39 lines
797 B
Text
39 lines
797 B
Text
mutual
|
|
@[simp] def isEven : Nat → Bool
|
|
| 0 => true
|
|
| n+1 => isOdd n
|
|
@[simp] def isOdd : Nat → Bool
|
|
| 0 => false
|
|
| n+1 => isEven n
|
|
end
|
|
termination_by measure fun
|
|
| Sum.inl n => n
|
|
| Sum.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
|
|
| succ x ih => simp [Nat.mul_succ, Nat.add_succ, ih]
|
|
|
|
def f (x : Nat) : Nat :=
|
|
match x with
|
|
| 0 => 1
|
|
| x + 1 => f x * 2
|
|
termination_by measure id
|
|
decreasing_by apply Nat.lt_succ_self
|
|
|
|
attribute [simp] f
|
|
|
|
theorem f_succ (x : Nat) : f (x+1) = f x * 2 := by
|
|
simp
|
|
|
|
theorem f_succ₂ (x : Nat) : f (x+1) = f x * 2 := by
|
|
simp [-f]
|
|
simp
|
|
|
|
attribute [-simp] f
|
|
|
|
theorem f_succ₃ (x : Nat) : f (x+1) = f x * 2 := by
|
|
simp
|
|
simp [f]
|