Merge pull request #64 from AeneasVerif/son/merge_back

Merge the forward/backward functions

1 files changed, 1006 insertions, 0 deletions

diff --git a/tests/fstar-split/betree/BetreeMain.Funs.fst b/tests/fstar-split/betree/BetreeMain.Funs.fst
new file mode 100644
index 00000000..33133236
--- /dev/null
+++ b/tests/fstar-split/betree/BetreeMain.Funs.fst
@@ -0,0 +1,1006 @@
+(** THIS FILE WAS AUTOMATICALLY GENERATED BY AENEAS *)
+(** [betree_main]: function definitions *)
+module BetreeMain.Funs
+open Primitives
+include BetreeMain.Types
+include BetreeMain.FunsExternal
+include BetreeMain.Clauses
+
+#set-options "--z3rlimit 50 --fuel 1 --ifuel 1"
+
+(** [betree_main::betree::load_internal_node]: forward function
+    Source: 'src/betree.rs', lines 36:0-36:52 *)
+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
+    Source: 'src/betree.rs', lines 41:0-41:60 *)
+let betree_store_internal_node
+  (id : u64) (content : betree_List_t (u64 & betree_Message_t)) (st : state) :
+  result (state & unit)
+  =
+  let* (st1, _) = betree_utils_store_internal_node id content st in
+  Return (st1, ())
+
+(** [betree_main::betree::load_leaf_node]: forward function
+    Source: 'src/betree.rs', lines 46:0-46:44 *)
+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
+    Source: 'src/betree.rs', lines 51:0-51:52 *)
+let betree_store_leaf_node
+  (id : u64) (content : betree_List_t (u64 & u64)) (st : state) :
+  result (state & unit)
+  =
+  let* (st1, _) = betree_utils_store_leaf_node id content st in
+  Return (st1, ())
+
+(** [betree_main::betree::fresh_node_id]: forward function
+    Source: 'src/betree.rs', lines 55:0-55:48 *)
+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
+    Source: 'src/betree.rs', lines 55:0-55:48 *)
+let betree_fresh_node_id_back (counter : u64) : result u64 =
+  u64_add counter 1
+
+(** [betree_main::betree::{betree_main::betree::NodeIdCounter}::new]: forward function
+    Source: 'src/betree.rs', lines 206:4-206:20 *)
+let betree_NodeIdCounter_new : result betree_NodeIdCounter_t =
+  Return { next_node_id = 0 }
+
+(** [betree_main::betree::{betree_main::betree::NodeIdCounter}::fresh_id]: forward function
+    Source: 'src/betree.rs', lines 210:4-210:36 *)
+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::{betree_main::betree::NodeIdCounter}::fresh_id]: backward function 0
+    Source: 'src/betree.rs', lines 210:4-210:36 *)
+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
+    Source: 'src/betree.rs', lines 234:0-234:70 *)
+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 _ -> Return 0
+    end
+  | Some prev1 ->
+    begin match st with
+    | Betree_UpsertFunState_Add v ->
+      let* margin = u64_sub core_u64_max prev1 in
+      if margin >= v then u64_add prev1 v else Return core_u64_max
+    | Betree_UpsertFunState_Sub v ->
+      if prev1 >= v then u64_sub prev1 v else Return 0
+    end
+  end
+
+(** [betree_main::betree::{betree_main::betree::List<T>#1}::len]: forward function
+    Source: 'src/betree.rs', lines 276:4-276:24 *)
+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 _ tl -> let* i = betree_List_len t tl in u64_add 1 i
+  | Betree_List_Nil -> Return 0
+  end
+
+(** [betree_main::betree::{betree_main::betree::List<T>#1}::split_at]: forward function
+    Source: 'src/betree.rs', lines 284:4-284:51 *)
+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
+      Return (Betree_List_Cons hd ls0, ls1)
+    | Betree_List_Nil -> Fail Failure
+    end
+
+(** [betree_main::betree::{betree_main::betree::List<T>#1}::push_front]: merged forward/backward function
+    (there is a single backward function, and the forward function returns ())
+    Source: 'src/betree.rs', lines 299:4-299:34 *)
+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
+  Return (Betree_List_Cons x tl)
+
+(** [betree_main::betree::{betree_main::betree::List<T>#1}::pop_front]: forward function
+    Source: 'src/betree.rs', lines 306:4-306:32 *)
+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 _ -> Return x
+  | Betree_List_Nil -> Fail Failure
+  end
+
+(** [betree_main::betree::{betree_main::betree::List<T>#1}::pop_front]: backward function 0
+    Source: 'src/betree.rs', lines 306:4-306:32 *)
+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 _ tl -> Return tl
+  | Betree_List_Nil -> Fail Failure
+  end
+
+(** [betree_main::betree::{betree_main::betree::List<T>#1}::hd]: forward function
+    Source: 'src/betree.rs', lines 318:4-318:22 *)
+let betree_List_hd (t : Type0) (self : betree_List_t t) : result t =
+  begin match self with
+  | Betree_List_Cons hd _ -> Return hd
+  | Betree_List_Nil -> Fail Failure
+  end
+
+(** [betree_main::betree::{betree_main::betree::List<(u64, T)>#2}::head_has_key]: forward function
+    Source: 'src/betree.rs', lines 327:4-327:44 *)
+let betree_ListTupleU64T_head_has_key
+  (t : Type0) (self : betree_List_t (u64 & t)) (key : u64) : result bool =
+  begin match self with
+  | Betree_List_Cons hd _ -> let (i, _) = hd in Return (i = key)
+  | Betree_List_Nil -> Return false
+  end
+
+(** [betree_main::betree::{betree_main::betree::List<(u64, T)>#2}::partition_at_pivot]: forward function
+    Source: 'src/betree.rs', lines 339:4-339:73 *)
+let rec betree_ListTupleU64T_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_ListTupleU64T_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_ListTupleU64T_partition_at_pivot t tl pivot in
+      let (ls0, ls1) = p in
+      Return (Betree_List_Cons (i, x) ls0, ls1)
+  | Betree_List_Nil -> Return (Betree_List_Nil, Betree_List_Nil)
+  end
+
+(** [betree_main::betree::{betree_main::betree::Leaf#3}::split]: forward function
+    Source: 'src/betree.rs', lines 359:4-364:17 *)
+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* p1 = betree_List_hd (u64 & u64) content1 in
+  let (pivot, _) = p1 in
+  let* id0 = betree_NodeIdCounter_fresh_id node_id_cnt in
+  let* node_id_cnt1 = betree_NodeIdCounter_fresh_id_back node_id_cnt in
+  let* id1 = betree_NodeIdCounter_fresh_id node_id_cnt1 in
+  let* (st1, _) = betree_store_leaf_node id0 content0 st in
+  let* (st2, _) = betree_store_leaf_node id1 content1 st1 in
+  let n = Betree_Node_Leaf { id = id0; size = params.split_size } in
+  let n1 = Betree_Node_Leaf { id = id1; size = params.split_size } in
+  Return (st2, { id = self.id; pivot = pivot; left = n; right = n1 })
+
+(** [betree_main::betree::{betree_main::betree::Leaf#3}::split]: backward function 2
+    Source: 'src/betree.rs', lines 359:4-364:17 *)
+let betree_Leaf_split_back
+  (self : betree_Leaf_t) (content : betree_List_t (u64 & u64))
+  (params : betree_Params_t) (node_id_cnt : betree_NodeIdCounter_t)
+  (st : state) :
+  result 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_cnt1 = betree_NodeIdCounter_fresh_id_back node_id_cnt in
+  let* id1 = betree_NodeIdCounter_fresh_id node_id_cnt1 in
+  let* (st1, _) = betree_store_leaf_node id0 content0 st in
+  let* _ = betree_store_leaf_node id1 content1 st1 in
+  betree_NodeIdCounter_fresh_id_back node_id_cnt1
+
+(** [betree_main::betree::{betree_main::betree::Node#5}::lookup_first_message_for_key]: forward function
+    Source: 'src/betree.rs', lines 789:4-792:34 *)
+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::{betree_main::betree::Node#5}::lookup_first_message_for_key]: backward function 0
+    Source: 'src/betree.rs', lines 789:4-792:34 *)
+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_msgs1 =
+        betree_Node_lookup_first_message_for_key_back key next_msgs ret in
+      Return (Betree_List_Cons (i, m) next_msgs1)
+  | Betree_List_Nil -> Return ret
+  end
+
+(** [betree_main::betree::{betree_main::betree::Node#5}::lookup_in_bindings]: forward function
+    Source: 'src/betree.rs', lines 636:4-636:80 *)
+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, i1) = hd in
+    if i = key
+    then Return (Some i1)
+    else if i > key then Return None else betree_Node_lookup_in_bindings key tl
+  | Betree_List_Nil -> Return None
+  end
+
+(** [betree_main::betree::{betree_main::betree::Node#5}::apply_upserts]: forward function
+    Source: 'src/betree.rs', lines 819:4-819:90 *)
+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_ListTupleU64T_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 _ -> Fail Failure
+    | Betree_Message_Delete -> Fail Failure
+    | Betree_Message_Upsert s ->
+      let* v = betree_upsert_update prev s in
+      let* msgs1 = betree_List_pop_front_back (u64 & betree_Message_t) msgs in
+      betree_Node_apply_upserts msgs1 (Some v) key st
+    end
+  else
+    let* (st1, 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 (st1, v)
+
+(** [betree_main::betree::{betree_main::betree::Node#5}::apply_upserts]: backward function 0
+    Source: 'src/betree.rs', lines 819:4-819:90 *)
+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_ListTupleU64T_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 _ -> Fail Failure
+    | Betree_Message_Delete -> Fail Failure
+    | Betree_Message_Upsert s ->
+      let* v = betree_upsert_update prev s in
+      let* msgs1 = betree_List_pop_front_back (u64 & betree_Message_t) msgs in
+      betree_Node_apply_upserts_back msgs1 (Some v) key st
+    end
+  else
+    let* (_, v) = core_option_Option_unwrap u64 prev st in
+    betree_List_push_front (u64 & betree_Message_t) msgs (key,
+      Betree_Message_Insert v)
+
+(** [betree_main::betree::{betree_main::betree::Internal#4}::lookup_in_children]: forward function
+    Source: 'src/betree.rs', lines 395:4-395:63 *)
+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::{betree_main::betree::Internal#4}::lookup_in_children]: backward function 0
+    Source: 'src/betree.rs', lines 395:4-395:63 *)
+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.pivot
+  then
+    let* n = betree_Node_lookup_back self.left key st in
+    Return { self with left = n }
+  else
+    let* n = betree_Node_lookup_back self.right key st in
+    Return { self with right = n }
+
+(** [betree_main::betree::{betree_main::betree::Node#5}::lookup]: forward function
+    Source: 'src/betree.rs', lines 709:4-709:58 *)
+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* (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* (st2, o) = betree_Internal_lookup_in_children node key st1 in
+        let* _ =
+          betree_Node_lookup_first_message_for_key_back key msgs
+            (Betree_List_Cons (k, msg) l) in
+        Return (st2, 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 (st1, 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 (st1, None)
+        | Betree_Message_Upsert ufs ->
+          let* (st2, v) = betree_Internal_lookup_in_children node key st1 in
+          let* (st3, v1) =
+            betree_Node_apply_upserts (Betree_List_Cons (k,
+              Betree_Message_Upsert ufs) l) v key st2 in
+          let* node1 = betree_Internal_lookup_in_children_back node key st1 in
+          let* pending1 =
+            betree_Node_apply_upserts_back (Betree_List_Cons (k,
+              Betree_Message_Upsert ufs) l) v key st2 in
+          let* msgs1 =
+            betree_Node_lookup_first_message_for_key_back key msgs pending1 in
+          let* (st4, _) = betree_store_internal_node node1.id msgs1 st3 in
+          Return (st4, Some v1)
+        end
+    | Betree_List_Nil ->
+      let* (st2, o) = betree_Internal_lookup_in_children node key st1 in
+      let* _ =
+        betree_Node_lookup_first_message_for_key_back key msgs Betree_List_Nil
+        in
+      Return (st2, o)
+    end
+  | Betree_Node_Leaf node ->
+    let* (st1, bindings) = betree_load_leaf_node node.id st in
+    let* o = betree_Node_lookup_in_bindings key bindings in
+    Return (st1, o)
+  end
+
+(** [betree_main::betree::{betree_main::betree::Node#5}::lookup]: backward function 0
+    Source: 'src/betree.rs', lines 709:4-709:58 *)
+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
+  | 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* node1 = betree_Internal_lookup_in_children_back node key st1 in
+        Return (Betree_Node_Internal node1)
+      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 (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 (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* node1 = betree_Internal_lookup_in_children_back node key st1 in
+          let* pending1 =
+            betree_Node_apply_upserts_back (Betree_List_Cons (k,
+              Betree_Message_Upsert ufs) l) v key st2 in
+          let* msgs1 =
+            betree_Node_lookup_first_message_for_key_back key msgs pending1 in
+          let* _ = betree_store_internal_node node1.id msgs1 st3 in
+          Return (Betree_Node_Internal node1)
+        end
+    | Betree_List_Nil ->
+      let* _ =
+        betree_Node_lookup_first_message_for_key_back key msgs Betree_List_Nil
+        in
+      let* node1 = betree_Internal_lookup_in_children_back node key st1 in
+      Return (Betree_Node_Internal node1)
+    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 (Betree_Node_Leaf node)
+  end
+
+(** [betree_main::betree::{betree_main::betree::Node#5}::filter_messages_for_key]: merged forward/backward function
+    (there is a single backward function, and the forward function returns ())
+    Source: 'src/betree.rs', lines 674:4-674:77 *)
+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* msgs1 =
+        betree_List_pop_front_back (u64 & betree_Message_t) (Betree_List_Cons
+          (k, m) l) in
+      betree_Node_filter_messages_for_key key msgs1
+    else Return (Betree_List_Cons (k, m) l)
+  | Betree_List_Nil -> Return Betree_List_Nil
+  end
+
+(** [betree_main::betree::{betree_main::betree::Node#5}::lookup_first_message_after_key]: forward function
+    Source: 'src/betree.rs', lines 689:4-692:34 *)
+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::{betree_main::betree::Node#5}::lookup_first_message_after_key]: backward function 0
+    Source: 'src/betree.rs', lines 689:4-692:34 *)
+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_msgs1 =
+        betree_Node_lookup_first_message_after_key_back key next_msgs ret in
+      Return (Betree_List_Cons (k, m) next_msgs1)
+    else Return ret
+  | Betree_List_Nil -> Return ret
+  end
+
+(** [betree_main::betree::{betree_main::betree::Node#5}::apply_to_internal]: merged forward/backward function
+    (there is a single backward function, and the forward function returns ())
+    Source: 'src/betree.rs', lines 521:4-521:89 *)
+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* msgs1 = betree_Node_lookup_first_message_for_key key msgs in
+  let* b = betree_ListTupleU64T_head_has_key betree_Message_t msgs1 key in
+  if b
+  then
+    begin match new_msg with
+    | Betree_Message_Insert i ->
+      let* msgs2 = betree_Node_filter_messages_for_key key msgs1 in
+      let* msgs3 =
+        betree_List_push_front (u64 & betree_Message_t) msgs2 (key,
+          Betree_Message_Insert i) in
+      betree_Node_lookup_first_message_for_key_back key msgs msgs3
+    | Betree_Message_Delete ->
+      let* msgs2 = betree_Node_filter_messages_for_key key msgs1 in
+      let* msgs3 =
+        betree_List_push_front (u64 & betree_Message_t) msgs2 (key,
+          Betree_Message_Delete) in
+      betree_Node_lookup_first_message_for_key_back key msgs msgs3
+    | Betree_Message_Upsert s ->
+      let* p = betree_List_hd (u64 & betree_Message_t) msgs1 in
+      let (_, m) = p in
+      begin match m with
+      | Betree_Message_Insert prev ->
+        let* v = betree_upsert_update (Some prev) s in
+        let* msgs2 = betree_List_pop_front_back (u64 & betree_Message_t) msgs1
+          in
+        let* msgs3 =
+          betree_List_push_front (u64 & betree_Message_t) msgs2 (key,
+            Betree_Message_Insert v) in
+        betree_Node_lookup_first_message_for_key_back key msgs msgs3
+      | Betree_Message_Delete ->
+        let* v = betree_upsert_update None s in
+        let* msgs2 = betree_List_pop_front_back (u64 & betree_Message_t) msgs1
+          in
+        let* msgs3 =
+          betree_List_push_front (u64 & betree_Message_t) msgs2 (key,
+            Betree_Message_Insert v) in
+        betree_Node_lookup_first_message_for_key_back key msgs msgs3
+      | Betree_Message_Upsert _ ->
+        let* msgs2 = betree_Node_lookup_first_message_after_key key msgs1 in
+        let* msgs3 =
+          betree_List_push_front (u64 & betree_Message_t) msgs2 (key,
+            Betree_Message_Upsert s) in
+        let* msgs4 =
+          betree_Node_lookup_first_message_after_key_back key msgs1 msgs3 in
+        betree_Node_lookup_first_message_for_key_back key msgs msgs4
+      end
+    end
+  else
+    let* msgs2 =
+      betree_List_push_front (u64 & betree_Message_t) msgs1 (key, new_msg) in
+    betree_Node_lookup_first_message_for_key_back key msgs msgs2
+
+(** [betree_main::betree::{betree_main::betree::Node#5}::apply_messages_to_internal]: merged forward/backward function
+    (there is a single backward function, and the forward function returns ())
+    Source: 'src/betree.rs', lines 502:4-505:5 *)
+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* msgs1 = betree_Node_apply_to_internal msgs i m in
+    betree_Node_apply_messages_to_internal msgs1 new_msgs_tl
+  | Betree_List_Nil -> Return msgs
+  end
+
+(** [betree_main::betree::{betree_main::betree::Node#5}::lookup_mut_in_bindings]: forward function
+    Source: 'src/betree.rs', lines 653:4-656:32 *)
+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, i1) = hd in
+    if i >= key
+    then Return (Betree_List_Cons (i, i1) tl)
+    else betree_Node_lookup_mut_in_bindings key tl
+  | Betree_List_Nil -> Return Betree_List_Nil
+  end
+
+(** [betree_main::betree::{betree_main::betree::Node#5}::lookup_mut_in_bindings]: backward function 0
+    Source: 'src/betree.rs', lines 653:4-656:32 *)
+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, i1) = hd in
+    if i >= key
+    then Return ret
+    else
+      let* tl1 = betree_Node_lookup_mut_in_bindings_back key tl ret in
+      Return (Betree_List_Cons (i, i1) tl1)
+  | Betree_List_Nil -> Return ret
+  end
+
+(** [betree_main::betree::{betree_main::betree::Node#5}::apply_to_leaf]: merged forward/backward function
+    (there is a single backward function, and the forward function returns ())
+    Source: 'src/betree.rs', lines 460:4-460:87 *)
+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* bindings1 = betree_Node_lookup_mut_in_bindings key bindings in
+  let* b = betree_ListTupleU64T_head_has_key u64 bindings1 key in
+  if b
+  then
+    let* hd = betree_List_pop_front (u64 & u64) bindings1 in
+    begin match new_msg with
+    | Betree_Message_Insert v ->
+      let* bindings2 = betree_List_pop_front_back (u64 & u64) bindings1 in
+      let* bindings3 = betree_List_push_front (u64 & u64) bindings2 (key, v) in
+      betree_Node_lookup_mut_in_bindings_back key bindings bindings3
+    | Betree_Message_Delete ->
+      let* bindings2 = betree_List_pop_front_back (u64 & u64) bindings1 in
+      betree_Node_lookup_mut_in_bindings_back key bindings bindings2
+    | Betree_Message_Upsert s ->
+      let (_, i) = hd in
+      let* v = betree_upsert_update (Some i) s in
+      let* bindings2 = betree_List_pop_front_back (u64 & u64) bindings1 in
+      let* bindings3 = betree_List_push_front (u64 & u64) bindings2 (key, v) in
+      betree_Node_lookup_mut_in_bindings_back key bindings bindings3
+    end
+  else
+    begin match new_msg with
+    | Betree_Message_Insert v ->
+      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 ->
+      betree_Node_lookup_mut_in_bindings_back key bindings bindings1
+    | Betree_Message_Upsert s ->
+      let* v = betree_upsert_update None s in
+      let* bindings2 = betree_List_push_front (u64 & u64) bindings1 (key, v) in
+      betree_Node_lookup_mut_in_bindings_back key bindings bindings2
+    end
+
+(** [betree_main::betree::{betree_main::betree::Node#5}::apply_messages_to_leaf]: merged forward/backward function
+    (there is a single backward function, and the forward function returns ())
+    Source: 'src/betree.rs', lines 444:4-447:5 *)
+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* bindings1 = betree_Node_apply_to_leaf bindings i m in
+    betree_Node_apply_messages_to_leaf bindings1 new_msgs_tl
+  | Betree_List_Nil -> Return bindings
+  end
+
+(** [betree_main::betree::{betree_main::betree::Internal#4}::flush]: forward function
+    Source: 'src/betree.rs', lines 410:4-415:26 *)
+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_ListTupleU64T_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* (_, node_id_cnt1) =
+      betree_Node_apply_messages_back self.left params node_id_cnt msgs_left st
+      in
+    let* len_right = betree_List_len (u64 & betree_Message_t) msgs_right in
+    if len_right >= params.min_flush_size
+    then
+      let* (st2, _) =
+        betree_Node_apply_messages self.right params node_id_cnt1 msgs_right
+          st1 in
+      let* _ =
+        betree_Node_apply_messages_back self.right params node_id_cnt1
+          msgs_right st1 in
+      Return (st2, Betree_List_Nil)
+    else Return (st1, msgs_right)
+  else
+    let* (st1, _) =
+      betree_Node_apply_messages self.right params node_id_cnt msgs_right st in
+    let* _ =
+      betree_Node_apply_messages_back self.right params node_id_cnt msgs_right
+        st in
+    Return (st1, msgs_left)
+
+(** [betree_main::betree::{betree_main::betree::Internal#4}::flush]: backward function 0
+    Source: 'src/betree.rs', lines 410:4-415:26 *)
+and betree_Internal_flush_back
+  (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 (betree_Internal_t & betree_NodeIdCounter_t))
+  (decreases (
+    betree_Internal_flush_decreases self params node_id_cnt content st))
+  =
+  let* p =
+    betree_ListTupleU64T_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* (n, node_id_cnt1) =
+      betree_Node_apply_messages_back self.left params node_id_cnt msgs_left st
+      in
+    let* len_right = betree_List_len (u64 & betree_Message_t) msgs_right in
+    if len_right >= params.min_flush_size
+    then
+      let* (n1, node_id_cnt2) =
+        betree_Node_apply_messages_back self.right params node_id_cnt1
+          msgs_right st1 in
+      Return ({ self with left = n; right = n1 }, node_id_cnt2)
+    else Return ({ self with left = n }, node_id_cnt1)
+  else
+    let* (n, node_id_cnt1) =
+      betree_Node_apply_messages_back self.right params node_id_cnt msgs_right
+        st in
+    Return ({ self with right = n }, node_id_cnt1)
+
+(** [betree_main::betree::{betree_main::betree::Node#5}::apply_messages]: forward function
+    Source: 'src/betree.rs', lines 588:4-593:5 *)
+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* (st1, content) = betree_load_internal_node node.id st in
+    let* content1 = betree_Node_apply_messages_to_internal content msgs in
+    let* num_msgs = betree_List_len (u64 & betree_Message_t) content1 in
+    if num_msgs >= params.min_flush_size
+    then
+      let* (st2, content2) =
+        betree_Internal_flush node params node_id_cnt content1 st1 in
+      let* (node1, _) =
+        betree_Internal_flush_back node params node_id_cnt content1 st1 in
+      let* (st3, _) = betree_store_internal_node node1.id content2 st2 in
+      Return (st3, ())
+    else
+      let* (st2, _) = betree_store_internal_node node.id content1 st1 in
+      Return (st2, ())
+  | Betree_Node_Leaf node ->
+    let* (st1, content) = betree_load_leaf_node node.id st in
+    let* content1 = betree_Node_apply_messages_to_leaf content msgs in
+    let* len = betree_List_len (u64 & u64) content1 in
+    let* i = u64_mul 2 params.split_size in
+    if len >= i
+    then
+      let* (st2, _) = betree_Leaf_split node content1 params node_id_cnt st1 in
+      let* (st3, _) = betree_store_leaf_node node.id Betree_List_Nil st2 in
+      Return (st3, ())
+    else
+      let* (st2, _) = betree_store_leaf_node node.id content1 st1 in
+      Return (st2, ())
+  end
+
+(** [betree_main::betree::{betree_main::betree::Node#5}::apply_messages]: backward function 0
+    Source: 'src/betree.rs', lines 588:4-593:5 *)
+and betree_Node_apply_messages_back
+  (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 (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* content1 = betree_Node_apply_messages_to_internal content msgs in
+    let* num_msgs = betree_List_len (u64 & betree_Message_t) content1 in
+    if num_msgs >= params.min_flush_size
+    then
+      let* (st2, content2) =
+        betree_Internal_flush node params node_id_cnt content1 st1 in
+      let* (node1, node_id_cnt1) =
+        betree_Internal_flush_back node params node_id_cnt content1 st1 in
+      let* _ = betree_store_internal_node node1.id content2 st2 in
+      Return (Betree_Node_Internal node1, node_id_cnt1)
+    else
+      let* _ = betree_store_internal_node node.id content1 st1 in
+      Return (Betree_Node_Internal node, node_id_cnt)
+  | Betree_Node_Leaf node ->
+    let* (st1, content) = betree_load_leaf_node node.id st in
+    let* content1 = betree_Node_apply_messages_to_leaf content msgs in
+    let* len = betree_List_len (u64 & u64) content1 in
+    let* i = u64_mul 2 params.split_size in
+    if len >= i
+    then
+      let* (st2, new_node) =
+        betree_Leaf_split node content1 params node_id_cnt st1 in
+      let* _ = betree_store_leaf_node node.id Betree_List_Nil st2 in
+      let* node_id_cnt1 =
+        betree_Leaf_split_back node content1 params node_id_cnt st1 in
+      Return (Betree_Node_Internal new_node, node_id_cnt1)
+    else
+      let* _ = betree_store_leaf_node node.id content1 st1 in
+      Return (Betree_Node_Leaf { node with size = len }, node_id_cnt)
+  end
+
+(** [betree_main::betree::{betree_main::betree::Node#5}::apply]: forward function
+    Source: 'src/betree.rs', lines 576:4-582:5 *)
+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* (st1, _) =
+    betree_Node_apply_messages self params node_id_cnt (Betree_List_Cons (key,
+      new_msg) Betree_List_Nil) st in
+  let* _ =
+    betree_Node_apply_messages_back self params node_id_cnt (Betree_List_Cons
+      (key, new_msg) Betree_List_Nil) st in
+  Return (st1, ())
+
+(** [betree_main::betree::{betree_main::betree::Node#5}::apply]: backward function 0
+    Source: 'src/betree.rs', lines 576:4-582:5 *)
+let betree_Node_apply_back
+  (self : betree_Node_t) (params : betree_Params_t)
+  (node_id_cnt : betree_NodeIdCounter_t) (key : u64)
+  (new_msg : betree_Message_t) (st : state) :
+  result (betree_Node_t & betree_NodeIdCounter_t)
+  =
+  betree_Node_apply_messages_back self params node_id_cnt (Betree_List_Cons
+    (key, new_msg) Betree_List_Nil) st
+
+(** [betree_main::betree::{betree_main::betree::BeTree#6}::new]: forward function
+    Source: 'src/betree.rs', lines 849:4-849:60 *)
+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* (st1, _) = betree_store_leaf_node id Betree_List_Nil st in
+  let* node_id_cnt1 = betree_NodeIdCounter_fresh_id_back node_id_cnt in
+  Return (st1,
+    {
+      params = { min_flush_size = min_flush_size; split_size = split_size };
+      node_id_cnt = node_id_cnt1;
+      root = (Betree_Node_Leaf { id = id; size = 0 })
+    })
+
+(** [betree_main::betree::{betree_main::betree::BeTree#6}::apply]: forward function
+    Source: 'src/betree.rs', lines 868:4-868:47 *)
+let betree_BeTree_apply
+  (self : betree_BeTree_t) (key : u64) (msg : betree_Message_t) (st : state) :
+  result (state & unit)
+  =
+  let* (st1, _) =
+    betree_Node_apply self.root self.params self.node_id_cnt key msg st in
+  let* _ =
+    betree_Node_apply_back self.root self.params self.node_id_cnt key msg st in
+  Return (st1, ())
+
+(** [betree_main::betree::{betree_main::betree::BeTree#6}::apply]: backward function 0
+    Source: 'src/betree.rs', lines 868:4-868:47 *)
+let betree_BeTree_apply_back
+  (self : betree_BeTree_t) (key : u64) (msg : betree_Message_t) (st : state) :
+  result betree_BeTree_t
+  =
+  let* (n, nic) =
+    betree_Node_apply_back self.root self.params self.node_id_cnt key msg st in
+  Return { self with node_id_cnt = nic; root = n }
+
+(** [betree_main::betree::{betree_main::betree::BeTree#6}::insert]: forward function
+    Source: 'src/betree.rs', lines 874:4-874:52 *)
+let betree_BeTree_insert
+  (self : betree_BeTree_t) (key : u64) (value : u64) (st : state) :
+  result (state & unit)
+  =
+  let* (st1, _) = betree_BeTree_apply self key (Betree_Message_Insert value) st
+    in
+  let* _ = betree_BeTree_apply_back self key (Betree_Message_Insert value) st
+    in
+  Return (st1, ())
+
+(** [betree_main::betree::{betree_main::betree::BeTree#6}::insert]: backward function 0
+    Source: 'src/betree.rs', lines 874:4-874:52 *)
+let betree_BeTree_insert_back
+  (self : betree_BeTree_t) (key : u64) (value : u64) (st : state) :
+  result betree_BeTree_t
+  =
+  betree_BeTree_apply_back self key (Betree_Message_Insert value) st
+
+(** [betree_main::betree::{betree_main::betree::BeTree#6}::delete]: forward function
+    Source: 'src/betree.rs', lines 880:4-880:38 *)
+let betree_BeTree_delete
+  (self : betree_BeTree_t) (key : u64) (st : state) : result (state & unit) =
+  let* (st1, _) = betree_BeTree_apply self key Betree_Message_Delete st in
+  let* _ = betree_BeTree_apply_back self key Betree_Message_Delete st in
+  Return (st1, ())
+
+(** [betree_main::betree::{betree_main::betree::BeTree#6}::delete]: backward function 0
+    Source: 'src/betree.rs', lines 880:4-880:38 *)
+let betree_BeTree_delete_back
+  (self : betree_BeTree_t) (key : u64) (st : state) : result betree_BeTree_t =
+  betree_BeTree_apply_back self key Betree_Message_Delete st
+
+(** [betree_main::betree::{betree_main::betree::BeTree#6}::upsert]: forward function
+    Source: 'src/betree.rs', lines 886:4-886:59 *)
+let betree_BeTree_upsert
+  (self : betree_BeTree_t) (key : u64) (upd : betree_UpsertFunState_t)
+  (st : state) :
+  result (state & unit)
+  =
+  let* (st1, _) = betree_BeTree_apply self key (Betree_Message_Upsert upd) st
+    in
+  let* _ = betree_BeTree_apply_back self key (Betree_Message_Upsert upd) st in
+  Return (st1, ())
+
+(** [betree_main::betree::{betree_main::betree::BeTree#6}::upsert]: backward function 0
+    Source: 'src/betree.rs', lines 886:4-886:59 *)
+let betree_BeTree_upsert_back
+  (self : betree_BeTree_t) (key : u64) (upd : betree_UpsertFunState_t)
+  (st : state) :
+  result betree_BeTree_t
+  =
+  betree_BeTree_apply_back self key (Betree_Message_Upsert upd) st
+
+(** [betree_main::betree::{betree_main::betree::BeTree#6}::lookup]: forward function
+    Source: 'src/betree.rs', lines 895:4-895:62 *)
+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_main::betree::BeTree#6}::lookup]: backward function 0
+    Source: 'src/betree.rs', lines 895:4-895:62 *)
+let betree_BeTree_lookup_back
+  (self : betree_BeTree_t) (key : u64) (st : state) : result betree_BeTree_t =
+  let* n = betree_Node_lookup_back self.root key st in
+  Return { self with root = n }
+
+(** [betree_main::main]: forward function
+    Source: 'src/betree_main.rs', lines 5:0-5:9 *)
+let main : result unit =
+  Return ()
+
+(** Unit test for [betree_main::main] *)
+let _ = assert_norm (main = Return ())
+