import data.nat
open nat tactic

definition fib : nat → nat
| 0        := 1
| 1        := 1
| (n+2)    := fib n + fib (n+1)

example (a : nat) : fib a > 0 :=
by do
  get_local `a >>= λ H, induction_core semireducible H `nat.rec_on [`n, `iH1],
  trace_state, trace "-------",
  unfold [`fib],
  trace_state, trace "-------",
  mk_const `zero_lt_one >>= apply,
  trace_state, trace "-------",
  get_local `n >>= cases,
  unfold [`fib],
  mk_const `zero_lt_one >>= apply,
  unfold [`fib],
  trace_state,
  mk_const `add_pos_of_nonneg_of_pos >>= apply,
  mk_const `nat.zero_le >>= apply,
  assumption