Linear Diaphontine Equations

We solve linear diaphontine equations of the form a * x + b * y = c where a, b, c are integers if they have a solution with proof. Otherwise, we return a proof that there is no solution.

inductive DiaphontineSolution (a : ℤ) (b : ℤ) (c : ℤ) : Type

• solution: {a b c : ℤ} → (x y : ℤ) → a * x + b * y = c → DiaphontineSolution a b c
• unsolvable: {a b c : ℤ} → (∀ (x y : ℤ), ¬a * x + b * y = c) → DiaphontineSolution a b c

Solution of the linear diaphontine equation a * x + b * y = c where a, b, c are integers or a proof that there is no solution.

This has a solution if and only if the gcd of a and b divides c.

• If the gcd of a and b divides c, by Bezout's Lemma there are integers x and y such that a * x + b * y = gcd a b. Further, as gcd a b divides c, we have an integer d such that (gcd a b) * d = c. Then x * d and y * d are integers such that a * (x * d) + b * (y * d) = c.
• The converse follows as gcd a b divides a and b, hence c = a * x + b * y.

The main results we need are in the library. Here are most of them:

#check Int.gcd_dvd_left -- ∀ (i j : ℤ), ↑(Int.gcd i j) ∣ i
#check Int.emod_add_ediv -- ∀ (a b : ℤ), a % b + b * (a / b) = a
#check Int.emod_eq_zero_of_dvd -- ∀ {a b : ℤ}, a ∣ b → b % a = 0
#check Int.dvd_mul_right -- ∀ (a b : ℤ), a ∣ a * b
#check Int.gcd_eq_gcd_ab -- ∀ (x y : ℤ), ↑(Int.gcd x y) = x * Int.gcdA x y + y * Int.gcdB x y
#check dvd_add /-∀ {α : Type u_1} [inst : Add α] [inst_1 : Semigroup α] [inst_2 : LeftDistribClass α] {a b c : α},
  a ∣ b → a ∣ c → a ∣ b + c-/

def dvdQuotient (a : ℤ) (b : ℤ) (h : b ∣ a) : { q // a = b * q }

Given a b : ℤ such that b ∣ a, we return an integer q such that a = b * q.

theorem eqn_solvable_divides (a : ℤ) (b : ℤ) (c : ℤ) : (∃ x y, a * x + b * y = c) → ↑(Int.gcd a b) ∣ c

If a * x + b * y = c has a solution, then gcd a b divides c.

def DiaphontineSolution.solve (a : ℤ) (b : ℤ) (c : ℤ) : DiaphontineSolution a b c

Solution or proof there is no solution for a * x + b * y = c