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

21 files changed, 3285 insertions, 239 deletions

diff --git a/.gitignore b/.gitignore
index 489b6f1c..4794d3be 100644
--- a/.gitignore
+++ b/.gitignore
@@ -35,6 +35,7 @@ rust-scripts/target/
 .depend
 *.hints
 tests/betree/obj/
+tests/betree_back_stateful/obj/
 tests/hashmap/obj/
 tests/hashmap_on_disk/obj/
 tests/misc/obj/
diff --git a/Makefile b/Makefile
index c9b0b566..a0ce6d21 100644
--- a/Makefile
+++ b/Makefile
@@ -54,7 +54,8 @@ clean:
 tests: build trans-no_nested_borrows trans-paper \
 	trans-hashmap trans-hashmap_main \
 	trans-external trans-constants \
-	trans-polonius-betree_polonius trans-polonius-betree_main
+	trans-polonius-betree_polonius trans-polonius-betree_main \
+	test-trans-polonius-betree_main
 
 # Verify the F* files generated by the translation
 .PHONY: verify
@@ -66,6 +67,12 @@ verify: build tests
 format:
 	cd compiler && dune promote
 
+# The commands to run Charon to generate the .llbc files
+CHARON_CMD = cd $(CHARON_HOME)/$(CHARON_TEST_SUBFOLDER) && NOT_ALL_TESTS=1 $(MAKE) test-$*
+
+# The command to run Aeneas on the proper llbc file
+AENEAS_CMD = cd compiler && dune exec -- ./$(AENEAS_DRIVER) ../$(CHARON_TESTS_DIR)/$(FILE).llbc -dest ../$(DEST_DIR)/$(SUBDIR) $(TRANS_OPTIONS)
+
 # Add specific options to some tests
 trans-no_nested_borrows trans-paper: \
 	TRANS_OPTIONS += -test-units -no-split-files -no-state -no-decreases-clauses
@@ -86,19 +93,32 @@ trans-constants: SUBDIR:=misc
 trans-external: TRANS_OPTIONS +=
 trans-external: SUBDIR:=misc
 
-trans-polonius-betree_main: TRANS_OPTIONS += -template-clauses
+BETREE_OPTIONS = -template-clauses
+trans-polonius-betree_main: TRANS_OPTIONS += $(BETREE_OPTIONS) -backward-no-state-update
 trans-polonius-betree_main: SUBDIR:=betree
 
+# Additional test on the betree: translate it without `-backward-no-state-update`.
+# This generates very ugly code, but is good to test the translation.
+test-trans-polonius-betree_main: trans-polonius-betree_main
+test-trans-polonius-betree_main: TRANS_OPTIONS += $(BETREE_OPTIONS)
+test-trans-polonius-betree_main: SUBDIR:=betree_back_stateful
+test-trans-polonius-betree_main: CHARON_TESTS_DIR = $(CHARON_HOME)/tests-polonius/llbc
+test-trans-polonius-betree_main: FILE = betree_main
+test-trans-polonius-betree_main:
+	$(AENEAS_CMD)
+
 # Generic rules to extract the LLBC from a rust file
 # We use the rules in Charon's Makefile to generate the .llbc files: the options
 # vary with the test files.
 .PHONY: gen-llbc-polonius-%
+gen-llbc-polonius-%: CHARON_TEST_SUBFOLDER = tests-polonius
 gen-llbc-polonius-%: build
-	cd $(CHARON_HOME)/tests-polonius && NOT_ALL_TESTS=1 $(MAKE) test-$*
+	$(CHARON_CMD)
 
 .PHONY: gen-llbc-%
+gen-llbc-%: CHARON_TEST_SUBFOLDER = tests
 gen-llbc-%: build
-	cd $(CHARON_HOME)/tests && NOT_ALL_TESTS=1 $(MAKE) test-$*
+	$(CHARON_CMD)
 
 # Generic rule to test the translation of an LLBC file.
 # Note that the files requiring the Polonius borrow-checker are generated
@@ -107,8 +127,10 @@ gen-llbc-%: build
 trans-%: CHARON_TESTS_DIR = $(CHARON_HOME)/tests/llbc
 trans-polonius-%: CHARON_TESTS_DIR = $(CHARON_HOME)/tests-polonius/llbc
 
+trans-polonius-%: FILE = $*
 trans-polonius-%: gen-llbc-polonius-%
-	cd compiler && dune exec -- ./$(AENEAS_DRIVER) ../$(CHARON_TESTS_DIR)/$*.llbc -dest ../$(DEST_DIR)/$(SUBDIR) $(TRANS_OPTIONS)
+	$(AENEAS_CMD)
 
+trans-%: FILE = $*
 trans-%: gen-llbc-%
-	cd compiler && dune exec -- ./$(AENEAS_DRIVER) ../$(CHARON_TESTS_DIR)/$*.llbc -dest ../$(DEST_DIR)/$(SUBDIR) $(TRANS_OPTIONS)
+	$(AENEAS_CMD)
diff --git a/compiler/Driver.ml b/compiler/Driver.ml
index d19aca93..6f0e8074 100644
--- a/compiler/Driver.ml
+++ b/compiler/Driver.ml
@@ -38,6 +38,8 @@ let () =
   let test_trans_units = ref false in
   let no_decreases_clauses = ref false in
   let no_state = ref false in
+  (* [backward_no_state_update]: see the comment for {!Translate.config.backward_no_state_update} *)
+  let backward_no_state_update = ref false in
   let template_decreases_clauses = ref false in
   let no_split_files = ref false in
   let no_check_inv = ref false in
@@ -78,6 +80,9 @@ let () =
       ( "-no-state",
         Arg.Set no_state,
         " Do not use state-error monads, simply use error monads" );
+      ( "-backward-no-state-update",
+        Arg.Set backward_no_state_update,
+        " Forbid backward functions from updating the state" );
       ( "-template-clauses",
         Arg.Set template_decreases_clauses,
         " Generate templates for the required decreases clauses, in a\n\
@@ -95,6 +100,8 @@ let () =
   in
   (* Sanity check: -template-clauses ==> not -no-decrease-clauses *)
   assert ((not !no_decreases_clauses) || not !template_decreases_clauses);
+  (* Sanity check: -backward-no-state-update ==> not -no-state *)
+  assert ((not !backward_no_state_update) || not !no_state);
 
   let spec = Arg.align spec in
   let filenames = ref [] in
@@ -110,10 +117,10 @@ let () =
     | [ f ] ->
         (* TODO: update the extension *)
         if not (Filename.check_suffix f ".llbc") then (
-          print_string "Unrecognized file extension";
+          print_string ("Unrecognized file extension: " ^ f ^ "\n");
           fail ())
         else if not (Sys.file_exists f) then (
-          print_string "File not found";
+          print_string ("File not found: " ^ f ^ "\n");
           fail ())
         else f
     | _ ->
@@ -198,6 +205,7 @@ let () =
           extract_decreases_clauses = not !no_decreases_clauses;
           extract_template_decreases_clauses = !template_decreases_clauses;
           use_state = not !no_state;
+          backward_no_state_update = !backward_no_state_update;
         }
       in
       Translate.translate_module filename dest_dir trans_config m;
diff --git a/compiler/ExtractToFStar.ml b/compiler/ExtractToFStar.ml
index 2a7d6a6c..a995d4a6 100644
--- a/compiler/ExtractToFStar.ml
+++ b/compiler/ExtractToFStar.ml
@@ -1451,17 +1451,19 @@ let extract_fun_decl (ctx : extraction_ctx) (fmt : F.formatter)
        * function (the additional input values "given back" to the
        * backward functions have no influence on termination: we thus
        * share the decrease clauses between the forward and the backward
-       * functions).
+       * functions - we also ignore the additional state received by the
+       * backward function, if there is one).
        *)
       let inputs_lvs =
         let all_inputs = (Option.get def.body).inputs_lvs in
         (* We have to count:
          * - the forward inputs
-         * - the state
+         * - the state (if there is one)
          *)
         let num_fwd_inputs = def.signature.info.num_fwd_inputs in
         let num_fwd_inputs =
-          if def.signature.info.effect_info.input_state then 1 + num_fwd_inputs
+          if def.signature.info.effect_info.stateful_group then
+            1 + num_fwd_inputs
           else num_fwd_inputs
         in
         Collections.List.prefix num_fwd_inputs all_inputs
diff --git a/compiler/FunsAnalysis.ml b/compiler/FunsAnalysis.ml
index 9413bd6a..4d33056b 100644
--- a/compiler/FunsAnalysis.ml
+++ b/compiler/FunsAnalysis.ml
@@ -16,7 +16,7 @@ module EU = ExpressionsUtils
  *)
 type fun_info = {
   can_fail : bool;
-  (* Not used yet: all the extracted functions use an error monad *)
+      (* Not used yet: all the extracted functions use an error monad *)
   stateful : bool;
   divergent : bool; (* Not used yet *)
 }
diff --git a/compiler/Interpreter.ml b/compiler/Interpreter.ml
index d3b3c7e6..e752594e 100644
--- a/compiler/Interpreter.ml
+++ b/compiler/Interpreter.ml
@@ -253,8 +253,15 @@ let evaluate_function_symbolic (config : C.partial_config) (synthesize : bool)
               cf_pop cf_return ctx
           | Some back_id ->
               (* Backward translation *)
-              evaluate_function_symbolic_synthesize_backward_from_return config
-                fdef inst_sg back_id ctx
+              let e =
+                evaluate_function_symbolic_synthesize_backward_from_return
+                  config fdef inst_sg back_id ctx
+              in
+              (* We insert a delimiter to indicate the point where we switch
+               * from the part which is common to all the functions (forwards
+               * and backwards) and the part specific to this backward function.
+               *)
+              S.synthesize_forward_end e
         else None
     | Panic ->
         (* Note that as we explore all the execution branches, one of
diff --git a/compiler/Pure.ml b/compiler/Pure.ml
index a50dd5f9..cc29469a 100644
--- a/compiler/Pure.ml
+++ b/compiler/Pure.ml
@@ -496,19 +496,26 @@ and meta =
 
 (** Information about the "effect" of a function *)
 type fun_effect_info = {
-  input_state : bool;  (** [true] if the function takes a state as input *)
-  output_state : bool;
-      (** [true] if the function outputs a state (it then lives
-                            in a state monad) *)
+  stateful_group : bool;
+      (** [true] if the function group is stateful. By *function group*, we mean
+          the set { forward function } U { backward functions }.
+
+          We need this because the option {!Translate.eval_config.backward_no_state_update}:
+          if it is [true], then in case of a backward function {!stateful} is [false],
+          but we might need to know whether the corresponding forward function
+          is stateful or not.
+       *)
+  stateful : bool;  (** [true] if the function is stateful (updates a state) *)
   can_fail : bool;  (** [true] if the return type is a [result] *)
 }
 
 (** Meta information about a function signature *)
 type fun_sig_info = {
   num_fwd_inputs : int;
-      (** The number of input types for forward computation *)
+      (** The number of input types for forward computation, ignoring the state *)
   num_back_inputs : int option;
-      (** The number of additional inputs for the backward computation (if pertinent) *)
+      (** The number of additional inputs for the backward computation (if pertinent),
+          ignoring the state *)
   effect_info : fun_effect_info;
 }
 
@@ -523,12 +530,18 @@ type fun_sig_info = {
       `in_ty0 -> ... -> in_tyn -> back_in0 -> ... back_inm -> (back_out0 & ... & back_outp)` (* pure function *)
       `in_ty0 -> ... -> in_tyn -> back_in0 -> ... back_inm ->
        result (back_out0 & ... & back_outp)` (* error-monad *)
-      `in_ty0 -> ... -> in_tyn -> state -> back_in0 -> ... back_inm ->
-       result (back_out0 & ... & back_outp)` (* state-error *)
+      `in_ty0 -> ... -> in_tyn -> state -> back_in0 -> ... back_inm -> state ->
+       result (state & (back_out0 & ... & back_outp))` (* state-error *)
        
-       Note that a backward function never returns (i.e., updates) a state: only
-       forward functions do so. Also, the state input parameter is *betwee*
-       the forward inputs and the backward inputs.
+       Note that a stateful backward function takes two states as inputs: the
+       state received by the associated forward function, and the state at which
+       the backward is called. This leads to code of the following shape:
+
+       {[
+         (st1, y)  <-- f_fwd x st0; // st0 is the state upon calling f_fwd
+         ... // the state may be updated
+         (st3, x') <-- f_back x st0 y' st2; // st2 is the state upon calling f_back
+       ]}
 
     The function's type should be given by `mk_arrows sig.inputs sig.output`.
     We provide additional meta-information:
diff --git a/compiler/PureUtils.ml b/compiler/PureUtils.ml
index ff379bf5..1ab3439c 100644
--- a/compiler/PureUtils.ml
+++ b/compiler/PureUtils.ml
@@ -456,3 +456,9 @@ let mk_result_return_pattern (v : typed_pattern) : typed_pattern =
 
 let opt_unmeta_mplace (e : texpression) : mplace option * texpression =
   match e.e with Meta (MPlace mp, e) -> (Some mp, e) | _ -> (None, e)
+
+let mk_state_var (vid : VarId.id) : var =
+  { id = vid; basename = Some ConstStrings.state_basename; ty = mk_state_ty }
+
+let mk_state_texpression (vid : VarId.id) : texpression =
+  { e = Var vid; ty = mk_state_ty }
diff --git a/compiler/SymbolicAst.ml b/compiler/SymbolicAst.ml
index 528d8255..9d9adf4f 100644
--- a/compiler/SymbolicAst.ml
+++ b/compiler/SymbolicAst.ml
@@ -77,6 +77,14 @@ type expression =
           We use it to compute meaningful names for the variables we introduce,
           to prettify the generated code.
        *)
+  | ForwardEnd of expression
+      (** We use this delimiter to indicate at which point we switch to the
+          generation of code specific to the backward function(s).
+
+          TODO: use this to factorize the generation of the forward and backward
+          functions (today we replay the *whole* symbolic execution once per
+          generated function).
+       *)
   | Meta of meta * expression  (** Meta information *)
 
 and expansion =
diff --git a/compiler/SymbolicToPure.ml b/compiler/SymbolicToPure.ml
index 6d01614d..9d249cfb 100644
--- a/compiler/SymbolicToPure.ml
+++ b/compiler/SymbolicToPure.ml
@@ -12,6 +12,10 @@ module FA = FunsAnalysis
 (** The local logger *)
 let log = L.symbolic_to_pure_log
 
+(* TODO: carrying configurations everywhere is super annoying.
+   Group everything in references in a [Config.ml] file (put aside the execution
+   mode, maybe).
+*)
 type config = {
   filter_useless_back_calls : bool;
       (** If [true], filter the useless calls to backward functions.
@@ -39,6 +43,12 @@ type config = {
           Note that we later filter the useless *forward* calls in the micro-passes,
           where it is more natural to do.
        *)
+  backward_no_state_update : bool;
+      (** Controls whether backward functions update the state, in case we use
+          a state ({!use_state}).
+
+          See {!Translate.config.backward_no_state_update}.
+   *)
 }
 
 type type_context = {
@@ -110,7 +120,23 @@ type bs_ctx = {
        *)
   var_counter : VarId.generator;
   state_var : VarId.id;
-      (** The current state variable, in case we use a state *)
+      (** The current state variable, in case the function is stateful *)
+  back_state_var : VarId.id;
+      (** The additional input state variable received by a stateful backward function.
+          When generating stateful functions, we generate code of the following
+          form:
+
+          {[
+            (st1, y)  <-- f_fwd x st0; // st0 is the state upon calling f_fwd
+            ... // the state may be updated
+            (st3, x') <-- f_back x st0 y' st2; // st2 is the state upon calling f_back
+          ]}
+
+          When translating a backward function, we need at some point to update
+          [state_var] with [back_state_var], to account for the fact that the
+          state may have been updated by the caller between the call to the
+          forward function and the call to the backward function.
+       *)
   forward_inputs : var list;
       (** The input parameters for the forward function *)
   backward_inputs : var list T.RegionGroupId.Map.t;
@@ -498,20 +524,26 @@ let list_ancestor_abstractions (ctx : bs_ctx) (abs : V.abs) :
   let abs_ids = list_ancestor_abstractions_ids ctx abs in
   List.map (fun id -> V.AbstractionId.Map.find id ctx.abstractions) abs_ids
 
-(** Small utility. *)
-let get_fun_effect_info (fun_infos : FA.fun_info A.FunDeclId.Map.t)
-    (fun_id : A.fun_id) (gid : T.RegionGroupId.id option) : fun_effect_info =
+(** Small utility.
+
+    [backward_no_state_update]: see {!config}
+ *)
+let get_fun_effect_info (backward_no_state_update : bool)
+    (fun_infos : FA.fun_info A.FunDeclId.Map.t) (fun_id : A.fun_id)
+    (gid : T.RegionGroupId.id option) : fun_effect_info =
   match fun_id with
   | A.Regular fid ->
       let info = A.FunDeclId.Map.find fid fun_infos in
-      let input_state = info.stateful in
-      let output_state = input_state && gid = None in
-      { can_fail = info.can_fail; input_state; output_state }
+      let stateful_group = info.stateful in
+      let stateful =
+        stateful_group && ((not backward_no_state_update) || gid = None)
+      in
+      { can_fail = info.can_fail; stateful_group; stateful }
   | A.Assumed aid ->
       {
         can_fail = Assumed.assumed_can_fail aid;
-        input_state = false;
-        output_state = false;
+        stateful_group = false;
+        stateful = false;
       }
 
 (** Translate a function signature.
@@ -519,10 +551,11 @@ let get_fun_effect_info (fun_infos : FA.fun_info A.FunDeclId.Map.t)
     Note that the function also takes a list of names for the inputs, and
     computes, for every output for the backward functions, a corresponding
     name (outputs for backward functions come from borrows in the inputs
-    of the forward function).
+    of the forward function) which we use as hints to generate pretty names.
  *)
-let translate_fun_sig (fun_infos : FA.fun_info A.FunDeclId.Map.t)
-    (fun_id : A.fun_id) (types_infos : TA.type_infos) (sg : A.fun_sig)
+let translate_fun_sig (backward_no_state_update : bool)
+    (fun_infos : FA.fun_info A.FunDeclId.Map.t) (fun_id : A.fun_id)
+    (types_infos : TA.type_infos) (sg : A.fun_sig)
     (input_names : string option list) (bid : T.RegionGroupId.id option) :
     fun_sig_named_outputs =
   (* Retrieve the list of parent backward functions *)
@@ -572,17 +605,42 @@ let translate_fun_sig (fun_infos : FA.fun_info A.FunDeclId.Map.t)
         *)
         List.filter_map (translate_back_ty_for_gid gid) [ sg.output ]
   in
-  (* Does the function take a state as input, does it return a state and can
-   * it fail? *)
-  let effect_info = get_fun_effect_info fun_infos fun_id bid in
-  (* *)
-  let state_ty = if effect_info.input_state then [ mk_state_ty ] else [] in
+  (* Is the function stateful, and can it fail? *)
+  let effect_info =
+    get_fun_effect_info backward_no_state_update fun_infos fun_id bid
+  in
+  (* If the function is stateful, the inputs are:
+     - forward:  [fwd_ty0, ..., fwd_tyn, state]
+     - backward:
+       - if config.no_backward_state: [fwd_ty0, ..., fwd_tyn, state, back_ty, state]
+       - otherwise: [fwd_ty0, ..., fwd_tyn, state, back_ty]
+
+       The backward takes the same state as input as the forward function,
+       together with the state at the point where it gets called, if it is
+       stateful.
+
+       See the comments for {!Translate.config.backward_no_state_update}
+  *)
+  let fwd_state_ty =
+    (* For the forward state, we check if the *whole group* is stateful.
+       See {!effect_info}. *)
+    if effect_info.stateful_group then [ mk_state_ty ] else []
+  in
+  let back_state_ty =
+    (* For the backward state, we check if the function is a backward function,
+       and it is stateful *)
+    if effect_info.stateful && Option.is_some gid then [ mk_state_ty ] else []
+  in
+
   (* Concatenate the inputs, in the following order:
    * - forward inputs
-   * - state input
+   * - forward state input
    * - backward inputs
+   * - backward state input
    *)
-  let inputs = List.concat [ fwd_inputs; state_ty; back_inputs ] in
+  let inputs =
+    List.concat [ fwd_inputs; fwd_state_ty; back_inputs; back_state_ty ]
+  in
   (* Outputs *)
   let output_names, doutputs =
     match gid with
@@ -620,7 +678,7 @@ let translate_fun_sig (fun_infos : FA.fun_info A.FunDeclId.Map.t)
     let output = mk_simpl_tuple_ty doutputs in
     (* Add the output state *)
     let output =
-      if effect_info.output_state then mk_simpl_tuple_ty [ mk_state_ty; output ]
+      if effect_info.stateful then mk_simpl_tuple_ty [ mk_state_ty; output ]
       else output
     in
     (* Wrap in a result type *)
@@ -1087,6 +1145,15 @@ let rec translate_expression (config : config) (e : S.expression) (ctx : bs_ctx)
   | Assertion (v, e) -> translate_assertion config v e ctx
   | Expansion (p, sv, exp) -> translate_expansion config p sv exp ctx
   | Meta (meta, e) -> translate_meta config meta e ctx
+  | ForwardEnd e ->
+      (* Update the current state with the additional state received by the backward
+         function, if needs be *)
+      let ctx =
+        match ctx.bid with
+        | None -> ctx
+        | Some _ -> { ctx with state_var = ctx.back_state_var }
+      in
+      translate_expression config e ctx
 
 and translate_panic (ctx : bs_ctx) : texpression =
   (* Here we use the function return type - note that it is ok because
@@ -1095,13 +1162,15 @@ and translate_panic (ctx : bs_ctx) : texpression =
   (* If we use a state monad, we need to add a lambda for the state variable *)
   (* Note that only forward functions return a state *)
   let output_ty = mk_simpl_tuple_ty ctx.sg.doutputs in
-  if ctx.sg.info.effect_info.output_state then
+  (* TODO: we should use a [Fail] function *)
+  if ctx.sg.info.effect_info.stateful then
     (* Create the [Fail] value *)
     let ret_ty = mk_simpl_tuple_ty [ mk_state_ty; output_ty ] in
     let ret_v = mk_result_fail_texpression ret_ty in
     ret_v
   else mk_result_fail_texpression output_ty
 
+(** [opt_v]: the value to return, in case we translate a forward function *)
 and translate_return (opt_v : V.typed_value option) (ctx : bs_ctx) : texpression
     =
   (* There are two cases:
@@ -1109,44 +1178,40 @@ and translate_return (opt_v : V.typed_value option) (ctx : bs_ctx) : texpression
        value should be [Some] (it is the returned value)
      - or we are translating a backward function, in which case it should be [None]
   *)
-  match ctx.bid with
-  | None ->
-      (* Forward function *)
-      let v = Option.get opt_v in
-      let v = typed_value_to_texpression ctx v in
-      (* We may need to return a state
-       * - error-monad: Return x
-       * - state-error: Return (state, x)
-       * *)
-      if ctx.sg.info.effect_info.output_state then
-        let state_var =
-          {
-            id = ctx.state_var;
-            basename = Some ConstStrings.state_basename;
-            ty = mk_state_ty;
-          }
+  (* Compute the values that we should return *without the state and the result
+   * wrapper* *)
+  let output =
+    match ctx.bid with
+    | None ->
+        (* Forward function *)
+        let v = Option.get opt_v in
+        typed_value_to_texpression ctx v
+    | Some bid ->
+        (* Backward function *)
+        (* Sanity check *)
+        assert (opt_v = None);
+        (* Group the variables in which we stored the values we need to give back.
+         * See the explanations for the [SynthInput] case in [translate_end_abstraction] *)
+        let backward_outputs =
+          T.RegionGroupId.Map.find bid ctx.backward_outputs
         in
-        let state_rvalue = mk_texpression_from_var state_var in
-        mk_result_return_texpression
-          (mk_simpl_tuple_texpression [ state_rvalue; v ])
-      else mk_result_return_texpression v
-  | Some bid ->
-      (* Backward function *)
-      (* Sanity check *)
-      assert (opt_v = None);
-      assert (not ctx.sg.info.effect_info.output_state);
-      (* We simply need to return the variables in which we stored the values
-       * we need to give back.
-       * See the explanations for the [SynthInput] case in [translate_end_abstraction] *)
-      let backward_outputs =
-        T.RegionGroupId.Map.find bid ctx.backward_outputs
-      in
-      let field_values = List.map mk_texpression_from_var backward_outputs in
-      (* Backward functions never return a state *)
-      (* TODO: we should use a [fail] function, it would be cleaner *)
-      let ret_value = mk_simpl_tuple_texpression field_values in
-      let ret_value = mk_result_return_texpression ret_value in
-      ret_value
+        let field_values = List.map mk_texpression_from_var backward_outputs in
+        mk_simpl_tuple_texpression field_values
+  in
+  (* We may need to return a state
+   * - error-monad: Return x
+   * - state-error: Return (state, x)
+   * *)
+  let effect_info = ctx.sg.info.effect_info in
+  let output =
+    if effect_info.stateful then
+      let state_rvalue = mk_state_texpression ctx.state_var in
+      mk_simpl_tuple_texpression [ state_rvalue; output ]
+    else output
+  in
+  (* Wrap in a result - TODO: check effect_info.can_fail to not always wrap *)
+  (* TODO: we should use a [Return] function *)
+  mk_result_return_texpression output
 
 and translate_function_call (config : config) (call : S.call) (e : S.expression)
     (ctx : bs_ctx) : texpression =
@@ -1171,29 +1236,26 @@ and translate_function_call (config : config) (call : S.call) (e : S.expression)
         (* Retrieve the effect information about this function (can fail,
          * takes a state as input, etc.) *)
         let effect_info =
-          get_fun_effect_info ctx.fun_context.fun_infos fid None
-        in
-        (* Add the state input argument *)
-        let args =
-          if effect_info.input_state then
-            let state_var = { e = Var ctx.state_var; ty = mk_state_ty } in
-            List.append args [ state_var ]
-          else args
+          get_fun_effect_info config.backward_no_state_update
+            ctx.fun_context.fun_infos fid None
         in
-        (* Generate a fresh state variable if the function call introduces
-         * a new variable *)
-        let ctx, out_state =
-          if effect_info.input_state then
-            let ctx, var = bs_ctx_fresh_state_var ctx in
-            (ctx, Some var)
-          else (ctx, None)
+        (* If the function is stateful:
+         * - add the state input argument
+         * - generate a fresh state variable for the returned state
+         *)
+        let args, ctx, out_state =
+          if effect_info.stateful then
+            let state_var = mk_state_texpression ctx.state_var in
+            let ctx, nstate_var = bs_ctx_fresh_state_var ctx in
+            (List.append args [ state_var ], ctx, Some nstate_var)
+          else (args, ctx, None)
         in
         (* Register the function call *)
         let ctx = bs_ctx_register_forward_call call_id call args ctx in
         (ctx, func, effect_info, args, out_state)
     | S.Unop E.Not ->
         let effect_info =
-          { can_fail = false; input_state = false; output_state = false }
+          { can_fail = false; stateful_group = false; stateful = false }
         in
         (ctx, Unop Not, effect_info, args, None)
     | S.Unop E.Neg -> (
@@ -1203,14 +1265,14 @@ and translate_function_call (config : config) (call : S.call) (e : S.expression)
             (* Note that negation can lead to an overflow and thus fail (it
              * is thus monadic) *)
             let effect_info =
-              { can_fail = true; input_state = false; output_state = false }
+              { can_fail = true; stateful_group = false; stateful = false }
             in
             (ctx, Unop (Neg int_ty), effect_info, args, None)
         | _ -> raise (Failure "Unreachable"))
     | S.Unop (E.Cast (src_ty, tgt_ty)) ->
         (* Note that cast can fail *)
         let effect_info =
-          { can_fail = true; input_state = false; output_state = false }
+          { can_fail = true; stateful_group = false; stateful = false }
         in
         (ctx, Unop (Cast (src_ty, tgt_ty)), effect_info, args, None)
     | S.Binop binop -> (
@@ -1222,8 +1284,8 @@ and translate_function_call (config : config) (call : S.call) (e : S.expression)
             let effect_info =
               {
                 can_fail = ExpressionsUtils.binop_can_fail binop;
-                input_state = false;
-                output_state = false;
+                stateful_group = false;
+                stateful = false;
               }
             in
             (ctx, Binop (binop, int_ty0), effect_info, args, None)
@@ -1307,6 +1369,17 @@ and translate_end_abstraction (config : config) (abs : V.abs) (e : S.expression)
   | V.FunCall ->
       let call_info = V.FunCallId.Map.find abs.call_id ctx.calls in
       let call = call_info.forward in
+      let fun_id =
+        match call.call_id with
+        | S.Fun (fun_id, _) -> fun_id
+        | Unop _ | Binop _ ->
+            (* Those don't have backward functions *)
+            raise (Failure "Unreachable")
+      in
+      let effect_info =
+        get_fun_effect_info config.backward_no_state_update
+          ctx.fun_context.fun_infos fun_id (Some abs.back_id)
+      in
       let type_args = List.map (ctx_translate_fwd_ty ctx) call.type_params in
       (* Retrieve the original call and the parent abstractions *)
       let _forward, backwards = get_abs_ancestors ctx abs in
@@ -1322,8 +1395,21 @@ and translate_end_abstraction (config : config) (abs : V.abs) (e : S.expression)
       (* Retrieve the values consumed upon ending the loans inside this
        * abstraction: those give us the remaining input values *)
       let back_inputs = abs_to_consumed ctx abs in
+      (* If the function is stateful:
+       * - add the state input argument
+       * - generate a fresh state variable for the returned state
+       *)
+      let back_state, ctx, nstate =
+        if effect_info.stateful then
+          let back_state = mk_state_texpression ctx.state_var in
+          let ctx, nstate = bs_ctx_fresh_state_var ctx in
+          ([ back_state ], ctx, Some nstate)
+        else ([], ctx, None)
+      in
+      (* Concatenate all the inpus *)
       let inputs =
-        List.concat [ fwd_inputs; back_ancestors_inputs; back_inputs ]
+        List.concat
+          [ fwd_inputs; back_ancestors_inputs; back_inputs; back_state ]
       in
       (* Retrieve the values given back by this function: those are the output
        * values. We rely on the fact that there are no nested borrows to use the
@@ -1333,43 +1419,42 @@ and translate_end_abstraction (config : config) (abs : V.abs) (e : S.expression)
         List.append (List.map translate_opt_mplace call.args_places) [ None ]
       in
       let ctx, outputs = abs_to_given_back output_mpl abs ctx in
-      (* Group the output values together (note that for now, backward functions
-       * never return an output state) *)
+      (* Group the output values together: first the updated inputs *)
       let output = mk_simpl_tuple_pattern outputs in
-      (* Sanity check: the inputs and outputs have the proper number and the proper type *)
-      let fun_id =
-        match call.call_id with
-        | S.Fun (fun_id, _) -> fun_id
-        | Unop _ | Binop _ ->
-            (* Those don't have backward functions *)
-            raise (Failure "Unreachable")
+      (* Add the returned state if the function is stateful *)
+      let output =
+        match nstate with
+        | None -> output
+        | Some nstate -> mk_simpl_tuple_pattern [ nstate; output ]
       in
-
-      let inst_sg =
-        get_instantiated_fun_sig fun_id (Some abs.back_id) type_args ctx
+      (* Sanity check: the inputs and outputs have the proper number and the proper type *)
+      let _ =
+        let inst_sg =
+          get_instantiated_fun_sig fun_id (Some abs.back_id) type_args ctx
+        in
+        log#ldebug
+          (lazy
+            ("\n- fun_id: " ^ A.show_fun_id fun_id ^ "\n- inputs ("
+            ^ string_of_int (List.length inputs)
+            ^ "): "
+            ^ String.concat ", " (List.map show_texpression inputs)
+            ^ "\n- inst_sg.inputs ("
+            ^ string_of_int (List.length inst_sg.inputs)
+            ^ "): "
+            ^ String.concat ", " (List.map show_ty inst_sg.inputs)));
+        List.iter
+          (fun (x, ty) -> assert ((x : texpression).ty = ty))
+          (List.combine inputs inst_sg.inputs);
+        log#ldebug
+          (lazy
+            ("\n- outputs: "
+            ^ string_of_int (List.length outputs)
+            ^ "\n- expected outputs: "
+            ^ string_of_int (List.length inst_sg.doutputs)));
+        List.iter
+          (fun (x, ty) -> assert ((x : typed_pattern).ty = ty))
+          (List.combine outputs inst_sg.doutputs)
       in
-      log#ldebug
-        (lazy
-          ("\n- fun_id: " ^ A.show_fun_id fun_id ^ "\n- inputs ("
-          ^ string_of_int (List.length inputs)
-          ^ "): "
-          ^ String.concat ", " (List.map show_texpression inputs)
-          ^ "\n- inst_sg.inputs ("
-          ^ string_of_int (List.length inst_sg.inputs)
-          ^ "): "
-          ^ String.concat ", " (List.map show_ty inst_sg.inputs)));
-      List.iter
-        (fun (x, ty) -> assert ((x : texpression).ty = ty))
-        (List.combine inputs inst_sg.inputs);
-      log#ldebug
-        (lazy
-          ("\n- outputs: "
-          ^ string_of_int (List.length outputs)
-          ^ "\n- expected outputs: "
-          ^ string_of_int (List.length inst_sg.doutputs)));
-      List.iter
-        (fun (x, ty) -> assert ((x : typed_pattern).ty = ty))
-        (List.combine outputs inst_sg.doutputs);
       (* Retrieve the function id, and register the function call in the context
        * if necessary *)
       let ctx, func = bs_ctx_register_backward_call abs back_inputs ctx in
@@ -1382,9 +1467,6 @@ and translate_end_abstraction (config : config) (abs : V.abs) (e : S.expression)
           (fun (arg, mp) -> mk_opt_mplace_texpression mp arg)
           (List.combine inputs args_mplaces)
       in
-      let effect_info =
-        get_fun_effect_info ctx.fun_context.fun_infos fun_id (Some abs.back_id)
-      in
       let input_tys = (List.map (fun (x : texpression) -> x.ty)) args in
       let ret_ty =
         if effect_info.can_fail then mk_result_ty output.ty else output.ty
@@ -1398,8 +1480,13 @@ and translate_end_abstraction (config : config) (abs : V.abs) (e : S.expression)
        * We do a small optimization here: if the backward function doesn't
        * have any output, we don't introduce any function call.
        * See the comment in [config].
+       *
+       * TODO: use an option to disallow backward functions from updating the state.
+       * TODO: a backward function which only gives back shared borrows shouldn't
+       * update the state (state updates should only be used for mutable borrows,
+       * with objects like Rc for instance.
        *)
-      if config.filter_useless_back_calls && outputs = [] then (
+      if config.filter_useless_back_calls && outputs = [] && nstate = None then (
         (* No outputs - we do a small sanity check: the backward function
          * should have exactly the same number of inputs as the forward:
          * this number can be different only if the forward function returned
@@ -1708,32 +1795,28 @@ let translate_fun_decl (config : config) (ctx : bs_ctx)
   (* Translate the declaration *)
   let def_id = def.A.def_id in
   let basename = def.name in
-  (* Lookup the signature *)
-  let signature = bs_ctx_lookup_local_function_sig def_id bid ctx in
+  (* Retrieve the signature *)
+  let signature = ctx.sg in
   (* Translate the body, if there is *)
   let body =
     match body with
     | None -> None
     | Some body ->
         let body = translate_expression config body ctx in
-        (* Sanity check *)
-        type_check_texpression ctx body;
-        (* Introduce the input state, if necessary *)
         let effect_info =
-          get_fun_effect_info ctx.fun_context.fun_infos (Regular def_id) bid
+          get_fun_effect_info config.backward_no_state_update
+            ctx.fun_context.fun_infos (Regular def_id) bid
         in
-        let input_state =
-          if effect_info.input_state then
-            [
-              {
-                id = ctx.state_var;
-                basename = Some ConstStrings.state_basename;
-                ty = mk_state_ty;
-              };
-            ]
+        (* Sanity check *)
+        type_check_texpression ctx body;
+        (* Introduce the forward input state (the state at call site of the
+         * *forward* function), if necessary. *)
+        let fwd_state =
+          (* We check if the *whole group* is stateful. See {!effect_info} *)
+          if effect_info.stateful_group then [ mk_state_var ctx.state_var ]
           else []
         in
-        (* Compute the list of (properly ordered) input variables *)
+        (* Compute the list of (properly ordered) backward input variables *)
         let backward_inputs : var list =
           match bid with
           | None -> []
@@ -1747,8 +1830,17 @@ let translate_fun_decl (config : config) (ctx : bs_ctx)
                    (fun id -> T.RegionGroupId.Map.find id ctx.backward_inputs)
                    backward_ids)
         in
+        (* Introduce the backward input state (the state at call site of the
+         * *backward* function), if necessary *)
+        let back_state =
+          if effect_info.stateful && Option.is_some bid then
+            [ mk_state_var ctx.back_state_var ]
+          else []
+        in
+        (* Group the inputs together *)
         let inputs =
-          List.concat [ ctx.forward_inputs; input_state; backward_inputs ]
+          List.concat
+            [ ctx.forward_inputs; fwd_state; backward_inputs; back_state ]
         in
         let inputs_lvs =
           List.map (fun v -> mk_typed_pattern_from_var v None) inputs
@@ -1756,12 +1848,18 @@ let translate_fun_decl (config : config) (ctx : bs_ctx)
         (* Sanity check *)
         log#ldebug
           (lazy
-            ("SymbolicToPure.translate_fun_decl:" ^ "\n- forward_inputs: "
+            ("SymbolicToPure.translate_fun_decl: "
+            ^ Print.fun_name_to_string def.A.name
+            ^ " ("
+            ^ Print.option_to_string T.RegionGroupId.to_string bid
+            ^ ")" ^ "\n- forward_inputs: "
             ^ String.concat ", " (List.map show_var ctx.forward_inputs)
-            ^ "\n- input_state: "
-            ^ String.concat ", " (List.map show_var input_state)
+            ^ "\n- fwd_state: "
+            ^ String.concat ", " (List.map show_var fwd_state)
             ^ "\n- backward_inputs: "
             ^ String.concat ", " (List.map show_var backward_inputs)
+            ^ "\n- back_state: "
+            ^ String.concat ", " (List.map show_var back_state)
             ^ "\n- signature.inputs: "
             ^ String.concat ", " (List.map show_ty signature.inputs)));
         assert (
@@ -1804,8 +1902,8 @@ let translate_type_decls (type_decls : T.type_decl list) : type_decl list =
     - optional names for the outputs values (we derive them for the backward
       functions)
  *)
-let translate_fun_signatures (fun_infos : FA.fun_info A.FunDeclId.Map.t)
-    (types_infos : TA.type_infos)
+let translate_fun_signatures (backward_no_state_update : bool)
+    (fun_infos : FA.fun_info A.FunDeclId.Map.t) (types_infos : TA.type_infos)
     (functions : (A.fun_id * string option list * A.fun_sig) list) :
     fun_sig_named_outputs RegularFunIdMap.t =
   (* For every function, translate the signatures of:
@@ -1816,7 +1914,8 @@ let translate_fun_signatures (fun_infos : FA.fun_info A.FunDeclId.Map.t)
       (sg : A.fun_sig) : (regular_fun_id * fun_sig_named_outputs) list =
     (* The forward function *)
     let fwd_sg =
-      translate_fun_sig fun_infos fun_id types_infos sg input_names None
+      translate_fun_sig backward_no_state_update fun_infos fun_id types_infos sg
+        input_names None
     in
     let fwd_id = (fun_id, None) in
     (* The backward functions *)
@@ -1824,8 +1923,8 @@ let translate_fun_signatures (fun_infos : FA.fun_info A.FunDeclId.Map.t)
       List.map
         (fun (rg : T.region_var_group) ->
           let tsg =
-            translate_fun_sig fun_infos fun_id types_infos sg input_names
-              (Some rg.id)
+            translate_fun_sig backward_no_state_update fun_infos fun_id
+              types_infos sg input_names (Some rg.id)
           in
           let id = (fun_id, Some rg.id) in
           (id, tsg))
diff --git a/compiler/SynthesizeSymbolic.ml b/compiler/SynthesizeSymbolic.ml
index c74a831e..8d4dac82 100644
--- a/compiler/SynthesizeSymbolic.ml
+++ b/compiler/SynthesizeSymbolic.ml
@@ -12,7 +12,7 @@ let mk_mplace (p : E.place) (ctx : Contexts.eval_ctx) : mplace =
 
 let mk_opt_mplace (p : E.place option) (ctx : Contexts.eval_ctx) : mplace option
     =
-  match p with None -> None | Some p -> Some (mk_mplace p ctx)
+  Option.map (fun p -> mk_mplace p ctx) p
 
 let mk_opt_place_from_op (op : E.operand) (ctx : Contexts.eval_ctx) :
     mplace option =
@@ -22,11 +22,11 @@ let mk_opt_place_from_op (op : E.operand) (ctx : Contexts.eval_ctx) :
 
 let synthesize_symbolic_expansion (sv : V.symbolic_value)
     (place : mplace option) (seel : V.symbolic_expansion option list)
-    (exprl : expression list option) : expression option =
-  match exprl with
+    (el : expression list option) : expression option =
+  match el with
   | None -> None
-  | Some exprl ->
-      let ls = List.combine seel exprl in
+  | Some el ->
+      let ls = List.combine seel el in
       (* Match on the symbolic value type to know which can of expansion happened *)
       let expansion =
         match sv.V.sv_ty with
@@ -89,19 +89,18 @@ let synthesize_symbolic_expansion (sv : V.symbolic_value)
       Some (Expansion (place, sv, expansion))
 
 let synthesize_symbolic_expansion_no_branching (sv : V.symbolic_value)
-    (place : mplace option) (see : V.symbolic_expansion)
-    (expr : expression option) : expression option =
-  let exprl = match expr with None -> None | Some expr -> Some [ expr ] in
-  synthesize_symbolic_expansion sv place [ Some see ] exprl
+    (place : mplace option) (see : V.symbolic_expansion) (e : expression option)
+    : expression option =
+  let el = Option.map (fun e -> [ e ]) e in
+  synthesize_symbolic_expansion sv place [ Some see ] el
 
 let synthesize_function_call (call_id : call_id)
     (abstractions : V.AbstractionId.id list) (type_params : T.ety list)
     (args : V.typed_value list) (args_places : mplace option list)
     (dest : V.symbolic_value) (dest_place : mplace option)
-    (expr : expression option) : expression option =
-  match expr with
-  | None -> None
-  | Some expr ->
+    (e : expression option) : expression option =
+  Option.map
+    (fun e ->
       let call =
         {
           call_id;
@@ -113,48 +112,45 @@ let synthesize_function_call (call_id : call_id)
           dest_place;
         }
       in
-      Some (FunCall (call, expr))
+      FunCall (call, e))
+    e
 
 let synthesize_global_eval (gid : A.GlobalDeclId.id) (dest : V.symbolic_value)
-    (expr : expression option) : expression option =
-  match expr with None -> None | Some e -> Some (EvalGlobal (gid, dest, e))
+    (e : expression option) : expression option =
+  Option.map (fun e -> EvalGlobal (gid, dest, e)) e
 
 let synthesize_regular_function_call (fun_id : A.fun_id)
     (call_id : V.FunCallId.id) (abstractions : V.AbstractionId.id list)
     (type_params : T.ety list) (args : V.typed_value list)
     (args_places : mplace option list) (dest : V.symbolic_value)
-    (dest_place : mplace option) (expr : expression option) : expression option
-    =
+    (dest_place : mplace option) (e : expression option) : expression option =
   synthesize_function_call
     (Fun (fun_id, call_id))
-    abstractions type_params args args_places dest dest_place expr
+    abstractions type_params args args_places dest dest_place e
 
 let synthesize_unary_op (unop : E.unop) (arg : V.typed_value)
     (arg_place : mplace option) (dest : V.symbolic_value)
-    (dest_place : mplace option) (expr : expression option) : expression option
-    =
+    (dest_place : mplace option) (e : expression option) : expression option =
   synthesize_function_call (Unop unop) [] [] [ arg ] [ arg_place ] dest
-    dest_place expr
+    dest_place e
 
 let synthesize_binary_op (binop : E.binop) (arg0 : V.typed_value)
     (arg0_place : mplace option) (arg1 : V.typed_value)
     (arg1_place : mplace option) (dest : V.symbolic_value)
-    (dest_place : mplace option) (expr : expression option) : expression option
-    =
+    (dest_place : mplace option) (e : expression option) : expression option =
   synthesize_function_call (Binop binop) [] [] [ arg0; arg1 ]
-    [ arg0_place; arg1_place ] dest dest_place expr
+    [ arg0_place; arg1_place ] dest dest_place e
 
-let synthesize_end_abstraction (abs : V.abs) (expr : expression option) :
+let synthesize_end_abstraction (abs : V.abs) (e : expression option) :
     expression option =
-  match expr with
-  | None -> None
-  | Some expr -> Some (EndAbstraction (abs, expr))
+  Option.map (fun e -> EndAbstraction (abs, e)) e
 
 let synthesize_assignment (lplace : mplace) (rvalue : V.typed_value)
-    (rplace : mplace option) (expr : expression option) : expression option =
-  match expr with
-  | None -> None
-  | Some expr -> Some (Meta (Assignment (lplace, rvalue, rplace), expr))
+    (rplace : mplace option) (e : expression option) : expression option =
+  Option.map (fun e -> Meta (Assignment (lplace, rvalue, rplace), e)) e
+
+let synthesize_assertion (v : V.typed_value) (e : expression option) =
+  Option.map (fun e -> Assertion (v, e)) e
 
-let synthesize_assertion (v : V.typed_value) (expr : expression option) =
-  match expr with None -> None | Some expr -> Some (Assertion (v, expr))
+let synthesize_forward_end (e : expression option) =
+  Option.map (fun e -> ForwardEnd e) e
diff --git a/compiler/Translate.ml b/compiler/Translate.ml
index d7cc9155..72322c73 100644
--- a/compiler/Translate.ml
+++ b/compiler/Translate.ml
@@ -18,6 +18,30 @@ type config = {
       (** Controls whether we need to use a state to model the external world
           (I/O, for instance).
        *)
+  backward_no_state_update : bool;
+      (** Controls whether backward functions update the state, in case we use
+          a state ({!use_state}).
+
+          If they update the state, we generate code of the following style:
+          {[
+             (st1, y)  <-- f_fwd x st0; // st0 is the state upon calling f_fwd
+             ...
+             (st3, x') <-- f_back x st0 y' st2; // st2 is the state upon calling f_back
+          }]
+
+          Otherwise, we generate code of the following shape:
+          {[
+             (st1, y)  <-- f_fwd x st0;
+             ...
+             x' <-- f_back x st0 y';
+          }]
+
+          The second format is easier to reason about, but the first one is
+          necessary to properly handle some Rust functions which use internal
+          mutability such as {{:https://doc.rust-lang.org/std/cell/struct.RefCell.html#method.try_borrow_mut} [RefCell::try_mut_borrow]}:
+          in order to model this behaviour we would need a state, and calling the backward
+          function would update the state by reinserting the updated value in it.
+       *)
   split_files : bool;
       (** Controls whether we split the generated definitions between different
           files for the types, clauses and functions, or if we group them in
@@ -96,7 +120,8 @@ let translate_function_to_symbolics (config : C.partial_config)
     TODO: maybe we should introduce a record for this.
 *)
 let translate_function_to_pure (config : C.partial_config)
-    (mp_config : Micro.config) (trans_ctx : trans_ctx)
+    (mp_config : Micro.config) (backward_no_state_update : bool)
+    (trans_ctx : trans_ctx)
     (fun_sigs : SymbolicToPure.fun_sig_named_outputs RegularFunIdMap.t)
     (pure_type_decls : Pure.type_decl Pure.TypeDeclId.Map.t) (fdef : A.fun_decl)
     : pure_fun_translation =
@@ -123,6 +148,7 @@ let translate_function_to_pure (config : C.partial_config)
   let sv_to_var = V.SymbolicValueId.Map.empty in
   let var_counter = Pure.VarId.generator_zero in
   let state_var, var_counter = Pure.VarId.fresh var_counter in
+  let back_state_var, var_counter = Pure.VarId.fresh var_counter in
   let calls = V.FunCallId.Map.empty in
   let abstractions = V.AbstractionId.Map.empty in
   let type_context =
@@ -151,6 +177,7 @@ let translate_function_to_pure (config : C.partial_config)
       sv_to_var;
       var_counter;
       state_var;
+      back_state_var;
       type_context;
       fun_context;
       global_context;
@@ -188,6 +215,7 @@ let translate_function_to_pure (config : C.partial_config)
     {
       SymbolicToPure.filter_useless_back_calls =
         mp_config.filter_useless_monadic_calls;
+      backward_no_state_update;
     }
   in
 
@@ -231,12 +259,22 @@ let translate_function_to_pure (config : C.partial_config)
         in
         (* We need to ignore the forward inputs, and the state input (if there is) *)
         let fun_info =
-          SymbolicToPure.get_fun_effect_info fun_context.fun_infos
-            (A.Regular def_id) (Some back_id)
+          SymbolicToPure.get_fun_effect_info backward_no_state_update
+            fun_context.fun_infos (A.Regular def_id) (Some back_id)
         in
-        let _, backward_inputs =
-          Collections.List.split_at backward_sg.sg.inputs
-            (num_forward_inputs + if fun_info.input_state then 1 else 0)
+        let backward_inputs =
+          (* We need to ignore the forward state and the backward state *)
+          (* TODO: this is ad-hoc and error-prone. We should group all this
+           * information in the signature information. *)
+          let fwd_state_n = if fun_info.stateful_group then 1 else 0 in
+          let num_forward_inputs = num_forward_inputs + fwd_state_n in
+          let back_state_n = if fun_info.stateful then 1 else 0 in
+          let num_back_inputs =
+            List.length backward_sg.sg.inputs
+            - num_forward_inputs - back_state_n
+          in
+          Collections.List.subslice backward_sg.sg.inputs num_forward_inputs
+            num_back_inputs
         in
         (* As we forbid nested borrows, the additional inputs for the backward
          * functions come from the borrows in the return value of the rust function:
@@ -285,7 +323,8 @@ let translate_function_to_pure (config : C.partial_config)
   (pure_forward, pure_backwards)
 
 let translate_module_to_pure (config : C.partial_config)
-    (mp_config : Micro.config) (use_state : bool) (crate : A.crate) :
+    (mp_config : Micro.config) (use_state : bool)
+    (backward_no_state_update : bool) (crate : A.crate) :
     trans_ctx * Pure.type_decl list * (bool * pure_fun_translation) list =
   (* Debug *)
   log#ldebug (lazy "translate_module_to_pure");
@@ -333,15 +372,15 @@ let translate_module_to_pure (config : C.partial_config)
   in
   let sigs = List.append assumed_sigs local_sigs in
   let fun_sigs =
-    SymbolicToPure.translate_fun_signatures fun_context.fun_infos
-      type_context.type_infos sigs
+    SymbolicToPure.translate_fun_signatures backward_no_state_update
+      fun_context.fun_infos type_context.type_infos sigs
   in
 
   (* Translate all the *transparent* functions *)
   let pure_translations =
     List.map
-      (translate_function_to_pure config mp_config trans_ctx fun_sigs
-         type_decls_map)
+      (translate_function_to_pure config mp_config backward_no_state_update
+         trans_ctx fun_sigs type_decls_map)
       crate.functions
   in
 
@@ -631,7 +670,7 @@ let translate_module (filename : string) (dest_dir : string) (config : config)
   (* Translate the module to the pure AST *)
   let trans_ctx, trans_types, trans_funs =
     translate_module_to_pure config.eval_config config.mp_config
-      config.use_state crate
+      config.use_state config.backward_no_state_update crate
   in
 
   (* Initialize the extraction context - for now we extract only to F*.
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