statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
of_c_eq_zero' : to_poly ⟨0, 0, 0, d⟩ = C d | of_c_eq_zero rfl rfl rfl | lemma | cubic.of_c_eq_zero' | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_d_eq_zero (ha : P.a = 0) (hb : P.b = 0) (hc : P.c = 0) (hd : P.d = 0) :
P.to_poly = 0 | by rw [of_c_eq_zero ha hb hc, hd, C_0] | lemma | cubic.of_d_eq_zero | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_d_eq_zero' : (⟨0, 0, 0, 0⟩ : cubic R).to_poly = 0 | of_d_eq_zero rfl rfl rfl rfl | lemma | cubic.of_d_eq_zero' | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [
"cubic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero : (0 : cubic R).to_poly = 0 | of_d_eq_zero' | lemma | cubic.zero | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [
"cubic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_poly_eq_zero_iff (P : cubic R) : P.to_poly = 0 ↔ P = 0 | by rw [← zero, to_poly_injective] | lemma | cubic.to_poly_eq_zero_iff | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [
"cubic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_zero (h0 : P.a ≠ 0 ∨ P.b ≠ 0 ∨ P.c ≠ 0 ∨ P.d ≠ 0) : P.to_poly ≠ 0 | by { contrapose! h0, rw [(to_poly_eq_zero_iff P).mp h0], exact ⟨rfl, rfl, rfl, rfl⟩ } | lemma | cubic.ne_zero | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [
"ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_zero_of_a_ne_zero (ha : P.a ≠ 0) : P.to_poly ≠ 0 | (or_imp_distrib.mp ne_zero).1 ha | lemma | cubic.ne_zero_of_a_ne_zero | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [
"ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_zero_of_b_ne_zero (hb : P.b ≠ 0) : P.to_poly ≠ 0 | (or_imp_distrib.mp (or_imp_distrib.mp ne_zero).2).1 hb | lemma | cubic.ne_zero_of_b_ne_zero | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [
"ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_zero_of_c_ne_zero (hc : P.c ≠ 0) : P.to_poly ≠ 0 | (or_imp_distrib.mp (or_imp_distrib.mp (or_imp_distrib.mp ne_zero).2).2).1 hc | lemma | cubic.ne_zero_of_c_ne_zero | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [
"ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_zero_of_d_ne_zero (hd : P.d ≠ 0) : P.to_poly ≠ 0 | (or_imp_distrib.mp (or_imp_distrib.mp (or_imp_distrib.mp ne_zero).2).2).2 hd | lemma | cubic.ne_zero_of_d_ne_zero | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [
"ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
leading_coeff_of_a_ne_zero (ha : P.a ≠ 0) : P.to_poly.leading_coeff = P.a | leading_coeff_cubic ha | lemma | cubic.leading_coeff_of_a_ne_zero | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
leading_coeff_of_a_ne_zero' (ha : a ≠ 0) : (to_poly ⟨a, b, c, d⟩).leading_coeff = a | leading_coeff_of_a_ne_zero ha | lemma | cubic.leading_coeff_of_a_ne_zero' | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
leading_coeff_of_b_ne_zero (ha : P.a = 0) (hb : P.b ≠ 0) :
P.to_poly.leading_coeff = P.b | by rw [of_a_eq_zero ha, leading_coeff_quadratic hb] | lemma | cubic.leading_coeff_of_b_ne_zero | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
leading_coeff_of_b_ne_zero' (hb : b ≠ 0) : (to_poly ⟨0, b, c, d⟩).leading_coeff = b | leading_coeff_of_b_ne_zero rfl hb | lemma | cubic.leading_coeff_of_b_ne_zero' | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
leading_coeff_of_c_ne_zero (ha : P.a = 0) (hb : P.b = 0) (hc : P.c ≠ 0) :
P.to_poly.leading_coeff = P.c | by rw [of_b_eq_zero ha hb, leading_coeff_linear hc] | lemma | cubic.leading_coeff_of_c_ne_zero | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
leading_coeff_of_c_ne_zero' (hc : c ≠ 0) : (to_poly ⟨0, 0, c, d⟩).leading_coeff = c | leading_coeff_of_c_ne_zero rfl rfl hc | lemma | cubic.leading_coeff_of_c_ne_zero' | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
leading_coeff_of_c_eq_zero (ha : P.a = 0) (hb : P.b = 0) (hc : P.c = 0) :
P.to_poly.leading_coeff = P.d | by rw [of_c_eq_zero ha hb hc, leading_coeff_C] | lemma | cubic.leading_coeff_of_c_eq_zero | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
leading_coeff_of_c_eq_zero' : (to_poly ⟨0, 0, 0, d⟩).leading_coeff = d | leading_coeff_of_c_eq_zero rfl rfl rfl | lemma | cubic.leading_coeff_of_c_eq_zero' | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monic_of_a_eq_one (ha : P.a = 1) : P.to_poly.monic | begin
nontriviality,
rw [monic, leading_coeff_of_a_ne_zero $ by { rw [ha], exact one_ne_zero }, ha]
end | lemma | cubic.monic_of_a_eq_one | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monic_of_a_eq_one' : (to_poly ⟨1, b, c, d⟩).monic | monic_of_a_eq_one rfl | lemma | cubic.monic_of_a_eq_one' | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monic_of_b_eq_one (ha : P.a = 0) (hb : P.b = 1) : P.to_poly.monic | begin
nontriviality,
rw [monic, leading_coeff_of_b_ne_zero ha $ by { rw [hb], exact one_ne_zero }, hb]
end | lemma | cubic.monic_of_b_eq_one | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monic_of_b_eq_one' : (to_poly ⟨0, 1, c, d⟩).monic | monic_of_b_eq_one rfl rfl | lemma | cubic.monic_of_b_eq_one' | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monic_of_c_eq_one (ha : P.a = 0) (hb : P.b = 0) (hc : P.c = 1) : P.to_poly.monic | begin
nontriviality,
rw [monic, leading_coeff_of_c_ne_zero ha hb $ by { rw [hc], exact one_ne_zero }, hc]
end | lemma | cubic.monic_of_c_eq_one | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monic_of_c_eq_one' : (to_poly ⟨0, 0, 1, d⟩).monic | monic_of_c_eq_one rfl rfl rfl | lemma | cubic.monic_of_c_eq_one' | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monic_of_d_eq_one (ha : P.a = 0) (hb : P.b = 0) (hc : P.c = 0) (hd : P.d = 1) :
P.to_poly.monic | by rw [monic, leading_coeff_of_c_eq_zero ha hb hc, hd] | lemma | cubic.monic_of_d_eq_one | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monic_of_d_eq_one' : (to_poly ⟨0, 0, 0, 1⟩).monic | monic_of_d_eq_one rfl rfl rfl rfl | lemma | cubic.monic_of_d_eq_one' | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv : cubic R ≃ {p : R[X] // p.degree ≤ 3} | { to_fun := λ P, ⟨P.to_poly, degree_cubic_le⟩,
inv_fun := λ f, ⟨coeff f 3, coeff f 2, coeff f 1, coeff f 0⟩,
left_inv := λ P, by ext; simp only [subtype.coe_mk, coeffs],
right_inv := λ f,
begin
ext (_ | _ | _ | _ | n); simp only [subtype.coe_mk, coeffs],
have h3 : 3 < n + 4 := by linarith only,
... | def | cubic.equiv | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [
"cubic",
"equiv",
"inv_fun",
"subtype.coe_mk"
] | The equivalence between cubic polynomials and polynomials of degree at most three. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
degree_of_a_ne_zero (ha : P.a ≠ 0) : P.to_poly.degree = 3 | degree_cubic ha | lemma | cubic.degree_of_a_ne_zero | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
degree_of_a_ne_zero' (ha : a ≠ 0) : (to_poly ⟨a, b, c, d⟩).degree = 3 | degree_of_a_ne_zero ha | lemma | cubic.degree_of_a_ne_zero' | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
degree_of_a_eq_zero (ha : P.a = 0) : P.to_poly.degree ≤ 2 | by simpa only [of_a_eq_zero ha] using degree_quadratic_le | lemma | cubic.degree_of_a_eq_zero | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
degree_of_a_eq_zero' : (to_poly ⟨0, b, c, d⟩).degree ≤ 2 | degree_of_a_eq_zero rfl | lemma | cubic.degree_of_a_eq_zero' | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
degree_of_b_ne_zero (ha : P.a = 0) (hb : P.b ≠ 0) : P.to_poly.degree = 2 | by rw [of_a_eq_zero ha, degree_quadratic hb] | lemma | cubic.degree_of_b_ne_zero | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
degree_of_b_ne_zero' (hb : b ≠ 0) : (to_poly ⟨0, b, c, d⟩).degree = 2 | degree_of_b_ne_zero rfl hb | lemma | cubic.degree_of_b_ne_zero' | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
degree_of_b_eq_zero (ha : P.a = 0) (hb : P.b = 0) : P.to_poly.degree ≤ 1 | by simpa only [of_b_eq_zero ha hb] using degree_linear_le | lemma | cubic.degree_of_b_eq_zero | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
degree_of_b_eq_zero' : (to_poly ⟨0, 0, c, d⟩).degree ≤ 1 | degree_of_b_eq_zero rfl rfl | lemma | cubic.degree_of_b_eq_zero' | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
degree_of_c_ne_zero (ha : P.a = 0) (hb : P.b = 0) (hc : P.c ≠ 0) :
P.to_poly.degree = 1 | by rw [of_b_eq_zero ha hb, degree_linear hc] | lemma | cubic.degree_of_c_ne_zero | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
degree_of_c_ne_zero' (hc : c ≠ 0) : (to_poly ⟨0, 0, c, d⟩).degree = 1 | degree_of_c_ne_zero rfl rfl hc | lemma | cubic.degree_of_c_ne_zero' | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
degree_of_c_eq_zero (ha : P.a = 0) (hb : P.b = 0) (hc : P.c = 0) : P.to_poly.degree ≤ 0 | by simpa only [of_c_eq_zero ha hb hc] using degree_C_le | lemma | cubic.degree_of_c_eq_zero | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
degree_of_c_eq_zero' : (to_poly ⟨0, 0, 0, d⟩).degree ≤ 0 | degree_of_c_eq_zero rfl rfl rfl | lemma | cubic.degree_of_c_eq_zero' | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
degree_of_d_ne_zero (ha : P.a = 0) (hb : P.b = 0) (hc : P.c = 0) (hd : P.d ≠ 0) :
P.to_poly.degree = 0 | by rw [of_c_eq_zero ha hb hc, degree_C hd] | lemma | cubic.degree_of_d_ne_zero | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
degree_of_d_ne_zero' (hd : d ≠ 0) : (to_poly ⟨0, 0, 0, d⟩).degree = 0 | degree_of_d_ne_zero rfl rfl rfl hd | lemma | cubic.degree_of_d_ne_zero' | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
degree_of_d_eq_zero (ha : P.a = 0) (hb : P.b = 0) (hc : P.c = 0) (hd : P.d = 0) :
P.to_poly.degree = ⊥ | by rw [of_d_eq_zero ha hb hc hd, degree_zero] | lemma | cubic.degree_of_d_eq_zero | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
degree_of_d_eq_zero' : (⟨0, 0, 0, 0⟩ : cubic R).to_poly.degree = ⊥ | degree_of_d_eq_zero rfl rfl rfl rfl | lemma | cubic.degree_of_d_eq_zero' | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [
"cubic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
degree_of_zero : (0 : cubic R).to_poly.degree = ⊥ | degree_of_d_eq_zero' | lemma | cubic.degree_of_zero | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [
"cubic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_degree_of_a_ne_zero (ha : P.a ≠ 0) : P.to_poly.nat_degree = 3 | nat_degree_cubic ha | lemma | cubic.nat_degree_of_a_ne_zero | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_degree_of_a_ne_zero' (ha : a ≠ 0) : (to_poly ⟨a, b, c, d⟩).nat_degree = 3 | nat_degree_of_a_ne_zero ha | lemma | cubic.nat_degree_of_a_ne_zero' | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_degree_of_a_eq_zero (ha : P.a = 0) : P.to_poly.nat_degree ≤ 2 | by simpa only [of_a_eq_zero ha] using nat_degree_quadratic_le | lemma | cubic.nat_degree_of_a_eq_zero | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_degree_of_a_eq_zero' : (to_poly ⟨0, b, c, d⟩).nat_degree ≤ 2 | nat_degree_of_a_eq_zero rfl | lemma | cubic.nat_degree_of_a_eq_zero' | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_degree_of_b_ne_zero (ha : P.a = 0) (hb : P.b ≠ 0) : P.to_poly.nat_degree = 2 | by rw [of_a_eq_zero ha, nat_degree_quadratic hb] | lemma | cubic.nat_degree_of_b_ne_zero | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_degree_of_b_ne_zero' (hb : b ≠ 0) : (to_poly ⟨0, b, c, d⟩).nat_degree = 2 | nat_degree_of_b_ne_zero rfl hb | lemma | cubic.nat_degree_of_b_ne_zero' | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_degree_of_b_eq_zero (ha : P.a = 0) (hb : P.b = 0) : P.to_poly.nat_degree ≤ 1 | by simpa only [of_b_eq_zero ha hb] using nat_degree_linear_le | lemma | cubic.nat_degree_of_b_eq_zero | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_degree_of_b_eq_zero' : (to_poly ⟨0, 0, c, d⟩).nat_degree ≤ 1 | nat_degree_of_b_eq_zero rfl rfl | lemma | cubic.nat_degree_of_b_eq_zero' | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_degree_of_c_ne_zero (ha : P.a = 0) (hb : P.b = 0) (hc : P.c ≠ 0) :
P.to_poly.nat_degree = 1 | by rw [of_b_eq_zero ha hb, nat_degree_linear hc] | lemma | cubic.nat_degree_of_c_ne_zero | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_degree_of_c_ne_zero' (hc : c ≠ 0) : (to_poly ⟨0, 0, c, d⟩).nat_degree = 1 | nat_degree_of_c_ne_zero rfl rfl hc | lemma | cubic.nat_degree_of_c_ne_zero' | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_degree_of_c_eq_zero (ha : P.a = 0) (hb : P.b = 0) (hc : P.c = 0) :
P.to_poly.nat_degree = 0 | by rw [of_c_eq_zero ha hb hc, nat_degree_C] | lemma | cubic.nat_degree_of_c_eq_zero | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_degree_of_c_eq_zero' : (to_poly ⟨0, 0, 0, d⟩).nat_degree = 0 | nat_degree_of_c_eq_zero rfl rfl rfl | lemma | cubic.nat_degree_of_c_eq_zero' | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_degree_of_zero : (0 : cubic R).to_poly.nat_degree = 0 | nat_degree_of_c_eq_zero' | lemma | cubic.nat_degree_of_zero | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [
"cubic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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 | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [
"cubic"
] | Map a cubic polynomial across a semiring homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_to_poly : (map φ P).to_poly = polynomial.map φ P.to_poly | by simp only [map, to_poly, map_C, map_X, polynomial.map_add, polynomial.map_mul,
polynomial.map_pow] | lemma | cubic.map_to_poly | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [
"polynomial.map",
"polynomial.map_add",
"polynomial.map_mul",
"polynomial.map_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
roots [is_domain R] (P : cubic R) : multiset R | P.to_poly.roots | def | cubic.roots | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [
"cubic",
"is_domain",
"multiset"
] | The roots of a cubic polynomial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_roots [is_domain S] : (map φ P).roots = (polynomial.map φ P.to_poly).roots | by rw [roots, map_to_poly] | lemma | cubic.map_roots | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [
"is_domain",
"polynomial.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_roots_iff [is_domain R] (h0 : P.to_poly ≠ 0) (x : R) :
x ∈ P.roots ↔ P.a * x ^ 3 + P.b * x ^ 2 + P.c * x + P.d = 0 | begin
rw [roots, mem_roots h0, is_root, to_poly],
simp only [eval_C, eval_X, eval_add, eval_mul, eval_pow]
end | theorem | cubic.mem_roots_iff | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [
"is_domain"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_roots_le [is_domain R] [decidable_eq R] : P.roots.to_finset.card ≤ 3 | begin
apply (to_finset_card_le P.to_poly.roots).trans,
by_cases hP : P.to_poly = 0,
{ exact (card_roots' P.to_poly).trans (by { rw [hP, nat_degree_zero], exact zero_le 3 }) },
{ exact with_bot.coe_le_coe.1 ((card_roots hP).trans degree_cubic_le) }
end | theorem | cubic.card_roots_le | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [
"is_domain"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
splits_iff_card_roots (ha : P.a ≠ 0) : splits φ P.to_poly ↔ (map φ P).roots.card = 3 | begin
replace ha : (map φ P).a ≠ 0 := (_root_.map_ne_zero φ).mpr ha,
nth_rewrite_lhs 0 [← ring_hom.id_comp φ],
rw [roots, ← splits_map_iff, ← map_to_poly, splits_iff_card_roots,
← ((degree_eq_iff_nat_degree_eq $ ne_zero_of_a_ne_zero ha).mp $
degree_of_a_ne_zero ha : _ = 3)]
end | theorem | cubic.splits_iff_card_roots | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [
"ring_hom.id_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
splits_iff_roots_eq_three (ha : P.a ≠ 0) :
splits φ P.to_poly ↔ ∃ 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 | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_prod_three_roots (ha : P.a ≠ 0) (h3 : (map φ P).roots = {x, y, z}) :
(map φ P).to_poly = C (φ P.a) * (X - C x) * (X - C y) * (X - C z) | begin
rw [map_to_poly, eq_prod_roots_of_splits $ (splits_iff_roots_eq_three ha).mpr $ exists.intro x $
exists.intro y $ exists.intro z h3, leading_coeff_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_ass... | theorem | cubic.eq_prod_three_roots | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [
"mul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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)⟩ | begin
apply_fun to_poly,
any_goals { exact λ P Q, (to_poly_injective P Q).mp },
rw [eq_prod_three_roots ha h3, C_mul_prod_X_sub_C_eq]
end | theorem | cubic.eq_sum_three_roots | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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 | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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 | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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 | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disc {R : Type*} [ring R] (P : cubic R) : R | P.b ^ 2 * P.c ^ 2 - 4 * P.a * P.c ^ 3 - 4 * P.b ^ 3 * P.d - 27 * P.a ^ 2 * P.d ^ 2
+ 18 * P.a * P.b * P.c * P.d | def | cubic.disc | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [
"cubic",
"ring"
] | The discriminant of a cubic polynomial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
disc_eq_prod_three_roots (ha : P.a ≠ 0) (h3 : (map φ P).roots = {x, y, z}) :
φ P.disc = (φ P.a * φ P.a * (x - y) * (x - z) * (y - z)) ^ 2 | begin
simp only [disc, ring_hom.map_add, ring_hom.map_sub, ring_hom.map_mul, map_pow],
simp only [ring_hom.map_one, map_bit0, map_bit1],
rw [b_eq_three_roots ha h3, c_eq_three_roots ha h3, d_eq_three_roots ha h3],
ring1
end | theorem | cubic.disc_eq_prod_three_roots | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [
"map_bit0",
"map_bit1",
"map_pow",
"ring_hom.map_add",
"ring_hom.map_mul",
"ring_hom.map_one",
"ring_hom.map_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disc_ne_zero_iff_roots_ne (ha : P.a ≠ 0) (h3 : (map φ P).roots = {x, y, z}) :
P.disc ≠ 0 ↔ x ≠ y ∧ x ≠ z ∧ y ≠ z | begin
rw [←_root_.map_ne_zero φ, disc_eq_prod_three_roots ha h3, pow_two],
simp_rw [mul_ne_zero_iff, sub_ne_zero, _root_.map_ne_zero, and_self, and_iff_right ha, and_assoc],
end | theorem | cubic.disc_ne_zero_iff_roots_ne | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [
"mul_ne_zero_iff",
"pow_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disc_ne_zero_iff_roots_nodup (ha : P.a ≠ 0) (h3 : (map φ P).roots = {x, y, z}) :
P.disc ≠ 0 ↔ (map φ P).roots.nodup | begin
rw [disc_ne_zero_iff_roots_ne ha h3, h3],
change _ ↔ (x ::ₘ y ::ₘ {z}).nodup,
rw [nodup_cons, nodup_cons, mem_cons, mem_singleton, mem_singleton],
simp only [nodup_singleton],
tautology
end | theorem | cubic.disc_ne_zero_iff_roots_nodup | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [
"mem_cons"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_roots_of_disc_ne_zero [decidable_eq K] (ha : P.a ≠ 0)
(h3 : (map φ P).roots = {x, y, z}) (hd : P.disc ≠ 0) : (map φ P).roots.to_finset.card = 3 | begin
rw [to_finset_card_of_nodup $ (disc_ne_zero_iff_roots_nodup ha h3).mp hd,
← splits_iff_card_roots ha, splits_iff_roots_eq_three ha],
exact ⟨x, ⟨y, ⟨z, h3⟩⟩⟩
end | theorem | cubic.card_roots_of_disc_ne_zero | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
directed_system (f : Π i j, i ≤ j → G i → G j) : Prop | (map_self [] : ∀ i x h, f i i h x = x)
(map_map [] : ∀ {i j k} hij hjk x, f j k hjk (f i j hij x) = f i k (le_trans hij hjk) x) | class | directed_system | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [] | A directed system is a functor from a category (directed poset) to another category. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
directed_system.map_self [directed_system G (λ i j h, f i j h)] (i x h) :
f i i h x = x | directed_system.map_self (λ i j h, f i j h) i x h | lemma | module.directed_system.map_self | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"directed_system"
] | A copy of `directed_system.map_self` specialized to linear maps, as otherwise the
`λ i j h, f i j h` can confuse the simplifier. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
directed_system.map_map [directed_system G (λ i j h, f i j h)] {i j k} (hij hjk x) :
f j k hjk (f i j hij x) = f i k (le_trans hij hjk) x | directed_system.map_map (λ i j h, f i j h) hij hjk x | lemma | module.directed_system.map_map | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"directed_system"
] | A copy of `directed_system.map_map` specialized to linear maps, as otherwise the
`λ i j h, f i j h` can confuse the simplifier. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
direct_limit : Type (max v w) | direct_sum ι G ⧸ (span R $ { a | ∃ (i j) (H : i ≤ j) x,
direct_sum.lof R ι G i x - direct_sum.lof R ι G j (f i j H x) = a }) | def | module.direct_limit | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"direct_sum",
"direct_sum.lof"
] | The direct limit of a directed system is the modules glued together along the maps. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of (i) : G i →ₗ[R] direct_limit G f | (mkq _).comp $ direct_sum.lof R ι G i | def | module.direct_limit.of | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"direct_sum.lof"
] | The canonical map from a component to the direct limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_f {i j hij x} : (of R ι G f j (f i j hij x)) = of R ι G f i x | eq.symm $ (submodule.quotient.eq _).2 $ subset_span ⟨i, j, hij, x, rfl⟩ | lemma | module.direct_limit.of_f | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"submodule.quotient.eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_of [nonempty ι] [is_directed ι (≤)] (z : direct_limit G f) :
∃ i x, of R ι G f i x = z | nonempty.elim (by apply_instance) $ assume ind : ι,
quotient.induction_on' z $ λ z, direct_sum.induction_on z
⟨ind, 0, linear_map.map_zero _⟩
(λ i x, ⟨i, x, rfl⟩)
(λ p q ⟨i, x, ihx⟩ ⟨j, y, ihy⟩, let ⟨k, hik, hjk⟩ := exists_ge_ge i j in
⟨k, f i k hik x + f j k hjk y, by rw [linear_map.map_add, of_f, of_f, ihx,... | theorem | module.direct_limit.exists_of | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"direct_sum.induction_on",
"exists_ge_ge",
"is_directed",
"linear_map.map_add",
"linear_map.map_zero",
"quotient.induction_on'"
] | Every element of the direct limit corresponds to some element in
some component of the directed system. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
induction_on [nonempty ι] [is_directed ι (≤)] {C : direct_limit G f → Prop}
(z : direct_limit G f)
(ih : ∀ i x, C (of R ι G f i x)) : C z | let ⟨i, x, h⟩ := exists_of z in h ▸ ih i x | theorem | module.direct_limit.induction_on | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"ih",
"is_directed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift : direct_limit G f →ₗ[R] P | liftq _ (direct_sum.to_module R ι P g)
(span_le.2 $ λ a ⟨i, j, hij, x, hx⟩, by rw [← hx, set_like.mem_coe, linear_map.sub_mem_ker_iff,
direct_sum.to_module_lof, direct_sum.to_module_lof, Hg]) | def | module.direct_limit.lift | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"direct_sum.to_module",
"direct_sum.to_module_lof",
"lift",
"linear_map.sub_mem_ker_iff",
"set_like.mem_coe"
] | The universal property of the direct limit: maps from the components to another module
that respect the directed system structure (i.e. make some diagram commute) give rise
to a unique map out of the direct limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_of {i} (x) : lift R ι G f g Hg (of R ι G f i x) = g i x | direct_sum.to_module_lof R _ _ | lemma | module.direct_limit.lift_of | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"direct_sum.to_module_lof",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_unique [nonempty ι] [is_directed ι (≤)] (F : direct_limit G f →ₗ[R] P) (x) :
F x = lift R ι G f (λ i, F.comp $ of R ι G f i)
(λ i j hij x, by rw [linear_map.comp_apply, of_f]; refl) x | direct_limit.induction_on x $ λ i x, by rw lift_of; refl | theorem | module.direct_limit.lift_unique | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"is_directed",
"lift",
"lift_unique",
"linear_map.comp_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
totalize (i j) : G i →ₗ[R] G j | if h : i ≤ j then f i j h else 0 | def | module.direct_limit.totalize | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [] | `totalize G f i j` is a linear map from `G i` to `G j`, for *every* `i` and `j`.
If `i ≤ j`, then it is the map `f i j` that comes with the directed system `G`,
and otherwise it is the zero map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
totalize_of_le {i j} (h : i ≤ j) : totalize G f i j = f i j h | dif_pos h | lemma | module.direct_limit.totalize_of_le | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
totalize_of_not_le {i j} (h : ¬(i ≤ j)) : totalize G f i j = 0 | dif_neg h | lemma | module.direct_limit.totalize_of_not_le | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_module_totalize_of_le {x : direct_sum ι G} {i j : ι}
(hij : i ≤ j) (hx : ∀ k ∈ x.support, k ≤ i) :
direct_sum.to_module R ι (G j) (λ k, totalize G f k j) x =
f i j hij (direct_sum.to_module R ι (G i) (λ k, totalize G f k i) x) | begin
rw [← @dfinsupp.sum_single ι G _ _ _ x],
unfold dfinsupp.sum,
simp only [linear_map.map_sum],
refine finset.sum_congr rfl (λ k hk, _),
rw [direct_sum.single_eq_lof R k (x k), direct_sum.to_module_lof, direct_sum.to_module_lof,
totalize_of_le (hx k hk), totalize_of_le (le_trans (hx k hk) hij), direct... | lemma | module.direct_limit.to_module_totalize_of_le | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"dfinsupp.sum_single",
"direct_sum",
"direct_sum.single_eq_lof",
"direct_sum.to_module",
"direct_sum.to_module_lof",
"linear_map.map_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of.zero_exact_aux [nonempty ι] [is_directed ι (≤)] {x : direct_sum ι G}
(H : submodule.quotient.mk x = (0 : direct_limit G f)) :
∃ j, (∀ k ∈ x.support, k ≤ j) ∧
direct_sum.to_module R ι (G j) (λ i, totalize G f i j) x = (0 : G j) | nonempty.elim (by apply_instance) $ assume ind : ι,
span_induction ((quotient.mk_eq_zero _).1 H)
(λ x ⟨i, j, hij, y, hxy⟩, let ⟨k, hik, hjk⟩ := exists_ge_ge i j in
⟨k, begin
clear_,
subst hxy,
split,
{ intros i0 hi0,
rw [dfinsupp.mem_support_iff, direct_sum.sub_apply, ← direct_sum.... | lemma | module.direct_limit.of.zero_exact_aux | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"dfinsupp.mem_support_iff",
"dfinsupp.single_apply",
"dfinsupp.support_add",
"direct_sum",
"direct_sum.apply_eq_component",
"direct_sum.component.of",
"direct_sum.single_eq_lof",
"direct_sum.sub_apply",
"direct_sum.support_smul",
"direct_sum.to_module",
"exists_ge_ge",
"finset.not_mem_empty",
... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of.zero_exact [is_directed ι (≤)] {i x} (H : of R ι G f i x = 0) :
∃ j hij, f i j hij x = (0 : G j) | by haveI : nonempty ι := ⟨i⟩; exact
let ⟨j, hj, hxj⟩ := of.zero_exact_aux H in
if hx0 : x = 0 then ⟨i, le_rfl, by simp [hx0]⟩
else
have hij : i ≤ j, from hj _ $
by simp [direct_sum.apply_eq_component, hx0],
⟨j, hij, by simpa [totalize_of_le hij] using hxj⟩ | theorem | module.direct_limit.of.zero_exact | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"direct_sum.apply_eq_component",
"is_directed",
"le_rfl"
] | A component that corresponds to zero in the direct limit is already zero in some
bigger module in the directed system. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
direct_limit (f : Π i j, i ≤ j → G i →+ G j) : Type* | @module.direct_limit ℤ _ ι _ _ G _ _
(λ i j hij, (f i j hij).to_int_linear_map) | def | add_comm_group.direct_limit | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"module.direct_limit"
] | The direct limit of a directed system is the abelian groups glued together along the maps. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
directed_system [h : directed_system G (λ i j h, f i j h)] :
directed_system G (λ i j hij, (f i j hij).to_int_linear_map) | h | lemma | add_comm_group.direct_limit.directed_system | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"directed_system"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of (i) : G i →ₗ[ℤ] direct_limit G f | module.direct_limit.of ℤ ι G (λ i j hij, (f i j hij).to_int_linear_map) i | def | add_comm_group.direct_limit.of | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"module.direct_limit.of"
] | The canonical map from a component to the direct limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_f {i j} (hij) (x) : of G f j (f i j hij x) = of G f i x | module.direct_limit.of_f | lemma | add_comm_group.direct_limit.of_f | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"module.direct_limit.of_f"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induction_on [nonempty ι] [is_directed ι (≤)] {C : direct_limit G f → Prop}
(z : direct_limit G f) (ih : ∀ i x, C (of G f i x)) : C z | module.direct_limit.induction_on z ih | theorem | add_comm_group.direct_limit.induction_on | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"ih",
"is_directed",
"module.direct_limit.induction_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of.zero_exact [is_directed ι (≤)] [directed_system G (λ i j h, f i j h)] (i x)
(h : of G f i x = 0) :
∃ j hij, f i j hij x = 0 | module.direct_limit.of.zero_exact h | theorem | add_comm_group.direct_limit.of.zero_exact | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"directed_system",
"is_directed",
"module.direct_limit.of.zero_exact"
] | A component that corresponds to zero in the direct limit is already zero in some
bigger module in the directed system. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift : direct_limit G f →ₗ[ℤ] P | module.direct_limit.lift ℤ ι G (λ i j hij, (f i j hij).to_int_linear_map)
(λ i, (g i).to_int_linear_map) Hg | def | add_comm_group.direct_limit.lift | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"lift",
"module.direct_limit.lift"
] | The universal property of the direct limit: maps from the components to another abelian group
that respect the directed system structure (i.e. make some diagram commute) give rise
to a unique map out of the direct limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_of (i x) : lift G f P g Hg (of G f i x) = g i x | module.direct_limit.lift_of _ _ _ | lemma | add_comm_group.direct_limit.lift_of | algebra | src/algebra/direct_limit.lean | [
"data.finset.order",
"algebra.direct_sum.module",
"ring_theory.free_comm_ring",
"ring_theory.ideal.quotient"
] | [
"lift",
"module.direct_limit.lift_of"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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