import data.nat data.prod open nat well_founded decidable prod eq.ops namespace playground -- Setup definition pair_nat.lt := lex nat.lt nat.lt definition pair_nat.lt.wf [instance] : well_founded pair_nat.lt := intro_k (prod.lex.wf lt.wf lt.wf) 20 -- the '20' is for being able to execute the examples... it means 20 recursive call without proof computation infixl `≺`:50 := pair_nat.lt -- Lemma for justifying recursive call private lemma lt₁ (x₁ y₁ : nat) : (x₁ - y₁, succ y₁) ≺ (succ x₁, succ y₁) := !lex.left (lt_succ_of_le (sub_le x₁ y₁)) -- Lemma for justifying recursive call private lemma lt₂ (x₁ y₁ : nat) : (succ x₁, y₁ - x₁) ≺ (succ x₁, succ y₁) := !lex.right (lt_succ_of_le (sub_le y₁ x₁)) definition gcd.F (p₁ : nat × nat) : (Π p₂ : nat × nat, p₂ ≺ p₁ → nat) → nat := prod.cases_on p₁ (λ (x y : nat), nat.cases_on x (λ f, y) -- x = 0 (λ x₁, nat.cases_on y (λ f, succ x₁) -- y = 0 (λ y₁ (f : (Π p₂ : nat × nat, p₂ ≺ (succ x₁, succ y₁) → nat)), if y₁ ≤ x₁ then f (x₁ - y₁, succ y₁) !lt₁ else f (succ x₁, y₁ - x₁) !lt₂))) definition gcd (x y : nat) := fix gcd.F (pair x y) theorem gcd_def_z_y (y : nat) : gcd 0 y = y := well_founded.fix_eq gcd.F (0, y) theorem gcd_def_sx_z (x : nat) : gcd (x+1) 0 = x+1 := well_founded.fix_eq gcd.F (x+1, 0) theorem gcd_def_sx_sy (x y : nat) : gcd (x+1) (y+1) = if y ≤ x then gcd (x-y) (y+1) else gcd (x+1) (y-x) := well_founded.fix_eq gcd.F (x+1, y+1) example : gcd 4 6 = 2 := rfl example : gcd 15 6 = 3 := rfl example : gcd 0 2 = 2 := rfl end playground