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