fact
stringlengths 6
3.84k
| type
stringclasses 11
values | library
stringclasses 32
values | imports
listlengths 1
14
| filename
stringlengths 20
95
| symbolic_name
stringlengths 1
90
| docstring
stringlengths 7
20k
⌀ |
|---|---|---|---|---|---|---|
infinite_of_charZero (R A : Type*) [CommRing R] [Ring A] [Algebra R A]
[CharZero A] : { x : A | IsAlgebraic R x }.Infinite := by
letI := MulActionWithZero.nontrivial R A
exact infinite_of_injective_forall_mem Nat.cast_injective isAlgebraic_nat
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Cardinal",
"Mathlib.RingTheory.Algebraic.Basic"
] |
Mathlib/Algebra/AlgebraicCard.lean
|
infinite_of_charZero
| null |
aleph0_le_cardinalMk_of_charZero (R A : Type*) [CommRing R] [Ring A]
[Algebra R A] [CharZero A] : ℵ₀ ≤ #{ x : A // IsAlgebraic R x } :=
infinite_iff.1 (Set.infinite_coe_iff.2 <| infinite_of_charZero R A)
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Cardinal",
"Mathlib.RingTheory.Algebraic.Basic"
] |
Mathlib/Algebra/AlgebraicCard.lean
|
aleph0_le_cardinalMk_of_charZero
| null |
cardinalMk_lift_le_mul :
Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by
rw [← mk_uLift, ← mk_uLift]
choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop
refine lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => ?_
rw [lift_le_aleph0, le_aleph0_iff_set_countable]
suffices MapsTo (↑) (g ⁻¹' {f}) (f.rootSet A) from
this.countable_of_injOn Subtype.coe_injective.injOn (f.rootSet_finite A).countable
rintro x (rfl : g x = f)
exact mem_rootSet.2 ⟨hg₁ x, hg₂ x⟩
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Cardinal",
"Mathlib.RingTheory.Algebraic.Basic"
] |
Mathlib/Algebra/AlgebraicCard.lean
|
cardinalMk_lift_le_mul
| null |
cardinalMk_lift_le_max :
Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ max (Cardinal.lift.{v} #R) ℵ₀ :=
(cardinalMk_lift_le_mul R A).trans <|
(mul_le_mul_right' (lift_le.2 cardinalMk_le_max) _).trans <| by simp
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Cardinal",
"Mathlib.RingTheory.Algebraic.Basic"
] |
Mathlib/Algebra/AlgebraicCard.lean
|
cardinalMk_lift_le_max
| null |
cardinalMk_lift_of_infinite [Infinite R] :
Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } = Cardinal.lift.{v} #R :=
((cardinalMk_lift_le_max R A).trans_eq (max_eq_left <| aleph0_le_mk _)).antisymm <|
lift_mk_le'.2 ⟨⟨fun x => ⟨algebraMap R A x, isAlgebraic_algebraMap _⟩, fun _ _ h =>
FaithfulSMul.algebraMap_injective R A (Subtype.ext_iff.1 h)⟩⟩
variable [Countable R]
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Cardinal",
"Mathlib.RingTheory.Algebraic.Basic"
] |
Mathlib/Algebra/AlgebraicCard.lean
|
cardinalMk_lift_of_infinite
| null |
protected countable : Set.Countable { x : A | IsAlgebraic R x } := by
rw [← le_aleph0_iff_set_countable, ← lift_le_aleph0]
apply (cardinalMk_lift_le_max R A).trans
simp
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Cardinal",
"Mathlib.RingTheory.Algebraic.Basic"
] |
Mathlib/Algebra/AlgebraicCard.lean
|
countable
| null |
cardinalMk_of_countable_of_charZero [CharZero A] :
#{ x : A // IsAlgebraic R x } = ℵ₀ :=
(Algebraic.countable R A).le_aleph0.antisymm (aleph0_le_cardinalMk_of_charZero R A)
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Cardinal",
"Mathlib.RingTheory.Algebraic.Basic"
] |
Mathlib/Algebra/AlgebraicCard.lean
|
cardinalMk_of_countable_of_charZero
| null |
cardinalMk_le_mul : #{ x : A // IsAlgebraic R x } ≤ #R[X] * ℵ₀ := by
rw [← lift_id #_, ← lift_id #R[X]]
exact cardinalMk_lift_le_mul R A
@[stacks 09GK]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Cardinal",
"Mathlib.RingTheory.Algebraic.Basic"
] |
Mathlib/Algebra/AlgebraicCard.lean
|
cardinalMk_le_mul
| null |
cardinalMk_le_max : #{ x : A // IsAlgebraic R x } ≤ max #R ℵ₀ := by
rw [← lift_id #_, ← lift_id #R]
exact cardinalMk_lift_le_max R A
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Cardinal",
"Mathlib.RingTheory.Algebraic.Basic"
] |
Mathlib/Algebra/AlgebraicCard.lean
|
cardinalMk_le_max
| null |
cardinalMk_of_infinite [Infinite R] : #{ x : A // IsAlgebraic R x } = #R :=
lift_inj.1 <| cardinalMk_lift_of_infinite R A
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Cardinal",
"Mathlib.RingTheory.Algebraic.Basic"
] |
Mathlib/Algebra/AlgebraicCard.lean
|
cardinalMk_of_infinite
| null |
@[ext]
Cubic (R : Type*) where
/-- The degree-3 coefficient -/
a : R
/-- The degree-2 coefficient -/
b : R
/-- The degree-1 coefficient -/
c : R
/-- The degree-0 coefficient -/
d : R
|
structure
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
Cubic
|
The structure representing a cubic polynomial.
|
toPoly (P : Cubic R) : R[X] :=
C P.a * X ^ 3 + C P.b * X ^ 2 + C P.c * X + C P.d
|
def
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
toPoly
|
Convert a cubic polynomial to a polynomial.
|
C_mul_prod_X_sub_C_eq [CommRing S] {w x y z : S} :
C w * (X - C x) * (X - C y) * (X - C z) =
toPoly ⟨w, w * -(x + y + z), w * (x * y + x * z + y * z), w * -(x * y * z)⟩ := by
simp only [toPoly, C_neg, C_add, C_mul]
ring1
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
C_mul_prod_X_sub_C_eq
| null |
prod_X_sub_C_eq [CommRing S] {x y z : S} :
(X - C x) * (X - C y) * (X - C z) =
toPoly ⟨1, -(x + y + z), x * y + x * z + y * z, -(x * y * z)⟩ := by
rw [← one_mul <| X - C x, ← C_1, C_mul_prod_X_sub_C_eq, one_mul, one_mul, one_mul]
/-! ### Coefficients -/
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
prod_X_sub_C_eq
| null |
private coeffs : (∀ n > 3, P.toPoly.coeff n = 0) ∧ P.toPoly.coeff 3 = P.a ∧
P.toPoly.coeff 2 = P.b ∧ P.toPoly.coeff 1 = P.c ∧ P.toPoly.coeff 0 = P.d := by
simp only [toPoly, coeff_add, coeff_C, coeff_C_mul_X, coeff_C_mul_X_pow]
norm_num
intro n hn
repeat' rw [if_neg]
any_goals cutsat
repeat' rw [zero_add]
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
coeffs
| null |
coeff_eq_zero {n : ℕ} (hn : 3 < n) : P.toPoly.coeff n = 0 :=
coeffs.1 n hn
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
coeff_eq_zero
| null |
coeff_eq_a : P.toPoly.coeff 3 = P.a :=
coeffs.2.1
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
coeff_eq_a
| null |
coeff_eq_b : P.toPoly.coeff 2 = P.b :=
coeffs.2.2.1
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
coeff_eq_b
| null |
coeff_eq_c : P.toPoly.coeff 1 = P.c :=
coeffs.2.2.2.1
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
coeff_eq_c
| null |
coeff_eq_d : P.toPoly.coeff 0 = P.d :=
coeffs.2.2.2.2
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
coeff_eq_d
| null |
a_of_eq (h : P.toPoly = Q.toPoly) : P.a = Q.a := by rw [← coeff_eq_a, h, coeff_eq_a]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
a_of_eq
| null |
b_of_eq (h : P.toPoly = Q.toPoly) : P.b = Q.b := by rw [← coeff_eq_b, h, coeff_eq_b]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
b_of_eq
| null |
c_of_eq (h : P.toPoly = Q.toPoly) : P.c = Q.c := by rw [← coeff_eq_c, h, coeff_eq_c]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
c_of_eq
| null |
d_of_eq (h : P.toPoly = Q.toPoly) : P.d = Q.d := by rw [← coeff_eq_d, h, coeff_eq_d]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
d_of_eq
| null |
toPoly_injective (P Q : Cubic R) : P.toPoly = Q.toPoly ↔ P = Q :=
⟨fun h ↦ Cubic.ext (a_of_eq h) (b_of_eq h) (c_of_eq h) (d_of_eq h), congr_arg toPoly⟩
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
toPoly_injective
| null |
of_a_eq_zero (ha : P.a = 0) : P.toPoly = C P.b * X ^ 2 + C P.c * X + C P.d := by
rw [toPoly, ha, C_0, zero_mul, zero_add]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
of_a_eq_zero
| null |
of_a_eq_zero' : toPoly ⟨0, b, c, d⟩ = C b * X ^ 2 + C c * X + C d :=
of_a_eq_zero rfl
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
of_a_eq_zero'
| null |
of_b_eq_zero (ha : P.a = 0) (hb : P.b = 0) : P.toPoly = C P.c * X + C P.d := by
rw [of_a_eq_zero ha, hb, C_0, zero_mul, zero_add]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
of_b_eq_zero
| null |
of_b_eq_zero' : toPoly ⟨0, 0, c, d⟩ = C c * X + C d :=
of_b_eq_zero rfl rfl
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
of_b_eq_zero'
| null |
of_c_eq_zero (ha : P.a = 0) (hb : P.b = 0) (hc : P.c = 0) : P.toPoly = C P.d := by
rw [of_b_eq_zero ha hb, hc, C_0, zero_mul, zero_add]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
of_c_eq_zero
| null |
of_c_eq_zero' : toPoly ⟨0, 0, 0, d⟩ = C d :=
of_c_eq_zero rfl rfl rfl
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
of_c_eq_zero'
| null |
of_d_eq_zero (ha : P.a = 0) (hb : P.b = 0) (hc : P.c = 0) (hd : P.d = 0) :
P.toPoly = 0 := by
rw [of_c_eq_zero ha hb hc, hd, C_0]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
of_d_eq_zero
| null |
of_d_eq_zero' : (⟨0, 0, 0, 0⟩ : Cubic R).toPoly = 0 :=
of_d_eq_zero rfl rfl rfl rfl
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
of_d_eq_zero'
| null |
zero : (0 : Cubic R).toPoly = 0 :=
of_d_eq_zero'
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
zero
| null |
toPoly_eq_zero_iff (P : Cubic R) : P.toPoly = 0 ↔ P = 0 := by
rw [← zero, toPoly_injective]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
toPoly_eq_zero_iff
| null |
private ne_zero (h0 : P.a ≠ 0 ∨ P.b ≠ 0 ∨ P.c ≠ 0 ∨ P.d ≠ 0) : P.toPoly ≠ 0 := by
contrapose! h0
rw [(toPoly_eq_zero_iff P).mp h0]
exact ⟨rfl, rfl, rfl, rfl⟩
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
ne_zero
| null |
ne_zero_of_a_ne_zero (ha : P.a ≠ 0) : P.toPoly ≠ 0 :=
(or_imp.mp ne_zero).1 ha
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
ne_zero_of_a_ne_zero
| null |
ne_zero_of_b_ne_zero (hb : P.b ≠ 0) : P.toPoly ≠ 0 :=
(or_imp.mp (or_imp.mp ne_zero).2).1 hb
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
ne_zero_of_b_ne_zero
| null |
ne_zero_of_c_ne_zero (hc : P.c ≠ 0) : P.toPoly ≠ 0 :=
(or_imp.mp (or_imp.mp (or_imp.mp ne_zero).2).2).1 hc
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
ne_zero_of_c_ne_zero
| null |
ne_zero_of_d_ne_zero (hd : P.d ≠ 0) : P.toPoly ≠ 0 :=
(or_imp.mp (or_imp.mp (or_imp.mp ne_zero).2).2).2 hd
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
ne_zero_of_d_ne_zero
| null |
leadingCoeff_of_a_ne_zero (ha : P.a ≠ 0) : P.toPoly.leadingCoeff = P.a :=
leadingCoeff_cubic ha
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
leadingCoeff_of_a_ne_zero
| null |
leadingCoeff_of_a_ne_zero' (ha : a ≠ 0) : (toPoly ⟨a, b, c, d⟩).leadingCoeff = a := by
simp [ha]
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
leadingCoeff_of_a_ne_zero'
| null |
leadingCoeff_of_b_ne_zero (ha : P.a = 0) (hb : P.b ≠ 0) : P.toPoly.leadingCoeff = P.b := by
rw [of_a_eq_zero ha, leadingCoeff_quadratic hb]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
leadingCoeff_of_b_ne_zero
| null |
leadingCoeff_of_b_ne_zero' (hb : b ≠ 0) : (toPoly ⟨0, b, c, d⟩).leadingCoeff = b := by
simp [hb]
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
leadingCoeff_of_b_ne_zero'
| null |
leadingCoeff_of_c_ne_zero (ha : P.a = 0) (hb : P.b = 0) (hc : P.c ≠ 0) :
P.toPoly.leadingCoeff = P.c := by
rw [of_b_eq_zero ha hb, leadingCoeff_linear hc]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
leadingCoeff_of_c_ne_zero
| null |
leadingCoeff_of_c_ne_zero' (hc : c ≠ 0) : (toPoly ⟨0, 0, c, d⟩).leadingCoeff = c := by
simp [hc]
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
leadingCoeff_of_c_ne_zero'
| null |
leadingCoeff_of_c_eq_zero (ha : P.a = 0) (hb : P.b = 0) (hc : P.c = 0) :
P.toPoly.leadingCoeff = P.d := by
rw [of_c_eq_zero ha hb hc, leadingCoeff_C]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
leadingCoeff_of_c_eq_zero
| null |
leadingCoeff_of_c_eq_zero' : (toPoly ⟨0, 0, 0, d⟩).leadingCoeff = d :=
leadingCoeff_of_c_eq_zero rfl rfl rfl
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
leadingCoeff_of_c_eq_zero'
| null |
monic_of_a_eq_one (ha : P.a = 1) : P.toPoly.Monic := by
nontriviality R
rw [Monic, leadingCoeff_of_a_ne_zero (ha ▸ one_ne_zero), ha]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
monic_of_a_eq_one
| null |
monic_of_a_eq_one' : (toPoly ⟨1, b, c, d⟩).Monic :=
monic_of_a_eq_one rfl
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
monic_of_a_eq_one'
| null |
monic_of_b_eq_one (ha : P.a = 0) (hb : P.b = 1) : P.toPoly.Monic := by
nontriviality R
rw [Monic, leadingCoeff_of_b_ne_zero ha (hb ▸ one_ne_zero), hb]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
monic_of_b_eq_one
| null |
monic_of_b_eq_one' : (toPoly ⟨0, 1, c, d⟩).Monic :=
monic_of_b_eq_one rfl rfl
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
monic_of_b_eq_one'
| null |
monic_of_c_eq_one (ha : P.a = 0) (hb : P.b = 0) (hc : P.c = 1) : P.toPoly.Monic := by
nontriviality R
rw [Monic, leadingCoeff_of_c_ne_zero ha hb (hc ▸ one_ne_zero), hc]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
monic_of_c_eq_one
| null |
monic_of_c_eq_one' : (toPoly ⟨0, 0, 1, d⟩).Monic :=
monic_of_c_eq_one rfl rfl rfl
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
monic_of_c_eq_one'
| null |
monic_of_d_eq_one (ha : P.a = 0) (hb : P.b = 0) (hc : P.c = 0) (hd : P.d = 1) :
P.toPoly.Monic := by
rw [Monic, leadingCoeff_of_c_eq_zero ha hb hc, hd]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
monic_of_d_eq_one
| null |
monic_of_d_eq_one' : (toPoly ⟨0, 0, 0, 1⟩).Monic :=
monic_of_d_eq_one rfl rfl rfl rfl
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
monic_of_d_eq_one'
| null |
@[simps]
equiv : Cubic R ≃ { p : R[X] // p.degree ≤ 3 } where
toFun P := ⟨P.toPoly, degree_cubic_le⟩
invFun f := ⟨coeff f 3, coeff f 2, coeff f 1, coeff f 0⟩
left_inv P := by ext <;> simp only [coeffs]
right_inv f := by
ext n
obtain hn | hn := le_or_gt n 3
· interval_cases n <;> simp only <;> ring_nf <;> try simp only [coeffs]
· rw [coeff_eq_zero hn, (degree_le_iff_coeff_zero (f : R[X]) 3).mp f.2]
simpa using hn
@[simp]
|
def
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
equiv
|
The equivalence between cubic polynomials and polynomials of degree at most three.
|
degree_of_a_ne_zero (ha : P.a ≠ 0) : P.toPoly.degree = 3 :=
degree_cubic ha
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
degree_of_a_ne_zero
| null |
degree_of_a_ne_zero' (ha : a ≠ 0) : (toPoly ⟨a, b, c, d⟩).degree = 3 := by
simp [ha]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
degree_of_a_ne_zero'
| null |
degree_of_a_eq_zero (ha : P.a = 0) : P.toPoly.degree ≤ 2 := by
simpa only [of_a_eq_zero ha] using degree_quadratic_le
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
degree_of_a_eq_zero
| null |
degree_of_a_eq_zero' : (toPoly ⟨0, b, c, d⟩).degree ≤ 2 :=
degree_of_a_eq_zero rfl
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
degree_of_a_eq_zero'
| null |
degree_of_b_ne_zero (ha : P.a = 0) (hb : P.b ≠ 0) : P.toPoly.degree = 2 := by
rw [of_a_eq_zero ha, degree_quadratic hb]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
degree_of_b_ne_zero
| null |
degree_of_b_ne_zero' (hb : b ≠ 0) : (toPoly ⟨0, b, c, d⟩).degree = 2 := by
simp [hb]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
degree_of_b_ne_zero'
| null |
degree_of_b_eq_zero (ha : P.a = 0) (hb : P.b = 0) : P.toPoly.degree ≤ 1 := by
simpa only [of_b_eq_zero ha hb] using degree_linear_le
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
degree_of_b_eq_zero
| null |
degree_of_b_eq_zero' : (toPoly ⟨0, 0, c, d⟩).degree ≤ 1 :=
degree_of_b_eq_zero rfl rfl
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
degree_of_b_eq_zero'
| null |
degree_of_c_ne_zero (ha : P.a = 0) (hb : P.b = 0) (hc : P.c ≠ 0) : P.toPoly.degree = 1 := by
rw [of_b_eq_zero ha hb, degree_linear hc]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
degree_of_c_ne_zero
| null |
degree_of_c_ne_zero' (hc : c ≠ 0) : (toPoly ⟨0, 0, c, d⟩).degree = 1 := by
simp [hc]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
degree_of_c_ne_zero'
| null |
degree_of_c_eq_zero (ha : P.a = 0) (hb : P.b = 0) (hc : P.c = 0) : P.toPoly.degree ≤ 0 := by
simpa only [of_c_eq_zero ha hb hc] using degree_C_le
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
degree_of_c_eq_zero
| null |
degree_of_c_eq_zero' : (toPoly ⟨0, 0, 0, d⟩).degree ≤ 0 :=
degree_of_c_eq_zero rfl rfl rfl
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
degree_of_c_eq_zero'
| null |
degree_of_d_ne_zero (ha : P.a = 0) (hb : P.b = 0) (hc : P.c = 0) (hd : P.d ≠ 0) :
P.toPoly.degree = 0 := by
rw [of_c_eq_zero ha hb hc, degree_C hd]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
degree_of_d_ne_zero
| null |
degree_of_d_ne_zero' (hd : d ≠ 0) : (toPoly ⟨0, 0, 0, d⟩).degree = 0 := by
simp [hd]
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
degree_of_d_ne_zero'
| null |
degree_of_d_eq_zero (ha : P.a = 0) (hb : P.b = 0) (hc : P.c = 0) (hd : P.d = 0) :
P.toPoly.degree = ⊥ := by
rw [of_d_eq_zero ha hb hc hd, degree_zero]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
degree_of_d_eq_zero
| null |
degree_of_d_eq_zero' : (⟨0, 0, 0, 0⟩ : Cubic R).toPoly.degree = ⊥ :=
degree_of_d_eq_zero rfl rfl rfl rfl
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
degree_of_d_eq_zero'
| null |
degree_of_zero : (0 : Cubic R).toPoly.degree = ⊥ :=
degree_of_d_eq_zero'
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
degree_of_zero
| null |
natDegree_of_a_ne_zero (ha : P.a ≠ 0) : P.toPoly.natDegree = 3 :=
natDegree_cubic ha
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
natDegree_of_a_ne_zero
| null |
natDegree_of_a_ne_zero' (ha : a ≠ 0) : (toPoly ⟨a, b, c, d⟩).natDegree = 3 := by
simp [ha]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
natDegree_of_a_ne_zero'
| null |
natDegree_of_a_eq_zero (ha : P.a = 0) : P.toPoly.natDegree ≤ 2 := by
simpa only [of_a_eq_zero ha] using natDegree_quadratic_le
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
natDegree_of_a_eq_zero
| null |
natDegree_of_a_eq_zero' : (toPoly ⟨0, b, c, d⟩).natDegree ≤ 2 :=
natDegree_of_a_eq_zero rfl
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
natDegree_of_a_eq_zero'
| null |
natDegree_of_b_ne_zero (ha : P.a = 0) (hb : P.b ≠ 0) : P.toPoly.natDegree = 2 := by
rw [of_a_eq_zero ha, natDegree_quadratic hb]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
natDegree_of_b_ne_zero
| null |
natDegree_of_b_ne_zero' (hb : b ≠ 0) : (toPoly ⟨0, b, c, d⟩).natDegree = 2 := by
simp [hb]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
natDegree_of_b_ne_zero'
| null |
natDegree_of_b_eq_zero (ha : P.a = 0) (hb : P.b = 0) : P.toPoly.natDegree ≤ 1 := by
simpa only [of_b_eq_zero ha hb] using natDegree_linear_le
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
natDegree_of_b_eq_zero
| null |
natDegree_of_b_eq_zero' : (toPoly ⟨0, 0, c, d⟩).natDegree ≤ 1 :=
natDegree_of_b_eq_zero rfl rfl
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
natDegree_of_b_eq_zero'
| null |
natDegree_of_c_ne_zero (ha : P.a = 0) (hb : P.b = 0) (hc : P.c ≠ 0) :
P.toPoly.natDegree = 1 := by
rw [of_b_eq_zero ha hb, natDegree_linear hc]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
natDegree_of_c_ne_zero
| null |
natDegree_of_c_ne_zero' (hc : c ≠ 0) : (toPoly ⟨0, 0, c, d⟩).natDegree = 1 := by
simp [hc]
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
natDegree_of_c_ne_zero'
| null |
natDegree_of_c_eq_zero (ha : P.a = 0) (hb : P.b = 0) (hc : P.c = 0) :
P.toPoly.natDegree = 0 := by
rw [of_c_eq_zero ha hb hc, natDegree_C]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
natDegree_of_c_eq_zero
| null |
natDegree_of_c_eq_zero' : (toPoly ⟨0, 0, 0, d⟩).natDegree = 0 :=
natDegree_of_c_eq_zero rfl rfl rfl
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
natDegree_of_c_eq_zero'
| null |
natDegree_of_zero : (0 : Cubic R).toPoly.natDegree = 0 :=
natDegree_of_c_eq_zero'
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
natDegree_of_zero
| null |
map (φ : R →+* S) (P : Cubic R) : Cubic S :=
⟨φ P.a, φ P.b, φ P.c, φ P.d⟩
|
def
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
map
|
Map a cubic polynomial across a semiring homomorphism.
|
map_toPoly : (map φ P).toPoly = Polynomial.map φ P.toPoly := by
simp only [map, toPoly, map_C, map_X, Polynomial.map_add, Polynomial.map_mul, Polynomial.map_pow]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
map_toPoly
| null |
roots [IsDomain R] (P : Cubic R) : Multiset R :=
P.toPoly.roots
|
def
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
roots
|
The roots of a cubic polynomial.
|
map_roots [IsDomain S] : (map φ P).roots = (Polynomial.map φ P.toPoly).roots := by
rw [roots, map_toPoly]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
map_roots
| null |
mem_roots_iff [IsDomain R] (h0 : P.toPoly ≠ 0) (x : R) :
x ∈ P.roots ↔ P.a * x ^ 3 + P.b * x ^ 2 + P.c * x + P.d = 0 := by
rw [roots, mem_roots h0, IsRoot, toPoly]
simp only [eval_C, eval_X, eval_add, eval_mul, eval_pow]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
mem_roots_iff
| null |
card_roots_le [IsDomain R] [DecidableEq R] : P.roots.toFinset.card ≤ 3 := by
apply (toFinset_card_le P.toPoly.roots).trans
by_cases hP : P.toPoly = 0
· exact (card_roots' P.toPoly).trans (by rw [hP, natDegree_zero]; exact zero_le 3)
· exact WithBot.coe_le_coe.1 ((card_roots hP).trans degree_cubic_le)
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
card_roots_le
| null |
splits_iff_card_roots (ha : P.a ≠ 0) :
Splits φ P.toPoly ↔ Multiset.card (map φ P).roots = 3 := by
replace ha : (map φ P).a ≠ 0 := (_root_.map_ne_zero φ).mpr ha
nth_rw 1 [← RingHom.id_comp φ]
rw [roots, ← splits_map_iff, ← map_toPoly, Polynomial.splits_iff_card_roots,
← ((degree_eq_iff_natDegree_eq <| ne_zero_of_a_ne_zero ha).1 <| degree_of_a_ne_zero ha : _ = 3)]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
splits_iff_card_roots
| null |
splits_iff_roots_eq_three (ha : P.a ≠ 0) :
Splits φ P.toPoly ↔ ∃ x y z : K, (map φ P).roots = {x, y, z} := by
rw [splits_iff_card_roots ha, card_eq_three]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
splits_iff_roots_eq_three
| null |
eq_prod_three_roots (ha : P.a ≠ 0) (h3 : (map φ P).roots = {x, y, z}) :
(map φ P).toPoly = C (φ P.a) * (X - C x) * (X - C y) * (X - C z) := by
rw [map_toPoly,
eq_prod_roots_of_splits <|
(splits_iff_roots_eq_three ha).mpr <| Exists.intro x <| Exists.intro y <| Exists.intro z h3,
leadingCoeff_of_a_ne_zero ha, ← map_roots, h3]
change C (φ P.a) * ((X - C x) ::ₘ (X - C y) ::ₘ {X - C z}).prod = _
rw [prod_cons, prod_cons, prod_singleton, mul_assoc, mul_assoc]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
eq_prod_three_roots
| null |
eq_sum_three_roots (ha : P.a ≠ 0) (h3 : (map φ P).roots = {x, y, z}) :
map φ P =
⟨φ P.a, φ P.a * -(x + y + z), φ P.a * (x * y + x * z + y * z), φ P.a * -(x * y * z)⟩ := by
apply_fun @toPoly _ _
· rw [eq_prod_three_roots ha h3, C_mul_prod_X_sub_C_eq]
· exact fun P Q ↦ (toPoly_injective P Q).mp
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
eq_sum_three_roots
| null |
b_eq_three_roots (ha : P.a ≠ 0) (h3 : (map φ P).roots = {x, y, z}) :
φ P.b = φ P.a * -(x + y + z) := by
injection eq_sum_three_roots ha h3
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
b_eq_three_roots
| null |
c_eq_three_roots (ha : P.a ≠ 0) (h3 : (map φ P).roots = {x, y, z}) :
φ P.c = φ P.a * (x * y + x * z + y * z) := by
injection eq_sum_three_roots ha h3
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
c_eq_three_roots
| null |
d_eq_three_roots (ha : P.a ≠ 0) (h3 : (map φ P).roots = {x, y, z}) :
φ P.d = φ P.a * -(x * y * z) := by
injection eq_sum_three_roots ha h3
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Polynomial.Splits",
"Mathlib.Tactic.IntervalCases"
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
d_eq_three_roots
| null |
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