path: root/tests/fstar/betree_back_stateful/BetreeMain.Funs.fst

(** THIS FILE WAS AUTOMATICALLY GENERATED BY AENEAS *)
(** [betree_main]: function definitions *)
module BetreeMain.Funs
open Primitives
include BetreeMain.Types
include BetreeMain.Opaque
include BetreeMain.Clauses

#set-options "--z3rlimit 50 --fuel 1 --ifuel 1"

(** [betree_main::betree::load_internal_node]: forward function *)
let betree_load_internal_node
  (id : u64) (st : state) :
  result (state & (betree_List_t (u64 & betree_Message_t)))
  =
  betree_utils_load_internal_node id st

(** [betree_main::betree::store_internal_node]: forward function *)
let betree_store_internal_node
  (id : u64) (content : betree_List_t (u64 & betree_Message_t)) (st : state) :
  result (state & unit)
  =
  let* (st0, _) = betree_utils_store_internal_node id content st in
  Return (st0, ())

(** [betree_main::betree::load_leaf_node]: forward function *)
let betree_load_leaf_node
  (id : u64) (st : state) : result (state & (betree_List_t (u64 & u64))) =
  betree_utils_load_leaf_node id st

(** [betree_main::betree::store_leaf_node]: forward function *)
let betree_store_leaf_node
  (id : u64) (content : betree_List_t (u64 & u64)) (st : state) :
  result (state & unit)
  =
  let* (st0, _) = betree_utils_store_leaf_node id content st in
  Return (st0, ())

(** [betree_main::betree::fresh_node_id]: forward function *)
let betree_fresh_node_id (counter : u64) : result u64 =
  let* _ = u64_add counter 1 in Return counter

(** [betree_main::betree::fresh_node_id]: backward function 0 *)
let betree_fresh_node_id_back (counter : u64) : result u64 =
  u64_add counter 1

(** [betree_main::betree::NodeIdCounter::{0}::new]: forward function *)
let betree_NodeIdCounter_new : result betree_NodeIdCounter_t =
  Return { next_node_id = 0 }

(** [betree_main::betree::NodeIdCounter::{0}::fresh_id]: forward function *)
let betree_NodeIdCounter_fresh_id
  (self : betree_NodeIdCounter_t) : result u64 =
  let* _ = u64_add self.next_node_id 1 in Return self.next_node_id

(** [betree_main::betree::NodeIdCounter::{0}::fresh_id]: backward function 0 *)
let betree_NodeIdCounter_fresh_id_back
  (self : betree_NodeIdCounter_t) : result betree_NodeIdCounter_t =
  let* i = u64_add self.next_node_id 1 in Return { next_node_id = i }

(** [betree_main::betree::upsert_update]: forward function *)
let betree_upsert_update
  (prev : option u64) (st : betree_UpsertFunState_t) : result u64 =
  begin match prev with
  | None ->
    begin match st with
    | Betree_UpsertFunState_Add v -> Return v
    | Betree_UpsertFunState_Sub i -> Return 0
    end
  | Some prev0 ->
    begin match st with
    | Betree_UpsertFunState_Add v ->
      let* margin = u64_sub core_u64_max prev0 in
      if margin >= v then u64_add prev0 v else Return core_u64_max
    | Betree_UpsertFunState_Sub v ->
      if prev0 >= v then u64_sub prev0 v else Return 0
    end
  end

(** [betree_main::betree::List::{1}::len]: forward function *)
let rec betree_List_len
  (t : Type0) (self : betree_List_t t) :
  Tot (result u64) (decreases (betree_List_len_decreases t self))
  =
  begin match self with
  | Betree_List_Cons x tl -> let* i = betree_List_len t tl in u64_add 1 i
  | Betree_List_Nil -> Return 0
  end

(** [betree_main::betree::List::{1}::split_at]: forward function *)
let rec betree_List_split_at
  (t : Type0) (self : betree_List_t t) (n : u64) :
  Tot (result ((betree_List_t t) & (betree_List_t t)))
  (decreases (betree_List_split_at_decreases t self n))
  =
  if n = 0
  then Return (Betree_List_Nil, self)
  else
    begin match self with
    | Betree_List_Cons hd tl ->
      let* i = u64_sub n 1 in
      let* p = betree_List_split_at t tl i in
      let (ls0, ls1) = p in
      let l = ls0 in
      Return (Betree_List_Cons hd l, ls1)
    | Betree_List_Nil -> Fail Failure
    end

(** [betree_main::betree::List::{1}::push_front]: merged forward/backward function
    (there is a single backward function, and the forward function returns ()) *)
let betree_List_push_front
  (t : Type0) (self : betree_List_t t) (x : t) : result (betree_List_t t) =
  let tl = core_mem_replace (betree_List_t t) self Betree_List_Nil in
  let l = tl in
  Return (Betree_List_Cons x l)

(** [betree_main::betree::List::{1}::pop_front]: forward function *)
let betree_List_pop_front (t : Type0) (self : betree_List_t t) : result t =
  let ls = core_mem_replace (betree_List_t t) self Betree_List_Nil in
  begin match ls with
  | Betree_List_Cons x tl -> Return x
  | Betree_List_Nil -> Fail Failure
  end

(** [betree_main::betree::List::{1}::pop_front]: backward function 0 *)
let betree_List_pop_front_back
  (t : Type0) (self : betree_List_t t) : result (betree_List_t t) =
  let ls = core_mem_replace (betree_List_t t) self Betree_List_Nil in
  begin match ls with
  | Betree_List_Cons x tl -> Return tl
  | Betree_List_Nil -> Fail Failure
  end

(** [betree_main::betree::List::{1}::hd]: forward function *)
let betree_List_hd (t : Type0) (self : betree_List_t t) : result t =
  begin match self with
  | Betree_List_Cons hd l -> Return hd
  | Betree_List_Nil -> Fail Failure
  end

(** [betree_main::betree::List::{2}::head_has_key]: forward function *)
let betree_List_head_has_key
  (t : Type0) (self : betree_List_t (u64 & t)) (key : u64) : result bool =
  begin match self with
  | Betree_List_Cons hd l -> let (i, _) = hd in Return (i = key)
  | Betree_List_Nil -> Return false
  end

(** [betree_main::betree::List::{2}::partition_at_pivot]: forward function *)
let rec betree_List_partition_at_pivot
  (t : Type0) (self : betree_List_t (u64 & t)) (pivot : u64) :
  Tot (result ((betree_List_t (u64 & t)) & (betree_List_t (u64 & t))))
  (decreases (betree_List_partition_at_pivot_decreases t self pivot))
  =
  begin match self with
  | Betree_List_Cons hd tl ->
    let (i, x) = hd in
    if i >= pivot
    then Return (Betree_List_Nil, Betree_List_Cons (i, x) tl)
    else
      let* p = betree_List_partition_at_pivot t tl pivot in
      let (ls0, ls1) = p in
      let l = ls0 in
      Return (Betree_List_Cons (i, x) l, ls1)
  | Betree_List_Nil -> Return (Betree_List_Nil, Betree_List_Nil)
  end

(** [betree_main::betree::Leaf::{3}::split]: forward function *)
let betree_Leaf_split
  (self : betree_Leaf_t) (content : betree_List_t (u64 & u64))
  (params : betree_Params_t) (node_id_cnt : betree_NodeIdCounter_t)
  (st : state) :
  result (state & betree_Internal_t)
  =
  let* p = betree_List_split_at (u64 & u64) content params.split_size in
  let (content0, content1) = p in
  let* p0 = betree_List_hd (u64 & u64) content1 in
  let (pivot, _) = p0 in
  let* id0 = betree_NodeIdCounter_fresh_id node_id_cnt in
  let* node_id_cnt0 = betree_NodeIdCounter_fresh_id_back node_id_cnt in
  let* id1 = betree_NodeIdCounter_fresh_id node_id_cnt0 in
  let* (st0, _) = betree_store_leaf_node id0 content0 st in
  let* (st1, _) = betree_store_leaf_node id1 content1 st0 in
  let n = Betree_Node_Leaf { id = id0; size = params.split_size } in
  let n0 = Betree_Node_Leaf { id = id1; size = params.split_size } in
  Return (st1, { id = self.id; pivot = pivot; left = n; right = n0 })

(** [betree_main::betree::Leaf::{3}::split]: backward function 0 *)
let betree_Leaf_split_back0
  (self : betree_Leaf_t) (content : betree_List_t (u64 & u64))
  (params : betree_Params_t) (node_id_cnt : betree_NodeIdCounter_t)
  (st : state) (st0 : state) :
  result (state & unit)
  =
  let* p = betree_List_split_at (u64 & u64) content params.split_size in
  let (content0, content1) = p in
  let* _ = betree_List_hd (u64 & u64) content1 in
  let* id0 = betree_NodeIdCounter_fresh_id node_id_cnt in
  let* node_id_cnt0 = betree_NodeIdCounter_fresh_id_back node_id_cnt in
  let* id1 = betree_NodeIdCounter_fresh_id node_id_cnt0 in
  let* (st1, _) = betree_store_leaf_node id0 content0 st in
  let* _ = betree_store_leaf_node id1 content1 st1 in
  Return (st0, ())

(** [betree_main::betree::Leaf::{3}::split]: backward function 1 *)
let betree_Leaf_split_back1
  (self : betree_Leaf_t) (content : betree_List_t (u64 & u64))
  (params : betree_Params_t) (node_id_cnt : betree_NodeIdCounter_t)
  (st : state) (st0 : state) :
  result (state & unit)
  =
  let* p = betree_List_split_at (u64 & u64) content params.split_size in
  let (content0, content1) = p in
  let* _ = betree_List_hd (u64 & u64) content1 in
  let* id0 = betree_NodeIdCounter_fresh_id node_id_cnt in
  let* node_id_cnt0 = betree_NodeIdCounter_fresh_id_back node_id_cnt in
  let* id1 = betree_NodeIdCounter_fresh_id node_id_cnt0 in
  let* (st1, _) = betree_store_leaf_node id0 content0 st in
  let* _ = betree_store_leaf_node id1 content1 st1 in
  Return (st0, ())

(** [betree_main::betree::Leaf::{3}::split]: backward function 2 *)
let betree_Leaf_split_back2
  (self : betree_Leaf_t) (content : betree_List_t (u64 & u64))
  (params : betree_Params_t) (node_id_cnt : betree_NodeIdCounter_t)
  (st : state) (st0 : state) :
  result (state & betree_NodeIdCounter_t)
  =
  let* p = betree_List_split_at (u64 & u64) content params.split_size in
  let (content0, content1) = p in
  let* _ = betree_List_hd (u64 & u64) content1 in
  let* id0 = betree_NodeIdCounter_fresh_id node_id_cnt in
  let* node_id_cnt0 = betree_NodeIdCounter_fresh_id_back node_id_cnt in
  let* id1 = betree_NodeIdCounter_fresh_id node_id_cnt0 in
  let* (st1, _) = betree_store_leaf_node id0 content0 st in
  let* _ = betree_store_leaf_node id1 content1 st1 in
  let* node_id_cnt1 = betree_NodeIdCounter_fresh_id_back node_id_cnt0 in
  Return (st0, node_id_cnt1)

(** [betree_main::betree::Node::{5}::lookup_first_message_for_key]: forward function *)
let rec betree_Node_lookup_first_message_for_key
  (key : u64) (msgs : betree_List_t (u64 & betree_Message_t)) :
  Tot (result (betree_List_t (u64 & betree_Message_t)))
  (decreases (betree_Node_lookup_first_message_for_key_decreases key msgs))
  =
  begin match msgs with
  | Betree_List_Cons x next_msgs ->
    let (i, m) = x in
    if i >= key
    then Return (Betree_List_Cons (i, m) next_msgs)
    else betree_Node_lookup_first_message_for_key key next_msgs
  | Betree_List_Nil -> Return Betree_List_Nil
  end

(** [betree_main::betree::Node::{5}::lookup_first_message_for_key]: backward function 0 *)
let rec betree_Node_lookup_first_message_for_key_back
  (key : u64) (msgs : betree_List_t (u64 & betree_Message_t))
  (ret : betree_List_t (u64 & betree_Message_t)) :
  Tot (result (betree_List_t (u64 & betree_Message_t)))
  (decreases (betree_Node_lookup_first_message_for_key_decreases key msgs))
  =
  begin match msgs with
  | Betree_List_Cons x next_msgs ->
    let (i, m) = x in
    if i >= key
    then Return ret
    else
      let* next_msgs0 =
        betree_Node_lookup_first_message_for_key_back key next_msgs ret in
      Return (Betree_List_Cons (i, m) next_msgs0)
  | Betree_List_Nil -> Return ret
  end

(** [betree_main::betree::Node::{5}::apply_upserts]: forward function *)
let rec betree_Node_apply_upserts
  (msgs : betree_List_t (u64 & betree_Message_t)) (prev : option u64)
  (key : u64) (st : state) :
  Tot (result (state & u64))
  (decreases (betree_Node_apply_upserts_decreases msgs prev key st))
  =
  let* b = betree_List_head_has_key betree_Message_t msgs key in
  if b
  then
    let* msg = betree_List_pop_front (u64 & betree_Message_t) msgs in
    let (_, m) = msg in
    begin match m with
    | Betree_Message_Insert i -> Fail Failure
    | Betree_Message_Delete -> Fail Failure
    | Betree_Message_Upsert s ->
      let* v = betree_upsert_update prev s in
      let* msgs0 = betree_List_pop_front_back (u64 & betree_Message_t) msgs in
      betree_Node_apply_upserts msgs0 (Some v) key st
    end
  else
    let* (st0, v) = core_option_Option_unwrap u64 prev st in
    let* _ =
      betree_List_push_front (u64 & betree_Message_t) msgs (key,
        Betree_Message_Insert v) in
    Return (st0, v)

(** [betree_main::betree::Node::{5}::apply_upserts]: backward function 0 *)
let rec betree_Node_apply_upserts_back
  (msgs : betree_List_t (u64 & betree_Message_t)) (prev : option u64)
  (key : u64) (st : state) (st0 : state) :
  Tot (result (state & (betree_List_t (u64 & betree_Message_t))))
  (decreases (betree_Node_apply_upserts_decreases msgs prev key st))
  =
  let* b = betree_List_head_has_key betree_Message_t msgs key in
  if b
  then
    let* msg = betree_List_pop_front (u64 & betree_Message_t) msgs in
    let (_, m) = msg in
    begin match m with
    | Betree_Message_Insert i -> Fail Failure
    | Betree_Message_Delete -> Fail Failure
    | Betree_Message_Upsert s ->
      let* v = betree_upsert_update prev s in
      let* msgs0 = betree_List_pop_front_back (u64 & betree_Message_t) msgs in
      betree_Node_apply_upserts_back msgs0 (Some v) key st st0
    end
  else
    let* (_, v) = core_option_Option_unwrap u64 prev st in
    let* msgs0 =
      betree_List_push_front (u64 & betree_Message_t) msgs (key,
        Betree_Message_Insert v) in
    Return (st0, msgs0)

(** [betree_main::betree::Node::{5}::lookup_in_bindings]: forward function *)
let rec betree_Node_lookup_in_bindings
  (key : u64) (bindings : betree_List_t (u64 & u64)) :
  Tot (result (option u64))
  (decreases (betree_Node_lookup_in_bindings_decreases key bindings))
  =
  begin match bindings with
  | Betree_List_Cons hd tl ->
    let (i, i0) = hd in
    if i = key
    then Return (Some i0)
    else if i > key then Return None else betree_Node_lookup_in_bindings key tl
  | Betree_List_Nil -> Return None
  end

(** [betree_main::betree::Internal::{4}::lookup_in_children]: forward function *)
let rec betree_Internal_lookup_in_children
  (self : betree_Internal_t) (key : u64) (st : state) :
  Tot (result (state & (option u64)))
  (decreases (betree_Internal_lookup_in_children_decreases self key st))
  =
  if key < self.pivot
  then betree_Node_lookup self.left key st
  else betree_Node_lookup self.right key st

(** [betree_main::betree::Internal::{4}::lookup_in_children]: backward function 0 *)
and betree_Internal_lookup_in_children_back
  (self : betree_Internal_t) (key : u64) (st : state) (st0 : state) :
  Tot (result (state & betree_Internal_t))
  (decreases (betree_Internal_lookup_in_children_decreases self key st))
  =
  if key < self.pivot
  then
    let* (st1, n) = betree_Node_lookup_back self.left key st st0 in
    Return (st1, { self with left = n })
  else
    let* (st1, n) = betree_Node_lookup_back self.right key st st0 in
    Return (st1, { self with right = n })

(** [betree_main::betree::Node::{5}::lookup]: forward function *)
and betree_Node_lookup
  (self : betree_Node_t) (key : u64) (st : state) :
  Tot (result (state & (option u64)))
  (decreases (betree_Node_lookup_decreases self key st))
  =
  begin match self with
  | Betree_Node_Internal node ->
    let* (st0, msgs) = betree_load_internal_node node.id st in
    let* pending = betree_Node_lookup_first_message_for_key key msgs in
    begin match pending with
    | Betree_List_Cons p l ->
      let (k, msg) = p in
      if k <> key
      then
        let* (st1, o) = betree_Internal_lookup_in_children node key st0 in
        let* _ =
          betree_Node_lookup_first_message_for_key_back key msgs
            (Betree_List_Cons (k, msg) l) in
        Return (st1, o)
      else
        begin match msg with
        | Betree_Message_Insert v ->
          let* _ =
            betree_Node_lookup_first_message_for_key_back key msgs
              (Betree_List_Cons (k, Betree_Message_Insert v) l) in
          Return (st0, Some v)
        | Betree_Message_Delete ->
          let* _ =
            betree_Node_lookup_first_message_for_key_back key msgs
              (Betree_List_Cons (k, Betree_Message_Delete) l) in
          Return (st0, None)
        | Betree_Message_Upsert ufs ->
          let* (st1, v) = betree_Internal_lookup_in_children node key st0 in
          let* (st2, v0) =
            betree_Node_apply_upserts (Betree_List_Cons (k,
              Betree_Message_Upsert ufs) l) v key st1 in
          let* (st3, node0) =
            betree_Internal_lookup_in_children_back node key st0 st2 in
          let* (st4, pending0) =
            betree_Node_apply_upserts_back (Betree_List_Cons (k,
              Betree_Message_Upsert ufs) l) v key st1 st3 in
          let* msgs0 =
            betree_Node_lookup_first_message_for_key_back key msgs pending0 in
          let* (st5, _) = betree_store_internal_node node0.id msgs0 st4 in
          Return (st5, Some v0)
        end
    | Betree_List_Nil ->
      let* (st1, o) = betree_Internal_lookup_in_children node key st0 in
      let* _ =
        betree_Node_lookup_first_message_for_key_back key msgs Betree_List_Nil
        in
      Return (st1, o)
    end
  | Betree_Node_Leaf node ->
    let* (st0, bindings) = betree_load_leaf_node node.id st in
    let* o = betree_Node_lookup_in_bindings key bindings in
    Return (st0, o)
  end

(** [betree_main::betree::Node::{5}::lookup]: backward function 0 *)
and betree_Node_lookup_back
  (self : betree_Node_t) (key : u64) (st : state) (st0 : state) :
  Tot (result (state & betree_Node_t))
  (decreases (betree_Node_lookup_decreases self key st))
  =
  begin match self with
  | Betree_Node_Internal node ->
    let* (st1, msgs) = betree_load_internal_node node.id st in
    let* pending = betree_Node_lookup_first_message_for_key key msgs in
    begin match pending with
    | Betree_List_Cons p l ->
      let (k, msg) = p in
      if k <> key
      then
        let* _ =
          betree_Node_lookup_first_message_for_key_back key msgs
            (Betree_List_Cons (k, msg) l) in
        let* (st2, node0) =
          betree_Internal_lookup_in_children_back node key st1 st0 in
        Return (st2, Betree_Node_Internal node0)
      else
        begin match msg with
        | Betree_Message_Insert v ->
          let* _ =
            betree_Node_lookup_first_message_for_key_back key msgs
              (Betree_List_Cons (k, Betree_Message_Insert v) l) in
          Return (st0, Betree_Node_Internal node)
        | Betree_Message_Delete ->
          let* _ =
            betree_Node_lookup_first_message_for_key_back key msgs
              (Betree_List_Cons (k, Betree_Message_Delete) l) in
          Return (st0, Betree_Node_Internal node)
        | Betree_Message_Upsert ufs ->
          let* (st2, v) = betree_Internal_lookup_in_children node key st1 in
          let* (st3, _) =
            betree_Node_apply_upserts (Betree_List_Cons (k,
              Betree_Message_Upsert ufs) l) v key st2 in
          let* (st4, node0) =
            betree_Internal_lookup_in_children_back node key st1 st3 in
          let* (st5, pending0) =
            betree_Node_apply_upserts_back (Betree_List_Cons (k,
              Betree_Message_Upsert ufs) l) v key st2 st4 in
          let* msgs0 =
            betree_Node_lookup_first_message_for_key_back key msgs pending0 in
          let* _ = betree_store_internal_node node0.id msgs0 st5 in
          Return (st0, Betree_Node_Internal node0)
        end
    | Betree_List_Nil ->
      let* _ =
        betree_Node_lookup_first_message_for_key_back key msgs Betree_List_Nil
        in
      let* (st2, node0) =
        betree_Internal_lookup_in_children_back node key st1 st0 in
      Return (st2, Betree_Node_Internal node0)
    end
  | Betree_Node_Leaf node ->
    let* (_, bindings) = betree_load_leaf_node node.id st in
    let* _ = betree_Node_lookup_in_bindings key bindings in
    Return (st0, Betree_Node_Leaf node)
  end

(** [betree_main::betree::Node::{5}::filter_messages_for_key]: merged forward/backward function
    (there is a single backward function, and the forward function returns ()) *)
let rec betree_Node_filter_messages_for_key
  (key : u64) (msgs : betree_List_t (u64 & betree_Message_t)) :
  Tot (result (betree_List_t (u64 & betree_Message_t)))
  (decreases (betree_Node_filter_messages_for_key_decreases key msgs))
  =
  begin match msgs with
  | Betree_List_Cons p l ->
    let (k, m) = p in
    if k = key
    then
      let* msgs0 =
        betree_List_pop_front_back (u64 & betree_Message_t) (Betree_List_Cons
          (k, m) l) in
      betree_Node_filter_messages_for_key key msgs0
    else Return (Betree_List_Cons (k, m) l)
  | Betree_List_Nil -> Return Betree_List_Nil
  end

(** [betree_main::betree::Node::{5}::lookup_first_message_after_key]: forward function *)
let rec betree_Node_lookup_first_message_after_key
  (key : u64) (msgs : betree_List_t (u64 & betree_Message_t)) :
  Tot (result (betree_List_t (u64 & betree_Message_t)))
  (decreases (betree_Node_lookup_first_message_after_key_decreases key msgs))
  =
  begin match msgs with
  | Betree_List_Cons p next_msgs ->
    let (k, m) = p in
    if k = key
    then betree_Node_lookup_first_message_after_key key next_msgs
    else Return (Betree_List_Cons (k, m) next_msgs)
  | Betree_List_Nil -> Return Betree_List_Nil
  end

(** [betree_main::betree::Node::{5}::lookup_first_message_after_key]: backward function 0 *)
let rec betree_Node_lookup_first_message_after_key_back
  (key : u64) (msgs : betree_List_t (u64 & betree_Message_t))
  (ret : betree_List_t (u64 & betree_Message_t)) :
  Tot (result (betree_List_t (u64 & betree_Message_t)))
  (decreases (betree_Node_lookup_first_message_after_key_decreases key msgs))
  =
  begin match msgs with
  | Betree_List_Cons p next_msgs ->
    let (k, m) = p in
    if k = key
    then
      let* next_msgs0 =
        betree_Node_lookup_first_message_after_key_back key next_msgs ret in
      Return (Betree_List_Cons (k, m) next_msgs0)
    else Return ret
  | Betree_List_Nil -> Return ret
  end

(** [betree_main::betree::Node::{5}::apply_to_internal]: merged forward/backward function
    (there is a single backward function, and the forward function returns ()) *)
let betree_Node_apply_to_internal
  (msgs : betree_List_t (u64 & betree_Message_t)) (key : u64)
  (new_msg : betree_Message_t) :
  result (betree_List_t (u64 & betree_Message_t))
  =
  let* msgs0 = betree_Node_lookup_first_message_for_key key msgs in
  let* b = betree_List_head_has_key betree_Message_t msgs0 key in
  if b
  then
    begin match new_msg with
    | Betree_Message_Insert i ->
      let* msgs1 = betree_Node_filter_messages_for_key key msgs0 in
      let* msgs2 =
        betree_List_push_front (u64 & betree_Message_t) msgs1 (key,
          Betree_Message_Insert i) in
      betree_Node_lookup_first_message_for_key_back key msgs msgs2
    | Betree_Message_Delete ->
      let* msgs1 = betree_Node_filter_messages_for_key key msgs0 in
      let* msgs2 =
        betree_List_push_front (u64 & betree_Message_t) msgs1 (key,
          Betree_Message_Delete) in
      betree_Node_lookup_first_message_for_key_back key msgs msgs2
    | Betree_Message_Upsert s ->
      let* p = betree_List_hd (u64 & betree_Message_t) msgs0 in
      let (_, m) = p in
      begin match m with
      | Betree_Message_Insert prev ->
        let* v = betree_upsert_update (Some prev) s in
        let* msgs1 = betree_List_pop_front_back (u64 & betree_Message_t) msgs0
          in
        let* msgs2 =
          betree_List_push_front (u64 & betree_Message_t) msgs1 (key,
            Betree_Message_Insert v) in
        betree_Node_lookup_first_message_for_key_back key msgs msgs2
      | Betree_Message_Delete ->
        let* v = betree_upsert_update None s in
        let* msgs1 = betree_List_pop_front_back (u64 & betree_Message_t) msgs0
          in
        let* msgs2 =
          betree_List_push_front (u64 & betree_Message_t) msgs1 (key,
            Betree_Message_Insert v) in
        betree_Node_lookup_first_message_for_key_back key msgs msgs2
      | Betree_Message_Upsert ufs ->
        let* msgs1 = betree_Node_lookup_first_message_after_key key msgs0 in
        let* msgs2 =
          betree_List_push_front (u64 & betree_Message_t) msgs1 (key,
            Betree_Message_Upsert s) in
        let* msgs3 =
          betree_Node_lookup_first_message_after_key_back key msgs0 msgs2 in
        betree_Node_lookup_first_message_for_key_back key msgs msgs3
      end
    end
  else
    let* msgs1 =
      betree_List_push_front (u64 & betree_Message_t) msgs0 (key, new_msg) in
    betree_Node_lookup_first_message_for_key_back key msgs msgs1

(** [betree_main::betree::Node::{5}::apply_messages_to_internal]: merged forward/backward function
    (there is a single backward function, and the forward function returns ()) *)
let rec betree_Node_apply_messages_to_internal
  (msgs : betree_List_t (u64 & betree_Message_t))
  (new_msgs : betree_List_t (u64 & betree_Message_t)) :
  Tot (result (betree_List_t (u64 & betree_Message_t)))
  (decreases (betree_Node_apply_messages_to_internal_decreases msgs new_msgs))
  =
  begin match new_msgs with
  | Betree_List_Cons new_msg new_msgs_tl ->
    let (i, m) = new_msg in
    let* msgs0 = betree_Node_apply_to_internal msgs i m in
    betree_Node_apply_messages_to_internal msgs0 new_msgs_tl
  | Betree_List_Nil -> Return msgs
  end

(** [betree_main::betree::Node::{5}::lookup_mut_in_bindings]: forward function *)
let rec betree_Node_lookup_mut_in_bindings
  (key : u64) (bindings : betree_List_t (u64 & u64)) :
  Tot (result (betree_List_t (u64 & u64)))
  (decreases (betree_Node_lookup_mut_in_bindings_decreases key bindings))
  =
  begin match bindings with
  | Betree_List_Cons hd tl ->
    let (i, i0) = hd in
    if i >= key
    then Return (Betree_List_Cons (i, i0) tl)
    else betree_Node_lookup_mut_in_bindings key tl
  | Betree_List_Nil -> Return Betree_List_Nil
  end

(** [betree_main::betree::Node::{5}::lookup_mut_in_bindings]: backward function 0 *)
let rec betree_Node_lookup_mut_in_bindings_back
  (key : u64) (bindings : betree_List_t (u64 & u64))
  (ret : betree_List_t (u64 & u64)) :
  Tot (result (betree_List_t (u64 & u64)))
  (decreases (betree_Node_lookup_mut_in_bindings_decreases key bindings))
  =
  begin match bindings with
  | Betree_List_Cons hd tl ->
    let (i, i0) = hd in
    if i >= key
    then Return ret
    else
      let* tl0 = betree_Node_lookup_mut_in_bindings_back key tl ret in
      Return (Betree_List_Cons (i, i0) tl0)
  | Betree_List_Nil -> Return ret
  end

(** [betree_main::betree::Node::{5}::apply_to_leaf]: merged forward/backward function
    (there is a single backward function, and the forward function returns ()) *)
let betree_Node_apply_to_leaf
  (bindings : betree_List_t (u64 & u64)) (key : u64)
  (new_msg : betree_Message_t) :
  result (betree_List_t (u64 & u64))
  =
  let* bindings0 = betree_Node_lookup_mut_in_bindings key bindings in
  let* b = betree_List_head_has_key u64 bindings0 key in
  if b
  then
    let* hd = betree_List_pop_front (u64 & u64) bindings0 in
    begin match new_msg with
    | Betree_Message_Insert v ->
      let* bindings1 = betree_List_pop_front_back (u64 & u64) bindings0 in
      let* bindings2 = betree_List_push_front (u64 & u64) bindings1 (key, v) in
      betree_Node_lookup_mut_in_bindings_back key bindings bindings2
    | Betree_Message_Delete ->
      let* bindings1 = betree_List_pop_front_back (u64 & u64) bindings0 in
      betree_Node_lookup_mut_in_bindings_back key bindings bindings1
    | Betree_Message_Upsert s ->
      let (_, i) = hd in
      let* v = betree_upsert_update (Some i) s in
      let* bindings1 = betree_List_pop_front_back (u64 & u64) bindings0 in
      let* bindings2 = betree_List_push_front (u64 & u64) bindings1 (key, v) in
      betree_Node_lookup_mut_in_bindings_back key bindings bindings2
    end
  else
    begin match new_msg with
    | Betree_Message_Insert v ->
      let* bindings1 = betree_List_push_front (u64 & u64) bindings0 (key, v) in
      betree_Node_lookup_mut_in_bindings_back key bindings bindings1
    | Betree_Message_Delete ->
      betree_Node_lookup_mut_in_bindings_back key bindings bindings0
    | Betree_Message_Upsert s ->
      let* v = betree_upsert_update None s in
      let* bindings1 = betree_List_push_front (u64 & u64) bindings0 (key, v) in
      betree_Node_lookup_mut_in_bindings_back key bindings bindings1
    end

(** [betree_main::betree::Node::{5}::apply_messages_to_leaf]: merged forward/backward function
    (there is a single backward function, and the forward function returns ()) *)
let rec betree_Node_apply_messages_to_leaf
  (bindings : betree_List_t (u64 & u64))
  (new_msgs : betree_List_t (u64 & betree_Message_t)) :
  Tot (result (betree_List_t (u64 & u64)))
  (decreases (betree_Node_apply_messages_to_leaf_decreases bindings new_msgs))
  =
  begin match new_msgs with
  | Betree_List_Cons new_msg new_msgs_tl ->
    let (i, m) = new_msg in
    let* bindings0 = betree_Node_apply_to_leaf bindings i m in
    betree_Node_apply_messages_to_leaf bindings0 new_msgs_tl
  | Betree_List_Nil -> Return bindings
  end

(** [betree_main::betree::Internal::{4}::flush]: forward function *)
let rec betree_Internal_flush
  (self : betree_Internal_t) (params : betree_Params_t)
  (node_id_cnt : betree_NodeIdCounter_t)
  (content : betree_List_t (u64 & betree_Message_t)) (st : state) :
  Tot (result (state & (betree_List_t (u64 & betree_Message_t))))
  (decreases (
    betree_Internal_flush_decreases self params node_id_cnt content st))
  =
  let* p = betree_List_partition_at_pivot betree_Message_t content self.pivot
    in
  let (msgs_left, msgs_right) = p in
  let* len_left = betree_List_len (u64 & betree_Message_t) msgs_left in
  if len_left >= params.min_flush_size
  then
    let* (st0, _) =
      betree_Node_apply_messages self.left params node_id_cnt msgs_left st in
    let* (st1, (_, node_id_cnt0)) =
      betree_Node_apply_messages_back'a self.left params node_id_cnt msgs_left
        st st0 in
    let* (st2, ()) =
      betree_Node_apply_messages_back1 self.left params node_id_cnt msgs_left
        st st1 in
    let* len_right = betree_List_len (u64 & betree_Message_t) msgs_right in
    if len_right >= params.min_flush_size
    then
      let* (st3, _) =
        betree_Node_apply_messages self.right params node_id_cnt0 msgs_right
          st2 in
      let* (st4, (_, _)) =
        betree_Node_apply_messages_back'a self.right params node_id_cnt0
          msgs_right st2 st3 in
      let* (st5, ()) =
        betree_Node_apply_messages_back1 self.right params node_id_cnt0
          msgs_right st2 st4 in
      Return (st5, Betree_List_Nil)
    else Return (st2, msgs_right)
  else
    let* (st0, _) =
      betree_Node_apply_messages self.right params node_id_cnt msgs_right st in
    let* (st1, (_, _)) =
      betree_Node_apply_messages_back'a self.right params node_id_cnt
        msgs_right st st0 in
    let* (st2, ()) =
      betree_Node_apply_messages_back1 self.right params node_id_cnt msgs_right
        st st1 in
    Return (st2, msgs_left)

(** [betree_main::betree::Internal::{4}::flush]: backward function 0 *)
and betree_Internal_flush_back'a
  (self : betree_Internal_t) (params : betree_Params_t)
  (node_id_cnt : betree_NodeIdCounter_t)
  (content : betree_List_t (u64 & betree_Message_t)) (st : state) (st0 : state)
  :
  Tot (result (state & (betree_Internal_t & betree_NodeIdCounter_t)))
  (decreases (
    betree_Internal_flush_decreases self params node_id_cnt content st))
  =
  let* p = betree_List_partition_at_pivot betree_Message_t content self.pivot
    in
  let (msgs_left, msgs_right) = p in
  let* len_left = betree_List_len (u64 & betree_Message_t) msgs_left in
  if len_left >= params.min_flush_size
  then
    let* (st1, _) =
      betree_Node_apply_messages self.left params node_id_cnt msgs_left st in
    let* (st2, (n, node_id_cnt0)) =
      betree_Node_apply_messages_back'a self.left params node_id_cnt msgs_left
        st st1 in
    let* (st3, ()) =
      betree_Node_apply_messages_back1 self.left params node_id_cnt msgs_left
        st st2 in
    let* len_right = betree_List_len (u64 & betree_Message_t) msgs_right in
    if len_right >= params.min_flush_size
    then
      let* (st4, _) =
        betree_Node_apply_messages self.right params node_id_cnt0 msgs_right
          st3 in
      let* (st5, (n0, node_id_cnt1)) =
        betree_Node_apply_messages_back'a self.right params node_id_cnt0
          msgs_right st3 st4 in
      let* _ =
        betree_Node_apply_messages_back1 self.right params node_id_cnt0
          msgs_right st3 st5 in
      Return (st0, ({ self with left = n; right = n0 }, node_id_cnt1))
    else Return (st0, ({ self with left = n }, node_id_cnt0))
  else
    let* (st1, _) =
      betree_Node_apply_messages self.right params node_id_cnt msgs_right st in
    let* (st2, (n, node_id_cnt0)) =
      betree_Node_apply_messages_back'a self.right params node_id_cnt
        msgs_right st st1 in
    let* _ =
      betree_Node_apply_messages_back1 self.right params node_id_cnt msgs_right
        st st2 in
    Return (st0, ({ self with right = n }, node_id_cnt0))

(** [betree_main::betree::Internal::{4}::flush]: backward function 1 *)
and betree_Internal_flush_back1
  (self : betree_Internal_t) (params : betree_Params_t)
  (node_id_cnt : betree_NodeIdCounter_t)
  (content : betree_List_t (u64 & betree_Message_t)) (st : state) (st0 : state)
  :
  Tot (result (state & unit))
  (decreases (
    betree_Internal_flush_decreases self params node_id_cnt content st))
  =
  let* p = betree_List_partition_at_pivot betree_Message_t content self.pivot
    in
  let (msgs_left, msgs_right) = p in
  let* len_left = betree_List_len (u64 & betree_Message_t) msgs_left in
  if len_left >= params.min_flush_size
  then
    let* (st1, _) =
      betree_Node_apply_messages self.left params node_id_cnt msgs_left st in
    let* (st2, (_, node_id_cnt0)) =
      betree_Node_apply_messages_back'a self.left params node_id_cnt msgs_left
        st st1 in
    let* (st3, ()) =
      betree_Node_apply_messages_back1 self.left params node_id_cnt msgs_left
        st st2 in
    let* len_right = betree_List_len (u64 & betree_Message_t) msgs_right in
    if len_right >= params.min_flush_size
    then
      let* (st4, _) =
        betree_Node_apply_messages self.right params node_id_cnt0 msgs_right
          st3 in
      let* (st5, (_, _)) =
        betree_Node_apply_messages_back'a self.right params node_id_cnt0
          msgs_right st3 st4 in
      let* _ =
        betree_Node_apply_messages_back1 self.right params node_id_cnt0
          msgs_right st3 st5 in
      Return (st0, ())
    else Return (st0, ())
  else
    let* (st1, _) =
      betree_Node_apply_messages self.right params node_id_cnt msgs_right st in
    let* (st2, (_, _)) =
      betree_Node_apply_messages_back'a self.right params node_id_cnt
        msgs_right st st1 in
    let* _ =
      betree_Node_apply_messages_back1 self.right params node_id_cnt msgs_right
        st st2 in
    Return (st0, ())

(** [betree_main::betree::Node::{5}::apply_messages]: forward function *)
and betree_Node_apply_messages
  (self : betree_Node_t) (params : betree_Params_t)
  (node_id_cnt : betree_NodeIdCounter_t)
  (msgs : betree_List_t (u64 & betree_Message_t)) (st : state) :
  Tot (result (state & unit))
  (decreases (
    betree_Node_apply_messages_decreases self params node_id_cnt msgs st))
  =
  begin match self with
  | Betree_Node_Internal node ->
    let* (st0, content) = betree_load_internal_node node.id st in
    let* content0 = betree_Node_apply_messages_to_internal content msgs in
    let* num_msgs = betree_List_len (u64 & betree_Message_t) content0 in
    if num_msgs >= params.min_flush_size
    then
      let* (st1, content1) =
        betree_Internal_flush node params node_id_cnt content0 st0 in
      let* (st2, (node0, _)) =
        betree_Internal_flush_back'a node params node_id_cnt content0 st0 st1
        in
      let* (st3, _) = betree_store_internal_node node0.id content1 st2 in
      Return (st3, ())
    else
      let* (st1, _) = betree_store_internal_node node.id content0 st0 in
      Return (st1, ())
  | Betree_Node_Leaf node ->
    let* (st0, content) = betree_load_leaf_node node.id st in
    let* content0 = betree_Node_apply_messages_to_leaf content msgs in
    let* len = betree_List_len (u64 & u64) content0 in
    let* i = u64_mul 2 params.split_size in
    if len >= i
    then
      let* (st1, _) = betree_Leaf_split node content0 params node_id_cnt st0 in
      let* (st2, _) = betree_store_leaf_node node.id Betree_List_Nil st1 in
      betree_Leaf_split_back0 node content0 params node_id_cnt st0 st2
    else
      let* (st1, _) = betree_store_leaf_node node.id content0 st0 in
      Return (st1, ())
  end

(** [betree_main::betree::Node::{5}::apply_messages]: backward function 0 *)
and betree_Node_apply_messages_back'a
  (self : betree_Node_t) (params : betree_Params_t)
  (node_id_cnt : betree_NodeIdCounter_t)
  (msgs : betree_List_t (u64 & betree_Message_t)) (st : state) (st0 : state) :
  Tot (result (state & (betree_Node_t & betree_NodeIdCounter_t)))
  (decreases (
    betree_Node_apply_messages_decreases self params node_id_cnt msgs st))
  =
  begin match self with
  | Betree_Node_Internal node ->
    let* (st1, content) = betree_load_internal_node node.id st in
    let* content0 = betree_Node_apply_messages_to_internal content msgs in
    let* num_msgs = betree_List_len (u64 & betree_Message_t) content0 in
    if num_msgs >= params.min_flush_size
    then
      let* (st2, content1) =
        betree_Internal_flush node params node_id_cnt content0 st1 in
      let* (st3, (node0, node_id_cnt0)) =
        betree_Internal_flush_back'a node params node_id_cnt content0 st1 st2
        in
      let* _ = betree_store_internal_node node0.id content1 st3 in
      Return (st0, (Betree_Node_Internal node0, node_id_cnt0))
    else
      let* _ = betree_store_internal_node node.id content0 st1 in
      Return (st0, (Betree_Node_Internal node, node_id_cnt))
  | Betree_Node_Leaf node ->
    let* (st1, content) = betree_load_leaf_node node.id st in
    let* content0 = betree_Node_apply_messages_to_leaf content msgs in
    let* len = betree_List_len (u64 & u64) content0 in
    let* i = u64_mul 2 params.split_size in
    if len >= i
    then
      let* (st2, new_node) =
        betree_Leaf_split node content0 params node_id_cnt st1 in
      let* (st3, _) = betree_store_leaf_node node.id Betree_List_Nil st2 in
      let* _ = betree_Leaf_split_back0 node content0 params node_id_cnt st1 st3
        in
      let* (st4, node_id_cnt0) =
        betree_Leaf_split_back2 node content0 params node_id_cnt st1 st0 in
      Return (st4, (Betree_Node_Internal new_node, node_id_cnt0))
    else
      let* _ = betree_store_leaf_node node.id content0 st1 in
      Return (st0, (Betree_Node_Leaf { node with size = len }, node_id_cnt))
  end

(** [betree_main::betree::Node::{5}::apply_messages]: backward function 1 *)
and betree_Node_apply_messages_back1
  (self : betree_Node_t) (params : betree_Params_t)
  (node_id_cnt : betree_NodeIdCounter_t)
  (msgs : betree_List_t (u64 & betree_Message_t)) (st : state) (st0 : state) :
  Tot (result (state & unit))
  (decreases (
    betree_Node_apply_messages_decreases self params node_id_cnt msgs st))
  =
  begin match self with
  | Betree_Node_Internal node ->
    let* (st1, content) = betree_load_internal_node node.id st in
    let* content0 = betree_Node_apply_messages_to_internal content msgs in
    let* num_msgs = betree_List_len (u64 & betree_Message_t) content0 in
    if num_msgs >= params.min_flush_size
    then
      let* (st2, content1) =
        betree_Internal_flush node params node_id_cnt content0 st1 in
      let* (st3, (node0, _)) =
        betree_Internal_flush_back'a node params node_id_cnt content0 st1 st2
        in
      let* _ = betree_store_internal_node node0.id content1 st3 in
      betree_Internal_flush_back1 node params node_id_cnt content0 st1 st0
    else
      let* _ = betree_store_internal_node node.id content0 st1 in
      Return (st0, ())
  | Betree_Node_Leaf node ->
    let* (st1, content) = betree_load_leaf_node node.id st in
    let* content0 = betree_Node_apply_messages_to_leaf content msgs in
    let* len = betree_List_len (u64 & u64) content0 in
    let* i = u64_mul 2 params.split_size in
    if len >= i
    then
      let* (st2, _) = betree_Leaf_split node content0 params node_id_cnt st1 in
      let* (st3, _) = betree_store_leaf_node node.id Betree_List_Nil st2 in
      let* _ = betree_Leaf_split_back0 node content0 params node_id_cnt st1 st3
        in
      betree_Leaf_split_back1 node content0 params node_id_cnt st1 st0
    else
      let* _ = betree_store_leaf_node node.id content0 st1 in Return (st0, ())
  end

(** [betree_main::betree::Node::{5}::apply]: forward function *)
let betree_Node_apply
  (self : betree_Node_t) (params : betree_Params_t)
  (node_id_cnt : betree_NodeIdCounter_t) (key : u64)
  (new_msg : betree_Message_t) (st : state) :
  result (state & unit)
  =
  let l = Betree_List_Nil in
  let* (st0, _) =
    betree_Node_apply_messages self params node_id_cnt (Betree_List_Cons (key,
      new_msg) l) st in
  let* (st1, (_, _)) =
    betree_Node_apply_messages_back'a self params node_id_cnt (Betree_List_Cons
      (key, new_msg) l) st st0 in
  betree_Node_apply_messages_back1 self params node_id_cnt (Betree_List_Cons
    (key, new_msg) l) st st1

(** [betree_main::betree::Node::{5}::apply]: backward function 0 *)
let betree_Node_apply_back'a
  (self : betree_Node_t) (params : betree_Params_t)
  (node_id_cnt : betree_NodeIdCounter_t) (key : u64)
  (new_msg : betree_Message_t) (st : state) (st0 : state) :
  result (state & (betree_Node_t & betree_NodeIdCounter_t))
  =
  let l = Betree_List_Nil in
  let* (st1, _) =
    betree_Node_apply_messages self params node_id_cnt (Betree_List_Cons (key,
      new_msg) l) st in
  let* (st2, (self0, node_id_cnt0)) =
    betree_Node_apply_messages_back'a self params node_id_cnt (Betree_List_Cons
      (key, new_msg) l) st st1 in
  let* _ =
    betree_Node_apply_messages_back1 self params node_id_cnt (Betree_List_Cons
      (key, new_msg) l) st st2 in
  Return (st0, (self0, node_id_cnt0))

(** [betree_main::betree::Node::{5}::apply]: backward function 1 *)
let betree_Node_apply_back1
  (self : betree_Node_t) (params : betree_Params_t)
  (node_id_cnt : betree_NodeIdCounter_t) (key : u64)
  (new_msg : betree_Message_t) (st : state) (st0 : state) :
  result (state & unit)
  =
  let l = Betree_List_Nil in
  let* (st1, _) =
    betree_Node_apply_messages self params node_id_cnt (Betree_List_Cons (key,
      new_msg) l) st in
  let* (st2, (_, _)) =
    betree_Node_apply_messages_back'a self params node_id_cnt (Betree_List_Cons
      (key, new_msg) l) st st1 in
  let* _ =
    betree_Node_apply_messages_back1 self params node_id_cnt (Betree_List_Cons
      (key, new_msg) l) st st2 in
  Return (st0, ())

(** [betree_main::betree::BeTree::{6}::new]: forward function *)
let betree_BeTree_new
  (min_flush_size : u64) (split_size : u64) (st : state) :
  result (state & betree_BeTree_t)
  =
  let* node_id_cnt = betree_NodeIdCounter_new in
  let* id = betree_NodeIdCounter_fresh_id node_id_cnt in
  let* (st0, _) = betree_store_leaf_node id Betree_List_Nil st in
  let* node_id_cnt0 = betree_NodeIdCounter_fresh_id_back node_id_cnt in
  Return (st0,
    {
      params = { min_flush_size = min_flush_size; split_size = split_size };
      node_id_cnt = node_id_cnt0;
      root = (Betree_Node_Leaf { id = id; size = 0 })
    })

(** [betree_main::betree::BeTree::{6}::apply]: forward function *)
let betree_BeTree_apply
  (self : betree_BeTree_t) (key : u64) (msg : betree_Message_t) (st : state) :
  result (state & unit)
  =
  let* (st0, _) =
    betree_Node_apply self.root self.params self.node_id_cnt key msg st in
  let* (st1, (_, _)) =
    betree_Node_apply_back'a self.root self.params self.node_id_cnt key msg st
      st0 in
  betree_Node_apply_back1 self.root self.params self.node_id_cnt key msg st st1

(** [betree_main::betree::BeTree::{6}::apply]: backward function 0 *)
let betree_BeTree_apply_back
  (self : betree_BeTree_t) (key : u64) (msg : betree_Message_t) (st : state)
  (st0 : state) :
  result (state & betree_BeTree_t)
  =
  let* (st1, _) =
    betree_Node_apply self.root self.params self.node_id_cnt key msg st in
  let* (st2, (n, nic)) =
    betree_Node_apply_back'a self.root self.params self.node_id_cnt key msg st
      st1 in
  let* _ =
    betree_Node_apply_back1 self.root self.params self.node_id_cnt key msg st
      st2 in
  Return (st0, { self with node_id_cnt = nic; root = n })

(** [betree_main::betree::BeTree::{6}::insert]: forward function *)
let betree_BeTree_insert
  (self : betree_BeTree_t) (key : u64) (value : u64) (st : state) :
  result (state & unit)
  =
  let* (st0, _) = betree_BeTree_apply self key (Betree_Message_Insert value) st
    in
  let* (st1, _) =
    betree_BeTree_apply_back self key (Betree_Message_Insert value) st st0 in
  Return (st1, ())

(** [betree_main::betree::BeTree::{6}::insert]: backward function 0 *)
let betree_BeTree_insert_back
  (self : betree_BeTree_t) (key : u64) (value : u64) (st : state) (st0 : state)
  :
  result (state & betree_BeTree_t)
  =
  let* (st1, _) = betree_BeTree_apply self key (Betree_Message_Insert value) st
    in
  let* (_, self0) =
    betree_BeTree_apply_back self key (Betree_Message_Insert value) st st1 in
  Return (st0, self0)

(** [betree_main::betree::BeTree::{6}::delete]: forward function *)
let betree_BeTree_delete
  (self : betree_BeTree_t) (key : u64) (st : state) : result (state & unit) =
  let* (st0, _) = betree_BeTree_apply self key Betree_Message_Delete st in
  let* (st1, _) =
    betree_BeTree_apply_back self key Betree_Message_Delete st st0 in
  Return (st1, ())

(** [betree_main::betree::BeTree::{6}::delete]: backward function 0 *)
let betree_BeTree_delete_back
  (self : betree_BeTree_t) (key : u64) (st : state) (st0 : state) :
  result (state & betree_BeTree_t)
  =
  let* (st1, _) = betree_BeTree_apply self key Betree_Message_Delete st in
  let* (_, self0) =
    betree_BeTree_apply_back self key Betree_Message_Delete st st1 in
  Return (st0, self0)

(** [betree_main::betree::BeTree::{6}::upsert]: forward function *)
let betree_BeTree_upsert
  (self : betree_BeTree_t) (key : u64) (upd : betree_UpsertFunState_t)
  (st : state) :
  result (state & unit)
  =
  let* (st0, _) = betree_BeTree_apply self key (Betree_Message_Upsert upd) st
    in
  let* (st1, _) =
    betree_BeTree_apply_back self key (Betree_Message_Upsert upd) st st0 in
  Return (st1, ())

(** [betree_main::betree::BeTree::{6}::upsert]: backward function 0 *)
let betree_BeTree_upsert_back
  (self : betree_BeTree_t) (key : u64) (upd : betree_UpsertFunState_t)
  (st : state) (st0 : state) :
  result (state & betree_BeTree_t)
  =
  let* (st1, _) = betree_BeTree_apply self key (Betree_Message_Upsert upd) st
    in
  let* (_, self0) =
    betree_BeTree_apply_back self key (Betree_Message_Upsert upd) st st1 in
  Return (st0, self0)

(** [betree_main::betree::BeTree::{6}::lookup]: forward function *)
let betree_BeTree_lookup
  (self : betree_BeTree_t) (key : u64) (st : state) :
  result (state & (option u64))
  =
  betree_Node_lookup self.root key st

(** [betree_main::betree::BeTree::{6}::lookup]: backward function 0 *)
let betree_BeTree_lookup_back
  (self : betree_BeTree_t) (key : u64) (st : state) (st0 : state) :
  result (state & betree_BeTree_t)
  =
  let* (st1, n) = betree_Node_lookup_back self.root key st st0 in
  Return (st1, { self with root = n })

(** [betree_main::main]: forward function *)
let main : result unit =
  Return ()

(** Unit test for [betree_main::main] *)
let _ = assert_norm (main = Return ())