lean2/tests/lean/run/fib_brec.lean

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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 poly_unit.star) poly_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 0), succ 0)
(λ (n₁ : nat), nat.cases_on n₁
(λ b₁ : nat.below (succ 0), 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