# mathlibdocumentation

This file defines the p-adic valuation and the p-adic norm on ℚ.

The p-adic valuation on ℚ is the difference of the multiplicities of p in the numerator and denominator of q. This function obeys the standard properties of a valuation, with the appropriate assumptions on p.

The valuation induces a norm on ℚ. This norm is a nonarchimedean absolute value. It takes values in {0} ∪ {1/p^k | k ∈ ℤ}.

## Notations #

This file uses the local notation /. for rat.mk.

## Implementation notes #

Much, but not all, of this file assumes that p is prime. This assumption is inferred automatically by taking [fact (prime p)] as a type class argument.

## Tags #

def padic_val_rat (p : ) (q : ) :

For p ≠ 1, the p-adic valuation of an integer z ≠ 0 is the largest natural number n such that p^n divides z.

padic_val_rat defines the valuation of a rational q to be the valuation of q.num minus the valuation of q.denom. If q = 0 or p = 1, then padic_val_rat p q defaults to 0.

Equations
theorem padic_val_rat_def (p : ) [hp : fact (nat.prime p)] {q : } (hq : q 0) :
= q.num).get _) - (q.denom)).get _)

A simplification of the definition of padic_val_rat p q when q ≠ 0 and p is prime.

@[protected, simp]
theorem padic_val_rat.neg {p : } (q : ) :
(-q) =

padic_val_rat p q is symmetric in q.

@[protected, simp]
theorem padic_val_rat.one {p : } :
= 0

padic_val_rat p 1 is 0 for any p.

@[simp]
= 1

For p ≠ 0, p ≠ 1,padic_val_rat p p is 1.

theorem padic_val_rat.padic_val_rat_of_int {p : } (z : ) (hp : p 1) (hz : z 0) :
= z).get _)

The p-adic value of an integer z ≠ 0 is the multiplicity of p in z.

def padic_val_nat (p n : ) :

A convenience function for the case of padic_val_rat when both inputs are natural numbers.

Equations
theorem zero_le_padic_val_rat_of_nat (p n : ) :
0

padic_val_nat is defined as an int.to_nat cast; this lemma ensures that the cast is well-behaved.

@[simp, norm_cast]
theorem padic_val_rat_of_nat (p n : ) :
n) =

padic_val_rat coincides with padic_val_nat.

theorem padic_val_nat_def {p : } [hp : fact (nat.prime p)] {n : } (hn : n 0) :
= n).get _

A simplification of padic_val_nat when one input is prime, by analogy with padic_val_rat_def.

theorem one_le_padic_val_nat_of_dvd {n p : } [prime : fact (nat.prime p)] (nonzero : n 0) (div : p n) :
1
@[simp]
theorem padic_val_nat_zero (m : ) :
= 0
@[simp]
theorem padic_val_nat_one (m : ) :
= 0
theorem padic_val_rat.finite_int_prime_iff {p : } [p_prime : fact (nat.prime p)] {a : } :
a 0

The multiplicity of p : ℕ in a : ℤ is finite exactly when a ≠ 0.

@[protected]
theorem padic_val_rat.defn (p : ) [p_prime : fact (nat.prime p)] {q : } {n d : } (hqz : q 0) (qdf : q = n /. d) :
= n).get _) - d).get _)

A rewrite lemma for padic_val_rat p q when q is expressed in terms of rat.mk.

@[protected]
theorem padic_val_rat.mul (p : ) [p_prime : fact (nat.prime p)] {q r : } (hq : q 0) (hr : r 0) :
(q * r) = +

A rewrite lemma for padic_val_rat p (q * r) with conditions q ≠ 0, r ≠ 0.

@[protected]
theorem padic_val_rat.pow (p : ) [p_prime : fact (nat.prime p)] {q : } (hq : q 0) {k : } :
(q ^ k) = (k) *

A rewrite lemma for padic_val_rat p (q^k) with conditionq ≠ 0.

@[protected]
theorem padic_val_rat.inv (p : ) [p_prime : fact (nat.prime p)] {q : } (hq : q 0) :
= -

A rewrite lemma for padic_val_rat p (q⁻¹) with condition q ≠ 0.

@[protected]
theorem padic_val_rat.div (p : ) [p_prime : fact (nat.prime p)] {q r : } (hq : q 0) (hr : r 0) :
(q / r) = -

A rewrite lemma for padic_val_rat p (q / r) with conditions q ≠ 0, r ≠ 0.

theorem padic_val_rat.padic_val_rat_le_padic_val_rat_iff (p : ) [p_prime : fact (nat.prime p)] {n₁ n₂ d₁ d₂ : } (hn₁ : n₁ 0) (hn₂ : n₂ 0) (hd₁ : d₁ 0) (hd₂ : d₂ 0) :
(n₁ /. d₁) (n₂ /. d₂) ∀ (n : ), p ^ n n₁ * d₂p ^ n n₂ * d₁

A condition for padic_val_rat p (n₁ / d₁) ≤ padic_val_rat p (n₂ / d₂), in terms of divisibility byp^n.

theorem padic_val_rat.le_padic_val_rat_add_of_le (p : ) [p_prime : fact (nat.prime p)] {q r : } (hq : q 0) (hr : r 0) (hqr : q + r 0) (h : ) :
(q + r)

Sufficient conditions to show that the p-adic valuation of q is less than or equal to the p-adic vlauation of q + r.

theorem padic_val_rat.min_le_padic_val_rat_add (p : ) [p_prime : fact (nat.prime p)] {q r : } (hq : q 0) (hr : r 0) (hqr : q + r 0) :
min q) r) (q + r)

The minimum of the valuations of q and r is less than or equal to the valuation of q + r.

theorem padic_val_rat.sum_pos_of_pos (p : ) [p_prime : fact (nat.prime p)] {n : } {F : } (hF : ∀ (i : ), i < n0 < (F i)) (hn0 : ∑ (i : ) in , F i 0) :
0 < (∑ (i : ) in , F i)

A finite sum of rationals with positive p-adic valuation has positive p-adic valuation (if the sum is non-zero).

@[protected]
theorem padic_val_nat.mul (p : ) [p_prime : fact (nat.prime p)] {q r : } (hq : q 0) (hr : r 0) :
(q * r) = +

A rewrite lemma for padic_val_nat p (q * r) with conditions q ≠ 0, r ≠ 0.

@[protected]
theorem padic_val_nat.div {p : } [p_prime : fact (nat.prime p)] {b : } (dvd : p b) :
(b / p) = - 1

Dividing out by a prime factor reduces the padic_val_nat by 1.

theorem padic_val_nat_of_not_dvd {p : } [fact (nat.prime p)] {n : } (not_dvd : ¬p n) :
= 0

If a prime doesn't appear in n, padic_val_nat p n is 0.

theorem dvd_of_one_le_padic_val_nat {n p : } [prime : fact (nat.prime p)] (hp : 1 ) :
p n
theorem padic_val_nat_primes {p q : } [p_prime : fact (nat.prime p)] [q_prime : fact (nat.prime q)] (neq : p q) :
= 0
@[protected]
theorem padic_val_nat.div' {p : } [p_prime : fact (nat.prime p)] {m : } (cpm : p.coprime m) {b : } (dvd : m b) :
(b / m) =
theorem padic_val_nat_eq_factors_count (p : ) [hp : fact (nat.prime p)] (n : ) :
=
theorem prod_pow_prime_padic_val_nat (n : ) (hn : n 0) (m : ) (pr : n < m) :
∏ (p : ) in , p ^ = n
def padic_norm (p : ) (q : ) :

If q ≠ 0, the p-adic norm of a rational q is p ^ (-(padic_val_rat p q)). If q = 0, the p-adic norm of q is 0.

Equations
@[protected, simp]
theorem padic_norm.eq_fpow_of_nonzero (p : ) {q : } (hq : q 0) :
q = p ^ -

Unfolds the definition of the p-adic norm of q when q ≠ 0.

@[protected]
theorem padic_norm.nonneg (p : ) (q : ) :
0 q

@[protected, simp]
theorem padic_norm.zero (p : ) :
0 = 0

The p-adic norm of 0 is 0.

@[protected, simp]
theorem padic_norm.one (p : ) :
1 = 1

The p-adic norm of 1 is 1.

p = 1 / p

The p-adic norm of p is 1/p if p > 1.

See also padic_norm.padic_norm_p_of_prime for a version that assumes p is prime.

@[simp]

The p-adic norm of p is 1/p if p is prime.

See also padic_norm.padic_norm_p for a version that assumes 1 < p.

theorem padic_norm.padic_norm_of_prime_of_ne {p q : } [p_prime : fact (nat.prime p)] [q_prime : fact (nat.prime q)] (neq : p q) :
q = 1

The p-adic norm of q is 1 if q is prime and not equal to p.

p < 1

The p-adic norm of p is less than 1 if 1 < p.

See also padic_norm.padic_norm_p_lt_one_of_prime for a version assuming prime p.

The p-adic norm of p is less than 1 if p is prime.

See also padic_norm.padic_norm_p_lt_one for a version assuming 1 < p.

@[protected]
theorem padic_norm.values_discrete (p : ) {q : } (hq : q 0) :
∃ (z : ), q = p ^ -z

padic_norm p q takes discrete values p ^ -z for z : ℤ.

@[protected, simp]
theorem padic_norm.neg (p : ) (q : ) :
(-q) = q

padic_norm p is symmetric.

@[protected]
theorem padic_norm.nonzero (p : ) [hp : fact (nat.prime p)] {q : } (hq : q 0) :
q 0

If q ≠ 0, then padic_norm p q ≠ 0.

theorem padic_norm.zero_of_padic_norm_eq_zero (p : ) [hp : fact (nat.prime p)] {q : } (h : q = 0) :
q = 0

If the p-adic norm of q is 0, then q is 0.

@[protected, simp]
theorem padic_norm.mul (p : ) [hp : fact (nat.prime p)] (q r : ) :
(q * r) = q) * r

@[protected, simp]
theorem padic_norm.div (p : ) [hp : fact (nat.prime p)] (q r : ) :
(q / r) = q / r

@[protected]
theorem padic_norm.of_int (p : ) [hp : fact (nat.prime p)] (z : ) :
z 1

The p-adic norm of an integer is at most 1.

@[protected]
theorem padic_norm.nonarchimedean (p : ) [hp : fact (nat.prime p)] {q r : } :
(q + r) max q) r)

The p-adic norm is nonarchimedean: the norm of p + q is at most the max of the norm of p and the norm of q.

theorem padic_norm.triangle_ineq (p : ) [hp : fact (nat.prime p)] (q r : ) :
(q + r) q + r

The p-adic norm respects the triangle inequality: the norm of p + q is at most the norm of p plus the norm of q.

@[protected]
theorem padic_norm.sub (p : ) [hp : fact (nat.prime p)] {q r : } :
(q - r) max q) r)

The p-adic norm of a difference is at most the max of each component. Restates the archimedean property of the p-adic norm.

theorem padic_norm.add_eq_max_of_ne (p : ) [hp : fact (nat.prime p)] {q r : } (hne : q r) :
(q + r) = max q) r)

If the p-adic norms of q and r are different, then the norm of q + r is equal to the max of the norms of q and r`.

@[protected, instance]

The p-adic norm is an absolute value: positive-definite and multiplicative, satisfying the triangle inequality.

theorem padic_norm.dvd_iff_norm_le {p : } [hp : fact (nat.prime p)] {n : } {z : } :
(p ^ n) z z p ^ -n