Datasets:

phanerozoic
/

Lean4-Mathlib

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

End of preview. Expand in Data Studio

Mathlib4

Structured dataset of mathematical formalizations from the Mathlib4 library for Lean 4.

Schema

Column	Type	Description
fact	string	Full declaration body
type	string	theorem, lemma, def, class, structure, instance, inductive, abbrev, axiom
library	string	Top-level Mathlib module (Algebra, Analysis, Topology, etc.)
imports	list	Import statements from source file
filename	string	Source file path within Mathlib
symbolic_name	string	Declaration identifier
docstring	string	Documentation comment (30% coverage)

Statistics

By Type

Type	Count	Percentage
theorem	111,804	57%
lemma	36,970	19%
def	27,965	14%
instance	10,322	5%
abbrev	2,570	1%
class	1,710	1%
structure	1,387	1%
inductive	310	<1%

By Library (Top 10)

Library	Count
Algebra	30,936
Analysis	20,343
Data	20,327
CategoryTheory	18,553
Topology	16,800
Order	13,686
RingTheory	12,566
LinearAlgebra	9,279
MeasureTheory	8,927
Combinatorics	4,949