lean2/tests/lean/run/nested_rec.lean

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open nat prod sigma
-- We will define the following example by well-foudned recursion
-- g 0 := 0
-- g (succ x) := g (g x)
definition g.F (x : nat) : (Π y, y < x → Σ r : nat, r ≤ y) → Σ r : nat, r ≤ x :=
nat.cases_on x
(λ f, (dpair zero (le.refl zero)))
(λ x₁ (f : Π y, y < succ x₁ → Σ r : nat, r ≤ y),
let p₁ := f x₁ (lt.base x₁) in
let gx₁ := dpr₁ p₁ in
let p₂ := f gx₁ (lt.of_le_of_lt (dpr₂ p₁) (lt.base x₁)) in
let ggx₁ := dpr₁ p₂ in
dpair ggx₁ (le.step (le.trans (dpr₂ p₂) (dpr₂ p₁))))
definition g (x : nat) : nat :=
dpr₁ (well_founded.fix g.F x)
example : g 3 = 0 :=
rfl
example : g 6 = 0 :=
rfl
theorem g_zero : g 0 = 0 :=
rfl
theorem g_succ (a : nat) : g (succ a) = g (g a) :=
have aux : well_founded.fix g.F (succ a) = dpair (g (g a)) _, from
well_founded.fix_eq g.F (succ a),
calc g (succ a) = dpr₁ (well_founded.fix g.F (succ a)) : rfl
... = dpr₁ (dpair (g (g a)) _) : aux
... = g (g a) : rfl
theorem g_all_zero (a : nat) : g a = zero :=
nat.induction_on a
g_zero
(λ a₁ (ih : g a₁ = zero), calc
g (succ a₁) = g (g a₁) : g_succ
... = g 0 : ih
... = 0 : g_zero)