51 lines
1.6 KiB
Text
51 lines
1.6 KiB
Text
import data.nat data.prod logic.wf_k
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open nat well_founded decidable prod eq.ops
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namespace playground
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-- Setup
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definition pair_nat.lt := lex lt lt
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definition pair_nat.lt.wf [instance] : well_founded pair_nat.lt :=
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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
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infixl `≺`:50 := pair_nat.lt
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-- Lemma for justifying recursive call
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private lemma lt₁ (x₁ y₁ : nat) : (x₁ - y₁, succ y₁) ≺ (succ x₁, succ y₁) :=
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!lex.left (le_imp_lt_succ (sub_le_self x₁ y₁))
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-- Lemma for justifying recursive call
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private lemma lt₂ (x₁ y₁ : nat) : (succ x₁, y₁ - x₁) ≺ (succ x₁, succ y₁) :=
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!lex.right (le_imp_lt_succ (sub_le_self y₁ x₁))
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definition gcd.F (p₁ : nat × nat) : (Π p₂ : nat × nat, p₂ ≺ p₁ → nat) → nat :=
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prod.cases_on p₁ (λ (x y : nat),
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nat.cases_on x
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(λ f, y) -- x = 0
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(λ x₁, nat.cases_on y
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(λ f, succ x₁) -- y = 0
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(λ y₁ (f : (Π p₂ : nat × nat, p₂ ≺ (succ x₁, succ y₁) → nat)),
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if y₁ ≤ x₁ then f (x₁ - y₁, succ y₁) !lt₁
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else f (succ x₁, y₁ - x₁) !lt₂)))
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definition gcd (x y : nat) :=
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fix gcd.F (pair x y)
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theorem gcd_def_z_y (y : nat) : gcd 0 y = y :=
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well_founded.fix_eq gcd.F (0, y)
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theorem gcd_def_sx_z (x : nat) : gcd (x+1) 0 = x+1 :=
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well_founded.fix_eq gcd.F (x+1, 0)
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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) :=
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well_founded.fix_eq gcd.F (x+1, y+1)
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example : gcd 4 6 = 2 :=
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rfl
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example : gcd 15 6 = 3 :=
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rfl
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example : gcd 0 2 = 2 :=
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rfl
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end playground
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