : K →+* ℂ} {σ₁ σ₂ : K ≃ₐ[k] K} (h₁ : NumberField.ComplexEmbedding.IsConj φ σ₁)
Goal: σ₁ = σ₂ ↔ NumberField.ComplexEmbedding.IsConj φ σ₂
-- NumberField.ComplexEmbedding.IsConj.isReal_comp
lemma IsConj.isReal_comp (h : IsConj φ σ) : IsReal (φ.comp (algebraMap k K)) 
Arguments: {K : Type u_1} [Field K] {k : Type u_2} [Field k] [Algebra k K] {φ : K →+* ℂ} {σ : K ≃ₐ[k] K} (h : NumberField.ComplexEmbedding.IsConj φ σ)
Goal: NumberField.ComplexEmbedding.IsReal (φ.comp (algebraMap k K))
-- NumberField.ComplexEmbedding.isConj_one_iff
lemma isConj_one_iff : IsConj φ (1 : K ≃ₐ[k] K) ↔ IsReal φ 
Arguments: {K : Type u_1} [Field K] {k : Type u_2} [Field k] [Algebra k K] {φ : K →+* ℂ}
Goal: NumberField.ComplexEmbedding.IsConj φ 1 ↔ NumberField.ComplexEmbedding.IsReal φ
-- NumberField.ComplexEmbedding.IsConj.symm
lemma IsConj.symm (hσ : IsConj φ σ) :
    IsConj φ σ.symm 
Arguments: {K : Type u_1} [Field K] {k : Type u_2} [Field k] [Algebra k K] {φ : K →+* ℂ} {σ : K ≃ₐ[k] K} (hσ : NumberField.ComplexEmbedding.IsConj φ σ)
Goal: NumberField.ComplexEmbedding.IsConj φ σ.symm
-- NumberField.ComplexEmbedding.isConj_symm
lemma isConj_symm : IsConj φ σ.symm ↔ IsConj φ σ 
Arguments: {K : Type u_1} [Field K] {k : Type u_2} [Field k] [Algebra k K] {φ : K →+* ℂ} {σ : K ≃ₐ[k] K}
Goal: NumberField.ComplexEmbedding.IsConj φ σ.symm ↔ NumberField.ComplexEmbedding.IsConj φ σ
-- NumberField.InfinitePlace.apply
@[simp]
theorem apply (φ : K →+* ℂ) (x : K) : (mk φ) x = Complex.abs (φ x) 
Arguments: {K : Type u_2} [Field K] (φ : K →+* ℂ) (x : K)
Goal: (NumberField.InfinitePlace.mk φ) x = Complex.abs (φ x)
-- NumberField.InfinitePlace.mk_embedding
@[simp]
theorem mk_embedding (w : InfinitePlace K) : mk (embedding w) = w 
Arguments: {K : Type u_2} [Field K] (w : NumberField.InfinitePlace K)
Goal: NumberField.InfinitePlace.mk w.embedding = w
-- NumberField.InfinitePlace.mk_conjugate_eq
@[simp]
theorem mk_conjugate_eq (φ : K →+* ℂ) : mk (ComplexEmbedding.conjugate φ) = mk φ 
Arguments: {K : Type u_2} [Field K] (φ : K →+* ℂ)
Goal: NumberField.InfinitePlace.mk (NumberField.ComplexEmbedding.conjugate φ) = NumberField.InfinitePlace.mk φ
-- NumberField.InfinitePlace.norm_embedding_eq
theorem norm_embedding_eq (w : InfinitePlace K) (x : K) :
    ‖(embedding w) x‖ = w x 
Arguments: {K : Type u_2} [Field K] (w : NumberField.InfinitePlace K) (x : K)
Goal: ‖w.embedding x‖ = w x
-- NumberField.InfinitePlace.eq_iff_eq
theorem eq_iff_eq (x : K) (r : ℝ) : (∀ w : InfinitePlace K, w x = r) ↔ ∀ φ : K →+* ℂ, ‖φ x‖ = r 
Arguments: {K : Type u_2} [Field K] (x : K) (r : ℝ)
Goal: (∀ (w : NumberField.InfinitePlace K), w x = r) ↔ ∀ (φ : K →+* ℂ), ‖φ x‖ = r
-- NumberField.InfinitePlace.le_iff_le
theorem le_iff_le (x : K) (r : ℝ) : (∀ w : InfinitePlace K, w x ≤ r) ↔ ∀ φ : K →+* ℂ, ‖φ x‖ ≤ r 
Arguments: {K : Type u_2} [Field K] (x : K) (r : ℝ)
Goal: (∀ (w : NumberField.InfinitePlace K), w x ≤ r) ↔ ∀ (φ : K →+* ℂ), ‖φ x‖ ≤ r
-- NumberField.InfinitePlace.pos_iff
theorem pos_iff {w : InfinitePlace K} {x : K} : 0 < w x ↔ x ≠ 0 
Arguments: {K : Type u_2} [Field K] {w : NumberField.InfinitePlace K} {x : K}
Goal: 0 < w x ↔ x ≠ 0
-- NumberField.InfinitePlace.mk_eq_iff
@[simp]
theorem mk_eq_iff {φ ψ : K →+* ℂ} : mk φ = mk ψ ↔ φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ 
Arguments: {K : Type u_2} [Field K] {φ ψ : K →+* ℂ}
Goal: NumberField.InfinitePlace.mk φ = NumberField.InfinitePlace.mk ψ ↔ φ = ψ ∨ NumberField.ComplexEmbedding.conjugate φ = ψ
-- NumberField.InfinitePlace.embedding_mk_eq
theorem embedding_mk_eq (φ : K →+* ℂ) :
    embedding (mk φ) = φ ∨ embedding (mk φ) = ComplexEmbedding.conjugate φ 
Arguments: {K : Type u_2} [Field K] (φ : K →+* ℂ)
Goal: (NumberField.InfinitePlace.mk φ).embedding = φ ∨
  (NumberField.InfinitePlace.mk φ).embedding = NumberField.ComplexEmbedding.conjugate φ
-- NumberField.InfinitePlace.embedding_mk_eq_of_isReal
@[simp]
theorem embedding_mk_eq_of_isReal {φ : K →+* ℂ} (h : ComplexEmbedding.IsReal φ) :
    embedding (mk φ) = φ 
Arguments: {K : Type u_2} [Field K] {φ : K →+* ℂ} (h : NumberField.ComplexEmbedding.IsReal φ)
Goal: (NumberField.InfinitePlace.mk φ).embedding = φ
-- NumberField.InfinitePlace.isReal_iff
theorem isReal_iff {w : InfinitePlace K} :
    IsReal w ↔ ComplexEmbedding.IsReal (embedding w) 
Arguments: {K : Type u_2} [Field K] {w : NumberField.InfinitePlace K}
Goal: w.IsReal ↔ NumberField.ComplexEmbedding.IsReal w.embedding
-- NumberField.InfinitePlace.isComplex_iff
theorem isComplex_iff {w : InfinitePlace K} :
    IsComplex w ↔ ¬ComplexEmbedding.IsReal (embedding w) 
Arguments: {K : Type u_2} [Field K] {w : NumberField.InfinitePlace K}
Goal: w.IsComplex ↔ ¬NumberField.ComplexEmbedding.IsReal w.embedding
-- NumberField.InfinitePlace.conjugate_embedding_eq_of_isReal
@[simp]
theorem conjugate_embedding_eq_of_isReal {w : InfinitePlace K} (h : IsReal w) :
    ComplexEmbedding.conjugate (embedding w) = embedding w 
Arguments: {K : Type u_2} [Field K] {w : NumberField.InfinitePlace K} (h : w.IsReal)
Goal: NumberField.ComplexEmbedding.conjugate w.embedding = w.embedding
-- NumberField.InfinitePlace.not_isReal_iff_isComplex
@[simp]
theorem not_isReal_iff_isComplex {w : InfinitePlace K} : ¬IsReal w ↔ IsComplex w 
Arguments: {K : Type u_2} [Field K] {w : NumberField.InfinitePlace K}
Goal: ¬w.IsReal ↔ w.IsComplex
-- NumberField.InfinitePlace.not_isComplex_iff_isReal
@[simp]
theorem not_isComplex_iff_isReal {w : InfinitePlace K} : ¬IsComplex w ↔ IsReal w 
Arguments: {K : Type u_2} [Field K] {w : NumberField.InfinitePlace K}
Goal: ¬w.IsComplex ↔ w.IsReal
-- NumberField.InfinitePlace.isReal_or_isComplex
theorem isReal_or_isComplex (w : InfinitePlace K) : IsReal w ∨ IsComplex w 
Arguments: {K : Type u_2} [Field K] (w : NumberField.InfinitePlace K)
Goal: w.IsReal ∨ w.IsComplex
-- NumberField.InfinitePlace.ne_of_isReal_isComplex
theorem ne_of_isReal_isComplex {w w' : InfinitePlace K} (h : IsReal w) (h' : IsComplex w') :
    w ≠ w' 
Arguments: {K : Type u_2} [Field K] {w w' : NumberField.InfinitePlace K} (h : w.IsReal) (h' : w'.IsComplex)
Goal: w ≠ w'
-- NumberField.InfinitePlace.disjoint_isReal_isComplex
theorem disjoint_isReal_isComplex :
    Disjoint {(w : InfinitePlace K) | IsReal w} {(w : InfinitePlace K) | IsComplex w} 
Arguments: (K : Type u_2) [Field K]
Goal: Disjoint {w | w.IsReal} {w | w.IsComplex}
-- NumberField.InfinitePlace.embedding_of_isReal_apply
@[simp]
theorem embedding_of_isReal_apply {w : InfinitePlace K} (hw : IsReal w) (x : K) :
    ((embedding_of_isReal hw) x : ℂ) = (embedding w) x 
Arguments: {K : Type u_2} [Field K] {w : NumberField.InfinitePlace K} (hw : w.IsReal) (x : K)
Goal: ↑((NumberField.InfinitePlace.embedding_of_isReal hw) x) = w.embedding x
-- NumberField.InfinitePlace.norm_embedding_of_isReal
theorem norm_embedding_of_isReal {w : InfinitePlace K} (hw : IsReal w) (x : K) :
    ‖embedding_of_isReal hw x‖ = w x 
Arguments: {K : Type u_2} [Field K] {w : NumberField.InfinitePlace K} (hw : w.IsReal) (x : K)
Goal: ‖(NumberField.InfinitePlace.embedding_of_isReal hw) x‖ = w x
-- NumberField.InfinitePlace.isReal_of_mk_isReal
@[simp]
theorem isReal_of_mk_isReal {φ : K →+* ℂ} (h : IsReal (mk φ)) :
    ComplexEmbedding.IsReal φ 
Arguments: {K : Type u_2} [Field K] {φ : K →+* ℂ} (h : (NumberField.InfinitePlace.mk φ).IsReal)
Goal: NumberField.ComplexEmbedding.IsReal φ
-- NumberField.InfinitePlace.isReal_mk_iff
lemma isReal_mk_iff {φ : K →+* ℂ} :
    IsReal (mk φ) ↔ ComplexEmbedding.IsReal φ 
Arguments: {K : Type u_2} [Field K] {φ : K →+* ℂ}
Goal: (NumberField.InfinitePlace.mk φ).IsReal ↔ NumberField.ComplexEmbedding.IsReal φ
-- NumberField.InfinitePlace.isComplex_mk_iff
lemma isComplex_mk_iff {φ : K →+* ℂ} :
    IsComplex (mk φ) ↔ ¬ ComplexEmbedding.IsReal φ 
Arguments: {K : Type u_2} [Field K] {φ : K →+* ℂ}
Goal: (NumberField.InfinitePlace.mk φ).IsComplex ↔ ¬NumberField.ComplexEmbedding.IsReal φ
-- NumberField.InfinitePlace.not_isReal_of_mk_isComplex
@[simp]
theorem not_isReal_of_mk_isComplex {φ : K →+* ℂ} (h : IsComplex (mk φ)) :
    ¬ ComplexEmbedding.IsReal φ 
Arguments: {K : Type u_2} [Field K] {φ : K →+* ℂ} (h : (NumberField.InfinitePlace.mk φ).IsComplex)
Goal: ¬NumberField.ComplexEmbedding.IsReal φ
-- NumberField.InfinitePlace.mult_pos
theorem mult_pos {w : InfinitePlace K} : 0 < mult w 
Arguments: {K : Type u_2} [Field K] {w : NumberField.InfinitePlace K}
Goal: 0 < w.mult
-- NumberField.InfinitePlace.mult_ne_zero
@[simp]
theorem mult_ne_zero {w : InfinitePlace K} : mult w ≠ 0 
Arguments: {K : Type u_2} [Field K] {w : NumberField.InfinitePlace K}
Goal: w.mult ≠ 0
-- NumberField.InfinitePlace.one_le_mult
theorem one_le_mult {w : InfinitePlace K} : (1 : ℝ) ≤ mult w 
Arguments: {K : Type u_2} [Field K] {w : NumberField.InfinitePlace K}
Goal: 1 ≤ ↑w.mult
-- NumberField.InfinitePlace.card_filter_mk_eq
theorem card_filter_mk_eq [NumberField K] (w : InfinitePlace K) :
    (Finset.univ.filter fun φ => mk φ = w).card = mult w 
Arguments: {K : Type u_2} [Field K] [NumberField K] (w : NumberField.InfinitePlace K)
Goal: (Finset.filter (fun φ => NumberField.InfinitePlace.mk φ = w) Finset.univ).card = w.mult
-- NumberField.InfinitePlace.sum_mult_eq
theorem sum_mult_eq [NumberField K] :
    ∑ w : InfinitePlace K, mult w = FiniteDimensional.finrank ℚ K 
Arguments: {K : Type u_2} [Field K] [NumberField K]
Goal: ∑ w : NumberField.InfinitePlace K, w.mult = FiniteDimensional.finrank ℚ K
-- NumberField.InfinitePlace.mkReal_coe
@[simp]
theorem mkReal_coe (φ : { φ : K →+* ℂ // ComplexEmbedding.IsReal φ }) :
    (mkReal φ : InfinitePlace K) = mk (φ : K →+* ℂ) 
Arguments: {K : Type u_2} [Field K] (φ : { φ // NumberField.ComplexEmbedding.IsReal φ })
Goal: ↑(NumberField.InfinitePlace.mkReal φ) = NumberField.InfinitePlace.mk ↑φ
-- NumberField.InfinitePlace.mkComplex_coe
@[simp]
theorem mkComplex_coe (φ : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ }) :
    (mkComplex φ : InfinitePlace K) = mk (φ : K →+* ℂ) 
Arguments: {K : Type u_2} [Field K] (φ : { φ // ¬NumberField.ComplexEmbedding.IsReal φ })
Goal: ↑(NumberField.InfinitePlace.mkComplex φ) = NumberField.InfinitePlace.mk ↑φ
-- NumberField.InfinitePlace.prod_eq_abs_norm
theorem prod_eq_abs_norm (x : K) :
    ∏ w : InfinitePlace K, w x ^ mult w = abs (Algebra.norm ℚ x) 
Arguments: {K : Type u_2} [Field K] [NumberField K] (x : K)
Goal: ∏ w : NumberField.InfinitePlace K, w x ^ w.mult = ↑|(Algebra.norm ℚ) x|
-- NumberField.InfinitePlace.one_le_of_lt_one
theorem one_le_of_lt_one {w : InfinitePlace K} {a : (𝓞 K)} (ha : a ≠ 0)
    (h : ∀ ⦃z⦄, z ≠ w → z a < 1) : 1 ≤ w a 
Arguments: {K : Type u_2} [Field K] [NumberField K] {w : NumberField.InfinitePlace K} {a : NumberField.RingOfIntegers K} (ha : a ≠ 0) (h : ∀ ⦃z : NumberField.InfinitePlace K⦄, z ≠ w → z ↑a < 1)
Goal: 1 ≤ w ↑a
-- NumberField.is_primitive_element_of_infinitePlace_lt
theorem _root_.NumberField.is_primitive_element_of_infinitePlace_lt {x : 𝓞 K}
    {w : InfinitePlace K} (h₁ : x ≠ 0) (h₂ : ∀ ⦃w'⦄, w' ≠ w → w' x < 1)
    (h₃ : IsReal w ∨ |(w.embedding x).re| < 1) : ℚ⟮(x : K)⟯ = ⊤ 
Arguments: {K : Type u_2} [Field K] [NumberField K] {x : NumberField.RingOfIntegers K} {w : NumberField.InfinitePlace K} (h₁ : x ≠ 0) (h₂ : ∀ ⦃w' : NumberField.InfinitePlace K⦄, w' ≠ w → w' ↑x < 1) (h₃ : w.IsReal ∨ |(w.embedding ↑x).re| < 1)
Goal: ℚ⟮↑x⟯ = ⊤