import data.nat.basic data.prod open prod namespace nat namespace manual definition brec_on {C : nat → Type} (n : nat) (F : Π (n : nat), @nat.below C n → C n) : C n := have general : C n × @nat.below C n, from nat.rec_on n (pair (F zero unit.star) unit.star) (λ (n₁ : nat) (r₁ : C n₁ × @nat.below C n₁), have b : @nat.below C (succ n₁), from r₁, have c : C (succ n₁), from F (succ n₁) b, pair c b), pr₁ general end manual definition fib (n : nat) := nat.brec_on n (λ (n : nat), nat.cases_on n (λ (b₀ : nat.below zero), succ zero) (λ (n₁ : nat), nat.cases_on n₁ (λ b₁ : nat.below (succ zero), succ zero) (λ (n₂ : nat) (b₂ : nat.below (succ (succ n₂))), pr₁ b₂ + pr₁ (pr₂ b₂)))) theorem fib_0 : fib 0 = 1 := rfl theorem fib_1 : fib 1 = 1 := rfl theorem fib_s_s (n : nat) : fib (succ (succ n)) = fib (succ n) + fib n := rfl example : fib 5 = 8 := rfl example : fib 9 = 55 := rfl end nat