26 lines
627 B
Text
26 lines
627 B
Text
mutual
|
|
def isEven : Nat → Bool
|
|
| 0 => true
|
|
| n+1 => isOdd n
|
|
def isOdd : Nat → Bool
|
|
| 0 => false
|
|
| n+1 => isEven n
|
|
end
|
|
termination_by' measure fun | PSum.inl n => n | PSum.inr n => n
|
|
decreasing_by apply Nat.lt_succ_self
|
|
|
|
theorem isEven_double (x : Nat) : isEven (2 * x) = true := by
|
|
induction x with
|
|
| zero => simp [isEven]
|
|
| succ x ih =>
|
|
unfold isEven
|
|
trace_state
|
|
rw [Nat.mul_succ, Nat.add_succ]
|
|
simp
|
|
unfold isOdd
|
|
trace_state
|
|
simp
|
|
exact ih
|
|
|
|
theorem isEven_succ_succ (x : Nat) : isEven (x + 2) = isEven x := by
|
|
conv => lhs; unfold isEven; simp; unfold isOdd; simp
|