Datasets:

phanerozoic
/

Lean4-Mathlib

statement stringlengths 1 2.93k	proof stringlengths 0 19.2k	type stringclasses 13 values	symbolic_name stringlengths 1 131	library stringlengths 4 62	filename stringlengths 20 95	imports listlengths 0 10	deps listlengths 0 64	docstring stringlengths 0 4.95k	source_url stringclasses 1 value	commit stringclasses 1 value
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	Algebraic.infinite_of_charZero	Algebra	Mathlib/Algebra/AlgebraicCard.lean	[]	[ "Algebra", "CharZero", "CommRing", "Infinite", "IsAlgebraic", "MulActionWithZero.nontrivial", "Nat.cast_injective", "Ring", "isAlgebraic_nat" ]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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	Algebraic.aleph0_le_cardinalMk_of_charZero	Algebra	Mathlib/Algebra/AlgebraicCard.lean	[]	[ "Algebra", "CharZero", "CommRing", "IsAlgebraic", "Ring" ]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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...	theorem	Algebraic.cardinalMk_lift_le_mul	Algebra	Mathlib/Algebra/AlgebraicCard.lean	[]	[ "IsAlgebraic" ]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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 <\| by grw [lift_le.2 cardinalMk_le_max]; simp	theorem	Algebraic.cardinalMk_lift_le_max	Algebra	Mathlib/Algebra/AlgebraicCard.lean	[]	[ "IsAlgebraic", "trans" ]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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)⟩⟩	theorem	Algebraic.cardinalMk_lift_of_infinite	Algebra	Mathlib/Algebra/AlgebraicCard.lean	[]	[ "FaithfulSMul.algebraMap_injective", "Infinite", "IsAlgebraic", "antisymm", "isAlgebraic_algebraMap" ]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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	theorem	Algebraic.countable	Algebra	Mathlib/Algebra/AlgebraicCard.lean	[]	[ "IsAlgebraic", "Set.Countable", "trans" ]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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	Algebraic.cardinalMk_of_countable_of_charZero	Algebra	Mathlib/Algebra/AlgebraicCard.lean	[]	[ "Algebraic.countable", "CharZero", "IsAlgebraic" ]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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	theorem	Algebraic.cardinalMk_le_mul	Algebra	Mathlib/Algebra/AlgebraicCard.lean	[]	[ "IsAlgebraic" ]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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	theorem	Algebraic.cardinalMk_le_max	Algebra	Mathlib/Algebra/AlgebraicCard.lean	[]	[ "IsAlgebraic" ]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
cardinalMk_of_infinite [Infinite R] : #{ x : A // IsAlgebraic R x } = #R	lift_inj.1 <\| cardinalMk_lift_of_infinite R A	theorem	Algebraic.cardinalMk_of_infinite	Algebra	Mathlib/Algebra/AlgebraicCard.lean	[]	[ "Infinite", "IsAlgebraic" ]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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	Cubic	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]	The structure representing a cubic polynomial.	https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
toPoly (P : Cubic R) : R[X]	C P.a * X ^ 3 + C P.b * X ^ 2 + C P.c * X + C P.d	def	Cubic.toPoly	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[ "Cubic" ]	Convert a cubic polynomial to a polynomial.	https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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	Cubic.C_mul_prod_X_sub_C_eq	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[ "CommRing" ]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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]	theorem	Cubic.prod_X_sub_C_eq	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[ "CommRing", "one_mul" ]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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 [Cubic.toPoly, Polynomial.coeff_add, Polynomial.coeff_C, Polynomial.coeff_C_mul_X, Polynomial.coeff_C_mul_X_pow] grind [zero_add]	theorem	Cubic.coeffs	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[ "Cubic.toPoly", "Polynomial.coeff_C", "Polynomial.coeff_C_mul_X", "Polynomial.coeff_C_mul_X_pow", "Polynomial.coeff_add" ]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
coeff_eq_zero {n : ℕ} (hn : 3 < n) : P.toPoly.coeff n = 0	coeffs.1 n hn	theorem	Cubic.coeff_eq_zero	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
coeff_eq_a : P.toPoly.coeff 3 = P.a	coeffs.2.1	theorem	Cubic.coeff_eq_a	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
coeff_eq_b : P.toPoly.coeff 2 = P.b	coeffs.2.2.1	theorem	Cubic.coeff_eq_b	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
coeff_eq_c : P.toPoly.coeff 1 = P.c	coeffs.2.2.2.1	theorem	Cubic.coeff_eq_c	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
coeff_eq_d : P.toPoly.coeff 0 = P.d	coeffs.2.2.2.2	theorem	Cubic.coeff_eq_d	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
a_of_eq (h : P.toPoly = Q.toPoly) : P.a = Q.a	by rw [← coeff_eq_a, h, coeff_eq_a]	theorem	Cubic.a_of_eq	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
b_of_eq (h : P.toPoly = Q.toPoly) : P.b = Q.b	by rw [← coeff_eq_b, h, coeff_eq_b]	theorem	Cubic.b_of_eq	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
c_of_eq (h : P.toPoly = Q.toPoly) : P.c = Q.c	by rw [← coeff_eq_c, h, coeff_eq_c]	theorem	Cubic.c_of_eq	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
d_of_eq (h : P.toPoly = Q.toPoly) : P.d = Q.d	by rw [← coeff_eq_d, h, coeff_eq_d]	theorem	Cubic.d_of_eq	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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	Cubic.toPoly_injective	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[ "Cubic" ]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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	Cubic.of_a_eq_zero	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
of_a_eq_zero' : toPoly ⟨0, b, c, d⟩ = C b * X ^ 2 + C c * X + C d	of_a_eq_zero rfl	theorem	Cubic.of_a_eq_zero'	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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	Cubic.of_b_eq_zero	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
of_b_eq_zero' : toPoly ⟨0, 0, c, d⟩ = C c * X + C d	of_b_eq_zero rfl rfl	theorem	Cubic.of_b_eq_zero'	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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	Cubic.of_c_eq_zero	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
of_c_eq_zero' : toPoly ⟨0, 0, 0, d⟩ = C d	of_c_eq_zero rfl rfl rfl	theorem	Cubic.of_c_eq_zero'	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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	Cubic.of_d_eq_zero	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
of_d_eq_zero' : (⟨0, 0, 0, 0⟩ : Cubic R).toPoly = 0	of_d_eq_zero rfl rfl rfl rfl	theorem	Cubic.of_d_eq_zero'	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[ "Cubic" ]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
zero : (0 : Cubic R).toPoly = 0	of_d_eq_zero'	theorem	Cubic.zero	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[ "Cubic" ]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
toPoly_eq_zero_iff (P : Cubic R) : P.toPoly = 0 ↔ P = 0	by rw [← zero, toPoly_injective]	theorem	Cubic.toPoly_eq_zero_iff	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[ "Cubic" ]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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	Cubic.ne_zero	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
ne_zero_of_a_ne_zero (ha : P.a ≠ 0) : P.toPoly ≠ 0	(or_imp.mp ne_zero).1 ha	theorem	Cubic.ne_zero_of_a_ne_zero	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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	Cubic.ne_zero_of_b_ne_zero	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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	Cubic.ne_zero_of_c_ne_zero	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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	theorem	Cubic.ne_zero_of_d_ne_zero	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
leadingCoeff_of_a_ne_zero (ha : P.a ≠ 0) : P.toPoly.leadingCoeff = P.a	leadingCoeff_cubic ha	theorem	Cubic.leadingCoeff_of_a_ne_zero	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
leadingCoeff_of_a_ne_zero' (ha : a ≠ 0) : (toPoly ⟨a, b, c, d⟩).leadingCoeff = a	by simp [ha]	theorem	Cubic.leadingCoeff_of_a_ne_zero'	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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	Cubic.leadingCoeff_of_b_ne_zero	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
leadingCoeff_of_b_ne_zero' (hb : b ≠ 0) : (toPoly ⟨0, b, c, d⟩).leadingCoeff = b	by simp [hb]	theorem	Cubic.leadingCoeff_of_b_ne_zero'	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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	Cubic.leadingCoeff_of_c_ne_zero	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
leadingCoeff_of_c_ne_zero' (hc : c ≠ 0) : (toPoly ⟨0, 0, c, d⟩).leadingCoeff = c	by simp [hc]	theorem	Cubic.leadingCoeff_of_c_ne_zero'	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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	Cubic.leadingCoeff_of_c_eq_zero	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
leadingCoeff_of_c_eq_zero' : (toPoly ⟨0, 0, 0, d⟩).leadingCoeff = d	leadingCoeff_of_c_eq_zero rfl rfl rfl	theorem	Cubic.leadingCoeff_of_c_eq_zero'	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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	Cubic.monic_of_a_eq_one	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[ "one_ne_zero" ]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
monic_of_a_eq_one' : (toPoly ⟨1, b, c, d⟩).Monic	monic_of_a_eq_one rfl	theorem	Cubic.monic_of_a_eq_one'	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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	Cubic.monic_of_b_eq_one	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[ "one_ne_zero" ]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
monic_of_b_eq_one' : (toPoly ⟨0, 1, c, d⟩).Monic	monic_of_b_eq_one rfl rfl	theorem	Cubic.monic_of_b_eq_one'	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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	Cubic.monic_of_c_eq_one	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[ "one_ne_zero" ]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
monic_of_c_eq_one' : (toPoly ⟨0, 0, 1, d⟩).Monic	monic_of_c_eq_one rfl rfl rfl	theorem	Cubic.monic_of_c_eq_one'	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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	Cubic.monic_of_d_eq_one	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
monic_of_d_eq_one' : (toPoly ⟨0, 0, 0, 1⟩).Monic	monic_of_d_eq_one rfl rfl rfl rfl	theorem	Cubic.monic_of_d_eq_one'	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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,...	def	Cubic.equiv	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[ "Cubic", "le_or_gt" ]	The equivalence between cubic polynomials and polynomials of degree at most three.	https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
degree_of_a_ne_zero (ha : P.a ≠ 0) : P.toPoly.degree = 3	degree_cubic ha	theorem	Cubic.degree_of_a_ne_zero	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
degree_of_a_ne_zero' (ha : a ≠ 0) : (toPoly ⟨a, b, c, d⟩).degree = 3	by simp [ha]	theorem	Cubic.degree_of_a_ne_zero'	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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	Cubic.degree_of_a_eq_zero	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
degree_of_a_eq_zero' : (toPoly ⟨0, b, c, d⟩).degree ≤ 2	degree_of_a_eq_zero rfl	theorem	Cubic.degree_of_a_eq_zero'	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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	Cubic.degree_of_b_ne_zero	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
degree_of_b_ne_zero' (hb : b ≠ 0) : (toPoly ⟨0, b, c, d⟩).degree = 2	by simp [hb]	theorem	Cubic.degree_of_b_ne_zero'	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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	Cubic.degree_of_b_eq_zero	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
degree_of_b_eq_zero' : (toPoly ⟨0, 0, c, d⟩).degree ≤ 1	degree_of_b_eq_zero rfl rfl	theorem	Cubic.degree_of_b_eq_zero'	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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	Cubic.degree_of_c_ne_zero	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
degree_of_c_ne_zero' (hc : c ≠ 0) : (toPoly ⟨0, 0, c, d⟩).degree = 1	by simp [hc]	theorem	Cubic.degree_of_c_ne_zero'	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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	Cubic.degree_of_c_eq_zero	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
degree_of_c_eq_zero' : (toPoly ⟨0, 0, 0, d⟩).degree ≤ 0	degree_of_c_eq_zero rfl rfl rfl	theorem	Cubic.degree_of_c_eq_zero'	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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	Cubic.degree_of_d_ne_zero	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
degree_of_d_ne_zero' (hd : d ≠ 0) : (toPoly ⟨0, 0, 0, d⟩).degree = 0	by simp [hd]	theorem	Cubic.degree_of_d_ne_zero'	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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	Cubic.degree_of_d_eq_zero	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
degree_of_d_eq_zero' : (⟨0, 0, 0, 0⟩ : Cubic R).toPoly.degree = ⊥	degree_of_d_eq_zero rfl rfl rfl rfl	theorem	Cubic.degree_of_d_eq_zero'	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[ "Cubic" ]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
degree_of_zero : (0 : Cubic R).toPoly.degree = ⊥	degree_of_d_eq_zero'	theorem	Cubic.degree_of_zero	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[ "Cubic" ]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
natDegree_of_a_ne_zero (ha : P.a ≠ 0) : P.toPoly.natDegree = 3	natDegree_cubic ha	theorem	Cubic.natDegree_of_a_ne_zero	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
natDegree_of_a_ne_zero' (ha : a ≠ 0) : (toPoly ⟨a, b, c, d⟩).natDegree = 3	by simp [ha]	theorem	Cubic.natDegree_of_a_ne_zero'	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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	Cubic.natDegree_of_a_eq_zero	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
natDegree_of_a_eq_zero' : (toPoly ⟨0, b, c, d⟩).natDegree ≤ 2	natDegree_of_a_eq_zero rfl	theorem	Cubic.natDegree_of_a_eq_zero'	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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	Cubic.natDegree_of_b_ne_zero	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
natDegree_of_b_ne_zero' (hb : b ≠ 0) : (toPoly ⟨0, b, c, d⟩).natDegree = 2	by simp [hb]	theorem	Cubic.natDegree_of_b_ne_zero'	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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	Cubic.natDegree_of_b_eq_zero	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
natDegree_of_b_eq_zero' : (toPoly ⟨0, 0, c, d⟩).natDegree ≤ 1	natDegree_of_b_eq_zero rfl rfl	theorem	Cubic.natDegree_of_b_eq_zero'	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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	Cubic.natDegree_of_c_ne_zero	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
natDegree_of_c_ne_zero' (hc : c ≠ 0) : (toPoly ⟨0, 0, c, d⟩).natDegree = 1	by simp [hc]	theorem	Cubic.natDegree_of_c_ne_zero'	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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	Cubic.natDegree_of_c_eq_zero	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
natDegree_of_c_eq_zero' : (toPoly ⟨0, 0, 0, d⟩).natDegree = 0	natDegree_of_c_eq_zero rfl rfl rfl	theorem	Cubic.natDegree_of_c_eq_zero'	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
natDegree_of_zero : (0 : Cubic R).toPoly.natDegree = 0	natDegree_of_c_eq_zero'	theorem	Cubic.natDegree_of_zero	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[ "Cubic" ]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
map (φ : R →+* S) (P : Cubic R) : Cubic S	⟨φ P.a, φ P.b, φ P.c, φ P.d⟩	def	Cubic.map	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[ "Cubic" ]	Map a cubic polynomial across a semiring homomorphism.	https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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	Cubic.map_toPoly	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[ "Polynomial.map", "Polynomial.map_add", "Polynomial.map_mul", "Polynomial.map_pow" ]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
roots [IsDomain R] (P : Cubic R) : Multiset R	P.toPoly.roots	def	Cubic.roots	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[ "Cubic", "IsDomain" ]	The roots of a cubic polynomial.	https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
map_roots [IsDomain S] : (map φ P).roots = (Polynomial.map φ P.toPoly).roots	by rw [roots, map_toPoly]	theorem	Cubic.map_roots	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[ "IsDomain", "Polynomial.map" ]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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	Cubic.mem_roots_iff	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[ "IsDomain" ]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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 · simp [hP] · exact WithBot.coe_le_coe.1 ((card_roots hP).trans degree_cubic_le)	theorem	Cubic.card_roots_le	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[ "IsDomain", "by_cases", "trans" ]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
splits_iff_card_roots (ha : P.a ≠ 0) : Splits (P.toPoly.map φ) ↔ (map φ P).roots.card = 3	by replace ha : (map φ P).a ≠ 0 := (map_ne_zero φ).mpr ha rw [roots, ← 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	Cubic.splits_iff_card_roots	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[ "Polynomial.splits_iff_card_roots", "map_ne_zero" ]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
splits_iff_roots_eq_three (ha : P.a ≠ 0) : Splits (P.toPoly.map φ) ↔ ∃ x y z : K, (map φ P).roots = {x, y, z}	by rw [splits_iff_card_roots ha, card_eq_three]	theorem	Cubic.splits_iff_roots_eq_three	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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, Splits.eq_prod_roots <\| (splits_iff_roots_eq_three ha).mpr <\| Exists.intro x <\| Exists.intro y <\| Exists.intro z h3, leadingCoeff_map, 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_...	theorem	Cubic.eq_prod_three_roots	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[ "mul_assoc" ]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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	Cubic.eq_sum_three_roots	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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	Cubic.b_eq_three_roots	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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	Cubic.c_eq_three_roots	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319
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	Cubic.d_eq_three_roots	Algebra	Mathlib/Algebra/CubicDiscriminant.lean	[]	[]		https://github.com/leanprover-community/mathlib4	b9f14353520df73472ae3825fb53f86559a01319

End of preview. Expand in Data Studio

Lean4-Mathlib

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

Source

Repository: https://github.com/leanprover-community/mathlib4
Commit: b9f14353520df73472ae3825fb53f86559a01319
Files: 8170
License: apache-2.0

Schema

Column	Type	Description
statement	string	Declaration signature/claim with the leading keyword removed (verbatim slice); the full declaration minus its proof
proof	string	Verbatim proof/body, empty if the declaration has none
type	string	Declaration keyword
symbolic_name	string	Declaration identifier
library	string	Sub-library
filename	string	Repository-relative source path
imports	list[string]	File-level `Require`/`Import` modules
deps	list[string]	Intra-corpus identifiers referenced
docstring	string	Preceding documentation comment, empty if absent
source_url	string	Upstream repository
commit	string	Upstream commit extracted

Statistics

Entries: 232,431
With proof: 228,810 (98.4%)
With docstring: 67,814 (29.2%)
Libraries: 1103

By type

Type	Count
theorem	125,463
lemma	56,488
def	31,961
instance	10,971
abbrev	3,379
class	1,937
structure	1,580
inductive	326
elab	168
macro	147
class inductive	6
class abbrev	3
opaque	2

Example

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

type: theorem | symbolic_name: Algebraic.infinite_of_charZero | Mathlib/Algebra/AlgebraicCard.lean

Use

Each declaration is split into a statement (signature/claim) and a proof (body) that are disjoint and together form the complete declaration, for proof modeling, autoformalization, retrieval, and dependency analysis via deps.

Citation

@misc{lean4_mathlib_dataset,
  title  = {Lean4-Mathlib},
  author = {Norton, Charles},
  year   = {2026},
  note   = {Extracted from https://github.com/leanprover-community/mathlib4, commit b9f14353520d},
  url    = {https://huggingface.co/datasets/phanerozoic/Lean4-Mathlib}
}

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Digesting proof assistant libraries for AI ingestion. • 103 items • Updated 1 day ago • 3