25 lines
744 B
Text
25 lines
744 B
Text
namespace List
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@[simp] theorem filter_nil {p : α → Bool} : filter p [] = [] := by
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simp [filter, filterAux, reverse, reverseAux]
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theorem cons_eq_append (a : α) (as : List α) : a :: as = [a] ++ as := rfl
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theorem filter_cons (a : α) (as : List α) :
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filter p (a :: as) = if p a then a :: filter p as else filter p as :=
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sorry
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@[simp] theorem filter_append {as bs : List α} {p : α → Bool} :
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filter p (as ++ bs) = filter p as ++ filter p bs :=
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match as with
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| [] => by simp
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| a :: as => by
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rw [filter_cons, cons_append, filter_cons]
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cases p a
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simp [filter_append]
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simp [filter_append]
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-- the previous contains a more complicated version of
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def f : Nat → Nat
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| 0 => 1
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| i+1 => (fun x => f x) i
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