```------------------------------------------------------------------------
-- The Agda standard library
--
-- Coprimality
------------------------------------------------------------------------

module Data.Nat.Coprimality where

open import Data.Empty
open import Data.Fin using (toℕ; fromℕ≤)
open import Data.Fin.Props as FinProp
open import Data.Nat
open import Data.Nat.Primality
import Data.Nat.Properties as NatProp
open import Data.Nat.Divisibility as Div
open import Data.Nat.GCD
open import Data.Nat.GCD.Lemmas
open import Data.Product as Prod
open import Function
open import Relation.Binary.PropositionalEquality as PropEq
using (_≡_; _≢_; refl)
open import Relation.Nullary
open import Relation.Binary
open import Algebra
private
module P  = Poset Div.poset
module CS = CommutativeSemiring NatProp.commutativeSemiring

-- Coprime m n is inhabited iff m and n are coprime (relatively
-- prime), i.e. if their only common divisor is 1.

Coprime : (m n : ℕ) → Set
Coprime m n = ∀ {i} → i ∣ m × i ∣ n → i ≡ 1

-- Coprime numbers have 1 as their gcd.

coprime-gcd : ∀ {m n} → Coprime m n → GCD m n 1
coprime-gcd {m} {n} c = GCD.is (1∣ m , 1∣ n) greatest
where
greatest : ∀ {d} → d ∣ m × d ∣ n → d ∣ 1
greatest      cd with c cd
greatest .{1} cd | refl = 1∣ 1

-- If two numbers have 1 as their gcd, then they are coprime.

gcd-coprime : ∀ {m n} → GCD m n 1 → Coprime m n
gcd-coprime g cd with GCD.greatest g cd
gcd-coprime g cd | divides q eq =
NatProp.i*j≡1⇒j≡1 q _ (PropEq.sym eq)

-- Coprime is decidable.

private
0≢1 : 0 ≢ 1
0≢1 ()

2+≢1 : ∀ {n} → suc (suc n) ≢ 1
2+≢1 ()

coprime? : Decidable Coprime
coprime? i j with gcd i j
... | (0           , g) = no  (0≢1  ∘ GCD.unique g ∘ coprime-gcd)
... | (1           , g) = yes (λ {i} → gcd-coprime g {i})
... | (suc (suc d) , g) = no  (2+≢1 ∘ GCD.unique g ∘ coprime-gcd)

-- The coprimality relation is symmetric.

sym : ∀ {m n} → Coprime m n → Coprime n m
sym c = c ∘ swap

-- Everything is coprime to 1.

1-coprimeTo : ∀ m → Coprime 1 m
1-coprimeTo m = ∣1⇒≡1 ∘ proj₁

-- Nothing except for 1 is coprime to 0.

0-coprimeTo-1 : ∀ {m} → Coprime 0 m → m ≡ 1
0-coprimeTo-1 {m} c = c (m ∣0 , P.refl)

-- If m and n are coprime, then n + m and n are also coprime.

coprime-+ : ∀ {m n} → Coprime m n → Coprime (n + m) n
coprime-+ c (d₁ , d₂) = c (∣-∸ d₁ d₂ , d₂)

-- If the "gcd" in Bézout's identity is non-zero, then the "other"
-- divisors are coprime.

Bézout-coprime : ∀ {i j d} →
Bézout.Identity (suc d) (i * suc d) (j * suc d) →
Coprime i j
Bézout-coprime (Bézout.+- x y eq) (divides q₁ refl , divides q₂ refl) =
lem₁₀ y q₂ x q₁ eq
Bézout-coprime (Bézout.-+ x y eq) (divides q₁ refl , divides q₂ refl) =
lem₁₀ x q₁ y q₂ eq

-- Coprime numbers satisfy Bézout's identity.

coprime-Bézout : ∀ {i j} → Coprime i j → Bézout.Identity 1 i j
coprime-Bézout = Bézout.identity ∘ coprime-gcd

-- If i divides jk and is coprime to j, then it divides k.

coprime-divisor : ∀ {k i j} → Coprime i j → i ∣ j * k → i ∣ k
coprime-divisor {k} c (divides q eq′) with coprime-Bézout c
... | Bézout.+- x y eq = divides (x * k ∸ y * q) (lem₈ x y eq eq′)
... | Bézout.-+ x y eq = divides (y * q ∸ x * k) (lem₉ x y eq eq′)

-- If d is a common divisor of mk and nk, and m and n are coprime,
-- then d divides k.

coprime-factors : ∀ {d m n k} →
Coprime m n → d ∣ m * k × d ∣ n * k → d ∣ k
coprime-factors c (divides q₁ eq₁ , divides q₂ eq₂) with coprime-Bézout c
... | Bézout.+- x y eq = divides (x * q₁ ∸ y * q₂) (lem₁₁ x y eq eq₁ eq₂)
... | Bézout.-+ x y eq = divides (y * q₂ ∸ x * q₁) (lem₁₁ y x eq eq₂ eq₁)

-- A variant of GCD.

data GCD′ : ℕ → ℕ → ℕ → Set where
gcd-* : ∀ {d} q₁ q₂ (c : Coprime q₁ q₂) →
GCD′ (q₁ * d) (q₂ * d) d

-- The two definitions are equivalent.

gcd-gcd′ : ∀ {d m n} → GCD m n d → GCD′ m n d
gcd-gcd′         g with GCD.commonDivisor g
gcd-gcd′ {zero}  g | (divides q₁ refl , divides q₂ refl)
with q₁ * 0 | CS.*-comm 0 q₁
| q₂ * 0 | CS.*-comm 0 q₂
...                  | .0 | refl | .0 | refl = gcd-* 1 1 (1-coprimeTo 1)
gcd-gcd′ {suc d} g | (divides q₁ refl , divides q₂ refl) =
gcd-* q₁ q₂ (Bézout-coprime (Bézout.identity g))

gcd′-gcd : ∀ {m n d} → GCD′ m n d → GCD m n d
gcd′-gcd (gcd-* q₁ q₂ c) = GCD.is (∣-* q₁ , ∣-* q₂) (coprime-factors c)

-- Calculates (the alternative representation of) the gcd of the
-- arguments.

gcd′ : ∀ m n → ∃ λ d → GCD′ m n d
gcd′ m n = Prod.map id gcd-gcd′ (gcd m n)

-- Primality implies coprimality.

prime⇒coprime : ∀ m → Prime m →
∀ n → 0 < n → n < m → Coprime m n
prime⇒coprime 0             () _ _  _     _
prime⇒coprime 1             () _ _  _     _
prime⇒coprime (suc (suc m)) _  0 () _     _
prime⇒coprime (suc (suc m)) _  _ _  _ {1} _                       = refl
prime⇒coprime (suc (suc m)) p  _ _  _ {0} (divides q 2+m≡q*0 , _) =
⊥-elim \$ NatProp.i+1+j≢i 0 (begin
2 + m  ≡⟨ 2+m≡q*0 ⟩
q * 0  ≡⟨ proj₂ CS.zero q ⟩
0      ∎)
where open PropEq.≡-Reasoning
prime⇒coprime (suc (suc m)) p (suc n) _ 1+n<2+m {suc (suc i)}
(2+i∣2+m , 2+i∣1+n) =
⊥-elim (p _ 2+i′∣2+m)
where
i<m : i < m
i<m = ≤-pred \$ ≤-pred (begin
3 + i  ≤⟨ s≤s (∣⇒≤ 2+i∣1+n) ⟩
2 + n  ≤⟨ 1+n<2+m ⟩
2 + m  ∎)
where open ≤-Reasoning

2+i′∣2+m : 2 + toℕ (fromℕ≤ i<m) ∣ 2 + m
2+i′∣2+m = PropEq.subst
(λ j → j ∣ 2 + m)
(PropEq.sym (PropEq.cong (_+_ 2) (FinProp.toℕ-fromℕ≤ i<m)))
2+i∣2+m
```