feat(library/data/nat): add coprime_primes theorem

library/data/nat/gcd.lean

@@ -366,6 +366,12 @@ coprime_swap (coprime_of_coprime_mul_left (coprime_swap H))
 theorem coprime_of_coprime_mul_right_right {k m n : ℕ} (H : coprime m (n * k)) : coprime m n :=
 coprime_of_coprime_mul_left_right (!mul.comm ▸ H)
 
+theorem comprime_one_left : ∀ n, coprime 1 n :=
+λ n, !gcd_one_left
+
+theorem comprime_one_right : ∀ n, coprime n 1 :=
+λ n, !gcd_one_right
+
 theorem exists_eq_prod_and_dvd_and_dvd {m n k} (H : k ∣ m * n) :
   ∃ m' n', k = m' * n' ∧ m' ∣ m ∧ n' ∣ n :=
 or.elim (eq_zero_or_pos (gcd k m))

library/data/nat/power.lean

@@ -5,7 +5,7 @@ Authors: Leonardo de Moura, Jeremy Avigad
 
 The power function on the natural numbers.
 -/
-import data.nat.basic data.nat.order data.nat.div algebra.group_power
+import data.nat.basic data.nat.order data.nat.div data.nat.gcd algebra.group_power
 
 namespace nat
 
@@ -61,4 +61,17 @@ lemma dvd_pow_of_dvd_of_pos : ∀ {i j n : nat}, i ∣ j → n > 0 → i ∣ j^n
 lemma pow_mod_eq_zero (i : nat) {n : nat} (h : n > 0) : (i^n) mod i = 0 :=
 iff.mp !dvd_iff_mod_eq_zero (dvd_pow i h)
 
+lemma coprime_pow_right {a b} : ∀ n, coprime b a → coprime b (a^n)
+| 0        h := !comprime_one_right
+| (succ n) h :=
+  begin
+    rewrite [pow_succ],
+    apply coprime_mul_right,
+      exact coprime_pow_right n h,
+      exact h
+  end
+
+lemma coprime_pow_left {a b} : ∀ n, coprime b a → coprime (b^n) a :=
+λ n h, coprime_swap (coprime_pow_right n (coprime_swap h))
+
 end nat

library/data/nat/primes.lean

@@ -159,4 +159,22 @@ lemma dvd_of_prime_of_dvd_pow {p m : nat} : ∀ {n}, prime p → p ∣ m^n → p
     (λ h, dvd_of_prime_of_dvd_pow hp h)
     (λ h, h)
 
+lemma coprime_pow_of_prime_of_not_dvd {p m a : nat} : prime p → ¬ p ∣ a → coprime a (p^m) :=
+λ h₁ h₂, coprime_pow_right m (coprime_swap (coprime_of_prime_of_not_dvd h₁ h₂))
+
+lemma coprime_primes {p q : nat} : prime p → prime q → p ≠ q → coprime p q :=
+λ hp hq hn,
+  assert d₁ : gcd p q ∣ p, from !gcd_dvd_left,
+  assert d₂ : gcd p q ∣ q, from !gcd_dvd_right,
+  or.elim (divisor_of_prime hp d₁)
+    (λ h : gcd p q = 1, h)
+    (λ h : gcd p q = p,
+      have d₃ : p ∣ q, by rewrite -h; exact d₂,
+      or.elim (divisor_of_prime hq d₃)
+        (λ h₁ : p = 1, by subst p; exact absurd hp not_prime_one)
+        (λ he : p = q, by contradiction))
+
+lemma coprime_pow_primes {p q : nat} (n m : nat) : prime p → prime q → p ≠ q → coprime (p^n) (q^m) :=
+λ hp hq hn, coprime_pow_right m (coprime_pow_left n (coprime_primes hp hq hn))
+
 end nat