apply div2.
Defined.
Compute getPath 8.
Definition p9 := getPath 9.
Compute getPath 12.
Definition p12 := getPath 12.
Fixpoint buildNodeLevelDict {A} (root: PBTree A) (currL : nat) : list (prod nat nat) :=
match root with
| Leaf idx val => [pair idx currL]
| Node idx val l r =>
[pair idx currL] ++ buildNodeLevelDict l (S currL)  ++ buildNodeLevelDict r (S currL)
end.
Compute buildNodeLevelDict PBTree1 0.
Fixpoint writeAtPath {A} (root : PBTree A)
(path : list Dir) (data : A) : PBTree A :=
match root with
| Leaf idx val =>
match path with
| [] => Leaf idx data
| _ => Leaf idx val (* path longer than height of tree*)
end
| Node idx val l r =>
match path with
| [] => Node idx data l r
| L :: path' => Node idx val (writeAtPath l path' data) r
| R :: path' => Node idx val l (writeAtPath r path' data)
end
end.
Compute writeAtPath PBTree1 p12 [1;2;3].
Compute writeAtPath PBTree1 p9 [4;5;6].
Compute writeAtPath PBTree1 (getPath 4) [5;5;5]. (* 4 and 5 seem to be flipped *)
Definition writeToNode {A} (root : PBTree A) (lfIdx : nat) (tgtl : nat) (data : A) : PBTree A :=
writeAtPath root (firstn tgtl (getPath lfIdx)) data.
Compute PBTree1.
Compute writeToNode PBTree1 9 2 [0; 1; 2].
(* Definition of rand comes from Yves Bertot *)
CoFixpoint rand (seed n1 n2 : Z) : Stream Z :=
let seed' := Zmod seed n2 in Cons seed' (rand (seed' * n1) n1 n2).
(* find a definition of one-way function with unique inputs *)
Fixpoint takeS {X} n (str : Stream X) : list X :=
match n with
| 0 => []
| S m => match str with
| Cons x str' => x :: takeS m str'
end
end.
Compute  takeS 60(rand 475 919 953).
Definition first60 := takeS 60(rand 475 919 953).
Definition md_tot := fun x => Zmod x 15.
Compute List.map Z.to_nat (List.map md_tot first60).
Definition randSeq := List.map Z.to_nat (List.map md_tot first60).
Fixpoint modNodesTotal (randList : list Z) : list nat :=
List.map Z.to_nat(List.map md_tot randList).
Fixpoint indexed_list {X} (start : nat) (l : list X) : list (nat * X) :=
match l with
| [] => []
| h :: t => (start, h) :: indexed_list (S start) t
end.
Compute indexed_list 0 randSeq.
Definition position_map := indexed_list 0 randSeq.
Fixpoint posMpLookUp {X} (psmp: list (nat * X)) (id: nat) : option X :=
match psmp with
| [] => None
| (k, v) :: t => if Nat.eqb k id
then Some v
else posMpLookUp t id
end.
Fixpoint aggregation_helper (key : nat)(val : nat)(l : list (nat * (list nat))):
list (nat * (list nat)) :=
match l with
| [] => [(key, [val])]
| (n, al) :: t => if Nat.eqb n key
then (key, val :: al) :: t
else (n, al) :: aggregation_helper key val t
end.
Fixpoint aggregation (l : list (nat * nat)): list (nat * (list nat)):=
match l with
| [] =>  []
| (b, n) :: t => aggregation_helper n b (aggregation t)
end.
Compute aggregation [(1, 3); (2,3); (8, 3); (4, 4); (5,4)].
Definition n_bl_pair := aggregation [(1, 3); (2,3); (8, 3); (4, 4); (5,4)].
Fixpoint makeNZeros (n : nat) : list nat :=
match n with
| O => []
| S n' => O :: makeNZeros n'
end.
Compute makeNZeros 10.
Fixpoint takeL {A} n (l : list A) : list A :=
match n with
| O => []
| S m => match l with
| [] => []
| h :: t => h :: takeL m t
end
end.
Compute takeL 4 [1;2;3].
Fixpoint takeFromIdx {A} (n : nat) (l : list A)  : list A :=
match l with
| [] => []
| h :: t => match n with
| O => h :: t
| S m => takeFromIdx m t
end
end.
Compute takeFromIdx 2 [1; 2; 3;4].
Fixpoint pairGen {A} (l : list(nat * list nat)) (rt : PBTree A ): list (nat * nat) :=
match l with
| [] => []
| (n, bs) :: t => match rt with
| Leaf idx _ => if Nat.eqb n idx
then List.combine bs (makeNZeros (List.length bs))
(* expand this n's bs *)
else pairGen t rt (* recurse  *)
| Node idx _ l r => if Nat.eqb n idx
then List.combine bs (makeNZeros (List.length bs))
(* expand this n's bs *)
else pairGen t rt  (* recurse  *)
end
end.
Compute pairGen n_bl_pair (Leaf 3 3).
Compute takeL 2 (pairGen n_bl_pair (Leaf 3 3)).
Check writeAtPath.
Definition initialT A := (PBTree A, list(nat * nat)).
Print option.
(* define type of the initialzation function; needs to talk to Big T*)
Inductive initialType A : Type :=
| TreeStash : PBTree A -> list(nat * nat) -> initialType A.
Fixpoint initializeTree (rt : PBTree (list (nat * nat))) (stsh : list (nat * nat))
(l : list(nat * list nat)): option(initialType (list (nat * nat))) :=
match rt with
| Leaf idx val =>
let data := pairGen l rt in
match data with
| [] => None
| h :: t => let dataH := takeL 4 data in
let dataT := takeFromIdx 4 data in
let newStsh := stsh ++ dataT in
Some(TreeStash (list(nat * nat))(writeAtPath rt (getPath idx) dataH) newStsh)
end
| Node idx val lc rc =>
let data := pairGen l rt in
match data with
| [] => let dataH := takeL 4 data in
let dataT := takeFromIdx 4 data in
let newStsh := stsh ++ dataT in
Some(TreeStash (list (nat * nat))(writeAtPath rt (getPath idx) dataH) newStsh)
| h :: t => match initializeTree lc stsh l with
| Some x => Some x
| None => initializeTree rc stsh l
end
end
end.
Program Fixpoint getPath' (lfIdx : nat) {measure lfIdx} : list nat :=
match lfIdx with
| 0 => [0]
| S idx => if Nat.even idx
then [S idx] ++ getPath' (PeanoNat.Nat.div2 idx)
else [S idx] ++ getPath' (PeanoNat.Nat.div2 idx)
end.
Next Obligation.
apply div2.
Qed.
Next Obligation.
apply div2.
Defined.
Compute rev(getPath' 11).
Fixpoint writeatPath {A} (root: PBTree A) (path : list nat) (data: A): PBTree A :=
match root with
| Leaf idx val =>
match path with
| [] => Leaf idx data
| _ => Leaf idx val (* path longer than height of tree*)
end
| Node idx val l r =>
match path with
| [] => Node idx data l r
| h :: t => if Nat.even h
then Node idx val l (writeatPath r t data)
else Node idx val (writeatPath l t data) r
end
end.
Fixpoint clearPath (rt: PBTree (list (nat * nat ))) (l : list nat): PBTree (list(nat * nat)) :=
match l with
| [] => rt
| h :: t => match rt with
| Leaf idx val => if Nat.eqb h idx
then Leaf idx []
else Leaf idx val
| Node idx val lc rc => if Nat.eqb h idx
then Node idx [] (clearPath lc t) (clearPath rc t)
else Node idx val (clearPath lc t) (clearPath rc t)
end
end.
Compute getPath' 5.
Compute rev(getPath' 5).
Compute writeatPath PBTree1 (rev (getPath' 5)) [5;5;5].
Compute PBTree1.
Compute clearPath PBTree1 (rev(getPath' 11)).
Compute rev(getPath' 11).
Inductive  NodeData: Type := Data:(nat * list BlockEntry) -> NodeData .
Fixpoint getNodeAtLevel (lvl: nat) (l: list nat) (rt: PBTree (list BlockEntry)): option NodeData:=
match lvl with
| O => match l with
| [] => None
| h :: t => match rt with
| Leaf idx val => if Nat.eqb idx h
then Some (Data (idx, val))
else None
| Node idx val lc rc => if Nat.eqb idx h
then Some (Data (idx, val))
else
match getNodeAtLevel lvl t lc with
| None => getNodeAtLevel lvl t rc
| Some x => Some x
end
end
end
| S m => match l with
| [] => None
| h :: t => match rt with
|Leaf idx val => if Nat.eqb idx h
then Some (Data (idx, val))
else None
|Node idx val lc rc => if Nat.eqb idx h
then Some (Data(idx, val))
else
match getNodeAtLevel lvl t lc with
| None => getNodeAtLevel lvl t rc
| Some x => Some x
end
end
end
end.
Fixpoint ReadnPopNodes (rt: PBTree (list BlockEntry)) (l: list nat) (stsh: list BlockEntry) : list BlockEntry :=
match l with
| [] => []
| h :: t => match rt with
| Leaf idx val => if Nat.eqb h idx
then stsh ++ val
else stsh
| Node idx val lc rc => if Nat.eqb h idx
then stsh ++ val
else stsh
end
end.
Scheme Equality for list.
Scheme Equality for prod.
Print list_beq.
(* Definition pairEqL A B := TODO *)
Fixpoint posMapLookUp (bID : nat) (posMap : list(nat * nat)) :option nat :=
match posMap with
| [] => None
| h :: t => if Nat.eqb (fst h) bID
then Some (snd h)
else posMapLookUp bID t
end.
Fixpoint retSomeVal (x : option nat) : nat :=
match x with
| None => 0
| Some n => n
end.
Fixpoint eqListPair (l1 : list BlockEntry) (l2 :  list BlockEntry) : bool :=
match l1 with
| [] => match l2 with
| [] => true
| h :: t => false
end
| x :: xs => match x with
| (BV f s) =>
match l2 with
| [] => false
| h :: t => match h with
| (BV f' s') => if (andb (Nat.eqb f f') (Nat.eqb s s'))
then eqListPair xs t
else false
end
end
end
end.
Fixpoint NodeDataEq (n1: NodeData) (n2: NodeData) : bool :=
match n1 with
| Data (x, y) =>
match n2 with
| Data (a, b) => if Nat.eqb x a
then eqListPair y b
else false
end
end.
Fixpoint getCandidateBlocksHelper (rt: PBTree(list BlockEntry)) (l: list nat)
(lvl: nat)(bID: nat)(stsh: list BlockEntry): option BlockEntry:=
match getNodeAtLevel lvl l rt with (* P(x, l) *)
| None => None
| Some val =>
match getNodeAtLevel lvl (getPath' (retSomeVal(posMapLookUp bID position_map))) rt with (* P(position[a'],l) *)
| None => None
| Some val' => if NodeDataEq val val'
then match readBlockFromStash stsh bID with
| Some n => Some(BV bID n)
| _ => None
end
else None
end
end.
Fixpoint getCandidateBlocks (rt: PBTree(list BlockEntry)) (l: list nat) (lvl: nat) (stsh: list BlockEntry) : list BlockEntry :=
match stsh with
| [] => []
| (BV bid bdata) :: t =>
match getCandidateBlocksHelper rt l lvl bid stsh with
| None =>  (getCandidateBlocks rt l lvl t)
| Some v => v :: (getCandidateBlocks rt l lvl t)
end
end.
End Tree.
Section PathORAM.
Definition nodesTotal := 60.
Definition nodeCap := 4.
(* Definition stashInit := []. *)
Inductive Op :=
| Rd
| Wr.
Fixpoint leb (n m : nat) : bool :=
match n with
| O =>true
| S n' =>
match m with
| O => false
| S m' =>leb n' m'
end
end.
Definition getWriteBackBlocks (rt : PBTree(list BlockEntry))(c: nat) (l: list nat) (lvl: nat)(stsh: list BlockEntry): list BlockEntry :=
match List.length(stsh) with
| O => let candidateBlocks := @nil BlockEntry in
let writeBackSize := O in
[]
| S m => let candidateBlocks := getCandidateBlocks rt l lvl stsh in
if  leb c (List.length(candidateBlocks))
then let writeBackSize := c in
takeL c candidateBlocks
else let writeBackSize := List.length(candidateBlocks) in
takeL c candidateBlocks
end.
Definition st_rand := prod(Stream Z)
(prod(list BlockEntry)
(PBTree(list BlockEntry))).
Definition triFunction leafIdx lIDs lvl (s : st_rand) : st_rand :=
match s with
| (stream,(stsh, tr)) =>
let writeBackBlocks := getWriteBackBlocks tr leafIdx lIDs lvl stsh in
let updateStash := popStash stsh writeBackBlocks in
let newTree := writeToNode tr leafIdx lvl writeBackBlocks in
(stream, (updateStash, newTree))
end.
Definition writeBacks leafIdx lIDs (lvl:nat) : state st_rand unit :=
let* s := get in
(let s' := triFunction leafIdx lIDs lvl s in
put s').
Fixpoint access_rec (leafIdx: nat) (lIDs : list nat) (lvl: nat): state st_rand unit :=
match lvl with
| O => writeBacks leafIdx lIDs O
| S m => let* _ := writeBacks leafIdx lIDs lvl in
access_rec leafIdx lIDs m
end.
Print writeBacks.
Print access_rec.
Definition LEVEL := 3.
Definition option_get {T} (o : option T) (defaultT : T) : T :=
match o with
| Some val => val
| None => defaultT
end.
Fixpoint posMapUpdate (posMap: list(nat * nat)) (nID: nat) (random: nat) : list(nat * nat) :=
match posMap with
| [] => []
| (b, n) :: t => if Nat.eqb b nID
then (b, random) :: (posMapUpdate t nID random)
else (b, n) :: (posMapUpdate t nID random)
end.
Definition get_random_nat (_ : unit) : state st_rand nat :=
let* (stream, (stsh, tr)) := get in
let (random, stream') :=
match stream with
| Cons r s => (r, s)
end
in
let* _ := put (stream', (stsh, tr)) in
ret (Z.to_nat (md_tot random)).
(* lfIdx is always some val move this to its own section*)
Definition access (op : Op) (bID : nat) (dataN : option nat) : state st_rand nat :=
let lfIdx_o := posMapLookUp bID position_map in
let lfidx := option_get lfIdx_o 0 in (* prove lfidx will not be None*)
let* random_nat := get_random_nat tt in
let position_map' := posMapUpdate position_map lfidx random_nat  in
let* (stream, (stsh, tr)) := get in
let* _ := put(stream, ((ReadnPopNodes tr (getPath' lfidx) stsh), tr)) in
let dataOld := readBlockFromStash stsh bID in
match op with
| Wr => let toPopstsh := updateStash bID (option_get dataN 0) stsh in
let* _ := put(stream, (toPopstsh, tr)) in
let* _ := access_rec lfidx (getPath' lfidx) LEVEL in
ret (option_get dataOld O)
| Rd => let* _ := access_rec lfidx (getPath' lfidx) LEVEL in
ret (option_get dataOld O)
end.
(* pos map was not recorded in the global state  *)
End PathORAM.
Section PathORAM.
Definition ValEq (a : nat) (b : nat): Prop := eq_nat a b.
Definition twoAccesses(blockId: nat ) (data: nat) : state st_rand nat :=
let* (stream, (stsh, tr)) := get in
let* _ := access Wr blockId (Some data) in
access Rd blockId None.
(* From QuickChick Require Import QuickChick. *)
(* Import QcDefaultNotation. Open Scope qc_scope. *)

Lemma reflect_eq: forall (bID: nat), (bID =? bID) = true.