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 |
|---|---|---|---|---|---|---|---|---|---|---|
monic.dvd_of_fraction_map_dvd_fraction_map [is_integrally_closed R] {p q : R[X]}
(hp : p.monic ) (hq : q.monic) (h : q.map (algebra_map R K) ∣ p.map (algebra_map R K)) : q ∣ p | begin
obtain ⟨r, hr⟩ := h,
obtain ⟨d', hr'⟩ := is_integrally_closed.eq_map_mul_C_of_dvd K hp (dvd_of_mul_left_eq _ hr.symm),
rw [monic.leading_coeff, C_1, mul_one] at hr',
rw [← hr', ← polynomial.map_mul] at hr,
exact dvd_of_mul_right_eq _ (polynomial.map_injective _ (is_fraction_ring.injective R K) hr.symm),... | theorem | polynomial.monic.dvd_of_fraction_map_dvd_fraction_map | ring_theory.polynomial | src/ring_theory/polynomial/gauss_lemma.lean | [
"field_theory.splitting_field.construction",
"ring_theory.int.basic",
"ring_theory.localization.integral",
"ring_theory.integrally_closed"
] | [
"algebra_map",
"is_fraction_ring.injective",
"is_integrally_closed",
"is_integrally_closed.eq_map_mul_C_of_dvd",
"mul_one",
"polynomial.map_injective",
"polynomial.map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monic.dvd_iff_fraction_map_dvd_fraction_map [is_integrally_closed R] {p q : R[X]}
(hp : p.monic ) (hq : q.monic) : q.map (algebra_map R K) ∣ p.map (algebra_map R K) ↔ q ∣ p | ⟨λ h, hp.dvd_of_fraction_map_dvd_fraction_map hq h,
λ ⟨a,b⟩, ⟨a.map (algebra_map R K), b.symm ▸ polynomial.map_mul (algebra_map R K)⟩⟩ | theorem | polynomial.monic.dvd_iff_fraction_map_dvd_fraction_map | ring_theory.polynomial | src/ring_theory/polynomial/gauss_lemma.lean | [
"field_theory.splitting_field.construction",
"ring_theory.int.basic",
"ring_theory.localization.integral",
"ring_theory.integrally_closed"
] | [
"algebra_map",
"is_integrally_closed",
"polynomial.map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit_or_eq_zero_of_is_unit_integer_normalization_prim_part
{p : K[X]} (h0 : p ≠ 0) (h : is_unit (integer_normalization R⁰ p).prim_part) :
is_unit p | begin
rcases is_unit_iff.1 h with ⟨_, ⟨u, rfl⟩, hu⟩,
obtain ⟨⟨c, c0⟩, hc⟩ := integer_normalization_map_to_map R⁰ p,
rw [subtype.coe_mk, algebra.smul_def, algebra_map_apply] at hc,
apply is_unit_of_mul_is_unit_right,
rw [← hc, (integer_normalization R⁰ p).eq_C_content_mul_prim_part, ← hu,
← ring_hom.map_mu... | lemma | polynomial.is_unit_or_eq_zero_of_is_unit_integer_normalization_prim_part | ring_theory.polynomial | src/ring_theory/polynomial/gauss_lemma.lean | [
"field_theory.splitting_field.construction",
"ring_theory.int.basic",
"ring_theory.localization.integral",
"ring_theory.integrally_closed"
] | [
"algebra.smul_def",
"algebra_map",
"algebra_map_apply",
"con",
"is_fraction_ring.injective",
"is_fraction_ring.integer_normalization_eq_zero_iff",
"is_unit",
"is_unit_of_mul_is_unit_right",
"mul_eq_zero",
"ring_hom.map_mul",
"subtype.coe_mk",
"units.ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_primitive.irreducible_iff_irreducible_map_fraction_map
{p : R[X]} (hp : p.is_primitive) :
irreducible p ↔ irreducible (p.map (algebra_map R K)) | begin
refine ⟨λ hi, ⟨λ h, hi.not_unit (hp.is_unit_iff_is_unit_map.2 h), λ a b hab, _⟩,
hp.irreducible_of_irreducible_map_of_injective (is_fraction_ring.injective _ _)⟩,
obtain ⟨⟨c, c0⟩, hc⟩ := integer_normalization_map_to_map R⁰ a,
obtain ⟨⟨d, d0⟩, hd⟩ := integer_normalization_map_to_map R⁰ b,
rw [algebra.s... | theorem | polynomial.is_primitive.irreducible_iff_irreducible_map_fraction_map | ring_theory.polynomial | src/ring_theory/polynomial/gauss_lemma.lean | [
"field_theory.splitting_field.construction",
"ring_theory.int.basic",
"ring_theory.localization.integral",
"ring_theory.integrally_closed"
] | [
"algebra.smul_def",
"algebra_map",
"algebra_map_apply",
"associated",
"con",
"dvd_dvd_iff_associated",
"irreducible",
"is_fraction_ring.injective",
"is_unit_of_mul_is_unit_right",
"mem_non_zero_divisors_iff_ne_zero",
"mul_assoc",
"mul_comm",
"mul_eq_zero",
"mul_left_cancel₀",
"mul_ne_zer... | **Gauss's Lemma** for GCD domains states that a primitive polynomial is irreducible iff it is
irreducible in the fraction field. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_primitive.dvd_of_fraction_map_dvd_fraction_map {p q : R[X]}
(hp : p.is_primitive) (hq : q.is_primitive)
(h_dvd : p.map (algebra_map R K) ∣ q.map (algebra_map R K)) : p ∣ q | begin
rcases h_dvd with ⟨r, hr⟩,
obtain ⟨⟨s, s0⟩, hs⟩ := integer_normalization_map_to_map R⁰ r,
rw [subtype.coe_mk, algebra.smul_def, algebra_map_apply] at hs,
have h : p ∣ q * C s,
{ use (integer_normalization R⁰ r),
apply map_injective (algebra_map R K) (is_fraction_ring.injective _ _),
rw [polynomi... | lemma | polynomial.is_primitive.dvd_of_fraction_map_dvd_fraction_map | ring_theory.polynomial | src/ring_theory/polynomial/gauss_lemma.lean | [
"field_theory.splitting_field.construction",
"ring_theory.int.basic",
"ring_theory.localization.integral",
"ring_theory.integrally_closed"
] | [
"algebra.smul_def",
"algebra_map",
"algebra_map_apply",
"associated.dvd_iff_dvd_right",
"is_fraction_ring.injective",
"mem_non_zero_divisors_iff_ne_zero",
"mul_assoc",
"mul_comm",
"mul_ne_zero",
"polynomial.map_mul",
"subtype.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_primitive.dvd_iff_fraction_map_dvd_fraction_map {p q : R[X]}
(hp : p.is_primitive) (hq : q.is_primitive) :
(p ∣ q) ↔ (p.map (algebra_map R K) ∣ q.map (algebra_map R K)) | ⟨λ ⟨a,b⟩, ⟨a.map (algebra_map R K), b.symm ▸ polynomial.map_mul (algebra_map R K)⟩,
λ h, hp.dvd_of_fraction_map_dvd_fraction_map hq h⟩ | lemma | polynomial.is_primitive.dvd_iff_fraction_map_dvd_fraction_map | ring_theory.polynomial | src/ring_theory/polynomial/gauss_lemma.lean | [
"field_theory.splitting_field.construction",
"ring_theory.int.basic",
"ring_theory.localization.integral",
"ring_theory.integrally_closed"
] | [
"algebra_map",
"polynomial.map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_primitive.int.irreducible_iff_irreducible_map_cast
{p : ℤ[X]} (hp : p.is_primitive) :
irreducible p ↔ irreducible (p.map (int.cast_ring_hom ℚ)) | hp.irreducible_iff_irreducible_map_fraction_map | theorem | polynomial.is_primitive.int.irreducible_iff_irreducible_map_cast | ring_theory.polynomial | src/ring_theory/polynomial/gauss_lemma.lean | [
"field_theory.splitting_field.construction",
"ring_theory.int.basic",
"ring_theory.localization.integral",
"ring_theory.integrally_closed"
] | [
"int.cast_ring_hom",
"irreducible"
] | **Gauss's Lemma** for `ℤ` states that a primitive integer polynomial is irreducible iff it is
irreducible over `ℚ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_primitive.int.dvd_iff_map_cast_dvd_map_cast (p q : ℤ[X])
(hp : p.is_primitive) (hq : q.is_primitive) :
(p ∣ q) ↔ (p.map (int.cast_ring_hom ℚ) ∣ q.map (int.cast_ring_hom ℚ)) | hp.dvd_iff_fraction_map_dvd_fraction_map ℚ hq | lemma | polynomial.is_primitive.int.dvd_iff_map_cast_dvd_map_cast | ring_theory.polynomial | src/ring_theory/polynomial/gauss_lemma.lean | [
"field_theory.splitting_field.construction",
"ring_theory.int.basic",
"ring_theory.localization.integral",
"ring_theory.integrally_closed"
] | [
"int.cast_ring_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_ring_equiv (R : Type*) [semiring R] : R[X]ᵐᵒᵖ ≃+* Rᵐᵒᵖ[X] | ((to_finsupp_iso R).op.trans add_monoid_algebra.op_ring_equiv).trans (to_finsupp_iso _).symm | def | polynomial.op_ring_equiv | ring_theory.polynomial | src/ring_theory/polynomial/opposites.lean | [
"data.polynomial.degree.definitions"
] | [
"add_monoid_algebra.op_ring_equiv",
"semiring"
] | Ring isomorphism between `R[X]ᵐᵒᵖ` and `Rᵐᵒᵖ[X]` sending each coefficient of a polynomial
to the corresponding element of the opposite ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_ring_equiv_op_monomial (n : ℕ) (r : R) :
op_ring_equiv R (op (monomial n r : R[X])) = monomial n (op r) | by simp only [op_ring_equiv, ring_equiv.trans_apply, ring_equiv.op_apply_apply,
ring_equiv.to_add_equiv_eq_coe, add_equiv.mul_op_apply, add_equiv.to_fun_eq_coe,
add_equiv.coe_trans, op_add_equiv_apply, ring_equiv.coe_to_add_equiv, op_add_equiv_symm_apply,
function.comp_app, unop_op, to_finsupp_iso_apply, to... | lemma | polynomial.op_ring_equiv_op_monomial | ring_theory.polynomial | src/ring_theory/polynomial/opposites.lean | [
"data.polynomial.degree.definitions"
] | [
"add_monoid_algebra.op_ring_equiv_single",
"ring_equiv.coe_to_add_equiv",
"ring_equiv.to_add_equiv_eq_coe",
"ring_equiv.trans_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_ring_equiv_op_C (a : R) :
op_ring_equiv R (op (C a)) = C (op a) | op_ring_equiv_op_monomial 0 a | lemma | polynomial.op_ring_equiv_op_C | ring_theory.polynomial | src/ring_theory/polynomial/opposites.lean | [
"data.polynomial.degree.definitions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_ring_equiv_op_X :
op_ring_equiv R (op (X : R[X])) = X | op_ring_equiv_op_monomial 1 1 | lemma | polynomial.op_ring_equiv_op_X | ring_theory.polynomial | src/ring_theory/polynomial/opposites.lean | [
"data.polynomial.degree.definitions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_ring_equiv_op_C_mul_X_pow (r : R) (n : ℕ) :
op_ring_equiv R (op (C r * X ^ n : R[X])) = C (op r) * X ^ n | by simp only [X_pow_mul, op_mul, op_pow, map_mul, map_pow, op_ring_equiv_op_X, op_ring_equiv_op_C] | lemma | polynomial.op_ring_equiv_op_C_mul_X_pow | ring_theory.polynomial | src/ring_theory/polynomial/opposites.lean | [
"data.polynomial.degree.definitions"
] | [
"map_mul",
"map_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_ring_equiv_symm_monomial (n : ℕ) (r : Rᵐᵒᵖ) :
(op_ring_equiv R).symm (monomial n r) = op (monomial n (unop r)) | (op_ring_equiv R).injective (by simp) | lemma | polynomial.op_ring_equiv_symm_monomial | ring_theory.polynomial | src/ring_theory/polynomial/opposites.lean | [
"data.polynomial.degree.definitions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_ring_equiv_symm_C (a : Rᵐᵒᵖ) :
(op_ring_equiv R).symm (C a) = op (C (unop a)) | op_ring_equiv_symm_monomial 0 a | lemma | polynomial.op_ring_equiv_symm_C | ring_theory.polynomial | src/ring_theory/polynomial/opposites.lean | [
"data.polynomial.degree.definitions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_ring_equiv_symm_X :
(op_ring_equiv R).symm (X : Rᵐᵒᵖ[X]) = op X | op_ring_equiv_symm_monomial 1 1 | lemma | polynomial.op_ring_equiv_symm_X | ring_theory.polynomial | src/ring_theory/polynomial/opposites.lean | [
"data.polynomial.degree.definitions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_ring_equiv_symm_C_mul_X_pow (r : Rᵐᵒᵖ) (n : ℕ) :
(op_ring_equiv R).symm (C r * X ^ n : Rᵐᵒᵖ[X]) = op (C (unop r) * X ^ n) | by rw [C_mul_X_pow_eq_monomial, op_ring_equiv_symm_monomial, ← C_mul_X_pow_eq_monomial] | lemma | polynomial.op_ring_equiv_symm_C_mul_X_pow | ring_theory.polynomial | src/ring_theory/polynomial/opposites.lean | [
"data.polynomial.degree.definitions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_op_ring_equiv (p : R[X]ᵐᵒᵖ) (n : ℕ) :
(op_ring_equiv R p).coeff n = op ((unop p).coeff n) | begin
induction p using mul_opposite.rec,
cases p,
refl
end | lemma | polynomial.coeff_op_ring_equiv | ring_theory.polynomial | src/ring_theory/polynomial/opposites.lean | [
"data.polynomial.degree.definitions"
] | [
"mul_opposite.rec"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_op_ring_equiv (p : R[X]ᵐᵒᵖ) :
(op_ring_equiv R p).support = (unop p).support | begin
induction p using mul_opposite.rec,
cases p,
exact finsupp.support_map_range_of_injective _ _ op_injective
end | lemma | polynomial.support_op_ring_equiv | ring_theory.polynomial | src/ring_theory/polynomial/opposites.lean | [
"data.polynomial.degree.definitions"
] | [
"finsupp.support_map_range_of_injective",
"mul_opposite.rec"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_degree_op_ring_equiv (p : R[X]ᵐᵒᵖ) :
(op_ring_equiv R p).nat_degree = (unop p).nat_degree | begin
by_cases p0 : p = 0,
{ simp only [p0, _root_.map_zero, nat_degree_zero, unop_zero] },
{ simp only [p0, nat_degree_eq_support_max', ne.def, add_equiv_class.map_eq_zero_iff,
not_false_iff, support_op_ring_equiv, unop_eq_zero_iff] }
end | lemma | polynomial.nat_degree_op_ring_equiv | ring_theory.polynomial | src/ring_theory/polynomial/opposites.lean | [
"data.polynomial.degree.definitions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
leading_coeff_op_ring_equiv (p : R[X]ᵐᵒᵖ) :
(op_ring_equiv R p).leading_coeff = op (unop p).leading_coeff | by rw [leading_coeff, coeff_op_ring_equiv, nat_degree_op_ring_equiv, leading_coeff] | lemma | polynomial.leading_coeff_op_ring_equiv | ring_theory.polynomial | src/ring_theory/polynomial/opposites.lean | [
"data.polynomial.degree.definitions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pochhammer : ℕ → S[X] | | 0 := 1
| (n+1) := X * (pochhammer n).comp (X + 1) | def | pochhammer | ring_theory.polynomial | src/ring_theory/polynomial/pochhammer.lean | [
"tactic.abel",
"data.polynomial.eval"
] | [] | `pochhammer S n` is the polynomial `X * (X+1) * ... * (X + n - 1)`,
with coefficients in the semiring `S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pochhammer_zero : pochhammer S 0 = 1 | rfl | lemma | pochhammer_zero | ring_theory.polynomial | src/ring_theory/polynomial/pochhammer.lean | [
"tactic.abel",
"data.polynomial.eval"
] | [
"pochhammer"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pochhammer_one : pochhammer S 1 = X | by simp [pochhammer] | lemma | pochhammer_one | ring_theory.polynomial | src/ring_theory/polynomial/pochhammer.lean | [
"tactic.abel",
"data.polynomial.eval"
] | [
"pochhammer"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pochhammer_succ_left (n : ℕ) : pochhammer S (n+1) = X * (pochhammer S n).comp (X+1) | by rw pochhammer | lemma | pochhammer_succ_left | ring_theory.polynomial | src/ring_theory/polynomial/pochhammer.lean | [
"tactic.abel",
"data.polynomial.eval"
] | [
"pochhammer"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pochhammer_map (f : S →+* T) (n : ℕ) : (pochhammer S n).map f = pochhammer T n | begin
induction n with n ih,
{ simp, },
{ simp [ih, pochhammer_succ_left, map_comp], },
end | lemma | pochhammer_map | ring_theory.polynomial | src/ring_theory/polynomial/pochhammer.lean | [
"tactic.abel",
"data.polynomial.eval"
] | [
"ih",
"map_comp",
"pochhammer",
"pochhammer_succ_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pochhammer_eval_cast (n k : ℕ) :
((pochhammer ℕ n).eval k : S) = (pochhammer S n).eval k | begin
rw [←pochhammer_map (algebra_map ℕ S), eval_map, ←eq_nat_cast (algebra_map ℕ S),
eval₂_at_nat_cast, nat.cast_id, eq_nat_cast],
end | lemma | pochhammer_eval_cast | ring_theory.polynomial | src/ring_theory/polynomial/pochhammer.lean | [
"tactic.abel",
"data.polynomial.eval"
] | [
"algebra_map",
"eq_nat_cast",
"nat.cast_id",
"pochhammer"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pochhammer_eval_zero {n : ℕ} : (pochhammer S n).eval 0 = if n = 0 then 1 else 0 | begin
cases n,
{ simp, },
{ simp [X_mul, nat.succ_ne_zero, pochhammer_succ_left], }
end | lemma | pochhammer_eval_zero | ring_theory.polynomial | src/ring_theory/polynomial/pochhammer.lean | [
"tactic.abel",
"data.polynomial.eval"
] | [
"pochhammer",
"pochhammer_succ_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pochhammer_zero_eval_zero : (pochhammer S 0).eval 0 = 1 | by simp | lemma | pochhammer_zero_eval_zero | ring_theory.polynomial | src/ring_theory/polynomial/pochhammer.lean | [
"tactic.abel",
"data.polynomial.eval"
] | [
"pochhammer"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pochhammer_ne_zero_eval_zero {n : ℕ} (h : n ≠ 0) : (pochhammer S n).eval 0 = 0 | by simp [pochhammer_eval_zero, h] | lemma | pochhammer_ne_zero_eval_zero | ring_theory.polynomial | src/ring_theory/polynomial/pochhammer.lean | [
"tactic.abel",
"data.polynomial.eval"
] | [
"pochhammer",
"pochhammer_eval_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pochhammer_succ_right (n : ℕ) : pochhammer S (n+1) = pochhammer S n * (X + n) | begin
suffices h : pochhammer ℕ (n+1) = pochhammer ℕ n * (X + n),
{ apply_fun polynomial.map (algebra_map ℕ S) at h,
simpa only [pochhammer_map, polynomial.map_mul, polynomial.map_add,
map_X, polynomial.map_nat_cast] using h },
induction n with n ih,
{ simp, },
{ conv_lhs
{ rw [pochhamme... | lemma | pochhammer_succ_right | ring_theory.polynomial | src/ring_theory/polynomial/pochhammer.lean | [
"tactic.abel",
"data.polynomial.eval"
] | [
"algebra_map",
"ih",
"nat.cast_succ",
"pochhammer",
"pochhammer_map",
"pochhammer_succ_left",
"polynomial.map",
"polynomial.map_add",
"polynomial.map_mul",
"polynomial.map_nat_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pochhammer_succ_eval {S : Type*} [semiring S] (n : ℕ) (k : S) :
(pochhammer S (n + 1)).eval k = (pochhammer S n).eval k * (k + n) | by rw [pochhammer_succ_right, mul_add, eval_add, eval_mul_X, ← nat.cast_comm, ← C_eq_nat_cast,
eval_C_mul, nat.cast_comm, ← mul_add] | lemma | pochhammer_succ_eval | ring_theory.polynomial | src/ring_theory/polynomial/pochhammer.lean | [
"tactic.abel",
"data.polynomial.eval"
] | [
"nat.cast_comm",
"pochhammer",
"pochhammer_succ_right",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pochhammer_succ_comp_X_add_one (n : ℕ) :
(pochhammer S (n + 1)).comp (X + 1) =
pochhammer S (n + 1) + (n + 1) • (pochhammer S n).comp (X + 1) | begin
suffices : (pochhammer ℕ (n + 1)).comp (X + 1) =
pochhammer ℕ (n + 1) + (n + 1) * (pochhammer ℕ n).comp (X + 1),
{ simpa [map_comp] using congr_arg (polynomial.map (nat.cast_ring_hom S)) this },
nth_rewrite 1 pochhammer_succ_left,
rw [← add_mul, pochhammer_succ_right ℕ n, mul_comp, mul_comm,... | lemma | pochhammer_succ_comp_X_add_one | ring_theory.polynomial | src/ring_theory/polynomial/pochhammer.lean | [
"tactic.abel",
"data.polynomial.eval"
] | [
"map_comp",
"mul_comm",
"nat.cast_ring_hom",
"pochhammer",
"pochhammer_succ_left",
"pochhammer_succ_right",
"polynomial.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
polynomial.mul_X_add_nat_cast_comp {p q : S[X]} {n : ℕ} :
(p * (X + n)).comp q = (p.comp q) * (q + n) | by rw [mul_add, add_comp, mul_X_comp, ←nat.cast_comm, nat_cast_mul_comp, nat.cast_comm, mul_add] | lemma | polynomial.mul_X_add_nat_cast_comp | ring_theory.polynomial | src/ring_theory/polynomial/pochhammer.lean | [
"tactic.abel",
"data.polynomial.eval"
] | [
"nat.cast_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pochhammer_mul (n m : ℕ) :
pochhammer S n * (pochhammer S m).comp (X + n) = pochhammer S (n + m) | begin
induction m with m ih,
{ simp, },
{ rw [pochhammer_succ_right, polynomial.mul_X_add_nat_cast_comp, ←mul_assoc, ih,
nat.succ_eq_add_one, ←add_assoc, pochhammer_succ_right, nat.cast_add, add_assoc], }
end | lemma | pochhammer_mul | ring_theory.polynomial | src/ring_theory/polynomial/pochhammer.lean | [
"tactic.abel",
"data.polynomial.eval"
] | [
"ih",
"nat.cast_add",
"pochhammer",
"pochhammer_succ_right",
"polynomial.mul_X_add_nat_cast_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pochhammer_nat_eq_asc_factorial (n : ℕ) :
∀ k, (pochhammer ℕ k).eval (n + 1) = n.asc_factorial k | | 0 := by erw [eval_one]; refl
| (t + 1) := begin
rw [pochhammer_succ_right, eval_mul, pochhammer_nat_eq_asc_factorial t],
suffices : n.asc_factorial t * (n + 1 + t) = n.asc_factorial (t + 1), by simpa,
rw [nat.asc_factorial_succ, add_right_comm, mul_comm]
end | lemma | pochhammer_nat_eq_asc_factorial | ring_theory.polynomial | src/ring_theory/polynomial/pochhammer.lean | [
"tactic.abel",
"data.polynomial.eval"
] | [
"mul_comm",
"nat.asc_factorial_succ",
"pochhammer",
"pochhammer_succ_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pochhammer_nat_eq_desc_factorial (a b : ℕ) :
(pochhammer ℕ b).eval a = (a + b - 1).desc_factorial b | begin
cases b,
{ rw [nat.desc_factorial_zero, pochhammer_zero, polynomial.eval_one] },
rw [nat.add_succ, nat.succ_sub_succ, tsub_zero],
cases a,
{ rw [pochhammer_ne_zero_eval_zero _ b.succ_ne_zero, zero_add,
nat.desc_factorial_of_lt b.lt_succ_self] },
{ rw [nat.succ_add, ←nat.add_succ, nat.add_desc_fact... | lemma | pochhammer_nat_eq_desc_factorial | ring_theory.polynomial | src/ring_theory/polynomial/pochhammer.lean | [
"tactic.abel",
"data.polynomial.eval"
] | [
"nat.add_desc_factorial_eq_asc_factorial",
"nat.desc_factorial_zero",
"pochhammer",
"pochhammer_nat_eq_asc_factorial",
"pochhammer_ne_zero_eval_zero",
"pochhammer_zero",
"polynomial.eval_one",
"tsub_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pochhammer_pos (n : ℕ) (s : S) (h : 0 < s) : 0 < (pochhammer S n).eval s | begin
induction n with n ih,
{ simp only [nat.nat_zero_eq_zero, pochhammer_zero, eval_one], exact zero_lt_one, },
{ rw [pochhammer_succ_right, mul_add, eval_add, ←nat.cast_comm, eval_nat_cast_mul, eval_mul_X,
nat.cast_comm, ←mul_add],
exact mul_pos ih
(lt_of_lt_of_le h ((le_add_iff_nonneg_right _)... | lemma | pochhammer_pos | ring_theory.polynomial | src/ring_theory/polynomial/pochhammer.lean | [
"tactic.abel",
"data.polynomial.eval"
] | [
"ih",
"nat.cast_comm",
"nat.cast_nonneg",
"pochhammer",
"pochhammer_succ_right",
"pochhammer_zero",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pochhammer_eval_one (S : Type*) [semiring S] (n : ℕ) :
(pochhammer S n).eval (1 : S) = (n! : S) | by rw_mod_cast [pochhammer_nat_eq_asc_factorial, nat.zero_asc_factorial] | lemma | pochhammer_eval_one | ring_theory.polynomial | src/ring_theory/polynomial/pochhammer.lean | [
"tactic.abel",
"data.polynomial.eval"
] | [
"nat.zero_asc_factorial",
"pochhammer",
"pochhammer_nat_eq_asc_factorial",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
factorial_mul_pochhammer (S : Type*) [semiring S] (r n : ℕ) :
(r! : S) * (pochhammer S n).eval (r + 1) = (r + n)! | by rw_mod_cast [pochhammer_nat_eq_asc_factorial, nat.factorial_mul_asc_factorial] | lemma | factorial_mul_pochhammer | ring_theory.polynomial | src/ring_theory/polynomial/pochhammer.lean | [
"tactic.abel",
"data.polynomial.eval"
] | [
"nat.factorial_mul_asc_factorial",
"pochhammer",
"pochhammer_nat_eq_asc_factorial",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pochhammer_nat_eval_succ (r : ℕ) :
∀ n : ℕ, n * (pochhammer ℕ r).eval (n + 1) = (n + r) * (pochhammer ℕ r).eval n | | 0 := begin
by_cases h : r = 0,
{ simp only [h, zero_mul, zero_add], },
{ simp only [pochhammer_eval_zero, zero_mul, if_neg h, mul_zero], }
end
| (k + 1) := by simp only [pochhammer_nat_eq_asc_factorial, nat.succ_asc_factorial, add_right_comm] | lemma | pochhammer_nat_eval_succ | ring_theory.polynomial | src/ring_theory/polynomial/pochhammer.lean | [
"tactic.abel",
"data.polynomial.eval"
] | [
"mul_zero",
"nat.succ_asc_factorial",
"pochhammer",
"pochhammer_eval_zero",
"pochhammer_nat_eq_asc_factorial",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pochhammer_eval_succ (r n : ℕ) :
(n : S) * (pochhammer S r).eval (n + 1 : S) = (n + r) * (pochhammer S r).eval n | by exact_mod_cast congr_arg nat.cast (pochhammer_nat_eval_succ r n) | lemma | pochhammer_eval_succ | ring_theory.polynomial | src/ring_theory/polynomial/pochhammer.lean | [
"tactic.abel",
"data.polynomial.eval"
] | [
"nat.cast",
"pochhammer",
"pochhammer_nat_eval_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_span_X_sub_C_alg_equiv (x : R) :
(R[X] ⧸ ideal.span ({X - C x} : set R[X])) ≃ₐ[R] R | (ideal.quotient_equiv_alg_of_eq R
(by exact ker_eval_ring_hom x : ring_hom.ker (aeval x).to_ring_hom = _)).symm.trans $
ideal.quotient_ker_alg_equiv_of_right_inverse $ λ _, eval_C | def | polynomial.quotient_span_X_sub_C_alg_equiv | ring_theory.polynomial | src/ring_theory/polynomial/quotient.lean | [
"data.polynomial.div",
"ring_theory.polynomial.basic",
"ring_theory.ideal.quotient_operations"
] | [
"ideal.quotient_equiv_alg_of_eq",
"ideal.quotient_ker_alg_equiv_of_right_inverse",
"ideal.span",
"ring_hom.ker"
] | For a commutative ring $R$, evaluating a polynomial at an element $x \in R$ induces an
isomorphism of $R$-algebras $R[X] / \langle X - x \rangle \cong R$. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_span_X_sub_C_alg_equiv_mk (x : R) (p : R[X]) :
quotient_span_X_sub_C_alg_equiv x (ideal.quotient.mk _ p) = p.eval x | rfl | lemma | polynomial.quotient_span_X_sub_C_alg_equiv_mk | ring_theory.polynomial | src/ring_theory/polynomial/quotient.lean | [
"data.polynomial.div",
"ring_theory.polynomial.basic",
"ring_theory.ideal.quotient_operations"
] | [
"ideal.quotient.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_span_X_sub_C_alg_equiv_symm_apply (x : R) (y : R) :
(quotient_span_X_sub_C_alg_equiv x).symm y = algebra_map R _ y | rfl | lemma | polynomial.quotient_span_X_sub_C_alg_equiv_symm_apply | ring_theory.polynomial | src/ring_theory/polynomial/quotient.lean | [
"data.polynomial.div",
"ring_theory.polynomial.basic",
"ring_theory.ideal.quotient_operations"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_span_C_X_sub_C_alg_equiv (x y : R) :
(R[X] ⧸ (ideal.span {C x, X - C y} : ideal R[X])) ≃ₐ[R] R ⧸ (ideal.span {x} : ideal R) | (ideal.quotient_equiv_alg_of_eq R $ by rw [ideal.span_insert, sup_comm]).trans $
(double_quot.quot_quot_equiv_quot_supₐ R _ _).symm.trans $
(ideal.quotient_equiv_alg _ _ (quotient_span_X_sub_C_alg_equiv y) rfl).trans $
ideal.quotient_equiv_alg_of_eq R $
by { simp only [ideal.map_span, set.image_sing... | def | polynomial.quotient_span_C_X_sub_C_alg_equiv | ring_theory.polynomial | src/ring_theory/polynomial/quotient.lean | [
"data.polynomial.div",
"ring_theory.polynomial.basic",
"ring_theory.ideal.quotient_operations"
] | [
"double_quot.quot_quot_equiv_quot_supₐ",
"ideal",
"ideal.map_span",
"ideal.quotient_equiv_alg",
"ideal.quotient_equiv_alg_of_eq",
"ideal.span",
"ideal.span_insert",
"set.image_singleton",
"sup_comm"
] | For a commutative ring $R$, evaluating a polynomial at an element $y \in R$ induces an
isomorphism of $R$-algebras $R[X] / \langle x, X - y \rangle \cong R / \langle x \rangle$. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_map_C_eq_zero {I : ideal R} :
∀ a ∈ I, ((quotient.mk (map (C : R →+* R[X]) I : ideal R[X])).comp C) a = 0 | begin
intros a ha,
rw [ring_hom.comp_apply, quotient.eq_zero_iff_mem],
exact mem_map_of_mem _ ha,
end | lemma | ideal.quotient_map_C_eq_zero | ring_theory.polynomial | src/ring_theory/polynomial/quotient.lean | [
"data.polynomial.div",
"ring_theory.polynomial.basic",
"ring_theory.ideal.quotient_operations"
] | [
"ideal",
"ring_hom.comp_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eval₂_C_mk_eq_zero {I : ideal R} :
∀ f ∈ (map (C : R →+* R[X]) I : ideal R[X]), eval₂_ring_hom (C.comp (quotient.mk I)) X f = 0 | begin
intros a ha,
rw ← sum_monomial_eq a,
dsimp,
rw eval₂_sum,
refine finset.sum_eq_zero (λ n hn, _),
dsimp,
rw eval₂_monomial (C.comp (quotient.mk I)) X,
refine mul_eq_zero_of_left (polynomial.ext (λ m, _)) (X ^ n),
erw coeff_C,
by_cases h : m = 0,
{ simpa [h] using quotient.eq_zero_iff_mem.2 ((... | lemma | ideal.eval₂_C_mk_eq_zero | ring_theory.polynomial | src/ring_theory/polynomial/quotient.lean | [
"data.polynomial.div",
"ring_theory.polynomial.basic",
"ring_theory.ideal.quotient_operations"
] | [
"ideal",
"mul_eq_zero_of_left",
"polynomial.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
polynomial_quotient_equiv_quotient_polynomial (I : ideal R) :
(R ⧸ I)[X] ≃+* R[X] ⧸ (map C I : ideal R[X]) | { to_fun := eval₂_ring_hom
(quotient.lift I ((quotient.mk (map C I : ideal R[X])).comp C) quotient_map_C_eq_zero)
((quotient.mk (map C I : ideal R[X]) X)),
inv_fun := quotient.lift (map C I : ideal R[X])
(eval₂_ring_hom (C.comp (quotient.mk I)) X) eval₂_C_mk_eq_zero,
map_mul' := λ f g, by simp only [coe... | def | ideal.polynomial_quotient_equiv_quotient_polynomial | ring_theory.polynomial | src/ring_theory/polynomial/quotient.lean | [
"data.polynomial.div",
"ring_theory.polynomial.basic",
"ring_theory.ideal.quotient_operations"
] | [
"ideal",
"inv_fun",
"polynomial.induction_on'",
"quotient.lift_mk",
"ring_hom.coe_comp",
"ring_hom.map_add",
"ring_hom.map_mul",
"ring_hom.map_pow",
"submodule.quotient.quot_mk_eq_mk"
] | If `I` is an ideal of `R`, then the ring polynomials over the quotient ring `I.quotient` is
isomorphic to the quotient of `R[X]` by the ideal `map C I`,
where `map C I` contains exactly the polynomials whose coefficients all lie in `I` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
polynomial_quotient_equiv_quotient_polynomial_symm_mk (I : ideal R) (f : R[X]) :
I.polynomial_quotient_equiv_quotient_polynomial.symm (quotient.mk _ f) = f.map (quotient.mk I) | by rw [polynomial_quotient_equiv_quotient_polynomial, ring_equiv.symm_mk, ring_equiv.coe_mk,
ideal.quotient.lift_mk, coe_eval₂_ring_hom, eval₂_eq_eval_map, ←polynomial.map_map,
←eval₂_eq_eval_map, polynomial.eval₂_C_X] | lemma | ideal.polynomial_quotient_equiv_quotient_polynomial_symm_mk | ring_theory.polynomial | src/ring_theory/polynomial/quotient.lean | [
"data.polynomial.div",
"ring_theory.polynomial.basic",
"ring_theory.ideal.quotient_operations"
] | [
"ideal",
"ideal.quotient.lift_mk",
"polynomial.eval₂_C_X",
"ring_equiv.coe_mk",
"ring_equiv.symm_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
polynomial_quotient_equiv_quotient_polynomial_map_mk (I : ideal R) (f : R[X]) :
I.polynomial_quotient_equiv_quotient_polynomial (f.map I^.quotient.mk) = quotient.mk _ f | begin
apply (polynomial_quotient_equiv_quotient_polynomial I).symm.injective,
rw [ring_equiv.symm_apply_apply, polynomial_quotient_equiv_quotient_polynomial_symm_mk],
end | lemma | ideal.polynomial_quotient_equiv_quotient_polynomial_map_mk | ring_theory.polynomial | src/ring_theory/polynomial/quotient.lean | [
"data.polynomial.div",
"ring_theory.polynomial.basic",
"ring_theory.ideal.quotient_operations"
] | [
"ideal",
"ring_equiv.symm_apply_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_domain_map_C_quotient {P : ideal R} (H : is_prime P) :
is_domain (R[X] ⧸ (map (C : R →+* R[X]) P : ideal R[X])) | ring_equiv.is_domain (polynomial (R ⧸ P))
(polynomial_quotient_equiv_quotient_polynomial P).symm | lemma | ideal.is_domain_map_C_quotient | ring_theory.polynomial | src/ring_theory/polynomial/quotient.lean | [
"data.polynomial.div",
"ring_theory.polynomial.basic",
"ring_theory.ideal.quotient_operations"
] | [
"ideal",
"is_domain",
"polynomial",
"ring_equiv.is_domain"
] | If `P` is a prime ideal of `R`, then `R[x]/(P)` is an integral domain. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_zero_of_polynomial_mem_map_range (I : ideal R[X])
(x : ((quotient.mk I).comp C).range)
(hx : C x ∈ (I.map (polynomial.map_ring_hom ((quotient.mk I).comp C).range_restrict))) :
x = 0 | begin
let i := ((quotient.mk I).comp C).range_restrict,
have hi' : (polynomial.map_ring_hom i).ker ≤ I,
{ refine λ f hf, polynomial_mem_ideal_of_coeff_mem_ideal I f (λ n, _),
rw [mem_comap, ← quotient.eq_zero_iff_mem, ← ring_hom.comp_apply],
rw [ring_hom.mem_ker, coe_map_ring_hom] at hf,
replace hf :=... | lemma | ideal.eq_zero_of_polynomial_mem_map_range | ring_theory.polynomial | src/ring_theory/polynomial/quotient.lean | [
"data.polynomial.div",
"ring_theory.polynomial.basic",
"ring_theory.ideal.quotient_operations"
] | [
"ideal",
"polynomial",
"polynomial.map_ring_hom",
"polynomial.map_surjective",
"ring_hom.coe_range_restrict",
"ring_hom.comp_apply",
"ring_hom.map_sub",
"ring_hom.mem_ker",
"ring_hom.mem_range",
"subtype.ext_iff",
"subtype.val_eq_coe"
] | Given any ring `R` and an ideal `I` of `R[X]`, we get a map `R → R[x] → R[x]/I`.
If we let `R` be the image of `R` in `R[x]/I` then we also have a map `R[x] → R'[x]`.
In particular we can map `I` across this map, to get `I'` and a new map `R' → R'[x] → R'[x]/I`.
This theorem shows `I'` will not contain any non-ze... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_map_C_eq_zero {I : ideal R} {i : R} (hi : i ∈ I) :
(ideal.quotient.mk (ideal.map (C : R →+* mv_polynomial σ R) I :
ideal (mv_polynomial σ R))).comp C i = 0 | begin
simp only [function.comp_app, ring_hom.coe_comp, ideal.quotient.eq_zero_iff_mem],
exact ideal.mem_map_of_mem _ hi
end | lemma | mv_polynomial.quotient_map_C_eq_zero | ring_theory.polynomial | src/ring_theory/polynomial/quotient.lean | [
"data.polynomial.div",
"ring_theory.polynomial.basic",
"ring_theory.ideal.quotient_operations"
] | [
"ideal",
"ideal.map",
"ideal.mem_map_of_mem",
"ideal.quotient.eq_zero_iff_mem",
"ideal.quotient.mk",
"mv_polynomial",
"ring_hom.coe_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eval₂_C_mk_eq_zero {I : ideal R} {a : mv_polynomial σ R}
(ha : a ∈ (ideal.map (C : R →+* mv_polynomial σ R) I : ideal (mv_polynomial σ R))) :
eval₂_hom (C.comp (ideal.quotient.mk I)) X a = 0 | begin
rw as_sum a,
rw [coe_eval₂_hom, eval₂_sum],
refine finset.sum_eq_zero (λ n hn, _),
simp only [eval₂_monomial, function.comp_app, ring_hom.coe_comp],
refine mul_eq_zero_of_left _ _,
suffices : coeff n a ∈ I,
{ rw [← @ideal.mk_ker R _ I, ring_hom.mem_ker] at this,
simp only [this, C_0] },
exact ... | lemma | mv_polynomial.eval₂_C_mk_eq_zero | ring_theory.polynomial | src/ring_theory/polynomial/quotient.lean | [
"data.polynomial.div",
"ring_theory.polynomial.basic",
"ring_theory.ideal.quotient_operations"
] | [
"ideal",
"ideal.map",
"ideal.mk_ker",
"ideal.quotient.mk",
"mul_eq_zero_of_left",
"mv_polynomial",
"ring_hom.coe_comp",
"ring_hom.mem_ker"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_equiv_quotient_mv_polynomial (I : ideal R) :
mv_polynomial σ (R ⧸ I) ≃ₐ[R]
mv_polynomial σ R ⧸ (ideal.map C I : ideal (mv_polynomial σ R)) | { to_fun := eval₂_hom (ideal.quotient.lift I ((ideal.quotient.mk (ideal.map C I : ideal
(mv_polynomial σ R))).comp C) (λ i hi, quotient_map_C_eq_zero hi))
(λ i, ideal.quotient.mk (ideal.map C I : ideal (mv_polynomial σ R)) (X i)),
inv_fun := ideal.quotient.lift (ideal.map C I : ideal (mv_polynomial σ R))
... | def | mv_polynomial.quotient_equiv_quotient_mv_polynomial | ring_theory.polynomial | src/ring_theory/polynomial/quotient.lean | [
"data.polynomial.div",
"ring_theory.polynomial.basic",
"ring_theory.ideal.quotient_operations"
] | [
"ideal",
"ideal.map",
"ideal.quotient.lift",
"ideal.quotient.lift_mk",
"ideal.quotient.mk",
"ideal.quotient.mk_eq_mk",
"inv_fun",
"mv_polynomial",
"mv_polynomial.C_inj",
"mv_polynomial.coe_eval₂_hom",
"mv_polynomial.eval₂_add",
"mv_polynomial.eval₂_hom_C",
"ring_hom.coe_comp",
"ring_hom.ma... | If `I` is an ideal of `R`, then the ring `mv_polynomial σ I.quotient` is isomorphic as an
`R`-algebra to the quotient of `mv_polynomial σ R` by the ideal generated by `I`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
scale_roots_aeval_eq_zero_of_aeval_mk'_eq_zero {p : A[X]} {r : A} {s : M}
(hr : aeval (mk' S r s) p = 0) :
aeval (algebra_map A S r) (scale_roots p s) = 0 | begin
convert scale_roots_eval₂_eq_zero (algebra_map A S) hr,
rw [aeval_def, mk'_spec' _ r s]
end | lemma | scale_roots_aeval_eq_zero_of_aeval_mk'_eq_zero | ring_theory.polynomial | src/ring_theory/polynomial/rational_root.lean | [
"ring_theory.integrally_closed",
"ring_theory.localization.num_denom",
"ring_theory.polynomial.scale_roots"
] | [
"algebra_map",
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
num_is_root_scale_roots_of_aeval_eq_zero
[unique_factorization_monoid A] {p : A[X]} {x : K} (hr : aeval x p = 0) :
is_root (scale_roots p (denom A x)) (num A x) | begin
apply is_root_of_eval₂_map_eq_zero (is_fraction_ring.injective A K),
refine scale_roots_aeval_eq_zero_of_aeval_mk'_eq_zero _,
rw mk'_num_denom,
exact hr
end | lemma | num_is_root_scale_roots_of_aeval_eq_zero | ring_theory.polynomial | src/ring_theory/polynomial/rational_root.lean | [
"ring_theory.integrally_closed",
"ring_theory.localization.num_denom",
"ring_theory.polynomial.scale_roots"
] | [
"is_fraction_ring.injective",
"num",
"scale_roots_aeval_eq_zero_of_aeval_mk'_eq_zero",
"unique_factorization_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
num_dvd_of_is_root {p : A[X]} {r : K} (hr : aeval r p = 0) :
num A r ∣ p.coeff 0 | begin
suffices : num A r ∣ (scale_roots p (denom A r)).coeff 0,
{ simp only [coeff_scale_roots, tsub_zero] at this,
haveI := classical.prop_decidable,
by_cases hr : num A r = 0,
{ obtain ⟨u, hu⟩ := (is_unit_denom_of_num_eq_zero hr).pow p.nat_degree,
rw ←hu at this,
exact units.dvd_mul_right.... | theorem | num_dvd_of_is_root | ring_theory.polynomial | src/ring_theory/polynomial/rational_root.lean | [
"ring_theory.integrally_closed",
"ring_theory.localization.num_denom",
"ring_theory.polynomial.scale_roots"
] | [
"dvd_mul_of_dvd_right",
"mul_one",
"num",
"num_is_root_scale_roots_of_aeval_eq_zero",
"pow_dvd_pow",
"pow_one",
"pow_zero",
"tsub_zero"
] | Rational root theorem part 1:
if `r : f.codomain` is a root of a polynomial over the ufd `A`,
then the numerator of `r` divides the constant coefficient | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
denom_dvd_of_is_root {p : A[X]} {r : K} (hr : aeval r p = 0) :
(denom A r : A) ∣ p.leading_coeff | begin
suffices : (denom A r : A) ∣ p.leading_coeff * num A r ^ p.nat_degree,
{ refine dvd_of_dvd_mul_left_of_no_prime_factors
(mem_non_zero_divisors_iff_ne_zero.mp (denom A r).2) _ this,
intros q dvd_denom dvd_num_pow hq,
apply hq.not_unit,
exact num_denom_reduced A r (hq.dvd_of_dvd_pow dvd_num_po... | theorem | denom_dvd_of_is_root | ring_theory.polynomial | src/ring_theory/polynomial/rational_root.lean | [
"ring_theory.integrally_closed",
"ring_theory.localization.num_denom",
"ring_theory.polynomial.scale_roots"
] | [
"dvd_mul_of_dvd_right",
"dvd_zero",
"num",
"num_is_root_scale_roots_of_aeval_eq_zero",
"pow_dvd_pow",
"pow_one",
"zero_mul"
] | Rational root theorem part 2:
if `r : f.codomain` is a root of a polynomial over the ufd `A`,
then the denominator of `r` divides the leading coefficient | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_integer_of_is_root_of_monic {p : A[X]} (hp : monic p) {r : K}
(hr : aeval r p = 0) : is_integer A r | is_integer_of_is_unit_denom (is_unit_of_dvd_one _ (hp ▸ denom_dvd_of_is_root hr)) | theorem | is_integer_of_is_root_of_monic | ring_theory.polynomial | src/ring_theory/polynomial/rational_root.lean | [
"ring_theory.integrally_closed",
"ring_theory.localization.num_denom",
"ring_theory.polynomial.scale_roots"
] | [
"denom_dvd_of_is_root",
"is_unit_of_dvd_one"
] | Integral root theorem:
if `r : f.codomain` is a root of a monic polynomial over the ufd `A`,
then `r` is an integer | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
integer_of_integral {x : K} :
is_integral A x → is_integer A x | λ ⟨p, hp, hx⟩, is_integer_of_is_root_of_monic hp hx | lemma | unique_factorization_monoid.integer_of_integral | ring_theory.polynomial | src/ring_theory/polynomial/rational_root.lean | [
"ring_theory.integrally_closed",
"ring_theory.localization.num_denom",
"ring_theory.polynomial.scale_roots"
] | [
"is_integer_of_is_root_of_monic",
"is_integral"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
scale_roots (p : R[X]) (s : R) : R[X] | ∑ i in p.support, monomial i (p.coeff i * s ^ (p.nat_degree - i)) | def | polynomial.scale_roots | ring_theory.polynomial | src/ring_theory/polynomial/scale_roots.lean | [
"ring_theory.non_zero_divisors",
"data.polynomial.algebra_map"
] | [] | `scale_roots p s` is a polynomial with root `r * s` for each root `r` of `p`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff_scale_roots (p : R[X]) (s : R) (i : ℕ) :
(scale_roots p s).coeff i = coeff p i * s ^ (p.nat_degree - i) | by simp [scale_roots, coeff_monomial] {contextual := tt} | lemma | polynomial.coeff_scale_roots | ring_theory.polynomial | src/ring_theory/polynomial/scale_roots.lean | [
"ring_theory.non_zero_divisors",
"data.polynomial.algebra_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_scale_roots_nat_degree (p : R[X]) (s : R) :
(scale_roots p s).coeff p.nat_degree = p.leading_coeff | by rw [leading_coeff, coeff_scale_roots, tsub_self, pow_zero, mul_one] | lemma | polynomial.coeff_scale_roots_nat_degree | ring_theory.polynomial | src/ring_theory/polynomial/scale_roots.lean | [
"ring_theory.non_zero_divisors",
"data.polynomial.algebra_map"
] | [
"mul_one",
"pow_zero",
"tsub_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_scale_roots (s : R) : scale_roots 0 s = 0 | by { ext, simp } | lemma | polynomial.zero_scale_roots | ring_theory.polynomial | src/ring_theory/polynomial/scale_roots.lean | [
"ring_theory.non_zero_divisors",
"data.polynomial.algebra_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
scale_roots_ne_zero {p : R[X]} (hp : p ≠ 0) (s : R) :
scale_roots p s ≠ 0 | begin
intro h,
have : p.coeff p.nat_degree ≠ 0 := mt leading_coeff_eq_zero.mp hp,
have : (scale_roots p s).coeff p.nat_degree = 0 :=
congr_fun (congr_arg (coeff : R[X] → ℕ → R) h) p.nat_degree,
rw [coeff_scale_roots_nat_degree] at this,
contradiction
end | lemma | polynomial.scale_roots_ne_zero | ring_theory.polynomial | src/ring_theory/polynomial/scale_roots.lean | [
"ring_theory.non_zero_divisors",
"data.polynomial.algebra_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_scale_roots_le (p : R[X]) (s : R) :
(scale_roots p s).support ≤ p.support | by { intro, simpa using left_ne_zero_of_mul } | lemma | polynomial.support_scale_roots_le | ring_theory.polynomial | src/ring_theory/polynomial/scale_roots.lean | [
"ring_theory.non_zero_divisors",
"data.polynomial.algebra_map"
] | [
"left_ne_zero_of_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_scale_roots_eq (p : R[X]) {s : R} (hs : s ∈ non_zero_divisors R) :
(scale_roots p s).support = p.support | le_antisymm (support_scale_roots_le p s)
begin
intro i,
simp only [coeff_scale_roots, polynomial.mem_support_iff],
intros p_ne_zero ps_zero,
have := pow_mem hs (p.nat_degree - i) _ ps_zero,
contradiction
end | lemma | polynomial.support_scale_roots_eq | ring_theory.polynomial | src/ring_theory/polynomial/scale_roots.lean | [
"ring_theory.non_zero_divisors",
"data.polynomial.algebra_map"
] | [
"non_zero_divisors",
"polynomial.mem_support_iff",
"pow_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
degree_scale_roots (p : R[X]) {s : R} :
degree (scale_roots p s) = degree p | begin
haveI := classical.prop_decidable,
by_cases hp : p = 0,
{ rw [hp, zero_scale_roots] },
have := scale_roots_ne_zero hp s,
refine le_antisymm (finset.sup_mono (support_scale_roots_le p s)) (degree_le_degree _),
rw coeff_scale_roots_nat_degree,
intro h,
have := leading_coeff_eq_zero.mp h,
contradic... | lemma | polynomial.degree_scale_roots | ring_theory.polynomial | src/ring_theory/polynomial/scale_roots.lean | [
"ring_theory.non_zero_divisors",
"data.polynomial.algebra_map"
] | [
"finset.sup_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_degree_scale_roots (p : R[X]) (s : R) :
nat_degree (scale_roots p s) = nat_degree p | by simp only [nat_degree, degree_scale_roots] | lemma | polynomial.nat_degree_scale_roots | ring_theory.polynomial | src/ring_theory/polynomial/scale_roots.lean | [
"ring_theory.non_zero_divisors",
"data.polynomial.algebra_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monic_scale_roots_iff {p : R[X]} (s : R) :
monic (scale_roots p s) ↔ monic p | by simp only [monic, leading_coeff, nat_degree_scale_roots, coeff_scale_roots_nat_degree] | lemma | polynomial.monic_scale_roots_iff | ring_theory.polynomial | src/ring_theory/polynomial/scale_roots.lean | [
"ring_theory.non_zero_divisors",
"data.polynomial.algebra_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
scale_roots_eval₂_mul {p : S[X]} (f : S →+* R)
(r : R) (s : S) :
eval₂ f (f s * r) (scale_roots p s) = f s ^ p.nat_degree * eval₂ f r p | calc eval₂ f (f s * r) (scale_roots p s) =
(scale_roots p s).support.sum (λ i, f (coeff p i * s ^ (p.nat_degree - i)) * (f s * r) ^ i) :
by simp [eval₂_eq_sum, sum_def]
... = p.support.sum (λ i, f (coeff p i * s ^ (p.nat_degree - i)) * (f s * r) ^ i) :
finset.sum_subset (support_scale_roots_le p s)
(λ i hi hi',... | lemma | polynomial.scale_roots_eval₂_mul | ring_theory.polynomial | src/ring_theory/polynomial/scale_roots.lean | [
"ring_theory.non_zero_divisors",
"data.polynomial.algebra_map"
] | [
"mul_assoc",
"mul_left_comm",
"mul_pow",
"pow_add",
"tsub_add_cancel_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
scale_roots_eval₂_eq_zero {p : S[X]} (f : S →+* R)
{r : R} {s : S} (hr : eval₂ f r p = 0) :
eval₂ f (f s * r) (scale_roots p s) = 0 | by rw [scale_roots_eval₂_mul, hr, _root_.mul_zero] | lemma | polynomial.scale_roots_eval₂_eq_zero | ring_theory.polynomial | src/ring_theory/polynomial/scale_roots.lean | [
"ring_theory.non_zero_divisors",
"data.polynomial.algebra_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
scale_roots_aeval_eq_zero [algebra S R] {p : S[X]}
{r : R} {s : S} (hr : aeval r p = 0) :
aeval (algebra_map S R s * r) (scale_roots p s) = 0 | scale_roots_eval₂_eq_zero (algebra_map S R) hr | lemma | polynomial.scale_roots_aeval_eq_zero | ring_theory.polynomial | src/ring_theory/polynomial/scale_roots.lean | [
"ring_theory.non_zero_divisors",
"data.polynomial.algebra_map"
] | [
"algebra",
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
scale_roots_eval₂_eq_zero_of_eval₂_div_eq_zero
{p : A[X]} {f : A →+* K} (hf : function.injective f)
{r s : A} (hr : eval₂ f (f r / f s) p = 0) (hs : s ∈ non_zero_divisors A) :
eval₂ f (f r) (scale_roots p s) = 0 | begin
convert scale_roots_eval₂_eq_zero f hr,
rw [←mul_div_assoc, mul_comm, mul_div_cancel],
exact map_ne_zero_of_mem_non_zero_divisors _ hf hs
end | lemma | polynomial.scale_roots_eval₂_eq_zero_of_eval₂_div_eq_zero | ring_theory.polynomial | src/ring_theory/polynomial/scale_roots.lean | [
"ring_theory.non_zero_divisors",
"data.polynomial.algebra_map"
] | [
"map_ne_zero_of_mem_non_zero_divisors",
"mul_comm",
"mul_div_cancel",
"non_zero_divisors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
scale_roots_aeval_eq_zero_of_aeval_div_eq_zero [algebra A K]
(inj : function.injective (algebra_map A K)) {p : A[X]} {r s : A}
(hr : aeval (algebra_map A K r / algebra_map A K s) p = 0) (hs : s ∈ non_zero_divisors A) :
aeval (algebra_map A K r) (scale_roots p s) = 0 | scale_roots_eval₂_eq_zero_of_eval₂_div_eq_zero inj hr hs | lemma | polynomial.scale_roots_aeval_eq_zero_of_aeval_div_eq_zero | ring_theory.polynomial | src/ring_theory/polynomial/scale_roots.lean | [
"ring_theory.non_zero_divisors",
"data.polynomial.algebra_map"
] | [
"algebra",
"algebra_map",
"non_zero_divisors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_scale_roots (p : R[X]) (x : R) (f : R →+* S) (h : f p.leading_coeff ≠ 0) :
(p.scale_roots x).map f = (p.map f).scale_roots (f x) | begin
ext,
simp [polynomial.nat_degree_map_of_leading_coeff_ne_zero _ h],
end | lemma | polynomial.map_scale_roots | ring_theory.polynomial | src/ring_theory/polynomial/scale_roots.lean | [
"ring_theory.non_zero_divisors",
"data.polynomial.algebra_map"
] | [
"polynomial.nat_degree_map_of_leading_coeff_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
X_pow_sub_X_sub_one_irreducible_aux (z : ℂ) : ¬ (z ^ n = z + 1 ∧ z ^ n + z ^ 2 = 0) | begin
rintros ⟨h1, h2⟩,
replace h3 : z ^ 3 = 1,
{ linear_combination (1 - z - z ^ 2 - z ^ n) * h1 + (z ^ n - 2) * h2 }, -- thanks polyrith!
have key : z ^ n = 1 ∨ z ^ n = z ∨ z ^ n = z ^ 2,
{ rw [←nat.mod_add_div n 3, pow_add, pow_mul, h3, one_pow, mul_one],
have : n % 3 < 3 := nat.mod_lt n zero_lt_three,... | lemma | polynomial.X_pow_sub_X_sub_one_irreducible_aux | ring_theory.polynomial | src/ring_theory/polynomial/selmer.lean | [
"data.polynomial.unit_trinomial",
"ring_theory.polynomial.gauss_lemma",
"tactic.linear_combination"
] | [
"add_self_eq_zero",
"mul_one",
"one_ne_zero",
"one_pow",
"pow_add",
"pow_eq_zero",
"pow_mul",
"pow_one",
"pow_zero",
"zero_lt_three",
"zero_ne_one",
"zero_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
X_pow_sub_X_sub_one_irreducible (hn1 : n ≠ 1) : irreducible (X ^ n - X - 1 : ℤ[X]) | begin
by_cases hn0 : n = 0,
{ rw [hn0, pow_zero, sub_sub, add_comm, ←sub_sub, sub_self, zero_sub],
exact associated.irreducible ⟨-1, mul_neg_one X⟩ irreducible_X },
have hn : 1 < n := nat.one_lt_iff_ne_zero_and_ne_one.mpr ⟨hn0, hn1⟩,
have hp : (X ^ n - X - 1 : ℤ[X]) = trinomial 0 1 n (-1) (-1) 1 :=
by s... | lemma | polynomial.X_pow_sub_X_sub_one_irreducible | ring_theory.polynomial | src/ring_theory/polynomial/selmer.lean | [
"data.polynomial.unit_trinomial",
"ring_theory.polynomial.gauss_lemma",
"tactic.linear_combination"
] | [
"associated.irreducible",
"irreducible",
"map_one",
"mul_assoc",
"mul_eq_zero_of_left",
"mul_neg_one",
"ne_zero",
"pow_zero",
"ring",
"units.coe_neg",
"units.coe_one",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
X_pow_sub_X_sub_one_irreducible_rat (hn1 : n ≠ 1) : irreducible (X ^ n - X - 1 : ℚ[X]) | begin
by_cases hn0 : n = 0,
{ rw [hn0, pow_zero, sub_sub, add_comm, ←sub_sub, sub_self, zero_sub],
exact associated.irreducible ⟨-1, mul_neg_one X⟩ irreducible_X },
have hp : (X ^ n - X - 1 : ℤ[X]) = trinomial 0 1 n (-1) (-1) 1 :=
by simp only [trinomial, C_neg, C_1]; ring,
have hn : 1 < n := nat.one_lt_i... | lemma | polynomial.X_pow_sub_X_sub_one_irreducible_rat | ring_theory.polynomial | src/ring_theory/polynomial/selmer.lean | [
"data.polynomial.unit_trinomial",
"ring_theory.polynomial.gauss_lemma",
"tactic.linear_combination"
] | [
"associated.irreducible",
"irreducible",
"mul_neg_one",
"polynomial.map_X",
"polynomial.map_one",
"polynomial.map_pow",
"polynomial.map_sub",
"pow_zero",
"ring",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aeval_map_algebra_map (x : B) (p : R[X]) :
aeval x (map (algebra_map R A) p) = aeval x p | by rw [aeval_def, aeval_def, eval₂_map, is_scalar_tower.algebra_map_eq R A B] | theorem | polynomial.aeval_map_algebra_map | ring_theory.polynomial | src/ring_theory/polynomial/tower.lean | [
"algebra.algebra.tower",
"data.polynomial.algebra_map"
] | [
"algebra_map",
"is_scalar_tower.algebra_map_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aeval_algebra_map_apply (x : A) (p : R[X]) :
aeval (algebra_map A B x) p = algebra_map A B (aeval x p) | by rw [aeval_def, aeval_def, hom_eval₂, ←is_scalar_tower.algebra_map_eq] | lemma | polynomial.aeval_algebra_map_apply | ring_theory.polynomial | src/ring_theory/polynomial/tower.lean | [
"algebra.algebra.tower",
"data.polynomial.algebra_map"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aeval_algebra_map_eq_zero_iff [no_zero_smul_divisors A B] [nontrivial B]
(x : A) (p : R[X]) :
aeval (algebra_map A B x) p = 0 ↔ aeval x p = 0 | by rw [aeval_algebra_map_apply, algebra.algebra_map_eq_smul_one, smul_eq_zero,
iff_false_intro (one_ne_zero' B), or_false] | lemma | polynomial.aeval_algebra_map_eq_zero_iff | ring_theory.polynomial | src/ring_theory/polynomial/tower.lean | [
"algebra.algebra.tower",
"data.polynomial.algebra_map"
] | [
"algebra.algebra_map_eq_smul_one",
"algebra_map",
"no_zero_smul_divisors",
"nontrivial",
"one_ne_zero'",
"smul_eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aeval_algebra_map_eq_zero_iff_of_injective
{x : A} {p : R[X]}
(h : function.injective (algebra_map A B)) :
aeval (algebra_map A B x) p = 0 ↔ aeval x p = 0 | by rw [aeval_algebra_map_apply, ← (algebra_map A B).map_zero, h.eq_iff] | lemma | polynomial.aeval_algebra_map_eq_zero_iff_of_injective | ring_theory.polynomial | src/ring_theory/polynomial/tower.lean | [
"algebra.algebra.tower",
"data.polynomial.algebra_map"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aeval_coe (S : subalgebra R A) (x : S) (p : R[X]) :
aeval (x : A) p = aeval x p | aeval_algebra_map_apply A x p | lemma | subalgebra.aeval_coe | ring_theory.polynomial | src/ring_theory/polynomial/tower.lean | [
"algebra.algebra.tower",
"data.polynomial.algebra_map"
] | [
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_X_add_C_eq_sum_esymm (s : multiset R) :
(s.map (λ r, X + C r)).prod =
∑ j in finset.range (s.card + 1), C (s.esymm j) * X ^ (s.card - j) | begin
classical,
rw [prod_map_add, antidiagonal_eq_map_powerset, map_map, ←bind_powerset_len, function.comp,
map_bind, sum_bind, finset.sum_eq_multiset_sum, finset.range_val, map_congr (eq.refl _)],
intros _ _,
rw [esymm, ←sum_hom', ←sum_map_mul_right, map_congr (eq.refl _)],
intros _ ht,
rw mem_powerse... | lemma | multiset.prod_X_add_C_eq_sum_esymm | ring_theory.polynomial | src/ring_theory/polynomial/vieta.lean | [
"data.polynomial.splits",
"ring_theory.mv_polynomial.symmetric"
] | [
"finset.range",
"finset.range_val",
"map_bind",
"map_congr",
"multiset"
] | A sum version of Vieta's formula for `multiset`: the product of the linear terms `X + λ` where
`λ` runs through a multiset `s` is equal to a linear combination of the symmetric functions
`esymm s` of the `λ`'s . | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_X_add_C_coeff (s : multiset R) {k : ℕ} (h : k ≤ s.card) :
(s.map (λ r, X + C r)).prod.coeff k = s.esymm (s.card - k) | begin
convert polynomial.ext_iff.mp (prod_X_add_C_eq_sum_esymm s) k,
simp_rw [finset_sum_coeff, coeff_C_mul_X_pow],
rw finset.sum_eq_single_of_mem (s.card - k) _,
{ rw if_pos (nat.sub_sub_self h).symm, },
{ intros j hj1 hj2,
suffices : k ≠ card s - j,
{ rw if_neg this, },
{ intro hn,
rw [hn,... | lemma | multiset.prod_X_add_C_coeff | ring_theory.polynomial | src/ring_theory/polynomial/vieta.lean | [
"data.polynomial.splits",
"ring_theory.mv_polynomial.symmetric"
] | [
"finset.mem_range",
"multiset"
] | Vieta's formula for the coefficients of the product of linear terms `X + λ` where `λ` runs
through a multiset `s` : the `k`th coefficient is the symmetric function `esymm (card s - k) s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_X_add_C_coeff' {σ} (s : multiset σ) (r : σ → R) {k : ℕ} (h : k ≤ s.card) :
(s.map (λ i, X + C (r i))).prod.coeff k = (s.map r).esymm (s.card - k) | by rw [← map_map (λ r, X + C r) r, prod_X_add_C_coeff]; rwa s.card_map r | lemma | multiset.prod_X_add_C_coeff' | ring_theory.polynomial | src/ring_theory/polynomial/vieta.lean | [
"data.polynomial.splits",
"ring_theory.mv_polynomial.symmetric"
] | [
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.finset.prod_X_add_C_coeff {σ} (s : finset σ) (r : σ → R) {k : ℕ} (h : k ≤ s.card) :
(∏ i in s, (X + C (r i))).coeff k = ∑ t in s.powerset_len (s.card - k), ∏ i in t, r i | by { rw [finset.prod, prod_X_add_C_coeff' _ r h, finset.esymm_map_val], refl } | lemma | finset.prod_X_add_C_coeff | ring_theory.polynomial | src/ring_theory/polynomial/vieta.lean | [
"data.polynomial.splits",
"ring_theory.mv_polynomial.symmetric"
] | [
"finset",
"finset.esymm_map_val",
"finset.prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
esymm_neg (s : multiset R) (k : ℕ) :
(map has_neg.neg s).esymm k = (-1) ^ k * esymm s k | begin
rw [esymm, esymm, ←multiset.sum_map_mul_left, multiset.powerset_len_map, multiset.map_map,
map_congr (eq.refl _)],
intros x hx,
rw [(by { exact (mem_powerset_len.mp hx).right.symm }), ←prod_replicate, ←multiset.map_const],
nth_rewrite 2 ←map_id' x,
rw [←prod_map_mul, map_congr (eq.refl _)],
exact ... | lemma | multiset.esymm_neg | ring_theory.polynomial | src/ring_theory/polynomial/vieta.lean | [
"data.polynomial.splits",
"ring_theory.mv_polynomial.symmetric"
] | [
"map_congr",
"multiset",
"multiset.map_map",
"multiset.powerset_len_map",
"neg_one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_X_sub_C_eq_sum_esymm (s : multiset R) :
(s.map (λ t, X - C t)).prod =
∑ j in finset.range (s.card + 1), (-1) ^ j * (C (s.esymm j) * X ^ (s.card - j)) | begin
conv_lhs { congr, congr, funext, rw sub_eq_add_neg, rw ←map_neg C _, },
convert prod_X_add_C_eq_sum_esymm (map (λ t, -t) s) using 1,
{ rwa map_map, },
{ simp only [esymm_neg, card_map, mul_assoc, map_mul, map_pow, map_neg, map_one], },
end | lemma | multiset.prod_X_sub_C_eq_sum_esymm | ring_theory.polynomial | src/ring_theory/polynomial/vieta.lean | [
"data.polynomial.splits",
"ring_theory.mv_polynomial.symmetric"
] | [
"finset.range",
"map_mul",
"map_one",
"map_pow",
"mul_assoc",
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_X_sub_C_coeff (s : multiset R) {k : ℕ} (h : k ≤ s.card) :
(s.map (λ t, X - C t)).prod.coeff k = (-1) ^ (s.card - k) * s.esymm (s.card - k) | begin
conv_lhs { congr, congr, congr, funext, rw sub_eq_add_neg, rw ←map_neg C _, },
convert prod_X_add_C_coeff (map (λ t, -t) s) _ using 1,
{ rwa map_map, },
{ rwa [esymm_neg, card_map] },
{ rwa card_map },
end | lemma | multiset.prod_X_sub_C_coeff | ring_theory.polynomial | src/ring_theory/polynomial/vieta.lean | [
"data.polynomial.splits",
"ring_theory.mv_polynomial.symmetric"
] | [
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.polynomial.coeff_eq_esymm_roots_of_card [is_domain R] {p : R[X]}
(hroots : p.roots.card = p.nat_degree) {k : ℕ} (h : k ≤ p.nat_degree) :
p.coeff k = p.leading_coeff * (-1) ^ (p.nat_degree - k) * p.roots.esymm (p.nat_degree - k) | begin
conv_lhs { rw ← C_leading_coeff_mul_prod_multiset_X_sub_C hroots },
rw [coeff_C_mul, mul_assoc], congr,
convert p.roots.prod_X_sub_C_coeff _ using 3; rw hroots, exact h,
end | theorem | polynomial.coeff_eq_esymm_roots_of_card | ring_theory.polynomial | src/ring_theory/polynomial/vieta.lean | [
"data.polynomial.splits",
"ring_theory.mv_polynomial.symmetric"
] | [
"is_domain",
"mul_assoc"
] | Vieta's formula for the coefficients and the roots of a polynomial over an integral domain
with as many roots as its degree. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.polynomial.coeff_eq_esymm_roots_of_splits {F} [field F] {p : F[X]}
(hsplit : p.splits (ring_hom.id F)) {k : ℕ} (h : k ≤ p.nat_degree) :
p.coeff k = p.leading_coeff * (-1) ^ (p.nat_degree - k) * p.roots.esymm (p.nat_degree - k) | polynomial.coeff_eq_esymm_roots_of_card (splits_iff_card_roots.1 hsplit) h | theorem | polynomial.coeff_eq_esymm_roots_of_splits | ring_theory.polynomial | src/ring_theory/polynomial/vieta.lean | [
"data.polynomial.splits",
"ring_theory.mv_polynomial.symmetric"
] | [
"field",
"polynomial.coeff_eq_esymm_roots_of_card",
"ring_hom.id"
] | Vieta's formula for split polynomials over a field. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mv_polynomial.prod_C_add_X_eq_sum_esymm :
∏ i : σ, (X + C (mv_polynomial.X i)) =
∑ j in range (card σ + 1), (C (mv_polynomial.esymm σ R j) * X ^ (card σ - j)) | begin
let s := finset.univ.val.map (λ i : σ, mv_polynomial.X i),
rw (_ : card σ = s.card),
{ simp_rw [mv_polynomial.esymm_eq_multiset_esymm σ R, finset.prod_eq_multiset_prod],
convert multiset.prod_X_add_C_eq_sum_esymm s,
rwa multiset.map_map, },
{ rw multiset.card_map, refl, }
end | lemma | mv_polynomial.prod_C_add_X_eq_sum_esymm | ring_theory.polynomial | src/ring_theory/polynomial/vieta.lean | [
"data.polynomial.splits",
"ring_theory.mv_polynomial.symmetric"
] | [
"finset.prod_eq_multiset_prod",
"multiset.card_map",
"multiset.map_map",
"multiset.prod_X_add_C_eq_sum_esymm",
"mv_polynomial.X",
"mv_polynomial.esymm",
"mv_polynomial.esymm_eq_multiset_esymm"
] | A sum version of Vieta's formula for `mv_polynomial`: viewing `X i` as variables,
the product of linear terms `λ + X i` is equal to a linear combination of
the symmetric polynomials `esymm σ R j`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mv_polynomial.prod_X_add_C_coeff (k : ℕ) (h : k ≤ card σ) :
(∏ i : σ, (X + C (mv_polynomial.X i))).coeff k = mv_polynomial.esymm σ R (card σ - k) | begin
let s := finset.univ.val.map (λ i, (mv_polynomial.X i : mv_polynomial σ R)),
rw (_ : card σ = s.card) at ⊢ h,
{ rw [mv_polynomial.esymm_eq_multiset_esymm σ R, finset.prod_eq_multiset_prod],
convert multiset.prod_X_add_C_coeff s h,
rwa multiset.map_map },
repeat { rw multiset.card_map, refl, },
end | lemma | mv_polynomial.prod_X_add_C_coeff | ring_theory.polynomial | src/ring_theory/polynomial/vieta.lean | [
"data.polynomial.splits",
"ring_theory.mv_polynomial.symmetric"
] | [
"finset.prod_eq_multiset_prod",
"multiset.card_map",
"multiset.map_map",
"multiset.prod_X_add_C_coeff",
"mv_polynomial",
"mv_polynomial.X",
"mv_polynomial.esymm",
"mv_polynomial.esymm_eq_multiset_esymm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cyclotomic' (n : ℕ) (R : Type*) [comm_ring R] [is_domain R] : R[X] | ∏ μ in primitive_roots n R, (X - C μ) | def | polynomial.cyclotomic' | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"comm_ring",
"is_domain",
"primitive_roots"
] | The modified `n`-th cyclotomic polynomial with coefficients in `R`, it is the usual cyclotomic
polynomial if there is a primitive `n`-th root of unity in `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cyclotomic'_zero
(R : Type*) [comm_ring R] [is_domain R] : cyclotomic' 0 R = 1 | by simp only [cyclotomic', finset.prod_empty, primitive_roots_zero] | lemma | polynomial.cyclotomic'_zero | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"comm_ring",
"finset.prod_empty",
"is_domain",
"primitive_roots_zero"
] | The zeroth modified cyclotomic polyomial is `1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cyclotomic'_one
(R : Type*) [comm_ring R] [is_domain R] : cyclotomic' 1 R = X - 1 | begin
simp only [cyclotomic', finset.prod_singleton, ring_hom.map_one,
is_primitive_root.primitive_roots_one]
end | lemma | polynomial.cyclotomic'_one | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"comm_ring",
"finset.prod_singleton",
"is_domain",
"is_primitive_root.primitive_roots_one",
"ring_hom.map_one"
] | The first modified cyclotomic polyomial is `X - 1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.