ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Module.Finite.finite_basis
lemma Module.Finite.finite_basis {R M} [Ring R] [Nontrivial R] [AddCommGroup M] [Module R M]
    {ι} [Module.Finite R M] (b : Basis ι R M) :
    _root_.Finite ι 
Arguments: {R : Type u_6} {M : Type u_7} [Ring R] [Nontrivial R] [AddCommGroup M] [Module R M] {ι : Type u_8} [Module.Finite R M] (b : Basis ι R M)
Goal: Finite ι
-- RingHom.Finite.id
theorem id : Finite (RingHom.id A) 
Arguments: (A : Type u_1) [CommRing A]
Goal: (RingHom.id A).Finite
-- RingHom.Finite.of_surjective
theorem of_surjective (f : A →+* B) (hf : Surjective f) : f.Finite 
Arguments: {A : Type u_1} {B : Type u_2} [CommRing A] [CommRing B] (f : A →+* B) (hf : Function.Surjective ⇑f)
Goal: f.Finite
-- RingHom.Finite.comp
theorem comp {g : B →+* C} {f : A →+* B} (hg : g.Finite) (hf : f.Finite) : (g.comp f).Finite 
Arguments: {A : Type u_1} {B : Type u_2} {C : Type u_3} [CommRing A] [CommRing B] [CommRing C] {g : B →+* C} {f : A →+* B} (hg : g.Finite) (hf : f.Finite)
Goal: (g.comp f).Finite
-- RingHom.Finite.of_comp_finite
theorem of_comp_finite {f : A →+* B} {g : B →+* C} (h : (g.comp f).Finite) : g.Finite 
Arguments: {A : Type u_1} {B : Type u_2} {C : Type u_3} [CommRing A] [CommRing B] [CommRing C] {f : A →+* B} {g : B →+* C} (h : (g.comp f).Finite)
Goal: g.Finite
-- AlgHom.Finite.id
theorem id : Finite (AlgHom.id R A) 
Arguments: (R : Type u_1) (A : Type u_2) [CommRing R] [CommRing A] [Algebra R A]
Goal: (AlgHom.id R A).Finite
-- AlgHom.Finite.comp
theorem comp {g : B →ₐ[R] C} {f : A →ₐ[R] B} (hg : g.Finite) (hf : f.Finite) : (g.comp f).Finite 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [CommRing R] [CommRing A] [CommRing B] [CommRing C] [Algebra R A] [Algebra R B] [Algebra R C] {g : B →ₐ[R] C} {f : A →ₐ[R] B} (hg : g.Finite) (hf : f.Finite)
Goal: (g.comp f).Finite
-- AlgHom.Finite.of_surjective
theorem of_surjective (f : A →ₐ[R] B) (hf : Surjective f) : f.Finite 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommRing R] [CommRing A] [CommRing B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) (hf : Function.Surjective ⇑f)
Goal: f.Finite
-- AlgHom.Finite.of_comp_finite
theorem of_comp_finite {f : A →ₐ[R] B} {g : B →ₐ[R] C} (h : (g.comp f).Finite) : g.Finite 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [CommRing R] [CommRing A] [CommRing B] [CommRing C] [Algebra R A] [Algebra R B] [Algebra R C] {f : A →ₐ[R] B} {g : B →ₐ[R] C} (h : (g.comp f).Finite)
Goal: g.Finite
-- PowerSeries.coeff_def
theorem coeff_def {s : Unit →₀ ℕ} {n : ℕ} (h : s () = n) : coeff R n = MvPowerSeries.coeff R s 
Arguments: {R : Type u_1} [Semiring R] {s : Unit →₀ ℕ} {n : ℕ} (h : s () = n)
Goal: PowerSeries.coeff R n = MvPowerSeries.coeff R s
-- PowerSeries.ext
@[ext]
theorem ext {φ ψ : R⟦X⟧} (h : ∀ n, coeff R n φ = coeff R n ψ) : φ = ψ 
Arguments: {R : Type u_1} [Semiring R] {φ ψ : PowerSeries R} (h : ∀ (n : ℕ), (PowerSeries.coeff R n) φ = (PowerSeries.coeff R n) ψ)
Goal: φ = ψ
-- PowerSeries.ext_iff
theorem ext_iff {φ ψ : R⟦X⟧} : φ = ψ ↔ ∀ n, coeff R n φ = coeff R n ψ 
Arguments: {R : Type u_1} [Semiring R] {φ ψ : PowerSeries R}
Goal: φ = ψ ↔ ∀ (n : ℕ), (PowerSeries.coeff R n) φ = (PowerSeries.coeff R n) ψ
-- PowerSeries.coeff_mk
@[simp]
theorem coeff_mk (n : ℕ) (f : ℕ → R) : coeff R n (mk f) = f n 
Arguments: {R : Type u_1} [Semiring R] (n : ℕ) (f : ℕ → R)
Goal: (PowerSeries.coeff R n) (PowerSeries.mk f) = f n
-- PowerSeries.coeff_monomial
theorem coeff_monomial (m n : ℕ) (a : R) : coeff R m (monomial R n a) = if m = n then a else 0 
Arguments: {R : Type u_1} [Semiring R] (m n : ℕ) (a : R)
Goal: (PowerSeries.coeff R m) ((PowerSeries.monomial R n) a) = if m = n then a else 0
-- PowerSeries.monomial_eq_mk
theorem monomial_eq_mk (n : ℕ) (a : R) : monomial R n a = mk fun m => if m = n then a else 0 
Arguments: {R : Type u_1} [Semiring R] (n : ℕ) (a : R)
Goal: (PowerSeries.monomial R n) a = PowerSeries.mk fun m => if m = n then a else 0
-- PowerSeries.coeff_monomial_same
@[simp]
theorem coeff_monomial_same (n : ℕ) (a : R) : coeff R n (monomial R n a) = a 
Arguments: {R : Type u_1} [Semiring R] (n : ℕ) (a : R)
Goal: (PowerSeries.coeff R n) ((PowerSeries.monomial R n) a) = a
-- PowerSeries.coeff_comp_monomial
@[simp]
theorem coeff_comp_monomial (n : ℕ) : (coeff R n).comp (monomial R n) = LinearMap.id 
Arguments: {R : Type u_1} [Semiring R] (n : ℕ)
Goal: PowerSeries.coeff R n ∘ₗ PowerSeries.monomial R n = LinearMap.id
-- PowerSeries.commute_X
theorem commute_X (φ : R⟦X⟧) : Commute φ X 
Arguments: {R : Type u_1} [Semiring R] (φ : PowerSeries R)
Goal: Commute φ PowerSeries.X
-- PowerSeries.coeff_zero_eq_constantCoeff
@[simp]
theorem coeff_zero_eq_constantCoeff : ⇑(coeff R 0) = constantCoeff R 
Arguments: {R : Type u_1} [Semiring R]
Goal: ⇑(PowerSeries.coeff R 0) = ⇑(PowerSeries.constantCoeff R)
-- PowerSeries.coeff_zero_eq_constantCoeff_apply
theorem coeff_zero_eq_constantCoeff_apply (φ : R⟦X⟧) : coeff R 0 φ = constantCoeff R φ 
Arguments: {R : Type u_1} [Semiring R] (φ : PowerSeries R)
Goal: (PowerSeries.coeff R 0) φ = (PowerSeries.constantCoeff R) φ
-- PowerSeries.monomial_zero_eq_C
@[simp]
theorem monomial_zero_eq_C : ⇑(monomial R 0) = C R 
Arguments: {R : Type u_1} [Semiring R]
Goal: ⇑(PowerSeries.monomial R 0) = ⇑(PowerSeries.C R)
-- PowerSeries.monomial_zero_eq_C_apply
theorem monomial_zero_eq_C_apply (a : R) : monomial R 0 a = C R a 
Arguments: {R : Type u_1} [Semiring R] (a : R)
Goal: (PowerSeries.monomial R 0) a = (PowerSeries.C R) a
-- PowerSeries.coeff_C
theorem coeff_C (n : ℕ) (a : R) : coeff R n (C R a : R⟦X⟧) = if n = 0 then a else 0 
Arguments: {R : Type u_1} [Semiring R] (n : ℕ) (a : R)
Goal: (PowerSeries.coeff R n) ((PowerSeries.C R) a) = if n = 0 then a else 0
-- PowerSeries.coeff_zero_C
@[simp]
theorem coeff_zero_C (a : R) : coeff R 0 (C R a) = a 
Arguments: {R : Type u_1} [Semiring R] (a : R)
Goal: (PowerSeries.coeff R 0) ((PowerSeries.C R) a) = a
-- PowerSeries.coeff_ne_zero_C
theorem coeff_ne_zero_C {a : R} {n : ℕ} (h : n ≠ 0) : coeff R n (C R a) = 0 
Arguments: {R : Type u_1} [Semiring R] {a : R} {n : ℕ} (h : n ≠ 0)
Goal: (PowerSeries.coeff R n) ((PowerSeries.C R) a) = 0
-- PowerSeries.coeff_succ_C
@[simp]
theorem coeff_succ_C {a : R} {n : ℕ} : coeff R (n + 1) (C R a) = 0 
Arguments: {R : Type u_1} [Semiring R] {a : R} {n : ℕ}
Goal: (PowerSeries.coeff R (n + 1)) ((PowerSeries.C R) a) = 0
-- PowerSeries.C_injective
theorem C_injective : Function.Injective (C R) 
Arguments: {R : Type u_1} [Semiring R]
Goal: Function.Injective ⇑(PowerSeries.C R)
-- PowerSeries.subsingleton_iff
protected theorem subsingleton_iff : Subsingleton R⟦X⟧ ↔ Subsingleton R 
Arguments: {R : Type u_1} [Semiring R]
Goal: Subsingleton (PowerSeries R) ↔ Subsingleton R
-- PowerSeries.X_eq
theorem X_eq : (X : R⟦X⟧) = monomial R 1 1 
Arguments: {R : Type u_1} [Semiring R]
Goal: PowerSeries.X = (PowerSeries.monomial R 1) 1
-- PowerSeries.coeff_X
theorem coeff_X (n : ℕ) : coeff R n (X : R⟦X⟧) = if n = 1 then 1 else 0 
Arguments: {R : Type u_1} [Semiring R] (n : ℕ)
Goal: (PowerSeries.coeff R n) PowerSeries.X = if n = 1 then 1 else 0
-- PowerSeries.coeff_zero_X
@[simp]
theorem coeff_zero_X : coeff R 0 (X : R⟦X⟧) = 0 
Arguments: {R : Type u_1} [Semiring R]
Goal: (PowerSeries.coeff R 0) PowerSeries.X = 0
-- PowerSeries.coeff_one_X
@[simp]
theorem coeff_one_X : coeff R 1 (X : R⟦X⟧) = 1 
Arguments: {R : Type u_1} [Semiring R]
Goal: (PowerSeries.coeff R 1) PowerSeries.X = 1
-- PowerSeries.X_ne_zero
@[simp]
theorem X_ne_zero [Nontrivial R] : (X : R⟦X⟧) ≠ 0 
Arguments: {R : Type u_1} [Semiring R] [Nontrivial R]
Goal: PowerSeries.X ≠ 0
-- PowerSeries.X_pow_eq
theorem X_pow_eq (n : ℕ) : (X : R⟦X⟧) ^ n = monomial R n 1 
Arguments: {R : Type u_1} [Semiring R] (n : ℕ)
Goal: PowerSeries.X ^ n = (PowerSeries.monomial R n) 1
-- PowerSeries.coeff_X_pow
theorem coeff_X_pow (m n : ℕ) : coeff R m ((X : R⟦X⟧) ^ n) = if m = n then 1 else 0 
Arguments: {R : Type u_1} [Semiring R] (m n : ℕ)
Goal: (PowerSeries.coeff R m) (PowerSeries.X ^ n) = if m = n then 1 else 0
-- PowerSeries.coeff_X_pow_self
@[simp]
theorem coeff_X_pow_self (n : ℕ) : coeff R n ((X : R⟦X⟧) ^ n) = 1 
Arguments: {R : Type u_1} [Semiring R] (n : ℕ)
Goal: (PowerSeries.coeff R n) (PowerSeries.X ^ n) = 1
-- PowerSeries.coeff_one
@[simp]
theorem coeff_one (n : ℕ) : coeff R n (1 : R⟦X⟧) = if n = 0 then 1 else 0 
Arguments: {R : Type u_1} [Semiring R] (n : ℕ)
Goal: (PowerSeries.coeff R n) 1 = if n = 0 then 1 else 0
-- PowerSeries.coeff_zero_one
theorem coeff_zero_one : coeff R 0 (1 : R⟦X⟧) = 1 
Arguments: {R : Type u_1} [Semiring R]
Goal: (PowerSeries.coeff R 0) 1 = 1
-- PowerSeries.coeff_mul
theorem coeff_mul (n : ℕ) (φ ψ : R⟦X⟧) :
    coeff R n (φ * ψ) = ∑ p ∈ antidiagonal n, coeff R p.1 φ * coeff R p.2 ψ 
Arguments: {R : Type u_1} [Semiring R] (n : ℕ) (φ ψ : PowerSeries R)
Goal: (PowerSeries.coeff R n) (φ * ψ) = ∑ p ∈ Finset.antidiagonal n, (PowerSeries.coeff R p.1) φ * (PowerSeries.coeff R p.2) ψ
-- PowerSeries.coeff_mul_C
@[simp]
theorem coeff_mul_C (n : ℕ) (φ : R⟦X⟧) (a : R) : coeff R n (φ * C R a) = coeff R n φ * a 
Arguments: {R : Type u_1} [Semiring R] (n : ℕ) (φ : PowerSeries R) (a : R)
Goal: (PowerSeries.coeff R n) (φ * (PowerSeries.C R) a) = (PowerSeries.coeff R n) φ * a
-- PowerSeries.coeff_C_mul
@[simp]
theorem coeff_C_mul (n : ℕ) (φ : R⟦X⟧) (a : R) : coeff R n (C R a * φ) = a * coeff R n φ 
Arguments: {R : Type u_1} [Semiring R] (n : ℕ) (φ : PowerSeries R) (a : R)
Goal: (PowerSeries.coeff R n) ((PowerSeries.C R) a * φ) = a * (PowerSeries.coeff R n) φ
-- PowerSeries.coeff_smul
@[simp]
theorem coeff_smul {S : Type*} [Semiring S] [Module R S] (n : ℕ) (φ : PowerSeries S) (a : R) :
    coeff S n (a • φ) = a • coeff S n φ 
Arguments: {R : Type u_1} [Semiring R] {S : Type u_2} [Semiring S] [Module R S] (n : ℕ) (φ : PowerSeries S) (a : R)
Goal: (PowerSeries.coeff S n) (a • φ) = a • (PowerSeries.coeff S n) φ
-- PowerSeries.constantCoeff_smul
@[simp]
theorem constantCoeff_smul {S : Type*} [Semiring S] [Module R S] (φ : PowerSeries S) (a : R) :
    constantCoeff S (a • φ) = a • constantCoeff S φ 
Arguments: {R : Type u_1} [Semiring R] {S : Type u_2} [Semiring S] [Module R S] (φ : PowerSeries S) (a : R)
Goal: (PowerSeries.constantCoeff S) (a • φ) = a • (PowerSeries.constantCoeff S) φ
-- PowerSeries.smul_eq_C_mul
theorem smul_eq_C_mul (f : R⟦X⟧) (a : R) : a • f = C R a * f 
Arguments: {R : Type u_1} [Semiring R] (f : PowerSeries R) (a : R)
Goal: a • f = (PowerSeries.C R) a * f
-- PowerSeries.coeff_succ_mul_X
@[simp]
theorem coeff_succ_mul_X (n : ℕ) (φ : R⟦X⟧) : coeff R (n + 1) (φ * X) = coeff R n φ 
Arguments: {R : Type u_1} [Semiring R] (n : ℕ) (φ : PowerSeries R)
Goal: (PowerSeries.coeff R (n + 1)) (φ * PowerSeries.X) = (PowerSeries.coeff R n) φ
-- PowerSeries.coeff_succ_X_mul
@[simp]
theorem coeff_succ_X_mul (n : ℕ) (φ : R⟦X⟧) : coeff R (n + 1) (X * φ) = coeff R n φ 
Arguments: {R : Type u_1} [Semiring R] (n : ℕ) (φ : PowerSeries R)
Goal: (PowerSeries.coeff R (n + 1)) (PowerSeries.X * φ) = (PowerSeries.coeff R n) φ
-- PowerSeries.constantCoeff_C
@[simp]
theorem constantCoeff_C (a : R) : constantCoeff R (C R a) = a 
Arguments: {R : Type u_1} [Semiring R] (a : R)
Goal: (PowerSeries.constantCoeff R) ((PowerSeries.C R) a) = a
-- PowerSeries.constantCoeff_comp_C
@[simp]
theorem constantCoeff_comp_C : (constantCoeff R).comp (C R) = RingHom.id R 
Arguments: {R : Type u_1} [Semiring R]
Goal: (PowerSeries.constantCoeff R).comp (PowerSeries.C R) = RingHom.id R
-- PowerSeries.constantCoeff_zero
theorem constantCoeff_zero : constantCoeff R 0 = 0 
Arguments: {R : Type u_1} [Semiring R]
Goal: (PowerSeries.constantCoeff R) 0 = 0
-- PowerSeries.constantCoeff_one
theorem constantCoeff_one : constantCoeff R 1 = 1 
Arguments: {R : Type u_1} [Semiring R]
Goal: (PowerSeries.constantCoeff R) 1 = 1
-- PowerSeries.constantCoeff_X
@[simp]
theorem constantCoeff_X : constantCoeff R X = 0 
Arguments: {R : Type u_1} [Semiring R]
Goal: (PowerSeries.constantCoeff R) PowerSeries.X = 0
-- PowerSeries.constantCoeff_mk
@[simp]
theorem constantCoeff_mk {f : ℕ → R} : constantCoeff R (mk f) = f 0 
Arguments: {R : Type u_1} [Semiring R] {f : ℕ → R}
Goal: (PowerSeries.constantCoeff R) (PowerSeries.mk f) = f 0
-- PowerSeries.coeff_zero_mul_X
theorem coeff_zero_mul_X (φ : R⟦X⟧) : coeff R 0 (φ * X) = 0 
Arguments: {R : Type u_1} [Semiring R] (φ : PowerSeries R)
Goal: (PowerSeries.coeff R 0) (φ * PowerSeries.X) = 0
-- PowerSeries.coeff_zero_X_mul
theorem coeff_zero_X_mul (φ : R⟦X⟧) : coeff R 0 (X * φ) = 0 
Arguments: {R : Type u_1} [Semiring R] (φ : PowerSeries R)
Goal: (PowerSeries.coeff R 0) (PowerSeries.X * φ) = 0
-- PowerSeries.constantCoeff_surj
theorem constantCoeff_surj : Function.Surjective (constantCoeff R) 
Arguments: {R : Type u_1} [Semiring R]
Goal: Function.Surjective ⇑(PowerSeries.constantCoeff R)
-- PowerSeries.coeff_C_mul_X_pow
theorem coeff_C_mul_X_pow (x : R) (k n : ℕ) :
    coeff R n (C R x * X ^ k : R⟦X⟧) = if n = k then x else 0 
Arguments: {R : Type u_1} [Semiring R] (x : R) (k n : ℕ)
Goal: (PowerSeries.coeff R n) ((PowerSeries.C R) x * PowerSeries.X ^ k) = if n = k then x else 0
-- PowerSeries.coeff_mul_X_pow
@[simp]
theorem coeff_mul_X_pow (p : R⟦X⟧) (n d : ℕ) :
    coeff R (d + n) (p * X ^ n) = coeff R d p 
Arguments: {R : Type u_1} [Semiring R] (p : PowerSeries R) (n d : ℕ)
Goal: (PowerSeries.coeff R (d + n)) (p * PowerSeries.X ^ n) = (PowerSeries.coeff R d) p
-- PowerSeries.coeff_X_pow_mul
@[simp]
theorem coeff_X_pow_mul (p : R⟦X⟧) (n d : ℕ) :
    coeff R (d + n) (X ^ n * p) = coeff R d p 
Arguments: {R : Type u_1} [Semiring R] (p : PowerSeries R) (n d : ℕ)
Goal: (PowerSeries.coeff R (d + n)) (PowerSeries.X ^ n * p) = (PowerSeries.coeff R d) p
-- PowerSeries.coeff_mul_X_pow'
theorem coeff_mul_X_pow' (p : R⟦X⟧) (n d : ℕ) :
    coeff R d (p * X ^ n) = ite (n ≤ d) (coeff R (d - n) p) 0 
Arguments: {R : Type u_1} [Semiring R] (p : PowerSeries R) (n d : ℕ)
Goal: (PowerSeries.coeff R d) (p * PowerSeries.X ^ n) = if n ≤ d then (PowerSeries.coeff R (d - n)) p else 0
-- PowerSeries.coeff_X_pow_mul'
theorem coeff_X_pow_mul' (p : R⟦X⟧) (n d : ℕ) :
    coeff R d (X ^ n * p) = ite (n ≤ d) (coeff R (d - n) p) 0 
Arguments: {R : Type u_1} [Semiring R] (p : PowerSeries R) (n d : ℕ)
Goal: (PowerSeries.coeff R d) (PowerSeries.X ^ n * p) = if n ≤ d then (PowerSeries.coeff R (d - n)) p else 0
-- PowerSeries.isUnit_constantCoeff
theorem isUnit_constantCoeff (φ : R⟦X⟧) (h : IsUnit φ) : IsUnit (constantCoeff R φ) 
Arguments: {R : Type u_1} [Semiring R] (φ : PowerSeries R) (h : IsUnit φ)
Goal: IsUnit ((PowerSeries.constantCoeff R) φ)
-- PowerSeries.eq_shift_mul_X_add_const
theorem eq_shift_mul_X_add_const (φ : R⟦X⟧) :
    φ = (mk fun p => coeff R (p + 1) φ) * X + C R (constantCoeff R φ) 
Arguments: {R : Type u_1} [Semiring R] (φ : PowerSeries R)
Goal: φ =
  (PowerSeries.mk fun p => (PowerSeries.coeff R (p + 1)) φ) * PowerSeries.X +
    (PowerSeries.C R) ((PowerSeries.constantCoeff R) φ)
-- PowerSeries.eq_X_mul_shift_add_const
theorem eq_X_mul_shift_add_const (φ : R⟦X⟧) :
    φ = (X * mk fun p => coeff R (p + 1) φ) + C R (constantCoeff R φ) 
Arguments: {R : Type u_1} [Semiring R] (φ : PowerSeries R)
Goal: φ =
  (PowerSeries.X * PowerSeries.mk fun p => (PowerSeries.coeff R (p + 1)) φ) +
    (PowerSeries.C R) ((PowerSeries.constantCoeff R) φ)
-- PowerSeries.map_id
@[simp]
theorem map_id : (map (RingHom.id R) : R⟦X⟧ → R⟦X⟧) = id 
Arguments: {R : Type u_1} [Semiring R]
Goal: ⇑(PowerSeries.map (RingHom.id R)) = id
-- PowerSeries.map_comp
theorem map_comp : map (g.comp f) = (map g).comp (map f) 
Arguments: {R : Type u_1} [Semiring R] {S : Type u_2} {T : Type u_3} [Semiring S] [Semiring T] (f : R →+* S) (g : S →+* T)
Goal: PowerSeries.map (g.comp f) = (PowerSeries.map g).comp (PowerSeries.map f)
-- PowerSeries.coeff_map
@[simp]
theorem coeff_map (n : ℕ) (φ : R⟦X⟧) : coeff S n (map f φ) = f (coeff R n φ) 
Arguments: {R : Type u_1} [Semiring R] {S : Type u_2} [Semiring S] (f : R →+* S) (n : ℕ) (φ : PowerSeries R)
Goal: (PowerSeries.coeff S n) ((PowerSeries.map f) φ) = f ((PowerSeries.coeff R n) φ)
-- PowerSeries.map_C
@[simp]
theorem map_C (r : R) : map f (C _ r) = C _ (f r) 
Arguments: {R : Type u_1} [Semiring R] {S : Type u_2} [Semiring S] (f : R →+* S) (r : R)
Goal: (PowerSeries.map f) ((PowerSeries.C R) r) = (PowerSeries.C S) (f r)
-- PowerSeries.map_X
@[simp]
theorem map_X : map f X = X 
Arguments: {R : Type u_1} [Semiring R] {S : Type u_2} [Semiring S] (f : R →+* S)
Goal: (PowerSeries.map f) PowerSeries.X = PowerSeries.X
-- PowerSeries.X_pow_dvd_iff
theorem X_pow_dvd_iff {n : ℕ} {φ : R⟦X⟧} :
    (X : R⟦X⟧) ^ n ∣ φ ↔ ∀ m, m < n → coeff R m φ = 0 
Arguments: {R : Type u_1} [Semiring R] {n : ℕ} {φ : PowerSeries R}
Goal: PowerSeries.X ^ n ∣ φ ↔ ∀ m < n, (PowerSeries.coeff R m) φ = 0
-- PowerSeries.X_dvd_iff
theorem X_dvd_iff {φ : R⟦X⟧} : (X : R⟦X⟧) ∣ φ ↔ constantCoeff R φ = 0 
Arguments: {R : Type u_1} [Semiring R] {φ : PowerSeries R}
Goal: PowerSeries.X ∣ φ ↔ (PowerSeries.constantCoeff R) φ = 0
-- PowerSeries.coeff_rescale
@[simp]
theorem coeff_rescale (f : R⟦X⟧) (a : R) (n : ℕ) :
    coeff R n (rescale a f) = a ^ n * coeff R n f 
Arguments: {R : Type u_1} [CommSemiring R] (f : PowerSeries R) (a : R) (n : ℕ)
Goal: (PowerSeries.coeff R n) ((PowerSeries.rescale a) f) = a ^ n * (PowerSeries.coeff R n) f
-- PowerSeries.rescale_zero
@[simp]
theorem rescale_zero : rescale 0 = (C R).comp (constantCoeff R) 
Arguments: {R : Type u_1} [CommSemiring R]
Goal: PowerSeries.rescale 0 = (PowerSeries.C R).comp (PowerSeries.constantCoeff R)
-- PowerSeries.rescale_zero_apply
theorem rescale_zero_apply : rescale 0 X = C R (constantCoeff R X) 
Arguments: {R : Type u_1} [CommSemiring R]
Goal: (PowerSeries.rescale 0) PowerSeries.X = (PowerSeries.C R) ((PowerSeries.constantCoeff R) PowerSeries.X)
-- PowerSeries.rescale_one
@[simp]
theorem rescale_one : rescale 1 = RingHom.id R⟦X⟧ 
Arguments: {R : Type u_1} [CommSemiring R]
Goal: PowerSeries.rescale 1 = RingHom.id (PowerSeries R)
-- PowerSeries.rescale_mk
theorem rescale_mk (f : ℕ → R) (a : R) : rescale a (mk f) = mk fun n : ℕ => a ^ n * f n 
Arguments: {R : Type u_1} [CommSemiring R] (f : ℕ → R) (a : R)
Goal: (PowerSeries.rescale a) (PowerSeries.mk f) = PowerSeries.mk fun n => a ^ n * f n
-- PowerSeries.rescale_rescale
theorem rescale_rescale (f : R⟦X⟧) (a b : R) :
    rescale b (rescale a f) = rescale (a * b) f 
Arguments: {R : Type u_1} [CommSemiring R] (f : PowerSeries R) (a b : R)
Goal: (PowerSeries.rescale b) ((PowerSeries.rescale a) f) = (PowerSeries.rescale (a * b)) f
-- PowerSeries.rescale_mul
theorem rescale_mul (a b : R) : rescale (a * b) = (rescale b).comp (rescale a) 
Arguments: {R : Type u_1} [CommSemiring R] (a b : R)
Goal: PowerSeries.rescale (a * b) = (PowerSeries.rescale b).comp (PowerSeries.rescale a)
-- PowerSeries.coeff_prod
theorem coeff_prod (f : ι → PowerSeries R) (d : ℕ) (s : Finset ι) :
    coeff R d (∏ j ∈ s, f j) = ∑ l ∈ finsuppAntidiag s d, ∏ i ∈ s, coeff R (l i) (f i) 
Arguments: {R : Type u_2} [CommSemiring R] {ι : Type u_3} [DecidableEq ι] (f : ι → PowerSeries R) (d : ℕ) (s : Finset ι)
Goal: (PowerSeries.coeff R d) (∏ j ∈ s, f j) = ∑ l ∈ s.finsuppAntidiag d, ∏ i ∈ s, (PowerSeries.coeff R (l i)) (f i)
-- PowerSeries.coeff_pow
lemma coeff_pow (k n : ℕ) (φ : R⟦X⟧) :
    coeff R n (φ ^ k) = ∑ l ∈ finsuppAntidiag (range k) n, ∏ i ∈ range k, coeff R (l i) φ 
Arguments: {R : Type u_2} [CommSemiring R] (k n : ℕ) (φ : PowerSeries R)
Goal: (PowerSeries.coeff R n) (φ ^ k) =
  ∑ l ∈ (Finset.range k).finsuppAntidiag n, ∏ i ∈ Finset.range k, (PowerSeries.coeff R (l i)) φ
-- PowerSeries.coeff_one_mul
lemma coeff_one_mul (φ ψ : R⟦X⟧) : coeff R 1 (φ * ψ) =
    coeff R 1 φ * constantCoeff R ψ + coeff R 1 ψ * constantCoeff R φ 
Arguments: {R : Type u_2} [CommSemiring R] (φ ψ : PowerSeries R)
Goal: (PowerSeries.coeff R 1) (φ * ψ) =
  (PowerSeries.coeff R 1) φ * (PowerSeries.constantCoeff R) ψ +
    (PowerSeries.coeff R 1) ψ * (PowerSeries.constantCoeff R) φ
-- PowerSeries.coeff_one_pow
lemma coeff_one_pow (n : ℕ) (φ : R⟦X⟧) :
    coeff R 1 (φ ^ n) = n * coeff R 1 φ * (constantCoeff R φ) ^ (n - 1) 
Arguments: {R : Type u_2} [CommSemiring R] (n : ℕ) (φ : PowerSeries R)
Goal: (PowerSeries.coeff R 1) (φ ^ n) = ↑n * (PowerSeries.coeff R 1) φ * (PowerSeries.constantCoeff R) φ ^ (n - 1)
-- PowerSeries.not_isField
theorem not_isField : ¬IsField A⟦X⟧ 
Arguments: {A : Type u_2} [CommRing A]
Goal: ¬IsField (PowerSeries A)
-- PowerSeries.rescale_X
@[simp]
theorem rescale_X (a : A) : rescale a X = C A a * X 
Arguments: {A : Type u_2} [CommRing A] (a : A)
Goal: (PowerSeries.rescale a) PowerSeries.X = (PowerSeries.C A) a * PowerSeries.X
-- PowerSeries.rescale_neg_one_X
theorem rescale_neg_one_X : rescale (-1 : A) X = -X 
Arguments: {A : Type u_2} [CommRing A]
Goal: (PowerSeries.rescale (-1)) PowerSeries.X = -PowerSeries.X
-- PowerSeries.evalNegHom_X
@[simp]
theorem evalNegHom_X : evalNegHom (X : A⟦X⟧) = -X 
Arguments: {A : Type u_2} [CommRing A]
Goal: PowerSeries.evalNegHom PowerSeries.X = -PowerSeries.X
-- PowerSeries.eq_zero_or_eq_zero_of_mul_eq_zero
theorem eq_zero_or_eq_zero_of_mul_eq_zero [NoZeroDivisors R] (φ ψ : R⟦X⟧) (h : φ * ψ = 0) :
    φ = 0 ∨ ψ = 0 
Arguments: {R : Type u_1} [Ring R] [NoZeroDivisors R] (φ ψ : PowerSeries R) (h : φ * ψ = 0)
Goal: φ = 0 ∨ ψ = 0
-- PowerSeries.span_X_isPrime
theorem span_X_isPrime : (Ideal.span ({X} : Set R⟦X⟧)).IsPrime 
Arguments: {R : Type u_1} [CommRing R] [IsDomain R]
Goal: (Ideal.span {PowerSeries.X}).IsPrime
-- PowerSeries.X_prime
theorem X_prime : Prime (X : R⟦X⟧) 
Arguments: {R : Type u_1} [CommRing R] [IsDomain R]
Goal: Prime PowerSeries.X
-- PowerSeries.X_irreducible
theorem X_irreducible : Irreducible (X : R⟦X⟧) 
Arguments: {R : Type u_1} [CommRing R] [IsDomain R]
Goal: Irreducible PowerSeries.X
-- PowerSeries.rescale_injective
theorem rescale_injective {a : R} (ha : a ≠ 0) : Function.Injective (rescale a) 
Arguments: {R : Type u_1} [CommRing R] [IsDomain R] {a : R} (ha : a ≠ 0)
Goal: Function.Injective ⇑(PowerSeries.rescale a)
-- PowerSeries.C_eq_algebraMap
theorem C_eq_algebraMap {r : R} : C R r = (algebraMap R R⟦X⟧) r 
Arguments: {R : Type u_1} [CommSemiring R] {r : R}
Goal: (PowerSeries.C R) r = (algebraMap R (PowerSeries R)) r
-- PowerSeries.algebraMap_apply
theorem algebraMap_apply {r : R} : algebraMap R A⟦X⟧ r = C A (algebraMap R A r) 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] {r : R}
Goal: (algebraMap R (PowerSeries A)) r = (PowerSeries.C A) ((algebraMap R A) r)
-- Polynomial.coe_def
theorem coe_def : (φ : PowerSeries R) = PowerSeries.mk (coeff φ) 
Arguments: {R : Type u_2} [CommSemiring R] (φ : Polynomial R)
Goal: ↑φ = PowerSeries.mk φ.coeff
-- Polynomial.coeff_coe
@[simp, norm_cast]
theorem coeff_coe (n) : PowerSeries.coeff R n φ = coeff φ n 
Arguments: {R : Type u_2} [CommSemiring R] (φ : Polynomial R) (n : ℕ)
Goal: (PowerSeries.coeff R n) ↑φ = φ.coeff n
-- Polynomial.coe_monomial
@[simp, norm_cast]
theorem coe_monomial (n : ℕ) (a : R) :
    (monomial n a : PowerSeries R) = PowerSeries.monomial R n a 
Arguments: {R : Type u_2} [CommSemiring R] (n : ℕ) (a : R)
Goal: ↑((Polynomial.monomial n) a) = (PowerSeries.monomial R n) a
-- Polynomial.coe_zero
@[simp, norm_cast]
theorem coe_zero : ((0 : R[X]) : PowerSeries R) = 0 
Arguments: {R : Type u_2} [CommSemiring R]
Goal: ↑0 = 0
-- Polynomial.coe_one
@[simp, norm_cast]
theorem coe_one : ((1 : R[X]) : PowerSeries R) = 1 
Arguments: {R : Type u_2} [CommSemiring R]
Goal: ↑1 = 1
-- Polynomial.coe_add
@[simp, norm_cast]
theorem coe_add : ((φ + ψ : R[X]) : PowerSeries R) = φ + ψ 
Arguments: {R : Type u_2} [CommSemiring R] (φ ψ : Polynomial R)
Goal: ↑(φ + ψ) = ↑φ + ↑ψ
-- Polynomial.coe_mul
@[simp, norm_cast]
theorem coe_mul : ((φ * ψ : R[X]) : PowerSeries R) = φ * ψ 
Arguments: {R : Type u_2} [CommSemiring R] (φ ψ : Polynomial R)
Goal: ↑(φ * ψ) = ↑φ * ↑ψ
-- Polynomial.coe_C
@[simp, norm_cast]
theorem coe_C (a : R) : ((C a : R[X]) : PowerSeries R) = PowerSeries.C R a 
Arguments: {R : Type u_2} [CommSemiring R] (a : R)
Goal: ↑(Polynomial.C a) = (PowerSeries.C R) a
-- Polynomial.coe_X
@[simp, norm_cast]
theorem coe_X : ((X : R[X]) : PowerSeries R) = PowerSeries.X 
Arguments: {R : Type u_2} [CommSemiring R]
Goal: ↑Polynomial.X = PowerSeries.X
-- Polynomial.constantCoeff_coe
@[simp]
theorem constantCoeff_coe : PowerSeries.constantCoeff R φ = φ.coeff 0 
Arguments: {R : Type u_2} [CommSemiring R] (φ : Polynomial R)
Goal: (PowerSeries.constantCoeff R) ↑φ = φ.coeff 0