25 lines
793 B
Text
25 lines
793 B
Text
def f' (n : Nat) : Option { r : Nat // r ≤ n } :=
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match n with
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| 0 => some ⟨0, Nat.le_refl _⟩
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| n+1 => match f' n with
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| some ⟨m, h₁⟩ =>
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have : m < n+1 := Nat.lt_of_le_of_lt h₁ (Nat.lt_succ_self _)
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match f' m with
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| some ⟨r, h₂⟩ => some ⟨r, Nat.le_trans h₂ (Nat.le_trans h₁ (Nat.le_succ _))⟩
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| none => none
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| none => none
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theorem f'_ne_none (n : Nat) : f' n ≠ none := by
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match n with
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| 0 => simp (config := { decide := false }) [f']; done
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| n+1 =>
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simp [f']
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have ih₁ := f'_ne_none n
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split
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next m h₁ he =>
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have : m < n+1 := Nat.lt_of_le_of_lt h₁ (Nat.lt_succ_self _)
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have ih₂ := f'_ne_none m
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split
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next => simp
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next h => contradiction
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next => contradiction
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