49 lines
1 KiB
Text
49 lines
1 KiB
Text
open nat
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definition foo (a : nat) : nat :=
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match a with
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| 0 := 0
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| (succ n) := n
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end
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example : foo 3 = 2 := rfl
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open decidable
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protected theorem dec_eq : ∀ x y : nat, decidable (x = y)
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| 0 0 := is_true rfl
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| (succ x) 0 := is_false (λ h, nat.no_confusion h)
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| 0 (succ y) := is_false (λ h, nat.no_confusion h)
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| (succ x) (succ y) :=
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match (dec_eq x y) with
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| (is_true H) := is_true (eq.rec_on H rfl)
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| (is_false H) := is_false (λ h : succ x = succ y, nat.no_confusion h (λ heq : x = y, absurd heq H))
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end
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section
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open list
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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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| nil := nil
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| (a :: l) :=
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match (H a) with
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| (is_true h) := a :: filter l
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| (is_false h) := filter l
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end
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theorem filter_nil : filter nil = nil :=
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rfl
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end
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definition sub2 (a : nat) : nat :=
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match a with
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| 0 := 0
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| 1 := 0
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| (b+2) := b
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end
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example (a : nat) : sub2 (succ (succ a)) = a := rfl
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