53 lines
1.1 KiB
Text
53 lines
1.1 KiB
Text
import data.list
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open nat
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definition foo (a : nat) : nat :=
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match a with
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| zero := zero
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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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| dec_eq zero zero := inl rfl
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| dec_eq (succ x) zero := inr (λ h, nat.no_confusion h)
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| dec_eq zero (succ y) := inr (λ h, nat.no_confusion h)
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| dec_eq (succ x) (succ y) :=
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match dec_eq x y with
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| inl H := inl (eq.rec_on H rfl)
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| inr H := inr (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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context
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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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| filter nil := nil
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| filter (a :: l) :=
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match H a with
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| inl h := a :: filter l
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| inr 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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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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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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