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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
Set Implicit Arguments.
Require Import BinPos.
Require Export List.
Require Export ListTactics.
Open Local 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 => tail l
| xO p => jump p (jump p l)
| xI p => jump p (jump p (tail 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 (tail l))
end.
Lemma jump_tl : forall j l, tail (jump j l) = jump j (tail l).
Proof.
induction j;simpl;intros.
repeat rewrite IHj;trivial.
repeat rewrite IHj;trivial.
trivial.
Qed.
Lemma jump_Psucc : forall j l,
(jump (Psucc j) l) = (jump 1 (jump j l)).
Proof.
induction j;simpl;intros.
repeat rewrite IHj;simpl;repeat rewrite jump_tl;trivial.
repeat rewrite jump_tl;trivial.
trivial.
Qed.
Lemma jump_Pplus : forall i j l,
(jump (i + j) l) = (jump i (jump j l)).
Proof.
induction i;intros.
rewrite xI_succ_xO;rewrite Pplus_one_succ_r.
rewrite <- Pplus_diag;repeat rewrite <- Pplus_assoc.
repeat rewrite IHi.
rewrite Pplus_comm;rewrite <- Pplus_one_succ_r;rewrite jump_Psucc;trivial.
rewrite <- Pplus_diag;repeat rewrite <- Pplus_assoc.
repeat rewrite IHi;trivial.
rewrite Pplus_comm;rewrite <- Pplus_one_succ_r;rewrite jump_Psucc;trivial.
Qed.
Lemma jump_Pdouble_minus_one : forall i l,
(jump (Pdouble_minus_one i) (tail l)) = (jump i (jump i l)).
Proof.
induction i;intros;simpl.
repeat rewrite jump_tl;trivial.
rewrite IHi. do 2 rewrite <- jump_tl;rewrite IHi;trivial.
trivial.
Qed.
Lemma nth_jump : forall p l, nth p (tail l) = hd default (jump p l).
Proof.
induction p;simpl;intros.
rewrite <-jump_tl;rewrite IHp;trivial.
rewrite <-jump_tl;rewrite IHp;trivial.
trivial.
Qed.
Lemma nth_Pdouble_minus_one :
forall p l, nth (Pdouble_minus_one p) (tail l) = nth p (jump p l).
Proof.
induction p;simpl;intros.
repeat rewrite jump_tl;trivial.
rewrite jump_Pdouble_minus_one.
repeat rewrite <- jump_tl;rewrite IHp;trivial.
trivial.
Qed.
End MakeBinList.
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