Induction principles involving factorizations #
Definitions #
def
Nat.recOnPosPrimePosCoprime
{P : ℕ → Sort u_1}
(hp : (p n : ℕ) → Prime p → 0 < n → P (p ^ n))
(h0 : P 0)
(h1 : P 1)
(h : (a b : ℕ) → 1 < a → 1 < b → a.Coprime b → P a → P b → P (a * b))
(a : ℕ)
:
P a
Given P 0
, P 1
, and P (p ^ n)
for positive prime powers, and a way to extend P a
and
P b
to P (a * b)
when a, b
are positive coprime, we can define P
for all natural numbers.
Equations
Instances For
def
Nat.recOnPrimeCoprime
{P : ℕ → Sort u_1}
(h0 : P 0)
(hp : (p n : ℕ) → Prime p → P (p ^ n))
(h : (a b : ℕ) → 1 < a → 1 < b → a.Coprime b → P a → P b → P (a * b))
(a : ℕ)
:
P a
Given P 0
, P (p ^ n)
for all prime powers, and a way to extend P a
and P b
to
P (a * b)
when a, b
are positive coprime, we can define P
for all natural numbers.
Equations
Instances For
Lemmas on multiplicative functions #
theorem
Nat.multiplicative_factorization
{β : Type u_1}
[CommMonoid β]
(f : ℕ → β)
(h_mult : ∀ (x y : ℕ), x.Coprime y → f (x * y) = f x * f y)
(hf : f 1 = 1)
{n : ℕ}
:
For any multiplicative function f
with f 1 = 1
and any n ≠ 0
,
we can evaluate f n
by evaluating f
at p ^ k
over the factorization of n
theorem
Nat.multiplicative_factorization'
{n : ℕ}
{β : Type u_1}
[CommMonoid β]
(f : ℕ → β)
(h_mult : ∀ (x y : ℕ), x.Coprime y → f (x * y) = f x * f y)
(hf0 : f 0 = 1)
(hf1 : f 1 = 1)
:
For any multiplicative function f
with f 1 = 1
and f 0 = 1
,
we can evaluate f n
by evaluating f
at p ^ k
over the factorization of n