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, ⟨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 := pr₁ p₁ in let p₂ := f gx₁ (lt_of_le_of_lt (pr₂ p₁) (lt.base x₁)) in let ggx₁ := pr₁ p₂ in ⟨ggx₁, le_succ_of_le (le.trans (pr₂ p₂) (pr₂ p₁))⟩) definition g (x : nat) : nat := pr₁ (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) = sigma.mk (g (g a)) _, from well_founded.fix_eq g.F (succ a), calc g (succ a) = pr₁ (well_founded.fix g.F (succ a)) : rfl ... = pr₁ (sigma.mk (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)