Reorganize the Lean tests and extract the Polonius tests to Lean

4 files changed, 1501 insertions, 0 deletions

diff --git a/tests/lean/betree/BetreeMain/Clauses/Template.lean b/tests/lean/betree/BetreeMain/Clauses/Template.lean
new file mode 100644
index 00000000..1d18174e
--- /dev/null
+++ b/tests/lean/betree/BetreeMain/Clauses/Template.lean
@@ -0,0 +1,185 @@
+-- THIS FILE WAS AUTOMATICALLY GENERATED BY AENEAS
+-- [betree_main]: templates for the decreases clauses
+import Base.Primitives
+import BetreeMain.Types
+
+/- [betree_main::betree::List::{1}::len]: termination measure -/
+@[simp]
+def betree_list_len_terminates (T : Type) (self : betree_list_t T) := self
+
+/- [betree_main::betree::List::{1}::len]: decreases_by tactic -/
+syntax "betree_list_len_decreases" term+ : tactic
+macro_rules
+| `(tactic| betree_list_len_decreases $self) =>`(tactic| sorry)
+
+/- [betree_main::betree::List::{1}::split_at]: termination measure -/
+@[simp]
+def betree_list_split_at_terminates (T : Type) (self : betree_list_t T)
+  (n : UInt64) :=
+  (self, n)
+
+/- [betree_main::betree::List::{1}::split_at]: decreases_by tactic -/
+syntax "betree_list_split_at_decreases" term+ : tactic
+macro_rules
+| `(tactic| betree_list_split_at_decreases $self $n) =>`(tactic| sorry)
+
+/- [betree_main::betree::List::{2}::partition_at_pivot]: termination measure -/
+@[simp]
+def betree_list_partition_at_pivot_terminates (T : Type)
+  (self : betree_list_t (UInt64 × T)) (pivot : UInt64) :=
+  (self, pivot)
+
+/- [betree_main::betree::List::{2}::partition_at_pivot]: decreases_by tactic -/
+syntax "betree_list_partition_at_pivot_decreases" term+ : tactic
+macro_rules
+| `(tactic| betree_list_partition_at_pivot_decreases $self $pivot) =>
+  `(tactic| sorry)
+
+/- [betree_main::betree::Node::{5}::lookup_in_bindings]: termination measure -/
+@[simp]
+def betree_node_lookup_in_bindings_terminates (key : UInt64)
+  (bindings : betree_list_t (UInt64 × UInt64)) :=
+  (key, bindings)
+
+/- [betree_main::betree::Node::{5}::lookup_in_bindings]: decreases_by tactic -/
+syntax "betree_node_lookup_in_bindings_decreases" term+ : tactic
+macro_rules
+| `(tactic| betree_node_lookup_in_bindings_decreases $key $bindings) =>
+  `(tactic| sorry)
+
+/- [betree_main::betree::Node::{5}::lookup_first_message_for_key]: termination measure -/
+@[simp]
+def betree_node_lookup_first_message_for_key_terminates (key : UInt64)
+  (msgs : betree_list_t (UInt64 × betree_message_t)) :=
+  (key, msgs)
+
+/- [betree_main::betree::Node::{5}::lookup_first_message_for_key]: decreases_by tactic -/
+syntax "betree_node_lookup_first_message_for_key_decreases" term+ : tactic
+macro_rules
+| `(tactic| betree_node_lookup_first_message_for_key_decreases $key $msgs) =>
+  `(tactic| sorry)
+
+/- [betree_main::betree::Node::{5}::apply_upserts]: termination measure -/
+@[simp]
+def betree_node_apply_upserts_terminates
+  (msgs : betree_list_t (UInt64 × betree_message_t)) (prev : Option UInt64)
+  (key : UInt64) (st : State) :=
+  (msgs, prev, key, st)
+
+/- [betree_main::betree::Node::{5}::apply_upserts]: decreases_by tactic -/
+syntax "betree_node_apply_upserts_decreases" term+ : tactic
+macro_rules
+| `(tactic| betree_node_apply_upserts_decreases $msgs $prev $key $st) =>
+  `(tactic| sorry)
+
+/- [betree_main::betree::Node::{5}::lookup]: termination measure -/
+@[simp]
+def betree_node_lookup_terminates (self : betree_node_t) (key : UInt64)
+  (st : State) :=
+  (self, key, st)
+
+/- [betree_main::betree::Node::{5}::lookup]: decreases_by tactic -/
+syntax "betree_node_lookup_decreases" term+ : tactic
+macro_rules
+| `(tactic| betree_node_lookup_decreases $self $key $st) =>`(tactic| sorry)
+
+/- [betree_main::betree::Internal::{4}::lookup_in_children]: termination measure -/
+@[simp]
+def betree_internal_lookup_in_children_terminates (self : betree_internal_t)
+  (key : UInt64) (st : State) :=
+  (self, key, st)
+
+/- [betree_main::betree::Internal::{4}::lookup_in_children]: decreases_by tactic -/
+syntax "betree_internal_lookup_in_children_decreases" term+ : tactic
+macro_rules
+| `(tactic| betree_internal_lookup_in_children_decreases $self $key $st) =>
+  `(tactic| sorry)
+
+/- [betree_main::betree::Node::{5}::lookup_mut_in_bindings]: termination measure -/
+@[simp]
+def betree_node_lookup_mut_in_bindings_terminates (key : UInt64)
+  (bindings : betree_list_t (UInt64 × UInt64)) :=
+  (key, bindings)
+
+/- [betree_main::betree::Node::{5}::lookup_mut_in_bindings]: decreases_by tactic -/
+syntax "betree_node_lookup_mut_in_bindings_decreases" term+ : tactic
+macro_rules
+| `(tactic| betree_node_lookup_mut_in_bindings_decreases $key $bindings) =>
+  `(tactic| sorry)
+
+/- [betree_main::betree::Node::{5}::apply_messages_to_leaf]: termination measure -/
+@[simp]
+def betree_node_apply_messages_to_leaf_terminates
+  (bindings : betree_list_t (UInt64 × UInt64))
+  (new_msgs : betree_list_t (UInt64 × betree_message_t)) :=
+  (bindings, new_msgs)
+
+/- [betree_main::betree::Node::{5}::apply_messages_to_leaf]: decreases_by tactic -/
+syntax "betree_node_apply_messages_to_leaf_decreases" term+ : tactic
+macro_rules
+| `(tactic| betree_node_apply_messages_to_leaf_decreases $bindings
+$new_msgs) =>`(tactic| sorry)
+
+/- [betree_main::betree::Node::{5}::filter_messages_for_key]: termination measure -/
+@[simp]
+def betree_node_filter_messages_for_key_terminates (key : UInt64)
+  (msgs : betree_list_t (UInt64 × betree_message_t)) :=
+  (key, msgs)
+
+/- [betree_main::betree::Node::{5}::filter_messages_for_key]: decreases_by tactic -/
+syntax "betree_node_filter_messages_for_key_decreases" term+ : tactic
+macro_rules
+| `(tactic| betree_node_filter_messages_for_key_decreases $key $msgs) =>
+  `(tactic| sorry)
+
+/- [betree_main::betree::Node::{5}::lookup_first_message_after_key]: termination measure -/
+@[simp]
+def betree_node_lookup_first_message_after_key_terminates (key : UInt64)
+  (msgs : betree_list_t (UInt64 × betree_message_t)) :=
+  (key, msgs)
+
+/- [betree_main::betree::Node::{5}::lookup_first_message_after_key]: decreases_by tactic -/
+syntax "betree_node_lookup_first_message_after_key_decreases" term+ : tactic
+macro_rules
+| `(tactic| betree_node_lookup_first_message_after_key_decreases $key $msgs) =>
+  `(tactic| sorry)
+
+/- [betree_main::betree::Node::{5}::apply_messages_to_internal]: termination measure -/
+@[simp]
+def betree_node_apply_messages_to_internal_terminates
+  (msgs : betree_list_t (UInt64 × betree_message_t))
+  (new_msgs : betree_list_t (UInt64 × betree_message_t)) :=
+  (msgs, new_msgs)
+
+/- [betree_main::betree::Node::{5}::apply_messages_to_internal]: decreases_by tactic -/
+syntax "betree_node_apply_messages_to_internal_decreases" term+ : tactic
+macro_rules
+| `(tactic| betree_node_apply_messages_to_internal_decreases $msgs
+$new_msgs) =>`(tactic| sorry)
+
+/- [betree_main::betree::Node::{5}::apply_messages]: termination measure -/
+@[simp]
+def betree_node_apply_messages_terminates (self : betree_node_t)
+  (params : betree_params_t) (node_id_cnt : betree_node_id_counter_t)
+  (msgs : betree_list_t (UInt64 × betree_message_t)) (st : State) :=
+  (self, params, node_id_cnt, msgs, st)
+
+/- [betree_main::betree::Node::{5}::apply_messages]: decreases_by tactic -/
+syntax "betree_node_apply_messages_decreases" term+ : tactic
+macro_rules
+| `(tactic| betree_node_apply_messages_decreases $self $params $node_id_cnt
+$msgs $st) =>`(tactic| sorry)
+
+/- [betree_main::betree::Internal::{4}::flush]: termination measure -/
+@[simp]
+def betree_internal_flush_terminates (self : betree_internal_t)
+  (params : betree_params_t) (node_id_cnt : betree_node_id_counter_t)
+  (content : betree_list_t (UInt64 × betree_message_t)) (st : State) :=
+  (self, params, node_id_cnt, content, st)
+
+/- [betree_main::betree::Internal::{4}::flush]: decreases_by tactic -/
+syntax "betree_internal_flush_decreases" term+ : tactic
+macro_rules
+| `(tactic| betree_internal_flush_decreases $self $params $node_id_cnt $content
+$st) =>`(tactic| sorry)
+
diff --git a/tests/lean/betree/BetreeMain/Funs.lean b/tests/lean/betree/BetreeMain/Funs.lean
new file mode 100644
index 00000000..e40ca4ca
--- /dev/null
+++ b/tests/lean/betree/BetreeMain/Funs.lean
@@ -0,0 +1,1222 @@
+-- THIS FILE WAS AUTOMATICALLY GENERATED BY AENEAS
+-- [betree_main]: function definitions
+import Base.Primitives
+import BetreeMain.Types
+import BetreeMain.Opaque
+import BetreeMain.Clauses.Clauses
+
+section variable (opaque_defs: OpaqueDefs)
+
+/- [betree_main::betree::load_internal_node] -/
+def betree_load_internal_node_fwd
+  (id : UInt64) (st : State) :
+  Result (State × (betree_list_t (UInt64 × betree_message_t)))
+  :=
+  opaque_defs.betree_utils_load_internal_node_fwd id st
+
+/- [betree_main::betree::store_internal_node] -/
+def betree_store_internal_node_fwd
+  (id : UInt64) (content : betree_list_t (UInt64 × betree_message_t))
+  (st : State) :
+  Result (State × Unit)
+  :=
+  do
+    let (st0, _) ←
+      opaque_defs.betree_utils_store_internal_node_fwd id content st
+    Result.ret (st0, ())
+
+/- [betree_main::betree::load_leaf_node] -/
+def betree_load_leaf_node_fwd
+  (id : UInt64) (st : State) :
+  Result (State × (betree_list_t (UInt64 × UInt64)))
+  :=
+  opaque_defs.betree_utils_load_leaf_node_fwd id st
+
+/- [betree_main::betree::store_leaf_node] -/
+def betree_store_leaf_node_fwd
+  (id : UInt64) (content : betree_list_t (UInt64 × UInt64)) (st : State) :
+  Result (State × Unit)
+  :=
+  do
+    let (st0, _) ← opaque_defs.betree_utils_store_leaf_node_fwd id content st
+    Result.ret (st0, ())
+
+/- [betree_main::betree::fresh_node_id] -/
+def betree_fresh_node_id_fwd (counter : UInt64) : Result UInt64 :=
+  do
+    let _ ← UInt64.checked_add counter (UInt64.ofNatCore 1 (by intlit))
+    Result.ret counter
+
+/- [betree_main::betree::fresh_node_id] -/
+def betree_fresh_node_id_back (counter : UInt64) : Result UInt64 :=
+  UInt64.checked_add counter (UInt64.ofNatCore 1 (by intlit))
+
+/- [betree_main::betree::NodeIdCounter::{0}::new] -/
+def betree_node_id_counter_new_fwd : Result betree_node_id_counter_t :=
+  Result.ret
+  { betree_node_id_counter_next_node_id := (UInt64.ofNatCore 0 (by intlit)) }
+
+/- [betree_main::betree::NodeIdCounter::{0}::fresh_id] -/
+def betree_node_id_counter_fresh_id_fwd
+  (self : betree_node_id_counter_t) : Result UInt64 :=
+  do
+    let _ ← UInt64.checked_add self.betree_node_id_counter_next_node_id
+      (UInt64.ofNatCore 1 (by intlit))
+    Result.ret self.betree_node_id_counter_next_node_id
+
+/- [betree_main::betree::NodeIdCounter::{0}::fresh_id] -/
+def betree_node_id_counter_fresh_id_back
+  (self : betree_node_id_counter_t) : Result betree_node_id_counter_t :=
+  do
+    let i ← UInt64.checked_add self.betree_node_id_counter_next_node_id
+      (UInt64.ofNatCore 1 (by intlit))
+    Result.ret { betree_node_id_counter_next_node_id := i }
+
+/- [core::num::u64::{10}::MAX] -/
+def core_num_u64_max_body : Result UInt64 :=
+  Result.ret (UInt64.ofNatCore 18446744073709551615 (by intlit))
+def core_num_u64_max_c : UInt64 := eval_global core_num_u64_max_body (by simp)
+
+/- [betree_main::betree::upsert_update] -/
+def betree_upsert_update_fwd
+  (prev : Option UInt64) (st : betree_upsert_fun_state_t) : Result UInt64 :=
+  match h: prev with
+  | Option.none =>
+    match h: st with
+    | betree_upsert_fun_state_t.BetreeUpsertFunStateAdd v => Result.ret v
+    | betree_upsert_fun_state_t.BetreeUpsertFunStateSub i =>
+      Result.ret (UInt64.ofNatCore 0 (by intlit))
+  | Option.some prev0 =>
+    match h: st with
+    | betree_upsert_fun_state_t.BetreeUpsertFunStateAdd v =>
+      do
+        let margin ← UInt64.checked_sub core_num_u64_max_c prev0
+        if h: margin >= v
+        then UInt64.checked_add prev0 v
+        else Result.ret core_num_u64_max_c
+    | betree_upsert_fun_state_t.BetreeUpsertFunStateSub v =>
+      if h: prev0 >= v
+      then UInt64.checked_sub prev0 v
+      else Result.ret (UInt64.ofNatCore 0 (by intlit))
+
+/- [betree_main::betree::List::{1}::len] -/
+def betree_list_len_fwd
+  (T : Type) (self : betree_list_t T) : (Result UInt64) :=
+  match h: self with
+  | betree_list_t.BetreeListCons t tl =>
+    do
+      let i ← betree_list_len_fwd T tl
+      UInt64.checked_add (UInt64.ofNatCore 1 (by intlit)) i
+  | betree_list_t.BetreeListNil => Result.ret (UInt64.ofNatCore 0 (by intlit))
+termination_by betree_list_len_fwd self => betree_list_len_terminates T self
+decreasing_by betree_list_len_decreases self
+
+/- [betree_main::betree::List::{1}::split_at] -/
+def betree_list_split_at_fwd
+  (T : Type) (self : betree_list_t T) (n : UInt64) :
+  (Result ((betree_list_t T) × (betree_list_t T)))
+  :=
+  if h: n = (UInt64.ofNatCore 0 (by intlit))
+  then Result.ret (betree_list_t.BetreeListNil, self)
+  else
+    match h: self with
+    | betree_list_t.BetreeListCons hd tl =>
+      do
+        let i ← UInt64.checked_sub n (UInt64.ofNatCore 1 (by intlit))
+        let p ← betree_list_split_at_fwd T tl i
+        let (ls0, ls1) := p
+        let l := ls0
+        Result.ret (betree_list_t.BetreeListCons hd l, ls1)
+    | betree_list_t.BetreeListNil => Result.fail Error.panic
+termination_by betree_list_split_at_fwd self n =>
+  betree_list_split_at_terminates T self n
+decreasing_by betree_list_split_at_decreases self n
+
+/- [betree_main::betree::List::{1}::push_front] -/
+def betree_list_push_front_fwd_back
+  (T : Type) (self : betree_list_t T) (x : T) : Result (betree_list_t T) :=
+  let tl := mem_replace_fwd (betree_list_t T) self betree_list_t.BetreeListNil
+  let l := tl
+  Result.ret (betree_list_t.BetreeListCons x l)
+
+/- [betree_main::betree::List::{1}::pop_front] -/
+def betree_list_pop_front_fwd (T : Type) (self : betree_list_t T) : Result T :=
+  let ls := mem_replace_fwd (betree_list_t T) self betree_list_t.BetreeListNil
+  match h: ls with
+  | betree_list_t.BetreeListCons x tl => Result.ret x
+  | betree_list_t.BetreeListNil => Result.fail Error.panic
+
+/- [betree_main::betree::List::{1}::pop_front] -/
+def betree_list_pop_front_back
+  (T : Type) (self : betree_list_t T) : Result (betree_list_t T) :=
+  let ls := mem_replace_fwd (betree_list_t T) self betree_list_t.BetreeListNil
+  match h: ls with
+  | betree_list_t.BetreeListCons x tl => Result.ret tl
+  | betree_list_t.BetreeListNil => Result.fail Error.panic
+
+/- [betree_main::betree::List::{1}::hd] -/
+def betree_list_hd_fwd (T : Type) (self : betree_list_t T) : Result T :=
+  match h: self with
+  | betree_list_t.BetreeListCons hd l => Result.ret hd
+  | betree_list_t.BetreeListNil => Result.fail Error.panic
+
+/- [betree_main::betree::List::{2}::head_has_key] -/
+def betree_list_head_has_key_fwd
+  (T : Type) (self : betree_list_t (UInt64 × T)) (key : UInt64) :
+  Result Bool
+  :=
+  match h: self with
+  | betree_list_t.BetreeListCons hd l => let (i, _) := hd
+                                         Result.ret (i = key)
+  | betree_list_t.BetreeListNil => Result.ret false
+
+/- [betree_main::betree::List::{2}::partition_at_pivot] -/
+def betree_list_partition_at_pivot_fwd
+  (T : Type) (self : betree_list_t (UInt64 × T)) (pivot : UInt64) :
+  (Result ((betree_list_t (UInt64 × T)) × (betree_list_t (UInt64 × T))))
+  :=
+  match h: self with
+  | betree_list_t.BetreeListCons hd tl =>
+    let (i, t) := hd
+    if h: i >= pivot
+    then
+      Result.ret (betree_list_t.BetreeListNil, betree_list_t.BetreeListCons (i,
+        t) tl)
+    else
+      do
+        let p ← betree_list_partition_at_pivot_fwd T tl pivot
+        let (ls0, ls1) := p
+        let l := ls0
+        Result.ret (betree_list_t.BetreeListCons (i, t) l, ls1)
+  | betree_list_t.BetreeListNil =>
+    Result.ret (betree_list_t.BetreeListNil, betree_list_t.BetreeListNil)
+termination_by betree_list_partition_at_pivot_fwd self pivot =>
+  betree_list_partition_at_pivot_terminates T self pivot
+decreasing_by betree_list_partition_at_pivot_decreases self pivot
+
+/- [betree_main::betree::Leaf::{3}::split] -/
+def betree_leaf_split_fwd
+  (self : betree_leaf_t) (content : betree_list_t (UInt64 × UInt64))
+  (params : betree_params_t) (node_id_cnt : betree_node_id_counter_t)
+  (st : State) :
+  Result (State × betree_internal_t)
+  :=
+  do
+    let p ←
+      betree_list_split_at_fwd (UInt64 × UInt64) content
+        params.betree_params_split_size
+    let (content0, content1) := p
+    let p0 ← betree_list_hd_fwd (UInt64 × UInt64) content1
+    let (pivot, _) := p0
+    let id0 ← betree_node_id_counter_fresh_id_fwd node_id_cnt
+    let node_id_cnt0 ← betree_node_id_counter_fresh_id_back node_id_cnt
+    let id1 ← betree_node_id_counter_fresh_id_fwd node_id_cnt0
+    let (st0, _) ← betree_store_leaf_node_fwd id0 content0 st
+    let (st1, _) ← betree_store_leaf_node_fwd id1 content1 st0
+    let n := betree_node_t.BetreeNodeLeaf
+      {
+        betree_leaf_id := id0,
+        betree_leaf_size := params.betree_params_split_size
+      }
+    let n0 := betree_node_t.BetreeNodeLeaf
+      {
+        betree_leaf_id := id1,
+        betree_leaf_size := params.betree_params_split_size
+      }
+    Result.ret (st1,
+      {
+        betree_internal_id := self.betree_leaf_id,
+        betree_internal_pivot := pivot,
+        betree_internal_left := n,
+        betree_internal_right := n0
+      })
+
+/- [betree_main::betree::Leaf::{3}::split] -/
+def betree_leaf_split_back
+  (self : betree_leaf_t) (content : betree_list_t (UInt64 × UInt64))
+  (params : betree_params_t) (node_id_cnt : betree_node_id_counter_t)
+  (st : State) :
+  Result betree_node_id_counter_t
+  :=
+  do
+    let p ←
+      betree_list_split_at_fwd (UInt64 × UInt64) content
+        params.betree_params_split_size
+    let (content0, content1) := p
+    let _ ← betree_list_hd_fwd (UInt64 × UInt64) content1
+    let id0 ← betree_node_id_counter_fresh_id_fwd node_id_cnt
+    let node_id_cnt0 ← betree_node_id_counter_fresh_id_back node_id_cnt
+    let id1 ← betree_node_id_counter_fresh_id_fwd node_id_cnt0
+    let (st0, _) ← betree_store_leaf_node_fwd id0 content0 st
+    let _ ← betree_store_leaf_node_fwd id1 content1 st0
+    betree_node_id_counter_fresh_id_back node_id_cnt0
+
+/- [betree_main::betree::Node::{5}::lookup_in_bindings] -/
+def betree_node_lookup_in_bindings_fwd
+  (key : UInt64) (bindings : betree_list_t (UInt64 × UInt64)) :
+  (Result (Option UInt64))
+  :=
+  match h: bindings with
+  | betree_list_t.BetreeListCons hd tl =>
+    let (i, i0) := hd
+    if h: i = key
+    then Result.ret (Option.some i0)
+    else
+      if h: i > key
+      then Result.ret Option.none
+      else betree_node_lookup_in_bindings_fwd key tl
+  | betree_list_t.BetreeListNil => Result.ret Option.none
+termination_by betree_node_lookup_in_bindings_fwd key bindings =>
+  betree_node_lookup_in_bindings_terminates key bindings
+decreasing_by betree_node_lookup_in_bindings_decreases key bindings
+
+/- [betree_main::betree::Node::{5}::lookup_first_message_for_key] -/
+def betree_node_lookup_first_message_for_key_fwd
+  (key : UInt64) (msgs : betree_list_t (UInt64 × betree_message_t)) :
+  (Result (betree_list_t (UInt64 × betree_message_t)))
+  :=
+  match h: msgs with
+  | betree_list_t.BetreeListCons x next_msgs =>
+    let (i, m) := x
+    if h: i >= key
+    then Result.ret (betree_list_t.BetreeListCons (i, m) next_msgs)
+    else betree_node_lookup_first_message_for_key_fwd key next_msgs
+  | betree_list_t.BetreeListNil => Result.ret betree_list_t.BetreeListNil
+termination_by betree_node_lookup_first_message_for_key_fwd key msgs =>
+  betree_node_lookup_first_message_for_key_terminates key msgs
+decreasing_by betree_node_lookup_first_message_for_key_decreases key msgs
+
+/- [betree_main::betree::Node::{5}::lookup_first_message_for_key] -/
+def betree_node_lookup_first_message_for_key_back
+  (key : UInt64) (msgs : betree_list_t (UInt64 × betree_message_t))
+  (ret0 : betree_list_t (UInt64 × betree_message_t)) :
+  (Result (betree_list_t (UInt64 × betree_message_t)))
+  :=
+  match h: msgs with
+  | betree_list_t.BetreeListCons x next_msgs =>
+    let (i, m) := x
+    if h: i >= key
+    then Result.ret ret0
+    else
+      do
+        let next_msgs0 ←
+          betree_node_lookup_first_message_for_key_back key next_msgs ret0
+        Result.ret (betree_list_t.BetreeListCons (i, m) next_msgs0)
+  | betree_list_t.BetreeListNil => Result.ret ret0
+termination_by betree_node_lookup_first_message_for_key_back key msgs ret0 =>
+  betree_node_lookup_first_message_for_key_terminates key msgs
+decreasing_by betree_node_lookup_first_message_for_key_decreases key msgs
+
+/- [betree_main::betree::Node::{5}::apply_upserts] -/
+def betree_node_apply_upserts_fwd
+  (msgs : betree_list_t (UInt64 × betree_message_t)) (prev : Option UInt64)
+  (key : UInt64) (st : State) :
+  (Result (State × UInt64))
+  :=
+  do
+    let b ← betree_list_head_has_key_fwd betree_message_t msgs key
+    if h: b
+    then
+      do
+        let msg ← betree_list_pop_front_fwd (UInt64 × betree_message_t) msgs
+        let (_, m) := msg
+        match h: m with
+        | betree_message_t.BetreeMessageInsert i => Result.fail Error.panic
+        | betree_message_t.BetreeMessageDelete => Result.fail Error.panic
+        | betree_message_t.BetreeMessageUpsert s =>
+          do
+            let v ← betree_upsert_update_fwd prev s
+            let msgs0 ←
+              betree_list_pop_front_back (UInt64 × betree_message_t) msgs
+            betree_node_apply_upserts_fwd msgs0 (Option.some v) key st
+    else
+      do
+        let (st0, v) ←
+          opaque_defs.core_option_option_unwrap_fwd UInt64 prev st
+        let _ ←
+          betree_list_push_front_fwd_back (UInt64 × betree_message_t) msgs
+            (key, betree_message_t.BetreeMessageInsert v)
+        Result.ret (st0, v)
+termination_by betree_node_apply_upserts_fwd msgs prev key st =>
+  betree_node_apply_upserts_terminates msgs prev key st
+decreasing_by betree_node_apply_upserts_decreases msgs prev key st
+
+/- [betree_main::betree::Node::{5}::apply_upserts] -/
+def betree_node_apply_upserts_back
+  (msgs : betree_list_t (UInt64 × betree_message_t)) (prev : Option UInt64)
+  (key : UInt64) (st : State) :
+  (Result (betree_list_t (UInt64 × betree_message_t)))
+  :=
+  do
+    let b ← betree_list_head_has_key_fwd betree_message_t msgs key
+    if h: b
+    then
+      do
+        let msg ← betree_list_pop_front_fwd (UInt64 × betree_message_t) msgs
+        let (_, m) := msg
+        match h: m with
+        | betree_message_t.BetreeMessageInsert i => Result.fail Error.panic
+        | betree_message_t.BetreeMessageDelete => Result.fail Error.panic
+        | betree_message_t.BetreeMessageUpsert s =>
+          do
+            let v ← betree_upsert_update_fwd prev s
+            let msgs0 ←
+              betree_list_pop_front_back (UInt64 × betree_message_t) msgs
+            betree_node_apply_upserts_back msgs0 (Option.some v) key st
+    else
+      do
+        let (_, v) ← opaque_defs.core_option_option_unwrap_fwd UInt64 prev st
+        betree_list_push_front_fwd_back (UInt64 × betree_message_t) msgs (key,
+          betree_message_t.BetreeMessageInsert v)
+termination_by betree_node_apply_upserts_back msgs prev key st =>
+  betree_node_apply_upserts_terminates msgs prev key st
+decreasing_by betree_node_apply_upserts_decreases msgs prev key st
+
+/- [betree_main::betree::Node::{5}::lookup] -/
+mutual def betree_node_lookup_fwd
+  (self : betree_node_t) (key : UInt64) (st : State) :
+  (Result (State × (Option UInt64)))
+  :=
+  match h: self with
+  | betree_node_t.BetreeNodeInternal node =>
+    do
+      let (st0, msgs) ←
+        betree_load_internal_node_fwd node.betree_internal_id st
+      let pending ← betree_node_lookup_first_message_for_key_fwd key msgs
+      match h: pending with
+      | betree_list_t.BetreeListCons p l =>
+        let (k, msg) := p
+        if h: k != key
+        then
+          do
+            let (st1, opt) ←
+              betree_internal_lookup_in_children_fwd node key st0
+            let _ ←
+              betree_node_lookup_first_message_for_key_back key msgs
+                (betree_list_t.BetreeListCons (k, msg) l)
+            Result.ret (st1, opt)
+        else
+          match h: msg with
+          | betree_message_t.BetreeMessageInsert v =>
+            do
+              let _ ←
+                betree_node_lookup_first_message_for_key_back key msgs
+                  (betree_list_t.BetreeListCons (k,
+                  betree_message_t.BetreeMessageInsert v) l)
+              Result.ret (st0, Option.some v)
+          | betree_message_t.BetreeMessageDelete =>
+            do
+              let _ ←
+                betree_node_lookup_first_message_for_key_back key msgs
+                  (betree_list_t.BetreeListCons (k,
+                  betree_message_t.BetreeMessageDelete) l)
+              Result.ret (st0, Option.none)
+          | betree_message_t.BetreeMessageUpsert ufs =>
+            do
+              let (st1, v) ←
+                betree_internal_lookup_in_children_fwd node key st0
+              let (st2, v0) ←
+                betree_node_apply_upserts_fwd (betree_list_t.BetreeListCons (k,
+                  betree_message_t.BetreeMessageUpsert ufs) l) v key st1
+              let node0 ←
+                betree_internal_lookup_in_children_back node key st0
+              let pending0 ←
+                betree_node_apply_upserts_back (betree_list_t.BetreeListCons
+                  (k, betree_message_t.BetreeMessageUpsert ufs) l) v key st1
+              let msgs0 ←
+                betree_node_lookup_first_message_for_key_back key msgs pending0
+              let (st3, _) ←
+                betree_store_internal_node_fwd node0.betree_internal_id msgs0
+                  st2
+              Result.ret (st3, Option.some v0)
+      | betree_list_t.BetreeListNil =>
+        do
+          let (st1, opt) ←
+            betree_internal_lookup_in_children_fwd node key st0
+          let _ ←
+            betree_node_lookup_first_message_for_key_back key msgs
+              betree_list_t.BetreeListNil
+          Result.ret (st1, opt)
+  | betree_node_t.BetreeNodeLeaf node =>
+    do
+      let (st0, bindings) ← betree_load_leaf_node_fwd node.betree_leaf_id st
+      let opt ← betree_node_lookup_in_bindings_fwd key bindings
+      Result.ret (st0, opt)
+termination_by betree_node_lookup_fwd self key st =>
+  betree_node_lookup_terminates self key st
+decreasing_by betree_node_lookup_decreases self key st
+
+/- [betree_main::betree::Node::{5}::lookup] -/
+def betree_node_lookup_back
+  (self : betree_node_t) (key : UInt64) (st : State) :
+  (Result betree_node_t)
+  :=
+  match h: self with
+  | betree_node_t.BetreeNodeInternal node =>
+    do
+      let (st0, msgs) ←
+        betree_load_internal_node_fwd node.betree_internal_id st
+      let pending ← betree_node_lookup_first_message_for_key_fwd key msgs
+      match h: pending with
+      | betree_list_t.BetreeListCons p l =>
+        let (k, msg) := p
+        if h: k != key
+        then
+          do
+            let _ ←
+              betree_node_lookup_first_message_for_key_back key msgs
+                (betree_list_t.BetreeListCons (k, msg) l)
+            let node0 ← betree_internal_lookup_in_children_back node key st0
+            Result.ret (betree_node_t.BetreeNodeInternal node0)
+        else
+          match h: msg with
+          | betree_message_t.BetreeMessageInsert v =>
+            do
+              let _ ←
+                betree_node_lookup_first_message_for_key_back key msgs
+                  (betree_list_t.BetreeListCons (k,
+                  betree_message_t.BetreeMessageInsert v) l)
+              Result.ret (betree_node_t.BetreeNodeInternal node)
+          | betree_message_t.BetreeMessageDelete =>
+            do
+              let _ ←
+                betree_node_lookup_first_message_for_key_back key msgs
+                  (betree_list_t.BetreeListCons (k,
+                  betree_message_t.BetreeMessageDelete) l)
+              Result.ret (betree_node_t.BetreeNodeInternal node)
+          | betree_message_t.BetreeMessageUpsert ufs =>
+            do
+              let (st1, v) ←
+                betree_internal_lookup_in_children_fwd node key st0
+              let (st2, _) ←
+                betree_node_apply_upserts_fwd (betree_list_t.BetreeListCons (k,
+                  betree_message_t.BetreeMessageUpsert ufs) l) v key st1
+              let node0 ←
+                betree_internal_lookup_in_children_back node key st0
+              let pending0 ←
+                betree_node_apply_upserts_back (betree_list_t.BetreeListCons
+                  (k, betree_message_t.BetreeMessageUpsert ufs) l) v key st1
+              let msgs0 ←
+                betree_node_lookup_first_message_for_key_back key msgs pending0
+              let _ ←
+                betree_store_internal_node_fwd node0.betree_internal_id msgs0
+                  st2
+              Result.ret (betree_node_t.BetreeNodeInternal node0)
+      | betree_list_t.BetreeListNil =>
+        do
+          let _ ←
+            betree_node_lookup_first_message_for_key_back key msgs
+              betree_list_t.BetreeListNil
+          let node0 ← betree_internal_lookup_in_children_back node key st0
+          Result.ret (betree_node_t.BetreeNodeInternal node0)
+  | betree_node_t.BetreeNodeLeaf node =>
+    do
+      let (_, bindings) ← betree_load_leaf_node_fwd node.betree_leaf_id st
+      let _ ← betree_node_lookup_in_bindings_fwd key bindings
+      Result.ret (betree_node_t.BetreeNodeLeaf node)
+termination_by betree_node_lookup_back self key st =>
+  betree_node_lookup_terminates self key st
+decreasing_by betree_node_lookup_decreases self key st
+
+/- [betree_main::betree::Internal::{4}::lookup_in_children] -/
+def betree_internal_lookup_in_children_fwd
+  (self : betree_internal_t) (key : UInt64) (st : State) :
+  (Result (State × (Option UInt64)))
+  :=
+  if h: key < self.betree_internal_pivot
+  then betree_node_lookup_fwd self.betree_internal_left key st
+  else betree_node_lookup_fwd self.betree_internal_right key st
+termination_by betree_internal_lookup_in_children_fwd self key st =>
+  betree_internal_lookup_in_children_terminates self key st
+decreasing_by betree_internal_lookup_in_children_decreases self key st
+
+/- [betree_main::betree::Internal::{4}::lookup_in_children] -/
+def betree_internal_lookup_in_children_back
+  (self : betree_internal_t) (key : UInt64) (st : State) :
+  (Result betree_internal_t)
+  :=
+  if h: key < self.betree_internal_pivot
+  then
+    do
+      let n ← betree_node_lookup_back self.betree_internal_left key st
+      Result.ret
+        {
+          betree_internal_id := self.betree_internal_id,
+          betree_internal_pivot := self.betree_internal_pivot,
+          betree_internal_left := n,
+          betree_internal_right := self.betree_internal_right
+        }
+  else
+    do
+      let n ← betree_node_lookup_back self.betree_internal_right key st
+      Result.ret
+        {
+          betree_internal_id := self.betree_internal_id,
+          betree_internal_pivot := self.betree_internal_pivot,
+          betree_internal_left := self.betree_internal_left,
+          betree_internal_right := n
+        }
+termination_by betree_internal_lookup_in_children_back self key st =>
+  betree_internal_lookup_in_children_terminates self key st
+decreasing_by betree_internal_lookup_in_children_decreases self key st
+
+/- [betree_main::betree::Node::{5}::lookup_mut_in_bindings] -/
+def betree_node_lookup_mut_in_bindings_fwd
+  (key : UInt64) (bindings : betree_list_t (UInt64 × UInt64)) :
+  (Result (betree_list_t (UInt64 × UInt64)))
+  :=
+  match h: bindings with
+  | betree_list_t.BetreeListCons hd tl =>
+    let (i, i0) := hd
+    if h: i >= key
+    then Result.ret (betree_list_t.BetreeListCons (i, i0) tl)
+    else betree_node_lookup_mut_in_bindings_fwd key tl
+  | betree_list_t.BetreeListNil => Result.ret betree_list_t.BetreeListNil
+termination_by betree_node_lookup_mut_in_bindings_fwd key bindings =>
+  betree_node_lookup_mut_in_bindings_terminates key bindings
+decreasing_by betree_node_lookup_mut_in_bindings_decreases key bindings
+
+/- [betree_main::betree::Node::{5}::lookup_mut_in_bindings] -/
+def betree_node_lookup_mut_in_bindings_back
+  (key : UInt64) (bindings : betree_list_t (UInt64 × UInt64))
+  (ret0 : betree_list_t (UInt64 × UInt64)) :
+  (Result (betree_list_t (UInt64 × UInt64)))
+  :=
+  match h: bindings with
+  | betree_list_t.BetreeListCons hd tl =>
+    let (i, i0) := hd
+    if h: i >= key
+    then Result.ret ret0
+    else
+      do
+        let tl0 ← betree_node_lookup_mut_in_bindings_back key tl ret0
+        Result.ret (betree_list_t.BetreeListCons (i, i0) tl0)
+  | betree_list_t.BetreeListNil => Result.ret ret0
+termination_by betree_node_lookup_mut_in_bindings_back key bindings ret0 =>
+  betree_node_lookup_mut_in_bindings_terminates key bindings
+decreasing_by betree_node_lookup_mut_in_bindings_decreases key bindings
+
+/- [betree_main::betree::Node::{5}::apply_to_leaf] -/
+def betree_node_apply_to_leaf_fwd_back
+  (bindings : betree_list_t (UInt64 × UInt64)) (key : UInt64)
+  (new_msg : betree_message_t) :
+  Result (betree_list_t (UInt64 × UInt64))
+  :=
+  do
+    let bindings0 ← betree_node_lookup_mut_in_bindings_fwd key bindings
+    let b ← betree_list_head_has_key_fwd UInt64 bindings0 key
+    if h: b
+    then
+      do
+        let hd ← betree_list_pop_front_fwd (UInt64 × UInt64) bindings0
+        match h: new_msg with
+        | betree_message_t.BetreeMessageInsert v =>
+          do
+            let bindings1 ←
+              betree_list_pop_front_back (UInt64 × UInt64) bindings0
+            let bindings2 ←
+              betree_list_push_front_fwd_back (UInt64 × UInt64) bindings1
+                (key, v)
+            betree_node_lookup_mut_in_bindings_back key bindings bindings2
+        | betree_message_t.BetreeMessageDelete =>
+          do
+            let bindings1 ←
+              betree_list_pop_front_back (UInt64 × UInt64) bindings0
+            betree_node_lookup_mut_in_bindings_back key bindings bindings1
+        | betree_message_t.BetreeMessageUpsert s =>
+          do
+            let (_, i) := hd
+            let v ← betree_upsert_update_fwd (Option.some i) s
+            let bindings1 ←
+              betree_list_pop_front_back (UInt64 × UInt64) bindings0
+            let bindings2 ←
+              betree_list_push_front_fwd_back (UInt64 × UInt64) bindings1
+                (key, v)
+            betree_node_lookup_mut_in_bindings_back key bindings bindings2
+    else
+      match h: new_msg with
+      | betree_message_t.BetreeMessageInsert v =>
+        do
+          let bindings1 ←
+            betree_list_push_front_fwd_back (UInt64 × UInt64) bindings0 (key,
+              v)
+          betree_node_lookup_mut_in_bindings_back key bindings bindings1
+      | betree_message_t.BetreeMessageDelete =>
+        betree_node_lookup_mut_in_bindings_back key bindings bindings0
+      | betree_message_t.BetreeMessageUpsert s =>
+        do
+          let v ← betree_upsert_update_fwd Option.none s
+          let bindings1 ←
+            betree_list_push_front_fwd_back (UInt64 × UInt64) bindings0 (key,
+              v)
+          betree_node_lookup_mut_in_bindings_back key bindings bindings1
+
+/- [betree_main::betree::Node::{5}::apply_messages_to_leaf] -/
+def betree_node_apply_messages_to_leaf_fwd_back
+  (bindings : betree_list_t (UInt64 × UInt64))
+  (new_msgs : betree_list_t (UInt64 × betree_message_t)) :
+  (Result (betree_list_t (UInt64 × UInt64)))
+  :=
+  match h: new_msgs with
+  | betree_list_t.BetreeListCons new_msg new_msgs_tl =>
+    do
+      let (i, m) := new_msg
+      let bindings0 ← betree_node_apply_to_leaf_fwd_back bindings i m
+      betree_node_apply_messages_to_leaf_fwd_back bindings0 new_msgs_tl
+  | betree_list_t.BetreeListNil => Result.ret bindings
+termination_by betree_node_apply_messages_to_leaf_fwd_back bindings new_msgs =>
+  betree_node_apply_messages_to_leaf_terminates bindings new_msgs
+decreasing_by betree_node_apply_messages_to_leaf_decreases bindings new_msgs
+
+/- [betree_main::betree::Node::{5}::filter_messages_for_key] -/
+def betree_node_filter_messages_for_key_fwd_back
+  (key : UInt64) (msgs : betree_list_t (UInt64 × betree_message_t)) :
+  (Result (betree_list_t (UInt64 × betree_message_t)))
+  :=
+  match h: msgs with
+  | betree_list_t.BetreeListCons p l =>
+    let (k, m) := p
+    if h: k = key
+    then
+      do
+        let msgs0 ←
+          betree_list_pop_front_back (UInt64 × betree_message_t)
+            (betree_list_t.BetreeListCons (k, m) l)
+        betree_node_filter_messages_for_key_fwd_back key msgs0
+    else Result.ret (betree_list_t.BetreeListCons (k, m) l)
+  | betree_list_t.BetreeListNil => Result.ret betree_list_t.BetreeListNil
+termination_by betree_node_filter_messages_for_key_fwd_back key msgs =>
+  betree_node_filter_messages_for_key_terminates key msgs
+decreasing_by betree_node_filter_messages_for_key_decreases key msgs
+
+/- [betree_main::betree::Node::{5}::lookup_first_message_after_key] -/
+def betree_node_lookup_first_message_after_key_fwd
+  (key : UInt64) (msgs : betree_list_t (UInt64 × betree_message_t)) :
+  (Result (betree_list_t (UInt64 × betree_message_t)))
+  :=
+  match h: msgs with
+  | betree_list_t.BetreeListCons p next_msgs =>
+    let (k, m) := p
+    if h: k = key
+    then betree_node_lookup_first_message_after_key_fwd key next_msgs
+    else Result.ret (betree_list_t.BetreeListCons (k, m) next_msgs)
+  | betree_list_t.BetreeListNil => Result.ret betree_list_t.BetreeListNil
+termination_by betree_node_lookup_first_message_after_key_fwd key msgs =>
+  betree_node_lookup_first_message_after_key_terminates key msgs
+decreasing_by betree_node_lookup_first_message_after_key_decreases key msgs
+
+/- [betree_main::betree::Node::{5}::lookup_first_message_after_key] -/
+def betree_node_lookup_first_message_after_key_back
+  (key : UInt64) (msgs : betree_list_t (UInt64 × betree_message_t))
+  (ret0 : betree_list_t (UInt64 × betree_message_t)) :
+  (Result (betree_list_t (UInt64 × betree_message_t)))
+  :=
+  match h: msgs with
+  | betree_list_t.BetreeListCons p next_msgs =>
+    let (k, m) := p
+    if h: k = key
+    then
+      do
+        let next_msgs0 ←
+          betree_node_lookup_first_message_after_key_back key next_msgs ret0
+        Result.ret (betree_list_t.BetreeListCons (k, m) next_msgs0)
+    else Result.ret ret0
+  | betree_list_t.BetreeListNil => Result.ret ret0
+termination_by betree_node_lookup_first_message_after_key_back key msgs ret0 =>
+  betree_node_lookup_first_message_after_key_terminates key msgs
+decreasing_by betree_node_lookup_first_message_after_key_decreases key msgs
+
+/- [betree_main::betree::Node::{5}::apply_to_internal] -/
+def betree_node_apply_to_internal_fwd_back
+  (msgs : betree_list_t (UInt64 × betree_message_t)) (key : UInt64)
+  (new_msg : betree_message_t) :
+  Result (betree_list_t (UInt64 × betree_message_t))
+  :=
+  do
+    let msgs0 ← betree_node_lookup_first_message_for_key_fwd key msgs
+    let b ← betree_list_head_has_key_fwd betree_message_t msgs0 key
+    if h: b
+    then
+      match h: new_msg with
+      | betree_message_t.BetreeMessageInsert i =>
+        do
+          let msgs1 ← betree_node_filter_messages_for_key_fwd_back key msgs0
+          let msgs2 ←
+            betree_list_push_front_fwd_back (UInt64 × betree_message_t) msgs1
+              (key, betree_message_t.BetreeMessageInsert i)
+          betree_node_lookup_first_message_for_key_back key msgs msgs2
+      | betree_message_t.BetreeMessageDelete =>
+        do
+          let msgs1 ← betree_node_filter_messages_for_key_fwd_back key msgs0
+          let msgs2 ←
+            betree_list_push_front_fwd_back (UInt64 × betree_message_t) msgs1
+              (key, betree_message_t.BetreeMessageDelete)
+          betree_node_lookup_first_message_for_key_back key msgs msgs2
+      | betree_message_t.BetreeMessageUpsert s =>
+        do
+          let p ← betree_list_hd_fwd (UInt64 × betree_message_t) msgs0
+          let (_, m) := p
+          match h: m with
+          | betree_message_t.BetreeMessageInsert prev =>
+            do
+              let v ← betree_upsert_update_fwd (Option.some prev) s
+              let msgs1 ←
+                betree_list_pop_front_back (UInt64 × betree_message_t) msgs0
+              let msgs2 ←
+                betree_list_push_front_fwd_back (UInt64 × betree_message_t)
+                  msgs1 (key, betree_message_t.BetreeMessageInsert v)
+              betree_node_lookup_first_message_for_key_back key msgs msgs2
+          | betree_message_t.BetreeMessageDelete =>
+            do
+              let v ← betree_upsert_update_fwd Option.none s
+              let msgs1 ←
+                betree_list_pop_front_back (UInt64 × betree_message_t) msgs0
+              let msgs2 ←
+                betree_list_push_front_fwd_back (UInt64 × betree_message_t)
+                  msgs1 (key, betree_message_t.BetreeMessageInsert v)
+              betree_node_lookup_first_message_for_key_back key msgs msgs2
+          | betree_message_t.BetreeMessageUpsert ufs =>
+            do
+              let msgs1 ←
+                betree_node_lookup_first_message_after_key_fwd key msgs0
+              let msgs2 ←
+                betree_list_push_front_fwd_back (UInt64 × betree_message_t)
+                  msgs1 (key, betree_message_t.BetreeMessageUpsert s)
+              let msgs3 ←
+                betree_node_lookup_first_message_after_key_back key msgs0 msgs2
+              betree_node_lookup_first_message_for_key_back key msgs msgs3
+    else
+      do
+        let msgs1 ←
+          betree_list_push_front_fwd_back (UInt64 × betree_message_t) msgs0
+            (key, new_msg)
+        betree_node_lookup_first_message_for_key_back key msgs msgs1
+
+/- [betree_main::betree::Node::{5}::apply_messages_to_internal] -/
+def betree_node_apply_messages_to_internal_fwd_back
+  (msgs : betree_list_t (UInt64 × betree_message_t))
+  (new_msgs : betree_list_t (UInt64 × betree_message_t)) :
+  (Result (betree_list_t (UInt64 × betree_message_t)))
+  :=
+  match h: new_msgs with
+  | betree_list_t.BetreeListCons new_msg new_msgs_tl =>
+    do
+      let (i, m) := new_msg
+      let msgs0 ← betree_node_apply_to_internal_fwd_back msgs i m
+      betree_node_apply_messages_to_internal_fwd_back msgs0 new_msgs_tl
+  | betree_list_t.BetreeListNil => Result.ret msgs
+termination_by betree_node_apply_messages_to_internal_fwd_back msgs new_msgs =>
+  betree_node_apply_messages_to_internal_terminates msgs new_msgs
+decreasing_by betree_node_apply_messages_to_internal_decreases msgs new_msgs
+
+/- [betree_main::betree::Node::{5}::apply_messages] -/
+mutual def betree_node_apply_messages_fwd
+  (self : betree_node_t) (params : betree_params_t)
+  (node_id_cnt : betree_node_id_counter_t)
+  (msgs : betree_list_t (UInt64 × betree_message_t)) (st : State) :
+  (Result (State × Unit))
+  :=
+  match h: self with
+  | betree_node_t.BetreeNodeInternal node =>
+    do
+      let (st0, content) ←
+        betree_load_internal_node_fwd node.betree_internal_id st
+      let content0 ←
+        betree_node_apply_messages_to_internal_fwd_back content msgs
+      let num_msgs ←
+        betree_list_len_fwd (UInt64 × betree_message_t) content0
+      if h: num_msgs >= params.betree_params_min_flush_size
+      then
+        do
+          let (st1, content1) ←
+            betree_internal_flush_fwd node params node_id_cnt content0 st0
+          let (node0, _) ←
+            betree_internal_flush_back node params node_id_cnt content0 st0
+          let (st2, _) ←
+            betree_store_internal_node_fwd node0.betree_internal_id content1
+              st1
+          Result.ret (st2, ())
+      else
+        do
+          let (st1, _) ←
+            betree_store_internal_node_fwd node.betree_internal_id content0 st0
+          Result.ret (st1, ())
+  | betree_node_t.BetreeNodeLeaf node =>
+    do
+      let (st0, content) ← betree_load_leaf_node_fwd node.betree_leaf_id st
+      let content0 ← betree_node_apply_messages_to_leaf_fwd_back content msgs
+      let len ← betree_list_len_fwd (UInt64 × UInt64) content0
+      let i ← UInt64.checked_mul (UInt64.ofNatCore 2 (by intlit))
+        params.betree_params_split_size
+      if h: len >= i
+      then
+        do
+          let (st1, _) ←
+            betree_leaf_split_fwd node content0 params node_id_cnt st0
+          let (st2, _) ←
+            betree_store_leaf_node_fwd node.betree_leaf_id
+              betree_list_t.BetreeListNil st1
+          Result.ret (st2, ())
+      else
+        do
+          let (st1, _) ←
+            betree_store_leaf_node_fwd node.betree_leaf_id content0 st0
+          Result.ret (st1, ())
+termination_by betree_node_apply_messages_fwd self params node_id_cnt msgs st
+  =>
+  betree_node_apply_messages_terminates self params node_id_cnt msgs st
+decreasing_by
+  betree_node_apply_messages_decreases self params node_id_cnt msgs st
+
+/- [betree_main::betree::Node::{5}::apply_messages] -/
+def betree_node_apply_messages_back
+  (self : betree_node_t) (params : betree_params_t)
+  (node_id_cnt : betree_node_id_counter_t)
+  (msgs : betree_list_t (UInt64 × betree_message_t)) (st : State) :
+  (Result (betree_node_t × betree_node_id_counter_t))
+  :=
+  match h: self with
+  | betree_node_t.BetreeNodeInternal node =>
+    do
+      let (st0, content) ←
+        betree_load_internal_node_fwd node.betree_internal_id st
+      let content0 ←
+        betree_node_apply_messages_to_internal_fwd_back content msgs
+      let num_msgs ←
+        betree_list_len_fwd (UInt64 × betree_message_t) content0
+      if h: num_msgs >= params.betree_params_min_flush_size
+      then
+        do
+          let (st1, content1) ←
+            betree_internal_flush_fwd node params node_id_cnt content0 st0
+          let (node0, node_id_cnt0) ←
+            betree_internal_flush_back node params node_id_cnt content0 st0
+          let _ ←
+            betree_store_internal_node_fwd node0.betree_internal_id content1
+              st1
+          Result.ret (betree_node_t.BetreeNodeInternal node0, node_id_cnt0)
+      else
+        do
+          let _ ←
+            betree_store_internal_node_fwd node.betree_internal_id content0 st0
+          Result.ret (betree_node_t.BetreeNodeInternal node, node_id_cnt)
+  | betree_node_t.BetreeNodeLeaf node =>
+    do
+      let (st0, content) ← betree_load_leaf_node_fwd node.betree_leaf_id st
+      let content0 ← betree_node_apply_messages_to_leaf_fwd_back content msgs
+      let len ← betree_list_len_fwd (UInt64 × UInt64) content0
+      let i ← UInt64.checked_mul (UInt64.ofNatCore 2 (by intlit))
+        params.betree_params_split_size
+      if h: len >= i
+      then
+        do
+          let (st1, new_node) ←
+            betree_leaf_split_fwd node content0 params node_id_cnt st0
+          let _ ←
+            betree_store_leaf_node_fwd node.betree_leaf_id
+              betree_list_t.BetreeListNil st1
+          let node_id_cnt0 ←
+            betree_leaf_split_back node content0 params node_id_cnt st0
+          Result.ret (betree_node_t.BetreeNodeInternal new_node, node_id_cnt0)
+      else
+        do
+          let _ ← betree_store_leaf_node_fwd node.betree_leaf_id content0 st0
+          Result.ret (betree_node_t.BetreeNodeLeaf
+            { betree_leaf_id := node.betree_leaf_id, betree_leaf_size := len },
+            node_id_cnt)
+termination_by betree_node_apply_messages_back self params node_id_cnt msgs st
+  =>
+  betree_node_apply_messages_terminates self params node_id_cnt msgs st
+decreasing_by
+  betree_node_apply_messages_decreases self params node_id_cnt msgs st
+
+/- [betree_main::betree::Internal::{4}::flush] -/
+def betree_internal_flush_fwd
+  (self : betree_internal_t) (params : betree_params_t)
+  (node_id_cnt : betree_node_id_counter_t)
+  (content : betree_list_t (UInt64 × betree_message_t)) (st : State) :
+  (Result (State × (betree_list_t (UInt64 × betree_message_t))))
+  :=
+  do
+    let p ←
+      betree_list_partition_at_pivot_fwd betree_message_t content
+        self.betree_internal_pivot
+    let (msgs_left, msgs_right) := p
+    let len_left ← betree_list_len_fwd (UInt64 × betree_message_t) msgs_left
+    if h: len_left >= params.betree_params_min_flush_size
+    then
+      do
+        let (st0, _) ←
+          betree_node_apply_messages_fwd self.betree_internal_left params
+            node_id_cnt msgs_left st
+        let (_, node_id_cnt0) ←
+          betree_node_apply_messages_back self.betree_internal_left params
+            node_id_cnt msgs_left st
+        let len_right ←
+          betree_list_len_fwd (UInt64 × betree_message_t) msgs_right
+        if h: len_right >= params.betree_params_min_flush_size
+        then
+          do
+            let (st1, _) ←
+              betree_node_apply_messages_fwd self.betree_internal_right params
+                node_id_cnt0 msgs_right st0
+            let _ ←
+              betree_node_apply_messages_back self.betree_internal_right params
+                node_id_cnt0 msgs_right st0
+            Result.ret (st1, betree_list_t.BetreeListNil)
+        else Result.ret (st0, msgs_right)
+    else
+      do
+        let (st0, _) ←
+          betree_node_apply_messages_fwd self.betree_internal_right params
+            node_id_cnt msgs_right st
+        let _ ←
+          betree_node_apply_messages_back self.betree_internal_right params
+            node_id_cnt msgs_right st
+        Result.ret (st0, msgs_left)
+termination_by betree_internal_flush_fwd self params node_id_cnt content st =>
+  betree_internal_flush_terminates self params node_id_cnt content st
+decreasing_by
+  betree_internal_flush_decreases self params node_id_cnt content st
+
+/- [betree_main::betree::Internal::{4}::flush] -/
+def betree_internal_flush_back
+  (self : betree_internal_t) (params : betree_params_t)
+  (node_id_cnt : betree_node_id_counter_t)
+  (content : betree_list_t (UInt64 × betree_message_t)) (st : State) :
+  (Result (betree_internal_t × betree_node_id_counter_t))
+  :=
+  do
+    let p ←
+      betree_list_partition_at_pivot_fwd betree_message_t content
+        self.betree_internal_pivot
+    let (msgs_left, msgs_right) := p
+    let len_left ← betree_list_len_fwd (UInt64 × betree_message_t) msgs_left
+    if h: len_left >= params.betree_params_min_flush_size
+    then
+      do
+        let (st0, _) ←
+          betree_node_apply_messages_fwd self.betree_internal_left params
+            node_id_cnt msgs_left st
+        let (n, node_id_cnt0) ←
+          betree_node_apply_messages_back self.betree_internal_left params
+            node_id_cnt msgs_left st
+        let len_right ←
+          betree_list_len_fwd (UInt64 × betree_message_t) msgs_right
+        if h: len_right >= params.betree_params_min_flush_size
+        then
+          do
+            let (n0, node_id_cnt1) ←
+              betree_node_apply_messages_back self.betree_internal_right params
+                node_id_cnt0 msgs_right st0
+            Result.ret
+              ({
+                 betree_internal_id := self.betree_internal_id,
+                 betree_internal_pivot := self.betree_internal_pivot,
+                 betree_internal_left := n,
+                 betree_internal_right := n0
+               }, node_id_cnt1)
+        else
+          Result.ret
+            ({
+               betree_internal_id := self.betree_internal_id,
+               betree_internal_pivot := self.betree_internal_pivot,
+               betree_internal_left := n,
+               betree_internal_right := self.betree_internal_right
+             }, node_id_cnt0)
+    else
+      do
+        let (n, node_id_cnt0) ←
+          betree_node_apply_messages_back self.betree_internal_right params
+            node_id_cnt msgs_right st
+        Result.ret
+          ({
+             betree_internal_id := self.betree_internal_id,
+             betree_internal_pivot := self.betree_internal_pivot,
+             betree_internal_left := self.betree_internal_left,
+             betree_internal_right := n
+           }, node_id_cnt0)
+termination_by betree_internal_flush_back self params node_id_cnt content st =>
+  betree_internal_flush_terminates self params node_id_cnt content st
+decreasing_by
+  betree_internal_flush_decreases self params node_id_cnt content st
+
+/- [betree_main::betree::Node::{5}::apply] -/
+def betree_node_apply_fwd
+  (self : betree_node_t) (params : betree_params_t)
+  (node_id_cnt : betree_node_id_counter_t) (key : UInt64)
+  (new_msg : betree_message_t) (st : State) :
+  Result (State × Unit)
+  :=
+  do
+    let l := betree_list_t.BetreeListNil
+    let (st0, _) ←
+      betree_node_apply_messages_fwd self params node_id_cnt
+        (betree_list_t.BetreeListCons (key, new_msg) l) st
+    let _ ←
+      betree_node_apply_messages_back self params node_id_cnt
+        (betree_list_t.BetreeListCons (key, new_msg) l) st
+    Result.ret (st0, ())
+
+/- [betree_main::betree::Node::{5}::apply] -/
+def betree_node_apply_back
+  (self : betree_node_t) (params : betree_params_t)
+  (node_id_cnt : betree_node_id_counter_t) (key : UInt64)
+  (new_msg : betree_message_t) (st : State) :
+  Result (betree_node_t × betree_node_id_counter_t)
+  :=
+  let l := betree_list_t.BetreeListNil
+  betree_node_apply_messages_back self params node_id_cnt
+    (betree_list_t.BetreeListCons (key, new_msg) l) st
+
+/- [betree_main::betree::BeTree::{6}::new] -/
+def betree_be_tree_new_fwd
+  (min_flush_size : UInt64) (split_size : UInt64) (st : State) :
+  Result (State × betree_be_tree_t)
+  :=
+  do
+    let node_id_cnt ← betree_node_id_counter_new_fwd
+    let id ← betree_node_id_counter_fresh_id_fwd node_id_cnt
+    let (st0, _) ←
+      betree_store_leaf_node_fwd id betree_list_t.BetreeListNil st
+    let node_id_cnt0 ← betree_node_id_counter_fresh_id_back node_id_cnt
+    Result.ret (st0,
+      {
+        betree_be_tree_params := {
+                                   betree_params_min_flush_size := min_flush_size,
+                                   betree_params_split_size := split_size
+                                 },
+        betree_be_tree_node_id_cnt := node_id_cnt0,
+        betree_be_tree_root := (betree_node_t.BetreeNodeLeaf
+                                 {
+                                   betree_leaf_id := id,
+                                   betree_leaf_size := (UInt64.ofNatCore 0 (by intlit))
+                                 })
+      })
+
+/- [betree_main::betree::BeTree::{6}::apply] -/
+def betree_be_tree_apply_fwd
+  (self : betree_be_tree_t) (key : UInt64) (msg : betree_message_t)
+  (st : State) :
+  Result (State × Unit)
+  :=
+  do
+    let (st0, _) ←
+      betree_node_apply_fwd self.betree_be_tree_root self.betree_be_tree_params
+        self.betree_be_tree_node_id_cnt key msg st
+    let _ ←
+      betree_node_apply_back self.betree_be_tree_root
+        self.betree_be_tree_params self.betree_be_tree_node_id_cnt key msg st
+    Result.ret (st0, ())
+
+/- [betree_main::betree::BeTree::{6}::apply] -/
+def betree_be_tree_apply_back
+  (self : betree_be_tree_t) (key : UInt64) (msg : betree_message_t)
+  (st : State) :
+  Result betree_be_tree_t
+  :=
+  do
+    let (n, nic) ←
+      betree_node_apply_back self.betree_be_tree_root
+        self.betree_be_tree_params self.betree_be_tree_node_id_cnt key msg st
+    Result.ret
+      {
+        betree_be_tree_params := self.betree_be_tree_params,
+        betree_be_tree_node_id_cnt := nic,
+        betree_be_tree_root := n
+      }
+
+/- [betree_main::betree::BeTree::{6}::insert] -/
+def betree_be_tree_insert_fwd
+  (self : betree_be_tree_t) (key : UInt64) (value : UInt64) (st : State) :
+  Result (State × Unit)
+  :=
+  do
+    let (st0, _) ←
+      betree_be_tree_apply_fwd self key (betree_message_t.BetreeMessageInsert
+        value) st
+    let _ ←
+      betree_be_tree_apply_back self key (betree_message_t.BetreeMessageInsert
+        value) st
+    Result.ret (st0, ())
+
+/- [betree_main::betree::BeTree::{6}::insert] -/
+def betree_be_tree_insert_back
+  (self : betree_be_tree_t) (key : UInt64) (value : UInt64) (st : State) :
+  Result betree_be_tree_t
+  :=
+  betree_be_tree_apply_back self key (betree_message_t.BetreeMessageInsert
+    value) st
+
+/- [betree_main::betree::BeTree::{6}::delete] -/
+def betree_be_tree_delete_fwd
+  (self : betree_be_tree_t) (key : UInt64) (st : State) :
+  Result (State × Unit)
+  :=
+  do
+    let (st0, _) ←
+      betree_be_tree_apply_fwd self key betree_message_t.BetreeMessageDelete st
+    let _ ←
+      betree_be_tree_apply_back self key betree_message_t.BetreeMessageDelete
+        st
+    Result.ret (st0, ())
+
+/- [betree_main::betree::BeTree::{6}::delete] -/
+def betree_be_tree_delete_back
+  (self : betree_be_tree_t) (key : UInt64) (st : State) :
+  Result betree_be_tree_t
+  :=
+  betree_be_tree_apply_back self key betree_message_t.BetreeMessageDelete st
+
+/- [betree_main::betree::BeTree::{6}::upsert] -/
+def betree_be_tree_upsert_fwd
+  (self : betree_be_tree_t) (key : UInt64) (upd : betree_upsert_fun_state_t)
+  (st : State) :
+  Result (State × Unit)
+  :=
+  do
+    let (st0, _) ←
+      betree_be_tree_apply_fwd self key (betree_message_t.BetreeMessageUpsert
+        upd) st
+    let _ ←
+      betree_be_tree_apply_back self key (betree_message_t.BetreeMessageUpsert
+        upd) st
+    Result.ret (st0, ())
+
+/- [betree_main::betree::BeTree::{6}::upsert] -/
+def betree_be_tree_upsert_back
+  (self : betree_be_tree_t) (key : UInt64) (upd : betree_upsert_fun_state_t)
+  (st : State) :
+  Result betree_be_tree_t
+  :=
+  betree_be_tree_apply_back self key (betree_message_t.BetreeMessageUpsert upd)
+    st
+
+/- [betree_main::betree::BeTree::{6}::lookup] -/
+def betree_be_tree_lookup_fwd
+  (self : betree_be_tree_t) (key : UInt64) (st : State) :
+  Result (State × (Option UInt64))
+  :=
+  betree_node_lookup_fwd self.betree_be_tree_root key st
+
+/- [betree_main::betree::BeTree::{6}::lookup] -/
+def betree_be_tree_lookup_back
+  (self : betree_be_tree_t) (key : UInt64) (st : State) :
+  Result betree_be_tree_t
+  :=
+  do
+    let n ← betree_node_lookup_back self.betree_be_tree_root key st
+    Result.ret
+      {
+        betree_be_tree_params := self.betree_be_tree_params,
+        betree_be_tree_node_id_cnt := self.betree_be_tree_node_id_cnt,
+        betree_be_tree_root := n
+      }
+
+/- [betree_main::main] -/
+def main_fwd : Result Unit :=
+  Result.ret ()
+
+/- Unit test for [betree_main::main] -/
+#assert (main_fwd == .ret ())
+
diff --git a/tests/lean/betree/BetreeMain/Opaque.lean b/tests/lean/betree/BetreeMain/Opaque.lean
new file mode 100644
index 00000000..b3db37c2
--- /dev/null
+++ b/tests/lean/betree/BetreeMain/Opaque.lean
@@ -0,0 +1,33 @@
+-- THIS FILE WAS AUTOMATICALLY GENERATED BY AENEAS
+-- [betree_main]: opaque function definitions
+import Base.Primitives
+import BetreeMain.Types
+
+structure OpaqueDefs where
+  
+  /- [betree_main::betree_utils::load_internal_node] -/
+  betree_utils_load_internal_node_fwd
+    :
+    UInt64 -> State -> Result (State × (betree_list_t (UInt64 ×
+      betree_message_t)))
+  
+  /- [betree_main::betree_utils::store_internal_node] -/
+  betree_utils_store_internal_node_fwd
+    :
+    UInt64 -> betree_list_t (UInt64 × betree_message_t) -> State -> Result
+      (State × Unit)
+  
+  /- [betree_main::betree_utils::load_leaf_node] -/
+  betree_utils_load_leaf_node_fwd
+    : UInt64 -> State -> Result (State × (betree_list_t (UInt64 × UInt64)))
+  
+  /- [betree_main::betree_utils::store_leaf_node] -/
+  betree_utils_store_leaf_node_fwd
+    :
+    UInt64 -> betree_list_t (UInt64 × UInt64) -> State -> Result (State ×
+      Unit)
+  
+  /- [core::option::Option::{0}::unwrap] -/
+  core_option_option_unwrap_fwd
+    (T : Type) : Option T -> State -> Result (State × T)
+  
diff --git a/tests/lean/betree/BetreeMain/Types.lean b/tests/lean/betree/BetreeMain/Types.lean
new file mode 100644
index 00000000..2726e1f0
--- /dev/null
+++ b/tests/lean/betree/BetreeMain/Types.lean
@@ -0,0 +1,61 @@
+-- THIS FILE WAS AUTOMATICALLY GENERATED BY AENEAS
+-- [betree_main]: type definitions
+import Base.Primitives
+
+/- [betree_main::betree::List] -/
+inductive betree_list_t (T : Type) :=
+| BetreeListCons : T -> betree_list_t T -> betree_list_t T
+| BetreeListNil : betree_list_t T
+
+/- [betree_main::betree::UpsertFunState] -/
+inductive betree_upsert_fun_state_t :=
+| BetreeUpsertFunStateAdd : UInt64 -> betree_upsert_fun_state_t
+| BetreeUpsertFunStateSub : UInt64 -> betree_upsert_fun_state_t
+
+/- [betree_main::betree::Message] -/
+inductive betree_message_t :=
+| BetreeMessageInsert : UInt64 -> betree_message_t
+| BetreeMessageDelete : betree_message_t
+| BetreeMessageUpsert : betree_upsert_fun_state_t -> betree_message_t
+
+/- [betree_main::betree::Leaf] -/
+structure betree_leaf_t where
+
+  betree_leaf_id : UInt64 betree_leaf_size : UInt64
+
+
+/- [betree_main::betree::Node] -/
+mutual inductive betree_node_t :=
+| BetreeNodeInternal : betree_internal_t -> betree_node_t
+| BetreeNodeLeaf : betree_leaf_t -> betree_node_t
+
+/- [betree_main::betree::Internal] -/
+inductive betree_internal_t :=
+
+  betree_internal_id : UInt64
+  betree_internal_pivot : UInt64
+  betree_internal_left : betree_node_t
+  betree_internal_right : betree_node_t
+
+
+/- [betree_main::betree::Params] -/
+structure betree_params_t where
+
+  betree_params_min_flush_size : UInt64 betree_params_split_size : UInt64
+
+
+/- [betree_main::betree::NodeIdCounter] -/
+structure betree_node_id_counter_t where
+
+  betree_node_id_counter_next_node_id : UInt64
+
+
+/- [betree_main::betree::BeTree] -/
+structure betree_be_tree_t where
+
+  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
+
+/- The state type used in the state-error monad -/ axiom State : Type
+