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
value |
|---|---|---|---|---|---|---|---|---|---|---|
coe_add :
((φ + ψ : R[X]) : power_series R) = φ + ψ | by { ext, simp } | lemma | polynomial.coe_add | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul :
((φ * ψ : R[X]) : power_series R) = φ * ψ | power_series.ext $ λ n,
by simp only [coeff_coe, power_series.coeff_mul, coeff_mul] | lemma | polynomial.coe_mul | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"power_series",
"power_series.coeff_mul",
"power_series.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_C (a : R) :
((C a : R[X]) : power_series R) = power_series.C R a | begin
have := coe_monomial 0 a,
rwa power_series.monomial_zero_eq_C_apply at this,
end | lemma | polynomial.coe_C | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"power_series",
"power_series.C",
"power_series.monomial_zero_eq_C_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_bit0 :
((bit0 φ : R[X]) : power_series R) = bit0 (φ : power_series R) | coe_add φ φ | lemma | polynomial.coe_bit0 | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_bit1 :
((bit1 φ : R[X]) : power_series R) = bit1 (φ : power_series R) | by rw [bit1, bit1, coe_add, coe_one, coe_bit0] | lemma | polynomial.coe_bit1 | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_X :
((X : R[X]) : power_series R) = power_series.X | coe_monomial _ _ | lemma | polynomial.coe_X | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"power_series",
"power_series.X"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
constant_coeff_coe : power_series.constant_coeff R φ = φ.coeff 0 | rfl | lemma | polynomial.constant_coeff_coe | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"power_series.constant_coeff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_injective : function.injective (coe : R[X] → power_series R) | λ x y h, by { ext, simp_rw [←coeff_coe, h] } | lemma | polynomial.coe_injective | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inj : (φ : power_series R) = ψ ↔ φ = ψ | (coe_injective R).eq_iff | lemma | polynomial.coe_inj | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_eq_zero_iff : (φ : power_series R) = 0 ↔ φ = 0 | by rw [←coe_zero, coe_inj] | lemma | polynomial.coe_eq_zero_iff | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_eq_one_iff : (φ : power_series R) = 1 ↔ φ = 1 | by rw [←coe_one, coe_inj] | lemma | polynomial.coe_eq_one_iff | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_power_series.ring_hom : R[X] →+* power_series R | { to_fun := (coe : R[X] → power_series R),
map_zero' := coe_zero,
map_one' := coe_one,
map_add' := coe_add,
map_mul' := coe_mul } | def | polynomial.coe_to_power_series.ring_hom | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"power_series"
] | The coercion from polynomials to power series
as a ring homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_power_series.ring_hom_apply : coe_to_power_series.ring_hom φ = φ | rfl | lemma | polynomial.coe_to_power_series.ring_hom_apply | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_pow (n : ℕ):
((φ ^ n : R[X]) : power_series R) = (φ : power_series R) ^ n | coe_to_power_series.ring_hom.map_pow _ _ | lemma | polynomial.coe_pow | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_power_series.alg_hom : R[X] →ₐ[R] power_series A | { commutes' := λ r, by simp [algebra_map_apply, power_series.algebra_map_apply],
..(power_series.map (algebra_map R A)).comp coe_to_power_series.ring_hom } | def | polynomial.coe_to_power_series.alg_hom | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"algebra_map",
"algebra_map_apply",
"power_series",
"power_series.algebra_map_apply",
"power_series.map"
] | The coercion from polynomials to power series
as an algebra homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_power_series.alg_hom_apply : (coe_to_power_series.alg_hom A φ) =
power_series.map (algebra_map R A) ↑φ | rfl | lemma | polynomial.coe_to_power_series.alg_hom_apply | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"algebra_map",
"power_series.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_polynomial : algebra R[X] (power_series A) | ring_hom.to_algebra (polynomial.coe_to_power_series.alg_hom A).to_ring_hom | instance | power_series.algebra_polynomial | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"algebra",
"polynomial.coe_to_power_series.alg_hom",
"power_series",
"ring_hom.to_algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_power_series : algebra (power_series R) (power_series A) | (map (algebra_map R A)).to_algebra | instance | power_series.algebra_power_series | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"algebra",
"algebra_map",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_polynomial' {A : Type*} [comm_semiring A] [algebra R A[X]] :
algebra R (power_series A) | ring_hom.to_algebra $ polynomial.coe_to_power_series.ring_hom.comp (algebra_map R A[X]) | instance | power_series.algebra_polynomial' | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"algebra",
"algebra_map",
"comm_semiring",
"power_series",
"ring_hom.to_algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_apply' (p : R[X]) :
algebra_map R[X] (power_series A) p = map (algebra_map R A) p | rfl | lemma | power_series.algebra_map_apply' | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"algebra_map",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_apply'' :
algebra_map (power_series R) (power_series A) f = map (algebra_map R A) f | rfl | lemma | power_series.algebra_map_apply'' | ring_theory.power_series | src/ring_theory/power_series/basic.lean | [
"data.finsupp.interval",
"data.mv_polynomial.basic",
"data.polynomial.algebra_map",
"data.polynomial.coeff",
"linear_algebra.std_basis",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"tactic.linarith"
] | [
"algebra_map",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_units_sub (u : Rˣ) : power_series R | mk $ λ n, 1 /ₚ u ^ (n + 1) | def | power_series.inv_units_sub | ring_theory.power_series | src/ring_theory/power_series/well_known.lean | [
"ring_theory.power_series.basic",
"data.nat.parity",
"algebra.big_operators.nat_antidiagonal"
] | [
"power_series"
] | The power series for `1 / (u - x)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff_inv_units_sub (u : Rˣ) (n : ℕ) :
coeff R n (inv_units_sub u) = 1 /ₚ u ^ (n + 1) | coeff_mk _ _ | lemma | power_series.coeff_inv_units_sub | ring_theory.power_series | src/ring_theory/power_series/well_known.lean | [
"ring_theory.power_series.basic",
"data.nat.parity",
"algebra.big_operators.nat_antidiagonal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
constant_coeff_inv_units_sub (u : Rˣ) :
constant_coeff R (inv_units_sub u) = 1 /ₚ u | by rw [← coeff_zero_eq_constant_coeff_apply, coeff_inv_units_sub, zero_add, pow_one] | lemma | power_series.constant_coeff_inv_units_sub | ring_theory.power_series | src/ring_theory/power_series/well_known.lean | [
"ring_theory.power_series.basic",
"data.nat.parity",
"algebra.big_operators.nat_antidiagonal"
] | [
"pow_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_units_sub_mul_X (u : Rˣ) :
inv_units_sub u * X = inv_units_sub u * C R u - 1 | begin
ext (_|n),
{ simp },
{ simp [n.succ_ne_zero, pow_succ] }
end | lemma | power_series.inv_units_sub_mul_X | ring_theory.power_series | src/ring_theory/power_series/well_known.lean | [
"ring_theory.power_series.basic",
"data.nat.parity",
"algebra.big_operators.nat_antidiagonal"
] | [
"pow_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_units_sub_mul_sub (u : Rˣ) : inv_units_sub u * (C R u - X) = 1 | by simp [mul_sub, sub_sub_cancel] | lemma | power_series.inv_units_sub_mul_sub | ring_theory.power_series | src/ring_theory/power_series/well_known.lean | [
"ring_theory.power_series.basic",
"data.nat.parity",
"algebra.big_operators.nat_antidiagonal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_inv_units_sub (f : R →+* S) (u : Rˣ) :
map f (inv_units_sub u) = inv_units_sub (units.map (f : R →* S) u) | by { ext, simp [← map_pow] } | lemma | power_series.map_inv_units_sub | ring_theory.power_series | src/ring_theory/power_series/well_known.lean | [
"ring_theory.power_series.basic",
"data.nat.parity",
"algebra.big_operators.nat_antidiagonal"
] | [
"map_pow",
"units.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp : power_series A | mk $ λ n, algebra_map ℚ A (1 / n!) | def | power_series.exp | ring_theory.power_series | src/ring_theory/power_series/well_known.lean | [
"ring_theory.power_series.basic",
"data.nat.parity",
"algebra.big_operators.nat_antidiagonal"
] | [
"algebra_map",
"exp",
"power_series"
] | Power series for the exponential function at zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sin : power_series A | mk $ λ n, if even n then 0 else algebra_map ℚ A ((-1) ^ (n / 2) / n!) | def | power_series.sin | ring_theory.power_series | src/ring_theory/power_series/well_known.lean | [
"ring_theory.power_series.basic",
"data.nat.parity",
"algebra.big_operators.nat_antidiagonal"
] | [
"algebra_map",
"power_series"
] | Power series for the sine function at zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cos : power_series A | mk $ λ n, if even n then algebra_map ℚ A ((-1) ^ (n / 2) / n!) else 0 | def | power_series.cos | ring_theory.power_series | src/ring_theory/power_series/well_known.lean | [
"ring_theory.power_series.basic",
"data.nat.parity",
"algebra.big_operators.nat_antidiagonal"
] | [
"algebra_map",
"power_series"
] | Power series for the cosine function at zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff_exp : coeff A n (exp A) = algebra_map ℚ A (1 / n!) | coeff_mk _ _ | lemma | power_series.coeff_exp | ring_theory.power_series | src/ring_theory/power_series/well_known.lean | [
"ring_theory.power_series.basic",
"data.nat.parity",
"algebra.big_operators.nat_antidiagonal"
] | [
"algebra_map",
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
constant_coeff_exp : constant_coeff A (exp A) = 1 | by { rw [← coeff_zero_eq_constant_coeff_apply, coeff_exp], simp } | lemma | power_series.constant_coeff_exp | ring_theory.power_series | src/ring_theory/power_series/well_known.lean | [
"ring_theory.power_series.basic",
"data.nat.parity",
"algebra.big_operators.nat_antidiagonal"
] | [
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_sin_bit0 : coeff A (bit0 n) (sin A) = 0 | by rw [sin, coeff_mk, if_pos (even_bit0 n)] | lemma | power_series.coeff_sin_bit0 | ring_theory.power_series | src/ring_theory/power_series/well_known.lean | [
"ring_theory.power_series.basic",
"data.nat.parity",
"algebra.big_operators.nat_antidiagonal"
] | [
"even_bit0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_sin_bit1 : coeff A (bit1 n) (sin A) = (-1) ^ n * coeff A (bit1 n) (exp A) | by rw [sin, coeff_mk, if_neg n.not_even_bit1, nat.bit1_div_two,
←mul_one_div, map_mul, map_pow, map_neg, map_one, coeff_exp] | lemma | power_series.coeff_sin_bit1 | ring_theory.power_series | src/ring_theory/power_series/well_known.lean | [
"ring_theory.power_series.basic",
"data.nat.parity",
"algebra.big_operators.nat_antidiagonal"
] | [
"exp",
"map_mul",
"map_one",
"map_pow",
"nat.bit1_div_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_cos_bit0 : coeff A (bit0 n) (cos A) = (-1) ^ n * coeff A (bit0 n) (exp A) | by rw [cos, coeff_mk, if_pos (even_bit0 n), nat.bit0_div_two,
←mul_one_div, map_mul, map_pow, map_neg, map_one, coeff_exp] | lemma | power_series.coeff_cos_bit0 | ring_theory.power_series | src/ring_theory/power_series/well_known.lean | [
"ring_theory.power_series.basic",
"data.nat.parity",
"algebra.big_operators.nat_antidiagonal"
] | [
"even_bit0",
"exp",
"map_mul",
"map_one",
"map_pow",
"nat.bit0_div_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_cos_bit1 : coeff A (bit1 n) (cos A) = 0 | by rw [cos, coeff_mk, if_neg n.not_even_bit1] | lemma | power_series.coeff_cos_bit1 | ring_theory.power_series | src/ring_theory/power_series/well_known.lean | [
"ring_theory.power_series.basic",
"data.nat.parity",
"algebra.big_operators.nat_antidiagonal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_exp : map (f : A →+* A') (exp A) = exp A' | by { ext, simp } | lemma | power_series.map_exp | ring_theory.power_series | src/ring_theory/power_series/well_known.lean | [
"ring_theory.power_series.basic",
"data.nat.parity",
"algebra.big_operators.nat_antidiagonal"
] | [
"exp",
"map_exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_sin : map f (sin A) = sin A' | by { ext, simp [sin, apply_ite f] } | lemma | power_series.map_sin | ring_theory.power_series | src/ring_theory/power_series/well_known.lean | [
"ring_theory.power_series.basic",
"data.nat.parity",
"algebra.big_operators.nat_antidiagonal"
] | [
"apply_ite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_cos : map f (cos A) = cos A' | by { ext, simp [cos, apply_ite f] } | lemma | power_series.map_cos | ring_theory.power_series | src/ring_theory/power_series/well_known.lean | [
"ring_theory.power_series.basic",
"data.nat.parity",
"algebra.big_operators.nat_antidiagonal"
] | [
"apply_ite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_mul_exp_eq_exp_add [algebra ℚ A] (a b : A) :
rescale a (exp A) * rescale b (exp A) = rescale (a + b) (exp A) | begin
ext,
simp only [coeff_mul, exp, rescale, coeff_mk, coe_mk, factorial,
nat.sum_antidiagonal_eq_sum_range_succ_mk, add_pow, sum_mul],
apply sum_congr rfl,
rintros x hx,
suffices : a^x * b^(n - x) * (algebra_map ℚ A (1 / ↑(x.factorial)) * algebra_map ℚ A
(1 / ↑((n - x).factorial))) =
a^x * b^(n... | theorem | power_series.exp_mul_exp_eq_exp_add | ring_theory.power_series | src/ring_theory/power_series/well_known.lean | [
"ring_theory.power_series.basic",
"data.nat.parity",
"algebra.big_operators.nat_antidiagonal"
] | [
"add_pow",
"algebra",
"algebra_map",
"div_eq_iff",
"div_mul_eq_mul_div",
"exp",
"mul_one_div",
"one_div_mul_one_div",
"one_mul",
"ring",
"ring_hom.congr_arg"
] | Shows that $e^{aX} * e^{bX} = e^{(a + b)X}$ | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exp_mul_exp_neg_eq_one [algebra ℚ A] : exp A * eval_neg_hom (exp A) = 1 | by convert exp_mul_exp_eq_exp_add (1 : A) (-1); simp | theorem | power_series.exp_mul_exp_neg_eq_one | ring_theory.power_series | src/ring_theory/power_series/well_known.lean | [
"ring_theory.power_series.basic",
"data.nat.parity",
"algebra.big_operators.nat_antidiagonal"
] | [
"algebra",
"exp"
] | Shows that $e^{x} * e^{-x} = 1$ | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exp_pow_eq_rescale_exp [algebra ℚ A] (k : ℕ) : (exp A)^k = rescale (k : A) (exp A) | begin
induction k with k h,
{ simp only [rescale_zero, constant_coeff_exp, function.comp_app, map_one, cast_zero,
pow_zero, coe_comp], },
simpa only [succ_eq_add_one, cast_add, ←exp_mul_exp_eq_exp_add (k : A), ←h, cast_one,
id_apply, rescale_one] using pow_succ' (exp A) k,
end | theorem | power_series.exp_pow_eq_rescale_exp | ring_theory.power_series | src/ring_theory/power_series/well_known.lean | [
"ring_theory.power_series.basic",
"data.nat.parity",
"algebra.big_operators.nat_antidiagonal"
] | [
"algebra",
"exp",
"map_one",
"pow_succ'",
"pow_zero"
] | Shows that $(e^{X})^k = e^{kX}$. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exp_pow_sum [algebra ℚ A] (n : ℕ) : (finset.range n).sum (λ k, (exp A)^k) =
power_series.mk (λ p, (finset.range n).sum (λ k, k^p * algebra_map ℚ A p.factorial⁻¹)) | begin
simp only [exp_pow_eq_rescale_exp, rescale],
ext,
simp only [one_div, coeff_mk, coe_mk, coeff_exp, factorial, linear_map.map_sum],
end | theorem | power_series.exp_pow_sum | ring_theory.power_series | src/ring_theory/power_series/well_known.lean | [
"ring_theory.power_series.basic",
"data.nat.parity",
"algebra.big_operators.nat_antidiagonal"
] | [
"algebra",
"algebra_map",
"exp",
"finset.range",
"linear_map.map_sum",
"one_div",
"power_series.mk"
] | Shows that
$\sum_{k = 0}^{n - 1} (e^{X})^k = \sum_{p = 0}^{\infty} \sum_{k = 0}^{n - 1} \frac{k^p}{p!}X^p$. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finite_stable_under_composition :
stable_under_composition @finite | by { introv R hf hg, exactI hg.comp hf } | lemma | ring_hom.finite_stable_under_composition | ring_theory.ring_hom | src/ring_theory/ring_hom/finite.lean | [
"ring_theory.ring_hom_properties"
] | [
"finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_respects_iso :
respects_iso @finite | begin
apply finite_stable_under_composition.respects_iso,
introsI,
exact finite.of_surjective _ e.to_equiv.surjective,
end | lemma | ring_hom.finite_respects_iso | ring_theory.ring_hom | src/ring_theory/ring_hom/finite.lean | [
"ring_theory.ring_hom_properties"
] | [
"finite",
"finite.of_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_stable_under_base_change :
stable_under_base_change @finite | begin
refine stable_under_base_change.mk _ finite_respects_iso _,
classical,
introv h,
resetI,
replace h : module.finite R T := by { convert h, ext, rw algebra.smul_def, refl },
suffices : module.finite S (S ⊗[R] T),
{ change module.finite _ _, convert this, ext, rw algebra.smul_def, refl },
exactI infe... | lemma | ring_hom.finite_stable_under_base_change | ring_theory.ring_hom | src/ring_theory/ring_hom/finite.lean | [
"ring_theory.ring_hom_properties"
] | [
"algebra.smul_def",
"finite",
"module.finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_type_stable_under_composition :
stable_under_composition @finite_type | by { introv R hf hg, exactI hg.comp hf } | lemma | ring_hom.finite_type_stable_under_composition | ring_theory.ring_hom | src/ring_theory/ring_hom/finite_type.lean | [
"ring_theory.local_properties",
"ring_theory.localization.inv_submonoid"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_type_holds_for_localization_away :
holds_for_localization_away @finite_type | begin
introv R _,
resetI,
suffices : algebra.finite_type R S,
{ change algebra.finite_type _ _, convert this, ext, rw algebra.smul_def, refl },
exact is_localization.finite_type_of_monoid_fg (submonoid.powers r) S,
end | lemma | ring_hom.finite_type_holds_for_localization_away | ring_theory.ring_hom | src/ring_theory/ring_hom/finite_type.lean | [
"ring_theory.local_properties",
"ring_theory.localization.inv_submonoid"
] | [
"algebra.finite_type",
"algebra.smul_def",
"is_localization.finite_type_of_monoid_fg",
"submonoid.powers"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_type_of_localization_span_target : of_localization_span_target @finite_type | begin
-- Setup algebra intances.
rw of_localization_span_target_iff_finite,
introv R hs H,
resetI,
classical,
letI := f.to_algebra,
replace H : ∀ r : s, algebra.finite_type R (localization.away (r : S)),
{ intro r, convert H r, ext, rw algebra.smul_def, refl },
replace H := λ r, (H r).1,
constructor... | lemma | ring_hom.finite_type_of_localization_span_target | ring_theory.ring_hom | src/ring_theory/ring_hom/finite_type.lean | [
"ring_theory.local_properties",
"ring_theory.localization.inv_submonoid"
] | [
"algebra.adjoin",
"algebra.finite_type",
"algebra.smul_def",
"algebra.subset_adjoin",
"eq_top_iff",
"finset.coe_bUnion",
"finset.coe_image",
"finset.coe_union",
"finsupp.mem_span_iff_total",
"ideal.span",
"is_localization.exists_smul_mem_of_mem_adjoin",
"is_localization.finset_integer_multiple... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_type_is_local :
property_is_local @finite_type | ⟨localization_finite_type, finite_type_of_localization_span_target,
finite_type_stable_under_composition, finite_type_holds_for_localization_away⟩ | lemma | ring_hom.finite_type_is_local | ring_theory.ring_hom | src/ring_theory/ring_hom/finite_type.lean | [
"ring_theory.local_properties",
"ring_theory.localization.inv_submonoid"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_type_respects_iso : ring_hom.respects_iso @ring_hom.finite_type | ring_hom.finite_type_is_local.respects_iso | lemma | ring_hom.finite_type_respects_iso | ring_theory.ring_hom | src/ring_theory/ring_hom/finite_type.lean | [
"ring_theory.local_properties",
"ring_theory.localization.inv_submonoid"
] | [
"ring_hom.finite_type",
"ring_hom.respects_iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_stable_under_composition :
stable_under_composition (λ R S _ _ f, by exactI f.is_integral) | by { introv R hf hg, exactI ring_hom.is_integral_trans _ _ hf hg } | lemma | ring_hom.is_integral_stable_under_composition | ring_theory.ring_hom | src/ring_theory/ring_hom/integral.lean | [
"ring_theory.ring_hom_properties",
"ring_theory.integral_closure"
] | [
"ring_hom.is_integral_trans"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_respects_iso :
respects_iso (λ R S _ _ f, by exactI f.is_integral) | begin
apply is_integral_stable_under_composition.respects_iso,
introv x,
resetI,
rw ← e.apply_symm_apply x,
apply ring_hom.is_integral_map
end | lemma | ring_hom.is_integral_respects_iso | ring_theory.ring_hom | src/ring_theory/ring_hom/integral.lean | [
"ring_theory.ring_hom_properties",
"ring_theory.integral_closure"
] | [
"ring_hom.is_integral_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_stable_under_base_change :
stable_under_base_change (λ R S _ _ f, by exactI f.is_integral) | begin
refine stable_under_base_change.mk _ is_integral_respects_iso _,
introv h x,
resetI,
apply tensor_product.induction_on x,
{ apply is_integral_zero },
{ intros x y, exact is_integral.tmul x (h y) },
{ intros x y hx hy, exact is_integral_add _ hx hy }
end | lemma | ring_hom.is_integral_stable_under_base_change | ring_theory.ring_hom | src/ring_theory/ring_hom/integral.lean | [
"ring_theory.ring_hom_properties",
"ring_theory.integral_closure"
] | [
"is_integral.tmul",
"is_integral_add",
"is_integral_zero",
"tensor_product.induction_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
surjective_stable_under_composition :
stable_under_composition surjective | by { introv R hf hg, exactI hg.comp hf } | lemma | ring_hom.surjective_stable_under_composition | ring_theory.ring_hom | src/ring_theory/ring_hom/surjective.lean | [
"ring_theory.local_properties"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
surjective_respects_iso :
respects_iso surjective | begin
apply surjective_stable_under_composition.respects_iso,
introsI,
exact e.surjective
end | lemma | ring_hom.surjective_respects_iso | ring_theory.ring_hom | src/ring_theory/ring_hom/surjective.lean | [
"ring_theory.local_properties"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
surjective_stable_under_base_change :
stable_under_base_change surjective | begin
refine stable_under_base_change.mk _ surjective_respects_iso _,
classical,
introv h x,
resetI,
induction x using tensor_product.induction_on with x y x y ex ey,
{ exact ⟨0, map_zero _⟩ },
{ obtain ⟨y, rfl⟩ := h y, use y • x, dsimp,
rw [tensor_product.smul_tmul, algebra.algebra_map_eq_smul_one] }... | lemma | ring_hom.surjective_stable_under_base_change | ring_theory.ring_hom | src/ring_theory/ring_hom/surjective.lean | [
"ring_theory.local_properties"
] | [
"algebra.algebra_map_eq_smul_one",
"tensor_product.induction_on",
"tensor_product.smul_tmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
surjective_of_localization_span :
of_localization_span surjective | begin
introv R hs H,
resetI,
letI := f.to_algebra,
show function.surjective (algebra.of_id R S),
rw [← algebra.range_top_iff_surjective, eq_top_iff],
rintro x -,
obtain ⟨l, hl⟩ :=
(finsupp.mem_span_iff_total R s 1).mp (show _ ∈ ideal.span s, by { rw hs, trivial }),
fapply subalgebra.mem_of_finset_su... | lemma | ring_hom.surjective_of_localization_span | ring_theory.ring_hom | src/ring_theory/ring_hom/surjective.lean | [
"ring_theory.local_properties"
] | [
"algebra.of_id",
"algebra.range_top_iff_surjective",
"eq_top_iff",
"finsupp.mem_span_iff_total",
"ideal.span",
"is_localization.eq",
"is_localization.map_mk'",
"is_localization.mk'",
"is_localization.mk'_surjective",
"map_pow",
"pow_add",
"set.mem_range_self",
"subalgebra.mem_of_finset_sum_e... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
roots_of_unity (k : ℕ+) (M : Type*) [comm_monoid M] : subgroup Mˣ | { carrier := { ζ | ζ ^ (k : ℕ) = 1 },
one_mem' := one_pow _,
mul_mem' := λ ζ ξ hζ hξ, by simp only [*, set.mem_set_of_eq, mul_pow, one_mul] at *,
inv_mem' := λ ζ hζ, by simp only [*, set.mem_set_of_eq, inv_pow, inv_one] at * } | def | roots_of_unity | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"comm_monoid",
"inv_one",
"inv_pow",
"mul_pow",
"one_mul",
"one_pow",
"subgroup"
] | `roots_of_unity k M` is the subgroup of elements `m : Mˣ` that satisfy `m ^ k = 1` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_roots_of_unity (k : ℕ+) (ζ : Mˣ) :
ζ ∈ roots_of_unity k M ↔ ζ ^ (k : ℕ) = 1 | iff.rfl | lemma | mem_roots_of_unity | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"roots_of_unity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_roots_of_unity' (k : ℕ+) (ζ : Mˣ) :
ζ ∈ roots_of_unity k M ↔ (ζ : M) ^ (k : ℕ) = 1 | by { rw [mem_roots_of_unity], norm_cast } | lemma | mem_roots_of_unity' | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"mem_roots_of_unity",
"roots_of_unity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
roots_of_unity.coe_injective {n : ℕ+} : function.injective (coe : (roots_of_unity n M) → M) | units.ext.comp (λ x y, subtype.ext) | lemma | roots_of_unity.coe_injective | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"roots_of_unity",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
roots_of_unity.mk_of_pow_eq (ζ : M) {n : ℕ+} (h : ζ ^ (n : ℕ) = 1) :
roots_of_unity n M | ⟨units.of_pow_eq_one ζ n h n.ne_zero, units.pow_of_pow_eq_one _ _⟩ | def | roots_of_unity.mk_of_pow_eq | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"roots_of_unity",
"units.pow_of_pow_eq_one"
] | Make an element of `roots_of_unity` from a member of the base ring, and a proof that it has
a positive power equal to one. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
roots_of_unity.coe_mk_of_pow_eq {ζ : M} {n : ℕ+}
(h : ζ ^ (n : ℕ) = 1) : (roots_of_unity.mk_of_pow_eq _ h : M) = ζ | rfl | lemma | roots_of_unity.coe_mk_of_pow_eq | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"roots_of_unity.mk_of_pow_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
roots_of_unity_le_of_dvd (h : k ∣ l) : roots_of_unity k M ≤ roots_of_unity l M | begin
obtain ⟨d, rfl⟩ := h,
intros ζ h,
simp only [mem_roots_of_unity, pnat.mul_coe, pow_mul, one_pow, *] at *,
end | lemma | roots_of_unity_le_of_dvd | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"mem_roots_of_unity",
"one_pow",
"pnat.mul_coe",
"pow_mul",
"roots_of_unity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_roots_of_unity (f : Mˣ →* Nˣ) (k : ℕ+) :
(roots_of_unity k M).map f ≤ roots_of_unity k N | begin
rintros _ ⟨ζ, h, rfl⟩,
simp only [←map_pow, *, mem_roots_of_unity, set_like.mem_coe, monoid_hom.map_one] at *
end | lemma | map_roots_of_unity | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"mem_roots_of_unity",
"monoid_hom.map_one",
"roots_of_unity",
"set_like.mem_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
roots_of_unity.coe_pow [comm_monoid R] (ζ : roots_of_unity k R) (m : ℕ) :
↑(ζ ^ m) = (ζ ^ m : R) | begin
change ↑(↑(ζ ^ m) : Rˣ) = ↑(ζ : Rˣ) ^ m,
rw [subgroup.coe_pow, units.coe_pow],
end | lemma | roots_of_unity.coe_pow | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"comm_monoid",
"roots_of_unity",
"subgroup.coe_pow",
"units.coe_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_roots_of_unity [ring_hom_class F R S] (σ : F) (n : ℕ+) :
roots_of_unity n R →* roots_of_unity n S | let h : ∀ ξ : roots_of_unity n R, (σ ξ) ^ (n : ℕ) = 1 := λ ξ, by
{ change (σ (ξ : Rˣ)) ^ (n : ℕ) = 1,
rw [←map_pow, ←units.coe_pow, show ((ξ : Rˣ) ^ (n : ℕ) = 1), from ξ.2,
units.coe_one, map_one σ] } in
{ to_fun := λ ξ, ⟨@unit_of_invertible _ _ _ (invertible_of_pow_eq_one _ _ (h ξ) n.ne_zero),
by { ext, rw... | def | restrict_roots_of_unity | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"invertible_of_pow_eq_one",
"map_mul",
"map_one",
"ring_hom_class",
"roots_of_unity",
"subgroup.coe_mul",
"unit_of_invertible",
"units.coe_mul",
"units.coe_one",
"units.coe_pow"
] | Restrict a ring homomorphism to the nth roots of unity | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restrict_roots_of_unity_coe_apply [ring_hom_class F R S] (σ : F)
(ζ : roots_of_unity k R) : ↑(restrict_roots_of_unity σ k ζ) = σ ↑ζ | rfl | lemma | restrict_roots_of_unity_coe_apply | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"restrict_roots_of_unity",
"ring_hom_class",
"roots_of_unity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_equiv.restrict_roots_of_unity (σ : R ≃+* S) (n : ℕ+) :
roots_of_unity n R ≃* roots_of_unity n S | { to_fun := restrict_roots_of_unity σ.to_ring_hom n,
inv_fun :=restrict_roots_of_unity σ.symm.to_ring_hom n,
left_inv := λ ξ, by { ext, exact σ.symm_apply_apply ξ },
right_inv := λ ξ, by { ext, exact σ.apply_symm_apply ξ },
map_mul' := (restrict_roots_of_unity _ n).map_mul } | def | ring_equiv.restrict_roots_of_unity | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"inv_fun",
"map_mul",
"restrict_roots_of_unity",
"roots_of_unity"
] | Restrict a ring isomorphism to the nth roots of unity | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_equiv.restrict_roots_of_unity_coe_apply (σ : R ≃+* S) (ζ : roots_of_unity k R) :
↑(σ.restrict_roots_of_unity k ζ) = σ ↑ζ | rfl | lemma | ring_equiv.restrict_roots_of_unity_coe_apply | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"roots_of_unity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_equiv.restrict_roots_of_unity_symm (σ : R ≃+* S) :
(σ.restrict_roots_of_unity k).symm = σ.symm.restrict_roots_of_unity k | rfl | lemma | ring_equiv.restrict_roots_of_unity_symm | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_roots_of_unity_iff_mem_nth_roots {ζ : Rˣ} :
ζ ∈ roots_of_unity k R ↔ (ζ : R) ∈ nth_roots k (1 : R) | by simp only [mem_roots_of_unity, mem_nth_roots k.pos, units.ext_iff, units.coe_one, units.coe_pow] | lemma | mem_roots_of_unity_iff_mem_nth_roots | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"mem_roots_of_unity",
"roots_of_unity",
"units.coe_one",
"units.coe_pow",
"units.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
roots_of_unity_equiv_nth_roots :
roots_of_unity k R ≃ {x // x ∈ nth_roots k (1 : R)} | begin
refine
{ to_fun := λ x, ⟨x, mem_roots_of_unity_iff_mem_nth_roots.mp x.2⟩,
inv_fun := λ x, ⟨⟨x, x ^ (k - 1 : ℕ), _, _⟩, _⟩,
left_inv := _,
right_inv := _ },
swap 4, { rintro ⟨x, hx⟩, ext, refl },
swap 4, { rintro ⟨x, hx⟩, ext, refl },
all_goals
{ rcases x with ⟨x, hx⟩, rw [mem_nth_roots k.p... | def | roots_of_unity_equiv_nth_roots | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"inv_fun",
"pow_succ",
"pow_succ'",
"roots_of_unity",
"subtype.coe_mk",
"tsub_add_cancel_of_le",
"units.coe_mk",
"units.coe_one",
"units.coe_pow",
"units.ext_iff"
] | Equivalence between the `k`-th roots of unity in `R` and the `k`-th roots of `1`.
This is implemented as equivalence of subtypes,
because `roots_of_unity` is a subgroup of the group of units,
whereas `nth_roots` is a multiset. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
roots_of_unity_equiv_nth_roots_apply (x : roots_of_unity k R) :
(roots_of_unity_equiv_nth_roots R k x : R) = x | rfl | lemma | roots_of_unity_equiv_nth_roots_apply | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"roots_of_unity",
"roots_of_unity_equiv_nth_roots"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
roots_of_unity_equiv_nth_roots_symm_apply (x : {x // x ∈ nth_roots k (1 : R)}) :
((roots_of_unity_equiv_nth_roots R k).symm x : R) = x | rfl | lemma | roots_of_unity_equiv_nth_roots_symm_apply | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"roots_of_unity_equiv_nth_roots"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
roots_of_unity.fintype : fintype (roots_of_unity k R) | fintype.of_equiv {x // x ∈ nth_roots k (1 : R)} $ (roots_of_unity_equiv_nth_roots R k).symm | instance | roots_of_unity.fintype | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"fintype",
"fintype.of_equiv",
"roots_of_unity",
"roots_of_unity_equiv_nth_roots"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
roots_of_unity.is_cyclic : is_cyclic (roots_of_unity k R) | is_cyclic_of_subgroup_is_domain ((units.coe_hom R).comp (roots_of_unity k R).subtype)
(units.ext.comp subtype.val_injective) | instance | roots_of_unity.is_cyclic | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"is_cyclic",
"is_cyclic_of_subgroup_is_domain",
"roots_of_unity",
"subtype.val_injective",
"units.coe_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_roots_of_unity : fintype.card (roots_of_unity k R) ≤ k | calc fintype.card (roots_of_unity k R)
= fintype.card {x // x ∈ nth_roots k (1 : R)} :
fintype.card_congr (roots_of_unity_equiv_nth_roots R k)
... ≤ (nth_roots k (1 : R)).attach.card : multiset.card_le_of_le (multiset.dedup_le _)
... = (nth_roots k (1 : R)).card : multiset.card... | lemma | card_roots_of_unity | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"fintype.card",
"fintype.card_congr",
"multiset.card_attach",
"multiset.card_le_of_le",
"multiset.dedup_le",
"roots_of_unity",
"roots_of_unity_equiv_nth_roots"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_root_of_unity_eq_pow_self [ring_hom_class F R R] (σ : F) (ζ : roots_of_unity k R) :
∃ m : ℕ, σ ζ = ζ ^ m | begin
obtain ⟨m, hm⟩ := monoid_hom.map_cyclic (restrict_roots_of_unity σ k),
rw [←restrict_roots_of_unity_coe_apply, hm, zpow_eq_mod_order_of, ←int.to_nat_of_nonneg
(m.mod_nonneg (int.coe_nat_ne_zero.mpr (pos_iff_ne_zero.mp (order_of_pos ζ)))),
zpow_coe_nat, roots_of_unity.coe_pow],
exact ⟨(m % (order... | lemma | map_root_of_unity_eq_pow_self | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"monoid_hom.map_cyclic",
"order_of",
"order_of_pos",
"restrict_roots_of_unity",
"ring_hom_class",
"roots_of_unity",
"roots_of_unity.coe_pow",
"to_nat",
"zpow_coe_nat",
"zpow_eq_mod_order_of"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_roots_of_unity_prime_pow_mul_iff (p k : ℕ) (m : ℕ+) [hp : fact p.prime]
[char_p R p] {ζ : Rˣ} :
ζ ∈ roots_of_unity (⟨p, hp.1.pos⟩ ^ k * m) R ↔ ζ ∈ roots_of_unity m R | by simp [mem_roots_of_unity'] | lemma | mem_roots_of_unity_prime_pow_mul_iff | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"char_p",
"fact",
"mem_roots_of_unity'",
"roots_of_unity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_primitive_root (ζ : M) (k : ℕ) : Prop | (pow_eq_one : ζ ^ (k : ℕ) = 1)
(dvd_of_pow_eq_one : ∀ l : ℕ, ζ ^ l = 1 → k ∣ l) | structure | is_primitive_root | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [] | An element `ζ` is a primitive `k`-th root of unity if `ζ ^ k = 1`,
and if `l` satisfies `ζ ^ l = 1` then `k ∣ l`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_primitive_root.to_roots_of_unity {μ : M} {n : ℕ+} (h : is_primitive_root μ n) :
roots_of_unity n M | roots_of_unity.mk_of_pow_eq μ h.pow_eq_one | def | is_primitive_root.to_roots_of_unity | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"is_primitive_root",
"roots_of_unity",
"roots_of_unity.mk_of_pow_eq"
] | Turn a primitive root μ into a member of the `roots_of_unity` subgroup. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
primitive_roots (k : ℕ) (R : Type*) [comm_ring R] [is_domain R] : finset R | (nth_roots k (1 : R)).to_finset.filter (λ ζ, is_primitive_root ζ k) | def | primitive_roots | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"comm_ring",
"finset",
"is_domain",
"is_primitive_root"
] | `primitive_roots k R` is the finset of primitive `k`-th roots of unity
in the integral domain `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_primitive_roots {ζ : R} (h0 : 0 < k) :
ζ ∈ primitive_roots k R ↔ is_primitive_root ζ k | begin
rw [primitive_roots, mem_filter, multiset.mem_to_finset, mem_nth_roots h0, and_iff_right_iff_imp],
exact is_primitive_root.pow_eq_one
end | lemma | mem_primitive_roots | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"and_iff_right_iff_imp",
"is_primitive_root",
"multiset.mem_to_finset",
"primitive_roots"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
primitive_roots_zero : primitive_roots 0 R = ∅ | by rw [primitive_roots, nth_roots_zero, multiset.to_finset_zero, finset.filter_empty] | lemma | primitive_roots_zero | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"finset.filter_empty",
"multiset.to_finset_zero",
"primitive_roots"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_primitive_root_of_mem_primitive_roots {ζ : R} (h : ζ ∈ primitive_roots k R) :
is_primitive_root ζ k | k.eq_zero_or_pos.elim (λ hk, false.elim $ by simpa [hk] using h)
(λ hk, (mem_primitive_roots hk).1 h) | lemma | is_primitive_root_of_mem_primitive_roots | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"is_primitive_root",
"mem_primitive_roots",
"primitive_roots"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iff_def (ζ : M) (k : ℕ) :
is_primitive_root ζ k ↔ (ζ ^ k = 1) ∧ (∀ l : ℕ, ζ ^ l = 1 → k ∣ l) | ⟨λ ⟨h1, h2⟩, ⟨h1, h2⟩, λ ⟨h1, h2⟩, ⟨h1, h2⟩⟩ | lemma | is_primitive_root.iff_def | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"iff_def",
"is_primitive_root"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_of_lt (ζ : M) (hk : 0 < k) (h1 : ζ ^ k = 1) (h : ∀ l : ℕ, 0 < l → l < k → ζ ^ l ≠ 1) :
is_primitive_root ζ k | begin
refine ⟨h1, λ l hl, _⟩,
suffices : k.gcd l = k, { exact this ▸ k.gcd_dvd_right l },
rw eq_iff_le_not_lt,
refine ⟨nat.le_of_dvd hk (k.gcd_dvd_left l), _⟩,
intro h', apply h _ (nat.gcd_pos_of_pos_left _ hk) h',
exact pow_gcd_eq_one _ h1 hl
end | lemma | is_primitive_root.mk_of_lt | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"eq_iff_le_not_lt",
"is_primitive_root",
"nat.gcd_pos_of_pos_left",
"pow_gcd_eq_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_subsingleton [subsingleton M] (x : M) : is_primitive_root x 1 | ⟨subsingleton.elim _ _, λ _ _, one_dvd _⟩ | lemma | is_primitive_root.of_subsingleton | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"is_primitive_root",
"one_dvd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_eq_one_iff_dvd (l : ℕ) : ζ ^ l = 1 ↔ k ∣ l | ⟨h.dvd_of_pow_eq_one l,
by { rintro ⟨i, rfl⟩, simp only [pow_mul, h.pow_eq_one, one_pow, pnat.mul_coe] }⟩ | lemma | is_primitive_root.pow_eq_one_iff_dvd | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"one_pow",
"pnat.mul_coe",
"pow_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit (h : is_primitive_root ζ k) (h0 : 0 < k) : is_unit ζ | begin
apply is_unit_of_mul_eq_one ζ (ζ ^ (k - 1)),
rw [← pow_succ, tsub_add_cancel_of_le h0.nat_succ_le, h.pow_eq_one]
end | lemma | is_primitive_root.is_unit | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"is_primitive_root",
"is_unit",
"is_unit_of_mul_eq_one",
"pow_succ",
"tsub_add_cancel_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_ne_one_of_pos_of_lt (h0 : 0 < l) (hl : l < k) : ζ ^ l ≠ 1 | mt (nat.le_of_dvd h0 ∘ h.dvd_of_pow_eq_one _) $ not_le_of_lt hl | lemma | is_primitive_root.pow_ne_one_of_pos_of_lt | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"not_le_of_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_one (hk : 1 < k) : ζ ≠ 1 | h.pow_ne_one_of_pos_of_lt zero_lt_one hk ∘ (pow_one ζ).trans | lemma | is_primitive_root.ne_one | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"pow_one",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_inj (h : is_primitive_root ζ k) ⦃i j : ℕ⦄ (hi : i < k) (hj : j < k) (H : ζ ^ i = ζ ^ j) :
i = j | begin
wlog hij : i ≤ j generalizing i j,
{ exact (this hj hi H.symm (le_of_not_le hij)).symm },
apply le_antisymm hij,
rw ← tsub_eq_zero_iff_le,
apply nat.eq_zero_of_dvd_of_lt _ (lt_of_le_of_lt tsub_le_self hj),
apply h.dvd_of_pow_eq_one,
rw [← ((h.is_unit (lt_of_le_of_lt (nat.zero_le _) hi)).pow i).mul_l... | lemma | is_primitive_root.pow_inj | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"is_primitive_root",
"mul_left_inj",
"nat.eq_zero_of_dvd_of_lt",
"one_mul",
"pow_add",
"tsub_add_cancel_of_le",
"tsub_eq_zero_iff_le",
"tsub_le_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one : is_primitive_root (1 : M) 1 | { pow_eq_one := pow_one _,
dvd_of_pow_eq_one := λ l hl, one_dvd _ } | lemma | is_primitive_root.one | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"is_primitive_root",
"one_dvd",
"pow_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_right_iff : is_primitive_root ζ 1 ↔ ζ = 1 | begin
split,
{ intro h, rw [← pow_one ζ, h.pow_eq_one] },
{ rintro rfl, exact one }
end | lemma | is_primitive_root.one_right_iff | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"is_primitive_root",
"pow_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_submonoid_class_iff {M B : Type*} [comm_monoid M] [set_like B M]
[submonoid_class B M] {N : B} {ζ : N} : is_primitive_root (ζ : M) k ↔ is_primitive_root ζ k | by simp [iff_def, ← submonoid_class.coe_pow] | lemma | is_primitive_root.coe_submonoid_class_iff | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"comm_monoid",
"iff_def",
"is_primitive_root",
"set_like",
"submonoid_class",
"submonoid_class.coe_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_units_iff {ζ : Mˣ} :
is_primitive_root (ζ : M) k ↔ is_primitive_root ζ k | by simp only [iff_def, units.ext_iff, units.coe_pow, units.coe_one] | lemma | is_primitive_root.coe_units_iff | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"iff_def",
"is_primitive_root",
"units.coe_one",
"units.coe_pow",
"units.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_of_coprime (h : is_primitive_root ζ k) (i : ℕ) (hi : i.coprime k) :
is_primitive_root (ζ ^ i) k | begin
by_cases h0 : k = 0,
{ subst k, simp only [*, pow_one, nat.coprime_zero_right] at * },
rcases h.is_unit (nat.pos_of_ne_zero h0) with ⟨ζ, rfl⟩,
rw [← units.coe_pow],
rw coe_units_iff at h ⊢,
refine
{ pow_eq_one := by rw [← pow_mul', pow_mul, h.pow_eq_one, one_pow],
dvd_of_pow_eq_one := _ },
int... | lemma | is_primitive_root.pow_of_coprime | ring_theory.roots_of_unity | src/ring_theory/roots_of_unity/basic.lean | [
"algebra.char_p.two",
"algebra.ne_zero",
"algebra.gcd_monoid.integrally_closed",
"data.polynomial.ring_division",
"field_theory.finite.basic",
"field_theory.separable",
"group_theory.specific_groups.cyclic",
"number_theory.divisors",
"ring_theory.integral_domain",
"tactic.zify"
] | [
"is_primitive_root",
"mul_pow",
"mul_right_comm",
"nat.coprime_zero_right",
"nat.gcd_eq_gcd_ab",
"one_mul",
"one_pow",
"one_zpow",
"pow_mul",
"pow_mul'",
"pow_one",
"units.coe_pow",
"zpow_add",
"zpow_coe_nat",
"zpow_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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