statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
coeff_zero_X_mul (φ : power_series R) : coeff R 0 (X * φ) = 0 | by simp | lemma | power_series.coeff_zero_X_mul | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_C_mul_X_pow (x : R) (k n : ℕ) :
coeff R n (C R x * X ^ k : power_series R) = if n = k then x else 0 | by simp [X_pow_eq, coeff_monomial] | lemma | power_series.coeff_C_mul_X_pow | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_mul_X_pow (p : power_series R) (n d : ℕ) :
coeff R (d + n) (p * X ^ n) = coeff R d p | begin
rw [coeff_mul, finset.sum_eq_single (d, n), coeff_X_pow, if_pos rfl, mul_one],
{ rintros ⟨i,j⟩ h1 h2, rw [coeff_X_pow, if_neg, mul_zero], rintro rfl, apply h2,
rw [finset.nat.mem_antidiagonal, add_right_cancel_iff] at h1, subst h1 },
{ exact λ h1, (h1 (finset.nat.mem_antidiagonal.2 rfl)).elim }
end | theorem | power_series.coeff_mul_X_pow | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"finset.nat.mem_antidiagonal",
"mul_one",
"mul_zero",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_X_pow_mul (p : power_series R) (n d : ℕ) :
coeff R (d + n) (X ^ n * p) = coeff R d p | begin
rw [coeff_mul, finset.sum_eq_single (n,d), coeff_X_pow, if_pos rfl, one_mul],
{ rintros ⟨i,j⟩ h1 h2, rw [coeff_X_pow, if_neg, zero_mul], rintro rfl, apply h2,
rw [finset.nat.mem_antidiagonal, add_comm, add_right_cancel_iff] at h1, subst h1 },
{ rw add_comm,
exact λ h1, (h1 (finset.nat.mem_antidiagon... | theorem | power_series.coeff_X_pow_mul | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"finset.nat.mem_antidiagonal",
"one_mul",
"power_series",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_mul_X_pow' (p : power_series R) (n d : ℕ) :
coeff R d (p * X ^ n) = ite (n ≤ d) (coeff R (d - n) p) 0 | begin
split_ifs,
{ rw [← tsub_add_cancel_of_le h, coeff_mul_X_pow, add_tsub_cancel_right] },
{ refine (coeff_mul _ _ _).trans (finset.sum_eq_zero (λ x hx, _)),
rw [coeff_X_pow, if_neg, mul_zero],
exact ((le_of_add_le_right (finset.nat.mem_antidiagonal.mp hx).le).trans_lt $ not_le.mp h).ne }
end | lemma | power_series.coeff_mul_X_pow' | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"add_tsub_cancel_right",
"mul_zero",
"power_series",
"tsub_add_cancel_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_X_pow_mul' (p : power_series R) (n d : ℕ) :
coeff R d (X ^ n * p) = ite (n ≤ d) (coeff R (d - n) p) 0 | begin
split_ifs,
{ rw [← tsub_add_cancel_of_le h, coeff_X_pow_mul], simp, },
{ refine (coeff_mul _ _ _).trans (finset.sum_eq_zero (λ x hx, _)),
rw [coeff_X_pow, if_neg, zero_mul],
have := finset.nat.mem_antidiagonal.mp hx,
rw add_comm at this,
exact ((le_of_add_le_right this.le).trans_lt $ not_le.... | lemma | power_series.coeff_X_pow_mul' | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"power_series",
"tsub_add_cancel_of_le",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit_constant_coeff (φ : power_series R) (h : is_unit φ) :
is_unit (constant_coeff R φ) | mv_power_series.is_unit_constant_coeff φ h | lemma | power_series.is_unit_constant_coeff | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"is_unit",
"mv_power_series.is_unit_constant_coeff",
"power_series"
] | If a formal power series is invertible, then so is its constant coefficient. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_shift_mul_X_add_const (φ : power_series R) :
φ = mk (λ p, coeff R (p + 1) φ) * X + C R (constant_coeff R φ) | begin
ext (_ | n),
{ simp only [ring_hom.map_add, constant_coeff_C, constant_coeff_X, coeff_zero_eq_constant_coeff,
zero_add, mul_zero, ring_hom.map_mul], },
{ simp only [coeff_succ_mul_X, coeff_mk, linear_map.map_add, coeff_C, n.succ_ne_zero, sub_zero,
if_false, add_zero], }
end | lemma | power_series.eq_shift_mul_X_add_const | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"linear_map.map_add",
"mul_zero",
"power_series",
"ring_hom.map_add",
"ring_hom.map_mul"
] | Split off the constant coefficient. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_X_mul_shift_add_const (φ : power_series R) :
φ = X * mk (λ p, coeff R (p + 1) φ) + C R (constant_coeff R φ) | begin
ext (_ | n),
{ simp only [ring_hom.map_add, constant_coeff_C, constant_coeff_X, coeff_zero_eq_constant_coeff,
zero_add, zero_mul, ring_hom.map_mul], },
{ simp only [coeff_succ_X_mul, coeff_mk, linear_map.map_add, coeff_C, n.succ_ne_zero, sub_zero,
if_false, add_zero], }
end | lemma | power_series.eq_X_mul_shift_add_const | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"linear_map.map_add",
"power_series",
"ring_hom.map_add",
"ring_hom.map_mul",
"zero_mul"
] | Split off the constant coefficient. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map : power_series R →+* power_series S | mv_power_series.map _ f | def | power_series.map | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_power_series.map",
"power_series"
] | The map between formal power series induced by a map on the coefficients. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_id : (map (ring_hom.id R) :
power_series R → power_series R) = id | rfl | lemma | power_series.map_id | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"map_id",
"power_series",
"ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comp : map (g.comp f) = (map g).comp (map f) | rfl | lemma | power_series.map_comp | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"map_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_map (n : ℕ) (φ : power_series R) :
coeff S n (map f φ) = f (coeff R n φ) | rfl | lemma | power_series.coeff_map | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_C (r : R) : map f (C _ r) = C _ (f r) | by { ext, simp [coeff_C, apply_ite f] } | lemma | power_series.map_C | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"apply_ite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_X : map f X = X | by { ext, simp [coeff_X, apply_ite f] } | lemma | power_series.map_X | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"apply_ite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
X_pow_dvd_iff {n : ℕ} {φ : power_series R} :
(X : power_series R)^n ∣ φ ↔ ∀ m, m < n → coeff R m φ = 0 | begin
convert @mv_power_series.X_pow_dvd_iff unit R _ () n φ, apply propext,
classical, split; intros h m hm,
{ rw finsupp.unique_single m, convert h _ hm },
{ apply h, simpa only [finsupp.single_eq_same] using hm }
end | lemma | power_series.X_pow_dvd_iff | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"finsupp.single_eq_same",
"finsupp.unique_single",
"mv_power_series.X_pow_dvd_iff",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
X_dvd_iff {φ : power_series R} :
(X : power_series R) ∣ φ ↔ constant_coeff R φ = 0 | begin
rw [← pow_one (X : power_series R), X_pow_dvd_iff, ← coeff_zero_eq_constant_coeff_apply],
split; intro h,
{ exact h 0 zero_lt_one },
{ intros m hm, rwa nat.eq_zero_of_le_zero (nat.le_of_succ_le_succ hm) }
end | lemma | power_series.X_dvd_iff | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"pow_one",
"power_series",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rescale (a : R) : power_series R →+* power_series R | { to_fun := λ f, power_series.mk $ λ n, a^n * (power_series.coeff R n f),
map_zero' := by { ext, simp only [linear_map.map_zero, power_series.coeff_mk, mul_zero], },
map_one' := by { ext1, simp only [mul_boole, power_series.coeff_mk, power_series.coeff_one],
split_ifs, { rw [h, pow_zero], }, refl, ... | def | power_series.rescale | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"finset.mul_sum",
"linear_map.map_zero",
"mul_boole",
"mul_mul_mul_comm",
"mul_zero",
"pow_add",
"pow_zero",
"power_series",
"power_series.coeff",
"power_series.coeff_mk",
"power_series.coeff_mul",
"power_series.coeff_one",
"power_series.mk"
] | The ring homomorphism taking a power series `f(X)` to `f(aX)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff_rescale (f : power_series R) (a : R) (n : ℕ) :
coeff R n (rescale a f) = a^n * coeff R n f | coeff_mk n _ | lemma | power_series.coeff_rescale | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rescale_zero : rescale 0 = (C R).comp (constant_coeff R) | begin
ext,
simp only [function.comp_app, ring_hom.coe_comp, rescale, ring_hom.coe_mk,
power_series.coeff_mk _ _, coeff_C],
split_ifs,
{ simp only [h, one_mul, coeff_zero_eq_constant_coeff, pow_zero], },
{ rw [zero_pow' n h, zero_mul], },
end | lemma | power_series.rescale_zero | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"one_mul",
"pow_zero",
"power_series.coeff_mk",
"ring_hom.coe_comp",
"ring_hom.coe_mk",
"zero_mul",
"zero_pow'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rescale_zero_apply : rescale 0 X = C R (constant_coeff R X) | by simp | lemma | power_series.rescale_zero_apply | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rescale_one : rescale 1 = ring_hom.id (power_series R) | by { ext, simp only [ring_hom.id_apply, rescale, one_pow, coeff_mk, one_mul,
ring_hom.coe_mk], } | lemma | power_series.rescale_one | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"one_mul",
"one_pow",
"power_series",
"ring_hom.coe_mk",
"ring_hom.id",
"ring_hom.id_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rescale_mk (f : ℕ → R) (a : R) :
rescale a (mk f) = mk (λ n : ℕ, a^n * (f n)) | by { ext, rw [coeff_rescale, coeff_mk, coeff_mk], } | lemma | power_series.rescale_mk | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rescale_rescale (f : power_series R) (a b : R) :
rescale b (rescale a f) = rescale (a * b) f | begin
ext,
repeat { rw coeff_rescale, },
rw [mul_pow, mul_comm _ (b^n), mul_assoc],
end | lemma | power_series.rescale_rescale | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mul_assoc",
"mul_comm",
"mul_pow",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rescale_mul (a b : R) : rescale (a * b) = (rescale b).comp (rescale a) | by { ext, simp [← rescale_rescale], } | lemma | power_series.rescale_mul | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trunc (n : ℕ) (φ : power_series R) : R[X] | ∑ m in Ico 0 n, polynomial.monomial m (coeff R m φ) | def | power_series.trunc | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"polynomial.monomial",
"power_series",
"trunc"
] | The `n`th truncation of a formal power series to a polynomial | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff_trunc (m) (n) (φ : power_series R) :
(trunc n φ).coeff m = if m < n then coeff R m φ else 0 | by simp [trunc, polynomial.coeff_sum, polynomial.coeff_monomial, nat.lt_succ_iff] | lemma | power_series.coeff_trunc | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"nat.lt_succ_iff",
"polynomial.coeff_monomial",
"polynomial.coeff_sum",
"power_series",
"trunc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trunc_zero (n) : trunc n (0 : power_series R) = 0 | polynomial.ext $ λ m,
begin
rw [coeff_trunc, linear_map.map_zero, polynomial.coeff_zero],
split_ifs; refl
end | lemma | power_series.trunc_zero | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"linear_map.map_zero",
"polynomial.coeff_zero",
"polynomial.ext",
"power_series",
"trunc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trunc_one (n) : trunc (n + 1) (1 : power_series R) = 1 | polynomial.ext $ λ m,
begin
rw [coeff_trunc, coeff_one],
split_ifs with H H' H'; rw [polynomial.coeff_one],
{ subst m, rw [if_pos rfl] },
{ symmetry, exact if_neg (ne.elim (ne.symm H')) },
{ symmetry, refine if_neg _,
rintro rfl, apply H, exact nat.zero_lt_succ _ }
end | lemma | power_series.trunc_one | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"polynomial.coeff_one",
"polynomial.ext",
"power_series",
"trunc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trunc_C (n) (a : R) : trunc (n + 1) (C R a) = polynomial.C a | polynomial.ext $ λ m,
begin
rw [coeff_trunc, coeff_C, polynomial.coeff_C],
split_ifs with H; refl <|> try {simp * at *}
end | lemma | power_series.trunc_C | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"polynomial.C",
"polynomial.coeff_C",
"polynomial.ext",
"trunc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trunc_add (n) (φ ψ : power_series R) :
trunc n (φ + ψ) = trunc n φ + trunc n ψ | polynomial.ext $ λ m,
begin
simp only [coeff_trunc, add_monoid_hom.map_add, polynomial.coeff_add],
split_ifs with H, {refl}, {rw [zero_add]}
end | lemma | power_series.trunc_add | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"polynomial.coeff_add",
"polynomial.ext",
"power_series",
"trunc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv.aux : R → power_series R → power_series R | mv_power_series.inv.aux | def | power_series.inv.aux | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_power_series.inv.aux",
"power_series"
] | Auxiliary function used for computing inverse of a power series | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff_inv_aux (n : ℕ) (a : R) (φ : power_series R) :
coeff R n (inv.aux a φ) = if n = 0 then a else
- a * ∑ x in finset.nat.antidiagonal n,
if x.2 < n then coeff R x.1 φ * coeff R x.2 (inv.aux a φ) else 0 | begin
rw [coeff, inv.aux, mv_power_series.coeff_inv_aux],
simp only [finsupp.single_eq_zero],
split_ifs, {refl},
congr' 1,
symmetry,
apply finset.sum_bij (λ (p : ℕ × ℕ) h, (single () p.1, single () p.2)),
{ rintros ⟨i,j⟩ hij, rw finset.nat.mem_antidiagonal at hij,
rw [finsupp.mem_antidiagonal, ← finsu... | lemma | power_series.coeff_inv_aux | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"finset.nat.antidiagonal",
"finset.nat.mem_antidiagonal",
"finsupp.add_apply",
"finsupp.mem_antidiagonal",
"finsupp.single_add",
"finsupp.single_eq_same",
"finsupp.single_eq_zero",
"finsupp.unique_single",
"finsupp.unique_single_eq_iff",
"mv_power_series.coeff_inv_aux",
"power_series",
"prod.m... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_of_unit (φ : power_series R) (u : Rˣ) : power_series R | mv_power_series.inv_of_unit φ u | def | power_series.inv_of_unit | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_power_series.inv_of_unit",
"power_series"
] | A formal power series is invertible if the constant coefficient is invertible. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff_inv_of_unit (n : ℕ) (φ : power_series R) (u : Rˣ) :
coeff R n (inv_of_unit φ u) = if n = 0 then ↑u⁻¹ else
- ↑u⁻¹ * ∑ x in finset.nat.antidiagonal n,
if x.2 < n then coeff R x.1 φ * coeff R x.2 (inv_of_unit φ u) else 0 | coeff_inv_aux n ↑u⁻¹ φ | lemma | power_series.coeff_inv_of_unit | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"finset.nat.antidiagonal",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
constant_coeff_inv_of_unit (φ : power_series R) (u : Rˣ) :
constant_coeff R (inv_of_unit φ u) = ↑u⁻¹ | by rw [← coeff_zero_eq_constant_coeff_apply, coeff_inv_of_unit, if_pos rfl] | lemma | power_series.constant_coeff_inv_of_unit | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_of_unit (φ : power_series R) (u : Rˣ) (h : constant_coeff R φ = u) :
φ * inv_of_unit φ u = 1 | mv_power_series.mul_inv_of_unit φ u $ h | lemma | power_series.mul_inv_of_unit | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_power_series.mul_inv_of_unit",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_const_eq_shift_mul_X (φ : power_series R) :
φ - C R (constant_coeff R φ) = power_series.mk (λ p, coeff R (p + 1) φ) * X | sub_eq_iff_eq_add.mpr (eq_shift_mul_X_add_const φ) | lemma | power_series.sub_const_eq_shift_mul_X | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"power_series",
"power_series.mk"
] | Two ways of removing the constant coefficient of a power series are the same. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sub_const_eq_X_mul_shift (φ : power_series R) :
φ - C R (constant_coeff R φ) = X * power_series.mk (λ p, coeff R (p + 1) φ) | sub_eq_iff_eq_add.mpr (eq_X_mul_shift_add_const φ) | lemma | power_series.sub_const_eq_X_mul_shift | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"power_series",
"power_series.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rescale_X (a : A) : rescale a X = C A a * X | begin
ext,
simp only [coeff_rescale, coeff_C_mul, coeff_X],
split_ifs with h;
simp [h],
end | lemma | power_series.rescale_X | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rescale_neg_one_X : rescale (-1 : A) X = -X | by rw [rescale_X, map_neg, map_one, neg_one_mul] | lemma | power_series.rescale_neg_one_X | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"map_one",
"neg_one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eval_neg_hom : power_series A →+* power_series A | rescale (-1 : A) | def | power_series.eval_neg_hom | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"power_series"
] | The ring homomorphism taking a power series `f(X)` to `f(-X)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eval_neg_hom_X : eval_neg_hom (X : power_series A) = -X | rescale_neg_one_X | lemma | power_series.eval_neg_hom_X | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_zero_or_eq_zero_of_mul_eq_zero [no_zero_divisors R] (φ ψ : power_series R)
(h : φ * ψ = 0) : φ = 0 ∨ ψ = 0 | begin
rw or_iff_not_imp_left, intro H,
have ex : ∃ m, coeff R m φ ≠ 0, { contrapose! H, exact ext H },
let m := nat.find ex,
have hm₁ : coeff R m φ ≠ 0 := nat.find_spec ex,
have hm₂ : ∀ k < m, ¬coeff R k φ ≠ 0 := λ k, nat.find_min ex,
ext n, rw (coeff R n).map_zero, apply nat.strong_induction_on n,
clear ... | lemma | power_series.eq_zero_or_eq_zero_of_mul_eq_zero | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"finset.nat.mem_antidiagonal",
"ih",
"linear_map.map_zero",
"mul_zero",
"no_zero_divisors",
"or_iff_not_imp_left",
"power_series",
"prod.mk.inj_iff",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_X_is_prime : (ideal.span ({X} : set (power_series R))).is_prime | begin
suffices : ideal.span ({X} : set (power_series R)) = (constant_coeff R).ker,
{ rw this, exact ring_hom.ker_is_prime _ },
apply ideal.ext, intro φ,
rw [ring_hom.mem_ker, ideal.mem_span_singleton, X_dvd_iff]
end | lemma | power_series.span_X_is_prime | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"ideal.ext",
"ideal.mem_span_singleton",
"ideal.span",
"power_series",
"ring_hom.ker_is_prime",
"ring_hom.mem_ker"
] | The ideal spanned by the variable in the power series ring
over an integral domain is a prime ideal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
X_prime : prime (X : power_series R) | begin
rw ← ideal.span_singleton_prime,
{ exact span_X_is_prime },
{ intro h, simpa using congr_arg (coeff R 1) h }
end | lemma | power_series.X_prime | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"ideal.span_singleton_prime",
"power_series",
"prime"
] | The variable of the power series ring over an integral domain is prime. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rescale_injective {a : R} (ha : a ≠ 0) : function.injective (rescale a) | begin
intros p q h,
rw power_series.ext_iff at *,
intros n,
specialize h n,
rw [coeff_rescale, coeff_rescale, mul_eq_mul_left_iff] at h,
apply h.resolve_right,
intro h',
exact ha (pow_eq_zero h'),
end | lemma | power_series.rescale_injective | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mul_eq_mul_left_iff",
"pow_eq_zero",
"power_series.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map.is_local_ring_hom : is_local_ring_hom (map f) | mv_power_series.map.is_local_ring_hom f | instance | power_series.map.is_local_ring_hom | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"is_local_ring_hom",
"mv_power_series.map.is_local_ring_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
C_eq_algebra_map {r : R} : C R r = (algebra_map R (power_series R)) r | rfl | theorem | power_series.C_eq_algebra_map | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"algebra_map",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_apply {r : R} :
algebra_map R (power_series A) r = C A (algebra_map R A r) | mv_power_series.algebra_map_apply | theorem | power_series.algebra_map_apply | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"algebra_map",
"algebra_map_apply",
"mv_power_series.algebra_map_apply",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv : power_series k → power_series k | mv_power_series.inv | def | power_series.inv | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_power_series.inv",
"power_series"
] | The inverse 1/f of a power series f defined over a field | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_eq_inv_aux (φ : power_series k) :
φ⁻¹ = inv.aux (constant_coeff k φ)⁻¹ φ | rfl | lemma | power_series.inv_eq_inv_aux | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_inv (n) (φ : power_series k) :
coeff k n (φ⁻¹) = if n = 0 then (constant_coeff k φ)⁻¹ else
- (constant_coeff k φ)⁻¹ * ∑ x in finset.nat.antidiagonal n,
if x.2 < n then coeff k x.1 φ * coeff k x.2 (φ⁻¹) else 0 | by rw [inv_eq_inv_aux, coeff_inv_aux n (constant_coeff k φ)⁻¹ φ] | lemma | power_series.coeff_inv | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"finset.nat.antidiagonal",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
constant_coeff_inv (φ : power_series k) :
constant_coeff k (φ⁻¹) = (constant_coeff k φ)⁻¹ | mv_power_series.constant_coeff_inv φ | lemma | power_series.constant_coeff_inv | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_power_series.constant_coeff_inv",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_eq_zero {φ : power_series k} :
φ⁻¹ = 0 ↔ constant_coeff k φ = 0 | mv_power_series.inv_eq_zero | lemma | power_series.inv_eq_zero | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"inv_eq_zero",
"mv_power_series.inv_eq_zero",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_inv : (0 : power_series k)⁻¹ = 0 | mv_power_series.zero_inv | lemma | power_series.zero_inv | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_power_series.zero_inv",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_of_unit_eq (φ : power_series k) (h : constant_coeff k φ ≠ 0) :
inv_of_unit φ (units.mk0 _ h) = φ⁻¹ | mv_power_series.inv_of_unit_eq _ _ | lemma | power_series.inv_of_unit_eq | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_power_series.inv_of_unit_eq",
"power_series",
"units.mk0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_of_unit_eq' (φ : power_series k) (u : units k) (h : constant_coeff k φ = u) :
inv_of_unit φ u = φ⁻¹ | mv_power_series.inv_of_unit_eq' φ _ h | lemma | power_series.inv_of_unit_eq' | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_power_series.inv_of_unit_eq'",
"power_series",
"units"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_cancel (φ : power_series k) (h : constant_coeff k φ ≠ 0) :
φ * φ⁻¹ = 1 | mv_power_series.mul_inv_cancel φ h | lemma | power_series.mul_inv_cancel | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mul_inv_cancel",
"mv_power_series.mul_inv_cancel",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul_cancel (φ : power_series k) (h : constant_coeff k φ ≠ 0) :
φ⁻¹ * φ = 1 | mv_power_series.inv_mul_cancel φ h | lemma | power_series.inv_mul_cancel | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"inv_mul_cancel",
"mv_power_series.inv_mul_cancel",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_mul_inv_iff_mul_eq {φ₁ φ₂ φ₃ : power_series k} (h : constant_coeff k φ₃ ≠ 0) :
φ₁ = φ₂ * φ₃⁻¹ ↔ φ₁ * φ₃ = φ₂ | mv_power_series.eq_mul_inv_iff_mul_eq h | lemma | power_series.eq_mul_inv_iff_mul_eq | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"eq_mul_inv_iff_mul_eq",
"mv_power_series.eq_mul_inv_iff_mul_eq",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_inv_iff_mul_eq_one {φ ψ : power_series k} (h : constant_coeff k ψ ≠ 0) :
φ = ψ⁻¹ ↔ φ * ψ = 1 | mv_power_series.eq_inv_iff_mul_eq_one h | lemma | power_series.eq_inv_iff_mul_eq_one | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"eq_inv_iff_mul_eq_one",
"mv_power_series.eq_inv_iff_mul_eq_one",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_eq_iff_mul_eq_one {φ ψ : power_series k} (h : constant_coeff k ψ ≠ 0) :
ψ⁻¹ = φ ↔ φ * ψ = 1 | mv_power_series.inv_eq_iff_mul_eq_one h | lemma | power_series.inv_eq_iff_mul_eq_one | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"inv_eq_iff_mul_eq_one",
"mv_power_series.inv_eq_iff_mul_eq_one",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_rev (φ ψ : power_series k) :
(φ * ψ)⁻¹ = ψ⁻¹ * φ⁻¹ | mv_power_series.mul_inv_rev _ _ | lemma | power_series.mul_inv_rev | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mul_inv_rev",
"mv_power_series.mul_inv_rev",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
C_inv (r : k) : (C k r)⁻¹ = C k r⁻¹ | mv_power_series.C_inv _ | lemma | power_series.C_inv | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_power_series.C_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
X_inv : (X : power_series k)⁻¹ = 0 | mv_power_series.X_inv _ | lemma | power_series.X_inv | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_power_series.X_inv",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_inv (r : k) (φ : power_series k) :
(r • φ)⁻¹ = r⁻¹ • φ⁻¹ | mv_power_series.smul_inv _ _ | lemma | power_series.smul_inv | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"mv_power_series.smul_inv",
"power_series",
"smul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_coeff_ne_zero_iff_ne_zero : (∃ (n : ℕ), coeff R n φ ≠ 0) ↔ φ ≠ 0 | begin
refine not_iff_not.mp _,
push_neg,
simp [power_series.ext_iff]
end | lemma | power_series.exists_coeff_ne_zero_iff_ne_zero | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"power_series.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order (φ : power_series R) : part_enat | if h : φ = 0 then ⊤ else nat.find (exists_coeff_ne_zero_iff_ne_zero.mpr h) | def | power_series.order | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"part_enat",
"power_series"
] | The order of a formal power series `φ` is the greatest `n : part_enat`
such that `X^n` divides `φ`. The order is `⊤` if and only if `φ = 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_zero : order (0 : power_series R) = ⊤ | dif_pos rfl | lemma | power_series.order_zero | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"power_series"
] | The order of the `0` power series is infinite. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_finite_iff_ne_zero : (order φ).dom ↔ φ ≠ 0 | begin
simp only [order],
split,
{ split_ifs with h h; intro H,
{ contrapose! H,
simpa [←part.eq_none_iff'] },
{ exact h } },
{ intro h,
simp [h] }
end | lemma | power_series.order_finite_iff_ne_zero | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_order (h : (order φ).dom) :
coeff R (φ.order.get h) φ ≠ 0 | begin
simp only [order, order_finite_iff_ne_zero.mp h, not_false_iff, dif_neg, part_enat.get_coe'],
generalize_proofs h,
exact nat.find_spec h
end | lemma | power_series.coeff_order | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"part_enat.get_coe'"
] | If the order of a formal power series is finite,
then the coefficient indexed by the order is nonzero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_le (n : ℕ) (h : coeff R n φ ≠ 0) :
order φ ≤ n | begin
have := exists.intro n h,
rw [order, dif_neg],
{ simp only [part_enat.coe_le_coe, nat.find_le_iff],
exact ⟨n, le_rfl, h⟩ },
{ exact exists_coeff_ne_zero_iff_ne_zero.mp ⟨n, h⟩ }
end | lemma | power_series.order_le | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"le_rfl",
"nat.find_le_iff",
"part_enat.coe_le_coe"
] | If the `n`th coefficient of a formal power series is nonzero,
then the order of the power series is less than or equal to `n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff_of_lt_order (n : ℕ) (h: ↑n < order φ) :
coeff R n φ = 0 | by { contrapose! h, exact order_le _ h } | lemma | power_series.coeff_of_lt_order | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [] | The `n`th coefficient of a formal power series is `0` if `n` is strictly
smaller than the order of the power series. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_eq_top {φ : power_series R} :
φ.order = ⊤ ↔ φ = 0 | begin
split,
{ intro h, ext n, rw [(coeff R n).map_zero, coeff_of_lt_order], simp [h] },
{ rintros rfl, exact order_zero }
end | lemma | power_series.order_eq_top | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"power_series"
] | The `0` power series is the unique power series with infinite order. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat_le_order (φ : power_series R) (n : ℕ) (h : ∀ i < n, coeff R i φ = 0) :
↑n ≤ order φ | begin
by_contra H, rw not_le at H,
have : (order φ).dom := part_enat.dom_of_le_coe H.le,
rw [← part_enat.coe_get this, part_enat.coe_lt_coe] at H,
exact coeff_order this (h _ H)
end | lemma | power_series.nat_le_order | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"by_contra",
"part_enat.coe_get",
"part_enat.coe_lt_coe",
"part_enat.dom_of_le_coe",
"power_series"
] | The order of a formal power series is at least `n` if
the `i`th coefficient is `0` for all `i < n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_order (φ : power_series R) (n : part_enat) (h : ∀ i : ℕ, ↑i < n → coeff R i φ = 0) :
n ≤ order φ | begin
induction n using part_enat.cases_on,
{ show _ ≤ _, rw [top_le_iff, order_eq_top],
ext i, exact h _ (part_enat.coe_lt_top i) },
{ apply nat_le_order, simpa only [part_enat.coe_lt_coe] using h }
end | lemma | power_series.le_order | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"part_enat",
"part_enat.cases_on",
"part_enat.coe_lt_coe",
"part_enat.coe_lt_top",
"power_series",
"top_le_iff"
] | The order of a formal power series is at least `n` if
the `i`th coefficient is `0` for all `i < n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_eq_nat {φ : power_series R} {n : ℕ} :
order φ = n ↔ (coeff R n φ ≠ 0) ∧ (∀ i, i < n → coeff R i φ = 0) | begin
rcases eq_or_ne φ 0 with rfl|hφ,
{ simpa using (part_enat.coe_ne_top _).symm },
simp [order, dif_neg hφ, nat.find_eq_iff]
end | lemma | power_series.order_eq_nat | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"eq_or_ne",
"nat.find_eq_iff",
"part_enat.coe_ne_top",
"power_series"
] | The order of a formal power series is exactly `n` if the `n`th coefficient is nonzero,
and the `i`th coefficient is `0` for all `i < n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_eq {φ : power_series R} {n : part_enat} :
order φ = n ↔ (∀ i:ℕ, ↑i = n → coeff R i φ ≠ 0) ∧ (∀ i:ℕ, ↑i < n → coeff R i φ = 0) | begin
induction n using part_enat.cases_on,
{ rw order_eq_top, split,
{ rintro rfl, split; intros,
{ exfalso, exact part_enat.coe_ne_top ‹_› ‹_› },
{ exact (coeff _ _).map_zero } },
{ rintro ⟨h₁, h₂⟩, ext i, exact h₂ i (part_enat.coe_lt_top i) } },
{ simpa [part_enat.coe_inj] using order_eq_na... | lemma | power_series.order_eq | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"part_enat",
"part_enat.cases_on",
"part_enat.coe_inj",
"part_enat.coe_lt_top",
"part_enat.coe_ne_top",
"power_series"
] | The order of a formal power series is exactly `n` if the `n`th coefficient is nonzero,
and the `i`th coefficient is `0` for all `i < n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_order_add (φ ψ : power_series R) :
min (order φ) (order ψ) ≤ order (φ + ψ) | begin
refine le_order _ _ _,
simp [coeff_of_lt_order] {contextual := tt}
end | lemma | power_series.le_order_add | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"power_series"
] | The order of the sum of two formal power series
is at least the minimum of their orders. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_add_of_order_eq.aux (φ ψ : power_series R)
(h : order φ ≠ order ψ) (H : order φ < order ψ) :
order (φ + ψ) ≤ order φ ⊓ order ψ | begin
suffices : order (φ + ψ) = order φ,
{ rw [le_inf_iff, this], exact ⟨le_rfl, le_of_lt H⟩ },
{ rw order_eq, split,
{ intros i hi, rw ←hi at H, rw [(coeff _ _).map_add, coeff_of_lt_order i H, add_zero],
exact (order_eq_nat.1 hi.symm).1 },
{ intros i hi,
rw [(coeff _ _).map_add, coeff_of_lt_... | lemma | power_series.order_add_of_order_eq.aux | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"le_inf_iff",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_add_of_order_eq (φ ψ : power_series R) (h : order φ ≠ order ψ) :
order (φ + ψ) = order φ ⊓ order ψ | begin
refine le_antisymm _ (le_order_add _ _),
by_cases H₁ : order φ < order ψ,
{ apply order_add_of_order_eq.aux _ _ h H₁ },
by_cases H₂ : order ψ < order φ,
{ simpa only [add_comm, inf_comm] using order_add_of_order_eq.aux _ _ h.symm H₂ },
exfalso, exact h (le_antisymm (not_lt.1 H₂) (not_lt.1 H₁))
end | lemma | power_series.order_add_of_order_eq | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"inf_comm",
"power_series"
] | The order of the sum of two formal power series
is the minimum of their orders if their orders differ. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_mul_ge (φ ψ : power_series R) :
order φ + order ψ ≤ order (φ * ψ) | begin
apply le_order,
intros n hn, rw [coeff_mul, finset.sum_eq_zero],
rintros ⟨i,j⟩ hij,
by_cases hi : ↑i < order φ,
{ rw [coeff_of_lt_order i hi, zero_mul] },
by_cases hj : ↑j < order ψ,
{ rw [coeff_of_lt_order j hj, mul_zero] },
rw not_lt at hi hj, rw finset.nat.mem_antidiagonal at hij,
exfalso,
... | lemma | power_series.order_mul_ge | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"finset.nat.mem_antidiagonal",
"mul_zero",
"nat.cast_add",
"power_series",
"zero_mul"
] | The order of the product of two formal power series
is at least the sum of their orders. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_monomial (n : ℕ) (a : R) [decidable (a = 0)] :
order (monomial R n a) = if a = 0 then ⊤ else n | begin
split_ifs with h,
{ rw [h, order_eq_top, linear_map.map_zero] },
{ rw [order_eq], split; intros i hi,
{ rw [part_enat.coe_inj] at hi, rwa [hi, coeff_monomial_same] },
{ rw [part_enat.coe_lt_coe] at hi, rw [coeff_monomial, if_neg], exact ne_of_lt hi } }
end | lemma | power_series.order_monomial | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"linear_map.map_zero",
"part_enat.coe_inj",
"part_enat.coe_lt_coe"
] | The order of the monomial `a*X^n` is infinite if `a = 0` and `n` otherwise. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_monomial_of_ne_zero (n : ℕ) (a : R) (h : a ≠ 0) :
order (monomial R n a) = n | by rw [order_monomial, if_neg h] | lemma | power_series.order_monomial_of_ne_zero | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [] | The order of the monomial `a*X^n` is `n` if `a ≠ 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff_mul_of_lt_order {φ ψ : power_series R} {n : ℕ} (h : ↑n < ψ.order) :
coeff R n (φ * ψ) = 0 | begin
suffices : coeff R n (φ * ψ) = ∑ p in finset.nat.antidiagonal n, 0,
rw [this, finset.sum_const_zero],
rw [coeff_mul],
apply finset.sum_congr rfl (λ x hx, _),
refine mul_eq_zero_of_right (coeff R x.fst φ) (coeff_of_lt_order x.snd (lt_of_le_of_lt _ h)),
rw finset.nat.mem_antidiagonal at hx,
norm_cas... | lemma | power_series.coeff_mul_of_lt_order | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"finset.nat.antidiagonal",
"finset.nat.mem_antidiagonal",
"mul_eq_zero_of_right",
"power_series"
] | If `n` is strictly smaller than the order of `ψ`, then the `n`th coefficient of its product
with any other power series is `0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff_mul_one_sub_of_lt_order {R : Type*} [comm_ring R] {φ ψ : power_series R}
(n : ℕ) (h : ↑n < ψ.order) :
coeff R n (φ * (1 - ψ)) = coeff R n φ | by simp [coeff_mul_of_lt_order h, mul_sub] | lemma | power_series.coeff_mul_one_sub_of_lt_order | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"comm_ring",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_mul_prod_one_sub_of_lt_order {R ι : Type*} [comm_ring R] (k : ℕ) (s : finset ι)
(φ : power_series R) (f : ι → power_series R) :
(∀ i ∈ s, ↑k < (f i).order) → coeff R k (φ * ∏ i in s, (1 - f i)) = coeff R k φ | begin
apply finset.induction_on s,
{ simp },
{ intros a s ha ih t,
simp only [finset.mem_insert, forall_eq_or_imp] at t,
rw [finset.prod_insert ha, ← mul_assoc, mul_right_comm, coeff_mul_one_sub_of_lt_order _ t.1],
exact ih t.2 },
end | lemma | power_series.coeff_mul_prod_one_sub_of_lt_order | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"comm_ring",
"finset",
"finset.induction_on",
"finset.mem_insert",
"finset.prod_insert",
"forall_eq_or_imp",
"ih",
"mul_assoc",
"mul_right_comm",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
X_pow_order_dvd (h : (order φ).dom) : X ^ ((order φ).get h) ∣ φ | begin
refine ⟨power_series.mk (λ n, coeff R (n + (order φ).get h) φ), _⟩,
ext n,
simp only [coeff_mul, coeff_X_pow, coeff_mk, boole_mul, finset.sum_ite,
finset.nat.filter_fst_eq_antidiagonal, finset.sum_const_zero, add_zero],
split_ifs with hn hn,
{ simp [tsub_add_cancel_of_le hn] },
{ simp onl... | lemma | power_series.X_pow_order_dvd | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"boole_mul",
"finset.nat.filter_fst_eq_antidiagonal",
"part_enat.coe_lt_iff",
"tsub_add_cancel_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_eq_multiplicity_X {R : Type*} [semiring R] (φ : power_series R) :
order φ = multiplicity X φ | begin
rcases eq_or_ne φ 0 with rfl|hφ,
{ simp },
induction ho : order φ using part_enat.cases_on with n,
{ simpa [hφ] using ho },
have hn : φ.order.get (order_finite_iff_ne_zero.mpr hφ) = n,
{ simp [ho] },
rw ←hn,
refine le_antisymm (le_multiplicity_of_pow_dvd $ X_pow_order_dvd
(order_finite_iff_ne_... | lemma | power_series.order_eq_multiplicity_X | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"eq_or_ne",
"multiplicity",
"part_enat.cases_on",
"part_enat.coe_lt_coe",
"part_enat.coe_lt_top",
"part_enat.find_le",
"power_series",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_one : order (1 : power_series R) = 0 | by simpa using order_monomial_of_ne_zero 0 (1:R) one_ne_zero | lemma | power_series.order_one | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"one_ne_zero",
"power_series"
] | The order of the formal power series `1` is `0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_X : order (X : power_series R) = 1 | by simpa only [nat.cast_one] using order_monomial_of_ne_zero 1 (1:R) one_ne_zero | lemma | power_series.order_X | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"nat.cast_one",
"one_ne_zero",
"power_series"
] | The order of the formal power series `X` is `1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_X_pow (n : ℕ) : order ((X : power_series R)^n) = n | by { rw [X_pow_eq, order_monomial_of_ne_zero], exact one_ne_zero } | lemma | power_series.order_X_pow | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"one_ne_zero",
"power_series"
] | The order of the formal power series `X^n` is `n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_mul (φ ψ : power_series R) :
order (φ * ψ) = order φ + order ψ | begin
simp_rw [order_eq_multiplicity_X],
exact multiplicity.mul X_prime
end | lemma | power_series.order_mul | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"multiplicity.mul",
"power_series"
] | The order of the product of two formal power series over an integral domain
is the sum of their orders. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_power_series : has_coe R[X] (power_series R) | ⟨λ φ, power_series.mk $ λ n, coeff φ n⟩ | instance | polynomial.coe_to_power_series | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"power_series",
"power_series.mk"
] | The natural inclusion from polynomials into formal power series. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_def : (φ : power_series R) = power_series.mk (coeff φ) | rfl | lemma | polynomial.coe_def | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"power_series",
"power_series.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_coe (n) :
power_series.coeff R n φ = coeff φ n | congr_arg (coeff φ) (finsupp.single_eq_same) | lemma | polynomial.coeff_coe | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"finsupp.single_eq_same",
"power_series.coeff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_monomial (n : ℕ) (a : R) :
(monomial n a : power_series R) = power_series.monomial R n a | by { ext, simp [coeff_coe, power_series.coeff_monomial, polynomial.coeff_monomial, eq_comm] } | lemma | polynomial.coe_monomial | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"polynomial.coeff_monomial",
"power_series",
"power_series.coeff_monomial",
"power_series.monomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zero : ((0 : R[X]) : power_series R) = 0 | rfl | lemma | polynomial.coe_zero | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_one : ((1 : R[X]) : power_series R) = 1 | begin
have := coe_monomial 0 (1:R),
rwa power_series.monomial_zero_eq_C_apply at this,
end | lemma | polynomial.coe_one | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"power_series",
"power_series.monomial_zero_eq_C_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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