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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2017 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
Require Import BinPos.
Require Export List.
Set Implicit Arguments.
Local Open Scope positive_scope.
Section MakeBinList.
Variable A : Type.
Variable default : A.
Fixpoint jump (p:positive) (l:list A) {struct p} : list A :=
match p with
| xH => tl l
| xO p => jump p (jump p l)
| xI p => jump p (jump p (tl l))
end.
Fixpoint nth (p:positive) (l:list A) {struct p} : A:=
match p with
| xH => hd default l
| xO p => nth p (jump p l)
| xI p => nth p (jump p (tl l))
end.
Lemma jump_tl : forall j l, tl (jump j l) = jump j (tl l).
Proof.
induction j;simpl;intros; now rewrite ?IHj.
Qed.
Lemma jump_succ : forall j l,
jump (Pos.succ j) l = jump 1 (jump j l).
Proof.
induction j;simpl;intros.
- rewrite !IHj; simpl; now rewrite !jump_tl.
- now rewrite !jump_tl.
- trivial.
Qed.
Lemma jump_add : forall i j l,
jump (i + j) l = jump i (jump j l).
Proof.
induction i using Pos.peano_ind; intros.
- now rewrite Pos.add_1_l, jump_succ.
- now rewrite Pos.add_succ_l, !jump_succ, IHi.
Qed.
Lemma jump_pred_double : forall i l,
jump (Pos.pred_double i) (tl l) = jump i (jump i l).
Proof.
induction i;intros;simpl.
- now rewrite !jump_tl.
- now rewrite IHi, <- 2 jump_tl, IHi.
- trivial.
Qed.
Lemma nth_jump : forall p l, nth p (tl l) = hd default (jump p l).
Proof.
induction p;simpl;intros.
- now rewrite <-jump_tl, IHp.
- now rewrite <-jump_tl, IHp.
- trivial.
Qed.
Lemma nth_pred_double :
forall p l, nth (Pos.pred_double p) (tl l) = nth p (jump p l).
Proof.
induction p;simpl;intros.
- now rewrite !jump_tl.
- now rewrite jump_pred_double, <- !jump_tl, IHp.
- trivial.
Qed.
End MakeBinList.
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