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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|---|---|---|---|---|---|---|---|---|---|---|
iterate_derivative_at_1_ne_zero [char_zero R] (n ν : ℕ) (h : ν ≤ n) :
(polynomial.derivative^[n-ν] (bernstein_polynomial R n ν)).eval 1 ≠ 0 | begin
rw [bernstein_polynomial.iterate_derivative_at_1 _ _ _ h, ne.def, neg_one_pow_mul_eq_zero_iff,
←nat.cast_succ, ←pochhammer_eval_cast, ←nat.cast_zero, nat.cast_inj],
exact (pochhammer_pos _ _ (nat.succ_pos ν)).ne',
end | lemma | bernstein_polynomial.iterate_derivative_at_1_ne_zero | ring_theory.polynomial | src/ring_theory/polynomial/bernstein.lean | [
"data.polynomial.derivative",
"data.nat.choose.sum",
"ring_theory.polynomial.pochhammer",
"data.polynomial.algebra_map",
"linear_algebra.linear_independent",
"data.mv_polynomial.pderiv"
] | [
"bernstein_polynomial",
"bernstein_polynomial.iterate_derivative_at_1",
"char_zero",
"nat.cast_inj",
"neg_one_pow_mul_eq_zero_iff",
"pochhammer_pos",
"polynomial.derivative"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_independent_aux (n k : ℕ) (h : k ≤ n + 1):
linear_independent ℚ (λ ν : fin k, bernstein_polynomial ℚ n ν) | begin
induction k with k ih,
{ apply linear_independent_empty_type, },
{ apply linear_independent_fin_succ'.mpr,
fsplit,
{ exact ih (le_of_lt h), },
{ -- The actual work!
-- We show that the (n-k)-th derivative at 1 doesn't vanish,
-- but vanishes for everything in the span.
clear ih... | lemma | bernstein_polynomial.linear_independent_aux | ring_theory.polynomial | src/ring_theory/polynomial/bernstein.lean | [
"data.polynomial.derivative",
"data.nat.choose.sum",
"ring_theory.polynomial.pochhammer",
"data.polynomial.algebra_map",
"linear_algebra.linear_independent",
"data.mv_polynomial.pderiv"
] | [
"bernstein_polynomial",
"fin.coe_last",
"fin.coe_mk",
"fin.init_def",
"ih",
"linear_independent",
"linear_independent_empty_type",
"linear_map.pow_apply",
"not_and",
"not_exists",
"polynomial.derivative",
"polynomial.eval",
"submodule.mem_map",
"submodule.span_image",
"tsub_lt_tsub_iff_l... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_independent (n : ℕ) :
linear_independent ℚ (λ ν : fin (n+1), bernstein_polynomial ℚ n ν) | linear_independent_aux n (n+1) le_rfl | lemma | bernstein_polynomial.linear_independent | ring_theory.polynomial | src/ring_theory/polynomial/bernstein.lean | [
"data.polynomial.derivative",
"data.nat.choose.sum",
"ring_theory.polynomial.pochhammer",
"data.polynomial.algebra_map",
"linear_algebra.linear_independent",
"data.mv_polynomial.pderiv"
] | [
"bernstein_polynomial",
"le_rfl",
"linear_independent"
] | The Bernstein polynomials are linearly independent.
We prove by induction that the collection of `bernstein_polynomial n ν` for `ν = 0, ..., k`
are linearly independent.
The inductive step relies on the observation that the `(n-k)`-th derivative, evaluated at 1,
annihilates `bernstein_polynomial n ν` for `ν < k`, but ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sum (n : ℕ) : ∑ ν in finset.range (n + 1), bernstein_polynomial R n ν = 1 | calc ∑ ν in finset.range (n + 1), bernstein_polynomial R n ν = (X + (1 - X)) ^ n :
by { rw add_pow, simp only [bernstein_polynomial, mul_comm, mul_assoc, mul_left_comm] }
... = 1 : by simp | lemma | bernstein_polynomial.sum | ring_theory.polynomial | src/ring_theory/polynomial/bernstein.lean | [
"data.polynomial.derivative",
"data.nat.choose.sum",
"ring_theory.polynomial.pochhammer",
"data.polynomial.algebra_map",
"linear_algebra.linear_independent",
"data.mv_polynomial.pderiv"
] | [
"add_pow",
"bernstein_polynomial",
"finset.range",
"mul_assoc",
"mul_comm",
"mul_left_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_smul (n : ℕ) :
∑ ν in finset.range (n + 1), ν • bernstein_polynomial R n ν = n • X | begin
-- We calculate the `x`-derivative of `(x+y)^n`, evaluated at `y=(1-x)`,
-- either directly or by using the binomial theorem.
-- We'll work in `mv_polynomial bool R`.
let x : mv_polynomial bool R := mv_polynomial.X tt,
let y : mv_polynomial bool R := mv_polynomial.X ff,
have pderiv_tt_x : pderiv tt... | lemma | bernstein_polynomial.sum_smul | ring_theory.polynomial | src/ring_theory/polynomial/bernstein.lean | [
"data.polynomial.derivative",
"data.nat.choose.sum",
"ring_theory.polynomial.pochhammer",
"data.polynomial.algebra_map",
"linear_algebra.linear_independent",
"data.mv_polynomial.pderiv"
] | [
"add_pow",
"algebra.id.smul_eq_mul",
"bernstein_polynomial",
"bool.cond_ff",
"bool.cond_tt",
"derivation.leibniz_pow",
"derivation.map_coe_nat",
"finset.range",
"finset.sum_mul",
"map_mul",
"map_nat_cast",
"map_pow",
"mul_one",
"mul_zero",
"mv_polynomial",
"mv_polynomial.X",
"mv_poly... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_mul_smul (n : ℕ) :
∑ ν in finset.range (n + 1), (ν * (ν-1)) • bernstein_polynomial R n ν = (n * (n-1)) • X^2 | begin
-- We calculate the second `x`-derivative of `(x+y)^n`, evaluated at `y=(1-x)`,
-- either directly or by using the binomial theorem.
-- We'll work in `mv_polynomial bool R`.
let x : mv_polynomial bool R := mv_polynomial.X tt,
let y : mv_polynomial bool R := mv_polynomial.X ff,
have pderiv_tt_x : pde... | lemma | bernstein_polynomial.sum_mul_smul | ring_theory.polynomial | src/ring_theory/polynomial/bernstein.lean | [
"data.polynomial.derivative",
"data.nat.choose.sum",
"ring_theory.polynomial.pochhammer",
"data.polynomial.algebra_map",
"linear_algebra.linear_independent",
"data.mv_polynomial.pderiv"
] | [
"add_pow",
"algebra.id.smul_eq_mul",
"bernstein_polynomial",
"bool.cond_ff",
"bool.cond_tt",
"derivation.leibniz",
"derivation.leibniz_pow",
"derivation.map_coe_nat",
"derivation.map_smul_of_tower",
"finset.range",
"finset.sum_mul",
"map_mul",
"map_nat_cast",
"map_nsmul",
"map_pow",
"m... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
variance (n : ℕ) :
∑ ν in finset.range (n+1), (n • polynomial.X - ν)^2 * bernstein_polynomial R n ν =
n • polynomial.X * (1 - polynomial.X) | begin
have p :
(finset.range (n+1)).sum (λ ν, (ν * (ν-1)) • bernstein_polynomial R n ν) +
(1 - (2 * n) • polynomial.X) * (finset.range (n+1)).sum (λ ν, ν • bernstein_polynomial R n ν) +
(n^2 • X^2) * (finset.range (n+1)).sum (λ ν, bernstein_polynomial R n ν) = _ := rfl,
conv at p { to_lhs,
rw [finse... | lemma | bernstein_polynomial.variance | ring_theory.polynomial | src/ring_theory/polynomial/bernstein.lean | [
"data.polynomial.derivative",
"data.nat.choose.sum",
"ring_theory.polynomial.pochhammer",
"data.polynomial.algebra_map",
"linear_algebra.linear_independent",
"data.mv_polynomial.pderiv"
] | [
"bernstein_polynomial",
"finset.mul_sum",
"finset.range",
"polynomial.X",
"ring"
] | A certain linear combination of the previous three identities,
which we'll want later. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
T : ℕ → R[X] | | 0 := 1
| 1 := X
| (n + 2) := 2 * X * T (n + 1) - T n | def | polynomial.chebyshev.T | ring_theory.polynomial | src/ring_theory/polynomial/chebyshev.lean | [
"data.polynomial.derivative",
"tactic.linear_combination"
] | [] | `T n` is the `n`-th Chebyshev polynomial of the first kind | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
T_zero : T R 0 = 1 | rfl | lemma | polynomial.chebyshev.T_zero | ring_theory.polynomial | src/ring_theory/polynomial/chebyshev.lean | [
"data.polynomial.derivative",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
T_one : T R 1 = X | rfl | lemma | polynomial.chebyshev.T_one | ring_theory.polynomial | src/ring_theory/polynomial/chebyshev.lean | [
"data.polynomial.derivative",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
T_add_two (n : ℕ) : T R (n + 2) = 2 * X * T R (n + 1) - T R n | by rw T | lemma | polynomial.chebyshev.T_add_two | ring_theory.polynomial | src/ring_theory/polynomial/chebyshev.lean | [
"data.polynomial.derivative",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
T_two : T R 2 = 2 * X ^ 2 - 1 | by simp only [T, sub_left_inj, sq, mul_assoc] | lemma | polynomial.chebyshev.T_two | ring_theory.polynomial | src/ring_theory/polynomial/chebyshev.lean | [
"data.polynomial.derivative",
"tactic.linear_combination"
] | [
"mul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
T_of_two_le (n : ℕ) (h : 2 ≤ n) : T R n = 2 * X * T R (n - 1) - T R (n - 2) | begin
obtain ⟨n, rfl⟩ := nat.exists_eq_add_of_le h,
rw add_comm,
exact T_add_two R n
end | lemma | polynomial.chebyshev.T_of_two_le | ring_theory.polynomial | src/ring_theory/polynomial/chebyshev.lean | [
"data.polynomial.derivative",
"tactic.linear_combination"
] | [
"nat.exists_eq_add_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
U : ℕ → R[X] | | 0 := 1
| 1 := 2 * X
| (n + 2) := 2 * X * U (n + 1) - U n | def | polynomial.chebyshev.U | ring_theory.polynomial | src/ring_theory/polynomial/chebyshev.lean | [
"data.polynomial.derivative",
"tactic.linear_combination"
] | [] | `U n` is the `n`-th Chebyshev polynomial of the second kind | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
U_zero : U R 0 = 1 | rfl | lemma | polynomial.chebyshev.U_zero | ring_theory.polynomial | src/ring_theory/polynomial/chebyshev.lean | [
"data.polynomial.derivative",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
U_one : U R 1 = 2 * X | rfl | lemma | polynomial.chebyshev.U_one | ring_theory.polynomial | src/ring_theory/polynomial/chebyshev.lean | [
"data.polynomial.derivative",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
U_add_two (n : ℕ) : U R (n + 2) = 2 * X * U R (n + 1) - U R n | by rw U | lemma | polynomial.chebyshev.U_add_two | ring_theory.polynomial | src/ring_theory/polynomial/chebyshev.lean | [
"data.polynomial.derivative",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
U_two : U R 2 = 4 * X ^ 2 - 1 | by { simp only [U], ring, } | lemma | polynomial.chebyshev.U_two | ring_theory.polynomial | src/ring_theory/polynomial/chebyshev.lean | [
"data.polynomial.derivative",
"tactic.linear_combination"
] | [
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
U_of_two_le (n : ℕ) (h : 2 ≤ n) : U R n = 2 * X * U R (n - 1) - U R (n - 2) | begin
obtain ⟨n, rfl⟩ := nat.exists_eq_add_of_le h,
rw add_comm,
exact U_add_two R n
end | lemma | polynomial.chebyshev.U_of_two_le | ring_theory.polynomial | src/ring_theory/polynomial/chebyshev.lean | [
"data.polynomial.derivative",
"tactic.linear_combination"
] | [
"nat.exists_eq_add_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
U_eq_X_mul_U_add_T : ∀ n : ℕ, U R (n + 1) = X * U R n + T R (n + 1) | | 0 := by { simp only [U_zero, U_one, T_one], ring }
| 1 := by { simp only [U_one, T_two, U_two], ring }
| (n + 2) :=
calc U R (n + 2 + 1) = 2 * X * (X * U R (n + 1) + T R (n + 2)) - (X * U R n + T R (n + 1)) :
by simp only [U_add_two, U_eq_X_mul_U_add_T n, U_eq_X_mul_U_add_T (n + 1)]
...... | lemma | polynomial.chebyshev.U_eq_X_mul_U_add_T | ring_theory.polynomial | src/ring_theory/polynomial/chebyshev.lean | [
"data.polynomial.derivative",
"tactic.linear_combination"
] | [
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
T_eq_U_sub_X_mul_U (n : ℕ) : T R (n + 1) = U R (n + 1) - X * U R n | by rw [U_eq_X_mul_U_add_T, add_comm (X * U R n), add_sub_cancel] | lemma | polynomial.chebyshev.T_eq_U_sub_X_mul_U | ring_theory.polynomial | src/ring_theory/polynomial/chebyshev.lean | [
"data.polynomial.derivative",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
T_eq_X_mul_T_sub_pol_U : ∀ n : ℕ, T R (n + 2) = X * T R (n + 1) - (1 - X ^ 2) * U R n | | 0 := by { simp only [T_one, T_two, U_zero], ring }
| 1 := by { simp only [T_add_two, T_zero, T_add_two,
U_one, T_one], ring }
| (n + 2) :=
calc T R (n + 2 + 2)
= 2 * X * T R (n + 2 + 1) - T R (n + 2) : T_add_two _ _
... = 2 * X * (X * T R (n + 2) - (1 - X ^ 2) * U R (n... | lemma | polynomial.chebyshev.T_eq_X_mul_T_sub_pol_U | ring_theory.polynomial | src/ring_theory/polynomial/chebyshev.lean | [
"data.polynomial.derivative",
"tactic.linear_combination"
] | [
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_sub_X_sq_mul_U_eq_pol_in_T (n : ℕ) :
(1 - X ^ 2) * U R n = X * T R (n + 1) - T R (n + 2) | by rw [T_eq_X_mul_T_sub_pol_U, ←sub_add, sub_self, zero_add] | lemma | polynomial.chebyshev.one_sub_X_sq_mul_U_eq_pol_in_T | ring_theory.polynomial | src/ring_theory/polynomial/chebyshev.lean | [
"data.polynomial.derivative",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_T (f : R →+* S) : ∀ n : ℕ, map f (T R n) = T S n | | 0 := by simp only [T_zero, polynomial.map_one]
| 1 := by simp only [T_one, map_X]
| (n + 2) :=
begin
simp only [T_add_two, polynomial.map_mul, polynomial.map_sub, map_X, bit0,
polynomial.map_add, polynomial.map_one],
rw [map_T (n + 1), map_T n],
end | lemma | polynomial.chebyshev.map_T | ring_theory.polynomial | src/ring_theory/polynomial/chebyshev.lean | [
"data.polynomial.derivative",
"tactic.linear_combination"
] | [
"polynomial.map_add",
"polynomial.map_mul",
"polynomial.map_one",
"polynomial.map_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_U (f : R →+* S) : ∀ n : ℕ, map f (U R n) = U S n | | 0 := by simp only [U_zero, polynomial.map_one]
| 1 :=
begin
simp only [U_one, map_X, polynomial.map_mul, polynomial.map_add, polynomial.map_one],
change map f (1 + 1) * X = 2 * X,
simpa only [polynomial.map_add, polynomial.map_one]
end
| (n + 2) :=
begin
simp only [U_add_two, polynomial.map_mul, p... | lemma | polynomial.chebyshev.map_U | ring_theory.polynomial | src/ring_theory/polynomial/chebyshev.lean | [
"data.polynomial.derivative",
"tactic.linear_combination"
] | [
"polynomial.map_add",
"polynomial.map_mul",
"polynomial.map_one",
"polynomial.map_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
T_derivative_eq_U : ∀ n : ℕ, derivative (T R (n + 1)) = (n + 1) * U R n | | 0 := by simp only [T_one, U_zero, derivative_X, nat.cast_zero, zero_add, mul_one]
| 1 := by { simp only [T_two, U_one, derivative_sub, derivative_one, derivative_mul,
derivative_X_pow, nat.cast_one, nat.cast_two],
norm_num }
| (n + 2) :=
calc derivati... | lemma | polynomial.chebyshev.T_derivative_eq_U | ring_theory.polynomial | src/ring_theory/polynomial/chebyshev.lean | [
"data.polynomial.derivative",
"tactic.linear_combination"
] | [
"bit0_zero",
"mul_one",
"nat.cast_one",
"nat.cast_two",
"nat.cast_zero",
"ring",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_sub_X_sq_mul_derivative_T_eq_poly_in_T (n : ℕ) :
(1 - X ^ 2) * (derivative (T R (n + 1))) = (n + 1) * (T R n - X * T R (n+1)) | calc
(1 - X ^ 2) * (derivative (T R (n + 1))) = (1 - X ^ 2) * ((n + 1) * U R n) :
by rw T_derivative_eq_U
... = (n + 1) * ((1 - X ^ 2) * U R n) : by ring
... = (n + 1) * (X * T R (n + 1) - (2 * X * T R (n + 1) - T R n)) :
by rw [one_sub_X_sq_mul_U_eq_pol_in_T, T_add_two]
... = (n + 1) * ... | lemma | polynomial.chebyshev.one_sub_X_sq_mul_derivative_T_eq_poly_in_T | ring_theory.polynomial | src/ring_theory/polynomial/chebyshev.lean | [
"data.polynomial.derivative",
"tactic.linear_combination"
] | [
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_one_mul_T_eq_poly_in_U (n : ℕ) :
((n : R[X]) + 1) * T R (n + 1) = X * U R n - (1 - X ^ 2) * derivative (U R n) | begin
have h : derivative (T R (n + 2)) = (U R (n + 1) - X * U R n) + X * derivative (T R (n + 1))
+ 2 * X * U R n - (1 - X ^ 2) * derivative (U R n),
{ conv_lhs { rw T_eq_X_mul_T_sub_pol_U },
simp only [derivative_sub, derivative_mul, derivative_X, derivative_one, derivati... | lemma | polynomial.chebyshev.add_one_mul_T_eq_poly_in_U | ring_theory.polynomial | src/ring_theory/polynomial/chebyshev.lean | [
"data.polynomial.derivative",
"tactic.linear_combination"
] | [
"nat.cast_bit0",
"nat.cast_one",
"one_mul",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k | | 0 := by simp [two_mul, add_mul]
| 1 := by simp [add_comm]
| (m + 2) := begin
intros k,
-- clean up the `T` nat indices in the goal
suffices : 2 * T R (m + 2) * T R (m + k + 2) = T R (2 * m + k + 4) + T R k,
{ have h_nat₁ : 2 * (m + 2) + k = 2 * m + k + 4 := by ring,
have h_nat₂ : m + 2 + k = m + k + 2 := ... | lemma | polynomial.chebyshev.mul_T | ring_theory.polynomial | src/ring_theory/polynomial/chebyshev.lean | [
"data.polynomial.derivative",
"tactic.linear_combination"
] | [
"ring",
"two_mul"
] | The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
T_mul : ∀ m n, T R (m * n) = (T R m).comp (T R n) | | 0 := by simp
| 1 := by simp
| (m + 2) := begin
intros n,
have : 2 * T R n * T R ((m + 1) * n) = T R ((m + 2) * n) + T R (m * n),
{ convert mul_T R n (m * n); ring },
simp [this, T_mul m, ← T_mul (m + 1)]
end | lemma | polynomial.chebyshev.T_mul | ring_theory.polynomial | src/ring_theory/polynomial/chebyshev.lean | [
"data.polynomial.derivative",
"tactic.linear_combination"
] | [
"ring"
] | The `(m * n)`-th Chebyshev polynomial is the composition of the `m`-th and `n`-th | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_primitive (p : R[X]) : Prop | ∀ (r : R), C r ∣ p → is_unit r | def | polynomial.is_primitive | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [
"is_unit"
] | A polynomial is primitive when the only constant polynomials dividing it are units | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_primitive_iff_is_unit_of_C_dvd {p : R[X]} :
p.is_primitive ↔ ∀ (r : R), C r ∣ p → is_unit r | iff.rfl | lemma | polynomial.is_primitive_iff_is_unit_of_C_dvd | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_primitive_one : is_primitive (1 : R[X]) | λ r h, is_unit_C.mp (is_unit_of_dvd_one (C r) h) | lemma | polynomial.is_primitive_one | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [
"is_unit_of_dvd_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monic.is_primitive {p : R[X]} (hp : p.monic) : p.is_primitive | begin
rintros r ⟨q, h⟩,
exact is_unit_of_mul_eq_one r (q.coeff p.nat_degree) (by rwa [←coeff_C_mul, ←h]),
end | lemma | polynomial.monic.is_primitive | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [
"is_unit_of_mul_eq_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_primitive.ne_zero [nontrivial R] {p : R[X]} (hp : p.is_primitive) : p ≠ 0 | begin
rintro rfl,
exact (hp 0 (dvd_zero (C 0))).ne_zero rfl,
end | lemma | polynomial.is_primitive.ne_zero | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [
"dvd_zero",
"ne_zero",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_primitive_of_dvd {p q : R[X]} (hp : is_primitive p) (hq : q ∣ p) : is_primitive q | λ a ha, is_primitive_iff_is_unit_of_C_dvd.mp hp a (dvd_trans ha hq) | lemma | polynomial.is_primitive_of_dvd | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [
"dvd_trans"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
content (p : R[X]) : R | (p.support).gcd p.coeff | def | polynomial.content | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [] | `p.content` is the `gcd` of the coefficients of `p`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
content_dvd_coeff {p : R[X]} (n : ℕ) : p.content ∣ p.coeff n | begin
by_cases h : n ∈ p.support,
{ apply finset.gcd_dvd h },
rw [mem_support_iff, not_not] at h,
rw h,
apply dvd_zero,
end | lemma | polynomial.content_dvd_coeff | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [
"dvd_zero",
"finset.gcd_dvd",
"not_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
content_C {r : R} : (C r).content = normalize r | begin
rw content,
by_cases h0 : r = 0,
{ simp [h0] },
have h : (C r).support = {0} := support_monomial _ h0,
simp [h],
end | lemma | polynomial.content_C | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [
"normalize"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
content_zero : content (0 : R[X]) = 0 | by rw [← C_0, content_C, normalize_zero] | lemma | polynomial.content_zero | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [
"normalize_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
content_one : content (1 : R[X]) = 1 | by rw [← C_1, content_C, normalize_one] | lemma | polynomial.content_one | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [
"normalize_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
content_X_mul {p : R[X]} : content (X * p) = content p | begin
rw [content, content, finset.gcd_def, finset.gcd_def],
refine congr rfl _,
have h : (X * p).support = p.support.map ⟨nat.succ, nat.succ_injective⟩,
{ ext a,
simp only [exists_prop, finset.mem_map, function.embedding.coe_fn_mk, ne.def,
mem_support_iff],
cases a,
{ simp [coeff_X_mul_zero, ... | lemma | polynomial.content_X_mul | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [
"exists_prop",
"finset.gcd_def",
"finset.map_val",
"finset.mem_map",
"function.embedding.coe_fn_mk",
"mul_comm",
"multiset.map_map",
"nat.succ_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
content_X_pow {k : ℕ} : content ((X : R[X]) ^ k) = 1 | begin
induction k with k hi,
{ simp },
rw [pow_succ, content_X_mul, hi]
end | lemma | polynomial.content_X_pow | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [
"pow_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
content_X : content (X : R[X]) = 1 | by { rw [← mul_one X, content_X_mul, content_one] } | lemma | polynomial.content_X | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
content_C_mul (r : R) (p : R[X]) : (C r * p).content = normalize r * p.content | begin
by_cases h0 : r = 0, { simp [h0] },
rw content, rw content, rw ← finset.gcd_mul_left,
refine congr (congr rfl _) _; ext; simp [h0, mem_support_iff]
end | lemma | polynomial.content_C_mul | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [
"finset.gcd_mul_left",
"normalize"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
content_monomial {r : R} {k : ℕ} : content (monomial k r) = normalize r | by rw [← C_mul_X_pow_eq_monomial, content_C_mul, content_X_pow, mul_one] | lemma | polynomial.content_monomial | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [
"mul_one",
"normalize"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
content_eq_zero_iff {p : R[X]} : content p = 0 ↔ p = 0 | begin
rw [content, finset.gcd_eq_zero_iff],
split; intro h,
{ ext n,
by_cases h0 : n ∈ p.support,
{ rw [h n h0, coeff_zero], },
{ rw mem_support_iff at h0,
push_neg at h0,
simp [h0] } },
{ intros x h0,
simp [h] }
end | lemma | polynomial.content_eq_zero_iff | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [
"finset.gcd_eq_zero_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normalize_content {p : R[X]} : normalize p.content = p.content | finset.normalize_gcd | lemma | polynomial.normalize_content | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [
"finset.normalize_gcd",
"normalize"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
content_eq_gcd_range_of_lt (p : R[X]) (n : ℕ) (h : p.nat_degree < n) :
p.content = (finset.range n).gcd p.coeff | begin
apply dvd_antisymm_of_normalize_eq normalize_content finset.normalize_gcd,
{ rw finset.dvd_gcd_iff,
intros i hi,
apply content_dvd_coeff _ },
{ apply finset.gcd_mono,
intro i,
simp only [nat.lt_succ_iff, mem_support_iff, ne.def, finset.mem_range],
contrapose!,
intro h1,
apply coe... | lemma | polynomial.content_eq_gcd_range_of_lt | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [
"dvd_antisymm_of_normalize_eq",
"finset.dvd_gcd_iff",
"finset.gcd_mono",
"finset.mem_range",
"finset.normalize_gcd",
"finset.range",
"nat.lt_succ_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
content_eq_gcd_range_succ (p : R[X]) :
p.content = (finset.range p.nat_degree.succ).gcd p.coeff | content_eq_gcd_range_of_lt _ _ (nat.lt_succ_self _) | lemma | polynomial.content_eq_gcd_range_succ | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [
"finset.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
content_eq_gcd_leading_coeff_content_erase_lead (p : R[X]) :
p.content = gcd_monoid.gcd p.leading_coeff (erase_lead p).content | begin
by_cases h : p = 0,
{ simp [h] },
rw [← leading_coeff_eq_zero, leading_coeff, ← ne.def, ← mem_support_iff] at h,
rw [content, ← finset.insert_erase h, finset.gcd_insert, leading_coeff, content,
erase_lead_support],
refine congr rfl (finset.gcd_congr rfl (λ i hi, _)),
rw finset.mem_erase at hi,
r... | lemma | polynomial.content_eq_gcd_leading_coeff_content_erase_lead | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [
"finset.gcd_congr",
"finset.gcd_insert",
"finset.insert_erase",
"finset.mem_erase"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_content_iff_C_dvd {p : R[X]} {r : R} : r ∣ p.content ↔ C r ∣ p | begin
rw C_dvd_iff_dvd_coeff,
split,
{ intros h i,
apply h.trans (content_dvd_coeff _) },
{ intro h,
rw [content, finset.dvd_gcd_iff],
intros i hi,
apply h i }
end | lemma | polynomial.dvd_content_iff_C_dvd | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [
"finset.dvd_gcd_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
C_content_dvd (p : R[X]) : C p.content ∣ p | dvd_content_iff_C_dvd.1 dvd_rfl | lemma | polynomial.C_content_dvd | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [
"dvd_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_primitive_iff_content_eq_one {p : R[X]} : p.is_primitive ↔ p.content = 1 | begin
rw [←normalize_content, normalize_eq_one, is_primitive],
simp_rw [←dvd_content_iff_C_dvd],
exact ⟨λ h, h p.content (dvd_refl p.content), λ h r hdvd, is_unit_of_dvd_unit hdvd h⟩,
end | lemma | polynomial.is_primitive_iff_content_eq_one | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [
"dvd_refl",
"is_unit_of_dvd_unit",
"normalize_eq_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_primitive.content_eq_one {p : R[X]} (hp : p.is_primitive) : p.content = 1 | is_primitive_iff_content_eq_one.mp hp | lemma | polynomial.is_primitive.content_eq_one | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prim_part (p : R[X]) : R[X] | if p = 0 then 1 else classical.some (C_content_dvd p) | def | polynomial.prim_part | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [] | The primitive part of a polynomial `p` is the primitive polynomial gained by dividing `p` by
`p.content`. If `p = 0`, then `p.prim_part = 1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_C_content_mul_prim_part (p : R[X]) : p = C p.content * p.prim_part | begin
by_cases h : p = 0, { simp [h] },
rw [prim_part, if_neg h, ← classical.some_spec (C_content_dvd p)],
end | lemma | polynomial.eq_C_content_mul_prim_part | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prim_part_zero : prim_part (0 : R[X]) = 1 | if_pos rfl | lemma | polynomial.prim_part_zero | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_primitive_prim_part (p : R[X]) : p.prim_part.is_primitive | begin
by_cases h : p = 0, { simp [h] },
rw ← content_eq_zero_iff at h,
rw is_primitive_iff_content_eq_one,
apply mul_left_cancel₀ h,
conv_rhs { rw [p.eq_C_content_mul_prim_part, mul_one, content_C_mul, normalize_content] }
end | lemma | polynomial.is_primitive_prim_part | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [
"mul_left_cancel₀",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
content_prim_part (p : R[X]) : p.prim_part.content = 1 | p.is_primitive_prim_part.content_eq_one | lemma | polynomial.content_prim_part | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prim_part_ne_zero (p : R[X]) : p.prim_part ≠ 0 | p.is_primitive_prim_part.ne_zero | lemma | polynomial.prim_part_ne_zero | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_degree_prim_part (p : R[X]) : p.prim_part.nat_degree = p.nat_degree | begin
by_cases h : C p.content = 0,
{ rw [C_eq_zero, content_eq_zero_iff] at h, simp [h] },
conv_rhs { rw [p.eq_C_content_mul_prim_part,
nat_degree_mul h p.prim_part_ne_zero, nat_degree_C, zero_add] },
end | lemma | polynomial.nat_degree_prim_part | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_primitive.prim_part_eq {p : R[X]} (hp : p.is_primitive) : p.prim_part = p | by rw [← one_mul p.prim_part, ← C_1, ← hp.content_eq_one, ← p.eq_C_content_mul_prim_part] | lemma | polynomial.is_primitive.prim_part_eq | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit_prim_part_C (r : R) : is_unit (C r).prim_part | begin
by_cases h0 : r = 0,
{ simp [h0] },
unfold is_unit,
refine ⟨⟨C ↑(norm_unit r)⁻¹, C ↑(norm_unit r),
by rw [← ring_hom.map_mul, units.inv_mul, C_1],
by rw [← ring_hom.map_mul, units.mul_inv, C_1]⟩, _⟩,
rw [← normalize_eq_zero, ← C_eq_zero] at h0,
apply mul_left_cancel₀ h0,
conv_rhs { rw [← con... | lemma | polynomial.is_unit_prim_part_C | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [
"is_unit",
"mul_assoc",
"mul_left_cancel₀",
"mul_one",
"normalize_apply",
"normalize_eq_zero",
"ring_hom.map_mul",
"units.coe_mk",
"units.inv_mul",
"units.mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prim_part_dvd (p : R[X]) : p.prim_part ∣ p | dvd.intro_left (C p.content) p.eq_C_content_mul_prim_part.symm | lemma | polynomial.prim_part_dvd | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [
"dvd.intro_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aeval_prim_part_eq_zero {S : Type*} [ring S] [is_domain S] [algebra R S]
[no_zero_smul_divisors R S] {p : R[X]} {s : S} (hpzero : p ≠ 0) (hp : aeval s p = 0) :
aeval s p.prim_part = 0 | begin
rw [eq_C_content_mul_prim_part p, map_mul, aeval_C] at hp,
have hcont : p.content ≠ 0 := λ h, hpzero (content_eq_zero_iff.1 h),
replace hcont := function.injective.ne (no_zero_smul_divisors.algebra_map_injective R S) hcont,
rw [map_zero] at hcont,
exact eq_zero_of_ne_zero_of_mul_left_eq_zero hcont hp
en... | lemma | polynomial.aeval_prim_part_eq_zero | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [
"algebra",
"eq_zero_of_ne_zero_of_mul_left_eq_zero",
"function.injective.ne",
"is_domain",
"map_mul",
"no_zero_smul_divisors",
"no_zero_smul_divisors.algebra_map_injective",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eval₂_prim_part_eq_zero {S : Type*} [comm_ring S] [is_domain S] {f : R →+* S}
(hinj : function.injective f) {p : R[X]} {s : S} (hpzero : p ≠ 0)
(hp : eval₂ f s p = 0) : eval₂ f s p.prim_part = 0 | begin
rw [eq_C_content_mul_prim_part p, eval₂_mul, eval₂_C] at hp,
have hcont : p.content ≠ 0 := λ h, hpzero (content_eq_zero_iff.1 h),
replace hcont := function.injective.ne hinj hcont,
rw [map_zero] at hcont,
exact eq_zero_of_ne_zero_of_mul_left_eq_zero hcont hp
end | lemma | polynomial.eval₂_prim_part_eq_zero | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [
"comm_ring",
"eq_zero_of_ne_zero_of_mul_left_eq_zero",
"function.injective.ne",
"is_domain"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_content_eq_of_dvd_sub {a : R} {p q : R[X]} (h : C a ∣ p - q) :
gcd_monoid.gcd a p.content = gcd_monoid.gcd a q.content | begin
rw content_eq_gcd_range_of_lt p (max p.nat_degree q.nat_degree).succ
(lt_of_le_of_lt (le_max_left _ _) (nat.lt_succ_self _)),
rw content_eq_gcd_range_of_lt q (max p.nat_degree q.nat_degree).succ
(lt_of_le_of_lt (le_max_right _ _) (nat.lt_succ_self _)),
apply finset.gcd_eq_of_dvd_sub,
intros x hx,
... | lemma | polynomial.gcd_content_eq_of_dvd_sub | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [
"finset.gcd_eq_of_dvd_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
content_mul_aux {p q : R[X]} :
gcd_monoid.gcd (p * q).erase_lead.content p.leading_coeff =
gcd_monoid.gcd (p.erase_lead * q).content p.leading_coeff | begin
rw [gcd_comm (content _) _, gcd_comm (content _) _],
apply gcd_content_eq_of_dvd_sub,
rw [← self_sub_C_mul_X_pow, ← self_sub_C_mul_X_pow, sub_mul, sub_sub, add_comm, sub_add,
sub_sub_cancel, leading_coeff_mul, ring_hom.map_mul, mul_assoc, mul_assoc],
apply dvd_sub (dvd.intro _ rfl) (dvd.intro _ rfl),
... | lemma | polynomial.content_mul_aux | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [
"dvd.intro",
"dvd_sub",
"gcd_comm",
"mul_assoc",
"ring_hom.map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
content_mul {p q : R[X]} : (p * q).content = p.content * q.content | begin
classical,
suffices h : ∀ (n : ℕ) (p q : R[X]), ((p * q).degree < n) →
(p * q).content = p.content * q.content,
{ apply h,
apply (lt_of_le_of_lt degree_le_nat_degree (with_bot.coe_lt_coe.2 (nat.lt_succ_self _))) },
intro n,
induction n with n ih,
{ intros p q hpq,
rw [with_bot.coe_zero, na... | theorem | polynomial.content_mul | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [
"gcd_comm",
"gcd_mul_dvd_mul_gcd",
"ih",
"is_unit_iff_dvd_one",
"mul_assoc",
"mul_comm",
"mul_eq_zero",
"mul_ne_zero",
"mul_one",
"nat.lt_succ_iff_lt_or_eq",
"nat.with_bot.lt_zero_iff",
"normalize_eq_one",
"one_mul",
"with_bot.add_lt_add_iff_left",
"with_bot.add_lt_add_iff_right",
"wit... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_primitive.mul {p q : R[X]} (hp : p.is_primitive) (hq : q.is_primitive) :
(p * q).is_primitive | by rw [is_primitive_iff_content_eq_one, content_mul, hp.content_eq_one, hq.content_eq_one, mul_one] | theorem | polynomial.is_primitive.mul | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prim_part_mul {p q : R[X]} (h0 : p * q ≠ 0) :
(p * q).prim_part = p.prim_part * q.prim_part | begin
rw [ne.def, ← content_eq_zero_iff, ← C_eq_zero] at h0,
apply mul_left_cancel₀ h0,
conv_lhs { rw [← (p * q).eq_C_content_mul_prim_part,
p.eq_C_content_mul_prim_part, q.eq_C_content_mul_prim_part] },
rw [content_mul, ring_hom.map_mul],
ring,
end | theorem | polynomial.prim_part_mul | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [
"mul_left_cancel₀",
"ring",
"ring_hom.map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_primitive.dvd_prim_part_iff_dvd {p q : R[X]}
(hp : p.is_primitive) (hq : q ≠ 0) :
p ∣ q.prim_part ↔ p ∣ q | begin
refine ⟨λ h, h.trans (dvd.intro_left _ q.eq_C_content_mul_prim_part.symm), λ h, _⟩,
rcases h with ⟨r, rfl⟩,
apply dvd.intro _,
rw [prim_part_mul hq, hp.prim_part_eq],
end | lemma | polynomial.is_primitive.dvd_prim_part_iff_dvd | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [
"dvd.intro",
"dvd.intro_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_primitive_lcm_of_is_primitive {p q : R[X]}
(hp : p.is_primitive) (hq : q.is_primitive) :
∃ r : R[X], r.is_primitive ∧ (∀ s : R[X], p ∣ s ∧ q ∣ s ↔ r ∣ s) | begin
classical,
have h : ∃ (n : ℕ) (r : R[X]), r.nat_degree = n ∧ r.is_primitive ∧ p ∣ r ∧ q ∣ r :=
⟨(p * q).nat_degree, p * q, rfl, hp.mul hq, dvd_mul_right _ _, dvd_mul_left _ _⟩,
rcases nat.find_spec h with ⟨r, rdeg, rprim, pr, qr⟩,
refine ⟨r, rprim, λ s, ⟨_, λ rs, ⟨pr.trans rs, qr.trans rs⟩⟩⟩,
suffic... | theorem | polynomial.exists_primitive_lcm_of_is_primitive | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [
"associated.symm",
"con",
"dvd.intro_left",
"dvd_add",
"dvd_mul_left",
"dvd_mul_right",
"mul_comm",
"mul_ne_zero",
"mul_one",
"normalize_eq_one",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_iff_content_dvd_content_and_prim_part_dvd_prim_part
{p q : R[X]} (hq : q ≠ 0) :
p ∣ q ↔ p.content ∣ q.content ∧ p.prim_part ∣ q.prim_part | begin
split; intro h,
{ rcases h with ⟨r, rfl⟩,
rw [content_mul, p.is_primitive_prim_part.dvd_prim_part_iff_dvd hq],
exact ⟨dvd.intro _ rfl, p.prim_part_dvd.trans (dvd.intro _ rfl)⟩ },
{ rw [p.eq_C_content_mul_prim_part, q.eq_C_content_mul_prim_part],
exact mul_dvd_mul (ring_hom.map_dvd C h.1) h.2 }
e... | lemma | polynomial.dvd_iff_content_dvd_content_and_prim_part_dvd_prim_part | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [
"dvd.intro",
"mul_dvd_mul",
"ring_hom.map_dvd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normalized_gcd_monoid : normalized_gcd_monoid R[X] | normalized_gcd_monoid_of_exists_lcm $ λ p q, begin
rcases exists_primitive_lcm_of_is_primitive p.is_primitive_prim_part q.is_primitive_prim_part
with ⟨r, rprim, hr⟩,
refine ⟨C (lcm p.content q.content) * r, λ s, _⟩,
by_cases hs : s = 0,
{ simp [hs] },
by_cases hpq : C (lcm p.content q.content) = 0,
{ rw... | instance | polynomial.normalized_gcd_monoid | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [
"is_unit.mul_left_dvd",
"lcm_dvd_iff",
"lcm_eq_zero_iff",
"mul_ne_zero",
"mul_one",
"normalize_lcm",
"normalized_gcd_monoid",
"normalized_gcd_monoid_of_exists_lcm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
degree_gcd_le_left {p : R[X]} (hp : p ≠ 0) (q) : (gcd p q).degree ≤ p.degree | begin
have := nat_degree_le_iff_degree_le.mp
(nat_degree_le_of_dvd (gcd_dvd_left p q) hp),
rwa degree_eq_nat_degree hp
end | lemma | polynomial.degree_gcd_le_left | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
degree_gcd_le_right (p) {q : R[X]} (hq : q ≠ 0) : (gcd p q).degree ≤ q.degree | by { rw [gcd_comm], exact degree_gcd_le_left hq p } | lemma | polynomial.degree_gcd_le_right | ring_theory.polynomial | src/ring_theory/polynomial/content.lean | [
"algebra.gcd_monoid.finset",
"data.polynomial.field_division",
"data.polynomial.erase_lead",
"data.polynomial.cancel_leads"
] | [
"gcd_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dickson : ℕ → R[X] | | 0 := 3 - k
| 1 := X
| (n + 2) := X * dickson (n + 1) - (C a) * dickson n | def | polynomial.dickson | ring_theory.polynomial | src/ring_theory/polynomial/dickson.lean | [
"algebra.char_p.invertible",
"data.zmod.basic",
"ring_theory.localization.fraction_ring",
"ring_theory.polynomial.chebyshev",
"ring_theory.ideal.local_ring"
] | [] | `dickson` is the `n`the (generalised) Dickson polynomial of the `k`-th kind associated to the
element `a ∈ R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dickson_zero : dickson k a 0 = 3 - k | rfl | lemma | polynomial.dickson_zero | ring_theory.polynomial | src/ring_theory/polynomial/dickson.lean | [
"algebra.char_p.invertible",
"data.zmod.basic",
"ring_theory.localization.fraction_ring",
"ring_theory.polynomial.chebyshev",
"ring_theory.ideal.local_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dickson_one : dickson k a 1 = X | rfl | lemma | polynomial.dickson_one | ring_theory.polynomial | src/ring_theory/polynomial/dickson.lean | [
"algebra.char_p.invertible",
"data.zmod.basic",
"ring_theory.localization.fraction_ring",
"ring_theory.polynomial.chebyshev",
"ring_theory.ideal.local_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dickson_two : dickson k a 2 = X ^ 2 - C a * (3 - k) | by simp only [dickson, sq] | lemma | polynomial.dickson_two | ring_theory.polynomial | src/ring_theory/polynomial/dickson.lean | [
"algebra.char_p.invertible",
"data.zmod.basic",
"ring_theory.localization.fraction_ring",
"ring_theory.polynomial.chebyshev",
"ring_theory.ideal.local_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dickson_add_two (n : ℕ) :
dickson k a (n + 2) = X * dickson k a (n + 1) - C a * dickson k a n | by rw dickson | lemma | polynomial.dickson_add_two | ring_theory.polynomial | src/ring_theory/polynomial/dickson.lean | [
"algebra.char_p.invertible",
"data.zmod.basic",
"ring_theory.localization.fraction_ring",
"ring_theory.polynomial.chebyshev",
"ring_theory.ideal.local_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dickson_of_two_le {n : ℕ} (h : 2 ≤ n) :
dickson k a n = X * dickson k a (n - 1) - C a * dickson k a (n - 2) | begin
obtain ⟨n, rfl⟩ := nat.exists_eq_add_of_le h,
rw add_comm,
exact dickson_add_two k a n
end | lemma | polynomial.dickson_of_two_le | ring_theory.polynomial | src/ring_theory/polynomial/dickson.lean | [
"algebra.char_p.invertible",
"data.zmod.basic",
"ring_theory.localization.fraction_ring",
"ring_theory.polynomial.chebyshev",
"ring_theory.ideal.local_ring"
] | [
"nat.exists_eq_add_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_dickson (f : R →+* S) :
∀ (n : ℕ), map f (dickson k a n) = dickson k (f a) n | | 0 := by simp only [dickson_zero, polynomial.map_sub, polynomial.map_nat_cast,
bit1, bit0, polynomial.map_add, polynomial.map_one]
| 1 := by simp only [dickson_one, map_X]
| (n + 2) :=
begin
simp only [dickson_add_two, polynomial.map_sub, polynomial.map_mul, map_X, map_C],
r... | lemma | polynomial.map_dickson | ring_theory.polynomial | src/ring_theory/polynomial/dickson.lean | [
"algebra.char_p.invertible",
"data.zmod.basic",
"ring_theory.localization.fraction_ring",
"ring_theory.polynomial.chebyshev",
"ring_theory.ideal.local_ring"
] | [
"polynomial.map_add",
"polynomial.map_mul",
"polynomial.map_nat_cast",
"polynomial.map_one",
"polynomial.map_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dickson_two_zero :
∀ (n : ℕ), dickson 2 (0 : R) n = X ^ n | | 0 := by { simp only [dickson_zero, pow_zero], norm_num }
| 1 := by simp only [dickson_one, pow_one]
| (n + 2) :=
begin
simp only [dickson_add_two, C_0, zero_mul, sub_zero],
rw [dickson_two_zero, pow_add X (n + 1) 1, mul_comm, pow_one]
end | lemma | polynomial.dickson_two_zero | ring_theory.polynomial | src/ring_theory/polynomial/dickson.lean | [
"algebra.char_p.invertible",
"data.zmod.basic",
"ring_theory.localization.fraction_ring",
"ring_theory.polynomial.chebyshev",
"ring_theory.ideal.local_ring"
] | [
"mul_comm",
"pow_add",
"pow_one",
"pow_zero",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dickson_one_one_eval_add_inv (x y : R) (h : x * y = 1) :
∀ n, (dickson 1 (1 : R) n).eval (x + y) = x ^ n + y ^ n | | 0 := by { simp only [bit0, eval_one, eval_add, pow_zero, dickson_zero], norm_num }
| 1 := by simp only [eval_X, dickson_one, pow_one]
| (n + 2) :=
begin
simp only [eval_sub, eval_mul, dickson_one_one_eval_add_inv, eval_X, dickson_add_two, C_1,
eval_one],
conv_lhs { simp only [pow_succ, add_mul, mu... | lemma | polynomial.dickson_one_one_eval_add_inv | ring_theory.polynomial | src/ring_theory/polynomial/dickson.lean | [
"algebra.char_p.invertible",
"data.zmod.basic",
"ring_theory.localization.fraction_ring",
"ring_theory.polynomial.chebyshev",
"ring_theory.ideal.local_ring"
] | [
"mul_assoc",
"mul_comm",
"one_mul",
"pow_one",
"pow_succ",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dickson_one_one_eq_chebyshev_T [invertible (2 : R)] :
∀ n, dickson 1 (1 : R) n = 2 * (chebyshev.T R n).comp (C (⅟2) * X) | | 0 := by { simp only [chebyshev.T_zero, mul_one, one_comp, dickson_zero], norm_num }
| 1 := by rw [dickson_one, chebyshev.T_one, X_comp, ← mul_assoc, ← C_1, ← C_bit0, ← C_mul,
mul_inv_of_self, C_1, one_mul]
| (n + 2) :=
begin
simp only [dickson_add_two, chebyshev.T_add_two, dickson_on... | lemma | polynomial.dickson_one_one_eq_chebyshev_T | ring_theory.polynomial | src/ring_theory/polynomial/dickson.lean | [
"algebra.char_p.invertible",
"data.zmod.basic",
"ring_theory.localization.fraction_ring",
"ring_theory.polynomial.chebyshev",
"ring_theory.ideal.local_ring"
] | [
"invertible",
"mul_assoc",
"mul_inv_of_self",
"mul_one",
"one_mul",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
chebyshev_T_eq_dickson_one_one [invertible (2 : R)] (n : ℕ) :
chebyshev.T R n = C (⅟2) * (dickson 1 1 n).comp (2 * X) | begin
rw dickson_one_one_eq_chebyshev_T,
simp only [comp_assoc, mul_comp, C_comp, X_comp, ← mul_assoc, ← C_1, ← C_bit0, ← C_mul],
rw [inv_of_mul_self, C_1, one_mul, one_mul, comp_X]
end | lemma | polynomial.chebyshev_T_eq_dickson_one_one | ring_theory.polynomial | src/ring_theory/polynomial/dickson.lean | [
"algebra.char_p.invertible",
"data.zmod.basic",
"ring_theory.localization.fraction_ring",
"ring_theory.polynomial.chebyshev",
"ring_theory.ideal.local_ring"
] | [
"inv_of_mul_self",
"invertible",
"mul_assoc",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dickson_one_one_mul (m n : ℕ) :
dickson 1 (1 : R) (m * n) = (dickson 1 1 m).comp (dickson 1 1 n) | begin
have h : (1 : R) = int.cast_ring_hom R (1),
simp only [eq_int_cast, int.cast_one],
rw h,
simp only [← map_dickson (int.cast_ring_hom R), ← map_comp],
congr' 1,
apply map_injective (int.cast_ring_hom ℚ) int.cast_injective,
simp only [map_dickson, map_comp, eq_int_cast, int.cast_one,
dickson_one... | lemma | polynomial.dickson_one_one_mul | ring_theory.polynomial | src/ring_theory/polynomial/dickson.lean | [
"algebra.char_p.invertible",
"data.zmod.basic",
"ring_theory.localization.fraction_ring",
"ring_theory.polynomial.chebyshev",
"ring_theory.ideal.local_ring"
] | [
"eq_int_cast",
"int.cast_injective",
"int.cast_one",
"int.cast_ring_hom",
"inv_of_mul_self",
"map_comp",
"mul_assoc",
"one_mul",
"two_mul"
] | The `(m * n)`-th Dickson polynomial of the first kind is the composition of the `m`-th and
`n`-th. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dickson_one_one_comp_comm (m n : ℕ) :
(dickson 1 (1 : R) m).comp (dickson 1 1 n) = (dickson 1 1 n).comp (dickson 1 1 m) | by rw [← dickson_one_one_mul, mul_comm, dickson_one_one_mul] | lemma | polynomial.dickson_one_one_comp_comm | ring_theory.polynomial | src/ring_theory/polynomial/dickson.lean | [
"algebra.char_p.invertible",
"data.zmod.basic",
"ring_theory.localization.fraction_ring",
"ring_theory.polynomial.chebyshev",
"ring_theory.ideal.local_ring"
] | [
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dickson_one_one_zmod_p (p : ℕ) [fact p.prime] :
dickson 1 (1 : zmod p) p = X ^ p | begin
-- Recall that `dickson_eval_add_inv` characterises `dickson 1 1 p`
-- as a polynomial that maps `x + x⁻¹` to `x ^ p + (x⁻¹) ^ p`.
-- Since `X ^ p` also satisfies this property in characteristic `p`,
-- we can use a variant on `polynomial.funext` to conclude that these polynomials are equal.
-- For this... | lemma | polynomial.dickson_one_one_zmod_p | ring_theory.polynomial | src/ring_theory/polynomial/dickson.lean | [
"algebra.char_p.invertible",
"data.zmod.basic",
"ring_theory.localization.fraction_ring",
"ring_theory.polynomial.chebyshev",
"ring_theory.ideal.local_ring"
] | [
"add_pow_char",
"algebra_map",
"char_p",
"dvd_refl",
"exists_eq_right",
"exists_prop",
"fact",
"field",
"finset.mem_coe",
"fraction_ring",
"infinite",
"infinite.of_injective",
"inv_mul_cancel",
"inv_pow",
"inv_zero",
"is_fraction_ring.injective",
"mul_inv_cancel",
"mul_left_inj'",
... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dickson_one_one_char_p (p : ℕ) [fact p.prime] [char_p R p] :
dickson 1 (1 : R) p = X ^ p | begin
have h : (1 : R) = zmod.cast_hom (dvd_refl p) R (1),
simp only [zmod.cast_hom_apply, zmod.cast_one'],
rw [h, ← map_dickson (zmod.cast_hom (dvd_refl p) R), dickson_one_one_zmod_p,
polynomial.map_pow, map_X]
end | lemma | polynomial.dickson_one_one_char_p | ring_theory.polynomial | src/ring_theory/polynomial/dickson.lean | [
"algebra.char_p.invertible",
"data.zmod.basic",
"ring_theory.localization.fraction_ring",
"ring_theory.polynomial.chebyshev",
"ring_theory.ideal.local_ring"
] | [
"char_p",
"dvd_refl",
"fact",
"polynomial.map_pow",
"zmod.cast_hom",
"zmod.cast_hom_apply",
"zmod.cast_one'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_closure.mem_lifts_of_monic_of_dvd_map
{f : R[X]} (hf : f.monic) {g : K[X]} (hg : g.monic) (hd : g ∣ f.map (algebra_map R K)) :
g ∈ lifts (algebra_map (integral_closure R K) K) | begin
have := mem_lift_of_splits_of_roots_mem_range (integral_closure R g.splitting_field)
((splits_id_iff_splits _).2 $ splitting_field.splits g) (hg.map _)
(λ a ha, (set_like.ext_iff.mp (integral_closure R g.splitting_field).range_algebra_map _).mpr $
roots_mem_integral_closure hf _),
{ rw [lifts_if... | theorem | integral_closure.mem_lifts_of_monic_of_dvd_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",
"integral_closure",
"is_scalar_tower.algebra_map_eq",
"map_dvd",
"multiset.mem_of_le",
"ne_zero",
"ring_hom.coe_range",
"ring_hom.injective",
"roots_mem_integral_closure",
"subalgebra.range_algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integrally_closed.eq_map_mul_C_of_dvd [is_integrally_closed R] {f : R[X]} (hf : f.monic)
{g : K[X]} (hg : g ∣ f.map (algebra_map R K)) :
∃ g' : R[X], (g'.map (algebra_map R K)) * (C $ leading_coeff g) = g | begin
have g_ne_0 : g ≠ 0 := ne_zero_of_dvd_ne_zero (monic.ne_zero $ hf.map (algebra_map R K)) hg,
suffices lem : ∃ g' : R[X], g'.map (algebra_map R K) = g * (C g.leading_coeff⁻¹),
{ obtain ⟨g', hg'⟩ := lem,
use g',
rw [hg', mul_assoc, ← C_mul, inv_mul_cancel (leading_coeff_ne_zero.mpr g_ne_0), C_1, mul_o... | lemma | is_integrally_closed.eq_map_mul_C_of_dvd | 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.bot_equiv_of_injective",
"algebra_map",
"associated",
"associated.dvd_iff_dvd_left",
"associated_mul_is_unit_left_iff",
"integral_closure",
"integral_closure.mem_lifts_of_monic_of_dvd_map",
"inv_mul_cancel",
"inv_ne_zero",
"is_fraction_ring.injective",
"is_integrally_closed",
"is_unit... | If `K = Frac(R)` and `g : K[X]` divides a monic polynomial with coefficients in `R`, then
`g * (C g.leading_coeff⁻¹)` has coefficients in `R` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_primitive.is_unit_iff_is_unit_map_of_injective :
is_unit f ↔ is_unit (map φ f) | begin
refine ⟨(map_ring_hom φ).is_unit_map, λ h, _⟩,
rcases is_unit_iff.1 h with ⟨_, ⟨u, rfl⟩, hu⟩,
have hdeg := degree_C u.ne_zero,
rw [hu, degree_map_eq_of_injective hinj] at hdeg,
rw [eq_C_of_degree_eq_zero hdeg] at hf ⊢,
exact is_unit_C.mpr (is_primitive_iff_is_unit_of_C_dvd.mp hf (f.coeff 0) dvd_rfl),
... | lemma | polynomial.is_primitive.is_unit_iff_is_unit_map_of_injective | 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"
] | [
"dvd_rfl",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_primitive.irreducible_of_irreducible_map_of_injective (h_irr : irreducible (map φ f)) :
irreducible f | begin
refine ⟨λ h, h_irr.not_unit (is_unit.map (map_ring_hom φ) h),
λ a b h, (h_irr.is_unit_or_is_unit $ by rw [h, polynomial.map_mul]).imp _ _⟩,
all_goals { apply ((is_primitive_of_dvd hf _).is_unit_iff_is_unit_map_of_injective hinj).mpr },
exacts [(dvd.intro _ h.symm), dvd.intro_left _ h.symm],
end | lemma | polynomial.is_primitive.irreducible_of_irreducible_map_of_injective | 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"
] | [
"dvd.intro",
"dvd.intro_left",
"irreducible",
"is_unit.map",
"polynomial.map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_primitive.is_unit_iff_is_unit_map {p : R[X]} (hp : p.is_primitive) :
is_unit p ↔ is_unit (p.map (algebra_map R K)) | hp.is_unit_iff_is_unit_map_of_injective (is_fraction_ring.injective _ _) | lemma | polynomial.is_primitive.is_unit_iff_is_unit_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_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monic.irreducible_iff_irreducible_map_fraction_map [is_integrally_closed R] {p : R[X]}
(h : p.monic) : irreducible p ↔ irreducible (p.map $ algebra_map R K) | begin
/- The ← direction follows from `is_primitive.irreducible_of_irreducible_map_of_injective`.
For the → direction, it is enought to show that if `(p.map $ algebra_map R K) = a * b` and
`a` is not a unit then `b` is a unit -/
refine ⟨λ hp, irreducible_iff.mpr ⟨hp.not_unit.imp h.is_primitive.is_unit_iff... | theorem | polynomial.monic.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_map",
"by_contra",
"irreducible",
"is_fraction_ring.injective",
"is_integrally_closed",
"is_unit",
"is_unit.map",
"is_unit.mul",
"mul_assoc",
"mul_comm",
"one_mul",
"polynomial.coe_map_ring_hom",
"polynomial.map_injective",
"polynomial.map_mul"
] | **Gauss's Lemma** for integrally closed domains states that a monic polynomial is irreducible
iff it is irreducible in the fraction field. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_integrally_closed_iff' : is_integrally_closed R ↔
∀ p : R[X], p.monic → (irreducible p ↔ irreducible (p.map $ algebra_map R K)) | begin
split,
{ intros hR p hp, letI := hR, exact monic.irreducible_iff_irreducible_map_fraction_map hp },
{ intro H,
refine (is_integrally_closed_iff K).mpr (λ x hx, ring_hom.mem_range.mp $
minpoly.mem_range_of_degree_eq_one R x _),
rw ← monic.degree_map (minpoly.monic hx) (algebra_map R K),
app... | theorem | polynomial.is_integrally_closed_iff' | 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",
"irreducible",
"is_integrally_closed",
"is_integrally_closed_iff",
"minpoly.aeval",
"minpoly.irreducible",
"minpoly.mem_range_of_degree_eq_one",
"minpoly.monic"
] | Integrally closed domains are precisely the domains for in which Gauss's lemma holds
for monic polynomials | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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