-- THIS FILE WAS AUTOMATICALLY GENERATED BY AENEAS
-- [betree_main]: function definitions
import Base.Primitives
import BetreeMain.Types
import BetreeMain.Opaque
import BetreeMain.Clauses.Clauses

section variable (opaque_defs: OpaqueDefs)

/- [betree_main::betree::load_internal_node] -/
def betree_load_internal_node_fwd
  (id : UInt64) (st : State) :
  Result (State × (betree_list_t (UInt64 × betree_message_t)))
  :=
  opaque_defs.betree_utils_load_internal_node_fwd id st

/- [betree_main::betree::store_internal_node] -/
def betree_store_internal_node_fwd
  (id : UInt64) (content : betree_list_t (UInt64 × betree_message_t))
  (st : State) :
  Result (State × Unit)
  :=
  do
    let (st0, _) ←
      opaque_defs.betree_utils_store_internal_node_fwd id content st
    Result.ret (st0, ())

/- [betree_main::betree::load_leaf_node] -/
def betree_load_leaf_node_fwd
  (id : UInt64) (st : State) :
  Result (State × (betree_list_t (UInt64 × UInt64)))
  :=
  opaque_defs.betree_utils_load_leaf_node_fwd id st

/- [betree_main::betree::store_leaf_node] -/
def betree_store_leaf_node_fwd
  (id : UInt64) (content : betree_list_t (UInt64 × UInt64)) (st : State) :
  Result (State × Unit)
  :=
  do
    let (st0, _) ← opaque_defs.betree_utils_store_leaf_node_fwd id content st
    Result.ret (st0, ())

/- [betree_main::betree::fresh_node_id] -/
def betree_fresh_node_id_fwd (counter : UInt64) : Result UInt64 :=
  do
    let _ ← UInt64.checked_add counter (UInt64.ofNatCore 1 (by intlit))
    Result.ret counter

/- [betree_main::betree::fresh_node_id] -/
def betree_fresh_node_id_back (counter : UInt64) : Result UInt64 :=
  UInt64.checked_add counter (UInt64.ofNatCore 1 (by intlit))

/- [betree_main::betree::NodeIdCounter::{0}::new] -/
def betree_node_id_counter_new_fwd : Result betree_node_id_counter_t :=
  Result.ret
  { betree_node_id_counter_next_node_id := (UInt64.ofNatCore 0 (by intlit)) }

/- [betree_main::betree::NodeIdCounter::{0}::fresh_id] -/
def betree_node_id_counter_fresh_id_fwd
  (self : betree_node_id_counter_t) : Result UInt64 :=
  do
    let _ ← UInt64.checked_add self.betree_node_id_counter_next_node_id
      (UInt64.ofNatCore 1 (by intlit))
    Result.ret self.betree_node_id_counter_next_node_id

/- [betree_main::betree::NodeIdCounter::{0}::fresh_id] -/
def betree_node_id_counter_fresh_id_back
  (self : betree_node_id_counter_t) : Result betree_node_id_counter_t :=
  do
    let i ← UInt64.checked_add self.betree_node_id_counter_next_node_id
      (UInt64.ofNatCore 1 (by intlit))
    Result.ret { betree_node_id_counter_next_node_id := i }

/- [core::num::u64::{10}::MAX] -/
def core_num_u64_max_body : Result UInt64 :=
  Result.ret (UInt64.ofNatCore 18446744073709551615 (by intlit))
def core_num_u64_max_c : UInt64 := eval_global core_num_u64_max_body (by simp)

/- [betree_main::betree::upsert_update] -/
def betree_upsert_update_fwd
  (prev : Option UInt64) (st : betree_upsert_fun_state_t) : Result UInt64 :=
  match h: prev with
  | Option.none =>
    match h: st with
    | betree_upsert_fun_state_t.Add v => Result.ret v
    | betree_upsert_fun_state_t.Sub i =>
      Result.ret (UInt64.ofNatCore 0 (by intlit))
  | Option.some prev0 =>
    match h: st with
    | betree_upsert_fun_state_t.Add v =>
      do
        let margin ← UInt64.checked_sub core_num_u64_max_c prev0
        if h: margin >= v
        then UInt64.checked_add prev0 v
        else Result.ret core_num_u64_max_c
    | betree_upsert_fun_state_t.Sub v =>
      if h: prev0 >= v
      then UInt64.checked_sub prev0 v
      else Result.ret (UInt64.ofNatCore 0 (by intlit))

/- [betree_main::betree::List::{1}::len] -/
def betree_list_len_fwd
  (T : Type) (self : betree_list_t T) : (Result UInt64) :=
  match h: self with
  | betree_list_t.Cons t tl =>
    do
      let i ← betree_list_len_fwd T tl
      UInt64.checked_add (UInt64.ofNatCore 1 (by intlit)) i
  | betree_list_t.Nil => Result.ret (UInt64.ofNatCore 0 (by intlit))
termination_by betree_list_len_fwd self => betree_list_len_terminates T self
decreasing_by betree_list_len_decreases self

/- [betree_main::betree::List::{1}::split_at] -/
def betree_list_split_at_fwd
  (T : Type) (self : betree_list_t T) (n : UInt64) :
  (Result ((betree_list_t T) × (betree_list_t T)))
  :=
  if h: n = (UInt64.ofNatCore 0 (by intlit))
  then Result.ret (betree_list_t.Nil, self)
  else
    match h: self with
    | betree_list_t.Cons hd tl =>
      do
        let i ← UInt64.checked_sub n (UInt64.ofNatCore 1 (by intlit))
        let p ← betree_list_split_at_fwd T tl i
        let (ls0, ls1) := p
        let l := ls0
        Result.ret (betree_list_t.Cons hd l, ls1)
    | betree_list_t.Nil => Result.fail Error.panic
termination_by betree_list_split_at_fwd self n =>
  betree_list_split_at_terminates T self n
decreasing_by betree_list_split_at_decreases self n

/- [betree_main::betree::List::{1}::push_front] -/
def betree_list_push_front_fwd_back
  (T : Type) (self : betree_list_t T) (x : T) : Result (betree_list_t T) :=
  let tl := mem_replace_fwd (betree_list_t T) self betree_list_t.Nil
  let l := tl
  Result.ret (betree_list_t.Cons x l)

/- [betree_main::betree::List::{1}::pop_front] -/
def betree_list_pop_front_fwd (T : Type) (self : betree_list_t T) : Result T :=
  let ls := mem_replace_fwd (betree_list_t T) self betree_list_t.Nil
  match h: ls with
  | betree_list_t.Cons x tl => Result.ret x
  | betree_list_t.Nil => Result.fail Error.panic

/- [betree_main::betree::List::{1}::pop_front] -/
def betree_list_pop_front_back
  (T : Type) (self : betree_list_t T) : Result (betree_list_t T) :=
  let ls := mem_replace_fwd (betree_list_t T) self betree_list_t.Nil
  match h: ls with
  | betree_list_t.Cons x tl => Result.ret tl
  | betree_list_t.Nil => Result.fail Error.panic

/- [betree_main::betree::List::{1}::hd] -/
def betree_list_hd_fwd (T : Type) (self : betree_list_t T) : Result T :=
  match h: self with
  | betree_list_t.Cons hd l => Result.ret hd
  | betree_list_t.Nil => Result.fail Error.panic

/- [betree_main::betree::List::{2}::head_has_key] -/
def betree_list_head_has_key_fwd
  (T : Type) (self : betree_list_t (UInt64 × T)) (key : UInt64) :
  Result Bool
  :=
  match h: self with
  | betree_list_t.Cons hd l => let (i, _) := hd
                               Result.ret (i = key)
  | betree_list_t.Nil => Result.ret false

/- [betree_main::betree::List::{2}::partition_at_pivot] -/
def betree_list_partition_at_pivot_fwd
  (T : Type) (self : betree_list_t (UInt64 × T)) (pivot : UInt64) :
  (Result ((betree_list_t (UInt64 × T)) × (betree_list_t (UInt64 × T))))
  :=
  match h: self with
  | betree_list_t.Cons hd tl =>
    let (i, t) := hd
    if h: i >= pivot
    then Result.ret (betree_list_t.Nil, betree_list_t.Cons (i, t) tl)
    else
      do
        let p ← betree_list_partition_at_pivot_fwd T tl pivot
        let (ls0, ls1) := p
        let l := ls0
        Result.ret (betree_list_t.Cons (i, t) l, ls1)
  | betree_list_t.Nil => Result.ret (betree_list_t.Nil, betree_list_t.Nil)
termination_by betree_list_partition_at_pivot_fwd self pivot =>
  betree_list_partition_at_pivot_terminates T self pivot
decreasing_by betree_list_partition_at_pivot_decreases self pivot

/- [betree_main::betree::Leaf::{3}::split] -/
def betree_leaf_split_fwd
  (self : betree_leaf_t) (content : betree_list_t (UInt64 × UInt64))
  (params : betree_params_t) (node_id_cnt : betree_node_id_counter_t)
  (st : State) :
  Result (State × betree_internal_t)
  :=
  do
    let p ←
      betree_list_split_at_fwd (UInt64 × UInt64) content
        params.betree_params_split_size
    let (content0, content1) := p
    let p0 ← betree_list_hd_fwd (UInt64 × UInt64) content1
    let (pivot, _) := p0
    let id0 ← betree_node_id_counter_fresh_id_fwd node_id_cnt
    let node_id_cnt0 ← betree_node_id_counter_fresh_id_back node_id_cnt
    let id1 ← betree_node_id_counter_fresh_id_fwd node_id_cnt0
    let (st0, _) ← betree_store_leaf_node_fwd id0 content0 st
    let (st1, _) ← betree_store_leaf_node_fwd id1 content1 st0
    let n := betree_node_t.Leaf
      {
        betree_leaf_id := id0,
        betree_leaf_size := params.betree_params_split_size
      }
    let n0 := betree_node_t.Leaf
      {
        betree_leaf_id := id1,
        betree_leaf_size := params.betree_params_split_size
      }
    Result.ret (st1, mkbetree_internal_t self.betree_leaf_id pivot n n0)

/- [betree_main::betree::Leaf::{3}::split] -/
def betree_leaf_split_back
  (self : betree_leaf_t) (content : betree_list_t (UInt64 × UInt64))
  (params : betree_params_t) (node_id_cnt : betree_node_id_counter_t)
  (st : State) :
  Result betree_node_id_counter_t
  :=
  do
    let p ←
      betree_list_split_at_fwd (UInt64 × UInt64) content
        params.betree_params_split_size
    let (content0, content1) := p
    let _ ← betree_list_hd_fwd (UInt64 × UInt64) content1
    let id0 ← betree_node_id_counter_fresh_id_fwd node_id_cnt
    let node_id_cnt0 ← betree_node_id_counter_fresh_id_back node_id_cnt
    let id1 ← betree_node_id_counter_fresh_id_fwd node_id_cnt0
    let (st0, _) ← betree_store_leaf_node_fwd id0 content0 st
    let _ ← betree_store_leaf_node_fwd id1 content1 st0
    betree_node_id_counter_fresh_id_back node_id_cnt0

/- [betree_main::betree::Node::{5}::lookup_in_bindings] -/
def betree_node_lookup_in_bindings_fwd
  (key : UInt64) (bindings : betree_list_t (UInt64 × UInt64)) :
  (Result (Option UInt64))
  :=
  match h: bindings with
  | betree_list_t.Cons hd tl =>
    let (i, i0) := hd
    if h: i = key
    then Result.ret (Option.some i0)
    else
      if h: i > key
      then Result.ret Option.none
      else betree_node_lookup_in_bindings_fwd key tl
  | betree_list_t.Nil => Result.ret Option.none
termination_by betree_node_lookup_in_bindings_fwd key bindings =>
  betree_node_lookup_in_bindings_terminates key bindings
decreasing_by betree_node_lookup_in_bindings_decreases key bindings

/- [betree_main::betree::Node::{5}::lookup_first_message_for_key] -/
def betree_node_lookup_first_message_for_key_fwd
  (key : UInt64) (msgs : betree_list_t (UInt64 × betree_message_t)) :
  (Result (betree_list_t (UInt64 × betree_message_t)))
  :=
  match h: msgs with
  | betree_list_t.Cons x next_msgs =>
    let (i, m) := x
    if h: i >= key
    then Result.ret (betree_list_t.Cons (i, m) next_msgs)
    else betree_node_lookup_first_message_for_key_fwd key next_msgs
  | betree_list_t.Nil => Result.ret betree_list_t.Nil
termination_by betree_node_lookup_first_message_for_key_fwd key msgs =>
  betree_node_lookup_first_message_for_key_terminates key msgs
decreasing_by betree_node_lookup_first_message_for_key_decreases key msgs

/- [betree_main::betree::Node::{5}::lookup_first_message_for_key] -/
def betree_node_lookup_first_message_for_key_back
  (key : UInt64) (msgs : betree_list_t (UInt64 × betree_message_t))
  (ret0 : betree_list_t (UInt64 × betree_message_t)) :
  (Result (betree_list_t (UInt64 × betree_message_t)))
  :=
  match h: msgs with
  | betree_list_t.Cons x next_msgs =>
    let (i, m) := x
    if h: i >= key
    then Result.ret ret0
    else
      do
        let next_msgs0 ←
          betree_node_lookup_first_message_for_key_back key next_msgs ret0
        Result.ret (betree_list_t.Cons (i, m) next_msgs0)
  | betree_list_t.Nil => Result.ret ret0
termination_by betree_node_lookup_first_message_for_key_back key msgs ret0 =>
  betree_node_lookup_first_message_for_key_terminates key msgs
decreasing_by betree_node_lookup_first_message_for_key_decreases key msgs

/- [betree_main::betree::Node::{5}::apply_upserts] -/
def betree_node_apply_upserts_fwd
  (msgs : betree_list_t (UInt64 × betree_message_t)) (prev : Option UInt64)
  (key : UInt64) (st : State) :
  (Result (State × UInt64))
  :=
  do
    let b ← betree_list_head_has_key_fwd betree_message_t msgs key
    if h: b
    then
      do
        let msg ← betree_list_pop_front_fwd (UInt64 × betree_message_t) msgs
        let (_, m) := msg
        match h: m with
        | betree_message_t.Insert i => Result.fail Error.panic
        | betree_message_t.Delete => Result.fail Error.panic
        | betree_message_t.Upsert s =>
          do
            let v ← betree_upsert_update_fwd prev s
            let msgs0 ←
              betree_list_pop_front_back (UInt64 × betree_message_t) msgs
            betree_node_apply_upserts_fwd msgs0 (Option.some v) key st
    else
      do
        let (st0, v) ←
          opaque_defs.core_option_option_unwrap_fwd UInt64 prev st
        let _ ←
          betree_list_push_front_fwd_back (UInt64 × betree_message_t) msgs
            (key, betree_message_t.Insert v)
        Result.ret (st0, v)
termination_by betree_node_apply_upserts_fwd msgs prev key st =>
  betree_node_apply_upserts_terminates msgs prev key st
decreasing_by betree_node_apply_upserts_decreases msgs prev key st

/- [betree_main::betree::Node::{5}::apply_upserts] -/
def betree_node_apply_upserts_back
  (msgs : betree_list_t (UInt64 × betree_message_t)) (prev : Option UInt64)
  (key : UInt64) (st : State) :
  (Result (betree_list_t (UInt64 × betree_message_t)))
  :=
  do
    let b ← betree_list_head_has_key_fwd betree_message_t msgs key
    if h: b
    then
      do
        let msg ← betree_list_pop_front_fwd (UInt64 × betree_message_t) msgs
        let (_, m) := msg
        match h: m with
        | betree_message_t.Insert i => Result.fail Error.panic
        | betree_message_t.Delete => Result.fail Error.panic
        | betree_message_t.Upsert s =>
          do
            let v ← betree_upsert_update_fwd prev s
            let msgs0 ←
              betree_list_pop_front_back (UInt64 × betree_message_t) msgs
            betree_node_apply_upserts_back msgs0 (Option.some v) key st
    else
      do
        let (_, v) ← opaque_defs.core_option_option_unwrap_fwd UInt64 prev st
        betree_list_push_front_fwd_back (UInt64 × betree_message_t) msgs (key,
          betree_message_t.Insert v)
termination_by betree_node_apply_upserts_back msgs prev key st =>
  betree_node_apply_upserts_terminates msgs prev key st
decreasing_by betree_node_apply_upserts_decreases msgs prev key st

/- [betree_main::betree::Node::{5}::lookup] -/
mutual def betree_node_lookup_fwd
  (self : betree_node_t) (key : UInt64) (st : State) :
  (Result (State × (Option UInt64)))
  :=
  match h: self with
  | betree_node_t.Internal node =>
    do
      let (mkbetree_internal_t i i0 n n0) := node
      let (st0, msgs) ← betree_load_internal_node_fwd i st
      let pending ← betree_node_lookup_first_message_for_key_fwd key msgs
      match h: pending with
      | betree_list_t.Cons p l =>
        let (k, msg) := p
        if h: k != key
        then
          do
            let (st1, opt) ←
              betree_internal_lookup_in_children_fwd (mkbetree_internal_t i i0
                n n0) key st0
            let _ ←
              betree_node_lookup_first_message_for_key_back key msgs
                (betree_list_t.Cons (k, msg) l)
            Result.ret (st1, opt)
        else
          match h: msg with
          | betree_message_t.Insert v =>
            do
              let _ ←
                betree_node_lookup_first_message_for_key_back key msgs
                  (betree_list_t.Cons (k, betree_message_t.Insert v) l)
              Result.ret (st0, Option.some v)
          | betree_message_t.Delete =>
            do
              let _ ←
                betree_node_lookup_first_message_for_key_back key msgs
                  (betree_list_t.Cons (k, betree_message_t.Delete) l)
              Result.ret (st0, Option.none)
          | betree_message_t.Upsert ufs =>
            do
              let (st1, v) ←
                betree_internal_lookup_in_children_fwd (mkbetree_internal_t i
                  i0 n n0) key st0
              let (st2, v0) ←
                betree_node_apply_upserts_fwd (betree_list_t.Cons (k,
                  betree_message_t.Upsert ufs) l) v key st1
              let node0 ←
                betree_internal_lookup_in_children_back (mkbetree_internal_t i
                  i0 n n0) key st0
              let (mkbetree_internal_t i1 _ _ _) := node0
              let pending0 ←
                betree_node_apply_upserts_back (betree_list_t.Cons (k,
                  betree_message_t.Upsert ufs) l) v key st1
              let msgs0 ←
                betree_node_lookup_first_message_for_key_back key msgs pending0
              let (st3, _) ← betree_store_internal_node_fwd i1 msgs0 st2
              Result.ret (st3, Option.some v0)
      | betree_list_t.Nil =>
        do
          let (st1, opt) ←
            betree_internal_lookup_in_children_fwd (mkbetree_internal_t i i0 n
              n0) key st0
          let _ ←
            betree_node_lookup_first_message_for_key_back key msgs
              betree_list_t.Nil
          Result.ret (st1, opt)
  | betree_node_t.Leaf node =>
    do
      let (st0, bindings) ← betree_load_leaf_node_fwd node.betree_leaf_id st
      let opt ← betree_node_lookup_in_bindings_fwd key bindings
      Result.ret (st0, opt)
termination_by betree_node_lookup_fwd self key st =>
  betree_node_lookup_terminates self key st
decreasing_by betree_node_lookup_decreases self key st

/- [betree_main::betree::Node::{5}::lookup] -/
def betree_node_lookup_back
  (self : betree_node_t) (key : UInt64) (st : State) :
  (Result betree_node_t)
  :=
  match h: self with
  | betree_node_t.Internal node =>
    do
      let (mkbetree_internal_t i i0 n n0) := node
      let (st0, msgs) ← betree_load_internal_node_fwd i st
      let pending ← betree_node_lookup_first_message_for_key_fwd key msgs
      match h: pending with
      | betree_list_t.Cons p l =>
        let (k, msg) := p
        if h: k != key
        then
          do
            let _ ←
              betree_node_lookup_first_message_for_key_back key msgs
                (betree_list_t.Cons (k, msg) l)
            let node0 ←
              betree_internal_lookup_in_children_back (mkbetree_internal_t i i0
                n n0) key st0
            Result.ret (betree_node_t.Internal node0)
        else
          match h: msg with
          | betree_message_t.Insert v =>
            do
              let _ ←
                betree_node_lookup_first_message_for_key_back key msgs
                  (betree_list_t.Cons (k, betree_message_t.Insert v) l)
              Result.ret (betree_node_t.Internal (mkbetree_internal_t i i0 n
                n0))
          | betree_message_t.Delete =>
            do
              let _ ←
                betree_node_lookup_first_message_for_key_back key msgs
                  (betree_list_t.Cons (k, betree_message_t.Delete) l)
              Result.ret (betree_node_t.Internal (mkbetree_internal_t i i0 n
                n0))
          | betree_message_t.Upsert ufs =>
            do
              let (st1, v) ←
                betree_internal_lookup_in_children_fwd (mkbetree_internal_t i
                  i0 n n0) key st0
              let (st2, _) ←
                betree_node_apply_upserts_fwd (betree_list_t.Cons (k,
                  betree_message_t.Upsert ufs) l) v key st1
              let node0 ←
                betree_internal_lookup_in_children_back (mkbetree_internal_t i
                  i0 n n0) key st0
              let (mkbetree_internal_t i1 i2 n1 n2) := node0
              let pending0 ←
                betree_node_apply_upserts_back (betree_list_t.Cons (k,
                  betree_message_t.Upsert ufs) l) v key st1
              let msgs0 ←
                betree_node_lookup_first_message_for_key_back key msgs pending0
              let _ ← betree_store_internal_node_fwd i1 msgs0 st2
              Result.ret (betree_node_t.Internal (mkbetree_internal_t i1 i2 n1
                n2))
      | betree_list_t.Nil =>
        do
          let _ ←
            betree_node_lookup_first_message_for_key_back key msgs
              betree_list_t.Nil
          let node0 ←
            betree_internal_lookup_in_children_back (mkbetree_internal_t i i0 n
              n0) key st0
          Result.ret (betree_node_t.Internal node0)
  | betree_node_t.Leaf node =>
    do
      let (_, bindings) ← betree_load_leaf_node_fwd node.betree_leaf_id st
      let _ ← betree_node_lookup_in_bindings_fwd key bindings
      Result.ret (betree_node_t.Leaf node)
termination_by betree_node_lookup_back self key st =>
  betree_node_lookup_terminates self key st
decreasing_by betree_node_lookup_decreases self key st

/- [betree_main::betree::Internal::{4}::lookup_in_children] -/
def betree_internal_lookup_in_children_fwd
  (self : betree_internal_t) (key : UInt64) (st : State) :
  (Result (State × (Option UInt64)))
  :=
  let (mkbetree_internal_t _ i n n0) := self
  if h: key < i
  then betree_node_lookup_fwd n key st
  else betree_node_lookup_fwd n0 key st
termination_by betree_internal_lookup_in_children_fwd self key st =>
  betree_internal_lookup_in_children_terminates self key st
decreasing_by betree_internal_lookup_in_children_decreases self key st

/- [betree_main::betree::Internal::{4}::lookup_in_children] -/
def betree_internal_lookup_in_children_back
  (self : betree_internal_t) (key : UInt64) (st : State) :
  (Result betree_internal_t)
  :=
  let (mkbetree_internal_t i i0 n n0) := self
  if h: key < i0
  then
    do
      let n1 ← betree_node_lookup_back n key st
      Result.ret (mkbetree_internal_t i i0 n1 n0)
  else
    do
      let n1 ← betree_node_lookup_back n0 key st
      Result.ret (mkbetree_internal_t i i0 n n1)
termination_by betree_internal_lookup_in_children_back self key st =>
  betree_internal_lookup_in_children_terminates self key st
decreasing_by betree_internal_lookup_in_children_decreases self key st

/- [betree_main::betree::Node::{5}::lookup_mut_in_bindings] -/
def betree_node_lookup_mut_in_bindings_fwd
  (key : UInt64) (bindings : betree_list_t (UInt64 × UInt64)) :
  (Result (betree_list_t (UInt64 × UInt64)))
  :=
  match h: bindings with
  | betree_list_t.Cons hd tl =>
    let (i, i0) := hd
    if h: i >= key
    then Result.ret (betree_list_t.Cons (i, i0) tl)
    else betree_node_lookup_mut_in_bindings_fwd key tl
  | betree_list_t.Nil => Result.ret betree_list_t.Nil
termination_by betree_node_lookup_mut_in_bindings_fwd key bindings =>
  betree_node_lookup_mut_in_bindings_terminates key bindings
decreasing_by betree_node_lookup_mut_in_bindings_decreases key bindings

/- [betree_main::betree::Node::{5}::lookup_mut_in_bindings] -/
def betree_node_lookup_mut_in_bindings_back
  (key : UInt64) (bindings : betree_list_t (UInt64 × UInt64))
  (ret0 : betree_list_t (UInt64 × UInt64)) :
  (Result (betree_list_t (UInt64 × UInt64)))
  :=
  match h: bindings with
  | betree_list_t.Cons hd tl =>
    let (i, i0) := hd
    if h: i >= key
    then Result.ret ret0
    else
      do
        let tl0 ← betree_node_lookup_mut_in_bindings_back key tl ret0
        Result.ret (betree_list_t.Cons (i, i0) tl0)
  | betree_list_t.Nil => Result.ret ret0
termination_by betree_node_lookup_mut_in_bindings_back key bindings ret0 =>
  betree_node_lookup_mut_in_bindings_terminates key bindings
decreasing_by betree_node_lookup_mut_in_bindings_decreases key bindings

/- [betree_main::betree::Node::{5}::apply_to_leaf] -/
def betree_node_apply_to_leaf_fwd_back
  (bindings : betree_list_t (UInt64 × UInt64)) (key : UInt64)
  (new_msg : betree_message_t) :
  Result (betree_list_t (UInt64 × UInt64))
  :=
  do
    let bindings0 ← betree_node_lookup_mut_in_bindings_fwd key bindings
    let b ← betree_list_head_has_key_fwd UInt64 bindings0 key
    if h: b
    then
      do
        let hd ← betree_list_pop_front_fwd (UInt64 × UInt64) bindings0
        match h: new_msg with
        | betree_message_t.Insert v =>
          do
            let bindings1 ←
              betree_list_pop_front_back (UInt64 × UInt64) bindings0
            let bindings2 ←
              betree_list_push_front_fwd_back (UInt64 × UInt64) bindings1
                (key, v)
            betree_node_lookup_mut_in_bindings_back key bindings bindings2
        | betree_message_t.Delete =>
          do
            let bindings1 ←
              betree_list_pop_front_back (UInt64 × UInt64) bindings0
            betree_node_lookup_mut_in_bindings_back key bindings bindings1
        | betree_message_t.Upsert s =>
          do
            let (_, i) := hd
            let v ← betree_upsert_update_fwd (Option.some i) s
            let bindings1 ←
              betree_list_pop_front_back (UInt64 × UInt64) bindings0
            let bindings2 ←
              betree_list_push_front_fwd_back (UInt64 × UInt64) bindings1
                (key, v)
            betree_node_lookup_mut_in_bindings_back key bindings bindings2
    else
      match h: new_msg with
      | betree_message_t.Insert v =>
        do
          let bindings1 ←
            betree_list_push_front_fwd_back (UInt64 × UInt64) bindings0 (key,
              v)
          betree_node_lookup_mut_in_bindings_back key bindings bindings1
      | betree_message_t.Delete =>
        betree_node_lookup_mut_in_bindings_back key bindings bindings0
      | betree_message_t.Upsert s =>
        do
          let v ← betree_upsert_update_fwd Option.none s
          let bindings1 ←
            betree_list_push_front_fwd_back (UInt64 × UInt64) bindings0 (key,
              v)
          betree_node_lookup_mut_in_bindings_back key bindings bindings1

/- [betree_main::betree::Node::{5}::apply_messages_to_leaf] -/
def betree_node_apply_messages_to_leaf_fwd_back
  (bindings : betree_list_t (UInt64 × UInt64))
  (new_msgs : betree_list_t (UInt64 × betree_message_t)) :
  (Result (betree_list_t (UInt64 × UInt64)))
  :=
  match h: new_msgs with
  | betree_list_t.Cons new_msg new_msgs_tl =>
    do
      let (i, m) := new_msg
      let bindings0 ← betree_node_apply_to_leaf_fwd_back bindings i m
      betree_node_apply_messages_to_leaf_fwd_back bindings0 new_msgs_tl
  | betree_list_t.Nil => Result.ret bindings
termination_by betree_node_apply_messages_to_leaf_fwd_back bindings new_msgs =>
  betree_node_apply_messages_to_leaf_terminates bindings new_msgs
decreasing_by betree_node_apply_messages_to_leaf_decreases bindings new_msgs

/- [betree_main::betree::Node::{5}::filter_messages_for_key] -/
def betree_node_filter_messages_for_key_fwd_back
  (key : UInt64) (msgs : betree_list_t (UInt64 × betree_message_t)) :
  (Result (betree_list_t (UInt64 × betree_message_t)))
  :=
  match h: msgs with
  | betree_list_t.Cons p l =>
    let (k, m) := p
    if h: k = key
    then
      do
        let msgs0 ←
          betree_list_pop_front_back (UInt64 × betree_message_t)
            (betree_list_t.Cons (k, m) l)
        betree_node_filter_messages_for_key_fwd_back key msgs0
    else Result.ret (betree_list_t.Cons (k, m) l)
  | betree_list_t.Nil => Result.ret betree_list_t.Nil
termination_by betree_node_filter_messages_for_key_fwd_back key msgs =>
  betree_node_filter_messages_for_key_terminates key msgs
decreasing_by betree_node_filter_messages_for_key_decreases key msgs

/- [betree_main::betree::Node::{5}::lookup_first_message_after_key] -/
def betree_node_lookup_first_message_after_key_fwd
  (key : UInt64) (msgs : betree_list_t (UInt64 × betree_message_t)) :
  (Result (betree_list_t (UInt64 × betree_message_t)))
  :=
  match h: msgs with
  | betree_list_t.Cons p next_msgs =>
    let (k, m) := p
    if h: k = key
    then betree_node_lookup_first_message_after_key_fwd key next_msgs
    else Result.ret (betree_list_t.Cons (k, m) next_msgs)
  | betree_list_t.Nil => Result.ret betree_list_t.Nil
termination_by betree_node_lookup_first_message_after_key_fwd key msgs =>
  betree_node_lookup_first_message_after_key_terminates key msgs
decreasing_by betree_node_lookup_first_message_after_key_decreases key msgs

/- [betree_main::betree::Node::{5}::lookup_first_message_after_key] -/
def betree_node_lookup_first_message_after_key_back
  (key : UInt64) (msgs : betree_list_t (UInt64 × betree_message_t))
  (ret0 : betree_list_t (UInt64 × betree_message_t)) :
  (Result (betree_list_t (UInt64 × betree_message_t)))
  :=
  match h: msgs with
  | betree_list_t.Cons p next_msgs =>
    let (k, m) := p
    if h: k = key
    then
      do
        let next_msgs0 ←
          betree_node_lookup_first_message_after_key_back key next_msgs ret0
        Result.ret (betree_list_t.Cons (k, m) next_msgs0)
    else Result.ret ret0
  | betree_list_t.Nil => Result.ret ret0
termination_by betree_node_lookup_first_message_after_key_back key msgs ret0 =>
  betree_node_lookup_first_message_after_key_terminates key msgs
decreasing_by betree_node_lookup_first_message_after_key_decreases key msgs

/- [betree_main::betree::Node::{5}::apply_to_internal] -/
def betree_node_apply_to_internal_fwd_back
  (msgs : betree_list_t (UInt64 × betree_message_t)) (key : UInt64)
  (new_msg : betree_message_t) :
  Result (betree_list_t (UInt64 × betree_message_t))
  :=
  do
    let msgs0 ← betree_node_lookup_first_message_for_key_fwd key msgs
    let b ← betree_list_head_has_key_fwd betree_message_t msgs0 key
    if h: b
    then
      match h: new_msg with
      | betree_message_t.Insert i =>
        do
          let msgs1 ← betree_node_filter_messages_for_key_fwd_back key msgs0
          let msgs2 ←
            betree_list_push_front_fwd_back (UInt64 × betree_message_t) msgs1
              (key, betree_message_t.Insert i)
          betree_node_lookup_first_message_for_key_back key msgs msgs2
      | betree_message_t.Delete =>
        do
          let msgs1 ← betree_node_filter_messages_for_key_fwd_back key msgs0
          let msgs2 ←
            betree_list_push_front_fwd_back (UInt64 × betree_message_t) msgs1
              (key, betree_message_t.Delete)
          betree_node_lookup_first_message_for_key_back key msgs msgs2
      | betree_message_t.Upsert s =>
        do
          let p ← betree_list_hd_fwd (UInt64 × betree_message_t) msgs0
          let (_, m) := p
          match h: m with
          | betree_message_t.Insert prev =>
            do
              let v ← betree_upsert_update_fwd (Option.some prev) s
              let msgs1 ←
                betree_list_pop_front_back (UInt64 × betree_message_t) msgs0
              let msgs2 ←
                betree_list_push_front_fwd_back (UInt64 × betree_message_t)
                  msgs1 (key, betree_message_t.Insert v)
              betree_node_lookup_first_message_for_key_back key msgs msgs2
          | betree_message_t.Delete =>
            do
              let v ← betree_upsert_update_fwd Option.none s
              let msgs1 ←
                betree_list_pop_front_back (UInt64 × betree_message_t) msgs0
              let msgs2 ←
                betree_list_push_front_fwd_back (UInt64 × betree_message_t)
                  msgs1 (key, betree_message_t.Insert v)
              betree_node_lookup_first_message_for_key_back key msgs msgs2
          | betree_message_t.Upsert ufs =>
            do
              let msgs1 ←
                betree_node_lookup_first_message_after_key_fwd key msgs0
              let msgs2 ←
                betree_list_push_front_fwd_back (UInt64 × betree_message_t)
                  msgs1 (key, betree_message_t.Upsert s)
              let msgs3 ←
                betree_node_lookup_first_message_after_key_back key msgs0 msgs2
              betree_node_lookup_first_message_for_key_back key msgs msgs3
    else
      do
        let msgs1 ←
          betree_list_push_front_fwd_back (UInt64 × betree_message_t) msgs0
            (key, new_msg)
        betree_node_lookup_first_message_for_key_back key msgs msgs1

/- [betree_main::betree::Node::{5}::apply_messages_to_internal] -/
def betree_node_apply_messages_to_internal_fwd_back
  (msgs : betree_list_t (UInt64 × betree_message_t))
  (new_msgs : betree_list_t (UInt64 × betree_message_t)) :
  (Result (betree_list_t (UInt64 × betree_message_t)))
  :=
  match h: new_msgs with
  | betree_list_t.Cons new_msg new_msgs_tl =>
    do
      let (i, m) := new_msg
      let msgs0 ← betree_node_apply_to_internal_fwd_back msgs i m
      betree_node_apply_messages_to_internal_fwd_back msgs0 new_msgs_tl
  | betree_list_t.Nil => Result.ret msgs
termination_by betree_node_apply_messages_to_internal_fwd_back msgs new_msgs =>
  betree_node_apply_messages_to_internal_terminates msgs new_msgs
decreasing_by betree_node_apply_messages_to_internal_decreases msgs new_msgs

/- [betree_main::betree::Node::{5}::apply_messages] -/
mutual def 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 (UInt64 × betree_message_t)) (st : State) :
  (Result (State × Unit))
  :=
  match h: self with
  | betree_node_t.Internal node =>
    do
      let (mkbetree_internal_t i i0 n n0) := node
      let (st0, content) ← betree_load_internal_node_fwd i st
      let content0 ←
        betree_node_apply_messages_to_internal_fwd_back content msgs
      let num_msgs ←
        betree_list_len_fwd (UInt64 × betree_message_t) content0
      if h: num_msgs >= params.betree_params_min_flush_size
      then
        do
          let (st1, content1) ←
            betree_internal_flush_fwd (mkbetree_internal_t i i0 n n0) params
              node_id_cnt content0 st0
          let (node0, _) ←
            betree_internal_flush_back (mkbetree_internal_t i i0 n n0) params
              node_id_cnt content0 st0
          let (mkbetree_internal_t i1 _ _ _) := node0
          let (st2, _) ← betree_store_internal_node_fwd i1 content1 st1
          Result.ret (st2, ())
      else
        do
          let (st1, _) ← betree_store_internal_node_fwd i content0 st0
          Result.ret (st1, ())
  | betree_node_t.Leaf node =>
    do
      let (st0, content) ← betree_load_leaf_node_fwd node.betree_leaf_id st
      let content0 ← betree_node_apply_messages_to_leaf_fwd_back content msgs
      let len ← betree_list_len_fwd (UInt64 × UInt64) content0
      let i ← UInt64.checked_mul (UInt64.ofNatCore 2 (by intlit))
        params.betree_params_split_size
      if h: len >= i
      then
        do
          let (st1, _) ←
            betree_leaf_split_fwd node content0 params node_id_cnt st0
          let (st2, _) ←
            betree_store_leaf_node_fwd node.betree_leaf_id betree_list_t.Nil
              st1
          Result.ret (st2, ())
      else
        do
          let (st1, _) ←
            betree_store_leaf_node_fwd node.betree_leaf_id content0 st0
          Result.ret (st1, ())
termination_by betree_node_apply_messages_fwd self params node_id_cnt msgs st
  =>
  betree_node_apply_messages_terminates self params node_id_cnt msgs st
decreasing_by
  betree_node_apply_messages_decreases self params node_id_cnt msgs st

/- [betree_main::betree::Node::{5}::apply_messages] -/
def 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 (UInt64 × betree_message_t)) (st : State) :
  (Result (betree_node_t × betree_node_id_counter_t))
  :=
  match h: self with
  | betree_node_t.Internal node =>
    do
      let (mkbetree_internal_t i i0 n n0) := node
      let (st0, content) ← betree_load_internal_node_fwd i st
      let content0 ←
        betree_node_apply_messages_to_internal_fwd_back content msgs
      let num_msgs ←
        betree_list_len_fwd (UInt64 × betree_message_t) content0
      if h: num_msgs >= params.betree_params_min_flush_size
      then
        do
          let (st1, content1) ←
            betree_internal_flush_fwd (mkbetree_internal_t i i0 n n0) params
              node_id_cnt content0 st0
          let (node0, node_id_cnt0) ←
            betree_internal_flush_back (mkbetree_internal_t i i0 n n0) params
              node_id_cnt content0 st0
          let (mkbetree_internal_t i1 i2 n1 n2) := node0
          let _ ← betree_store_internal_node_fwd i1 content1 st1
          Result.ret (betree_node_t.Internal (mkbetree_internal_t i1 i2 n1 n2),
            node_id_cnt0)
      else
        do
          let _ ← betree_store_internal_node_fwd i content0 st0
          Result.ret (betree_node_t.Internal (mkbetree_internal_t i i0 n n0),
            node_id_cnt)
  | betree_node_t.Leaf node =>
    do
      let (st0, content) ← betree_load_leaf_node_fwd node.betree_leaf_id st
      let content0 ← betree_node_apply_messages_to_leaf_fwd_back content msgs
      let len ← betree_list_len_fwd (UInt64 × UInt64) content0
      let i ← UInt64.checked_mul (UInt64.ofNatCore 2 (by intlit))
        params.betree_params_split_size
      if h: len >= i
      then
        do
          let (st1, new_node) ←
            betree_leaf_split_fwd node content0 params node_id_cnt st0
          let _ ←
            betree_store_leaf_node_fwd node.betree_leaf_id betree_list_t.Nil
              st1
          let node_id_cnt0 ←
            betree_leaf_split_back node content0 params node_id_cnt st0
          Result.ret (betree_node_t.Internal new_node, node_id_cnt0)
      else
        do
          let _ ← betree_store_leaf_node_fwd node.betree_leaf_id content0 st0
          Result.ret (betree_node_t.Leaf { node with betree_leaf_size := len },
            node_id_cnt)
termination_by betree_node_apply_messages_back self params node_id_cnt msgs st
  =>
  betree_node_apply_messages_terminates self params node_id_cnt msgs st
decreasing_by
  betree_node_apply_messages_decreases self params node_id_cnt msgs st

/- [betree_main::betree::Internal::{4}::flush] -/
def betree_internal_flush_fwd
  (self : betree_internal_t) (params : betree_params_t)
  (node_id_cnt : betree_node_id_counter_t)
  (content : betree_list_t (UInt64 × betree_message_t)) (st : State) :
  (Result (State × (betree_list_t (UInt64 × betree_message_t))))
  :=
  do
    let (mkbetree_internal_t _ i n n0) := self
    let p ← betree_list_partition_at_pivot_fwd betree_message_t content i
    let (msgs_left, msgs_right) := p
    let len_left ← betree_list_len_fwd (UInt64 × betree_message_t) msgs_left
    if h: len_left >= params.betree_params_min_flush_size
    then
      do
        let (st0, _) ←
          betree_node_apply_messages_fwd n params node_id_cnt msgs_left st
        let (_, node_id_cnt0) ←
          betree_node_apply_messages_back n params node_id_cnt msgs_left st
        let len_right ←
          betree_list_len_fwd (UInt64 × betree_message_t) msgs_right
        if h: len_right >= params.betree_params_min_flush_size
        then
          do
            let (st1, _) ←
              betree_node_apply_messages_fwd n0 params node_id_cnt0 msgs_right
                st0
            let _ ←
              betree_node_apply_messages_back n0 params node_id_cnt0 msgs_right
                st0
            Result.ret (st1, betree_list_t.Nil)
        else Result.ret (st0, msgs_right)
    else
      do
        let (st0, _) ←
          betree_node_apply_messages_fwd n0 params node_id_cnt msgs_right st
        let _ ←
          betree_node_apply_messages_back n0 params node_id_cnt msgs_right st
        Result.ret (st0, msgs_left)
termination_by betree_internal_flush_fwd self params node_id_cnt content st =>
  betree_internal_flush_terminates self params node_id_cnt content st
decreasing_by
  betree_internal_flush_decreases self params node_id_cnt content st

/- [betree_main::betree::Internal::{4}::flush] -/
def betree_internal_flush_back
  (self : betree_internal_t) (params : betree_params_t)
  (node_id_cnt : betree_node_id_counter_t)
  (content : betree_list_t (UInt64 × betree_message_t)) (st : State) :
  (Result (betree_internal_t × betree_node_id_counter_t))
  :=
  do
    let (mkbetree_internal_t i i0 n n0) := self
    let p ← betree_list_partition_at_pivot_fwd betree_message_t content i0
    let (msgs_left, msgs_right) := p
    let len_left ← betree_list_len_fwd (UInt64 × betree_message_t) msgs_left
    if h: len_left >= params.betree_params_min_flush_size
    then
      do
        let (st0, _) ←
          betree_node_apply_messages_fwd n params node_id_cnt msgs_left st
        let (n1, node_id_cnt0) ←
          betree_node_apply_messages_back n params node_id_cnt msgs_left st
        let len_right ←
          betree_list_len_fwd (UInt64 × betree_message_t) msgs_right
        if h: len_right >= params.betree_params_min_flush_size
        then
          do
            let (n2, node_id_cnt1) ←
              betree_node_apply_messages_back n0 params node_id_cnt0 msgs_right
                st0
            Result.ret (mkbetree_internal_t i i0 n1 n2, node_id_cnt1)
        else Result.ret (mkbetree_internal_t i i0 n1 n0, node_id_cnt0)
    else
      do
        let (n1, node_id_cnt0) ←
          betree_node_apply_messages_back n0 params node_id_cnt msgs_right st
        Result.ret (mkbetree_internal_t i i0 n n1, node_id_cnt0)
termination_by betree_internal_flush_back self params node_id_cnt content st =>
  betree_internal_flush_terminates self params node_id_cnt content st
decreasing_by
  betree_internal_flush_decreases self params node_id_cnt content st

/- [betree_main::betree::Node::{5}::apply] -/
def betree_node_apply_fwd
  (self : betree_node_t) (params : betree_params_t)
  (node_id_cnt : betree_node_id_counter_t) (key : UInt64)
  (new_msg : betree_message_t) (st : State) :
  Result (State × Unit)
  :=
  do
    let l := betree_list_t.Nil
    let (st0, _) ←
      betree_node_apply_messages_fwd self params node_id_cnt
        (betree_list_t.Cons (key, new_msg) l) st
    let _ ←
      betree_node_apply_messages_back self params node_id_cnt
        (betree_list_t.Cons (key, new_msg) l) st
    Result.ret (st0, ())

/- [betree_main::betree::Node::{5}::apply] -/
def betree_node_apply_back
  (self : betree_node_t) (params : betree_params_t)
  (node_id_cnt : betree_node_id_counter_t) (key : UInt64)
  (new_msg : betree_message_t) (st : State) :
  Result (betree_node_t × betree_node_id_counter_t)
  :=
  let l := betree_list_t.Nil
  betree_node_apply_messages_back self params node_id_cnt (betree_list_t.Cons
    (key, new_msg) l) st

/- [betree_main::betree::BeTree::{6}::new] -/
def betree_be_tree_new_fwd
  (min_flush_size : UInt64) (split_size : UInt64) (st : State) :
  Result (State × betree_be_tree_t)
  :=
  do
    let node_id_cnt ← betree_node_id_counter_new_fwd
    let id ← betree_node_id_counter_fresh_id_fwd node_id_cnt
    let (st0, _) ← betree_store_leaf_node_fwd id betree_list_t.Nil st
    let node_id_cnt0 ← betree_node_id_counter_fresh_id_back node_id_cnt
    Result.ret (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 :=
          (betree_node_t.Leaf
            {
              betree_leaf_id := id,
              betree_leaf_size := (UInt64.ofNatCore 0 (by intlit))
            })
      })

/- [betree_main::betree::BeTree::{6}::apply] -/
def betree_be_tree_apply_fwd
  (self : betree_be_tree_t) (key : UInt64) (msg : betree_message_t)
  (st : State) :
  Result (State × Unit)
  :=
  do
    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
    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
    Result.ret (st0, ())

/- [betree_main::betree::BeTree::{6}::apply] -/
def betree_be_tree_apply_back
  (self : betree_be_tree_t) (key : UInt64) (msg : betree_message_t)
  (st : State) :
  Result betree_be_tree_t
  :=
  do
    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
    Result.ret
      { self with betree_be_tree_node_id_cnt := nic, betree_be_tree_root := n }

/- [betree_main::betree::BeTree::{6}::insert] -/
def betree_be_tree_insert_fwd
  (self : betree_be_tree_t) (key : UInt64) (value : UInt64) (st : State) :
  Result (State × Unit)
  :=
  do
    let (st0, _) ←
      betree_be_tree_apply_fwd self key (betree_message_t.Insert value) st
    let _ ←
      betree_be_tree_apply_back self key (betree_message_t.Insert value) st
    Result.ret (st0, ())

/- [betree_main::betree::BeTree::{6}::insert] -/
def betree_be_tree_insert_back
  (self : betree_be_tree_t) (key : UInt64) (value : UInt64) (st : State) :
  Result betree_be_tree_t
  :=
  betree_be_tree_apply_back self key (betree_message_t.Insert value) st

/- [betree_main::betree::BeTree::{6}::delete] -/
def betree_be_tree_delete_fwd
  (self : betree_be_tree_t) (key : UInt64) (st : State) :
  Result (State × Unit)
  :=
  do
    let (st0, _) ←
      betree_be_tree_apply_fwd self key betree_message_t.Delete st
    let _ ← betree_be_tree_apply_back self key betree_message_t.Delete st
    Result.ret (st0, ())

/- [betree_main::betree::BeTree::{6}::delete] -/
def betree_be_tree_delete_back
  (self : betree_be_tree_t) (key : UInt64) (st : State) :
  Result betree_be_tree_t
  :=
  betree_be_tree_apply_back self key betree_message_t.Delete st

/- [betree_main::betree::BeTree::{6}::upsert] -/
def betree_be_tree_upsert_fwd
  (self : betree_be_tree_t) (key : UInt64) (upd : betree_upsert_fun_state_t)
  (st : State) :
  Result (State × Unit)
  :=
  do
    let (st0, _) ←
      betree_be_tree_apply_fwd self key (betree_message_t.Upsert upd) st
    let _ ←
      betree_be_tree_apply_back self key (betree_message_t.Upsert upd) st
    Result.ret (st0, ())

/- [betree_main::betree::BeTree::{6}::upsert] -/
def betree_be_tree_upsert_back
  (self : betree_be_tree_t) (key : UInt64) (upd : betree_upsert_fun_state_t)
  (st : State) :
  Result betree_be_tree_t
  :=
  betree_be_tree_apply_back self key (betree_message_t.Upsert upd) st

/- [betree_main::betree::BeTree::{6}::lookup] -/
def betree_be_tree_lookup_fwd
  (self : betree_be_tree_t) (key : UInt64) (st : State) :
  Result (State × (Option UInt64))
  :=
  betree_node_lookup_fwd self.betree_be_tree_root key st

/- [betree_main::betree::BeTree::{6}::lookup] -/
def betree_be_tree_lookup_back
  (self : betree_be_tree_t) (key : UInt64) (st : State) :
  Result betree_be_tree_t
  :=
  do
    let n ← betree_node_lookup_back self.betree_be_tree_root key st
    Result.ret { self with betree_be_tree_root := n }

/- [betree_main::main] -/
def main_fwd : Result Unit :=
  Result.ret ()

/- Unit test for [betree_main::main] -/
#assert (main_fwd == .ret ())