ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- IsUnramifiedAtInfinitePlaces.card_infinitePlace
lemma IsUnramifiedAtInfinitePlaces.card_infinitePlace [NumberField k] [NumberField K]
    [IsGalois k K] [IsUnramifiedAtInfinitePlaces k K] :
    Fintype.card (InfinitePlace K) = Fintype.card (InfinitePlace k) * finrank k K 
Arguments: (k : Type u_1) [Field k] (K : Type u_2) [Field K] [Algebra k K] [NumberField k] [NumberField K] [IsGalois k K] [IsUnramifiedAtInfinitePlaces k K]
Goal: Fintype.card (NumberField.InfinitePlace K) = Fintype.card (NumberField.InfinitePlace k) * FiniteDimensional.finrank k K
-- IsPrimitiveRoot.nrRealPlaces_eq_zero_of_two_lt
theorem nrRealPlaces_eq_zero_of_two_lt (hk : 2 < k) (hζ : IsPrimitiveRoot ζ k) :
    NumberField.InfinitePlace.NrRealPlaces K = 0 
Arguments: {K : Type u_1} [Field K] [NumberField K] {ζ : K} {k : ℕ} (hk : 2 < k) (hζ : IsPrimitiveRoot ζ k)
Goal: NumberField.InfinitePlace.NrRealPlaces K = 0
-- IsCyclotomicExtension.Rat.nrRealPlaces_eq_zero
theorem nrRealPlaces_eq_zero [IsCyclotomicExtension {n} ℚ K]
    (hn : 2 < n) :
    haveI := IsCyclotomicExtension.numberField {n} ℚ K
    NrRealPlaces K = 0 
Arguments: {n : ℕ+} (K : Type u) [Field K] [CharZero K] [IsCyclotomicExtension {n} ℚ K] (hn : 2 < n)
Goal: NumberField.InfinitePlace.NrRealPlaces K = 0
-- IsCyclotomicExtension.Rat.nrComplexPlaces_eq_totient_div_two
theorem nrComplexPlaces_eq_totient_div_two [h : IsCyclotomicExtension {n} ℚ K] :
    haveI := IsCyclotomicExtension.numberField {n} ℚ K
    NrComplexPlaces K = φ n / 2 
Arguments: (n : ℕ+) (K : Type u) [Field K] [CharZero K] [h : IsCyclotomicExtension {n} ℚ K]
Goal: NumberField.InfinitePlace.NrComplexPlaces K = (↑n).totient / 2
-- IsPrimitiveRoot.autToPow_injective
theorem autToPow_injective : Function.Injective <| hμ.autToPow K 
Arguments: {n : ℕ+} (K : Type u_1) [Field K] {L : Type u_2} {μ : L} [CommRing L] [IsDomain L] (hμ : IsPrimitiveRoot μ ↑n) [Algebra K L] [IsCyclotomicExtension {n} K L]
Goal: Function.Injective ⇑(IsPrimitiveRoot.autToPow K hμ)
-- IsCyclotomicExtension.autEquivPow_apply
@[simps]
noncomputable def autEquivPow : (L ≃ₐ[K] L) ≃* (ZMod n)ˣ 
Arguments: {n : ℕ+} {K : Type u_1} [Field K] (L : Type u_2) [CommRing L] [IsDomain L] [Algebra K L] [IsCyclotomicExtension {n} K L] (h : Irreducible (Polynomial.cyclotomic (↑n) K))
Goal: ∀ (a : L ≃ₐ[K] L), (IsCyclotomicExtension.autEquivPow L h) a = (↑(IsPrimitiveRoot.autToPow K ⋯)).toFun a
-- IsCyclotomicExtension.autEquivPow_symm_apply
@[simps]
noncomputable def autEquivPow : (L ≃ₐ[K] L) ≃* (ZMod n)ˣ 
Arguments: {n : ℕ+} {K : Type u_1} [Field K] (L : Type u_2) [CommRing L] [IsDomain L] [Algebra K L] [IsCyclotomicExtension {n} K L] (h : Irreducible (Polynomial.cyclotomic (↑n) K)) (t : (ZMod ↑n)ˣ)
Goal: (IsCyclotomicExtension.autEquivPow L h).symm t =
  (IsPrimitiveRoot.powerBasis K ⋯).equivOfMinpoly (IsPrimitiveRoot.powerBasis K ⋯) ⋯
-- IsCyclotomicExtension.fromZetaAut_spec
theorem fromZetaAut_spec : fromZetaAut hμ h (zeta n K L) = μ 
Arguments: {n : ℕ+} {K : Type u_1} [Field K] {L : Type u_2} {μ : L} [CommRing L] [IsDomain L] (hμ : IsPrimitiveRoot μ ↑n) [Algebra K L] [IsCyclotomicExtension {n} K L] (h : Irreducible (Polynomial.cyclotomic (↑n) K))
Goal: (IsCyclotomicExtension.fromZetaAut hμ h) (IsCyclotomicExtension.zeta n K L) = μ
-- PadicInt.modPart_lt_p
theorem modPart_lt_p : modPart p r < p 
Arguments: {p : ℕ} [hp_prime : Fact (Nat.Prime p)] (r : ℚ)
Goal: PadicInt.modPart p r < ↑p
-- PadicInt.modPart_nonneg
theorem modPart_nonneg : 0 ≤ modPart p r 
Arguments: {p : ℕ} [hp_prime : Fact (Nat.Prime p)] (r : ℚ)
Goal: 0 ≤ PadicInt.modPart p r
-- PadicInt.isUnit_den
theorem isUnit_den (r : ℚ) (h : ‖(r : ℚ_[p])‖ ≤ 1) : IsUnit (r.den : ℤ_[p]) 
Arguments: {p : ℕ} [hp_prime : Fact (Nat.Prime p)] (r : ℚ) (h : ‖↑r‖ ≤ 1)
Goal: IsUnit ↑r.den
-- PadicInt.norm_sub_modPart_aux
theorem norm_sub_modPart_aux (r : ℚ) (h : ‖(r : ℚ_[p])‖ ≤ 1) :
    ↑p ∣ r.num - r.num * r.den.gcdA p % p * ↑r.den 
Arguments: {p : ℕ} [hp_prime : Fact (Nat.Prime p)] (r : ℚ) (h : ‖↑r‖ ≤ 1)
Goal: ↑p ∣ r.num - r.num * r.den.gcdA p % ↑p * ↑r.den
-- PadicInt.norm_sub_modPart
theorem norm_sub_modPart (h : ‖(r : ℚ_[p])‖ ≤ 1) : ‖(⟨r, h⟩ - modPart p r : ℤ_[p])‖ < 1 
Arguments: {p : ℕ} [hp_prime : Fact (Nat.Prime p)] (r : ℚ) (h : ‖↑r‖ ≤ 1)
Goal: ‖⟨↑r, h⟩ - ↑(PadicInt.modPart p r)‖ < 1
-- PadicInt.exists_mem_range_of_norm_rat_le_one
theorem exists_mem_range_of_norm_rat_le_one (h : ‖(r : ℚ_[p])‖ ≤ 1) :
    ∃ n : ℤ, 0 ≤ n ∧ n < p ∧ ‖(⟨r, h⟩ - n : ℤ_[p])‖ < 1 
Arguments: {p : ℕ} [hp_prime : Fact (Nat.Prime p)] (r : ℚ) (h : ‖↑r‖ ≤ 1)
Goal: ∃ n, 0 ≤ n ∧ n < ↑p ∧ ‖⟨↑r, h⟩ - ↑n‖ < 1
-- PadicInt.zmod_congr_of_sub_mem_span_aux
theorem zmod_congr_of_sub_mem_span_aux (n : ℕ) (x : ℤ_[p]) (a b : ℤ)
    (ha : x - a ∈ (Ideal.span {(p : ℤ_[p]) ^ n}))
    (hb : x - b ∈ (Ideal.span {(p : ℤ_[p]) ^ n})) : (a : ZMod (p ^ n)) = b 
Arguments: {p : ℕ} [hp_prime : Fact (Nat.Prime p)] (n : ℕ) (x : ℤ_[p]) (a b : ℤ) (ha : x - ↑a ∈ Ideal.span {↑p ^ n}) (hb : x - ↑b ∈ Ideal.span {↑p ^ n})
Goal: ↑a = ↑b
-- PadicInt.zmod_congr_of_sub_mem_span
theorem zmod_congr_of_sub_mem_span (n : ℕ) (x : ℤ_[p]) (a b : ℕ)
    (ha : x - a ∈ (Ideal.span {(p : ℤ_[p]) ^ n}))
    (hb : x - b ∈ (Ideal.span {(p : ℤ_[p]) ^ n})) : (a : ZMod (p ^ n)) = b 
Arguments: {p : ℕ} [hp_prime : Fact (Nat.Prime p)] (n : ℕ) (x : ℤ_[p]) (a b : ℕ) (ha : x - ↑a ∈ Ideal.span {↑p ^ n}) (hb : x - ↑b ∈ Ideal.span {↑p ^ n})
Goal: ↑a = ↑b
-- PadicInt.zmod_congr_of_sub_mem_max_ideal
theorem zmod_congr_of_sub_mem_max_ideal (x : ℤ_[p]) (m n : ℕ) (hm : x - m ∈ maximalIdeal ℤ_[p])
    (hn : x - n ∈ maximalIdeal ℤ_[p]) : (m : ZMod p) = n 
Arguments: {p : ℕ} [hp_prime : Fact (Nat.Prime p)] (x : ℤ_[p]) (m n : ℕ) (hm : x - ↑m ∈ LocalRing.maximalIdeal ℤ_[p]) (hn : x - ↑n ∈ LocalRing.maximalIdeal ℤ_[p])
Goal: ↑m = ↑n
-- PadicInt.exists_mem_range
theorem exists_mem_range : ∃ n : ℕ, n < p ∧ x - n ∈ maximalIdeal ℤ_[p] 
Arguments: {p : ℕ} [hp_prime : Fact (Nat.Prime p)] (x : ℤ_[p])
Goal: ∃ n < p, x - ↑n ∈ LocalRing.maximalIdeal ℤ_[p]
-- PadicInt.zmodRepr_spec
theorem zmodRepr_spec : zmodRepr x < p ∧ x - zmodRepr x ∈ maximalIdeal ℤ_[p] 
Arguments: {p : ℕ} [hp_prime : Fact (Nat.Prime p)] (x : ℤ_[p])
Goal: x.zmodRepr < p ∧ x - ↑x.zmodRepr ∈ LocalRing.maximalIdeal ℤ_[p]
-- PadicInt.zmodRepr_lt_p
theorem zmodRepr_lt_p : zmodRepr x < p 
Arguments: {p : ℕ} [hp_prime : Fact (Nat.Prime p)] (x : ℤ_[p])
Goal: x.zmodRepr < p
-- PadicInt.sub_zmodRepr_mem
theorem sub_zmodRepr_mem : x - zmodRepr x ∈ maximalIdeal ℤ_[p] 
Arguments: {p : ℕ} [hp_prime : Fact (Nat.Prime p)] (x : ℤ_[p])
Goal: x - ↑x.zmodRepr ∈ LocalRing.maximalIdeal ℤ_[p]
-- PadicInt.toZMod_spec
theorem toZMod_spec : x - (ZMod.cast (toZMod x) : ℤ_[p]) ∈ maximalIdeal ℤ_[p] 
Arguments: {p : ℕ} [hp_prime : Fact (Nat.Prime p)] (x : ℤ_[p])
Goal: x - (PadicInt.toZMod x).cast ∈ LocalRing.maximalIdeal ℤ_[p]
-- PadicInt.ker_toZMod
theorem ker_toZMod : RingHom.ker (toZMod : ℤ_[p] →+* ZMod p) = maximalIdeal ℤ_[p] 
Arguments: {p : ℕ} [hp_prime : Fact (Nat.Prime p)]
Goal: RingHom.ker PadicInt.toZMod = LocalRing.maximalIdeal ℤ_[p]
-- PadicInt.appr_lt
theorem appr_lt (x : ℤ_[p]) (n : ℕ) : x.appr n < p ^ n 
Arguments: {p : ℕ} [hp_prime : Fact (Nat.Prime p)] (x : ℤ_[p]) (n : ℕ)
Goal: x.appr n < p ^ n
-- PadicInt.appr_mono
theorem appr_mono (x : ℤ_[p]) : Monotone x.appr 
Arguments: {p : ℕ} [hp_prime : Fact (Nat.Prime p)] (x : ℤ_[p])
Goal: Monotone x.appr
-- PadicInt.dvd_appr_sub_appr
theorem dvd_appr_sub_appr (x : ℤ_[p]) (m n : ℕ) (h : m ≤ n) : p ^ m ∣ x.appr n - x.appr m 
Arguments: {p : ℕ} [hp_prime : Fact (Nat.Prime p)] (x : ℤ_[p]) (m n : ℕ) (h : m ≤ n)
Goal: p ^ m ∣ x.appr n - x.appr m
-- PadicInt.appr_spec
theorem appr_spec (n : ℕ) : ∀ x : ℤ_[p], x - appr x n ∈ Ideal.span {(p : ℤ_[p]) ^ n} 
Arguments: {p : ℕ} [hp_prime : Fact (Nat.Prime p)] (n : ℕ) (x : ℤ_[p])
Goal: x - ↑(x.appr n) ∈ Ideal.span {↑p ^ n}
-- PadicInt.ker_toZModPow
theorem ker_toZModPow (n : ℕ) :
    RingHom.ker (toZModPow n : ℤ_[p] →+* ZMod (p ^ n)) = Ideal.span {(p : ℤ_[p]) ^ n} 
Arguments: {p : ℕ} [hp_prime : Fact (Nat.Prime p)] (n : ℕ)
Goal: RingHom.ker (PadicInt.toZModPow n) = Ideal.span {↑p ^ n}
-- PadicInt.zmod_cast_comp_toZModPow
theorem zmod_cast_comp_toZModPow (m n : ℕ) (h : m ≤ n) :
    (ZMod.castHom (pow_dvd_pow p h) (ZMod (p ^ m))).comp (@toZModPow p _ n) = @toZModPow p _ m 
Arguments: {p : ℕ} [hp_prime : Fact (Nat.Prime p)] (m n : ℕ) (h : m ≤ n)
Goal: (ZMod.castHom ⋯ (ZMod (p ^ m))).comp (PadicInt.toZModPow n) = PadicInt.toZModPow m
-- PadicInt.cast_toZModPow
@[simp]
theorem cast_toZModPow (m n : ℕ) (h : m ≤ n) (x : ℤ_[p]) :
    ZMod.cast (toZModPow n x) = toZModPow m x 
Arguments: {p : ℕ} [hp_prime : Fact (Nat.Prime p)] (m n : ℕ) (h : m ≤ n) (x : ℤ_[p])
Goal: ((PadicInt.toZModPow n) x).cast = (PadicInt.toZModPow m) x
-- PadicInt.denseRange_natCast
theorem denseRange_natCast : DenseRange (Nat.cast : ℕ → ℤ_[p]) 
Arguments: {p : ℕ} [hp_prime : Fact (Nat.Prime p)]
Goal: DenseRange Nat.cast
-- PadicInt.denseRange_intCast
theorem denseRange_intCast : DenseRange (Int.cast : ℤ → ℤ_[p]) 
Arguments: {p : ℕ} [hp_prime : Fact (Nat.Prime p)]
Goal: DenseRange Int.cast
-- PadicInt.nthHom_zero
@[simp]
theorem nthHom_zero : nthHom f 0 = 0 
Arguments: {p : ℕ} {R : Type u_1} [NonAssocSemiring R] (f : (k : ℕ) → R →+* ZMod (p ^ k))
Goal: PadicInt.nthHom f 0 = 0
-- PadicInt.pow_dvd_nthHom_sub
theorem pow_dvd_nthHom_sub (r : R) (i j : ℕ) (h : i ≤ j) :
    (p : ℤ) ^ i ∣ nthHom f r j - nthHom f r i 
Arguments: {p : ℕ} [hp_prime : Fact (Nat.Prime p)] {R : Type u_1} [NonAssocSemiring R] {f : (k : ℕ) → R →+* ZMod (p ^ k)} (f_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), (ZMod.castHom ⋯ (ZMod (p ^ k1))).comp (f k2) = f k1) (r : R) (i j : ℕ) (h : i ≤ j)
Goal: ↑p ^ i ∣ PadicInt.nthHom f r j - PadicInt.nthHom f r i
-- PadicInt.isCauSeq_nthHom
theorem isCauSeq_nthHom (r : R) : IsCauSeq (padicNorm p) fun n => nthHom f r n 
Arguments: {p : ℕ} [hp_prime : Fact (Nat.Prime p)] {R : Type u_1} [NonAssocSemiring R] {f : (k : ℕ) → R →+* ZMod (p ^ k)} (f_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), (ZMod.castHom ⋯ (ZMod (p ^ k1))).comp (f k2) = f k1) (r : R)
Goal: IsCauSeq (padicNorm p) fun n => ↑(PadicInt.nthHom f r n)
-- PadicInt.nthHomSeq_one
theorem nthHomSeq_one : nthHomSeq f_compat 1 ≈ 1 
Arguments: {p : ℕ} [hp_prime : Fact (Nat.Prime p)] {R : Type u_1} [NonAssocSemiring R] {f : (k : ℕ) → R →+* ZMod (p ^ k)} (f_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), (ZMod.castHom ⋯ (ZMod (p ^ k1))).comp (f k2) = f k1)
Goal: PadicInt.nthHomSeq f_compat 1 ≈ 1
-- PadicInt.nthHomSeq_add
theorem nthHomSeq_add (r s : R) :
    nthHomSeq f_compat (r + s) ≈ nthHomSeq f_compat r + nthHomSeq f_compat s 
Arguments: {p : ℕ} [hp_prime : Fact (Nat.Prime p)] {R : Type u_1} [NonAssocSemiring R] {f : (k : ℕ) → R →+* ZMod (p ^ k)} (f_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), (ZMod.castHom ⋯ (ZMod (p ^ k1))).comp (f k2) = f k1) (r s : R)
Goal: PadicInt.nthHomSeq f_compat (r + s) ≈ PadicInt.nthHomSeq f_compat r + PadicInt.nthHomSeq f_compat s
-- PadicInt.nthHomSeq_mul
theorem nthHomSeq_mul (r s : R) :
    nthHomSeq f_compat (r * s) ≈ nthHomSeq f_compat r * nthHomSeq f_compat s 
Arguments: {p : ℕ} [hp_prime : Fact (Nat.Prime p)] {R : Type u_1} [NonAssocSemiring R] {f : (k : ℕ) → R →+* ZMod (p ^ k)} (f_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), (ZMod.castHom ⋯ (ZMod (p ^ k1))).comp (f k2) = f k1) (r s : R)
Goal: PadicInt.nthHomSeq f_compat (r * s) ≈ PadicInt.nthHomSeq f_compat r * PadicInt.nthHomSeq f_compat s
-- PadicInt.limNthHom_spec
theorem limNthHom_spec (r : R) :
    ∀ ε : ℝ, 0 < ε → ∃ N : ℕ, ∀ n ≥ N, ‖limNthHom f_compat r - nthHom f r n‖ < ε 
Arguments: {p : ℕ} [hp_prime : Fact (Nat.Prime p)] {R : Type u_1} [NonAssocSemiring R] {f : (k : ℕ) → R →+* ZMod (p ^ k)} (f_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), (ZMod.castHom ⋯ (ZMod (p ^ k1))).comp (f k2) = f k1) (r : R) (ε : ℝ)
Goal: 0 < ε → ∃ N, ∀ n ≥ N, ‖PadicInt.limNthHom f_compat r - ↑(PadicInt.nthHom f r n)‖ < ε
-- PadicInt.limNthHom_zero
theorem limNthHom_zero : limNthHom f_compat 0 = 0 
Arguments: {p : ℕ} [hp_prime : Fact (Nat.Prime p)] {R : Type u_1} [NonAssocSemiring R] {f : (k : ℕ) → R →+* ZMod (p ^ k)} (f_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), (ZMod.castHom ⋯ (ZMod (p ^ k1))).comp (f k2) = f k1)
Goal: PadicInt.limNthHom f_compat 0 = 0
-- PadicInt.limNthHom_one
theorem limNthHom_one : limNthHom f_compat 1 = 1 
Arguments: {p : ℕ} [hp_prime : Fact (Nat.Prime p)] {R : Type u_1} [NonAssocSemiring R] {f : (k : ℕ) → R →+* ZMod (p ^ k)} (f_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), (ZMod.castHom ⋯ (ZMod (p ^ k1))).comp (f k2) = f k1)
Goal: PadicInt.limNthHom f_compat 1 = 1
-- PadicInt.limNthHom_add
theorem limNthHom_add (r s : R) :
    limNthHom f_compat (r + s) = limNthHom f_compat r + limNthHom f_compat s 
Arguments: {p : ℕ} [hp_prime : Fact (Nat.Prime p)] {R : Type u_1} [NonAssocSemiring R] {f : (k : ℕ) → R →+* ZMod (p ^ k)} (f_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), (ZMod.castHom ⋯ (ZMod (p ^ k1))).comp (f k2) = f k1) (r s : R)
Goal: PadicInt.limNthHom f_compat (r + s) = PadicInt.limNthHom f_compat r + PadicInt.limNthHom f_compat s
-- PadicInt.limNthHom_mul
theorem limNthHom_mul (r s : R) :
    limNthHom f_compat (r * s) = limNthHom f_compat r * limNthHom f_compat s 
Arguments: {p : ℕ} [hp_prime : Fact (Nat.Prime p)] {R : Type u_1} [NonAssocSemiring R] {f : (k : ℕ) → R →+* ZMod (p ^ k)} (f_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), (ZMod.castHom ⋯ (ZMod (p ^ k1))).comp (f k2) = f k1) (r s : R)
Goal: PadicInt.limNthHom f_compat (r * s) = PadicInt.limNthHom f_compat r * PadicInt.limNthHom f_compat s
-- PadicInt.lift_sub_val_mem_span
theorem lift_sub_val_mem_span (r : R) (n : ℕ) :
    lift f_compat r - (f n r).val ∈ (Ideal.span {(p : ℤ_[p]) ^ n}) 
Arguments: {p : ℕ} [hp_prime : Fact (Nat.Prime p)] {R : Type u_1} [NonAssocSemiring R] {f : (k : ℕ) → R →+* ZMod (p ^ k)} (f_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), (ZMod.castHom ⋯ (ZMod (p ^ k1))).comp (f k2) = f k1) (r : R) (n : ℕ)
Goal: (PadicInt.lift f_compat) r - ↑((f n) r).val ∈ Ideal.span {↑p ^ n}
-- PadicInt.lift_spec
theorem lift_spec (n : ℕ) : (toZModPow n).comp (lift f_compat) = f n 
Arguments: {p : ℕ} [hp_prime : Fact (Nat.Prime p)] {R : Type u_1} [NonAssocSemiring R] {f : (k : ℕ) → R →+* ZMod (p ^ k)} (f_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), (ZMod.castHom ⋯ (ZMod (p ^ k1))).comp (f k2) = f k1) (n : ℕ)
Goal: (PadicInt.toZModPow n).comp (PadicInt.lift f_compat) = f n
-- PadicInt.lift_unique
theorem lift_unique (g : R →+* ℤ_[p]) (hg : ∀ n, (toZModPow n).comp g = f n) :
    lift f_compat = g 
Arguments: {p : ℕ} [hp_prime : Fact (Nat.Prime p)] {R : Type u_1} [NonAssocSemiring R] {f : (k : ℕ) → R →+* ZMod (p ^ k)} (f_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), (ZMod.castHom ⋯ (ZMod (p ^ k1))).comp (f k2) = f k1) (g : R →+* ℤ_[p]) (hg : ∀ (n : ℕ), (PadicInt.toZModPow n).comp g = f n)
Goal: PadicInt.lift f_compat = g
-- PadicInt.lift_self
@[simp]
theorem lift_self (z : ℤ_[p]) : @lift p _ ℤ_[p] _ toZModPow zmod_cast_comp_toZModPow z = z 
Arguments: {p : ℕ} [hp_prime : Fact (Nat.Prime p)] (z : ℤ_[p])
Goal: (PadicInt.lift ⋯) z = z
-- PadicInt.ext_of_toZModPow
theorem ext_of_toZModPow {x y : ℤ_[p]} : (∀ n, toZModPow n x = toZModPow n y) ↔ x = y 
Arguments: {p : ℕ} [hp_prime : Fact (Nat.Prime p)] {x y : ℤ_[p]}
Goal: (∀ (n : ℕ), (PadicInt.toZModPow n) x = (PadicInt.toZModPow n) y) ↔ x = y
-- PadicInt.toZModPow_eq_iff_ext
theorem toZModPow_eq_iff_ext {R : Type*} [NonAssocSemiring R] {g g' : R →+* ℤ_[p]} :
    (∀ n, (toZModPow n).comp g = (toZModPow n).comp g') ↔ g = g' 
Arguments: {p : ℕ} [hp_prime : Fact (Nat.Prime p)] {R : Type u_1} [NonAssocSemiring R] {g g' : R →+* ℤ_[p]}
Goal: (∀ (n : ℕ), (PadicInt.toZModPow n).comp g = (PadicInt.toZModPow n).comp g') ↔ g = g'
-- HurwitzZeta.evenKernel_def
lemma evenKernel_def (a x : ℝ) :
    ↑(evenKernel ↑a x) = cexp (-π * a ^ 2 * x) * jacobiTheta₂ (a * I * x) (I * x) 
Arguments: (a x : ℝ)
Goal: ↑(HurwitzZeta.evenKernel (↑a) x) = cexp (-↑π * ↑a ^ 2 * ↑x) * jacobiTheta₂ (↑a * Complex.I * ↑x) (Complex.I * ↑x)
-- HurwitzZeta.evenKernel_undef
lemma evenKernel_undef (a : UnitAddCircle) {x : ℝ} (hx : x ≤ 0) : evenKernel a x = 0 
Arguments: (a : UnitAddCircle) {x : ℝ} (hx : x ≤ 0)
Goal: HurwitzZeta.evenKernel a x = 0
-- HurwitzZeta.cosKernel_def
lemma cosKernel_def (a x : ℝ) : ↑(cosKernel ↑a x) = jacobiTheta₂ a (I * x) 
Arguments: (a x : ℝ)
Goal: ↑(HurwitzZeta.cosKernel (↑a) x) = jacobiTheta₂ (↑a) (Complex.I * ↑x)
-- HurwitzZeta.cosKernel_undef
lemma cosKernel_undef (a : UnitAddCircle) {x : ℝ} (hx : x ≤ 0) : cosKernel a x = 0 
Arguments: (a : UnitAddCircle) {x : ℝ} (hx : x ≤ 0)
Goal: HurwitzZeta.cosKernel a x = 0
-- HurwitzZeta.evenKernel_eq_cosKernel_of_zero
lemma evenKernel_eq_cosKernel_of_zero : evenKernel 0 = cosKernel 0 
Goal: HurwitzZeta.evenKernel 0 = HurwitzZeta.cosKernel 0
-- HurwitzZeta.evenKernel_neg
lemma evenKernel_neg (a : UnitAddCircle) (x : ℝ) : evenKernel (-a) x = evenKernel a x 
Arguments: (a : UnitAddCircle) (x : ℝ)
Goal: HurwitzZeta.evenKernel (-a) x = HurwitzZeta.evenKernel a x
-- HurwitzZeta.cosKernel_neg
lemma cosKernel_neg (a : UnitAddCircle) (x : ℝ) : cosKernel (-a) x = cosKernel a x 
Arguments: (a : UnitAddCircle) (x : ℝ)
Goal: HurwitzZeta.cosKernel (-a) x = HurwitzZeta.cosKernel a x
-- HurwitzZeta.continuousOn_evenKernel
lemma continuousOn_evenKernel (a : UnitAddCircle) : ContinuousOn (evenKernel a) (Ioi 0) 
Arguments: (a : UnitAddCircle)
Goal: ContinuousOn (HurwitzZeta.evenKernel a) (Set.Ioi 0)
-- HurwitzZeta.continuousOn_cosKernel
lemma continuousOn_cosKernel (a : UnitAddCircle) : ContinuousOn (cosKernel a) (Ioi 0) 
Arguments: (a : UnitAddCircle)
Goal: ContinuousOn (HurwitzZeta.cosKernel a) (Set.Ioi 0)
-- HurwitzZeta.evenKernel_functional_equation
lemma evenKernel_functional_equation (a : UnitAddCircle) (x : ℝ) :
    evenKernel a x = 1 / x ^ (1 / 2 : ℝ) * cosKernel a (1 / x) 
Arguments: (a : UnitAddCircle) (x : ℝ)
Goal: HurwitzZeta.evenKernel a x = 1 / x ^ (1 / 2) * HurwitzZeta.cosKernel a (1 / x)
-- HurwitzZeta.hasSum_int_evenKernel
lemma hasSum_int_evenKernel (a : ℝ) {t : ℝ} (ht : 0 < t) :
    HasSum (fun n : ℤ ↦ rexp (-π * (n + a) ^ 2 * t)) (evenKernel a t) 
Arguments: (a : ℝ) {t : ℝ} (ht : 0 < t)
Goal: HasSum (fun n => rexp (-π * (↑n + a) ^ 2 * t)) (HurwitzZeta.evenKernel (↑a) t)
-- HurwitzZeta.hasSum_int_cosKernel
lemma hasSum_int_cosKernel (a : ℝ) {t : ℝ} (ht : 0 < t) :
    HasSum (fun n : ℤ ↦ cexp (2 * π * I * a * n) * rexp (-π * n ^ 2 * t)) ↑(cosKernel a t) 
Arguments: (a : ℝ) {t : ℝ} (ht : 0 < t)
Goal: HasSum (fun n => cexp (2 * ↑π * Complex.I * ↑a * ↑n) * ↑(rexp (-π * ↑n ^ 2 * t))) ↑(HurwitzZeta.cosKernel (↑a) t)
-- HurwitzZeta.hasSum_int_evenKernel₀
lemma hasSum_int_evenKernel₀ (a : ℝ) {t : ℝ} (ht : 0 < t) :
    HasSum (fun n : ℤ ↦ if n + a = 0 then 0 else rexp (-π * (n + a) ^ 2 * t))
    (evenKernel a t - if (a : UnitAddCircle) = 0 then 1 else 0) 
Arguments: (a : ℝ) {t : ℝ} (ht : 0 < t)
Goal: HasSum (fun n => if ↑n + a = 0 then 0 else rexp (-π * (↑n + a) ^ 2 * t))
  (HurwitzZeta.evenKernel (↑a) t - if ↑a = 0 then 1 else 0)
-- HurwitzZeta.hasSum_int_cosKernel₀
lemma hasSum_int_cosKernel₀ (a : ℝ) {t : ℝ} (ht : 0 < t) :
    HasSum (fun n : ℤ ↦ if n = 0 then 0 else cexp (2 * π * I * a * n) * rexp (-π * n ^ 2 * t))
    (↑(cosKernel a t) - 1) 
Arguments: (a : ℝ) {t : ℝ} (ht : 0 < t)
Goal: HasSum (fun n => if n = 0 then 0 else cexp (2 * ↑π * Complex.I * ↑a * ↑n) * ↑(rexp (-π * ↑n ^ 2 * t)))
  (↑(HurwitzZeta.cosKernel (↑a) t) - 1)
-- HurwitzZeta.hasSum_nat_cosKernel₀
lemma hasSum_nat_cosKernel₀ (a : ℝ) {t : ℝ} (ht : 0 < t) :
    HasSum (fun n : ℕ ↦ 2 * Real.cos (2 * π * a * (n + 1)) * rexp (-π * (n + 1) ^ 2 * t))
    (cosKernel a t - 1) 
Arguments: (a : ℝ) {t : ℝ} (ht : 0 < t)
Goal: HasSum (fun n => 2 * Real.cos (2 * π * a * (↑n + 1)) * rexp (-π * (↑n + 1) ^ 2 * t)) (HurwitzZeta.cosKernel (↑a) t - 1)
-- HurwitzZeta.isBigO_atTop_evenKernel_sub
lemma isBigO_atTop_evenKernel_sub (a : UnitAddCircle) : ∃ p : ℝ, 0 < p ∧
    (evenKernel a · - (if a = 0 then 1 else 0)) =O[atTop] (rexp <| -p * ·) 
Arguments: (a : UnitAddCircle)
Goal: ∃ p, 0 < p ∧ (fun x => HurwitzZeta.evenKernel a x - if a = 0 then 1 else 0) =O[Filter.atTop] fun x => rexp (-p * x)
-- HurwitzZeta.isBigO_atTop_cosKernel_sub
lemma isBigO_atTop_cosKernel_sub (a : UnitAddCircle) :
    ∃ p, 0 < p ∧ IsBigO atTop (cosKernel a · - 1) (fun x ↦ Real.exp (-p * x)) 
Arguments: (a : UnitAddCircle)
Goal: ∃ p, 0 < p ∧ (fun x => HurwitzZeta.cosKernel a x - 1) =O[Filter.atTop] fun x => rexp (-p * x)
-- HurwitzZeta.hurwitzEvenFEPair_zero_symm
lemma hurwitzEvenFEPair_zero_symm :
    (hurwitzEvenFEPair 0).symm = hurwitzEvenFEPair 0 
Goal: (HurwitzZeta.hurwitzEvenFEPair 0).symm = HurwitzZeta.hurwitzEvenFEPair 0
-- HurwitzZeta.hurwitzEvenFEPair_neg
lemma hurwitzEvenFEPair_neg (a : UnitAddCircle) : hurwitzEvenFEPair (-a) = hurwitzEvenFEPair a 
Arguments: (a : UnitAddCircle)
Goal: HurwitzZeta.hurwitzEvenFEPair (-a) = HurwitzZeta.hurwitzEvenFEPair a
-- HurwitzZeta.completedHurwitzZetaEven_eq
lemma completedHurwitzZetaEven_eq (a : UnitAddCircle) (s : ℂ) :
    completedHurwitzZetaEven a s =
    completedHurwitzZetaEven₀ a s - (if a = 0 then 1 else 0) / s - 1 / (1 - s) 
Arguments: (a : UnitAddCircle) (s : ℂ)
Goal: HurwitzZeta.completedHurwitzZetaEven a s =
  HurwitzZeta.completedHurwitzZetaEven₀ a s - (if a = 0 then 1 else 0) / s - 1 / (1 - s)
-- HurwitzZeta.completedCosZeta_eq
lemma completedCosZeta_eq (a : UnitAddCircle) (s : ℂ) :
    completedCosZeta a s =
    completedCosZeta₀ a s - 1 / s - (if a = 0 then 1 else 0) / (1 - s) 
Arguments: (a : UnitAddCircle) (s : ℂ)
Goal: HurwitzZeta.completedCosZeta a s = HurwitzZeta.completedCosZeta₀ a s - 1 / s - (if a = 0 then 1 else 0) / (1 - s)
-- HurwitzZeta.completedHurwitzZetaEven_neg
lemma completedHurwitzZetaEven_neg (a : UnitAddCircle) (s : ℂ) :
    completedHurwitzZetaEven (-a) s = completedHurwitzZetaEven a s 
Arguments: (a : UnitAddCircle) (s : ℂ)
Goal: HurwitzZeta.completedHurwitzZetaEven (-a) s = HurwitzZeta.completedHurwitzZetaEven a s
-- HurwitzZeta.completedHurwitzZetaEven₀_neg
lemma completedHurwitzZetaEven₀_neg (a : UnitAddCircle) (s : ℂ) :
    completedHurwitzZetaEven₀ (-a) s = completedHurwitzZetaEven₀ a s 
Arguments: (a : UnitAddCircle) (s : ℂ)
Goal: HurwitzZeta.completedHurwitzZetaEven₀ (-a) s = HurwitzZeta.completedHurwitzZetaEven₀ a s
-- HurwitzZeta.completedCosZeta_neg
lemma completedCosZeta_neg (a : UnitAddCircle) (s : ℂ) :
    completedCosZeta (-a) s = completedCosZeta a s 
Arguments: (a : UnitAddCircle) (s : ℂ)
Goal: HurwitzZeta.completedCosZeta (-a) s = HurwitzZeta.completedCosZeta a s
-- HurwitzZeta.completedCosZeta₀_neg
lemma completedCosZeta₀_neg (a : UnitAddCircle) (s : ℂ) :
    completedCosZeta₀ (-a) s = completedCosZeta₀ a s 
Arguments: (a : UnitAddCircle) (s : ℂ)
Goal: HurwitzZeta.completedCosZeta₀ (-a) s = HurwitzZeta.completedCosZeta₀ a s
-- HurwitzZeta.completedHurwitzZetaEven_one_sub
lemma completedHurwitzZetaEven_one_sub (a : UnitAddCircle) (s : ℂ) :
    completedHurwitzZetaEven a (1 - s) = completedCosZeta a s 
Arguments: (a : UnitAddCircle) (s : ℂ)
Goal: HurwitzZeta.completedHurwitzZetaEven a (1 - s) = HurwitzZeta.completedCosZeta a s
-- HurwitzZeta.completedHurwitzZetaEven₀_one_sub
lemma completedHurwitzZetaEven₀_one_sub (a : UnitAddCircle) (s : ℂ) :
    completedHurwitzZetaEven₀ a (1 - s) = completedCosZeta₀ a s 
Arguments: (a : UnitAddCircle) (s : ℂ)
Goal: HurwitzZeta.completedHurwitzZetaEven₀ a (1 - s) = HurwitzZeta.completedCosZeta₀ a s
-- HurwitzZeta.completedCosZeta_one_sub
lemma completedCosZeta_one_sub (a : UnitAddCircle) (s : ℂ) :
    completedCosZeta a (1 - s) = completedHurwitzZetaEven a s 
Arguments: (a : UnitAddCircle) (s : ℂ)
Goal: HurwitzZeta.completedCosZeta a (1 - s) = HurwitzZeta.completedHurwitzZetaEven a s
-- HurwitzZeta.completedCosZeta₀_one_sub
lemma completedCosZeta₀_one_sub (a : UnitAddCircle) (s : ℂ) :
    completedCosZeta₀ a (1 - s) = completedHurwitzZetaEven₀ a s 
Arguments: (a : UnitAddCircle) (s : ℂ)
Goal: HurwitzZeta.completedCosZeta₀ a (1 - s) = HurwitzZeta.completedHurwitzZetaEven₀ a s
-- HurwitzZeta.differentiableAt_completedHurwitzZetaEven
lemma differentiableAt_completedHurwitzZetaEven
    (a : UnitAddCircle) {s : ℂ} (hs : s ≠ 0 ∨ a ≠ 0) (hs' : s ≠ 1) :
    DifferentiableAt ℂ (completedHurwitzZetaEven a) s 
Arguments: (a : UnitAddCircle) {s : ℂ} (hs : s ≠ 0 ∨ a ≠ 0) (hs' : s ≠ 1)
Goal: DifferentiableAt ℂ (HurwitzZeta.completedHurwitzZetaEven a) s