31 lines
775 B
Text
31 lines
775 B
Text
import data.list
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open nat list
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section
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parameter {A : Type}
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parameter (p : A → Prop)
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parameter [H : decidable_pred p]
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include H
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definition filter : list A → list A
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| filter nil := nil
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| filter (a :: l) := if p a then a :: filter l else filter l
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theorem filter_nil : filter nil = nil :=
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rfl
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theorem filter_cons (a : A) (l : list A) : filter (a :: l) = if p a then a :: filter l else filter l :=
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rfl
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open eq.ops
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theorem filter_cons_of_pos {a : A} (l : list A) (h : p a) : filter (a :: l) = a :: filter l :=
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if_pos h ▸ filter_cons a l
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theorem filter_cons_of_neg {a : A} (l : list A) (h : ¬ p a) : filter (a :: l) = filter l :=
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if_neg h ▸ filter_cons a l
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end
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check @filter
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check @filter_cons_of_pos
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check @filter_cons_of_neg
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