37 lines
766 B
Text
37 lines
766 B
Text
mutual
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@[simp] def isEven : Nat → Bool
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| 0 => true
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| n+1 => isOdd n
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decreasing_by apply Nat.lt_succ_self
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@[simp] def isOdd : Nat → Bool
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| 0 => false
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| n+1 => isEven n
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decreasing_by apply Nat.lt_succ_self
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end
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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 => simp [Nat.mul_succ, ih]
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def f (x : Nat) : Nat :=
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match x with
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| 0 => 1
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| x + 1 => f x * 2
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decreasing_by apply Nat.lt_succ_self
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attribute [simp] f
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theorem f_succ (x : Nat) : f (x+1) = f x * 2 := by
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simp
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theorem f_succ₂ (x : Nat) : f (x+1) = f x * 2 := by
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fail_if_success simp [-f]
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simp
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attribute [-simp] f
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theorem f_succ₃ (x : Nat) : f (x+1) = f x * 2 := by
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fail_if_success simp
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simp [f]
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