ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- constantCoeff_wittStructureInt
theorem constantCoeff_wittStructureInt (Φ : MvPolynomial idx ℤ) (h : constantCoeff Φ = 0) (n : ℕ) :
    constantCoeff (wittStructureInt p Φ n) = 0 
Arguments: (p : ℕ) {idx : Type u_2} [hp : Fact (Nat.Prime p)] (Φ : MvPolynomial idx ℤ) (h : MvPolynomial.constantCoeff Φ = 0) (n : ℕ)
Goal: MvPolynomial.constantCoeff (wittStructureInt p Φ n) = 0
-- wittStructureRat_vars
theorem wittStructureRat_vars [Fintype idx] (Φ : MvPolynomial idx ℚ) (n : ℕ) :
    (wittStructureRat p Φ n).vars ⊆ Finset.univ ×ˢ Finset.range (n + 1) 
Arguments: (p : ℕ) {idx : Type u_2} [hp : Fact (Nat.Prime p)] [Fintype idx] (Φ : MvPolynomial idx ℚ) (n : ℕ)
Goal: (wittStructureRat p Φ n).vars ⊆ Finset.univ ×ˢ Finset.range (n + 1)
-- wittStructureInt_vars
theorem wittStructureInt_vars [Fintype idx] (Φ : MvPolynomial idx ℤ) (n : ℕ) :
    (wittStructureInt p Φ n).vars ⊆ Finset.univ ×ˢ Finset.range (n + 1) 
Arguments: (p : ℕ) {idx : Type u_2} [hp : Fact (Nat.Prime p)] [Fintype idx] (Φ : MvPolynomial idx ℤ) (n : ℕ)
Goal: (wittStructureInt p Φ n).vars ⊆ Finset.univ ×ˢ Finset.range (n + 1)
-- Algebra.PreSubmersivePresentation.map_inj
@[nolint checkUnivs]
structure PreSubmersivePresentation extends Algebra.Presentation.{t, w} R S where
    map : rels → vars
  map_inj : Function.Injective map
  relations_finite : Finite rels
Arguments: {R : Type u} [CommRing R] {S : Type v} [CommRing S] [Algebra R S] (self : Algebra.PreSubmersivePresentation R S)
Goal: Function.Injective self.map
-- Algebra.PreSubmersivePresentation.relations_finite
@[nolint checkUnivs]
structure PreSubmersivePresentation extends Algebra.Presentation.{t, w} R S where
    map : rels → vars
  map_inj : Function.Injective map
  relations_finite : Finite rels
Arguments: {R : Type u} [CommRing R] {S : Type v} [CommRing S] [Algebra R S] (self : Algebra.PreSubmersivePresentation R S)
Goal: Finite self.rels
-- Algebra.PreSubmersivePresentation.card_relations_le_card_vars_of_isFinite
lemma card_relations_le_card_vars_of_isFinite [P.IsFinite] :
    Nat.card P.rels ≤ Nat.card P.vars 
Arguments: {R : Type u} [CommRing R] {S : Type v} [CommRing S] [Algebra R S] (P : Algebra.PreSubmersivePresentation R S) [P.IsFinite]
Goal: Nat.card P.rels ≤ Nat.card P.vars
-- Algebra.PreSubmersivePresentation.jacobian_eq_jacobiMatrix_det
lemma jacobian_eq_jacobiMatrix_det : P.jacobian = algebraMap P.Ring S P.jacobiMatrix.det 
Arguments: {R : Type u} [CommRing R] {S : Type v} [CommRing S] [Algebra R S] (P : Algebra.PreSubmersivePresentation R S) [Fintype P.rels] [DecidableEq P.rels]
Goal: P.jacobian = (algebraMap P.Ring S) P.jacobiMatrix.det
-- Algebra.PreSubmersivePresentation.jacobiMatrix_apply
lemma jacobiMatrix_apply (i j : P.rels) :
    P.jacobiMatrix i j = MvPolynomial.pderiv (P.map i) (P.relation j) 
Arguments: {R : Type u} [CommRing R] {S : Type v} [CommRing S] [Algebra R S] (P : Algebra.PreSubmersivePresentation R S) [Fintype P.rels] [DecidableEq P.rels] (i j : P.rels)
Goal: P.jacobiMatrix i j = (MvPolynomial.pderiv (P.map i)) (P.relation j)
-- Algebra.SubmersivePresentation.jacobian_isUnit
@[nolint checkUnivs]
structure SubmersivePresentation extends PreSubmersivePresentation.{t, w} R S where
  jacobian_isUnit : IsUnit toPreSubmersivePresentation.jacobian
  isFinite : toPreSubmersivePresentation.IsFinite := by infer_instance
Arguments: {R : Type u} [CommRing R] {S : Type v} [CommRing S] [Algebra R S] (self : Algebra.SubmersivePresentation R S)
Goal: IsUnit self.jacobian
-- Algebra.SubmersivePresentation.isFinite
@[nolint checkUnivs]
structure SubmersivePresentation extends PreSubmersivePresentation.{t, w} R S where
  jacobian_isUnit : IsUnit toPreSubmersivePresentation.jacobian
  isFinite : toPreSubmersivePresentation.IsFinite := by infer_instance
Arguments: {R : Type u} [CommRing R] {S : Type v} [CommRing S] [Algebra R S] (self : Algebra.SubmersivePresentation R S)
Goal: self.IsFinite
-- Algebra.IsStandardSmooth.out
class IsStandardSmooth : Prop where
  out : Nonempty (SubmersivePresentation.{t, w} R S)
Arguments: {R : Type u} [CommRing R] {S : Type v} [CommRing S] [Algebra R S] [self : Algebra.IsStandardSmooth R S]
Goal: Nonempty (Algebra.SubmersivePresentation R S)
-- Algebra.IsStandardSmoothOfRelativeDimension.out
class IsStandardSmoothOfRelativeDimension : Prop where
  out : ∃ P : SubmersivePresentation.{t, w} R S, P.dimension = n
Arguments: {n : ℕ} {R : Type u} [CommRing R] {S : Type v} [CommRing S] [Algebra R S] [self : Algebra.IsStandardSmoothOfRelativeDimension n R S]
Goal: ∃ P, P.dimension = n
-- Algebra.IsStandardSmoothOfRelativeDimension.isStandardSmooth
lemma IsStandardSmoothOfRelativeDimension.isStandardSmooth
    [IsStandardSmoothOfRelativeDimension.{t, w} n R S] :
    IsStandardSmooth.{t, w} R S 
Arguments: (n : ℕ) {R : Type u} [CommRing R] {S : Type v} [CommRing S] [Algebra R S] [Algebra.IsStandardSmoothOfRelativeDimension n R S]
Goal: Algebra.IsStandardSmooth R S
-- RingHom.IsStandardSmoothOfRelativeDimension.isStandardSmooth
lemma IsStandardSmoothOfRelativeDimension.isStandardSmooth (f : R →+* S)
    (hf : IsStandardSmoothOfRelativeDimension.{t, w} n f) :
    IsStandardSmooth.{t, w} f 
Arguments: (n : ℕ) {R : Type u} [CommRing R] {S : Type v} [CommRing S] (f : R →+* S) (hf : RingHom.IsStandardSmoothOfRelativeDimension n f)
Goal: f.IsStandardSmooth
-- WittVector.bind₁_frobeniusPolyRat_wittPolynomial
theorem bind₁_frobeniusPolyRat_wittPolynomial (n : ℕ) :
    bind₁ (frobeniusPolyRat p) (wittPolynomial p ℚ n) = wittPolynomial p ℚ (n + 1) 
Arguments: (p : ℕ) [hp : Fact (Nat.Prime p)] (n : ℕ)
Goal: (MvPolynomial.bind₁ (WittVector.frobeniusPolyRat p)) (wittPolynomial p ℚ n) = wittPolynomial p ℚ (n + 1)
-- WittVector.frobeniusPolyAux_eq
theorem frobeniusPolyAux_eq (n : ℕ) :
    frobeniusPolyAux p n =
      X (n + 1) - ∑ i ∈ range n,
          ∑ j ∈ range (p ^ (n - i)),
            (X i ^ p) ^ (p ^ (n - i) - (j + 1)) * frobeniusPolyAux p i ^ (j + 1) *
              C ↑((p ^ (n - i)).choose (j + 1) / p ^ (n - i - v p ⟨j + 1, Nat.succ_pos j⟩) *
                ↑p ^ (j - v p ⟨j + 1, Nat.succ_pos j⟩) : ℕ) 
Arguments: (p : ℕ) [hp : Fact (Nat.Prime p)] (n : ℕ)
Goal: WittVector.frobeniusPolyAux p n =
  MvPolynomial.X (n + 1) -
    ∑ i ∈ Finset.range n,
      ∑ j ∈ Finset.range (p ^ (n - i)),
        (MvPolynomial.X i ^ p) ^ (p ^ (n - i) - (j + 1)) * WittVector.frobeniusPolyAux p i ^ (j + 1) *
          MvPolynomial.C
            ↑((p ^ (n - i)).choose (j + 1) / p ^ (n - i - WittVector.pnat_multiplicity p ⟨j + 1, ⋯⟩) *
                p ^ (j - WittVector.pnat_multiplicity p ⟨j + 1, ⋯⟩))
-- WittVector.map_frobeniusPoly.key₁
theorem map_frobeniusPoly.key₁ (n j : ℕ) (hj : j < p ^ n) :
    p ^ (n - v p ⟨j + 1, j.succ_pos⟩) ∣ (p ^ n).choose (j + 1) 
Arguments: (p : ℕ) [hp : Fact (Nat.Prime p)] (n j : ℕ) (hj : j < p ^ n)
Goal: p ^ (n - WittVector.pnat_multiplicity p ⟨j + 1, ⋯⟩) ∣ (p ^ n).choose (j + 1)
-- WittVector.map_frobeniusPoly.key₂
theorem map_frobeniusPoly.key₂ {n i j : ℕ} (hi : i ≤ n) (hj : j < p ^ (n - i)) :
    j - v p ⟨j + 1, j.succ_pos⟩ + n = i + j + (n - i - v p ⟨j + 1, j.succ_pos⟩) 
Arguments: (p : ℕ) [hp : Fact (Nat.Prime p)] {n i j : ℕ} (hi : i ≤ n) (hj : j < p ^ (n - i))
Goal: j - WittVector.pnat_multiplicity p ⟨j + 1, ⋯⟩ + n = i + j + (n - i - WittVector.pnat_multiplicity p ⟨j + 1, ⋯⟩)
-- WittVector.map_frobeniusPoly
theorem map_frobeniusPoly (n : ℕ) :
    MvPolynomial.map (Int.castRingHom ℚ) (frobeniusPoly p n) = frobeniusPolyRat p n 
Arguments: (p : ℕ) [hp : Fact (Nat.Prime p)] (n : ℕ)
Goal: (MvPolynomial.map (Int.castRingHom ℚ)) (WittVector.frobeniusPoly p n) = WittVector.frobeniusPolyRat p n
-- WittVector.frobeniusPoly_zmod
theorem frobeniusPoly_zmod (n : ℕ) :
    MvPolynomial.map (Int.castRingHom (ZMod p)) (frobeniusPoly p n) = X n ^ p 
Arguments: (p : ℕ) [hp : Fact (Nat.Prime p)] (n : ℕ)
Goal: (MvPolynomial.map (Int.castRingHom (ZMod p))) (WittVector.frobeniusPoly p n) = MvPolynomial.X n ^ p
-- WittVector.bind₁_frobeniusPoly_wittPolynomial
@[simp]
theorem bind₁_frobeniusPoly_wittPolynomial (n : ℕ) :
    bind₁ (frobeniusPoly p) (wittPolynomial p ℤ n) = wittPolynomial p ℤ (n + 1) 
Arguments: (p : ℕ) [hp : Fact (Nat.Prime p)] (n : ℕ)
Goal: (MvPolynomial.bind₁ (WittVector.frobeniusPoly p)) (wittPolynomial p ℤ n) = wittPolynomial p ℤ (n + 1)
-- WittVector.coeff_frobeniusFun
theorem coeff_frobeniusFun (x : 𝕎 R) (n : ℕ) :
    coeff (frobeniusFun x) n = MvPolynomial.aeval x.coeff (frobeniusPoly p n) 
Arguments: {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] (x : WittVector p R) (n : ℕ)
Goal: x.frobeniusFun.coeff n = (MvPolynomial.aeval x.coeff) (WittVector.frobeniusPoly p n)
-- WittVector.ghostComponent_frobeniusFun
@[ghost_simps]
theorem ghostComponent_frobeniusFun (n : ℕ) (x : 𝕎 R) :
    ghostComponent n (frobeniusFun x) = ghostComponent (n + 1) x 
Arguments: {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] (n : ℕ) (x : WittVector p R)
Goal: (WittVector.ghostComponent n) x.frobeniusFun = (WittVector.ghostComponent (n + 1)) x
-- WittVector.coeff_frobenius
theorem coeff_frobenius (x : 𝕎 R) (n : ℕ) :
    coeff (frobenius x) n = MvPolynomial.aeval x.coeff (frobeniusPoly p n) 
Arguments: {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] (x : WittVector p R) (n : ℕ)
Goal: (WittVector.frobenius x).coeff n = (MvPolynomial.aeval x.coeff) (WittVector.frobeniusPoly p n)
-- WittVector.ghostComponent_frobenius
@[ghost_simps]
theorem ghostComponent_frobenius (n : ℕ) (x : 𝕎 R) :
    ghostComponent n (frobenius x) = ghostComponent (n + 1) x 
Arguments: {p : ℕ} {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] (n : ℕ) (x : WittVector p R)
Goal: (WittVector.ghostComponent n) (WittVector.frobenius x) = (WittVector.ghostComponent (n + 1)) x
-- WittVector.coeff_frobenius_charP
@[simp]
theorem coeff_frobenius_charP (x : 𝕎 R) (n : ℕ) : coeff (frobenius x) n = x.coeff n ^ p 
Arguments: (p : ℕ) {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] [CharP R p] (x : WittVector p R) (n : ℕ)
Goal: (WittVector.frobenius x).coeff n = x.coeff n ^ p
-- WittVector.frobenius_eq_map_frobenius
theorem frobenius_eq_map_frobenius : @frobenius p R _ _ = map (_root_.frobenius R p) 
Arguments: (p : ℕ) {R : Type u_1} [hp : Fact (Nat.Prime p)] [CommRing R] [CharP R p]
Goal: WittVector.frobenius = WittVector.map (frobenius R p)
-- WittVector.frobenius_zmodp
@[simp]
theorem frobenius_zmodp (x : 𝕎 (ZMod p)) : frobenius x = x 
Arguments: (p : ℕ) [hp : Fact (Nat.Prime p)] (x : WittVector p (ZMod p))
Goal: WittVector.frobenius x = x
-- WittVector.frobeniusEquiv_apply
@[simps (config := .asFn)]
def frobeniusEquiv [PerfectRing R p] : WittVector p R ≃+* WittVector p R 
Arguments: (p : ℕ) (R : Type u_1) [hp : Fact (Nat.Prime p)] [CommRing R] [CharP R p] [PerfectRing R p]
Goal: ⇑(WittVector.frobeniusEquiv p R) = ⇑WittVector.frobenius
-- WittVector.frobeniusEquiv_symm_apply
@[simps (config := .asFn)]
def frobeniusEquiv [PerfectRing R p] : WittVector p R ≃+* WittVector p R 
Arguments: (p : ℕ) (R : Type u_1) [hp : Fact (Nat.Prime p)] [CommRing R] [CharP R p] [PerfectRing R p]
Goal: ⇑(WittVector.frobeniusEquiv p R).symm = ⇑(WittVector.map ↑(frobeniusEquiv R p).symm)
-- WittVector.frobenius_bijective
theorem frobenius_bijective [PerfectRing R p] :
    Function.Bijective (@WittVector.frobenius p R _ _) 
Arguments: (p : ℕ) (R : Type u_1) [hp : Fact (Nat.Prime p)] [CommRing R] [CharP R p] [PerfectRing R p]
Goal: Function.Bijective ⇑WittVector.frobenius
-- MvPowerSeries.ext
@[ext]
theorem ext {φ ψ} (h : ∀ n : σ →₀ ℕ, coeff R n φ = coeff R n ψ) : φ = ψ 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R] {φ ψ : MvPowerSeries σ R} (h : ∀ (n : σ →₀ ℕ), (MvPowerSeries.coeff R n) φ = (MvPowerSeries.coeff R n) ψ)
Goal: φ = ψ
-- MvPowerSeries.ext_iff
theorem ext_iff {φ ψ : MvPowerSeries σ R} : φ = ψ ↔ ∀ n : σ →₀ ℕ, coeff R n φ = coeff R n ψ 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R] {φ ψ : MvPowerSeries σ R}
Goal: φ = ψ ↔ ∀ (n : σ →₀ ℕ), (MvPowerSeries.coeff R n) φ = (MvPowerSeries.coeff R n) ψ
-- MvPowerSeries.monomial_def
theorem monomial_def [DecidableEq σ] (n : σ →₀ ℕ) :
    (monomial R n) = LinearMap.stdBasis R (fun _ ↦ R) n 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R] [DecidableEq σ] (n : σ →₀ ℕ)
Goal: MvPowerSeries.monomial R n = LinearMap.stdBasis R (fun x => R) n
-- MvPowerSeries.coeff_monomial
theorem coeff_monomial [DecidableEq σ] (m n : σ →₀ ℕ) (a : R) :
    coeff R m (monomial R n a) = if m = n then a else 0 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R] [DecidableEq σ] (m n : σ →₀ ℕ) (a : R)
Goal: (MvPowerSeries.coeff R m) ((MvPowerSeries.monomial R n) a) = if m = n then a else 0
-- MvPowerSeries.coeff_monomial_same
@[simp]
theorem coeff_monomial_same (n : σ →₀ ℕ) (a : R) : coeff R n (monomial R n a) = a 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R] (n : σ →₀ ℕ) (a : R)
Goal: (MvPowerSeries.coeff R n) ((MvPowerSeries.monomial R n) a) = a
-- MvPowerSeries.coeff_monomial_ne
theorem coeff_monomial_ne {m n : σ →₀ ℕ} (h : m ≠ n) (a : R) : coeff R m (monomial R n a) = 0 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R] {m n : σ →₀ ℕ} (h : m ≠ n) (a : R)
Goal: (MvPowerSeries.coeff R m) ((MvPowerSeries.monomial R n) a) = 0
-- MvPowerSeries.eq_of_coeff_monomial_ne_zero
theorem eq_of_coeff_monomial_ne_zero {m n : σ →₀ ℕ} {a : R} (h : coeff R m (monomial R n a) ≠ 0) :
    m = n 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R] {m n : σ →₀ ℕ} {a : R} (h : (MvPowerSeries.coeff R m) ((MvPowerSeries.monomial R n) a) ≠ 0)
Goal: m = n
-- MvPowerSeries.coeff_comp_monomial
@[simp]
theorem coeff_comp_monomial (n : σ →₀ ℕ) : (coeff R n).comp (monomial R n) = LinearMap.id 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R] (n : σ →₀ ℕ)
Goal: MvPowerSeries.coeff R n ∘ₗ MvPowerSeries.monomial R n = LinearMap.id
-- MvPowerSeries.coeff_zero
theorem coeff_zero (n : σ →₀ ℕ) : coeff R n (0 : MvPowerSeries σ R) = 0 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R] (n : σ →₀ ℕ)
Goal: (MvPowerSeries.coeff R n) 0 = 0
-- MvPowerSeries.coeff_one
theorem coeff_one [DecidableEq σ] : coeff R n (1 : MvPowerSeries σ R) = if n = 0 then 1 else 0 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R] (n : σ →₀ ℕ) [DecidableEq σ]
Goal: (MvPowerSeries.coeff R n) 1 = if n = 0 then 1 else 0
-- MvPowerSeries.coeff_zero_one
theorem coeff_zero_one : coeff R (0 : σ →₀ ℕ) 1 = 1 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R]
Goal: (MvPowerSeries.coeff R 0) 1 = 1
-- MvPowerSeries.monomial_zero_one
theorem monomial_zero_one : monomial R (0 : σ →₀ ℕ) 1 = 1 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R]
Goal: (MvPowerSeries.monomial R 0) 1 = 1
-- MvPowerSeries.coeff_mul
theorem coeff_mul [DecidableEq σ] :
    coeff R n (φ * ψ) = ∑ p ∈ antidiagonal n, coeff R p.1 φ * coeff R p.2 ψ 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R] (n : σ →₀ ℕ) (φ ψ : MvPowerSeries σ R) [DecidableEq σ]
Goal: (MvPowerSeries.coeff R n) (φ * ψ) =
  ∑ p ∈ Finset.antidiagonal n, (MvPowerSeries.coeff R p.1) φ * (MvPowerSeries.coeff R p.2) ψ
-- MvPowerSeries.zero_mul
protected theorem zero_mul : (0 : MvPowerSeries σ R) * φ = 0 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R] (φ : MvPowerSeries σ R)
Goal: 0 * φ = 0
-- MvPowerSeries.mul_zero
protected theorem mul_zero : φ * 0 = 0 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R] (φ : MvPowerSeries σ R)
Goal: φ * 0 = 0
-- MvPowerSeries.coeff_monomial_mul
theorem coeff_monomial_mul (a : R) :
    coeff R m (monomial R n a * φ) = if n ≤ m then a * coeff R (m - n) φ else 0 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R] (m n : σ →₀ ℕ) (φ : MvPowerSeries σ R) (a : R)
Goal: (MvPowerSeries.coeff R m) ((MvPowerSeries.monomial R n) a * φ) =
  if n ≤ m then a * (MvPowerSeries.coeff R (m - n)) φ else 0
-- MvPowerSeries.coeff_mul_monomial
theorem coeff_mul_monomial (a : R) :
    coeff R m (φ * monomial R n a) = if n ≤ m then coeff R (m - n) φ * a else 0 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R] (m n : σ →₀ ℕ) (φ : MvPowerSeries σ R) (a : R)
Goal: (MvPowerSeries.coeff R m) (φ * (MvPowerSeries.monomial R n) a) =
  if n ≤ m then (MvPowerSeries.coeff R (m - n)) φ * a else 0
-- MvPowerSeries.coeff_add_monomial_mul
theorem coeff_add_monomial_mul (a : R) :
    coeff R (m + n) (monomial R m a * φ) = a * coeff R n φ 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R] (m n : σ →₀ ℕ) (φ : MvPowerSeries σ R) (a : R)
Goal: (MvPowerSeries.coeff R (m + n)) ((MvPowerSeries.monomial R m) a * φ) = a * (MvPowerSeries.coeff R n) φ
-- MvPowerSeries.coeff_add_mul_monomial
theorem coeff_add_mul_monomial (a : R) :
    coeff R (m + n) (φ * monomial R n a) = coeff R m φ * a 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R] (m n : σ →₀ ℕ) (φ : MvPowerSeries σ R) (a : R)
Goal: (MvPowerSeries.coeff R (m + n)) (φ * (MvPowerSeries.monomial R n) a) = (MvPowerSeries.coeff R m) φ * a
-- MvPowerSeries.commute_monomial
@[simp]
theorem commute_monomial {a : R} {n} :
    Commute φ (monomial R n a) ↔ ∀ m, Commute (coeff R m φ) a 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R] (φ : MvPowerSeries σ R) {a : R} {n : σ →₀ ℕ}
Goal: Commute φ ((MvPowerSeries.monomial R n) a) ↔ ∀ (m : σ →₀ ℕ), Commute ((MvPowerSeries.coeff R m) φ) a
-- MvPowerSeries.one_mul
protected theorem one_mul : (1 : MvPowerSeries σ R) * φ = φ 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R] (φ : MvPowerSeries σ R)
Goal: 1 * φ = φ
-- MvPowerSeries.mul_one
protected theorem mul_one : φ * 1 = φ 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R] (φ : MvPowerSeries σ R)
Goal: φ * 1 = φ
-- MvPowerSeries.mul_add
protected theorem mul_add (φ₁ φ₂ φ₃ : MvPowerSeries σ R) : φ₁ * (φ₂ + φ₃) = φ₁ * φ₂ + φ₁ * φ₃ 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R] (φ₁ φ₂ φ₃ : MvPowerSeries σ R)
Goal: φ₁ * (φ₂ + φ₃) = φ₁ * φ₂ + φ₁ * φ₃
-- MvPowerSeries.add_mul
protected theorem add_mul (φ₁ φ₂ φ₃ : MvPowerSeries σ R) : (φ₁ + φ₂) * φ₃ = φ₁ * φ₃ + φ₂ * φ₃ 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R] (φ₁ φ₂ φ₃ : MvPowerSeries σ R)
Goal: (φ₁ + φ₂) * φ₃ = φ₁ * φ₃ + φ₂ * φ₃
-- MvPowerSeries.mul_assoc
protected theorem mul_assoc (φ₁ φ₂ φ₃ : MvPowerSeries σ R) : φ₁ * φ₂ * φ₃ = φ₁ * (φ₂ * φ₃) 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R] (φ₁ φ₂ φ₃ : MvPowerSeries σ R)
Goal: φ₁ * φ₂ * φ₃ = φ₁ * (φ₂ * φ₃)
-- MvPowerSeries.monomial_mul_monomial
theorem monomial_mul_monomial (m n : σ →₀ ℕ) (a b : R) :
    monomial R m a * monomial R n b = monomial R (m + n) (a * b) 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R] (m n : σ →₀ ℕ) (a b : R)
Goal: (MvPowerSeries.monomial R m) a * (MvPowerSeries.monomial R n) b = (MvPowerSeries.monomial R (m + n)) (a * b)
-- MvPowerSeries.monomial_zero_eq_C
@[simp]
theorem monomial_zero_eq_C : ⇑(monomial R (0 : σ →₀ ℕ)) = C σ R 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R]
Goal: ⇑(MvPowerSeries.monomial R 0) = ⇑(MvPowerSeries.C σ R)
-- MvPowerSeries.monomial_zero_eq_C_apply
theorem monomial_zero_eq_C_apply (a : R) : monomial R (0 : σ →₀ ℕ) a = C σ R a 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R] (a : R)
Goal: (MvPowerSeries.monomial R 0) a = (MvPowerSeries.C σ R) a
-- MvPowerSeries.coeff_C
theorem coeff_C [DecidableEq σ] (n : σ →₀ ℕ) (a : R) :
    coeff R n (C σ R a) = if n = 0 then a else 0 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R] [DecidableEq σ] (n : σ →₀ ℕ) (a : R)
Goal: (MvPowerSeries.coeff R n) ((MvPowerSeries.C σ R) a) = if n = 0 then a else 0
-- MvPowerSeries.coeff_zero_C
theorem coeff_zero_C (a : R) : coeff R (0 : σ →₀ ℕ) (C σ R a) = a 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R] (a : R)
Goal: (MvPowerSeries.coeff R 0) ((MvPowerSeries.C σ R) a) = a
-- MvPowerSeries.coeff_X
theorem coeff_X [DecidableEq σ] (n : σ →₀ ℕ) (s : σ) :
    coeff R n (X s : MvPowerSeries σ R) = if n = single s 1 then 1 else 0 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R] [DecidableEq σ] (n : σ →₀ ℕ) (s : σ)
Goal: (MvPowerSeries.coeff R n) (MvPowerSeries.X s) = if n = Finsupp.single s 1 then 1 else 0
-- MvPowerSeries.coeff_index_single_X
theorem coeff_index_single_X [DecidableEq σ] (s t : σ) :
    coeff R (single t 1) (X s : MvPowerSeries σ R) = if t = s then 1 else 0 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R] [DecidableEq σ] (s t : σ)
Goal: (MvPowerSeries.coeff R (Finsupp.single t 1)) (MvPowerSeries.X s) = if t = s then 1 else 0
-- MvPowerSeries.coeff_index_single_self_X
@[simp]
theorem coeff_index_single_self_X (s : σ) : coeff R (single s 1) (X s : MvPowerSeries σ R) = 1 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R] (s : σ)
Goal: (MvPowerSeries.coeff R (Finsupp.single s 1)) (MvPowerSeries.X s) = 1
-- MvPowerSeries.coeff_zero_X
theorem coeff_zero_X (s : σ) : coeff R (0 : σ →₀ ℕ) (X s : MvPowerSeries σ R) = 0 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R] (s : σ)
Goal: (MvPowerSeries.coeff R 0) (MvPowerSeries.X s) = 0
-- MvPowerSeries.commute_X
theorem commute_X (φ : MvPowerSeries σ R) (s : σ) : Commute φ (X s) 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R] (φ : MvPowerSeries σ R) (s : σ)
Goal: Commute φ (MvPowerSeries.X s)
-- MvPowerSeries.X_def
theorem X_def (s : σ) : X s = monomial R (single s 1) 1 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R] (s : σ)
Goal: MvPowerSeries.X s = (MvPowerSeries.monomial R (Finsupp.single s 1)) 1
-- MvPowerSeries.X_pow_eq
theorem X_pow_eq (s : σ) (n : ℕ) : (X s : MvPowerSeries σ R) ^ n = monomial R (single s n) 1 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R] (s : σ) (n : ℕ)
Goal: MvPowerSeries.X s ^ n = (MvPowerSeries.monomial R (Finsupp.single s n)) 1
-- MvPowerSeries.coeff_X_pow
theorem coeff_X_pow [DecidableEq σ] (m : σ →₀ ℕ) (s : σ) (n : ℕ) :
    coeff R m ((X s : MvPowerSeries σ R) ^ n) = if m = single s n then 1 else 0 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R] [DecidableEq σ] (m : σ →₀ ℕ) (s : σ) (n : ℕ)
Goal: (MvPowerSeries.coeff R m) (MvPowerSeries.X s ^ n) = if m = Finsupp.single s n then 1 else 0
-- MvPowerSeries.coeff_mul_C
@[simp]
theorem coeff_mul_C (n : σ →₀ ℕ) (φ : MvPowerSeries σ R) (a : R) :
    coeff R n (φ * C σ R a) = coeff R n φ * a 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R] (n : σ →₀ ℕ) (φ : MvPowerSeries σ R) (a : R)
Goal: (MvPowerSeries.coeff R n) (φ * (MvPowerSeries.C σ R) a) = (MvPowerSeries.coeff R n) φ * a
-- MvPowerSeries.coeff_C_mul
@[simp]
theorem coeff_C_mul (n : σ →₀ ℕ) (φ : MvPowerSeries σ R) (a : R) :
    coeff R n (C σ R a * φ) = a * coeff R n φ 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R] (n : σ →₀ ℕ) (φ : MvPowerSeries σ R) (a : R)
Goal: (MvPowerSeries.coeff R n) ((MvPowerSeries.C σ R) a * φ) = a * (MvPowerSeries.coeff R n) φ
-- MvPowerSeries.coeff_zero_mul_X
theorem coeff_zero_mul_X (φ : MvPowerSeries σ R) (s : σ) : coeff R (0 : σ →₀ ℕ) (φ * X s) = 0 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R] (φ : MvPowerSeries σ R) (s : σ)
Goal: (MvPowerSeries.coeff R 0) (φ * MvPowerSeries.X s) = 0
-- MvPowerSeries.coeff_zero_X_mul
theorem coeff_zero_X_mul (φ : MvPowerSeries σ R) (s : σ) : coeff R (0 : σ →₀ ℕ) (X s * φ) = 0 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R] (φ : MvPowerSeries σ R) (s : σ)
Goal: (MvPowerSeries.coeff R 0) (MvPowerSeries.X s * φ) = 0
-- MvPowerSeries.coeff_zero_eq_constantCoeff
@[simp]
theorem coeff_zero_eq_constantCoeff : ⇑(coeff R (0 : σ →₀ ℕ)) = constantCoeff σ R 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R]
Goal: ⇑(MvPowerSeries.coeff R 0) = ⇑(MvPowerSeries.constantCoeff σ R)
-- MvPowerSeries.coeff_zero_eq_constantCoeff_apply
theorem coeff_zero_eq_constantCoeff_apply (φ : MvPowerSeries σ R) :
    coeff R (0 : σ →₀ ℕ) φ = constantCoeff σ R φ 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R] (φ : MvPowerSeries σ R)
Goal: (MvPowerSeries.coeff R 0) φ = (MvPowerSeries.constantCoeff σ R) φ
-- MvPowerSeries.constantCoeff_C
@[simp]
theorem constantCoeff_C (a : R) : constantCoeff σ R (C σ R a) = a 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R] (a : R)
Goal: (MvPowerSeries.constantCoeff σ R) ((MvPowerSeries.C σ R) a) = a
-- MvPowerSeries.constantCoeff_comp_C
@[simp]
theorem constantCoeff_comp_C : (constantCoeff σ R).comp (C σ R) = RingHom.id R 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R]
Goal: (MvPowerSeries.constantCoeff σ R).comp (MvPowerSeries.C σ R) = RingHom.id R
-- MvPowerSeries.constantCoeff_zero
theorem constantCoeff_zero : constantCoeff σ R 0 = 0 
Arguments: {σ : Type u_1} {R : Type u_2} [Semiring R]
Goal: (MvPowerSeries.constantCoeff σ R) 0 = 0