Program Fixpoint getPath (lfIdx : nat) {measure lfIdx} : list Dir :=
    match lfIdx with
    | 0 => []
    | S idx => if Nat.even idx
              then [L] ++ getPath (PeanoNat.Nat.div2 idx)
              else [R] ++ getPath (PeanoNat.Nat.div2 idx)
    end.Definition p9 := getPath 9.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.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.(* 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.(* 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.Definition first60 := takeS 60(rand 475 919 953).Definition md_tot := fun x => Zmod x 15.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.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.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.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.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.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.Definition initialT A := (PBTree A, list(nat * nat)).(* define type of the initialzation function; needs to talk to Big T*)
  Inductive initialType A : Type :=
  | TreeStash : PBTree A -> list(nat * nat) -> initialType A.(* 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.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.Inductive  NodeData: Type := Data:(nat * list BlockEntry) -> NodeData .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.(* 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.Definition nodesTotal := 60.Definition nodeCap := 4.(* Definition stashInit := []. *)

  Inductive Op :=
  | Rd
  | Wr.(* Definition stashInit := []. *)

  Inductive Op :=
  | Rd
  | Wr.(* 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 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 ValEq (a : nat) (b : nat): Prop := eq_nat a b.

Lemma posMapLookUp_some: forall (posMap: list(nat * nat)) (bID: nat) (kv: (nat * nat)), eq_nat (fst kv) bID -> In kv posMap -> exists v, posMapLookUp bID posMap = Some v.