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
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|---|---|---|---|---|---|---|---|---|---|---|
_root_.polynomial.monic.is_eisenstein_at_of_mem_of_not_mem (hf : f.monic) (h : 𝓟 ≠ ⊤)
(hmem : ∀ {n}, n < f.nat_degree → f.coeff n ∈ 𝓟) (hnot_mem : f.coeff 0 ∉ 𝓟 ^ 2) :
f.is_eisenstein_at 𝓟 | { leading := hf.leading_coeff_not_mem h,
mem := λ n hn, hmem hn,
not_mem := hnot_mem } | lemma | polynomial.monic.is_eisenstein_at_of_mem_of_not_mem | ring_theory.polynomial.eisenstein | src/ring_theory/polynomial/eisenstein/basic.lean | [
"ring_theory.eisenstein_criterion",
"ring_theory.polynomial.scale_roots"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_weakly_eisenstein_at : is_weakly_eisenstein_at f 𝓟 | ⟨λ _, hf.mem⟩ | lemma | polynomial.is_eisenstein_at.is_weakly_eisenstein_at | ring_theory.polynomial.eisenstein | src/ring_theory/polynomial/eisenstein/basic.lean | [
"ring_theory.eisenstein_criterion",
"ring_theory.polynomial.scale_roots"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_mem {n : ℕ} (hn : n ≠ f.nat_degree) : f.coeff n ∈ 𝓟 | begin
cases ne_iff_lt_or_gt.1 hn,
{ exact hf.mem h },
{ rw [coeff_eq_zero_of_nat_degree_lt h],
exact ideal.zero_mem _}
end | lemma | polynomial.is_eisenstein_at.coeff_mem | ring_theory.polynomial.eisenstein | src/ring_theory/polynomial/eisenstein/basic.lean | [
"ring_theory.eisenstein_criterion",
"ring_theory.polynomial.scale_roots"
] | [
"ideal.zero_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
irreducible (hprime : 𝓟.is_prime) (hu : f.is_primitive)
(hfd0 : 0 < f.nat_degree) : irreducible f | irreducible_of_eisenstein_criterion hprime hf.leading (λ n hn, hf.mem (coe_lt_degree.1 hn))
(nat_degree_pos_iff_degree_pos.1 hfd0) hf.not_mem hu | lemma | polynomial.is_eisenstein_at.irreducible | ring_theory.polynomial.eisenstein | src/ring_theory/polynomial/eisenstein/basic.lean | [
"ring_theory.eisenstein_criterion",
"ring_theory.polynomial.scale_roots"
] | [
"irreducible"
] | If a primitive `f` satisfies `f.is_eisenstein_at 𝓟`, where `𝓟.is_prime`, then `f` is
irreducible. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cyclotomic_comp_X_add_one_is_eisenstein_at [hp : fact p.prime] :
((cyclotomic p ℤ).comp (X + 1)).is_eisenstein_at 𝓟 | begin
refine monic.is_eisenstein_at_of_mem_of_not_mem _
(ideal.is_prime.ne_top $(ideal.span_singleton_prime (by exact_mod_cast hp.out.ne_zero)).2 $
nat.prime_iff_prime_int.1 hp.out) (λ i hi, _) _,
{ rw [show (X + 1 : ℤ[X]) = X + C 1, by simp],
refine ((cyclotomic.monic p ℤ).comp (monic_X_add_C 1) (λ h, ... | lemma | cyclotomic_comp_X_add_one_is_eisenstein_at | ring_theory.polynomial.eisenstein | src/ring_theory/polynomial/eisenstein/is_integral.lean | [
"data.nat.choose.dvd",
"ring_theory.integrally_closed",
"ring_theory.norm",
"ring_theory.polynomial.cyclotomic.expand"
] | [
"dvd.intro",
"fact",
"ideal.is_prime.ne_top",
"ideal.mem_span_singleton",
"ideal.span_singleton_pow",
"ideal.span_singleton_prime",
"ideal.submodule_span_eq",
"int.coe_nat_dvd",
"linear_map.map_sum",
"mul_assoc",
"mul_one",
"mul_right_inj'",
"nat.prime.not_dvd_one",
"nat.totient_prime",
... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cyclotomic_prime_pow_comp_X_add_one_is_eisenstein_at [hp : fact p.prime] (n : ℕ) :
((cyclotomic (p ^ (n + 1)) ℤ).comp (X + 1)).is_eisenstein_at 𝓟 | begin
refine monic.is_eisenstein_at_of_mem_of_not_mem _
(ideal.is_prime.ne_top $(ideal.span_singleton_prime (by exact_mod_cast hp.out.ne_zero)).2 $
nat.prime_iff_prime_int.1 hp.out) _ _,
{ rw [show (X + 1 : ℤ[X]) = X + C 1, by simp],
refine ((cyclotomic.monic _ ℤ).comp (monic_X_add_C 1) (λ h, _)),
r... | lemma | cyclotomic_prime_pow_comp_X_add_one_is_eisenstein_at | ring_theory.polynomial.eisenstein | src/ring_theory/polynomial/eisenstein/is_integral.lean | [
"data.nat.choose.dvd",
"ring_theory.integrally_closed",
"ring_theory.norm",
"ring_theory.polynomial.cyclotomic.expand"
] | [
"cyclotomic_comp_X_add_one_is_eisenstein_at",
"dvd.intro",
"fact",
"ideal.is_prime.ne_top",
"ideal.mem_span_singleton",
"ideal.span_singleton_pow",
"ideal.span_singleton_prime",
"ideal.submodule_span_eq",
"int.coe_cast_ring_hom",
"ite_eq_right_iff",
"lt_of_mul_lt_mul_left'",
"map_comp",
"mul... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_is_eiseinstein_at {B : power_basis K L}
(hp : prime p) (hBint : is_integral R B.gen) {z : L} {Q : R[X]}
(hQ : aeval B.gen Q = p • z) (hzint : is_integral R z)
(hei : (minpoly R B.gen).is_eisenstein_at 𝓟) : p ∣ Q.coeff 0 | begin
-- First define some abbreviations.
letI := B.finite_dimensional,
let P := minpoly R B.gen,
obtain ⟨n , hn⟩ := nat.exists_eq_succ_of_ne_zero B.dim_pos.ne',
have finrank_K_L : finite_dimensional.finrank K L = B.dim := B.finrank,
have deg_K_P : (minpoly K B.gen).nat_degree = B.dim := B.nat_degree_minpol... | lemma | dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_is_eiseinstein_at | ring_theory.polynomial.eisenstein | src/ring_theory/polynomial/eisenstein/is_integral.lean | [
"data.nat.choose.dvd",
"ring_theory.integrally_closed",
"ring_theory.norm",
"ring_theory.polynomial.cyclotomic.expand"
] | [
"adjoin_le_integral_closure",
"algebra.norm_algebra_map",
"algebra.smul_def",
"algebra.smul_mul_assoc",
"algebra_map",
"algebra_map_apply",
"aux",
"finite_dimensional.finrank",
"is_fraction_ring.injective",
"is_integral",
"is_integral.pow",
"is_integral.sum",
"is_integral_mul",
"is_integra... | Let `K` be the field of fraction of an integrally closed domain `R` and let `L` be a separable
extension of `K`, generated by an integral power basis `B` such that the minimal polynomial of
`B.gen` is Eisenstein at `p`. Given `z : L` integral over `R`, if `Q : R[X]` is such that
`aeval B.gen Q = p • z`, then `p ∣ Q.coe... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_adjoin_of_dvd_coeff_of_dvd_aeval {A B : Type*} [comm_semiring A] [comm_ring B]
[algebra A B] [no_zero_smul_divisors A B] {Q : A[X]} {p : A} {x z : B} (hp : p ≠ 0)
(hQ : ∀ i ∈ range (Q.nat_degree + 1), p ∣ Q.coeff i) (hz : aeval x Q = p • z) :
z ∈ adjoin A ({x} : set B) | begin
choose! f hf using hQ,
rw [aeval_eq_sum_range, sum_range] at hz,
conv_lhs at hz { congr, skip, funext,
rw [hf i (mem_range.2 (fin.is_lt i)), ← smul_smul] },
rw [← smul_sum] at hz,
rw [← smul_right_injective _ hp hz],
exact subalgebra.sum_mem _ (λ _ _, subalgebra.smul_mem _
(subalgebra.pow_mem ... | lemma | mem_adjoin_of_dvd_coeff_of_dvd_aeval | ring_theory.polynomial.eisenstein | src/ring_theory/polynomial/eisenstein/is_integral.lean | [
"data.nat.choose.dvd",
"ring_theory.integrally_closed",
"ring_theory.norm",
"ring_theory.polynomial.cyclotomic.expand"
] | [
"algebra",
"comm_ring",
"comm_semiring",
"fin.is_lt",
"no_zero_smul_divisors",
"set.mem_singleton",
"smul_right_injective",
"smul_smul",
"subalgebra.pow_mem",
"subalgebra.smul_mem",
"subalgebra.sum_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_adjoin_of_smul_prime_smul_of_minpoly_is_eiseinstein_at {B : power_basis K L}
(hp : prime p) (hBint : is_integral R B.gen) {z : L} (hzint : is_integral R z)
(hz : p • z ∈ adjoin R ({B.gen} : set L)) (hei : (minpoly R B.gen).is_eisenstein_at 𝓟) :
z ∈ adjoin R ({B.gen} : set L) | begin
-- First define some abbreviations.
have hndiv : ¬ p ^ 2 ∣ ((minpoly R B.gen)).coeff 0 := λ h,
hei.not_mem ((span_singleton_pow p 2).symm ▸ (ideal.mem_span_singleton.2 h)),
letI := finite_dimensional B,
set P := minpoly R B.gen with hP,
obtain ⟨n , hn⟩ := nat.exists_eq_succ_of_ne_zero B.dim_pos.ne',... | lemma | mem_adjoin_of_smul_prime_smul_of_minpoly_is_eiseinstein_at | ring_theory.polynomial.eisenstein | src/ring_theory/polynomial/eisenstein/is_integral.lean | [
"data.nat.choose.dvd",
"ring_theory.integrally_closed",
"ring_theory.norm",
"ring_theory.polynomial.cyclotomic.expand"
] | [
"adjoin_le_integral_closure",
"algebra.norm_algebra_map",
"algebra.smul_def",
"algebra_map",
"algebra_map_apply",
"dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_is_eiseinstein_at",
"dvd_mul_of_dvd_left",
"dvd_rfl",
"finite_dimensional",
"is_fraction_ring.injective",
"is_integral",
"is_integ... | Let `K` be the field of fraction of an integrally closed domain `R` and let `L` be a separable
extension of `K`, generated by an integral power basis `B` such that the minimal polynomial of
`B.gen` is Eisenstein at `p`. Given `z : L` integral over `R`, if `p • z ∈ adjoin R {B.gen}`, then
`z ∈ adjoin R {B.gen}`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_adjoin_of_smul_prime_pow_smul_of_minpoly_is_eiseinstein_at {B : power_basis K L}
(hp : prime p) (hBint : is_integral R B.gen) {n : ℕ} {z : L} (hzint : is_integral R z)
(hz : p ^ n • z ∈ adjoin R ({B.gen} : set L)) (hei : (minpoly R B.gen).is_eisenstein_at 𝓟) :
z ∈ adjoin R ({B.gen} : set L) | begin
induction n with n hn,
{ simpa using hz },
{ rw [pow_succ, mul_smul] at hz,
exact hn (mem_adjoin_of_smul_prime_smul_of_minpoly_is_eiseinstein_at
hp hBint (is_integral_smul _ hzint) hz hei) }
end | lemma | mem_adjoin_of_smul_prime_pow_smul_of_minpoly_is_eiseinstein_at | ring_theory.polynomial.eisenstein | src/ring_theory/polynomial/eisenstein/is_integral.lean | [
"data.nat.choose.dvd",
"ring_theory.integrally_closed",
"ring_theory.norm",
"ring_theory.polynomial.cyclotomic.expand"
] | [
"is_integral",
"is_integral_smul",
"mem_adjoin_of_smul_prime_smul_of_minpoly_is_eiseinstein_at",
"minpoly",
"pow_succ",
"power_basis",
"prime"
] | Let `K` be the field of fraction of an integrally closed domain `R` and let `L` be a separable
extension of `K`, generated by an integral power basis `B` such that the minimal polynomial of
`B.gen` is Eisenstein at `p`. Given `z : L` integral over `R`, if `p ^ n • z ∈ adjoin R {B.gen}`,
then `z ∈ adjoin R {B.gen}`. Tog... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hermite : ℕ → polynomial ℤ | | 0 := 1
| (n+1) := X * (hermite n) - (hermite n).derivative | def | polynomial.hermite | ring_theory.polynomial.hermite | src/ring_theory/polynomial/hermite/basic.lean | [
"data.polynomial.derivative",
"data.nat.parity",
"data.nat.factorial.double_factorial"
] | [
"polynomial"
] | the nth probabilist's Hermite polynomial | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hermite_succ (n : ℕ) : hermite (n+1) = X * (hermite n) - (hermite n).derivative | by rw hermite | lemma | polynomial.hermite_succ | ring_theory.polynomial.hermite | src/ring_theory/polynomial/hermite/basic.lean | [
"data.polynomial.derivative",
"data.nat.parity",
"data.nat.factorial.double_factorial"
] | [] | The recursion `hermite (n+1) = (x - d/dx) (hermite n)` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hermite_eq_iterate (n : ℕ) : hermite n = ((λ p, X*p - p.derivative)^[n] 1) | begin
induction n with n ih,
{ refl },
{ rw [function.iterate_succ_apply', ← ih, hermite_succ] }
end | lemma | polynomial.hermite_eq_iterate | ring_theory.polynomial.hermite | src/ring_theory/polynomial/hermite/basic.lean | [
"data.polynomial.derivative",
"data.nat.parity",
"data.nat.factorial.double_factorial"
] | [
"function.iterate_succ_apply'",
"ih"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hermite_zero : hermite 0 = C 1 | rfl | lemma | polynomial.hermite_zero | ring_theory.polynomial.hermite | src/ring_theory/polynomial/hermite/basic.lean | [
"data.polynomial.derivative",
"data.nat.parity",
"data.nat.factorial.double_factorial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hermite_one : hermite 1 = X | begin
rw [hermite_succ, hermite_zero],
simp only [map_one, mul_one, derivative_one, sub_zero]
end | lemma | polynomial.hermite_one | ring_theory.polynomial.hermite | src/ring_theory/polynomial/hermite/basic.lean | [
"data.polynomial.derivative",
"data.nat.parity",
"data.nat.factorial.double_factorial"
] | [
"map_one",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_hermite_succ_zero (n : ℕ) :
coeff (hermite (n + 1)) 0 = -(coeff (hermite n) 1) | by simp [coeff_derivative] | lemma | polynomial.coeff_hermite_succ_zero | ring_theory.polynomial.hermite | src/ring_theory/polynomial/hermite/basic.lean | [
"data.polynomial.derivative",
"data.nat.parity",
"data.nat.factorial.double_factorial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_hermite_succ_succ (n k : ℕ) :
coeff (hermite (n + 1)) (k + 1) = coeff (hermite n) k - (k + 2) * (coeff (hermite n) (k + 2)) | begin
rw [hermite_succ, coeff_sub, coeff_X_mul, coeff_derivative, mul_comm],
norm_cast
end | lemma | polynomial.coeff_hermite_succ_succ | ring_theory.polynomial.hermite | src/ring_theory/polynomial/hermite/basic.lean | [
"data.polynomial.derivative",
"data.nat.parity",
"data.nat.factorial.double_factorial"
] | [
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_hermite_of_lt {n k : ℕ} (hnk : n < k) : coeff (hermite n) k = 0 | begin
obtain ⟨k, rfl⟩ := nat.exists_eq_add_of_lt hnk,
clear hnk,
induction n with n ih generalizing k,
{ apply coeff_C },
{ have : n + k + 1 + 2 = n + (k + 2) + 1 := by ring,
rw [nat.succ_eq_add_one, coeff_hermite_succ_succ, add_right_comm, this, ih k, ih (k + 2),
mul_zero, sub_zero] }
end | lemma | polynomial.coeff_hermite_of_lt | ring_theory.polynomial.hermite | src/ring_theory/polynomial/hermite/basic.lean | [
"data.polynomial.derivative",
"data.nat.parity",
"data.nat.factorial.double_factorial"
] | [
"ih",
"mul_zero",
"nat.exists_eq_add_of_lt",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_hermite_self (n : ℕ) : coeff (hermite n) n = 1 | begin
induction n with n ih,
{ apply coeff_C },
{ rw [coeff_hermite_succ_succ, ih, coeff_hermite_of_lt, mul_zero, sub_zero],
simp }
end | lemma | polynomial.coeff_hermite_self | ring_theory.polynomial.hermite | src/ring_theory/polynomial/hermite/basic.lean | [
"data.polynomial.derivative",
"data.nat.parity",
"data.nat.factorial.double_factorial"
] | [
"ih",
"mul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
degree_hermite (n : ℕ) : (hermite n).degree = n | begin
rw degree_eq_of_le_of_coeff_ne_zero,
simp_rw [degree_le_iff_coeff_zero, with_bot.coe_lt_coe],
{ intro m,
exact coeff_hermite_of_lt },
{ simp [coeff_hermite_self n] }
end | lemma | polynomial.degree_hermite | ring_theory.polynomial.hermite | src/ring_theory/polynomial/hermite/basic.lean | [
"data.polynomial.derivative",
"data.nat.parity",
"data.nat.factorial.double_factorial"
] | [
"with_bot.coe_lt_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_degree_hermite {n : ℕ} : (hermite n).nat_degree = n | nat_degree_eq_of_degree_eq_some (degree_hermite n) | lemma | polynomial.nat_degree_hermite | ring_theory.polynomial.hermite | src/ring_theory/polynomial/hermite/basic.lean | [
"data.polynomial.derivative",
"data.nat.parity",
"data.nat.factorial.double_factorial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
leading_coeff_hermite (n : ℕ) : (hermite n).leading_coeff = 1 | begin
rw [← coeff_nat_degree, nat_degree_hermite, coeff_hermite_self],
end | lemma | polynomial.leading_coeff_hermite | ring_theory.polynomial.hermite | src/ring_theory/polynomial/hermite/basic.lean | [
"data.polynomial.derivative",
"data.nat.parity",
"data.nat.factorial.double_factorial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hermite_monic (n : ℕ) : (hermite n).monic | leading_coeff_hermite n | lemma | polynomial.hermite_monic | ring_theory.polynomial.hermite | src/ring_theory/polynomial/hermite/basic.lean | [
"data.polynomial.derivative",
"data.nat.parity",
"data.nat.factorial.double_factorial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_hermite_of_odd_add {n k : ℕ} (hnk : odd (n + k)) : coeff (hermite n) k = 0 | begin
induction n with n ih generalizing k,
{ rw zero_add at hnk,
exact coeff_hermite_of_lt hnk.pos },
{ cases k,
{ rw nat.succ_add_eq_succ_add at hnk,
rw [coeff_hermite_succ_zero, ih hnk, neg_zero] },
{ rw [coeff_hermite_succ_succ, ih, ih, mul_zero, sub_zero],
{ rwa [nat.succ_add_eq_succ_... | lemma | polynomial.coeff_hermite_of_odd_add | ring_theory.polynomial.hermite | src/ring_theory/polynomial/hermite/basic.lean | [
"data.polynomial.derivative",
"data.nat.parity",
"data.nat.factorial.double_factorial"
] | [
"even_two",
"ih",
"mul_zero",
"odd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_hermite_explicit :
∀ (n k : ℕ), coeff (hermite (2 * n + k)) k = (-1)^n * (2 * n - 1)‼ * nat.choose (2 * n + k) k | | 0 _ := by simp
| (n + 1) 0 := begin
convert coeff_hermite_succ_zero (2 * n + 1) using 1,
rw [coeff_hermite_explicit n 1,
(by ring_nf : 2 * (n + 1) - 1 = 2 * n + 1), nat.double_factorial_add_one,
nat.choose_zero_right, nat.choose_one_right, pow_succ],
push_cast,
ring,
end
| (n + 1) (k + 1) := begin... | lemma | polynomial.coeff_hermite_explicit | ring_theory.polynomial.hermite | src/ring_theory/polynomial/hermite/basic.lean | [
"data.polynomial.derivative",
"data.nat.parity",
"data.nat.factorial.double_factorial"
] | [
"mul_assoc",
"mul_comm",
"nat.choose",
"nat.choose_one_right",
"nat.choose_succ_right_eq",
"nat.choose_zero_right",
"nat.double_factorial_add_one",
"neg_eq_neg_one_mul",
"pow_succ",
"ring"
] | Because of `coeff_hermite_of_odd_add`, every nonzero coefficient is described as follows. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff_hermite_of_even_add {n k : ℕ} (hnk : even (n + k)) :
coeff (hermite n) k = (-1)^((n - k) / 2) * (n - k - 1)‼ * nat.choose n k | begin
cases le_or_lt k n with h_le h_lt,
{ rw [nat.even_add, ← (nat.even_sub h_le)] at hnk,
obtain ⟨m, hm⟩ := hnk,
rw [(by linarith : n = 2 * m + k), nat.add_sub_cancel,
nat.mul_div_cancel_left _ (nat.succ_pos 1), coeff_hermite_explicit] },
{ simp [nat.choose_eq_zero_of_lt h_lt, coeff_hermite_of_l... | lemma | polynomial.coeff_hermite_of_even_add | ring_theory.polynomial.hermite | src/ring_theory/polynomial/hermite/basic.lean | [
"data.polynomial.derivative",
"data.nat.parity",
"data.nat.factorial.double_factorial"
] | [
"nat.choose",
"nat.choose_eq_zero_of_lt",
"nat.even_add",
"nat.even_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_hermite (n k : ℕ) :
coeff (hermite n) k =
if even (n + k) then (-1)^((n - k) / 2) * (n - k - 1)‼ * nat.choose n k else 0 | begin
split_ifs with h,
exact coeff_hermite_of_even_add h,
exact coeff_hermite_of_odd_add (nat.odd_iff_not_even.mpr h),
end | lemma | polynomial.coeff_hermite | ring_theory.polynomial.hermite | src/ring_theory/polynomial/hermite/basic.lean | [
"data.polynomial.derivative",
"data.nat.parity",
"data.nat.factorial.double_factorial"
] | [
"nat.choose"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_gaussian_eq_hermite_mul_gaussian (n : ℕ) (x : ℝ) :
deriv^[n] (λ y, real.exp (-(y^2 / 2))) x =
(-1 : ℝ)^n * aeval x (hermite n) * real.exp (-(x^2 / 2)) | begin
rw mul_assoc,
induction n with n ih generalizing x,
{ rw [function.iterate_zero_apply, pow_zero, one_mul, hermite_zero, C_1, map_one, one_mul] },
{ replace ih : (deriv^[n] _) = _ := _root_.funext ih,
have deriv_gaussian : deriv (λ y, real.exp (-(y^2 / 2))) x = (-x) * real.exp (-(x^2 / 2)),
{ simp ... | lemma | polynomial.deriv_gaussian_eq_hermite_mul_gaussian | ring_theory.polynomial.hermite | src/ring_theory/polynomial/hermite/gaussian.lean | [
"ring_theory.polynomial.hermite.basic",
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.add",
"analysis.special_functions.exp",
"analysis.special_functions.exp_deriv"
] | [
"deriv",
"deriv_const_mul_field",
"deriv_mul",
"function.iterate_succ_apply'",
"function.iterate_zero_apply",
"ih",
"map_mul",
"map_one",
"mul_assoc",
"mul_comm",
"neg_mul",
"one_mul",
"polynomial.deriv_aeval",
"polynomial.differentiable_aeval",
"pow_succ",
"pow_zero",
"real.exp",
... | `hermite n` is (up to sign) the factor appearing in `deriv^[n]` of a gaussian | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hermite_eq_deriv_gaussian (n : ℕ) (x : ℝ) :
aeval x (hermite n) =
(-1 : ℝ)^n * (deriv^[n] (λ y, real.exp (-(y^2 / 2))) x) / real.exp (-(x^2 / 2)) | begin
rw deriv_gaussian_eq_hermite_mul_gaussian,
field_simp [real.exp_ne_zero],
rw [← @smul_eq_mul ℝ _ ((-1)^n), ← inv_smul_eq_iff₀, mul_assoc, smul_eq_mul,
← inv_pow, ← neg_inv, inv_one],
exact pow_ne_zero _ (by norm_num),
end | lemma | polynomial.hermite_eq_deriv_gaussian | ring_theory.polynomial.hermite | src/ring_theory/polynomial/hermite/gaussian.lean | [
"ring_theory.polynomial.hermite.basic",
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.add",
"analysis.special_functions.exp",
"analysis.special_functions.exp_deriv"
] | [
"deriv",
"inv_one",
"inv_pow",
"inv_smul_eq_iff₀",
"mul_assoc",
"neg_inv",
"pow_ne_zero",
"real.exp",
"real.exp_ne_zero",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hermite_eq_deriv_gaussian' (n : ℕ) (x : ℝ) :
aeval x (hermite n) =
(-1 : ℝ)^n * (deriv^[n] (λ y, real.exp (-(y^2 / 2))) x) * real.exp (x^2 / 2) | begin
rw [hermite_eq_deriv_gaussian, real.exp_neg],
field_simp [real.exp_ne_zero],
end | lemma | polynomial.hermite_eq_deriv_gaussian' | ring_theory.polynomial.hermite | src/ring_theory/polynomial/hermite/gaussian.lean | [
"ring_theory.polynomial.hermite.basic",
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.add",
"analysis.special_functions.exp",
"analysis.special_functions.exp_deriv"
] | [
"deriv",
"real.exp",
"real.exp_ne_zero",
"real.exp_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mv_power_series (σ : Type*) (R : Type*) | (σ →₀ ℕ) → R | def | mv_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"
] | [] | Multivariate formal power series, where `σ` is the index set of the variables
and `R` is the coefficient ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monomial (n : σ →₀ ℕ) : R →ₗ[R] mv_power_series σ R | linear_map.std_basis R _ n | def | mv_power_series.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.std_basis",
"mv_power_series"
] | The `n`th monomial with coefficient `a` as multivariate formal power series. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff (n : σ →₀ ℕ) : (mv_power_series σ R) →ₗ[R] R | linear_map.proj n | def | mv_power_series.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"
] | [
"linear_map.proj",
"mv_power_series"
] | The `n`th coefficient of a multivariate formal power series. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext {φ ψ} (h : ∀ (n : σ →₀ ℕ), coeff R n φ = coeff R n ψ) :
φ = ψ | funext h | lemma | mv_power_series.ext | 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"
] | [] | Two multivariate formal power series are equal if all their coefficients are equal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext_iff {φ ψ : mv_power_series σ R} :
φ = ψ ↔ (∀ (n : σ →₀ ℕ), coeff R n φ = coeff R n ψ) | function.funext_iff | lemma | mv_power_series.ext_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"
] | [
"function.funext_iff",
"mv_power_series"
] | Two multivariate formal power series are equal
if and only if all their coefficients are equal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monomial_def [decidable_eq σ] (n : σ →₀ ℕ) :
monomial R n = linear_map.std_basis R _ n | by convert rfl | lemma | mv_power_series.monomial_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"
] | [
"linear_map.std_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_monomial [decidable_eq σ] (m n : σ →₀ ℕ) (a : R) :
coeff R m (monomial R n a) = if m = n then a else 0 | by rw [coeff, monomial_def, linear_map.proj_apply, linear_map.std_basis_apply,
function.update_apply, pi.zero_apply] | lemma | mv_power_series.coeff_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.proj_apply",
"linear_map.std_basis_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_monomial_same (n : σ →₀ ℕ) (a : R) :
coeff R n (monomial R n a) = a | linear_map.std_basis_same R _ n a | lemma | mv_power_series.coeff_monomial_same | 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.std_basis_same"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_monomial_ne {m n : σ →₀ ℕ} (h : m ≠ n) (a : R) :
coeff R m (monomial R n a) = 0 | linear_map.std_basis_ne R _ _ _ h a | lemma | mv_power_series.coeff_monomial_ne | 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.std_basis_ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_coeff_monomial_ne_zero {m n : σ →₀ ℕ} {a : R} (h : coeff R m (monomial R n a) ≠ 0) :
m = n | by_contra $ λ h', h $ coeff_monomial_ne h' a | lemma | mv_power_series.eq_of_coeff_monomial_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"
] | [
"by_contra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_comp_monomial (n : σ →₀ ℕ) :
(coeff R n).comp (monomial R n) = linear_map.id | linear_map.ext $ coeff_monomial_same n | lemma | mv_power_series.coeff_comp_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.ext",
"linear_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_zero (n : σ →₀ ℕ) : coeff R n (0 : mv_power_series σ R) = 0 | rfl | lemma | mv_power_series.coeff_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"
] | [
"mv_power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_one [decidable_eq σ] :
coeff R n (1 : mv_power_series σ R) = if n = 0 then 1 else 0 | coeff_monomial _ _ _ | lemma | mv_power_series.coeff_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"
] | [
"mv_power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_zero_one : coeff R (0 : σ →₀ ℕ) 1 = 1 | coeff_monomial_same 0 1 | lemma | mv_power_series.coeff_zero_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monomial_zero_one : monomial R (0 : σ →₀ ℕ) 1 = 1 | rfl | lemma | mv_power_series.monomial_zero_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_mul : coeff R n (φ * ψ) =
∑ p in finsupp.antidiagonal n, coeff R p.1 φ * coeff R p.2 ψ | rfl | lemma | mv_power_series.coeff_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"
] | [
"finsupp.antidiagonal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_mul : (0 : mv_power_series σ R) * φ = 0 | ext $ λ n, by simp [coeff_mul] | lemma | mv_power_series.zero_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"
] | [
"mv_power_series",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_zero : φ * 0 = 0 | ext $ λ n, by simp [coeff_mul] | lemma | mv_power_series.mul_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"
] | [
"mul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_monomial_mul (a : R) :
coeff R m (monomial R n a * φ) = if n ≤ m then a * coeff R (m - n) φ else 0 | begin
have : ∀ p ∈ antidiagonal m,
coeff R (p : (σ →₀ ℕ) × (σ →₀ ℕ)).1 (monomial R n a) * coeff R p.2 φ ≠ 0 → p.1 = n :=
λ p _ hp, eq_of_coeff_monomial_ne_zero (left_ne_zero_of_mul hp),
rw [coeff_mul, ← finset.sum_filter_of_ne this, antidiagonal_filter_fst_eq,
finset.sum_ite_index],
simp only [finset.... | lemma | mv_power_series.coeff_monomial_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"
] | [
"left_ne_zero_of_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_mul_monomial (a : R) :
coeff R m (φ * monomial R n a) = if n ≤ m then coeff R (m - n) φ * a else 0 | begin
have : ∀ p ∈ antidiagonal m,
coeff R (p : (σ →₀ ℕ) × (σ →₀ ℕ)).1 φ * coeff R p.2 (monomial R n a) ≠ 0 → p.2 = n :=
λ p _ hp, eq_of_coeff_monomial_ne_zero (right_ne_zero_of_mul hp),
rw [coeff_mul, ← finset.sum_filter_of_ne this, antidiagonal_filter_snd_eq,
finset.sum_ite_index],
simp only [finset... | lemma | mv_power_series.coeff_mul_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"
] | [
"right_ne_zero_of_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_add_monomial_mul (a : R) :
coeff R (m + n) (monomial R m a * φ) = a * coeff R n φ | begin
rw [coeff_monomial_mul, if_pos, add_tsub_cancel_left],
exact le_add_right le_rfl
end | lemma | mv_power_series.coeff_add_monomial_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"
] | [
"add_tsub_cancel_left",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_add_mul_monomial (a : R) :
coeff R (m + n) (φ * monomial R n a) = coeff R m φ * a | begin
rw [coeff_mul_monomial, if_pos, add_tsub_cancel_right],
exact le_add_left le_rfl
end | lemma | mv_power_series.coeff_add_mul_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"
] | [
"add_tsub_cancel_right",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commute_monomial {a : R} {n} :
commute φ (monomial R n a) ↔ ∀ m, commute (coeff R m φ) a | begin
refine ext_iff.trans ⟨λ h m, _, λ h m, _⟩,
{ have := h (m + n),
rwa [coeff_add_mul_monomial, add_comm, coeff_add_monomial_mul] at this },
{ rw [coeff_mul_monomial, coeff_monomial_mul],
split_ifs; [apply h, refl] }
end | lemma | mv_power_series.commute_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"
] | [
"commute"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_mul : (1 : mv_power_series σ R) * φ = φ | ext $ λ n, by simpa using coeff_add_monomial_mul 0 n φ 1 | lemma | mv_power_series.one_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"
] | [
"mv_power_series",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_one : φ * 1 = φ | ext $ λ n, by simpa using coeff_add_mul_monomial n 0 φ 1 | lemma | mv_power_series.mul_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"
] | [
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_add (φ₁ φ₂ φ₃ : mv_power_series σ R) :
φ₁ * (φ₂ + φ₃) = φ₁ * φ₂ + φ₁ * φ₃ | ext $ λ n, by simp only [coeff_mul, mul_add, finset.sum_add_distrib, linear_map.map_add] | lemma | mv_power_series.mul_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"
] | [
"linear_map.map_add",
"mv_power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_mul (φ₁ φ₂ φ₃ : mv_power_series σ R) :
(φ₁ + φ₂) * φ₃ = φ₁ * φ₃ + φ₂ * φ₃ | ext $ λ n, by simp only [coeff_mul, add_mul, finset.sum_add_distrib, linear_map.map_add] | lemma | mv_power_series.add_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"
] | [
"linear_map.map_add",
"mv_power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_assoc (φ₁ φ₂ φ₃ : mv_power_series σ R) :
(φ₁ * φ₂) * φ₃ = φ₁ * (φ₂ * φ₃) | begin
ext1 n,
simp only [coeff_mul, finset.sum_mul, finset.mul_sum, finset.sum_sigma'],
refine finset.sum_bij (λ p _, ⟨(p.2.1, p.2.2 + p.1.2), (p.2.2, p.1.2)⟩) _ _ _ _;
simp only [mem_antidiagonal, finset.mem_sigma, heq_iff_eq, prod.mk.inj_iff, and_imp,
exists_prop],
{ rintros ⟨⟨i,j⟩, ⟨k,l⟩⟩, dsimp on... | lemma | mv_power_series.mul_assoc | 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"
] | [
"and_imp",
"exists_prop",
"finset.mem_sigma",
"finset.mul_sum",
"finset.sum_mul",
"heq_iff_eq",
"mul_assoc",
"mv_power_series",
"prod.mk.inj_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monomial_mul_monomial (m n : σ →₀ ℕ) (a b : R) :
monomial R m a * monomial R n b = monomial R (m + n) (a * b) | begin
ext k,
simp only [coeff_mul_monomial, coeff_monomial],
split_ifs with h₁ h₂ h₃ h₃ h₂; try { refl },
{ rw [← h₂, tsub_add_cancel_of_le h₁] at h₃, exact (h₃ rfl).elim },
{ rw [h₃, add_tsub_cancel_right] at h₂, exact (h₂ rfl).elim },
{ exact zero_mul b },
{ rw h₂ at h₁, exact (h₁ $ le_add_left le_rfl).... | lemma | mv_power_series.monomial_mul_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"
] | [
"add_tsub_cancel_right",
"le_rfl",
"tsub_add_cancel_of_le",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
C : R →+* mv_power_series σ R | { map_one' := rfl,
map_mul' := λ a b, (monomial_mul_monomial 0 0 a b).symm,
map_zero' := (monomial R (0 : _)).map_zero,
.. monomial R (0 : σ →₀ ℕ) } | def | mv_power_series.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"
] | [
"mv_power_series"
] | The constant multivariate formal power series. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monomial_zero_eq_C : ⇑(monomial R (0 : σ →₀ ℕ)) = C σ R | rfl | lemma | mv_power_series.monomial_zero_eq_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monomial_zero_eq_C_apply (a : R) : monomial R (0 : σ →₀ ℕ) a = C σ R a | rfl | lemma | mv_power_series.monomial_zero_eq_C_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 | |
coeff_C [decidable_eq σ] (n : σ →₀ ℕ) (a : R) :
coeff R n (C σ R a) = if n = 0 then a else 0 | coeff_monomial _ _ _ | lemma | mv_power_series.coeff_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_zero_C (a : R) : coeff R (0 : σ →₀ℕ) (C σ R a) = a | coeff_monomial_same 0 a | lemma | mv_power_series.coeff_zero_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
X (s : σ) : mv_power_series σ R | monomial R (single s 1) 1 | def | mv_power_series.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"
] | [
"mv_power_series"
] | The variables of the multivariate formal power series ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff_X [decidable_eq σ] (n : σ →₀ ℕ) (s : σ) :
coeff R n (X s : mv_power_series σ R) = if n = (single s 1) then 1 else 0 | coeff_monomial _ _ _ | lemma | mv_power_series.coeff_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"
] | [
"mv_power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_index_single_X [decidable_eq σ] (s t : σ) :
coeff R (single t 1) (X s : mv_power_series σ R) = if t = s then 1 else 0 | by simp only [coeff_X, single_left_inj one_ne_zero] | lemma | mv_power_series.coeff_index_single_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"
] | [
"mv_power_series",
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_index_single_self_X (s : σ) :
coeff R (single s 1) (X s : mv_power_series σ R) = 1 | coeff_monomial_same _ _ | lemma | mv_power_series.coeff_index_single_self_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"
] | [
"mv_power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_zero_X (s : σ) : coeff R (0 : σ →₀ ℕ) (X s : mv_power_series σ R) = 0 | by { rw [coeff_X, if_neg], intro h, exact one_ne_zero (single_eq_zero.mp h.symm) } | lemma | mv_power_series.coeff_zero_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"
] | [
"mv_power_series",
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commute_X (φ : mv_power_series σ R) (s : σ) : commute φ (X s) | φ.commute_monomial.mpr $ λ m, commute.one_right _ | lemma | mv_power_series.commute_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"
] | [
"commute",
"commute.one_right",
"mv_power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
X_def (s : σ) : X s = monomial R (single s 1) 1 | rfl | lemma | mv_power_series.X_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
X_pow_eq (s : σ) (n : ℕ) :
(X s : mv_power_series σ R)^n = monomial R (single s n) 1 | begin
induction n with n ih,
{ rw [pow_zero, finsupp.single_zero, monomial_zero_one] },
{ rw [pow_succ', ih, nat.succ_eq_add_one, finsupp.single_add, X, monomial_mul_monomial, one_mul] }
end | lemma | mv_power_series.X_pow_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"
] | [
"finsupp.single_add",
"finsupp.single_zero",
"ih",
"mv_power_series",
"one_mul",
"pow_succ'",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_X_pow [decidable_eq σ] (m : σ →₀ ℕ) (s : σ) (n : ℕ) :
coeff R m ((X s : mv_power_series σ R)^n) = if m = single s n then 1 else 0 | by rw [X_pow_eq s n, coeff_monomial] | lemma | mv_power_series.coeff_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"
] | [
"mv_power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_mul_C (n : σ →₀ ℕ) (φ : mv_power_series σ R) (a : R) :
coeff R n (φ * C σ R a) = coeff R n φ * a | by simpa using coeff_add_mul_monomial n 0 φ a | lemma | mv_power_series.coeff_mul_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"
] | [
"mv_power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_C_mul (n : σ →₀ ℕ) (φ : mv_power_series σ R) (a : R) :
coeff R n (C σ R a * φ) = a * coeff R n φ | by simpa using coeff_add_monomial_mul 0 n φ a | lemma | mv_power_series.coeff_C_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"
] | [
"mv_power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_zero_mul_X (φ : mv_power_series σ R) (s : σ) :
coeff R (0 : σ →₀ ℕ) (φ * X s) = 0 | begin
have : ¬single s 1 ≤ 0, from λ h, by simpa using h s,
simp only [X, coeff_mul_monomial, if_neg this]
end | lemma | mv_power_series.coeff_zero_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"
] | [
"mv_power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_zero_X_mul (φ : mv_power_series σ R) (s : σ) :
coeff R (0 : σ →₀ ℕ) (X s * φ) = 0 | by rw [← (φ.commute_X s).eq, coeff_zero_mul_X] | lemma | mv_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"
] | [
"mv_power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
constant_coeff : (mv_power_series σ R) →+* R | { to_fun := coeff R (0 : σ →₀ ℕ),
map_one' := coeff_zero_one,
map_mul' := λ φ ψ, by simp [coeff_mul, support_single_ne_zero],
map_zero' := linear_map.map_zero _,
.. coeff R (0 : σ →₀ ℕ) } | def | mv_power_series.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"
] | [
"linear_map.map_zero",
"mv_power_series"
] | The constant coefficient of a formal power series. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff_zero_eq_constant_coeff :
⇑(coeff R (0 : σ →₀ ℕ)) = constant_coeff σ R | rfl | lemma | mv_power_series.coeff_zero_eq_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_zero_eq_constant_coeff_apply (φ : mv_power_series σ R) :
coeff R (0 : σ →₀ ℕ) φ = constant_coeff σ R φ | rfl | lemma | mv_power_series.coeff_zero_eq_constant_coeff_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"
] | [
"mv_power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
constant_coeff_C (a : R) : constant_coeff σ R (C σ R a) = a | rfl | lemma | mv_power_series.constant_coeff_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
constant_coeff_comp_C :
(constant_coeff σ R).comp (C σ R) = ring_hom.id R | rfl | lemma | mv_power_series.constant_coeff_comp_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"
] | [
"ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
constant_coeff_zero : constant_coeff σ R 0 = 0 | rfl | lemma | mv_power_series.constant_coeff_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 | |
constant_coeff_one : constant_coeff σ R 1 = 1 | rfl | lemma | mv_power_series.constant_coeff_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
constant_coeff_X (s : σ) : constant_coeff σ R (X s) = 0 | coeff_zero_X s | lemma | mv_power_series.constant_coeff_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 | |
is_unit_constant_coeff (φ : mv_power_series σ R) (h : is_unit φ) :
is_unit (constant_coeff σ R φ) | h.map _ | lemma | mv_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"
] | If a multivariate formal power series is invertible,
then so is its constant coefficient. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff_smul (f : mv_power_series σ R) (n) (a : R) :
coeff _ n (a • f) = a * coeff _ n f | rfl | lemma | mv_power_series.coeff_smul | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_eq_C_mul (f : mv_power_series σ R) (a : R) :
a • f = C σ R a * f | by { ext, simp } | lemma | mv_power_series.smul_eq_C_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"
] | [
"mv_power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
X_inj [nontrivial R] {s t : σ} : (X s : mv_power_series σ R) = X t ↔ s = t | ⟨begin
intro h, replace h := congr_arg (coeff R (single s 1)) h, rw [coeff_X, if_pos rfl, coeff_X] at h,
split_ifs at h with H,
{ rw finsupp.single_eq_single_iff at H,
cases H, { exact H.1 }, { exfalso, exact one_ne_zero H.1 } },
{ exfalso, exact one_ne_zero h }
end, congr_arg X⟩ | lemma | mv_power_series.X_inj | 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_single_iff",
"mv_power_series",
"nontrivial",
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map : mv_power_series σ R →+* mv_power_series σ S | { to_fun := λ φ n, f $ coeff R n φ,
map_zero' := ext $ λ n, f.map_zero,
map_one' := ext $ λ n, show f ((coeff R n) 1) = (coeff S n) 1,
by { rw [coeff_one, coeff_one], split_ifs; simp [f.map_one, f.map_zero] },
map_add' := λ φ ψ, ext $ λ n,
show f ((coeff R n) (φ + ψ)) = f ((coeff R n) φ) + f ((coeff R n) ... | def | mv_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"
] | The map between multivariate formal power series induced by a map on the coefficients. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_id : map σ (ring_hom.id R) = ring_hom.id _ | rfl | lemma | mv_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",
"ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comp : map σ (g.comp f) = (map σ g).comp (map σ f) | rfl | lemma | mv_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 : σ →₀ ℕ) (φ : mv_power_series σ R) :
coeff S n (map σ f φ) = f (coeff R n φ) | rfl | lemma | mv_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"
] | [
"mv_power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
constant_coeff_map (φ : mv_power_series σ R) :
constant_coeff σ S (map σ f φ) = f (constant_coeff σ R φ) | rfl | lemma | mv_power_series.constant_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"
] | [
"mv_power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_monomial (n : σ →₀ ℕ) (a : R) :
map σ f (monomial R n a) = monomial S n (f a) | by { ext m, simp [coeff_monomial, apply_ite f] } | lemma | mv_power_series.map_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"
] | [
"apply_ite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_C (a : R) : map σ f (C σ R a) = C σ S (f a) | map_monomial _ _ _ | lemma | mv_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_X (s : σ) : map σ f (X s) = X s | by simp [mv_power_series.X] | lemma | mv_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"
] | [
"mv_power_series.X"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
C_eq_algebra_map : C σ R = (algebra_map R (mv_power_series σ R)) | rfl | theorem | mv_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",
"mv_power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_apply {r : R} :
algebra_map R (mv_power_series σ A) r = C σ A (algebra_map R A r) | begin
change (mv_power_series.map σ (algebra_map R A)).comp (C σ R) r = _,
simp,
end | theorem | mv_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",
"mv_power_series.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trunc_fun (φ : mv_power_series σ R) : mv_polynomial σ R | ∑ m in finset.Iio n, mv_polynomial.monomial m (coeff R m φ) | def | mv_power_series.trunc_fun | 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.Iio",
"mv_polynomial",
"mv_polynomial.monomial",
"mv_power_series"
] | Auxiliary definition for the truncation function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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