test(tests/lean/run): fibonacci using below_rec_on (aka brec_on)

tests/lean/run/fib_brec.lean

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+import data.nat.basic data.prod
+open prod
+
+namespace nat
+  definition below.{l} {C : nat → Type.{l}} (n : nat) :=
+  rec_on n unit.{max 1 l} (λ (n₁ : nat) (r₁ : Type.{max 1 l}), C n₁ × r₁)
+
+  definition brec_on {C : nat → Type} (n : nat) (F : Π (n : nat), @below C n → C n) : C n :=
+  have general : C n × @below C n, from
+    rec_on n
+      (pair (F zero unit.star) unit.star)
+      (λ (n₁ : nat) (r₁ : C n₁ × @below C n₁),
+         have b : @below C (succ n₁), from
+           r₁,
+         have c : C (succ n₁), from
+           F (succ n₁) b,
+         pair c b),
+  pr₁ general
+
+  definition fib (n : nat) :=
+  brec_on n (λ (n : nat),
+    cases_on n
+      (λ (b₀ : below zero), succ zero)
+      (λ (n₁ : nat), cases_on n₁
+          (λ b₁ : below (succ zero), succ zero)
+          (λ (n₂ : nat) (b₂ : 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