ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- NumberField.adjoin_eq_top_of_infinitePlace_lt
theorem _root_.NumberField.adjoin_eq_top_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) :
    Algebra.adjoin 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: Algebra.adjoin ℚ {↑x} = ⊤
-- NumberField.InfinitePlace.card_real_embeddings
theorem card_real_embeddings :
    card { φ : K →+* ℂ // ComplexEmbedding.IsReal φ } = NrRealPlaces K 
Arguments: (K : Type u_2) [Field K] [NumberField K]
Goal: Fintype.card { φ // NumberField.ComplexEmbedding.IsReal φ } = NumberField.InfinitePlace.NrRealPlaces K
-- NumberField.InfinitePlace.card_eq_nrRealPlaces_add_nrComplexPlaces
theorem card_eq_nrRealPlaces_add_nrComplexPlaces :
    Fintype.card (InfinitePlace K) = NrRealPlaces K + NrComplexPlaces K 
Arguments: (K : Type u_2) [Field K] [NumberField K]
Goal: Fintype.card (NumberField.InfinitePlace K) =
  NumberField.InfinitePlace.NrRealPlaces K + NumberField.InfinitePlace.NrComplexPlaces K
-- NumberField.InfinitePlace.card_complex_embeddings
theorem card_complex_embeddings :
    card { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } = 2 * NrComplexPlaces K 
Arguments: (K : Type u_2) [Field K] [NumberField K]
Goal: Fintype.card { φ // ¬NumberField.ComplexEmbedding.IsReal φ } = 2 * NumberField.InfinitePlace.NrComplexPlaces K
-- NumberField.InfinitePlace.card_add_two_mul_card_eq_rank
theorem card_add_two_mul_card_eq_rank :
    NrRealPlaces K + 2 * NrComplexPlaces K = finrank ℚ K 
Arguments: (K : Type u_2) [Field K] [NumberField K]
Goal: NumberField.InfinitePlace.NrRealPlaces K + 2 * NumberField.InfinitePlace.NrComplexPlaces K =
  FiniteDimensional.finrank ℚ K
-- NumberField.InfinitePlace.nrComplexPlaces_eq_zero_of_finrank_eq_one
theorem nrComplexPlaces_eq_zero_of_finrank_eq_one (h : finrank ℚ K = 1) :
    NrComplexPlaces K = 0 
Arguments: {K : Type u_2} [Field K] [NumberField K] (h : FiniteDimensional.finrank ℚ K = 1)
Goal: NumberField.InfinitePlace.NrComplexPlaces K = 0
-- NumberField.InfinitePlace.nrRealPlaces_eq_one_of_finrank_eq_one
theorem nrRealPlaces_eq_one_of_finrank_eq_one (h : finrank ℚ K = 1) :
    NrRealPlaces K = 1 
Arguments: {K : Type u_2} [Field K] [NumberField K] (h : FiniteDimensional.finrank ℚ K = 1)
Goal: NumberField.InfinitePlace.NrRealPlaces K = 1
-- NumberField.InfinitePlace.comap_mk
@[simp]
lemma comap_mk (φ : K →+* ℂ) (f : k →+* K) : (mk φ).comap f = mk (φ.comp f) 
Arguments: {k : Type u_1} [Field k] {K : Type u_2} [Field K] (φ : K →+* ℂ) (f : k →+* K)
Goal: (NumberField.InfinitePlace.mk φ).comap f = NumberField.InfinitePlace.mk (φ.comp f)
-- NumberField.InfinitePlace.comap_id
lemma comap_id (w : InfinitePlace K) : w.comap (RingHom.id K) = w 
Arguments: {K : Type u_2} [Field K] (w : NumberField.InfinitePlace K)
Goal: w.comap (RingHom.id K) = w
-- NumberField.InfinitePlace.comap_comp
lemma comap_comp (w : InfinitePlace K) (f : F →+* K) (g : k →+* F) :
    w.comap (f.comp g) = (w.comap f).comap g 
Arguments: {k : Type u_1} [Field k] {K : Type u_2} [Field K] {F : Type u_3} [Field F] (w : NumberField.InfinitePlace K) (f : F →+* K) (g : k →+* F)
Goal: w.comap (f.comp g) = (w.comap f).comap g
-- NumberField.InfinitePlace.IsReal.comap
lemma IsReal.comap (f : k →+* K) {w : InfinitePlace K} (hφ : IsReal w) :
    IsReal (w.comap f) 
Arguments: {k : Type u_1} [Field k] {K : Type u_2} [Field K] (f : k →+* K) {w : NumberField.InfinitePlace K} (hφ : w.IsReal)
Goal: (w.comap f).IsReal
-- NumberField.InfinitePlace.isReal_comap_iff
lemma isReal_comap_iff (f : k ≃+* K) {w : InfinitePlace K} :
    IsReal (w.comap (f : k →+* K)) ↔ IsReal w 
Arguments: {k : Type u_1} [Field k] {K : Type u_2} [Field K] (f : k ≃+* K) {w : NumberField.InfinitePlace K}
Goal: (w.comap ↑f).IsReal ↔ w.IsReal
-- NumberField.InfinitePlace.comap_surjective
lemma comap_surjective [Algebra k K] [Algebra.IsAlgebraic k K] :
    Function.Surjective (comap · (algebraMap k K)) 
Arguments: {k : Type u_1} [Field k] {K : Type u_2} [Field K] [Algebra k K] [Algebra.IsAlgebraic k K]
Goal: Function.Surjective fun x => x.comap (algebraMap k K)
-- NumberField.InfinitePlace.mult_comap_le
lemma mult_comap_le (f : k →+* K) (w : InfinitePlace K) : mult (w.comap f) ≤ mult w 
Arguments: {k : Type u_1} [Field k] {K : Type u_2} [Field K] (f : k →+* K) (w : NumberField.InfinitePlace K)
Goal: (w.comap f).mult ≤ w.mult
-- NumberField.InfinitePlace.card_mono
lemma card_mono [NumberField k] [NumberField K] :
    card (InfinitePlace k) ≤ card (InfinitePlace K) 
Arguments: (k : Type u_1) [Field k] (K : Type u_2) [Field K] [Algebra k K] [NumberField k] [NumberField K]
Goal: Fintype.card (NumberField.InfinitePlace k) ≤ Fintype.card (NumberField.InfinitePlace K)
-- NumberField.InfinitePlace.instMulActionAlgEquiv_smul_coe_apply
@[simps! smul_coe_apply]
instance : MulAction (K ≃ₐ[k] K) (InfinitePlace K) 
Arguments: {k : Type u_1} [Field k] {K : Type u_2} [Field K] [Algebra k K] (σ : K ≃ₐ[k] K) (w : NumberField.InfinitePlace K)
Goal: ∀ (a : K), ↑(SMul.smul σ w) a = ↑w (σ.symm a)
-- NumberField.InfinitePlace.smul_eq_comap
lemma smul_eq_comap : σ • w = w.comap σ.symm 
Arguments: {k : Type u_1} [Field k] {K : Type u_2} [Field K] [Algebra k K] (σ : K ≃ₐ[k] K) (w : NumberField.InfinitePlace K)
Goal: σ • w = w.comap ↑σ.symm
-- NumberField.InfinitePlace.smul_apply
@[simp] lemma smul_apply (x) : (σ • w) x = w (σ.symm x) 
Arguments: {k : Type u_1} [Field k] {K : Type u_2} [Field K] [Algebra k K] (σ : K ≃ₐ[k] K) (w : NumberField.InfinitePlace K) (x : K)
Goal: (σ • w) x = w (σ.symm x)
-- NumberField.InfinitePlace.smul_mk
@[simp] lemma smul_mk (φ : K →+* ℂ) : σ • mk φ = mk (φ.comp σ.symm) 
Arguments: {k : Type u_1} [Field k] {K : Type u_2} [Field K] [Algebra k K] (σ : K ≃ₐ[k] K) (φ : K →+* ℂ)
Goal: σ • NumberField.InfinitePlace.mk φ = NumberField.InfinitePlace.mk (φ.comp ↑σ.symm)
-- NumberField.InfinitePlace.comap_smul
lemma comap_smul {f : F →+* K} : (σ • w).comap f = w.comap (RingHom.comp σ.symm f) 
Arguments: {k : Type u_1} [Field k] {K : Type u_2} [Field K] {F : Type u_3} [Field F] [Algebra k K] (σ : K ≃ₐ[k] K) (w : NumberField.InfinitePlace K) {f : F →+* K}
Goal: (σ • w).comap f = w.comap ((↑σ.symm).comp f)
-- NumberField.InfinitePlace.isReal_smul_iff
lemma isReal_smul_iff : IsReal (σ • w) ↔ IsReal w 
Arguments: {k : Type u_1} [Field k] {K : Type u_2} [Field K] [Algebra k K] {σ : K ≃ₐ[k] K} {w : NumberField.InfinitePlace K}
Goal: (σ • w).IsReal ↔ w.IsReal
-- NumberField.InfinitePlace.isComplex_smul_iff
lemma isComplex_smul_iff : IsComplex (σ • w) ↔ IsComplex w 
Arguments: {k : Type u_1} [Field k] {K : Type u_2} [Field K] [Algebra k K] {σ : K ≃ₐ[k] K} {w : NumberField.InfinitePlace K}
Goal: (σ • w).IsComplex ↔ w.IsComplex
-- NumberField.InfinitePlace.ComplexEmbedding.exists_comp_symm_eq_of_comp_eq
lemma ComplexEmbedding.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.InfinitePlace.exists_smul_eq_of_comap_eq
lemma exists_smul_eq_of_comap_eq [Algebra k K] [IsGalois k K] {w w' : InfinitePlace K}
    (h : w.comap (algebraMap k K) = w'.comap (algebraMap k K)) : ∃ σ : K ≃ₐ[k] K, σ • w = w' 
Arguments: {k : Type u_1} [Field k] {K : Type u_2} [Field K] [Algebra k K] [IsGalois k K] {w w' : NumberField.InfinitePlace K} (h : w.comap (algebraMap k K) = w'.comap (algebraMap k K))
Goal: ∃ σ, σ • w = w'
-- NumberField.InfinitePlace.mem_orbit_iff
lemma mem_orbit_iff [IsGalois k K] {w w' : InfinitePlace K} :
    w' ∈ MulAction.orbit (K ≃ₐ[k] K) w ↔ w.comap (algebraMap k K) = w'.comap (algebraMap k K) 
Arguments: {k : Type u_1} [Field k] {K : Type u_2} [Field K] [Algebra k K] [IsGalois k K] {w w' : NumberField.InfinitePlace K}
Goal: w' ∈ MulAction.orbit (K ≃ₐ[k] K) w ↔ w.comap (algebraMap k K) = w'.comap (algebraMap k K)
-- NumberField.InfinitePlace.orbitRelEquiv_apply_mk''
lemma orbitRelEquiv_apply_mk'' [IsGalois k K] (w : InfinitePlace K) :
    orbitRelEquiv (Quotient.mk'' w) = comap w (algebraMap k K) 
Arguments: {k : Type u_1} [Field k] {K : Type u_2} [Field K] [Algebra k K] [IsGalois k K] (w : NumberField.InfinitePlace K)
Goal: NumberField.InfinitePlace.orbitRelEquiv (Quotient.mk'' w) = w.comap (algebraMap k K)
-- NumberField.InfinitePlace.isUnramified_self
lemma isUnramified_self : IsUnramified K w 
Arguments: {K : Type u_2} [Field K] (w : NumberField.InfinitePlace K)
Goal: NumberField.InfinitePlace.IsUnramified K w
-- NumberField.InfinitePlace.IsUnramified.eq
lemma IsUnramified.eq (h : IsUnramified k w) : mult (w.comap (algebraMap k K)) = mult w 
Arguments: {k : Type u_1} [Field k] {K : Type u_2} [Field K] [Algebra k K] {w : NumberField.InfinitePlace K} (h : NumberField.InfinitePlace.IsUnramified k w)
Goal: (w.comap (algebraMap k K)).mult = w.mult
-- NumberField.InfinitePlace.isUnramified_iff_mult_le
lemma isUnramified_iff_mult_le :
    IsUnramified k w ↔ mult w ≤ mult (w.comap (algebraMap k K)) 
Arguments: {k : Type u_1} [Field k] {K : Type u_2} [Field K] [Algebra k K] {w : NumberField.InfinitePlace K}
Goal: NumberField.InfinitePlace.IsUnramified k w ↔ w.mult ≤ (w.comap (algebraMap k K)).mult
-- NumberField.InfinitePlace.IsUnramified.comap_algHom
lemma IsUnramified.comap_algHom {w : InfinitePlace F} (h : IsUnramified k w) (f : K →ₐ[k] F) :
    IsUnramified k (w.comap (f : K →+* F)) 
Arguments: {k : Type u_1} [Field k] {K : Type u_2} [Field K] {F : Type u_3} [Field F] [Algebra k K] [Algebra k F] {w : NumberField.InfinitePlace F} (h : NumberField.InfinitePlace.IsUnramified k w) (f : K →ₐ[k] F)
Goal: NumberField.InfinitePlace.IsUnramified k (w.comap ↑f)
-- NumberField.InfinitePlace.IsUnramified.of_restrictScalars
lemma IsUnramified.of_restrictScalars {w : InfinitePlace F} (h : IsUnramified k w) :
    IsUnramified K w 
Arguments: {k : Type u_1} [Field k] (K : Type u_2) [Field K] {F : Type u_3} [Field F] [Algebra k K] [Algebra k F] [Algebra K F] [IsScalarTower k K F] {w : NumberField.InfinitePlace F} (h : NumberField.InfinitePlace.IsUnramified k w)
Goal: NumberField.InfinitePlace.IsUnramified K w
-- NumberField.InfinitePlace.IsUnramified.comap
lemma IsUnramified.comap {w : InfinitePlace F} (h : IsUnramified k w) :
    IsUnramified k (w.comap (algebraMap K F)) 
Arguments: {k : Type u_1} [Field k] (K : Type u_2) [Field K] {F : Type u_3} [Field F] [Algebra k K] [Algebra k F] [Algebra K F] [IsScalarTower k K F] {w : NumberField.InfinitePlace F} (h : NumberField.InfinitePlace.IsUnramified k w)
Goal: NumberField.InfinitePlace.IsUnramified k (w.comap (algebraMap K F))
-- NumberField.InfinitePlace.not_isUnramified_iff
lemma not_isUnramified_iff :
    ¬ IsUnramified k w ↔ IsComplex w ∧ IsReal (w.comap (algebraMap k K)) 
Arguments: {k : Type u_1} [Field k] {K : Type u_2} [Field K] [Algebra k K] {w : NumberField.InfinitePlace K}
Goal: ¬NumberField.InfinitePlace.IsUnramified k w ↔ w.IsComplex ∧ (w.comap (algebraMap k K)).IsReal
-- NumberField.InfinitePlace.isUnramified_iff
lemma isUnramified_iff :
    IsUnramified k w ↔ IsReal w ∨ IsComplex (w.comap (algebraMap k K)) 
Arguments: {k : Type u_1} [Field k] {K : Type u_2} [Field K] [Algebra k K] {w : NumberField.InfinitePlace K}
Goal: NumberField.InfinitePlace.IsUnramified k w ↔ w.IsReal ∨ (w.comap (algebraMap k K)).IsComplex
-- NumberField.InfinitePlace.IsReal.isUnramified
lemma IsReal.isUnramified (h : IsReal w) : IsUnramified k w 
Arguments: (k : Type u_1) [Field k] {K : Type u_2} [Field K] [Algebra k K] {w : NumberField.InfinitePlace K} (h : w.IsReal)
Goal: NumberField.InfinitePlace.IsUnramified k w
-- NumberField.ComplexEmbedding.IsConj.isUnramified_mk_iff
lemma _root_.NumberField.ComplexEmbedding.IsConj.isUnramified_mk_iff
    {φ : K →+* ℂ} (h : ComplexEmbedding.IsConj φ σ) :
    IsUnramified k (mk φ) ↔ σ = 1 
Arguments: {k : Type u_1} [Field k] {K : Type u_2} [Field K] [Algebra k K] {σ : K ≃ₐ[k] K} {φ : K →+* ℂ} (h : NumberField.ComplexEmbedding.IsConj φ σ)
Goal: NumberField.InfinitePlace.IsUnramified k (NumberField.InfinitePlace.mk φ) ↔ σ = 1
-- NumberField.InfinitePlace.isUnramified_mk_iff_forall_isConj
lemma isUnramified_mk_iff_forall_isConj [IsGalois k K] {φ : K →+* ℂ} :
    IsUnramified k (mk φ) ↔ ∀ σ : K ≃ₐ[k] K, ComplexEmbedding.IsConj φ σ → σ = 1 
Arguments: {k : Type u_1} [Field k] {K : Type u_2} [Field K] [Algebra k K] [IsGalois k K] {φ : K →+* ℂ}
Goal: NumberField.InfinitePlace.IsUnramified k (NumberField.InfinitePlace.mk φ) ↔
  ∀ (σ : K ≃ₐ[k] K), NumberField.ComplexEmbedding.IsConj φ σ → σ = 1
-- NumberField.InfinitePlace.mem_stabilizer_mk_iff
lemma mem_stabilizer_mk_iff (φ : K →+* ℂ) (σ : K ≃ₐ[k] K) :
    σ ∈ Stab (mk φ) ↔ σ = 1 ∨ ComplexEmbedding.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: σ ∈ MulAction.stabilizer (K ≃ₐ[k] K) (NumberField.InfinitePlace.mk φ) ↔ σ = 1 ∨ NumberField.ComplexEmbedding.IsConj φ σ
-- NumberField.InfinitePlace.IsUnramified.stabilizer_eq_bot
lemma IsUnramified.stabilizer_eq_bot (h : IsUnramified k w) : Stab w = ⊥ 
Arguments: {k : Type u_1} [Field k] {K : Type u_2} [Field K] [Algebra k K] {w : NumberField.InfinitePlace K} (h : NumberField.InfinitePlace.IsUnramified k w)
Goal: MulAction.stabilizer (K ≃ₐ[k] K) w = ⊥
-- NumberField.ComplexEmbedding.IsConj.coe_stabilzer_mk
lemma _root_.NumberField.ComplexEmbedding.IsConj.coe_stabilzer_mk
    {φ : K →+* ℂ} (h : ComplexEmbedding.IsConj φ σ) :
    (Stab (mk φ) : Set (K ≃ₐ[k] K)) = {1, σ} 
Arguments: {k : Type u_1} [Field k] {K : Type u_2} [Field K] [Algebra k K] {σ : K ≃ₐ[k] K} {φ : K →+* ℂ} (h : NumberField.ComplexEmbedding.IsConj φ σ)
Goal: ↑(MulAction.stabilizer (K ≃ₐ[k] K) (NumberField.InfinitePlace.mk φ)) = {1, σ}
-- NumberField.InfinitePlace.nat_card_stabilizer_eq_one_or_two
lemma nat_card_stabilizer_eq_one_or_two :
    Nat.card (Stab w) = 1 ∨ Nat.card (Stab w) = 2 
Arguments: (k : Type u_1) [Field k] {K : Type u_2} [Field K] [Algebra k K] (w : NumberField.InfinitePlace K)
Goal: Nat.card ↥(MulAction.stabilizer (K ≃ₐ[k] K) w) = 1 ∨ Nat.card ↥(MulAction.stabilizer (K ≃ₐ[k] K) w) = 2
-- NumberField.InfinitePlace.isUnramified_iff_stabilizer_eq_bot
lemma isUnramified_iff_stabilizer_eq_bot [IsGalois k K] : IsUnramified k w ↔ Stab w = ⊥ 
Arguments: {k : Type u_1} [Field k] {K : Type u_2} [Field K] [Algebra k K] {w : NumberField.InfinitePlace K} [IsGalois k K]
Goal: NumberField.InfinitePlace.IsUnramified k w ↔ MulAction.stabilizer (K ≃ₐ[k] K) w = ⊥
-- NumberField.InfinitePlace.isUnramified_iff_card_stabilizer_eq_one
lemma isUnramified_iff_card_stabilizer_eq_one [IsGalois k K] :
    IsUnramified k w ↔ Nat.card (Stab w) = 1 
Arguments: {k : Type u_1} [Field k] {K : Type u_2} [Field K] [Algebra k K] {w : NumberField.InfinitePlace K} [IsGalois k K]
Goal: NumberField.InfinitePlace.IsUnramified k w ↔ Nat.card ↥(MulAction.stabilizer (K ≃ₐ[k] K) w) = 1
-- NumberField.InfinitePlace.not_isUnramified_iff_card_stabilizer_eq_two
lemma not_isUnramified_iff_card_stabilizer_eq_two [IsGalois k K] :
    ¬ IsUnramified k w ↔ Nat.card (Stab w) = 2 
Arguments: {k : Type u_1} [Field k] {K : Type u_2} [Field K] [Algebra k K] {w : NumberField.InfinitePlace K} [IsGalois k K]
Goal: ¬NumberField.InfinitePlace.IsUnramified k w ↔ Nat.card ↥(MulAction.stabilizer (K ≃ₐ[k] K) w) = 2
-- NumberField.InfinitePlace.card_stabilizer
lemma card_stabilizer [IsGalois k K] :
    Nat.card (Stab w) = if IsUnramified k w then 1 else 2 
Arguments: {k : Type u_1} [Field k] {K : Type u_2} [Field K] [Algebra k K] {w : NumberField.InfinitePlace K} [IsGalois k K]
Goal: Nat.card ↥(MulAction.stabilizer (K ≃ₐ[k] K) w) = if NumberField.InfinitePlace.IsUnramified k w then 1 else 2
-- NumberField.InfinitePlace.even_nat_card_aut_of_not_isUnramified
lemma even_nat_card_aut_of_not_isUnramified [IsGalois k K] (hw : ¬ IsUnramified k w) :
    Even (Nat.card <| K ≃ₐ[k] K) 
Arguments: {k : Type u_1} [Field k] {K : Type u_2} [Field K] [Algebra k K] {w : NumberField.InfinitePlace K} [IsGalois k K] (hw : ¬NumberField.InfinitePlace.IsUnramified k w)
Goal: Even (Nat.card (K ≃ₐ[k] K))
-- NumberField.InfinitePlace.even_card_aut_of_not_isUnramified
lemma even_card_aut_of_not_isUnramified [IsGalois k K] [FiniteDimensional k K]
    (hw : ¬ IsUnramified k w) :
    Even (Fintype.card <| K ≃ₐ[k] K) 
Arguments: {k : Type u_1} [Field k] {K : Type u_2} [Field K] [Algebra k K] {w : NumberField.InfinitePlace K} [IsGalois k K] [FiniteDimensional k K] (hw : ¬NumberField.InfinitePlace.IsUnramified k w)
Goal: Even (Fintype.card (K ≃ₐ[k] K))
-- NumberField.InfinitePlace.even_finrank_of_not_isUnramified
lemma even_finrank_of_not_isUnramified [IsGalois k K]
    (hw : ¬ IsUnramified k w) : Even (finrank k K) 
Arguments: {k : Type u_1} [Field k] {K : Type u_2} [Field K] [Algebra k K] {w : NumberField.InfinitePlace K} [IsGalois k K] (hw : ¬NumberField.InfinitePlace.IsUnramified k w)
Goal: Even (FiniteDimensional.finrank k K)
-- NumberField.InfinitePlace.isUnramified_smul_iff
lemma isUnramified_smul_iff :
    IsUnramified k (σ • w) ↔ IsUnramified k w 
Arguments: {k : Type u_1} [Field k] {K : Type u_2} [Field K] [Algebra k K] {σ : K ≃ₐ[k] K} {w : NumberField.InfinitePlace K}
Goal: NumberField.InfinitePlace.IsUnramified k (σ • w) ↔ NumberField.InfinitePlace.IsUnramified k w
-- NumberField.InfinitePlace.isUnramifiedIn_comap
lemma isUnramifiedIn_comap [IsGalois k K] {w : InfinitePlace K} :
    (w.comap (algebraMap k K)).IsUnramifiedIn K ↔ w.IsUnramified k 
Arguments: {k : Type u_1} [Field k] {K : Type u_2} [Field K] [Algebra k K] [IsGalois k K] {w : NumberField.InfinitePlace K}
Goal: NumberField.InfinitePlace.IsUnramifiedIn K (w.comap (algebraMap k K)) ↔ NumberField.InfinitePlace.IsUnramified k w
-- NumberField.InfinitePlace.even_card_aut_of_not_isUnramifiedIn
lemma even_card_aut_of_not_isUnramifiedIn [IsGalois k K] [FiniteDimensional k K]
    {w : InfinitePlace k} (hw : ¬ w.IsUnramifiedIn K) :
    Even (Fintype.card <| K ≃ₐ[k] K) 
Arguments: {k : Type u_1} [Field k] {K : Type u_2} [Field K] [Algebra k K] [IsGalois k K] [FiniteDimensional k K] {w : NumberField.InfinitePlace k} (hw : ¬NumberField.InfinitePlace.IsUnramifiedIn K w)
Goal: Even (Fintype.card (K ≃ₐ[k] K))
-- NumberField.InfinitePlace.even_finrank_of_not_isUnramifiedIn
lemma even_finrank_of_not_isUnramifiedIn
    [IsGalois k K] {w : InfinitePlace k} (hw : ¬ w.IsUnramifiedIn K) :
    Even (finrank k K) 
Arguments: {k : Type u_1} [Field k] {K : Type u_2} [Field K] [Algebra k K] [IsGalois k K] {w : NumberField.InfinitePlace k} (hw : ¬NumberField.InfinitePlace.IsUnramifiedIn K w)
Goal: Even (FiniteDimensional.finrank k K)
-- NumberField.InfinitePlace.card_isUnramified
lemma card_isUnramified [NumberField k] [IsGalois k K] :
    Finset.card (univ.filter <| IsUnramified k (K := K)) =
      Finset.card (univ.filter <| IsUnramifiedIn K (k := k)) * (finrank k K) 
Arguments: (k : Type u_1) [Field k] (K : Type u_2) [Field K] [NumberField K] [Algebra k K] [NumberField k] [IsGalois k K]
Goal: (Finset.filter (NumberField.InfinitePlace.IsUnramified k) Finset.univ).card =
  (Finset.filter (NumberField.InfinitePlace.IsUnramifiedIn K) Finset.univ).card * FiniteDimensional.finrank k K
-- NumberField.InfinitePlace.card_isUnramified_compl
lemma card_isUnramified_compl [NumberField k] [IsGalois k K] :
    Finset.card (univ.filter <| IsUnramified k (K := K))ᶜ =
      Finset.card (univ.filter <| IsUnramifiedIn K (k := k))ᶜ * (finrank k K / 2) 
Arguments: (k : Type u_1) [Field k] (K : Type u_2) [Field K] [NumberField K] [Algebra k K] [NumberField k] [IsGalois k K]
Goal: (Finset.filter (NumberField.InfinitePlace.IsUnramified k) Finset.univ)ᶜ.card =
  (Finset.filter (NumberField.InfinitePlace.IsUnramifiedIn K) Finset.univ)ᶜ.card * (FiniteDimensional.finrank k K / 2)
-- NumberField.InfinitePlace.card_eq_card_isUnramifiedIn
lemma card_eq_card_isUnramifiedIn [NumberField k] [IsGalois k K] :
    Fintype.card (InfinitePlace K) =
      Finset.card (Finset.univ.filter <| IsUnramifiedIn K (k := k)) * finrank k K +
      Finset.card (Finset.univ.filter <| IsUnramifiedIn K (k := k))ᶜ * (finrank k K / 2) 
Arguments: (k : Type u_1) [Field k] (K : Type u_2) [Field K] [NumberField K] [Algebra k K] [NumberField k] [IsGalois k K]
Goal: Fintype.card (NumberField.InfinitePlace K) =
  (Finset.filter (NumberField.InfinitePlace.IsUnramifiedIn K) Finset.univ).card * FiniteDimensional.finrank k K +
    (Finset.filter (NumberField.InfinitePlace.IsUnramifiedIn K) Finset.univ)ᶜ.card * (FiniteDimensional.finrank k K / 2)
-- IsUnramifiedAtInfinitePlaces.isUnramified
class IsUnramifiedAtInfinitePlaces : Prop where
  isUnramified : ∀ w : InfinitePlace K, w.IsUnramified k
Arguments: {k : Type u_1} [Field k] {K : Type u_2} [Field K] [Algebra k K] [self : IsUnramifiedAtInfinitePlaces k K] (w : NumberField.InfinitePlace K)
Goal: NumberField.InfinitePlace.IsUnramified k w
-- IsUnramifiedAtInfinitePlaces.trans
lemma IsUnramifiedAtInfinitePlaces.trans
    [h₁ : IsUnramifiedAtInfinitePlaces k K] [h₂ : IsUnramifiedAtInfinitePlaces K F] :
    IsUnramifiedAtInfinitePlaces k F 
Arguments: (k : Type u_1) [Field k] (K : Type u_2) [Field K] (F : Type u_3) [Field F] [Algebra k K] [Algebra k F] [Algebra K F] [IsScalarTower k K F] [h₁ : IsUnramifiedAtInfinitePlaces k K] [h₂ : IsUnramifiedAtInfinitePlaces K F]
Goal: IsUnramifiedAtInfinitePlaces k F
-- IsUnramifiedAtInfinitePlaces.top
lemma IsUnramifiedAtInfinitePlaces.top [h : IsUnramifiedAtInfinitePlaces k F] :
    IsUnramifiedAtInfinitePlaces K F 
Arguments: (k : Type u_1) [Field k] (K : Type u_2) [Field K] (F : Type u_3) [Field F] [Algebra k K] [Algebra k F] [Algebra K F] [IsScalarTower k K F] [h : IsUnramifiedAtInfinitePlaces k F]
Goal: IsUnramifiedAtInfinitePlaces K F
-- IsUnramifiedAtInfinitePlaces.bot
lemma IsUnramifiedAtInfinitePlaces.bot [h₁ : IsUnramifiedAtInfinitePlaces k F]
    [Algebra.IsAlgebraic K F] :
    IsUnramifiedAtInfinitePlaces k K 
Arguments: (k : Type u_1) [Field k] (K : Type u_2) [Field K] (F : Type u_3) [Field F] [Algebra k K] [Algebra k F] [Algebra K F] [IsScalarTower k K F] [h₁ : IsUnramifiedAtInfinitePlaces k F] [Algebra.IsAlgebraic K F]
Goal: IsUnramifiedAtInfinitePlaces k K
-- NumberField.InfinitePlace.isUnramified
lemma NumberField.InfinitePlace.isUnramified [IsUnramifiedAtInfinitePlaces k K]
    (w : InfinitePlace K) : IsUnramified k w 
Arguments: (k : Type u_1) [Field k] {K : Type u_2} [Field K] [Algebra k K] [IsUnramifiedAtInfinitePlaces k K] (w : NumberField.InfinitePlace K)
Goal: NumberField.InfinitePlace.IsUnramified k w
-- NumberField.InfinitePlace.isUnramifiedIn
lemma NumberField.InfinitePlace.isUnramifiedIn [IsUnramifiedAtInfinitePlaces k K]
    (w : InfinitePlace k) : IsUnramifiedIn K w 
Arguments: {k : Type u_1} [Field k] (K : Type u_2) [Field K] [Algebra k K] [IsUnramifiedAtInfinitePlaces k K] (w : NumberField.InfinitePlace k)
Goal: NumberField.InfinitePlace.IsUnramifiedIn K w
-- IsUnramifiedAtInfinitePlaces_of_odd_card_aut
lemma IsUnramifiedAtInfinitePlaces_of_odd_card_aut [IsGalois k K] [FiniteDimensional k K]
    (h : Odd (Fintype.card <| K ≃ₐ[k] K)) : IsUnramifiedAtInfinitePlaces k K 
Arguments: {k : Type u_1} [Field k] {K : Type u_2} [Field K] [Algebra k K] [IsGalois k K] [FiniteDimensional k K] (h : Odd (Fintype.card (K ≃ₐ[k] K)))
Goal: IsUnramifiedAtInfinitePlaces k K
-- IsUnramifiedAtInfinitePlaces_of_odd_finrank
lemma IsUnramifiedAtInfinitePlaces_of_odd_finrank [IsGalois k K]
    (h : Odd (FiniteDimensional.finrank k K)) : IsUnramifiedAtInfinitePlaces k K 
Arguments: {k : Type u_1} [Field k] {K : Type u_2} [Field K] [Algebra k K] [IsGalois k K] (h : Odd (FiniteDimensional.finrank k K))
Goal: IsUnramifiedAtInfinitePlaces k K