30 lines
1.1 KiB
Text
30 lines
1.1 KiB
Text
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--- Copyright (c) 2014 Parikshit Khanna. All rights reserved.
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--- Released under Apache 2.0 license as described in the file LICENSE.
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--- Authors: Parikshit Khanna, Jeremy Avigad
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----------------------------------------------------------------------------------------------------
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-- Theory List
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-- ===========
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--
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-- Basic properties of Lists.
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open nat
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inductive List (T : Type) : Type
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| nil {} : List
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| cons : T → List → List
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namespace List
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theorem List_induction_on {T : Type} {P : List T → Prop} (l : List T) (Hnil : P nil)
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(Hind : forall x : T, forall l : List T, forall H : P l, P (cons x l)) : P l :=
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List.rec Hnil Hind l
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definition concat {T : Type} (s t : List T) : List T :=
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List.rec t (fun x : T, fun l : List T, fun u : List T, cons x u) s
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attribute concat [reducible]
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theorem concat_nil {T : Type} (t : List T) : concat t nil = t :=
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List_induction_on t (eq.refl (concat nil nil))
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(take (x : T) (l : List T) (H : concat l nil = l),
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H ▸ (eq.refl (concat (cons x l) nil)))
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end List
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