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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import data.nat
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open nat eq_ops
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inductive list (T : Type) : Type :=
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nil {} : list T,
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cons : T → list T → list T
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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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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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