9cc1164305
We can run the `simpNF` environment linter from Batteries. Nearly all its advice is good.
89 lines
2.2 KiB
Text
89 lines
2.2 KiB
Text
namespace Ex1
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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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end Ex1
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namespace Ex2
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def f (n : Nat) : Option { r : Nat // r ≤ n } :=
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if h : n = 0 then
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some ⟨0, h ▸ Nat.le_refl _⟩
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else
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match f (n-1) with
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| some ⟨m, h₁⟩ =>
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have : m < n := Nat.lt_of_le_of_lt h₁ (Nat.pred_lt h)
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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.pred_le _))⟩
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| none => none
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| none => none
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end Ex2
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namespace Ex3
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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 [f']
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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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def f (n : Nat) : Option Nat :=
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match f' n with
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| some r => some r.1
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| none => none
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theorem f_eq (n : Nat) :
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f n = match n with
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| 0 => some 0
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| n => match f (n - 1) with
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| some m => f m
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| none => none := by
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unfold f
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split
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next r h =>
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revert h
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split <;> try simp [f']
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next => intro h; subst h; simp
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next hne =>
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cases n <;> simp [f']
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next => contradiction
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next n _ =>
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split <;> try simp
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next r hrn h₁ =>
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split <;> simp
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next => intro he; subst he; simp [*]
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next h_eq_none =>
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have hne := f'_ne_none n
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contradiction
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end Ex3
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