41 lines
1.5 KiB
Text
41 lines
1.5 KiB
Text
open nat well_founded decidable prod
|
||
|
||
namespace playground
|
||
|
||
-- Setup
|
||
definition pair_nat.lt := lex nat.lt nat.lt
|
||
definition pair_nat.lt.wf : well_founded pair_nat.lt :=
|
||
prod.lex_wf lt_wf lt_wf
|
||
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 pair_nat.lt.wf gcd.F (x, y)
|
||
|
||
theorem gcd_def_z_y (y : nat) : gcd 0 y = y :=
|
||
well_founded.fix_eq pair_nat.lt.wf gcd.F (0, y)
|
||
|
||
theorem gcd_def_sx_z (x : nat) : gcd (x+1) 0 = x+1 :=
|
||
well_founded.fix_eq pair_nat.lt.wf 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 pair_nat.lt.wf gcd.F (x+1, y+1)
|
||
|
||
end playground
|