(** 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_fwd
  (id : u64) (st : state) :
  result (state & (betree_list_t (u64 & betree_message_t)))
  =
  betree_utils_load_internal_node_fwd id st

(** [betree_main::betree::store_internal_node]: forward function *)
let betree_store_internal_node_fwd
  (id : u64) (content : betree_list_t (u64 & betree_message_t)) (st : state) :
  result (state & unit)
  =
  let* (st0, _) = betree_utils_store_internal_node_fwd id content st in
  Return (st0, ())

(** [betree_main::betree::load_leaf_node]: forward function *)
let betree_load_leaf_node_fwd
  (id : u64) (st : state) : result (state & (betree_list_t (u64 & u64))) =
  betree_utils_load_leaf_node_fwd id st

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

(** [betree_main::betree::fresh_node_id]: forward function *)
let betree_fresh_node_id_fwd (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_node_id_counter_new_fwd : result betree_node_id_counter_t =
  Return { betree_node_id_counter_next_node_id = 0 }

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

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

(** [core::num::u64::{9}::MAX] *)
let core_num_u64_max_body : result u64 = Return 18446744073709551615
let core_num_u64_max_c : u64 = eval_global core_num_u64_max_body

(** [betree_main::betree::upsert_update]: forward function *)
let betree_upsert_update_fwd
  (prev : option u64) (st : betree_upsert_fun_state_t) : result u64 =
  begin match prev with
  | None ->
    begin match st with
    | BetreeUpsertFunStateAdd v -> Return v
    | BetreeUpsertFunStateSub i -> Return 0
    end
  | Some prev0 ->
    begin match st with
    | BetreeUpsertFunStateAdd v ->
      let* margin = u64_sub core_num_u64_max_c prev0 in
      if margin >= v then u64_add prev0 v else Return core_num_u64_max_c
    | BetreeUpsertFunStateSub 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_fwd
  (t : Type0) (self : betree_list_t t) :
  Tot (result u64) (decreases (betree_list_len_decreases t self))
  =
  begin match self with
  | BetreeListCons x tl -> let* i = betree_list_len_fwd t tl in u64_add 1 i
  | BetreeListNil -> Return 0
  end

(** [betree_main::betree::List::{1}::split_at]: forward function *)
let rec betree_list_split_at_fwd
  (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 (BetreeListNil, self)
  else
    begin match self with
    | BetreeListCons hd tl ->
      let* i = u64_sub n 1 in
      let* p = betree_list_split_at_fwd t tl i in
      let (ls0, ls1) = p in
      let l = ls0 in
      Return (BetreeListCons hd l, ls1)
    | BetreeListNil -> 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_fwd_back
  (t : Type0) (self : betree_list_t t) (x : t) : result (betree_list_t t) =
  let tl = mem_replace_fwd (betree_list_t t) self BetreeListNil in
  let l = tl in
  Return (BetreeListCons x l)

(** [betree_main::betree::List::{1}::pop_front]: forward function *)
let betree_list_pop_front_fwd (t : Type0) (self : betree_list_t t) : result t =
  let ls = mem_replace_fwd (betree_list_t t) self BetreeListNil in
  begin match ls with
  | BetreeListCons x tl -> Return x
  | BetreeListNil -> 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 = mem_replace_fwd (betree_list_t t) self BetreeListNil in
  begin match ls with
  | BetreeListCons x tl -> Return tl
  | BetreeListNil -> Fail Failure
  end

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

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

(** [betree_main::betree::List::{2}::partition_at_pivot]: forward function *)
let rec betree_list_partition_at_pivot_fwd
  (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
  | BetreeListCons hd tl ->
    let (i, x) = hd in
    if i >= pivot
    then Return (BetreeListNil, BetreeListCons (i, x) tl)
    else
      let* p = betree_list_partition_at_pivot_fwd t tl pivot in
      let (ls0, ls1) = p in
      let l = ls0 in
      Return (BetreeListCons (i, x) l, ls1)
  | BetreeListNil -> Return (BetreeListNil, BetreeListNil)
  end

(** [betree_main::betree::Leaf::{3}::split]: forward function *)
let betree_leaf_split_fwd
  (self : betree_leaf_t) (content : betree_list_t (u64 & u64))
  (params : betree_params_t) (node_id_cnt : betree_node_id_counter_t)
  (st : state) :
  result (state & betree_internal_t)
  =
  let* p =
    betree_list_split_at_fwd (u64 & u64) content
      params.betree_params_split_size in
  let (content0, content1) = p in
  let* p0 = betree_list_hd_fwd (u64 & u64) content1 in
  let (pivot, _) = p0 in
  let* id0 = betree_node_id_counter_fresh_id_fwd node_id_cnt in
  let* node_id_cnt0 = betree_node_id_counter_fresh_id_back node_id_cnt in
  let* id1 = betree_node_id_counter_fresh_id_fwd node_id_cnt0 in
  let* (st0, _) = betree_store_leaf_node_fwd id0 content0 st in
  let* (st1, _) = betree_store_leaf_node_fwd id1 content1 st0 in
  let n = BetreeNodeLeaf
    { betree_leaf_id = id0; betree_leaf_size = params.betree_params_split_size
    } in
  let n0 = BetreeNodeLeaf
    { betree_leaf_id = id1; betree_leaf_size = params.betree_params_split_size
    } in
  Return (st1,
    {
      betree_internal_id = self.betree_leaf_id;
      betree_internal_pivot = pivot;
      betree_internal_left = n;
      betree_internal_right = n0
    })

(** [betree_main::betree::Leaf::{3}::split]: backward function 2 *)
let betree_leaf_split_back
  (self : betree_leaf_t) (content : betree_list_t (u64 & u64))
  (params : betree_params_t) (node_id_cnt : betree_node_id_counter_t)
  (st : state) :
  result betree_node_id_counter_t
  =
  let* p =
    betree_list_split_at_fwd (u64 & u64) content
      params.betree_params_split_size in
  let (content0, content1) = p in
  let* _ = betree_list_hd_fwd (u64 & u64) content1 in
  let* id0 = betree_node_id_counter_fresh_id_fwd node_id_cnt in
  let* node_id_cnt0 = betree_node_id_counter_fresh_id_back node_id_cnt in
  let* id1 = betree_node_id_counter_fresh_id_fwd node_id_cnt0 in
  let* (st0, _) = betree_store_leaf_node_fwd id0 content0 st in
  let* _ = betree_store_leaf_node_fwd id1 content1 st0 in
  betree_node_id_counter_fresh_id_back node_id_cnt0

(** [betree_main::betree::Node::{5}::lookup_first_message_for_key]: forward function *)
let rec betree_node_lookup_first_message_for_key_fwd
  (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
  | BetreeListCons x next_msgs ->
    let (i, m) = x in
    if i >= key
    then Return (BetreeListCons (i, m) next_msgs)
    else betree_node_lookup_first_message_for_key_fwd key next_msgs
  | BetreeListNil -> Return BetreeListNil
  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
  | BetreeListCons 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 (BetreeListCons (i, m) next_msgs0)
  | BetreeListNil -> Return ret
  end

(** [betree_main::betree::Node::{5}::apply_upserts]: forward function *)
let rec betree_node_apply_upserts_fwd
  (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_fwd betree_message_t msgs key in
  if b
  then
    let* msg = betree_list_pop_front_fwd (u64 & betree_message_t) msgs in
    let (_, m) = msg in
    begin match m with
    | BetreeMessageInsert i -> Fail Failure
    | BetreeMessageDelete -> Fail Failure
    | BetreeMessageUpsert s ->
      let* v = betree_upsert_update_fwd prev s in
      let* msgs0 = betree_list_pop_front_back (u64 & betree_message_t) msgs in
      betree_node_apply_upserts_fwd msgs0 (Some v) key st
    end
  else
    let* (st0, v) = core_option_option_unwrap_fwd u64 prev st in
    let* _ =
      betree_list_push_front_fwd_back (u64 & betree_message_t) msgs (key,
        BetreeMessageInsert 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) :
  Tot (result (betree_list_t (u64 & betree_message_t)))
  (decreases (betree_node_apply_upserts_decreases msgs prev key st))
  =
  let* b = betree_list_head_has_key_fwd betree_message_t msgs key in
  if b
  then
    let* msg = betree_list_pop_front_fwd (u64 & betree_message_t) msgs in
    let (_, m) = msg in
    begin match m with
    | BetreeMessageInsert i -> Fail Failure
    | BetreeMessageDelete -> Fail Failure
    | BetreeMessageUpsert s ->
      let* v = betree_upsert_update_fwd 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
    end
  else
    let* (_, v) = core_option_option_unwrap_fwd u64 prev st in
    betree_list_push_front_fwd_back (u64 & betree_message_t) msgs (key,
      BetreeMessageInsert v)

(** [betree_main::betree::Node::{5}::lookup_in_bindings]: forward function *)
let rec betree_node_lookup_in_bindings_fwd
  (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
  | BetreeListCons 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_fwd key tl
  | BetreeListNil -> Return None
  end

(** [betree_main::betree::Internal::{4}::lookup_in_children]: forward function *)
let rec betree_internal_lookup_in_children_fwd
  (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.betree_internal_pivot
  then betree_node_lookup_fwd self.betree_internal_left key st
  else betree_node_lookup_fwd self.betree_internal_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) :
  Tot (result betree_internal_t)
  (decreases (betree_internal_lookup_in_children_decreases self key st))
  =
  if key < self.betree_internal_pivot
  then
    let* n = betree_node_lookup_back self.betree_internal_left key st in
    Return { self with betree_internal_left = n }
  else
    let* n = betree_node_lookup_back self.betree_internal_right key st in
    Return { self with betree_internal_right = n }

(** [betree_main::betree::Node::{5}::lookup]: forward function *)
and betree_node_lookup_fwd
  (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
  | BetreeNodeInternal node ->
    let* (st0, msgs) = betree_load_internal_node_fwd node.betree_internal_id st
      in
    let* pending = betree_node_lookup_first_message_for_key_fwd key msgs in
    begin match pending with
    | BetreeListCons p l ->
      let (k, msg) = p in
      if k <> key
      then
        let* (st1, opt) = betree_internal_lookup_in_children_fwd node key st0
          in
        let* _ =
          betree_node_lookup_first_message_for_key_back key msgs
            (BetreeListCons (k, msg) l) in
        Return (st1, opt)
      else
        begin match msg with
        | BetreeMessageInsert v ->
          let* _ =
            betree_node_lookup_first_message_for_key_back key msgs
              (BetreeListCons (k, BetreeMessageInsert v) l) in
          Return (st0, Some v)
        | BetreeMessageDelete ->
          let* _ =
            betree_node_lookup_first_message_for_key_back key msgs
              (BetreeListCons (k, BetreeMessageDelete) l) in
          Return (st0, None)
        | BetreeMessageUpsert ufs ->
          let* (st1, v) = betree_internal_lookup_in_children_fwd node key st0
            in
          let* (st2, v0) =
            betree_node_apply_upserts_fwd (BetreeListCons (k,
              BetreeMessageUpsert ufs) l) v key st1 in
          let* node0 = betree_internal_lookup_in_children_back node key st0 in
          let* pending0 =
            betree_node_apply_upserts_back (BetreeListCons (k,
              BetreeMessageUpsert ufs) l) v key st1 in
          let* msgs0 =
            betree_node_lookup_first_message_for_key_back key msgs pending0 in
          let* (st3, _) =
            betree_store_internal_node_fwd node0.betree_internal_id msgs0 st2
            in
          Return (st3, Some v0)
        end
    | BetreeListNil ->
      let* (st1, opt) = betree_internal_lookup_in_children_fwd node key st0 in
      let* _ =
        betree_node_lookup_first_message_for_key_back key msgs BetreeListNil in
      Return (st1, opt)
    end
  | BetreeNodeLeaf node ->
    let* (st0, bindings) = betree_load_leaf_node_fwd node.betree_leaf_id st in
    let* opt = betree_node_lookup_in_bindings_fwd key bindings in
    Return (st0, opt)
  end

(** [betree_main::betree::Node::{5}::lookup]: backward function 0 *)
and betree_node_lookup_back
  (self : betree_node_t) (key : u64) (st : state) :
  Tot (result betree_node_t)
  (decreases (betree_node_lookup_decreases self key st))
  =
  begin match self with
  | BetreeNodeInternal node ->
    let* (st0, msgs) = betree_load_internal_node_fwd node.betree_internal_id st
      in
    let* pending = betree_node_lookup_first_message_for_key_fwd key msgs in
    begin match pending with
    | BetreeListCons p l ->
      let (k, msg) = p in
      if k <> key
      then
        let* _ =
          betree_node_lookup_first_message_for_key_back key msgs
            (BetreeListCons (k, msg) l) in
        let* node0 = betree_internal_lookup_in_children_back node key st0 in
        Return (BetreeNodeInternal node0)
      else
        begin match msg with
        | BetreeMessageInsert v ->
          let* _ =
            betree_node_lookup_first_message_for_key_back key msgs
              (BetreeListCons (k, BetreeMessageInsert v) l) in
          Return (BetreeNodeInternal node)
        | BetreeMessageDelete ->
          let* _ =
            betree_node_lookup_first_message_for_key_back key msgs
              (BetreeListCons (k, BetreeMessageDelete) l) in
          Return (BetreeNodeInternal node)
        | BetreeMessageUpsert ufs ->
          let* (st1, v) = betree_internal_lookup_in_children_fwd node key st0
            in
          let* (st2, _) =
            betree_node_apply_upserts_fwd (BetreeListCons (k,
              BetreeMessageUpsert ufs) l) v key st1 in
          let* node0 = betree_internal_lookup_in_children_back node key st0 in
          let* pending0 =
            betree_node_apply_upserts_back (BetreeListCons (k,
              BetreeMessageUpsert ufs) l) v key st1 in
          let* msgs0 =
            betree_node_lookup_first_message_for_key_back key msgs pending0 in
          let* _ =
            betree_store_internal_node_fwd node0.betree_internal_id msgs0 st2
            in
          Return (BetreeNodeInternal node0)
        end
    | BetreeListNil ->
      let* _ =
        betree_node_lookup_first_message_for_key_back key msgs BetreeListNil in
      let* node0 = betree_internal_lookup_in_children_back node key st0 in
      Return (BetreeNodeInternal node0)
    end
  | BetreeNodeLeaf node ->
    let* (_, bindings) = betree_load_leaf_node_fwd node.betree_leaf_id st in
    let* _ = betree_node_lookup_in_bindings_fwd key bindings in
    Return (BetreeNodeLeaf 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_fwd_back
  (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
  | BetreeListCons p l ->
    let (k, m) = p in
    if k = key
    then
      let* msgs0 =
        betree_list_pop_front_back (u64 & betree_message_t) (BetreeListCons (k,
          m) l) in
      betree_node_filter_messages_for_key_fwd_back key msgs0
    else Return (BetreeListCons (k, m) l)
  | BetreeListNil -> Return BetreeListNil
  end

(** [betree_main::betree::Node::{5}::lookup_first_message_after_key]: forward function *)
let rec betree_node_lookup_first_message_after_key_fwd
  (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
  | BetreeListCons p next_msgs ->
    let (k, m) = p in
    if k = key
    then betree_node_lookup_first_message_after_key_fwd key next_msgs
    else Return (BetreeListCons (k, m) next_msgs)
  | BetreeListNil -> Return BetreeListNil
  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
  | BetreeListCons 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 (BetreeListCons (k, m) next_msgs0)
    else Return ret
  | BetreeListNil -> 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_fwd_back
  (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_fwd key msgs in
  let* b = betree_list_head_has_key_fwd betree_message_t msgs0 key in
  if b
  then
    begin match new_msg with
    | BetreeMessageInsert i ->
      let* msgs1 = betree_node_filter_messages_for_key_fwd_back key msgs0 in
      let* msgs2 =
        betree_list_push_front_fwd_back (u64 & betree_message_t) msgs1 (key,
          BetreeMessageInsert i) in
      betree_node_lookup_first_message_for_key_back key msgs msgs2
    | BetreeMessageDelete ->
      let* msgs1 = betree_node_filter_messages_for_key_fwd_back key msgs0 in
      let* msgs2 =
        betree_list_push_front_fwd_back (u64 & betree_message_t) msgs1 (key,
          BetreeMessageDelete) in
      betree_node_lookup_first_message_for_key_back key msgs msgs2
    | BetreeMessageUpsert s ->
      let* p = betree_list_hd_fwd (u64 & betree_message_t) msgs0 in
      let (_, m) = p in
      begin match m with
      | BetreeMessageInsert prev ->
        let* v = betree_upsert_update_fwd (Some prev) s in
        let* msgs1 = betree_list_pop_front_back (u64 & betree_message_t) msgs0
          in
        let* msgs2 =
          betree_list_push_front_fwd_back (u64 & betree_message_t) msgs1 (key,
            BetreeMessageInsert v) in
        betree_node_lookup_first_message_for_key_back key msgs msgs2
      | BetreeMessageDelete ->
        let* v = betree_upsert_update_fwd None s in
        let* msgs1 = betree_list_pop_front_back (u64 & betree_message_t) msgs0
          in
        let* msgs2 =
          betree_list_push_front_fwd_back (u64 & betree_message_t) msgs1 (key,
            BetreeMessageInsert v) in
        betree_node_lookup_first_message_for_key_back key msgs msgs2
      | BetreeMessageUpsert ufs ->
        let* msgs1 = betree_node_lookup_first_message_after_key_fwd key msgs0
          in
        let* msgs2 =
          betree_list_push_front_fwd_back (u64 & betree_message_t) msgs1 (key,
            BetreeMessageUpsert 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_fwd_back (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_fwd_back
  (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
  | BetreeListCons new_msg new_msgs_tl ->
    let (i, m) = new_msg in
    let* msgs0 = betree_node_apply_to_internal_fwd_back msgs i m in
    betree_node_apply_messages_to_internal_fwd_back msgs0 new_msgs_tl
  | BetreeListNil -> Return msgs
  end

(** [betree_main::betree::Node::{5}::lookup_mut_in_bindings]: forward function *)
let rec betree_node_lookup_mut_in_bindings_fwd
  (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
  | BetreeListCons hd tl ->
    let (i, i0) = hd in
    if i >= key
    then Return (BetreeListCons (i, i0) tl)
    else betree_node_lookup_mut_in_bindings_fwd key tl
  | BetreeListNil -> Return BetreeListNil
  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
  | BetreeListCons 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 (BetreeListCons (i, i0) tl0)
  | BetreeListNil -> 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_fwd_back
  (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_fwd key bindings in
  let* b = betree_list_head_has_key_fwd u64 bindings0 key in
  if b
  then
    let* hd = betree_list_pop_front_fwd (u64 & u64) bindings0 in
    begin match new_msg with
    | BetreeMessageInsert v ->
      let* bindings1 = betree_list_pop_front_back (u64 & u64) bindings0 in
      let* bindings2 =
        betree_list_push_front_fwd_back (u64 & u64) bindings1 (key, v) in
      betree_node_lookup_mut_in_bindings_back key bindings bindings2
    | BetreeMessageDelete ->
      let* bindings1 = betree_list_pop_front_back (u64 & u64) bindings0 in
      betree_node_lookup_mut_in_bindings_back key bindings bindings1
    | BetreeMessageUpsert s ->
      let (_, i) = hd in
      let* v = betree_upsert_update_fwd (Some i) s in
      let* bindings1 = betree_list_pop_front_back (u64 & u64) bindings0 in
      let* bindings2 =
        betree_list_push_front_fwd_back (u64 & u64) bindings1 (key, v) in
      betree_node_lookup_mut_in_bindings_back key bindings bindings2
    end
  else
    begin match new_msg with
    | BetreeMessageInsert v ->
      let* bindings1 =
        betree_list_push_front_fwd_back (u64 & u64) bindings0 (key, v) in
      betree_node_lookup_mut_in_bindings_back key bindings bindings1
    | BetreeMessageDelete ->
      betree_node_lookup_mut_in_bindings_back key bindings bindings0
    | BetreeMessageUpsert s ->
      let* v = betree_upsert_update_fwd None s in
      let* bindings1 =
        betree_list_push_front_fwd_back (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_fwd_back
  (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
  | BetreeListCons new_msg new_msgs_tl ->
    let (i, m) = new_msg in
    let* bindings0 = betree_node_apply_to_leaf_fwd_back bindings i m in
    betree_node_apply_messages_to_leaf_fwd_back bindings0 new_msgs_tl
  | BetreeListNil -> Return bindings
  end

(** [betree_main::betree::Internal::{4}::flush]: forward function *)
let rec betree_internal_flush_fwd
  (self : betree_internal_t) (params : betree_params_t)
  (node_id_cnt : betree_node_id_counter_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_fwd betree_message_t content
      self.betree_internal_pivot in
  let (msgs_left, msgs_right) = p in
  let* len_left = betree_list_len_fwd (u64 & betree_message_t) msgs_left in
  if len_left >= params.betree_params_min_flush_size
  then
    let* (st0, _) =
      betree_node_apply_messages_fwd self.betree_internal_left params
        node_id_cnt msgs_left st in
    let* (_, node_id_cnt0) =
      betree_node_apply_messages_back self.betree_internal_left params
        node_id_cnt msgs_left st in
    let* len_right = betree_list_len_fwd (u64 & betree_message_t) msgs_right in
    if len_right >= params.betree_params_min_flush_size
    then
      let* (st1, _) =
        betree_node_apply_messages_fwd self.betree_internal_right params
          node_id_cnt0 msgs_right st0 in
      let* _ =
        betree_node_apply_messages_back self.betree_internal_right params
          node_id_cnt0 msgs_right st0 in
      Return (st1, BetreeListNil)
    else Return (st0, msgs_right)
  else
    let* (st0, _) =
      betree_node_apply_messages_fwd self.betree_internal_right params
        node_id_cnt msgs_right st in
    let* _ =
      betree_node_apply_messages_back self.betree_internal_right params
        node_id_cnt msgs_right st in
    Return (st0, msgs_left)

(** [betree_main::betree::Internal::{4}::flush]: backward function 0 *)
and betree_internal_flush_back
  (self : betree_internal_t) (params : betree_params_t)
  (node_id_cnt : betree_node_id_counter_t)
  (content : betree_list_t (u64 & betree_message_t)) (st : state) :
  Tot (result (betree_internal_t & betree_node_id_counter_t))
  (decreases (
    betree_internal_flush_decreases self params node_id_cnt content st))
  =
  let* p =
    betree_list_partition_at_pivot_fwd betree_message_t content
      self.betree_internal_pivot in
  let (msgs_left, msgs_right) = p in
  let* len_left = betree_list_len_fwd (u64 & betree_message_t) msgs_left in
  if len_left >= params.betree_params_min_flush_size
  then
    let* (st0, _) =
      betree_node_apply_messages_fwd self.betree_internal_left params
        node_id_cnt msgs_left st in
    let* (n, node_id_cnt0) =
      betree_node_apply_messages_back self.betree_internal_left params
        node_id_cnt msgs_left st in
    let* len_right = betree_list_len_fwd (u64 & betree_message_t) msgs_right in
    if len_right >= params.betree_params_min_flush_size
    then
      let* (n0, node_id_cnt1) =
        betree_node_apply_messages_back self.betree_internal_right params
          node_id_cnt0 msgs_right st0 in
      Return
        ({ self with betree_internal_left = n; betree_internal_right = n0 },
        node_id_cnt1)
    else Return ({ self with betree_internal_left = n }, node_id_cnt0)
  else
    let* (n, node_id_cnt0) =
      betree_node_apply_messages_back self.betree_internal_right params
        node_id_cnt msgs_right st in
    Return ({ self with betree_internal_right = n }, node_id_cnt0)

(** [betree_main::betree::Node::{5}::apply_messages]: forward function *)
and betree_node_apply_messages_fwd
  (self : betree_node_t) (params : betree_params_t)
  (node_id_cnt : betree_node_id_counter_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
  | BetreeNodeInternal node ->
    let* (st0, content) =
      betree_load_internal_node_fwd node.betree_internal_id st in
    let* content0 =
      betree_node_apply_messages_to_internal_fwd_back content msgs in
    let* num_msgs = betree_list_len_fwd (u64 & betree_message_t) content0 in
    if num_msgs >= params.betree_params_min_flush_size
    then
      let* (st1, content1) =
        betree_internal_flush_fwd node params node_id_cnt content0 st0 in
      let* (node0, _) =
        betree_internal_flush_back node params node_id_cnt content0 st0 in
      let* (st2, _) =
        betree_store_internal_node_fwd node0.betree_internal_id content1 st1 in
      Return (st2, ())
    else
      let* (st1, _) =
        betree_store_internal_node_fwd node.betree_internal_id content0 st0 in
      Return (st1, ())
  | BetreeNodeLeaf node ->
    let* (st0, content) = betree_load_leaf_node_fwd node.betree_leaf_id st in
    let* content0 = betree_node_apply_messages_to_leaf_fwd_back content msgs in
    let* len = betree_list_len_fwd (u64 & u64) content0 in
    let* i = u64_mul 2 params.betree_params_split_size in
    if len >= i
    then
      let* (st1, _) =
        betree_leaf_split_fwd node content0 params node_id_cnt st0 in
      let* (st2, _) =
        betree_store_leaf_node_fwd node.betree_leaf_id BetreeListNil st1 in
      Return (st2, ())
    else
      let* (st1, _) =
        betree_store_leaf_node_fwd node.betree_leaf_id content0 st0 in
      Return (st1, ())
  end

(** [betree_main::betree::Node::{5}::apply_messages]: backward function 0 *)
and betree_node_apply_messages_back
  (self : betree_node_t) (params : betree_params_t)
  (node_id_cnt : betree_node_id_counter_t)
  (msgs : betree_list_t (u64 & betree_message_t)) (st : state) :
  Tot (result (betree_node_t & betree_node_id_counter_t))
  (decreases (
    betree_node_apply_messages_decreases self params node_id_cnt msgs st))
  =
  begin match self with
  | BetreeNodeInternal node ->
    let* (st0, content) =
      betree_load_internal_node_fwd node.betree_internal_id st in
    let* content0 =
      betree_node_apply_messages_to_internal_fwd_back content msgs in
    let* num_msgs = betree_list_len_fwd (u64 & betree_message_t) content0 in
    if num_msgs >= params.betree_params_min_flush_size
    then
      let* (st1, content1) =
        betree_internal_flush_fwd node params node_id_cnt content0 st0 in
      let* (node0, node_id_cnt0) =
        betree_internal_flush_back node params node_id_cnt content0 st0 in
      let* _ =
        betree_store_internal_node_fwd node0.betree_internal_id content1 st1 in
      Return (BetreeNodeInternal node0, node_id_cnt0)
    else
      let* _ =
        betree_store_internal_node_fwd node.betree_internal_id content0 st0 in
      Return (BetreeNodeInternal node, node_id_cnt)
  | BetreeNodeLeaf node ->
    let* (st0, content) = betree_load_leaf_node_fwd node.betree_leaf_id st in
    let* content0 = betree_node_apply_messages_to_leaf_fwd_back content msgs in
    let* len = betree_list_len_fwd (u64 & u64) content0 in
    let* i = u64_mul 2 params.betree_params_split_size in
    if len >= i
    then
      let* (st1, new_node) =
        betree_leaf_split_fwd node content0 params node_id_cnt st0 in
      let* _ = betree_store_leaf_node_fwd node.betree_leaf_id BetreeListNil st1
        in
      let* node_id_cnt0 =
        betree_leaf_split_back node content0 params node_id_cnt st0 in
      Return (BetreeNodeInternal new_node, node_id_cnt0)
    else
      let* _ = betree_store_leaf_node_fwd node.betree_leaf_id content0 st0 in
      Return (BetreeNodeLeaf { node with betree_leaf_size = len }, node_id_cnt)
  end

(** [betree_main::betree::Node::{5}::apply]: forward function *)
let betree_node_apply_fwd
  (self : betree_node_t) (params : betree_params_t)
  (node_id_cnt : betree_node_id_counter_t) (key : u64)
  (new_msg : betree_message_t) (st : state) :
  result (state & unit)
  =
  let l = BetreeListNil in
  let* (st0, _) =
    betree_node_apply_messages_fwd self params node_id_cnt (BetreeListCons
      (key, new_msg) l) st in
  let* _ =
    betree_node_apply_messages_back self params node_id_cnt (BetreeListCons
      (key, new_msg) l) st in
  Return (st0, ())

(** [betree_main::betree::Node::{5}::apply]: backward function 0 *)
let betree_node_apply_back
  (self : betree_node_t) (params : betree_params_t)
  (node_id_cnt : betree_node_id_counter_t) (key : u64)
  (new_msg : betree_message_t) (st : state) :
  result (betree_node_t & betree_node_id_counter_t)
  =
  let l = BetreeListNil in
  betree_node_apply_messages_back self params node_id_cnt (BetreeListCons (key,
    new_msg) l) st

(** [betree_main::betree::BeTree::{6}::new]: forward function *)
let betree_be_tree_new_fwd
  (min_flush_size : u64) (split_size : u64) (st : state) :
  result (state & betree_be_tree_t)
  =
  let* node_id_cnt = betree_node_id_counter_new_fwd in
  let* id = betree_node_id_counter_fresh_id_fwd node_id_cnt in
  let* (st0, _) = betree_store_leaf_node_fwd id BetreeListNil st in
  let* node_id_cnt0 = betree_node_id_counter_fresh_id_back node_id_cnt in
  Return (st0,
    {
      betree_be_tree_params =
        {
          betree_params_min_flush_size = min_flush_size;
          betree_params_split_size = split_size
        };
      betree_be_tree_node_id_cnt = node_id_cnt0;
      betree_be_tree_root =
        (BetreeNodeLeaf { betree_leaf_id = id; betree_leaf_size = 0 })
    })

(** [betree_main::betree::BeTree::{6}::apply]: forward function *)
let betree_be_tree_apply_fwd
  (self : betree_be_tree_t) (key : u64) (msg : betree_message_t) (st : state) :
  result (state & unit)
  =
  let* (st0, _) =
    betree_node_apply_fwd self.betree_be_tree_root self.betree_be_tree_params
      self.betree_be_tree_node_id_cnt key msg st in
  let* _ =
    betree_node_apply_back self.betree_be_tree_root self.betree_be_tree_params
      self.betree_be_tree_node_id_cnt key msg st in
  Return (st0, ())

(** [betree_main::betree::BeTree::{6}::apply]: backward function 0 *)
let betree_be_tree_apply_back
  (self : betree_be_tree_t) (key : u64) (msg : betree_message_t) (st : state) :
  result betree_be_tree_t
  =
  let* (n, nic) =
    betree_node_apply_back self.betree_be_tree_root self.betree_be_tree_params
      self.betree_be_tree_node_id_cnt key msg st in
  Return
    { self with betree_be_tree_node_id_cnt = nic; betree_be_tree_root = n }

(** [betree_main::betree::BeTree::{6}::insert]: forward function *)
let betree_be_tree_insert_fwd
  (self : betree_be_tree_t) (key : u64) (value : u64) (st : state) :
  result (state & unit)
  =
  let* (st0, _) =
    betree_be_tree_apply_fwd self key (BetreeMessageInsert value) st in
  let* _ = betree_be_tree_apply_back self key (BetreeMessageInsert value) st in
  Return (st0, ())

(** [betree_main::betree::BeTree::{6}::insert]: backward function 0 *)
let betree_be_tree_insert_back
  (self : betree_be_tree_t) (key : u64) (value : u64) (st : state) :
  result betree_be_tree_t
  =
  betree_be_tree_apply_back self key (BetreeMessageInsert value) st

(** [betree_main::betree::BeTree::{6}::delete]: forward function *)
let betree_be_tree_delete_fwd
  (self : betree_be_tree_t) (key : u64) (st : state) : result (state & unit) =
  let* (st0, _) = betree_be_tree_apply_fwd self key BetreeMessageDelete st in
  let* _ = betree_be_tree_apply_back self key BetreeMessageDelete st in
  Return (st0, ())

(** [betree_main::betree::BeTree::{6}::delete]: backward function 0 *)
let betree_be_tree_delete_back
  (self : betree_be_tree_t) (key : u64) (st : state) :
  result betree_be_tree_t
  =
  betree_be_tree_apply_back self key BetreeMessageDelete st

(** [betree_main::betree::BeTree::{6}::upsert]: forward function *)
let betree_be_tree_upsert_fwd
  (self : betree_be_tree_t) (key : u64) (upd : betree_upsert_fun_state_t)
  (st : state) :
  result (state & unit)
  =
  let* (st0, _) =
    betree_be_tree_apply_fwd self key (BetreeMessageUpsert upd) st in
  let* _ = betree_be_tree_apply_back self key (BetreeMessageUpsert upd) st in
  Return (st0, ())

(** [betree_main::betree::BeTree::{6}::upsert]: backward function 0 *)
let betree_be_tree_upsert_back
  (self : betree_be_tree_t) (key : u64) (upd : betree_upsert_fun_state_t)
  (st : state) :
  result betree_be_tree_t
  =
  betree_be_tree_apply_back self key (BetreeMessageUpsert upd) st

(** [betree_main::betree::BeTree::{6}::lookup]: forward function *)
let betree_be_tree_lookup_fwd
  (self : betree_be_tree_t) (key : u64) (st : state) :
  result (state & (option u64))
  =
  betree_node_lookup_fwd self.betree_be_tree_root key st

(** [betree_main::betree::BeTree::{6}::lookup]: backward function 0 *)
let betree_be_tree_lookup_back
  (self : betree_be_tree_t) (key : u64) (st : state) :
  result betree_be_tree_t
  =
  let* n = betree_node_lookup_back self.betree_be_tree_root key st in
  Return { self with betree_be_tree_root = n }

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

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