Implement the generation of stateful backward functions (controlled by an option)

9 files changed, 2870 insertions, 25 deletions

diff --git a/tests/betree_back_stateful/BetreeMain.Clauses.Template.fst b/tests/betree_back_stateful/BetreeMain.Clauses.Template.fst
new file mode 100644
index 00000000..d48213d3
--- /dev/null
+++ b/tests/betree_back_stateful/BetreeMain.Clauses.Template.fst
@@ -0,0 +1,106 @@
+(** THIS FILE WAS AUTOMATICALLY GENERATED BY AENEAS *)
+(** [betree_main]: templates for the decreases clauses *)
+module BetreeMain.Clauses.Template
+open Primitives
+open BetreeMain.Types
+
+#set-options "--z3rlimit 50 --fuel 1 --ifuel 1"
+
+(** [core::num::u64::{10}::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::List::{1}::len]: decreases clause *)
+unfold
+let betree_list_len_decreases (t : Type0) (self : betree_list_t t) : nat =
+  admit ()
+
+(** [betree_main::betree::List::{1}::split_at]: decreases clause *)
+unfold
+let betree_list_split_at_decreases (t : Type0) (self : betree_list_t t)
+  (n : u64) : nat =
+  admit ()
+
+(** [betree_main::betree::List::{2}::partition_at_pivot]: decreases clause *)
+unfold
+let betree_list_partition_at_pivot_decreases (t : Type0)
+  (self : betree_list_t (u64 & t)) (pivot : u64) : nat =
+  admit ()
+
+(** [betree_main::betree::Node::{5}::lookup_in_bindings]: decreases clause *)
+unfold
+let betree_node_lookup_in_bindings_decreases (key : u64)
+  (bindings : betree_list_t (u64 & u64)) : nat =
+  admit ()
+
+(** [betree_main::betree::Node::{5}::lookup_first_message_for_key]: decreases clause *)
+unfold
+let betree_node_lookup_first_message_for_key_decreases (key : u64)
+  (msgs : betree_list_t (u64 & betree_message_t)) : nat =
+  admit ()
+
+(** [betree_main::betree::Node::{5}::apply_upserts]: decreases clause *)
+unfold
+let betree_node_apply_upserts_decreases
+  (msgs : betree_list_t (u64 & betree_message_t)) (prev : option u64)
+  (key : u64) (st : state) : nat =
+  admit ()
+
+(** [betree_main::betree::Node::{5}::lookup]: decreases clause *)
+unfold
+let betree_node_lookup_decreases (self : betree_node_t) (key : u64)
+  (st : state) : nat =
+  admit ()
+
+(** [betree_main::betree::Internal::{4}::lookup_in_children]: decreases clause *)
+unfold
+let betree_internal_lookup_in_children_decreases (self : betree_internal_t)
+  (key : u64) (st : state) : nat =
+  admit ()
+
+(** [betree_main::betree::Node::{5}::lookup_mut_in_bindings]: decreases clause *)
+unfold
+let betree_node_lookup_mut_in_bindings_decreases (key : u64)
+  (bindings : betree_list_t (u64 & u64)) : nat =
+  admit ()
+
+(** [betree_main::betree::Node::{5}::apply_messages_to_leaf]: decreases clause *)
+unfold
+let betree_node_apply_messages_to_leaf_decreases
+  (bindings : betree_list_t (u64 & u64))
+  (new_msgs : betree_list_t (u64 & betree_message_t)) : nat =
+  admit ()
+
+(** [betree_main::betree::Node::{5}::filter_messages_for_key]: decreases clause *)
+unfold
+let betree_node_filter_messages_for_key_decreases (key : u64)
+  (msgs : betree_list_t (u64 & betree_message_t)) : nat =
+  admit ()
+
+(** [betree_main::betree::Node::{5}::lookup_first_message_after_key]: decreases clause *)
+unfold
+let betree_node_lookup_first_message_after_key_decreases (key : u64)
+  (msgs : betree_list_t (u64 & betree_message_t)) : nat =
+  admit ()
+
+(** [betree_main::betree::Node::{5}::apply_messages_to_internal]: decreases clause *)
+unfold
+let betree_node_apply_messages_to_internal_decreases
+  (msgs : betree_list_t (u64 & betree_message_t))
+  (new_msgs : betree_list_t (u64 & betree_message_t)) : nat =
+  admit ()
+
+(** [betree_main::betree::Node::{5}::apply_messages]: decreases clause *)
+unfold
+let betree_node_apply_messages_decreases (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) : nat =
+  admit ()
+
+(** [betree_main::betree::Internal::{4}::flush]: decreases clause *)
+unfold
+let betree_internal_flush_decreases (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) : nat =
+  admit ()
+
diff --git a/tests/betree_back_stateful/BetreeMain.Clauses.fst b/tests/betree_back_stateful/BetreeMain.Clauses.fst
new file mode 100644
index 00000000..07484711
--- /dev/null
+++ b/tests/betree_back_stateful/BetreeMain.Clauses.fst
@@ -0,0 +1,210 @@
+(** [betree_main]: templates for the decreases clauses *)
+module BetreeMain.Clauses
+open Primitives
+open BetreeMain.Types
+
+#set-options "--z3rlimit 50 --fuel 0 --ifuel 1"
+
+(*** Well-founded relations *)
+
+(* We had a few issues when proving termination of the mutually recursive functions:
+ * - betree_internal_flush
+ * - betree_node_apply_messages
+ *
+ * The quantity which effectively decreases is:
+ *   (betree_size, messages_length)
+ *   where messages_length is 0 when there are no messages
+ *   (and where we use the lexicographic ordering, of course)
+ *
+ * However, the `%[...]` and `{:well-founded ...} notations are not available outside
+ * of `decrease` clauses.
+ * 
+ * We thus resorted to writing and proving correct a well-founded relation over
+ * pairs of natural numbers. The trick is that `<<` can be used outside of decrease
+ * clauses, and can be used to trigger SMT patterns.
+ *
+ * What follows is adapted from:
+ * https://www.fstar-lang.org/tutorial/book/part2/part2_well_founded.html
+ *
+ * Also, the following PR might make things easier:
+ * https://github.com/FStarLang/FStar/pull/2561
+ *)
+
+module P = FStar.Preorder
+module W = FStar.WellFounded
+module L = FStar.LexicographicOrdering
+
+let lt_nat (x y:nat) : Type = x < y == true
+let rec wf_lt_nat (x:nat)
+  : W.acc lt_nat x
+  = W.AccIntro (fun y _ -> wf_lt_nat y)
+
+// A type abbreviation for a pair of nats
+let nat_pair = (x:nat & nat)
+
+// Making a lexicographic ordering from a pair of nat ordering
+let lex_order_nat_pair  : P.relation nat_pair =
+  L.lex_t lt_nat (fun _ -> lt_nat)
+
+// The lex order on nat pairs is well-founded, using our general proof
+// of lexicographic composition of well-founded orders
+let lex_order_nat_pair_wf : W.well_founded lex_order_nat_pair =
+  L.lex_t_wf wf_lt_nat (fun _ -> wf_lt_nat)
+
+// A utility to introduce lt_nat
+let mk_lt_nat (x:nat) (y:nat { x < y }) : lt_nat x y =
+  let _ : equals (x < y) true = Refl in
+  ()
+    
+// A utility to make a lex ordering of nat pairs
+let mk_lex_order_nat_pair (xy0:nat_pair) 
+                          (xy1:nat_pair {
+                            let (|x0, y0|) = xy0 in
+                            let (|x1, y1|) = xy1 in
+                            x0 < x1 \/ (x0 == x1 /\ y0 < y1)
+                          }) : lex_order_nat_pair xy0 xy1 =
+  let (|x0, y0|) = xy0 in
+  let (|x1, y1|) = xy1 in
+  if x0 < x1 then L.Left_lex x0 x1 y0 y1 (mk_lt_nat x0 x1)
+  else L.Right_lex x0 y0 y1 (mk_lt_nat y0 y1)
+
+let rec coerce #a #r #x (p:W.acc #a r x) : Tot (W.acc r x) (decreases p) =
+  W.AccIntro (fun y r -> coerce (p.access_smaller y r))
+
+let coerce_wf #a #r (p: (x:a -> W.acc r x)) : x:a -> W.acc r x =
+  fun x -> coerce (p x)
+
+(* We need this axiom, which comes from the following discussion:
+ * https://github.com/FStarLang/FStar/issues/1916
+ * An issue here is that the `{well-founded ... }` notation
+ *)
+assume
+val axiom_well_founded (a : Type) (rel : a -> a -> Type0)
+  (rwf : W.well_founded #a rel) (x y : a) :
+  Lemma (requires (rel x y)) (ensures (x << y))
+
+(* This lemma has a pattern (which makes it work) *)
+let wf_nat_pair_lem (p0 p1 : nat_pair) :
+  Lemma
+  (requires (
+    let (|x0, y0|) = p0 in
+    let (|x1, y1|) = p1 in
+    x0 < x1 || (x0 = x1 && y0 < y1)))
+  (ensures (p0 << p1))
+  [SMTPat (p0 << p1)] =
+  let rel = lex_order_nat_pair in
+  let rel_wf = lex_order_nat_pair_wf in
+  let _ = mk_lex_order_nat_pair p0 p1 in
+  assert(rel p0 p1);
+  axiom_well_founded nat_pair rel rel_wf p0 p1
+
+(*** Decrease clauses *)
+/// "Standard" decrease clauses
+
+(** [betree_main::betree::List::{1}::len]: decreases clause *)
+unfold
+let betree_list_len_decreases (t : Type0) (self : betree_list_t t) : betree_list_t t =
+  self
+
+(** [betree_main::betree::List::{1}::split_at]: decreases clause *)
+unfold
+let betree_list_split_at_decreases (t : Type0) (self : betree_list_t t)
+  (n : u64) : nat =
+  n
+
+(** [betree_main::betree::List::{2}::partition_at_pivot]: decreases clause *)
+unfold
+let betree_list_partition_at_pivot_decreases (t : Type0)
+  (self : betree_list_t (u64 & t)) (pivot : u64) : betree_list_t (u64 & t) =
+  self
+
+(** [betree_main::betree::Node::{5}::lookup_in_bindings]: decreases clause *)
+unfold
+let betree_node_lookup_in_bindings_decreases (key : u64)
+  (bindings : betree_list_t (u64 & u64)) : betree_list_t (u64 & u64) =
+  bindings
+
+(** [betree_main::betree::Node::{5}::lookup_first_message_for_key]: decreases clause *)
+unfold
+let betree_node_lookup_first_message_for_key_decreases (key : u64)
+  (msgs : betree_list_t (u64 & betree_message_t)) : betree_list_t (u64 & betree_message_t) =
+  msgs
+
+(** [betree_main::betree::Node::{5}::apply_upserts]: decreases clause *)
+unfold
+let betree_node_apply_upserts_decreases
+  (msgs : betree_list_t (u64 & betree_message_t)) (prev : option u64)
+  (key : u64) (st : state) : betree_list_t (u64 & betree_message_t) =
+  msgs
+
+(** [betree_main::betree::Internal::{4}::lookup_in_children]: decreases clause *)
+unfold
+let betree_internal_lookup_in_children_decreases (self : betree_internal_t)
+  (key : u64) (st : state) : betree_internal_t =
+  self
+
+(** [betree_main::betree::Node::{5}::lookup]: decreases clause *)
+unfold
+let betree_node_lookup_decreases (self : betree_node_t) (key : u64)
+  (st : state) : betree_node_t =
+  self
+
+(** [betree_main::betree::Node::{5}::lookup_mut_in_bindings]: decreases clause *)
+unfold
+let betree_node_lookup_mut_in_bindings_decreases (key : u64)
+  (bindings : betree_list_t (u64 & u64)) : betree_list_t (u64 & u64) =
+  bindings
+
+unfold
+let betree_node_apply_messages_to_leaf_decreases
+  (bindings : betree_list_t (u64 & u64))
+  (new_msgs : betree_list_t (u64 & betree_message_t)) : betree_list_t (u64 & betree_message_t) =
+  new_msgs
+
+(** [betree_main::betree::Node::{5}::filter_messages_for_key]: decreases clause *)
+unfold
+let betree_node_filter_messages_for_key_decreases (key : u64)
+  (msgs : betree_list_t (u64 & betree_message_t)) : betree_list_t (u64 & betree_message_t) =
+  msgs
+
+(** [betree_main::betree::Node::{5}::lookup_first_message_after_key]: decreases clause *)
+unfold
+let betree_node_lookup_first_message_after_key_decreases (key : u64)
+  (msgs : betree_list_t (u64 & betree_message_t)) : betree_list_t (u64 & betree_message_t) =
+  msgs
+
+let betree_node_apply_messages_to_internal_decreases
+  (msgs : betree_list_t (u64 & betree_message_t))
+  (new_msgs : betree_list_t (u64 & betree_message_t)) : betree_list_t (u64 & betree_message_t) =
+  new_msgs
+
+(*** Decrease clauses - nat_pair *)
+/// The following decrease clauses use the [nat_pair] definition and the well-founded
+/// relation proven above.
+
+let rec betree_size (bt : betree_node_t) : nat =
+  match bt with
+  | BetreeNodeInternal node -> 1 + betree_internal_size node
+  | BetreeNodeLeaf _ -> 1
+
+and betree_internal_size (node : betree_internal_t) : nat =
+  1 + betree_size node.betree_internal_left + betree_size node.betree_internal_right
+
+let rec betree_list_len (#a : Type0) (ls : betree_list_t a) : nat =
+  match ls with
+  | BetreeListCons _ tl -> 1 + betree_list_len tl
+  | BetreeListNil -> 0
+
+(** [betree_main::betree::Internal::{4}::flush]: decreases clause *)
+unfold
+let betree_internal_flush_decreases (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) : nat_pair =
+  (|betree_internal_size self, 0|)
+
+(** [betree_main::betree::Node::{5}::apply_messages]: decreases clause *)
+unfold
+let betree_node_apply_messages_decreases (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) : nat_pair =
+  (|betree_size self, betree_list_len msgs|)
diff --git a/tests/betree_back_stateful/BetreeMain.Funs.fst b/tests/betree_back_stateful/BetreeMain.Funs.fst
new file mode 100644
index 00000000..ea8344fa
--- /dev/null
+++ b/tests/betree_back_stateful/BetreeMain.Funs.fst
@@ -0,0 +1,2085 @@
+(** 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] *)
+let betree_load_internal_node_fwd
+  (id : u64) (st : state) :
+  result (state & (betree_list_t (u64 & betree_message_t)))
+  =
+  begin match betree_utils_load_internal_node_fwd id st with
+  | Fail -> Fail
+  | Return (st0, l) -> Return (st0, l)
+  end
+
+(** [betree_main::betree::store_internal_node] *)
+let betree_store_internal_node_fwd
+  (id : u64) (content : betree_list_t (u64 & betree_message_t)) (st : state) :
+  result (state & unit)
+  =
+  begin match betree_utils_store_internal_node_fwd id content st with
+  | Fail -> Fail
+  | Return (st0, _) -> Return (st0, ())
+  end
+
+(** [betree_main::betree::load_leaf_node] *)
+let betree_load_leaf_node_fwd
+  (id : u64) (st : state) : result (state & (betree_list_t (u64 & u64))) =
+  begin match betree_utils_load_leaf_node_fwd id st with
+  | Fail -> Fail
+  | Return (st0, l) -> Return (st0, l)
+  end
+
+(** [betree_main::betree::store_leaf_node] *)
+let betree_store_leaf_node_fwd
+  (id : u64) (content : betree_list_t (u64 & u64)) (st : state) :
+  result (state & unit)
+  =
+  begin match betree_utils_store_leaf_node_fwd id content st with
+  | Fail -> Fail
+  | Return (st0, _) -> Return (st0, ())
+  end
+
+(** [betree_main::betree::fresh_node_id] *)
+let betree_fresh_node_id_fwd (counter : u64) : result u64 =
+  begin match u64_add counter 1 with
+  | Fail -> Fail
+  | Return _ -> Return counter
+  end
+
+(** [betree_main::betree::fresh_node_id] *)
+let betree_fresh_node_id_back (counter : u64) : result u64 =
+  begin match u64_add counter 1 with
+  | Fail -> Fail
+  | Return counter0 -> Return counter0
+  end
+
+(** [betree_main::betree::NodeIdCounter::{0}::new] *)
+let betree_node_id_counter_new_fwd : result betree_node_id_counter_t =
+  Return (Mkbetree_node_id_counter_t 0)
+
+(** [betree_main::betree::NodeIdCounter::{0}::fresh_id] *)
+let betree_node_id_counter_fresh_id_fwd
+  (self : betree_node_id_counter_t) : result u64 =
+  begin match u64_add self.betree_node_id_counter_next_node_id 1 with
+  | Fail -> Fail
+  | Return _ -> Return self.betree_node_id_counter_next_node_id
+  end
+
+(** [betree_main::betree::NodeIdCounter::{0}::fresh_id] *)
+let betree_node_id_counter_fresh_id_back
+  (self : betree_node_id_counter_t) : result betree_node_id_counter_t =
+  begin match u64_add self.betree_node_id_counter_next_node_id 1 with
+  | Fail -> Fail
+  | Return i -> Return (Mkbetree_node_id_counter_t i)
+  end
+
+(** [core::num::u64::{10}::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] *)
+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 ->
+      begin match u64_sub core_num_u64_max_c prev0 with
+      | Fail -> Fail
+      | Return margin ->
+        if margin >= v
+        then
+          begin match u64_add prev0 v with
+          | Fail -> Fail
+          | Return i -> Return i
+          end
+        else Return core_num_u64_max_c
+      end
+    | BetreeUpsertFunStateSub v ->
+      if prev0 >= v
+      then
+        begin match u64_sub prev0 v with
+        | Fail -> Fail
+        | Return i -> Return i
+        end
+      else Return 0
+    end
+  end
+
+(** [betree_main::betree::List::{1}::len] *)
+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 ->
+    begin match betree_list_len_fwd t tl with
+    | Fail -> Fail
+    | Return i ->
+      begin match u64_add 1 i with | Fail -> Fail | Return i0 -> Return i0 end
+    end
+  | BetreeListNil -> Return 0
+  end
+
+(** [betree_main::betree::List::{1}::split_at] *)
+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 ->
+      begin match u64_sub n 1 with
+      | Fail -> Fail
+      | Return i ->
+        begin match betree_list_split_at_fwd t tl i with
+        | Fail -> Fail
+        | Return p ->
+          let (ls0, ls1) = p in
+          let l = ls0 in Return (BetreeListCons hd l, ls1)
+        end
+      end
+    | BetreeListNil -> Fail
+    end
+
+(** [betree_main::betree::List::{1}::push_front] *)
+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] *)
+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
+  end
+
+(** [betree_main::betree::List::{1}::pop_front] *)
+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
+  end
+
+(** [betree_main::betree::List::{1}::hd] *)
+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
+  end
+
+(** [betree_main::betree::List::{2}::head_has_key] *)
+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] *)
+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
+      begin match betree_list_partition_at_pivot_fwd t tl pivot with
+      | Fail -> Fail
+      | Return p ->
+        let (ls0, ls1) = p in
+        let l = ls0 in Return (BetreeListCons (i, x) l, ls1)
+      end
+  | BetreeListNil -> Return (BetreeListNil, BetreeListNil)
+  end
+
+(** [betree_main::betree::Leaf::{3}::split] *)
+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)
+  =
+  begin match
+    betree_list_split_at_fwd (u64 & u64) content
+      params.betree_params_split_size with
+  | Fail -> Fail
+  | Return p ->
+    let (content0, content1) = p in
+    begin match betree_list_hd_fwd (u64 & u64) content1 with
+    | Fail -> Fail
+    | Return p0 ->
+      let (pivot, _) = p0 in
+      begin match betree_node_id_counter_fresh_id_fwd node_id_cnt with
+      | Fail -> Fail
+      | Return id0 ->
+        begin match betree_node_id_counter_fresh_id_back node_id_cnt with
+        | Fail -> Fail
+        | Return node_id_cnt0 ->
+          begin match betree_node_id_counter_fresh_id_fwd node_id_cnt0 with
+          | Fail -> Fail
+          | Return id1 ->
+            begin match betree_store_leaf_node_fwd id0 content0 st with
+            | Fail -> Fail
+            | Return (st0, _) ->
+              begin match betree_store_leaf_node_fwd id1 content1 st0 with
+              | Fail -> Fail
+              | Return (st1, _) ->
+                let n = BetreeNodeLeaf (Mkbetree_leaf_t id0
+                  params.betree_params_split_size) in
+                let n0 = BetreeNodeLeaf (Mkbetree_leaf_t id1
+                  params.betree_params_split_size) in
+                Return
+                (st1,
+                Mkbetree_internal_t
+                self.betree_leaf_id
+                pivot
+                n
+                n0)
+              end
+            end
+          end
+        end
+      end
+    end
+  end
+
+(** [betree_main::betree::Leaf::{3}::split] *)
+let betree_leaf_split_back0
+  (self : betree_leaf_t) (content : betree_list_t (u64 & u64))
+  (params : betree_params_t) (node_id_cnt : betree_node_id_counter_t)
+  (st : state) (st0 : state) :
+  result (state & unit)
+  =
+  begin match
+    betree_list_split_at_fwd (u64 & u64) content
+      params.betree_params_split_size with
+  | Fail -> Fail
+  | Return p ->
+    let (content0, content1) = p in
+    begin match betree_list_hd_fwd (u64 & u64) content1 with
+    | Fail -> Fail
+    | Return _ ->
+      begin match betree_node_id_counter_fresh_id_fwd node_id_cnt with
+      | Fail -> Fail
+      | Return id0 ->
+        begin match betree_node_id_counter_fresh_id_back node_id_cnt with
+        | Fail -> Fail
+        | Return node_id_cnt0 ->
+          begin match betree_node_id_counter_fresh_id_fwd node_id_cnt0 with
+          | Fail -> Fail
+          | Return id1 ->
+            begin match betree_store_leaf_node_fwd id0 content0 st with
+            | Fail -> Fail
+            | Return (st1, _) ->
+              begin match betree_store_leaf_node_fwd id1 content1 st1 with
+              | Fail -> Fail
+              | Return (_, _) -> Return (st0, ())
+              end
+            end
+          end
+        end
+      end
+    end
+  end
+
+(** [betree_main::betree::Leaf::{3}::split] *)
+let betree_leaf_split_back1
+  (self : betree_leaf_t) (content : betree_list_t (u64 & u64))
+  (params : betree_params_t) (node_id_cnt : betree_node_id_counter_t)
+  (st : state) (st0 : state) :
+  result (state & unit)
+  =
+  begin match
+    betree_list_split_at_fwd (u64 & u64) content
+      params.betree_params_split_size with
+  | Fail -> Fail
+  | Return p ->
+    let (content0, content1) = p in
+    begin match betree_list_hd_fwd (u64 & u64) content1 with
+    | Fail -> Fail
+    | Return _ ->
+      begin match betree_node_id_counter_fresh_id_fwd node_id_cnt with
+      | Fail -> Fail
+      | Return id0 ->
+        begin match betree_node_id_counter_fresh_id_back node_id_cnt with
+        | Fail -> Fail
+        | Return node_id_cnt0 ->
+          begin match betree_node_id_counter_fresh_id_fwd node_id_cnt0 with
+          | Fail -> Fail
+          | Return id1 ->
+            begin match betree_store_leaf_node_fwd id0 content0 st with
+            | Fail -> Fail
+            | Return (st1, _) ->
+              begin match betree_store_leaf_node_fwd id1 content1 st1 with
+              | Fail -> Fail
+              | Return (_, _) -> Return (st0, ())
+              end
+            end
+          end
+        end
+      end
+    end
+  end
+
+(** [betree_main::betree::Leaf::{3}::split] *)
+let betree_leaf_split_back2
+  (self : betree_leaf_t) (content : betree_list_t (u64 & u64))
+  (params : betree_params_t) (node_id_cnt : betree_node_id_counter_t)
+  (st : state) (st0 : state) :
+  result (state & betree_node_id_counter_t)
+  =
+  begin match
+    betree_list_split_at_fwd (u64 & u64) content
+      params.betree_params_split_size with
+  | Fail -> Fail
+  | Return p ->
+    let (content0, content1) = p in
+    begin match betree_list_hd_fwd (u64 & u64) content1 with
+    | Fail -> Fail
+    | Return _ ->
+      begin match betree_node_id_counter_fresh_id_fwd node_id_cnt with
+      | Fail -> Fail
+      | Return id0 ->
+        begin match betree_node_id_counter_fresh_id_back node_id_cnt with
+        | Fail -> Fail
+        | Return node_id_cnt0 ->
+          begin match betree_node_id_counter_fresh_id_fwd node_id_cnt0 with
+          | Fail -> Fail
+          | Return id1 ->
+            begin match betree_store_leaf_node_fwd id0 content0 st with
+            | Fail -> Fail
+            | Return (st1, _) ->
+              begin match betree_store_leaf_node_fwd id1 content1 st1 with
+              | Fail -> Fail
+              | Return (_, _) ->
+                begin match betree_node_id_counter_fresh_id_back node_id_cnt0
+                  with
+                | Fail -> Fail
+                | Return node_id_cnt1 -> Return (st0, node_id_cnt1)
+                end
+              end
+            end
+          end
+        end
+      end
+    end
+  end
+
+(** [betree_main::betree::Node::{5}::lookup_in_bindings] *)
+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
+        begin match betree_node_lookup_in_bindings_fwd key tl with
+        | Fail -> Fail
+        | Return opt -> Return opt
+        end
+  | BetreeListNil -> Return None
+  end
+
+(** [betree_main::betree::Node::{5}::lookup_first_message_for_key] *)
+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
+      begin match betree_node_lookup_first_message_for_key_fwd key next_msgs
+        with
+      | Fail -> Fail
+      | Return l -> Return l
+      end
+  | BetreeListNil -> Return BetreeListNil
+  end
+
+(** [betree_main::betree::Node::{5}::lookup_first_message_for_key] *)
+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
+      begin match
+        betree_node_lookup_first_message_for_key_back key next_msgs ret with
+      | Fail -> Fail
+      | Return next_msgs0 -> Return (BetreeListCons (i, m) next_msgs0)
+      end
+  | BetreeListNil -> Return ret
+  end
+
+(** [betree_main::betree::Node::{5}::apply_upserts] *)
+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))
+  =
+  begin match betree_list_head_has_key_fwd betree_message_t msgs key with
+  | Fail -> Fail
+  | Return b ->
+    if b
+    then
+      begin match betree_list_pop_front_fwd (u64 & betree_message_t) msgs with
+      | Fail -> Fail
+      | Return msg ->
+        let (_, m) = msg in
+        begin match m with
+        | BetreeMessageInsert i -> Fail
+        | BetreeMessageDelete -> Fail
+        | BetreeMessageUpsert s ->
+          begin match betree_upsert_update_fwd prev s with
+          | Fail -> Fail
+          | Return v ->
+            begin match
+              betree_list_pop_front_back (u64 & betree_message_t) msgs with
+            | Fail -> Fail
+            | Return msgs0 ->
+              begin match betree_node_apply_upserts_fwd msgs0 (Some v) key st
+                with
+              | Fail -> Fail
+              | Return (st0, i) -> Return (st0, i)
+              end
+            end
+          end
+        end
+      end
+    else
+      begin match core_option_option_unwrap_fwd u64 prev st with
+      | Fail -> Fail
+      | Return (st0, v) ->
+        begin match
+          betree_list_push_front_fwd_back (u64 & betree_message_t) msgs (key,
+            BetreeMessageInsert v) with
+        | Fail -> Fail
+        | Return _ -> Return (st0, v)
+        end
+      end
+  end
+
+(** [betree_main::betree::Node::{5}::apply_upserts] *)
+let rec betree_node_apply_upserts_back
+  (msgs : betree_list_t (u64 & betree_message_t)) (prev : option u64)
+  (key : u64) (st : state) (st0 : state) :
+  Tot (result (state & (betree_list_t (u64 & betree_message_t))))
+  (decreases (betree_node_apply_upserts_decreases msgs prev key st))
+  =
+  begin match betree_list_head_has_key_fwd betree_message_t msgs key with
+  | Fail -> Fail
+  | Return b ->
+    if b
+    then
+      begin match betree_list_pop_front_fwd (u64 & betree_message_t) msgs with
+      | Fail -> Fail
+      | Return msg ->
+        let (_, m) = msg in
+        begin match m with
+        | BetreeMessageInsert i -> Fail
+        | BetreeMessageDelete -> Fail
+        | BetreeMessageUpsert s ->
+          begin match betree_upsert_update_fwd prev s with
+          | Fail -> Fail
+          | Return v ->
+            begin match
+              betree_list_pop_front_back (u64 & betree_message_t) msgs with
+            | Fail -> Fail
+            | Return msgs0 ->
+              begin match
+                betree_node_apply_upserts_back msgs0 (Some v) key st st0 with
+              | Fail -> Fail
+              | Return (st1, msgs1) -> Return (st1, msgs1)
+              end
+            end
+          end
+        end
+      end
+    else
+      begin match core_option_option_unwrap_fwd u64 prev st with
+      | Fail -> Fail
+      | Return (_, v) ->
+        begin match
+          betree_list_push_front_fwd_back (u64 & betree_message_t) msgs (key,
+            BetreeMessageInsert v) with
+        | Fail -> Fail
+        | Return msgs0 -> Return (st0, msgs0)
+        end
+      end
+  end
+
+(** [betree_main::betree::Node::{5}::lookup] *)
+let rec 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 ->
+    begin match betree_load_internal_node_fwd node.betree_internal_id st with
+    | Fail -> Fail
+    | Return (st0, msgs) ->
+      begin match betree_node_lookup_first_message_for_key_fwd key msgs with
+      | Fail -> Fail
+      | Return pending ->
+        begin match pending with
+        | BetreeListCons p l ->
+          let (k, msg) = p in
+          if k <> key
+          then
+            begin match betree_internal_lookup_in_children_fwd node key st0
+              with
+            | Fail -> Fail
+            | Return (st1, opt) ->
+              begin match
+                betree_node_lookup_first_message_for_key_back key msgs
+                  (BetreeListCons (k, msg) l) with
+              | Fail -> Fail
+              | Return _ -> Return (st1, opt)
+              end
+            end
+          else
+            begin match msg with
+            | BetreeMessageInsert v ->
+              begin match
+                betree_node_lookup_first_message_for_key_back key msgs
+                  (BetreeListCons (k, BetreeMessageInsert v) l) with
+              | Fail -> Fail
+              | Return _ -> Return (st0, Some v)
+              end
+            | BetreeMessageDelete ->
+              begin match
+                betree_node_lookup_first_message_for_key_back key msgs
+                  (BetreeListCons (k, BetreeMessageDelete) l) with
+              | Fail -> Fail
+              | Return _ -> Return (st0, None)
+              end
+            | BetreeMessageUpsert ufs ->
+              begin match betree_internal_lookup_in_children_fwd node key st0
+                with
+              | Fail -> Fail
+              | Return (st1, v) ->
+                begin match
+                  betree_node_apply_upserts_fwd (BetreeListCons (k,
+                    BetreeMessageUpsert ufs) l) v key st1 with
+                | Fail -> Fail
+                | Return (st2, v0) ->
+                  begin match
+                    betree_internal_lookup_in_children_back node key st0 st2
+                    with
+                  | Fail -> Fail
+                  | Return (st3, node0) ->
+                    begin match
+                      betree_node_apply_upserts_back (BetreeListCons (k,
+                        BetreeMessageUpsert ufs) l) v key st1 st3 with
+                    | Fail -> Fail
+                    | Return (st4, pending0) ->
+                      begin match
+                        betree_node_lookup_first_message_for_key_back key msgs
+                          pending0 with
+                      | Fail -> Fail
+                      | Return msgs0 ->
+                        begin match
+                          betree_store_internal_node_fwd
+                            node0.betree_internal_id msgs0 st4 with
+                        | Fail -> Fail
+                        | Return (st5, _) -> Return (st5, Some v0)
+                        end
+                      end
+                    end
+                  end
+                end
+              end
+            end
+        | BetreeListNil ->
+          begin match betree_internal_lookup_in_children_fwd node key st0 with
+          | Fail -> Fail
+          | Return (st1, opt) ->
+            begin match
+              betree_node_lookup_first_message_for_key_back key msgs
+                BetreeListNil with
+            | Fail -> Fail
+            | Return _ -> Return (st1, opt)
+            end
+          end
+        end
+      end
+    end
+  | BetreeNodeLeaf node ->
+    begin match betree_load_leaf_node_fwd node.betree_leaf_id st with
+    | Fail -> Fail
+    | Return (st0, bindings) ->
+      begin match betree_node_lookup_in_bindings_fwd key bindings with
+      | Fail -> Fail
+      | Return opt -> Return (st0, opt)
+      end
+    end
+  end
+
+(** [betree_main::betree::Node::{5}::lookup] *)
+and betree_node_lookup_back
+  (self : betree_node_t) (key : u64) (st : state) (st0 : state) :
+  Tot (result (state & betree_node_t))
+  (decreases (betree_node_lookup_decreases self key st))
+  =
+  begin match self with
+  | BetreeNodeInternal node ->
+    begin match betree_load_internal_node_fwd node.betree_internal_id st with
+    | Fail -> Fail
+    | Return (st1, msgs) ->
+      begin match betree_node_lookup_first_message_for_key_fwd key msgs with
+      | Fail -> Fail
+      | Return pending ->
+        begin match pending with
+        | BetreeListCons p l ->
+          let (k, msg) = p in
+          if k <> key
+          then
+            begin match
+              betree_node_lookup_first_message_for_key_back key msgs
+                (BetreeListCons (k, msg) l) with
+            | Fail -> Fail
+            | Return _ ->
+              begin match
+                betree_internal_lookup_in_children_back node key st1 st0 with
+              | Fail -> Fail
+              | Return (st2, node0) -> Return (st2, BetreeNodeInternal node0)
+              end
+            end
+          else
+            begin match msg with
+            | BetreeMessageInsert v ->
+              begin match
+                betree_node_lookup_first_message_for_key_back key msgs
+                  (BetreeListCons (k, BetreeMessageInsert v) l) with
+              | Fail -> Fail
+              | Return _ -> Return (st0, BetreeNodeInternal node)
+              end
+            | BetreeMessageDelete ->
+              begin match
+                betree_node_lookup_first_message_for_key_back key msgs
+                  (BetreeListCons (k, BetreeMessageDelete) l) with
+              | Fail -> Fail
+              | Return _ -> Return (st0, BetreeNodeInternal node)
+              end
+            | BetreeMessageUpsert ufs ->
+              begin match betree_internal_lookup_in_children_fwd node key st1
+                with
+              | Fail -> Fail
+              | Return (st2, v) ->
+                begin match
+                  betree_node_apply_upserts_fwd (BetreeListCons (k,
+                    BetreeMessageUpsert ufs) l) v key st2 with
+                | Fail -> Fail
+                | Return (st3, _) ->
+                  begin match
+                    betree_internal_lookup_in_children_back node key st1 st3
+                    with
+                  | Fail -> Fail
+                  | Return (st4, node0) ->
+                    begin match
+                      betree_node_apply_upserts_back (BetreeListCons (k,
+                        BetreeMessageUpsert ufs) l) v key st2 st4 with
+                    | Fail -> Fail
+                    | Return (st5, pending0) ->
+                      begin match
+                        betree_node_lookup_first_message_for_key_back key msgs
+                          pending0 with
+                      | Fail -> Fail
+                      | Return msgs0 ->
+                        begin match
+                          betree_store_internal_node_fwd
+                            node0.betree_internal_id msgs0 st5 with
+                        | Fail -> Fail
+                        | Return (_, _) ->
+                          Return (st0, BetreeNodeInternal node0)
+                        end
+                      end
+                    end
+                  end
+                end
+              end
+            end
+        | BetreeListNil ->
+          begin match
+            betree_node_lookup_first_message_for_key_back key msgs
+              BetreeListNil with
+          | Fail -> Fail
+          | Return _ ->
+            begin match
+              betree_internal_lookup_in_children_back node key st1 st0 with
+            | Fail -> Fail
+            | Return (st2, node0) -> Return (st2, BetreeNodeInternal node0)
+            end
+          end
+        end
+      end
+    end
+  | BetreeNodeLeaf node ->
+    begin match betree_load_leaf_node_fwd node.betree_leaf_id st with
+    | Fail -> Fail
+    | Return (_, bindings) ->
+      begin match betree_node_lookup_in_bindings_fwd key bindings with
+      | Fail -> Fail
+      | Return _ -> Return (st0, BetreeNodeLeaf node)
+      end
+    end
+  end
+
+(** [betree_main::betree::Internal::{4}::lookup_in_children] *)
+and 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
+    begin match betree_node_lookup_fwd self.betree_internal_left key st with
+    | Fail -> Fail
+    | Return (st0, opt) -> Return (st0, opt)
+    end
+  else
+    begin match betree_node_lookup_fwd self.betree_internal_right key st with
+    | Fail -> Fail
+    | Return (st0, opt) -> Return (st0, opt)
+    end
+
+(** [betree_main::betree::Internal::{4}::lookup_in_children] *)
+and betree_internal_lookup_in_children_back
+  (self : betree_internal_t) (key : u64) (st : state) (st0 : state) :
+  Tot (result (state & betree_internal_t))
+  (decreases (betree_internal_lookup_in_children_decreases self key st))
+  =
+  if key < self.betree_internal_pivot
+  then
+    begin match betree_node_lookup_back self.betree_internal_left key st st0
+      with
+    | Fail -> Fail
+    | Return (st1, n) ->
+      Return (st1, Mkbetree_internal_t self.betree_internal_id
+        self.betree_internal_pivot n self.betree_internal_right)
+    end
+  else
+    begin match betree_node_lookup_back self.betree_internal_right key st st0
+      with
+    | Fail -> Fail
+    | Return (st1, n) ->
+      Return (st1, Mkbetree_internal_t self.betree_internal_id
+        self.betree_internal_pivot self.betree_internal_left n)
+    end
+
+(** [betree_main::betree::Node::{5}::lookup_mut_in_bindings] *)
+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
+      begin match betree_node_lookup_mut_in_bindings_fwd key tl with
+      | Fail -> Fail
+      | Return l -> Return l
+      end
+  | BetreeListNil -> Return BetreeListNil
+  end
+
+(** [betree_main::betree::Node::{5}::lookup_mut_in_bindings] *)
+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
+      begin match betree_node_lookup_mut_in_bindings_back key tl ret with
+      | Fail -> Fail
+      | Return tl0 -> Return (BetreeListCons (i, i0) tl0)
+      end
+  | BetreeListNil -> Return ret
+  end
+
+(** [betree_main::betree::Node::{5}::apply_to_leaf] *)
+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))
+  =
+  begin match betree_node_lookup_mut_in_bindings_fwd key bindings with
+  | Fail -> Fail
+  | Return bindings0 ->
+    begin match betree_list_head_has_key_fwd u64 bindings0 key with
+    | Fail -> Fail
+    | Return b ->
+      if b
+      then
+        begin match betree_list_pop_front_fwd (u64 & u64) bindings0 with
+        | Fail -> Fail
+        | Return hd ->
+          begin match new_msg with
+          | BetreeMessageInsert v ->
+            begin match betree_list_pop_front_back (u64 & u64) bindings0 with
+            | Fail -> Fail
+            | Return bindings1 ->
+              begin match
+                betree_list_push_front_fwd_back (u64 & u64) bindings1 (key, v)
+                with
+              | Fail -> Fail
+              | Return bindings2 ->
+                begin match
+                  betree_node_lookup_mut_in_bindings_back key bindings
+                    bindings2 with
+                | Fail -> Fail
+                | Return bindings3 -> Return bindings3
+                end
+              end
+            end
+          | BetreeMessageDelete ->
+            begin match betree_list_pop_front_back (u64 & u64) bindings0 with
+            | Fail -> Fail
+            | Return bindings1 ->
+              begin match
+                betree_node_lookup_mut_in_bindings_back key bindings bindings1
+                with
+              | Fail -> Fail
+              | Return bindings2 -> Return bindings2
+              end
+            end
+          | BetreeMessageUpsert s ->
+            let (_, i) = hd in
+            begin match betree_upsert_update_fwd (Some i) s with
+            | Fail -> Fail
+            | Return v ->
+              begin match betree_list_pop_front_back (u64 & u64) bindings0 with
+              | Fail -> Fail
+              | Return bindings1 ->
+                begin match
+                  betree_list_push_front_fwd_back (u64 & u64) bindings1 (key,
+                    v) with
+                | Fail -> Fail
+                | Return bindings2 ->
+                  begin match
+                    betree_node_lookup_mut_in_bindings_back key bindings
+                      bindings2 with
+                  | Fail -> Fail
+                  | Return bindings3 -> Return bindings3
+                  end
+                end
+              end
+            end
+          end
+        end
+      else
+        begin match new_msg with
+        | BetreeMessageInsert v ->
+          begin match
+            betree_list_push_front_fwd_back (u64 & u64) bindings0 (key, v) with
+          | Fail -> Fail
+          | Return bindings1 ->
+            begin match
+              betree_node_lookup_mut_in_bindings_back key bindings bindings1
+              with
+            | Fail -> Fail
+            | Return bindings2 -> Return bindings2
+            end
+          end
+        | BetreeMessageDelete ->
+          begin match
+            betree_node_lookup_mut_in_bindings_back key bindings bindings0 with
+          | Fail -> Fail
+          | Return bindings1 -> Return bindings1
+          end
+        | BetreeMessageUpsert s ->
+          begin match betree_upsert_update_fwd None s with
+          | Fail -> Fail
+          | Return v ->
+            begin match
+              betree_list_push_front_fwd_back (u64 & u64) bindings0 (key, v)
+              with
+            | Fail -> Fail
+            | Return bindings1 ->
+              begin match
+                betree_node_lookup_mut_in_bindings_back key bindings bindings1
+                with
+              | Fail -> Fail
+              | Return bindings2 -> Return bindings2
+              end
+            end
+          end
+        end
+    end
+  end
+
+(** [betree_main::betree::Node::{5}::apply_messages_to_leaf] *)
+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
+    begin match betree_node_apply_to_leaf_fwd_back bindings i m with
+    | Fail -> Fail
+    | Return bindings0 ->
+      begin match
+        betree_node_apply_messages_to_leaf_fwd_back bindings0 new_msgs_tl with
+      | Fail -> Fail
+      | Return bindings1 -> Return bindings1
+      end
+    end
+  | BetreeListNil -> Return bindings
+  end
+
+(** [betree_main::betree::Node::{5}::filter_messages_for_key] *)
+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
+      begin match
+        betree_list_pop_front_back (u64 & betree_message_t) (BetreeListCons (k,
+          m) l) with
+      | Fail -> Fail
+      | Return msgs0 ->
+        begin match betree_node_filter_messages_for_key_fwd_back key msgs0 with
+        | Fail -> Fail
+        | Return msgs1 -> Return msgs1
+        end
+      end
+    else Return (BetreeListCons (k, m) l)
+  | BetreeListNil -> Return BetreeListNil
+  end
+
+(** [betree_main::betree::Node::{5}::lookup_first_message_after_key] *)
+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
+      begin match betree_node_lookup_first_message_after_key_fwd key next_msgs
+        with
+      | Fail -> Fail
+      | Return l -> Return l
+      end
+    else Return (BetreeListCons (k, m) next_msgs)
+  | BetreeListNil -> Return BetreeListNil
+  end
+
+(** [betree_main::betree::Node::{5}::lookup_first_message_after_key] *)
+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
+      begin match
+        betree_node_lookup_first_message_after_key_back key next_msgs ret with
+      | Fail -> Fail
+      | Return next_msgs0 -> Return (BetreeListCons (k, m) next_msgs0)
+      end
+    else Return ret
+  | BetreeListNil -> Return ret
+  end
+
+(** [betree_main::betree::Node::{5}::apply_to_internal] *)
+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))
+  =
+  begin match betree_node_lookup_first_message_for_key_fwd key msgs with
+  | Fail -> Fail
+  | Return msgs0 ->
+    begin match betree_list_head_has_key_fwd betree_message_t msgs0 key with
+    | Fail -> Fail
+    | Return b ->
+      if b
+      then
+        begin match new_msg with
+        | BetreeMessageInsert i ->
+          begin match betree_node_filter_messages_for_key_fwd_back key msgs0
+            with
+          | Fail -> Fail
+          | Return msgs1 ->
+            begin match
+              betree_list_push_front_fwd_back (u64 & betree_message_t) msgs1
+                (key, BetreeMessageInsert i) with
+            | Fail -> Fail
+            | Return msgs2 ->
+              begin match
+                betree_node_lookup_first_message_for_key_back key msgs msgs2
+                with
+              | Fail -> Fail
+              | Return msgs3 -> Return msgs3
+              end
+            end
+          end
+        | BetreeMessageDelete ->
+          begin match betree_node_filter_messages_for_key_fwd_back key msgs0
+            with
+          | Fail -> Fail
+          | Return msgs1 ->
+            begin match
+              betree_list_push_front_fwd_back (u64 & betree_message_t) msgs1
+                (key, BetreeMessageDelete) with
+            | Fail -> Fail
+            | Return msgs2 ->
+              begin match
+                betree_node_lookup_first_message_for_key_back key msgs msgs2
+                with
+              | Fail -> Fail
+              | Return msgs3 -> Return msgs3
+              end
+            end
+          end
+        | BetreeMessageUpsert s ->
+          begin match betree_list_hd_fwd (u64 & betree_message_t) msgs0 with
+          | Fail -> Fail
+          | Return p ->
+            let (_, m) = p in
+            begin match m with
+            | BetreeMessageInsert prev ->
+              begin match betree_upsert_update_fwd (Some prev) s with
+              | Fail -> Fail
+              | Return v ->
+                begin match
+                  betree_list_pop_front_back (u64 & betree_message_t) msgs0
+                  with
+                | Fail -> Fail
+                | Return msgs1 ->
+                  begin match
+                    betree_list_push_front_fwd_back (u64 & betree_message_t)
+                      msgs1 (key, BetreeMessageInsert v) with
+                  | Fail -> Fail
+                  | Return msgs2 ->
+                    begin match
+                      betree_node_lookup_first_message_for_key_back key msgs
+                        msgs2 with
+                    | Fail -> Fail
+                    | Return msgs3 -> Return msgs3
+                    end
+                  end
+                end
+              end
+            | BetreeMessageDelete ->
+              begin match betree_upsert_update_fwd None s with
+              | Fail -> Fail
+              | Return v ->
+                begin match
+                  betree_list_pop_front_back (u64 & betree_message_t) msgs0
+                  with
+                | Fail -> Fail
+                | Return msgs1 ->
+                  begin match
+                    betree_list_push_front_fwd_back (u64 & betree_message_t)
+                      msgs1 (key, BetreeMessageInsert v) with
+                  | Fail -> Fail
+                  | Return msgs2 ->
+                    begin match
+                      betree_node_lookup_first_message_for_key_back key msgs
+                        msgs2 with
+                    | Fail -> Fail
+                    | Return msgs3 -> Return msgs3
+                    end
+                  end
+                end
+              end
+            | BetreeMessageUpsert ufs ->
+              begin match
+                betree_node_lookup_first_message_after_key_fwd key msgs0 with
+              | Fail -> Fail
+              | Return msgs1 ->
+                begin match
+                  betree_list_push_front_fwd_back (u64 & betree_message_t)
+                    msgs1 (key, BetreeMessageUpsert s) with
+                | Fail -> Fail
+                | Return msgs2 ->
+                  begin match
+                    betree_node_lookup_first_message_after_key_back key msgs0
+                      msgs2 with
+                  | Fail -> Fail
+                  | Return msgs3 ->
+                    begin match
+                      betree_node_lookup_first_message_for_key_back key msgs
+                        msgs3 with
+                    | Fail -> Fail
+                    | Return msgs4 -> Return msgs4
+                    end
+                  end
+                end
+              end
+            end
+          end
+        end
+      else
+        begin match
+          betree_list_push_front_fwd_back (u64 & betree_message_t) msgs0 (key,
+            new_msg) with
+        | Fail -> Fail
+        | Return msgs1 ->
+          begin match
+            betree_node_lookup_first_message_for_key_back key msgs msgs1 with
+          | Fail -> Fail
+          | Return msgs2 -> Return msgs2
+          end
+        end
+    end
+  end
+
+(** [betree_main::betree::Node::{5}::apply_messages_to_internal] *)
+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
+    begin match betree_node_apply_to_internal_fwd_back msgs i m with
+    | Fail -> Fail
+    | Return msgs0 ->
+      begin match
+        betree_node_apply_messages_to_internal_fwd_back msgs0 new_msgs_tl with
+      | Fail -> Fail
+      | Return msgs1 -> Return msgs1
+      end
+    end
+  | BetreeListNil -> Return msgs
+  end
+
+(** [betree_main::betree::Node::{5}::apply_messages] *)
+let rec 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 ->
+    begin match betree_load_internal_node_fwd node.betree_internal_id st with
+    | Fail -> Fail
+    | Return (st0, content) ->
+      begin match betree_node_apply_messages_to_internal_fwd_back content msgs
+        with
+      | Fail -> Fail
+      | Return content0 ->
+        begin match betree_list_len_fwd (u64 & betree_message_t) content0 with
+        | Fail -> Fail
+        | Return num_msgs ->
+          if num_msgs >= params.betree_params_min_flush_size
+          then
+            begin match
+              betree_internal_flush_fwd node params node_id_cnt content0 st0
+              with
+            | Fail -> Fail
+            | Return (st1, content1) ->
+              begin match
+                betree_internal_flush_back'a node params node_id_cnt content0
+                  st0 st1 with
+              | Fail -> Fail
+              | Return (st2, (node0, _)) ->
+                begin match
+                  betree_store_internal_node_fwd node0.betree_internal_id
+                    content1 st2 with
+                | Fail -> Fail
+                | Return (st3, _) -> Return (st3, ())
+                end
+              end
+            end
+          else
+            begin match
+              betree_store_internal_node_fwd node.betree_internal_id content0
+                st0 with
+            | Fail -> Fail
+            | Return (st1, _) -> Return (st1, ())
+            end
+        end
+      end
+    end
+  | BetreeNodeLeaf node ->
+    begin match betree_load_leaf_node_fwd node.betree_leaf_id st with
+    | Fail -> Fail
+    | Return (st0, content) ->
+      begin match betree_node_apply_messages_to_leaf_fwd_back content msgs with
+      | Fail -> Fail
+      | Return content0 ->
+        begin match betree_list_len_fwd (u64 & u64) content0 with
+        | Fail -> Fail
+        | Return len ->
+          begin match u64_mul 2 params.betree_params_split_size with
+          | Fail -> Fail
+          | Return i ->
+            if len >= i
+            then
+              begin match
+                betree_leaf_split_fwd node content0 params node_id_cnt st0 with
+              | Fail -> Fail
+              | Return (st1, _) ->
+                begin match
+                  betree_store_leaf_node_fwd node.betree_leaf_id BetreeListNil
+                    st1 with
+                | Fail -> Fail
+                | Return (st2, _) ->
+                  begin match
+                    betree_leaf_split_back0 node content0 params node_id_cnt
+                      st0 st2 with
+                  | Fail -> Fail
+                  | Return (st3, ()) -> Return (st3, ())
+                  end
+                end
+              end
+            else
+              begin match
+                betree_store_leaf_node_fwd node.betree_leaf_id content0 st0
+                with
+              | Fail -> Fail
+              | Return (st1, _) -> Return (st1, ())
+              end
+          end
+        end
+      end
+    end
+  end
+
+(** [betree_main::betree::Node::{5}::apply_messages] *)
+and betree_node_apply_messages_back'a
+  (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) (st0 : state) :
+  Tot (result (state & (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 ->
+    begin match betree_load_internal_node_fwd node.betree_internal_id st with
+    | Fail -> Fail
+    | Return (st1, content) ->
+      begin match betree_node_apply_messages_to_internal_fwd_back content msgs
+        with
+      | Fail -> Fail
+      | Return content0 ->
+        begin match betree_list_len_fwd (u64 & betree_message_t) content0 with
+        | Fail -> Fail
+        | Return num_msgs ->
+          if num_msgs >= params.betree_params_min_flush_size
+          then
+            begin match
+              betree_internal_flush_fwd node params node_id_cnt content0 st1
+              with
+            | Fail -> Fail
+            | Return (st2, content1) ->
+              begin match
+                betree_internal_flush_back'a node params node_id_cnt content0
+                  st1 st2 with
+              | Fail -> Fail
+              | Return (st3, (node0, node_id_cnt0)) ->
+                begin match
+                  betree_store_internal_node_fwd node0.betree_internal_id
+                    content1 st3 with
+                | Fail -> Fail
+                | Return (_, _) ->
+                  Return (st0, (BetreeNodeInternal node0, node_id_cnt0))
+                end
+              end
+            end
+          else
+            begin match
+              betree_store_internal_node_fwd node.betree_internal_id content0
+                st1 with
+            | Fail -> Fail
+            | Return (_, _) ->
+              Return (st0, (BetreeNodeInternal node, node_id_cnt))
+            end
+        end
+      end
+    end
+  | BetreeNodeLeaf node ->
+    begin match betree_load_leaf_node_fwd node.betree_leaf_id st with
+    | Fail -> Fail
+    | Return (st1, content) ->
+      begin match betree_node_apply_messages_to_leaf_fwd_back content msgs with
+      | Fail -> Fail
+      | Return content0 ->
+        begin match betree_list_len_fwd (u64 & u64) content0 with
+        | Fail -> Fail
+        | Return len ->
+          begin match u64_mul 2 params.betree_params_split_size with
+          | Fail -> Fail
+          | Return i ->
+            if len >= i
+            then
+              begin match
+                betree_leaf_split_fwd node content0 params node_id_cnt st1 with
+              | Fail -> Fail
+              | Return (st2, new_node) ->
+                begin match
+                  betree_store_leaf_node_fwd node.betree_leaf_id BetreeListNil
+                    st2 with
+                | Fail -> Fail
+                | Return (st3, _) ->
+                  begin match
+                    betree_leaf_split_back0 node content0 params node_id_cnt
+                      st1 st3 with
+                  | Fail -> Fail
+                  | Return (_, ()) ->
+                    begin match
+                      betree_leaf_split_back2 node content0 params node_id_cnt
+                        st1 st0 with
+                    | Fail -> Fail
+                    | Return (st4, node_id_cnt0) ->
+                      Return (st4, (BetreeNodeInternal new_node, node_id_cnt0))
+                    end
+                  end
+                end
+              end
+            else
+              begin match
+                betree_store_leaf_node_fwd node.betree_leaf_id content0 st1
+                with
+              | Fail -> Fail
+              | Return (_, _) ->
+                Return (st0, (BetreeNodeLeaf (Mkbetree_leaf_t
+                  node.betree_leaf_id len), node_id_cnt))
+              end
+          end
+        end
+      end
+    end
+  end
+
+(** [betree_main::betree::Node::{5}::apply_messages] *)
+and betree_node_apply_messages_back1
+  (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) (st0 : state) :
+  Tot (result (state & unit))
+  (decreases (betree_node_apply_messages_decreases self params node_id_cnt msgs
+  st))
+  =
+  begin match self with
+  | BetreeNodeInternal node ->
+    begin match betree_load_internal_node_fwd node.betree_internal_id st with
+    | Fail -> Fail
+    | Return (st1, content) ->
+      begin match betree_node_apply_messages_to_internal_fwd_back content msgs
+        with
+      | Fail -> Fail
+      | Return content0 ->
+        begin match betree_list_len_fwd (u64 & betree_message_t) content0 with
+        | Fail -> Fail
+        | Return num_msgs ->
+          if num_msgs >= params.betree_params_min_flush_size
+          then
+            begin match
+              betree_internal_flush_fwd node params node_id_cnt content0 st1
+              with
+            | Fail -> Fail
+            | Return (st2, content1) ->
+              begin match
+                betree_internal_flush_back'a node params node_id_cnt content0
+                  st1 st2 with
+              | Fail -> Fail
+              | Return (st3, (node0, _)) ->
+                begin match
+                  betree_store_internal_node_fwd node0.betree_internal_id
+                    content1 st3 with
+                | Fail -> Fail
+                | Return (_, _) ->
+                  begin match
+                    betree_internal_flush_back1 node params node_id_cnt
+                      content0 st1 st0 with
+                  | Fail -> Fail
+                  | Return (st4, ()) -> Return (st4, ())
+                  end
+                end
+              end
+            end
+          else
+            begin match
+              betree_store_internal_node_fwd node.betree_internal_id content0
+                st1 with
+            | Fail -> Fail
+            | Return (_, _) -> Return (st0, ())
+            end
+        end
+      end
+    end
+  | BetreeNodeLeaf node ->
+    begin match betree_load_leaf_node_fwd node.betree_leaf_id st with
+    | Fail -> Fail
+    | Return (st1, content) ->
+      begin match betree_node_apply_messages_to_leaf_fwd_back content msgs with
+      | Fail -> Fail
+      | Return content0 ->
+        begin match betree_list_len_fwd (u64 & u64) content0 with
+        | Fail -> Fail
+        | Return len ->
+          begin match u64_mul 2 params.betree_params_split_size with
+          | Fail -> Fail
+          | Return i ->
+            if len >= i
+            then
+              begin match
+                betree_leaf_split_fwd node content0 params node_id_cnt st1 with
+              | Fail -> Fail
+              | Return (st2, _) ->
+                begin match
+                  betree_store_leaf_node_fwd node.betree_leaf_id BetreeListNil
+                    st2 with
+                | Fail -> Fail
+                | Return (st3, _) ->
+                  begin match
+                    betree_leaf_split_back0 node content0 params node_id_cnt
+                      st1 st3 with
+                  | Fail -> Fail
+                  | Return (_, ()) ->
+                    begin match
+                      betree_leaf_split_back1 node content0 params node_id_cnt
+                        st1 st0 with
+                    | Fail -> Fail
+                    | Return (st4, ()) -> Return (st4, ())
+                    end
+                  end
+                end
+              end
+            else
+              begin match
+                betree_store_leaf_node_fwd node.betree_leaf_id content0 st1
+                with
+              | Fail -> Fail
+              | Return (_, _) -> Return (st0, ())
+              end
+          end
+        end
+      end
+    end
+  end
+
+(** [betree_main::betree::Internal::{4}::flush] *)
+and 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))
+  =
+  begin match
+    betree_list_partition_at_pivot_fwd betree_message_t content
+      self.betree_internal_pivot with
+  | Fail -> Fail
+  | Return p ->
+    let (msgs_left, msgs_right) = p in
+    begin match betree_list_len_fwd (u64 & betree_message_t) msgs_left with
+    | Fail -> Fail
+    | Return len_left ->
+      if len_left >= params.betree_params_min_flush_size
+      then
+        begin match
+          betree_node_apply_messages_fwd self.betree_internal_left params
+            node_id_cnt msgs_left st with
+        | Fail -> Fail
+        | Return (st0, _) ->
+          begin match
+            betree_node_apply_messages_back'a self.betree_internal_left params
+              node_id_cnt msgs_left st st0 with
+          | Fail -> Fail
+          | Return (st1, (_, node_id_cnt0)) ->
+            begin match
+              betree_node_apply_messages_back1 self.betree_internal_left params
+                node_id_cnt msgs_left st st1 with
+            | Fail -> Fail
+            | Return (st2, ()) ->
+              begin match
+                betree_list_len_fwd (u64 & betree_message_t) msgs_right with
+              | Fail -> Fail
+              | Return len_right ->
+                if len_right >= params.betree_params_min_flush_size
+                then
+                  begin match
+                    betree_node_apply_messages_fwd self.betree_internal_right
+                      params node_id_cnt0 msgs_right st2 with
+                  | Fail -> Fail
+                  | Return (st3, _) ->
+                    begin match
+                      betree_node_apply_messages_back'a
+                        self.betree_internal_right params node_id_cnt0
+                        msgs_right st2 st3 with
+                    | Fail -> Fail
+                    | Return (st4, (_, _)) ->
+                      begin match
+                        betree_node_apply_messages_back1
+                          self.betree_internal_right params node_id_cnt0
+                          msgs_right st2 st4 with
+                      | Fail -> Fail
+                      | Return (st5, ()) -> Return (st5, BetreeListNil)
+                      end
+                    end
+                  end
+                else Return (st2, msgs_right)
+              end
+            end
+          end
+        end
+      else
+        begin match
+          betree_node_apply_messages_fwd self.betree_internal_right params
+            node_id_cnt msgs_right st with
+        | Fail -> Fail
+        | Return (st0, _) ->
+          begin match
+            betree_node_apply_messages_back'a self.betree_internal_right params
+              node_id_cnt msgs_right st st0 with
+          | Fail -> Fail
+          | Return (st1, (_, _)) ->
+            begin match
+              betree_node_apply_messages_back1 self.betree_internal_right
+                params node_id_cnt msgs_right st st1 with
+            | Fail -> Fail
+            | Return (st2, ()) -> Return (st2, msgs_left)
+            end
+          end
+        end
+    end
+  end
+
+(** [betree_main::betree::Internal::{4}::flush] *)
+and betree_internal_flush_back'a
+  (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) (st0 : state) :
+  Tot (result (state & (betree_internal_t & betree_node_id_counter_t)))
+  (decreases (betree_internal_flush_decreases self params node_id_cnt content
+  st))
+  =
+  begin match
+    betree_list_partition_at_pivot_fwd betree_message_t content
+      self.betree_internal_pivot with
+  | Fail -> Fail
+  | Return p ->
+    let (msgs_left, msgs_right) = p in
+    begin match betree_list_len_fwd (u64 & betree_message_t) msgs_left with
+    | Fail -> Fail
+    | Return len_left ->
+      if len_left >= params.betree_params_min_flush_size
+      then
+        begin match
+          betree_node_apply_messages_fwd self.betree_internal_left params
+            node_id_cnt msgs_left st with
+        | Fail -> Fail
+        | Return (st1, _) ->
+          begin match
+            betree_node_apply_messages_back'a self.betree_internal_left params
+              node_id_cnt msgs_left st st1 with
+          | Fail -> Fail
+          | Return (st2, (n, node_id_cnt0)) ->
+            begin match
+              betree_node_apply_messages_back1 self.betree_internal_left params
+                node_id_cnt msgs_left st st2 with
+            | Fail -> Fail
+            | Return (st3, ()) ->
+              begin match
+                betree_list_len_fwd (u64 & betree_message_t) msgs_right with
+              | Fail -> Fail
+              | Return len_right ->
+                if len_right >= params.betree_params_min_flush_size
+                then
+                  begin match
+                    betree_node_apply_messages_fwd self.betree_internal_right
+                      params node_id_cnt0 msgs_right st3 with
+                  | Fail -> Fail
+                  | Return (st4, _) ->
+                    begin match
+                      betree_node_apply_messages_back'a
+                        self.betree_internal_right params node_id_cnt0
+                        msgs_right st3 st4 with
+                    | Fail -> Fail
+                    | Return (st5, (n0, node_id_cnt1)) ->
+                      begin match
+                        betree_node_apply_messages_back1
+                          self.betree_internal_right params node_id_cnt0
+                          msgs_right st3 st5 with
+                      | Fail -> Fail
+                      | Return (_, ()) ->
+                        Return (st0, (Mkbetree_internal_t
+                          self.betree_internal_id self.betree_internal_pivot n
+                          n0, node_id_cnt1))
+                      end
+                    end
+                  end
+                else
+                  Return (st0, (Mkbetree_internal_t self.betree_internal_id
+                    self.betree_internal_pivot n self.betree_internal_right,
+                    node_id_cnt0))
+              end
+            end
+          end
+        end
+      else
+        begin match
+          betree_node_apply_messages_fwd self.betree_internal_right params
+            node_id_cnt msgs_right st with
+        | Fail -> Fail
+        | Return (st1, _) ->
+          begin match
+            betree_node_apply_messages_back'a self.betree_internal_right params
+              node_id_cnt msgs_right st st1 with
+          | Fail -> Fail
+          | Return (st2, (n, node_id_cnt0)) ->
+            begin match
+              betree_node_apply_messages_back1 self.betree_internal_right
+                params node_id_cnt msgs_right st st2 with
+            | Fail -> Fail
+            | Return (_, ()) ->
+              Return (st0, (Mkbetree_internal_t self.betree_internal_id
+                self.betree_internal_pivot self.betree_internal_left n,
+                node_id_cnt0))
+            end
+          end
+        end
+    end
+  end
+
+(** [betree_main::betree::Internal::{4}::flush] *)
+and betree_internal_flush_back1
+  (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) (st0 : state) :
+  Tot (result (state & unit))
+  (decreases (betree_internal_flush_decreases self params node_id_cnt content
+  st))
+  =
+  begin match
+    betree_list_partition_at_pivot_fwd betree_message_t content
+      self.betree_internal_pivot with
+  | Fail -> Fail
+  | Return p ->
+    let (msgs_left, msgs_right) = p in
+    begin match betree_list_len_fwd (u64 & betree_message_t) msgs_left with
+    | Fail -> Fail
+    | Return len_left ->
+      if len_left >= params.betree_params_min_flush_size
+      then
+        begin match
+          betree_node_apply_messages_fwd self.betree_internal_left params
+            node_id_cnt msgs_left st with
+        | Fail -> Fail
+        | Return (st1, _) ->
+          begin match
+            betree_node_apply_messages_back'a self.betree_internal_left params
+              node_id_cnt msgs_left st st1 with
+          | Fail -> Fail
+          | Return (st2, (_, node_id_cnt0)) ->
+            begin match
+              betree_node_apply_messages_back1 self.betree_internal_left params
+                node_id_cnt msgs_left st st2 with
+            | Fail -> Fail
+            | Return (st3, ()) ->
+              begin match
+                betree_list_len_fwd (u64 & betree_message_t) msgs_right with
+              | Fail -> Fail
+              | Return len_right ->
+                if len_right >= params.betree_params_min_flush_size
+                then
+                  begin match
+                    betree_node_apply_messages_fwd self.betree_internal_right
+                      params node_id_cnt0 msgs_right st3 with
+                  | Fail -> Fail
+                  | Return (st4, _) ->
+                    begin match
+                      betree_node_apply_messages_back'a
+                        self.betree_internal_right params node_id_cnt0
+                        msgs_right st3 st4 with
+                    | Fail -> Fail
+                    | Return (st5, (_, _)) ->
+                      begin match
+                        betree_node_apply_messages_back1
+                          self.betree_internal_right params node_id_cnt0
+                          msgs_right st3 st5 with
+                      | Fail -> Fail
+                      | Return (_, ()) -> Return (st0, ())
+                      end
+                    end
+                  end
+                else Return (st0, ())
+              end
+            end
+          end
+        end
+      else
+        begin match
+          betree_node_apply_messages_fwd self.betree_internal_right params
+            node_id_cnt msgs_right st with
+        | Fail -> Fail
+        | Return (st1, _) ->
+          begin match
+            betree_node_apply_messages_back'a self.betree_internal_right params
+              node_id_cnt msgs_right st st1 with
+          | Fail -> Fail
+          | Return (st2, (_, _)) ->
+            begin match
+              betree_node_apply_messages_back1 self.betree_internal_right
+                params node_id_cnt msgs_right st st2 with
+            | Fail -> Fail
+            | Return (_, ()) -> Return (st0, ())
+            end
+          end
+        end
+    end
+  end
+
+(** [betree_main::betree::Node::{5}::apply] *)
+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
+  begin match
+    betree_node_apply_messages_fwd self params node_id_cnt (BetreeListCons
+      (key, new_msg) l) st with
+  | Fail -> Fail
+  | Return (st0, _) ->
+    begin match
+      betree_node_apply_messages_back'a self params node_id_cnt (BetreeListCons
+        (key, new_msg) l) st st0 with
+    | Fail -> Fail
+    | Return (st1, (_, _)) ->
+      begin match
+        betree_node_apply_messages_back1 self params node_id_cnt
+          (BetreeListCons (key, new_msg) l) st st1 with
+      | Fail -> Fail
+      | Return (st2, ()) -> Return (st2, ())
+      end
+    end
+  end
+
+(** [betree_main::betree::Node::{5}::apply] *)
+let betree_node_apply_back'a
+  (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) (st0 : state) :
+  result (state & (betree_node_t & betree_node_id_counter_t))
+  =
+  let l = BetreeListNil in
+  begin match
+    betree_node_apply_messages_fwd self params node_id_cnt (BetreeListCons
+      (key, new_msg) l) st with
+  | Fail -> Fail
+  | Return (st1, _) ->
+    begin match
+      betree_node_apply_messages_back'a self params node_id_cnt (BetreeListCons
+        (key, new_msg) l) st st1 with
+    | Fail -> Fail
+    | Return (st2, (self0, node_id_cnt0)) ->
+      begin match
+        betree_node_apply_messages_back1 self params node_id_cnt
+          (BetreeListCons (key, new_msg) l) st st2 with
+      | Fail -> Fail
+      | Return (_, ()) -> Return (st0, (self0, node_id_cnt0))
+      end
+    end
+  end
+
+(** [betree_main::betree::Node::{5}::apply] *)
+let betree_node_apply_back1
+  (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) (st0 : state) :
+  result (state & unit)
+  =
+  let l = BetreeListNil in
+  begin match
+    betree_node_apply_messages_fwd self params node_id_cnt (BetreeListCons
+      (key, new_msg) l) st with
+  | Fail -> Fail
+  | Return (st1, _) ->
+    begin match
+      betree_node_apply_messages_back'a self params node_id_cnt (BetreeListCons
+        (key, new_msg) l) st st1 with
+    | Fail -> Fail
+    | Return (st2, (_, _)) ->
+      begin match
+        betree_node_apply_messages_back1 self params node_id_cnt
+          (BetreeListCons (key, new_msg) l) st st2 with
+      | Fail -> Fail
+      | Return (_, ()) -> Return (st0, ())
+      end
+    end
+  end
+
+(** [betree_main::betree::BeTree::{6}::new] *)
+let betree_be_tree_new_fwd
+  (min_flush_size : u64) (split_size : u64) (st : state) :
+  result (state & betree_be_tree_t)
+  =
+  begin match betree_node_id_counter_new_fwd with
+  | Fail -> Fail
+  | Return node_id_cnt ->
+    begin match betree_node_id_counter_fresh_id_fwd node_id_cnt with
+    | Fail -> Fail
+    | Return id ->
+      begin match betree_store_leaf_node_fwd id BetreeListNil st with
+      | Fail -> Fail
+      | Return (st0, _) ->
+        begin match betree_node_id_counter_fresh_id_back node_id_cnt with
+        | Fail -> Fail
+        | Return node_id_cnt0 ->
+          Return (st0, Mkbetree_be_tree_t (Mkbetree_params_t min_flush_size
+            split_size) node_id_cnt0 (BetreeNodeLeaf (Mkbetree_leaf_t id 0)))
+        end
+      end
+    end
+  end
+
+(** [betree_main::betree::BeTree::{6}::apply] *)
+let betree_be_tree_apply_fwd
+  (self : betree_be_tree_t) (key : u64) (msg : betree_message_t) (st : state) :
+  result (state & unit)
+  =
+  begin match
+    betree_node_apply_fwd self.betree_be_tree_root self.betree_be_tree_params
+      self.betree_be_tree_node_id_cnt key msg st with
+  | Fail -> Fail
+  | Return (st0, _) ->
+    begin match
+      betree_node_apply_back'a self.betree_be_tree_root
+        self.betree_be_tree_params self.betree_be_tree_node_id_cnt key msg st
+        st0 with
+    | Fail -> Fail
+    | Return (st1, (_, _)) ->
+      begin match
+        betree_node_apply_back1 self.betree_be_tree_root
+          self.betree_be_tree_params self.betree_be_tree_node_id_cnt key msg st
+          st1 with
+      | Fail -> Fail
+      | Return (st2, ()) -> Return (st2, ())
+      end
+    end
+  end
+
+(** [betree_main::betree::BeTree::{6}::apply] *)
+let betree_be_tree_apply_back
+  (self : betree_be_tree_t) (key : u64) (msg : betree_message_t) (st : state)
+  (st0 : state) :
+  result (state & betree_be_tree_t)
+  =
+  begin match
+    betree_node_apply_fwd self.betree_be_tree_root self.betree_be_tree_params
+      self.betree_be_tree_node_id_cnt key msg st with
+  | Fail -> Fail
+  | Return (st1, _) ->
+    begin match
+      betree_node_apply_back'a self.betree_be_tree_root
+        self.betree_be_tree_params self.betree_be_tree_node_id_cnt key msg st
+        st1 with
+    | Fail -> Fail
+    | Return (st2, (n, nic)) ->
+      begin match
+        betree_node_apply_back1 self.betree_be_tree_root
+          self.betree_be_tree_params self.betree_be_tree_node_id_cnt key msg st
+          st2 with
+      | Fail -> Fail
+      | Return (_, ()) ->
+        Return (st0, Mkbetree_be_tree_t self.betree_be_tree_params nic n)
+      end
+    end
+  end
+
+(** [betree_main::betree::BeTree::{6}::insert] *)
+let betree_be_tree_insert_fwd
+  (self : betree_be_tree_t) (key : u64) (value : u64) (st : state) :
+  result (state & unit)
+  =
+  begin match betree_be_tree_apply_fwd self key (BetreeMessageInsert value) st
+    with
+  | Fail -> Fail
+  | Return (st0, _) ->
+    begin match
+      betree_be_tree_apply_back self key (BetreeMessageInsert value) st st0
+      with
+    | Fail -> Fail
+    | Return (st1, _) -> Return (st1, ())
+    end
+  end
+
+(** [betree_main::betree::BeTree::{6}::insert] *)
+let betree_be_tree_insert_back
+  (self : betree_be_tree_t) (key : u64) (value : u64) (st : state)
+  (st0 : state) :
+  result (state & betree_be_tree_t)
+  =
+  begin match betree_be_tree_apply_fwd self key (BetreeMessageInsert value) st
+    with
+  | Fail -> Fail
+  | Return (st1, _) ->
+    begin match
+      betree_be_tree_apply_back self key (BetreeMessageInsert value) st st1
+      with
+    | Fail -> Fail
+    | Return (_, self0) -> Return (st0, self0)
+    end
+  end
+
+(** [betree_main::betree::BeTree::{6}::delete] *)
+let betree_be_tree_delete_fwd
+  (self : betree_be_tree_t) (key : u64) (st : state) : result (state & unit) =
+  begin match betree_be_tree_apply_fwd self key BetreeMessageDelete st with
+  | Fail -> Fail
+  | Return (st0, _) ->
+    begin match betree_be_tree_apply_back self key BetreeMessageDelete st st0
+      with
+    | Fail -> Fail
+    | Return (st1, _) -> Return (st1, ())
+    end
+  end
+
+(** [betree_main::betree::BeTree::{6}::delete] *)
+let betree_be_tree_delete_back
+  (self : betree_be_tree_t) (key : u64) (st : state) (st0 : state) :
+  result (state & betree_be_tree_t)
+  =
+  begin match betree_be_tree_apply_fwd self key BetreeMessageDelete st with
+  | Fail -> Fail
+  | Return (st1, _) ->
+    begin match betree_be_tree_apply_back self key BetreeMessageDelete st st1
+      with
+    | Fail -> Fail
+    | Return (_, self0) -> Return (st0, self0)
+    end
+  end
+
+(** [betree_main::betree::BeTree::{6}::upsert] *)
+let betree_be_tree_upsert_fwd
+  (self : betree_be_tree_t) (key : u64) (upd : betree_upsert_fun_state_t)
+  (st : state) :
+  result (state & unit)
+  =
+  begin match betree_be_tree_apply_fwd self key (BetreeMessageUpsert upd) st
+    with
+  | Fail -> Fail
+  | Return (st0, _) ->
+    begin match
+      betree_be_tree_apply_back self key (BetreeMessageUpsert upd) st st0 with
+    | Fail -> Fail
+    | Return (st1, _) -> Return (st1, ())
+    end
+  end
+
+(** [betree_main::betree::BeTree::{6}::upsert] *)
+let betree_be_tree_upsert_back
+  (self : betree_be_tree_t) (key : u64) (upd : betree_upsert_fun_state_t)
+  (st : state) (st0 : state) :
+  result (state & betree_be_tree_t)
+  =
+  begin match betree_be_tree_apply_fwd self key (BetreeMessageUpsert upd) st
+    with
+  | Fail -> Fail
+  | Return (st1, _) ->
+    begin match
+      betree_be_tree_apply_back self key (BetreeMessageUpsert upd) st st1 with
+    | Fail -> Fail
+    | Return (_, self0) -> Return (st0, self0)
+    end
+  end
+
+(** [betree_main::betree::BeTree::{6}::lookup] *)
+let betree_be_tree_lookup_fwd
+  (self : betree_be_tree_t) (key : u64) (st : state) :
+  result (state & (option u64))
+  =
+  begin match betree_node_lookup_fwd self.betree_be_tree_root key st with
+  | Fail -> Fail
+  | Return (st0, opt) -> Return (st0, opt)
+  end
+
+(** [betree_main::betree::BeTree::{6}::lookup] *)
+let betree_be_tree_lookup_back
+  (self : betree_be_tree_t) (key : u64) (st : state) (st0 : state) :
+  result (state & betree_be_tree_t)
+  =
+  begin match betree_node_lookup_back self.betree_be_tree_root key st st0 with
+  | Fail -> Fail
+  | Return (st1, n) ->
+    Return (st1, Mkbetree_be_tree_t self.betree_be_tree_params
+      self.betree_be_tree_node_id_cnt n)
+  end
+
+(** [betree_main::main] *)
+let main_fwd : result unit = Return ()
+
+(** Unit test for [betree_main::main] *)
+let _ = assert_norm (main_fwd = Return ())
+
diff --git a/tests/betree_back_stateful/BetreeMain.Opaque.fsti b/tests/betree_back_stateful/BetreeMain.Opaque.fsti
new file mode 100644
index 00000000..dc49601a
--- /dev/null
+++ b/tests/betree_back_stateful/BetreeMain.Opaque.fsti
@@ -0,0 +1,30 @@
+(** THIS FILE WAS AUTOMATICALLY GENERATED BY AENEAS *)
+(** [betree_main]: opaque function definitions *)
+module BetreeMain.Opaque
+open Primitives
+include BetreeMain.Types
+
+#set-options "--z3rlimit 50 --fuel 1 --ifuel 1"
+
+(** [betree_main::betree_utils::load_internal_node] *)
+val betree_utils_load_internal_node_fwd
+  : u64 -> state -> result (state & (betree_list_t (u64 & betree_message_t)))
+
+(** [betree_main::betree_utils::store_internal_node] *)
+val betree_utils_store_internal_node_fwd
+  :
+  u64 -> betree_list_t (u64 & betree_message_t) -> state -> result (state &
+    unit)
+
+(** [betree_main::betree_utils::load_leaf_node] *)
+val betree_utils_load_leaf_node_fwd
+  : u64 -> state -> result (state & (betree_list_t (u64 & u64)))
+
+(** [betree_main::betree_utils::store_leaf_node] *)
+val betree_utils_store_leaf_node_fwd
+  : u64 -> betree_list_t (u64 & u64) -> state -> result (state & unit)
+
+(** [core::option::Option::{0}::unwrap] *)
+val core_option_option_unwrap_fwd
+  (t : Type0) : option t -> state -> result (state & t)
+
diff --git a/tests/betree_back_stateful/BetreeMain.Types.fsti b/tests/betree_back_stateful/BetreeMain.Types.fsti
new file mode 100644
index 00000000..c81e3302
--- /dev/null
+++ b/tests/betree_back_stateful/BetreeMain.Types.fsti
@@ -0,0 +1,64 @@
+(** THIS FILE WAS AUTOMATICALLY GENERATED BY AENEAS *)
+(** [betree_main]: type definitions *)
+module BetreeMain.Types
+open Primitives
+
+#set-options "--z3rlimit 50 --fuel 1 --ifuel 1"
+
+(** [betree_main::betree::List] *)
+type betree_list_t (t : Type0) =
+| BetreeListCons : t -> betree_list_t t -> betree_list_t t
+| BetreeListNil : betree_list_t t
+
+(** [betree_main::betree::UpsertFunState] *)
+type betree_upsert_fun_state_t =
+| BetreeUpsertFunStateAdd : u64 -> betree_upsert_fun_state_t
+| BetreeUpsertFunStateSub : u64 -> betree_upsert_fun_state_t
+
+(** [betree_main::betree::Message] *)
+type betree_message_t =
+| BetreeMessageInsert : u64 -> betree_message_t
+| BetreeMessageDelete : betree_message_t
+| BetreeMessageUpsert : betree_upsert_fun_state_t -> betree_message_t
+
+(** [betree_main::betree::Leaf] *)
+type betree_leaf_t = { betree_leaf_id : u64; betree_leaf_size : u64; }
+
+(** [betree_main::betree::Node] *)
+type betree_node_t =
+| BetreeNodeInternal : betree_internal_t -> betree_node_t
+| BetreeNodeLeaf : betree_leaf_t -> betree_node_t
+
+(** [betree_main::betree::Internal] *)
+and betree_internal_t =
+{
+  betree_internal_id : u64;
+  betree_internal_pivot : u64;
+  betree_internal_left : betree_node_t;
+  betree_internal_right : betree_node_t;
+}
+
+(** [betree_main::betree::Params] *)
+type betree_params_t =
+{
+  betree_params_min_flush_size : u64; betree_params_split_size : u64;
+}
+
+(** [betree_main::betree::NodeIdCounter] *)
+type betree_node_id_counter_t = { betree_node_id_counter_next_node_id : u64; }
+
+(** [betree_main::betree::BeTree] *)
+type betree_be_tree_t =
+{
+  betree_be_tree_params : betree_params_t;
+  betree_be_tree_node_id_cnt : betree_node_id_counter_t;
+  betree_be_tree_root : betree_node_t;
+}
+
+(** [core::num::u64::{10}::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
+
+(** The state type used in the state-error monad *)
+val state : Type0
+
diff --git a/tests/betree_back_stateful/Makefile b/tests/betree_back_stateful/Makefile
new file mode 100644
index 00000000..a16b0edb
--- /dev/null
+++ b/tests/betree_back_stateful/Makefile
@@ -0,0 +1,47 @@
+INCLUDE_DIRS = .
+
+FSTAR_INCLUDES = $(addprefix --include ,$(INCLUDE_DIRS))
+
+FSTAR_HINTS ?= --use_hints --use_hint_hashes --record_hints
+
+FSTAR_OPTIONS = $(FSTAR_HINTS) \
+  --cache_checked_modules $(FSTAR_INCLUDES) --cmi \
+  --warn_error '+241@247+285-274' \
+
+FSTAR_NO_FLAGS = fstar.exe --already_cached 'Prims FStar LowStar Steel' --odir obj --cache_dir obj
+
+FSTAR = $(FSTAR_NO_FLAGS) $(FSTAR_OPTIONS)
+
+# The F* roots are used to compute the dependency graph, and generate the .depend file
+FSTAR_ROOTS ?= $(wildcard *.fst *.fsti)
+
+# Build all the files
+all: $(addprefix obj/,$(addsuffix .checked,$(FSTAR_ROOTS)))
+
+# This is the right way to ensure the .depend file always gets re-built.
+ifeq (,$(filter %-in,$(MAKECMDGOALS)))
+ifndef NODEPEND
+ifndef MAKE_RESTARTS
+.depend: .FORCE
+	$(FSTAR_NO_FLAGS) --dep full $(notdir $(FSTAR_ROOTS)) > $@
+
+.PHONY: .FORCE
+.FORCE:
+endif
+endif
+
+include .depend
+endif
+
+# For the interactive mode
+%.fst-in %.fsti-in:
+	@echo $(FSTAR_OPTIONS)
+
+# Generete the .checked files in batch mode
+%.checked:
+	$(FSTAR) $(FSTAR_OPTIONS) $< && \
+	touch -c $@
+
+.PHONY: clean
+clean:
+	rm -f obj/*
diff --git a/tests/betree_back_stateful/Primitives.fst b/tests/betree_back_stateful/Primitives.fst
new file mode 100644
index 00000000..b44fe9d1
--- /dev/null
+++ b/tests/betree_back_stateful/Primitives.fst
@@ -0,0 +1,286 @@
+/// This file lists primitive and assumed functions and types
+module Primitives
+open FStar.Mul
+open FStar.List.Tot
+
+#set-options "--z3rlimit 15 --fuel 0 --ifuel 1"
+
+(*** Utilities *)
+val list_update (#a : Type0) (ls : list a) (i : nat{i < length ls}) (x : a) :
+  ls':list a{
+    length ls' = length ls /\
+    index ls' i == x
+  }
+#push-options "--fuel 1"
+let rec list_update #a ls i x =
+  match ls with
+  | x' :: ls -> if i = 0 then x :: ls else x' :: list_update ls (i-1) x
+#pop-options
+
+(*** Result *)
+type result (a : Type0) : Type0 =
+| Return : v:a -> result a
+| Fail : result a
+
+// Monadic bind and return.
+// Re-definining those allows us to customize the result of the monadic notations
+// like: `y <-- f x;`
+let return (#a : Type0) (x:a) : result a = Return x
+let bind (#a #b : Type0) (m : result a) (f : a -> result b) : result b =
+    match m with
+    | Return x -> f x
+    | Fail -> Fail
+
+// Monadic assert(...)
+let massert (b:bool) : result unit = if b then Return () else Fail
+
+// Normalize and unwrap a successful result (used for globals).
+let eval_global (#a : Type0) (x : result a{Return? (normalize_term x)}) : a = Return?.v x
+
+(*** Misc *)
+type char = FStar.Char.char
+type string = string
+
+let mem_replace_fwd (a : Type0) (x : a) (y : a) : a = x
+let mem_replace_back (a : Type0) (x : a) (y : a) : a = y
+
+(*** Scalars *)
+/// Rk.: most of the following code was at least partially generated
+
+let isize_min : int = -9223372036854775808
+let isize_max : int = 9223372036854775807
+let i8_min : int = -128
+let i8_max : int = 127
+let i16_min : int = -32768
+let i16_max : int = 32767
+let i32_min : int = -2147483648
+let i32_max : int = 2147483647
+let i64_min : int = -9223372036854775808
+let i64_max : int = 9223372036854775807
+let i128_min : int = -170141183460469231731687303715884105728
+let i128_max : int = 170141183460469231731687303715884105727
+let usize_min : int = 0
+let usize_max : int = 4294967295 // being conservative here: [u32_max] instead of [u64_max]
+let u8_min : int = 0
+let u8_max : int = 255
+let u16_min : int = 0
+let u16_max : int = 65535
+let u32_min : int = 0
+let u32_max : int = 4294967295
+let u64_min : int = 0
+let u64_max : int = 18446744073709551615
+let u128_min : int = 0
+let u128_max : int = 340282366920938463463374607431768211455
+
+type scalar_ty =
+| Isize
+| I8
+| I16
+| I32
+| I64
+| I128
+| Usize
+| U8
+| U16
+| U32
+| U64
+| U128
+
+let scalar_min (ty : scalar_ty) : int =
+  match ty with
+  | Isize -> isize_min
+  | I8 -> i8_min
+  | I16 -> i16_min
+  | I32 -> i32_min
+  | I64 -> i64_min
+  | I128 -> i128_min
+  | Usize -> usize_min
+  | U8 -> u8_min
+  | U16 -> u16_min
+  | U32 -> u32_min
+  | U64 -> u64_min
+  | U128 -> u128_min
+
+let scalar_max (ty : scalar_ty) : int =
+  match ty with
+  | Isize -> isize_max
+  | I8 -> i8_max
+  | I16 -> i16_max
+  | I32 -> i32_max
+  | I64 -> i64_max
+  | I128 -> i128_max
+  | Usize -> usize_max
+  | U8 -> u8_max
+  | U16 -> u16_max
+  | U32 -> u32_max
+  | U64 -> u64_max
+  | U128 -> u128_max
+
+type scalar (ty : scalar_ty) : eqtype = x:int{scalar_min ty <= x && x <= scalar_max ty}
+
+let mk_scalar (ty : scalar_ty) (x : int) : result (scalar ty) =
+  if scalar_min ty <= x && scalar_max ty >= x then Return x else Fail
+
+let scalar_neg (#ty : scalar_ty) (x : scalar ty) : result (scalar ty) = mk_scalar ty (-x)
+
+let scalar_div (#ty : scalar_ty) (x : scalar ty) (y : scalar ty) : result (scalar ty) =
+  if y <> 0 then mk_scalar ty (x / y) else Fail
+
+/// The remainder operation
+let int_rem (x : int) (y : int{y <> 0}) : int =
+  if x >= 0 then (x % y) else -(x % y)
+
+(* Checking consistency with Rust *)
+let _ = assert_norm(int_rem 1 2 = 1)
+let _ = assert_norm(int_rem (-1) 2 = -1)
+let _ = assert_norm(int_rem 1 (-2) = 1)
+let _ = assert_norm(int_rem (-1) (-2) = -1)
+
+let scalar_rem (#ty : scalar_ty) (x : scalar ty) (y : scalar ty) : result (scalar ty) =
+  if y <> 0 then mk_scalar ty (int_rem x y) else Fail
+
+let scalar_add (#ty : scalar_ty) (x : scalar ty) (y : scalar ty) : result (scalar ty) =
+  mk_scalar ty (x + y)
+
+let scalar_sub (#ty : scalar_ty) (x : scalar ty) (y : scalar ty) : result (scalar ty) =
+  mk_scalar ty (x - y)
+
+let scalar_mul (#ty : scalar_ty) (x : scalar ty) (y : scalar ty) : result (scalar ty) =
+  mk_scalar ty (x * y)
+
+(** Cast an integer from a [src_ty] to a [tgt_ty] *)
+let scalar_cast (src_ty : scalar_ty) (tgt_ty : scalar_ty) (x : scalar src_ty) : result (scalar tgt_ty) =
+  mk_scalar tgt_ty x
+
+/// The scalar types
+type isize : eqtype = scalar Isize
+type i8 : eqtype = scalar I8
+type i16 : eqtype = scalar I16
+type i32 : eqtype = scalar I32
+type i64 : eqtype = scalar I64
+type i128 : eqtype = scalar I128
+type usize : eqtype = scalar Usize
+type u8 : eqtype = scalar U8
+type u16 : eqtype = scalar U16
+type u32 : eqtype = scalar U32
+type u64 : eqtype = scalar U64
+type u128 : eqtype = scalar U128
+
+/// Negation
+let isize_neg = scalar_neg #Isize
+let i8_neg = scalar_neg #I8
+let i16_neg = scalar_neg #I16
+let i32_neg = scalar_neg #I32
+let i64_neg = scalar_neg #I64
+let i128_neg = scalar_neg #I128
+
+/// Division
+let isize_div = scalar_div #Isize
+let i8_div = scalar_div #I8
+let i16_div = scalar_div #I16
+let i32_div = scalar_div #I32
+let i64_div = scalar_div #I64
+let i128_div = scalar_div #I128
+let usize_div = scalar_div #Usize
+let u8_div = scalar_div #U8
+let u16_div = scalar_div #U16
+let u32_div = scalar_div #U32
+let u64_div = scalar_div #U64
+let u128_div = scalar_div #U128
+
+/// Remainder
+let isize_rem = scalar_rem #Isize
+let i8_rem = scalar_rem #I8
+let i16_rem = scalar_rem #I16
+let i32_rem = scalar_rem #I32
+let i64_rem = scalar_rem #I64
+let i128_rem = scalar_rem #I128
+let usize_rem = scalar_rem #Usize
+let u8_rem = scalar_rem #U8
+let u16_rem = scalar_rem #U16
+let u32_rem = scalar_rem #U32
+let u64_rem = scalar_rem #U64
+let u128_rem = scalar_rem #U128
+
+/// Addition
+let isize_add = scalar_add #Isize
+let i8_add = scalar_add #I8
+let i16_add = scalar_add #I16
+let i32_add = scalar_add #I32
+let i64_add = scalar_add #I64
+let i128_add = scalar_add #I128
+let usize_add = scalar_add #Usize
+let u8_add = scalar_add #U8
+let u16_add = scalar_add #U16
+let u32_add = scalar_add #U32
+let u64_add = scalar_add #U64
+let u128_add = scalar_add #U128
+
+/// Substraction
+let isize_sub = scalar_sub #Isize
+let i8_sub = scalar_sub #I8
+let i16_sub = scalar_sub #I16
+let i32_sub = scalar_sub #I32
+let i64_sub = scalar_sub #I64
+let i128_sub = scalar_sub #I128
+let usize_sub = scalar_sub #Usize
+let u8_sub = scalar_sub #U8
+let u16_sub = scalar_sub #U16
+let u32_sub = scalar_sub #U32
+let u64_sub = scalar_sub #U64
+let u128_sub = scalar_sub #U128
+
+/// Multiplication
+let isize_mul = scalar_mul #Isize
+let i8_mul = scalar_mul #I8
+let i16_mul = scalar_mul #I16
+let i32_mul = scalar_mul #I32
+let i64_mul = scalar_mul #I64
+let i128_mul = scalar_mul #I128
+let usize_mul = scalar_mul #Usize
+let u8_mul = scalar_mul #U8
+let u16_mul = scalar_mul #U16
+let u32_mul = scalar_mul #U32
+let u64_mul = scalar_mul #U64
+let u128_mul = scalar_mul #U128
+
+(*** Vector *)
+type vec (a : Type0) = v:list a{length v <= usize_max}
+
+let vec_new (a  : Type0) : vec a = assert_norm(length #a [] == 0); []
+let vec_len (a : Type0) (v : vec a) : usize = length v
+
+// The **forward** function shouldn't be used
+let vec_push_fwd (a  : Type0) (v : vec a) (x : a) : unit = ()
+let vec_push_back (a  : Type0) (v : vec a) (x : a) :
+  Pure (result (vec a))
+  (requires True)
+  (ensures (fun res ->
+    match res with
+    | Fail -> True
+    | Return v' -> length v' = length v + 1)) =
+  if length v < usize_max then begin
+    (**) assert_norm(length [x] == 1);
+    (**) append_length v [x];
+    (**) assert(length (append v [x]) = length v + 1);
+    Return (append v [x])
+    end
+  else Fail
+
+// The **forward** function shouldn't be used
+let vec_insert_fwd (a : Type0) (v : vec a) (i : usize) (x : a) : result unit =
+  if i < length v then Return () else Fail
+let vec_insert_back (a : Type0) (v : vec a) (i : usize) (x : a) : result (vec a) =
+  if i < length v then Return (list_update v i x) else Fail
+
+// The **backward** function shouldn't be used
+let vec_index_fwd (a : Type0) (v : vec a) (i : usize) : result a =
+  if i < length v then Return (index v i) else Fail
+let vec_index_back (a : Type0) (v : vec a) (i : usize) (x : a) : result unit =
+  if i < length v then Return () else Fail
+
+let vec_index_mut_fwd (a : Type0) (v : vec a) (i : usize) : result a =
+  if i < length v then Return (index v i) else Fail
+let vec_index_mut_back (a : Type0) (v : vec a) (i : usize) (nx : a) : result (vec a) =
+  if i < length v then Return (list_update v i nx) else Fail
+
diff --git a/tests/misc/External.Funs.fst b/tests/misc/External.Funs.fst
index 874b739c..68a0061e 100644
--- a/tests/misc/External.Funs.fst
+++ b/tests/misc/External.Funs.fst
@@ -12,24 +12,31 @@ let swap_fwd (t : Type0) (x : t) (y : t) (st : state) : result (state & unit) =
   begin match core_mem_swap_fwd t x y st with
   | Fail -> Fail
   | Return (st0, _) ->
-    begin match core_mem_swap_back0 t x y st with
+    begin match core_mem_swap_back0 t x y st st0 with
     | Fail -> Fail
-    | Return _ ->
-      begin match core_mem_swap_back1 t x y st with
+    | Return (st1, _) ->
+      begin match core_mem_swap_back1 t x y st st1 with
       | Fail -> Fail
-      | Return _ -> Return (st0, ())
+      | Return (st2, _) -> Return (st2, ())
       end
     end
   end
 
 (** [external::swap] *)
-let swap_back (t : Type0) (x : t) (y : t) (st : state) : result (t & t) =
-  begin match core_mem_swap_back0 t x y st with
+let swap_back
+  (t : Type0) (x : t) (y : t) (st : state) (st0 : state) :
+  result (state & (t & t))
+  =
+  begin match core_mem_swap_fwd t x y st with
   | Fail -> Fail
-  | Return x0 ->
-    begin match core_mem_swap_back1 t x y st with
+  | Return (st1, _) ->
+    begin match core_mem_swap_back0 t x y st st1 with
     | Fail -> Fail
-    | Return y0 -> Return (x0, y0)
+    | Return (st2, x0) ->
+      begin match core_mem_swap_back1 t x y st st2 with
+      | Fail -> Fail
+      | Return (_, y0) -> Return (st0, (x0, y0))
+      end
     end
   end
 
@@ -64,25 +71,31 @@ let custom_swap_fwd
   begin match core_mem_swap_fwd t x y st with
   | Fail -> Fail
   | Return (st0, _) ->
-    begin match core_mem_swap_back0 t x y st with
+    begin match core_mem_swap_back0 t x y st st0 with
     | Fail -> Fail
-    | Return x0 ->
-      begin match core_mem_swap_back1 t x y st with
+    | Return (st1, x0) ->
+      begin match core_mem_swap_back1 t x y st st1 with
       | Fail -> Fail
-      | Return _ -> Return (st0, x0)
+      | Return (st2, _) -> Return (st2, x0)
       end
     end
   end
 
 (** [external::custom_swap] *)
 let custom_swap_back
-  (t : Type0) (x : t) (y : t) (st : state) (ret : t) : result (t & t) =
-  begin match core_mem_swap_back0 t x y st with
+  (t : Type0) (x : t) (y : t) (st : state) (ret : t) (st0 : state) :
+  result (state & (t & t))
+  =
+  begin match core_mem_swap_fwd t x y st with
   | Fail -> Fail
-  | Return _ ->
-    begin match core_mem_swap_back1 t x y st with
+  | Return (st1, _) ->
+    begin match core_mem_swap_back0 t x y st st1 with
     | Fail -> Fail
-    | Return y0 -> Return (ret, y0)
+    | Return (st2, _) ->
+      begin match core_mem_swap_back1 t x y st st2 with
+      | Fail -> Fail
+      | Return (_, y0) -> Return (st0, (ret, y0))
+      end
     end
   end
 
@@ -96,10 +109,12 @@ let test_custom_swap_fwd
 
 (** [external::test_custom_swap] *)
 let test_custom_swap_back
-  (x : u32) (y : u32) (st : state) : result (u32 & u32) =
-  begin match custom_swap_back u32 x y st 1 with
+  (x : u32) (y : u32) (st : state) (st0 : state) :
+  result (state & (u32 & u32))
+  =
+  begin match custom_swap_back u32 x y st 1 st0 with
   | Fail -> Fail
-  | Return (x0, y0) -> Return (x0, y0)
+  | Return (st1, (x0, y0)) -> Return (st1, (x0, y0))
   end
 
 (** [external::test_swap_non_zero] *)
@@ -107,9 +122,9 @@ let test_swap_non_zero_fwd (x : u32) (st : state) : result (state & u32) =
   begin match swap_fwd u32 x 0 st with
   | Fail -> Fail
   | Return (st0, _) ->
-    begin match swap_back u32 x 0 st with
+    begin match swap_back u32 x 0 st st0 with
     | Fail -> Fail
-    | Return (x0, _) -> if x0 = 0 then Fail else Return (st0, x0)
+    | Return (st1, (x0, _)) -> if x0 = 0 then Fail else Return (st1, x0)
     end
   end
 
diff --git a/tests/misc/External.Opaque.fsti b/tests/misc/External.Opaque.fsti
index 7c241c7c..7d86405a 100644
--- a/tests/misc/External.Opaque.fsti
+++ b/tests/misc/External.Opaque.fsti
@@ -10,10 +10,12 @@ include External.Types
 val core_mem_swap_fwd (t : Type0) : t -> t -> state -> result (state & unit)
 
 (** [core::mem::swap] *)
-val core_mem_swap_back0 (t : Type0) : t -> t -> state -> result t
+val core_mem_swap_back0
+  (t : Type0) : t -> t -> state -> state -> result (state & t)
 
 (** [core::mem::swap] *)
-val core_mem_swap_back1 (t : Type0) : t -> t -> state -> result t
+val core_mem_swap_back1
+  (t : Type0) : t -> t -> state -> state -> result (state & t)
 
 (** [core::num::nonzero::NonZeroU32::{14}::new] *)
 val core_num_nonzero_non_zero_u32_new_fwd