Make minor modifications and regenerate the Lean files

21 files changed, 1600 insertions, 1489 deletions

diff --git a/backends/lean/Primitives.lean b/backends/lean/Primitives.lean
index e558e476..346e4b89 100644
--- a/backends/lean/Primitives.lean
+++ b/backends/lean/Primitives.lean
@@ -21,7 +21,7 @@ open Error
 
 inductive Result (α : Type u) where
   | ret (v: α): Result α
-  | fail (e: Error): Result α 
+  | fail (e: Error): Result α
 deriving Repr, BEq
 
 open Result
diff --git a/tests/lean/hashmap/Base/Primitives.lean b/tests/lean/hashmap/Base/Primitives.lean
index 79958d94..5b64e908 100644
--- a/tests/lean/hashmap/Base/Primitives.lean
+++ b/tests/lean/hashmap/Base/Primitives.lean
@@ -9,74 +9,79 @@ import Mathlib.Tactic.RunCmd
 
 -- Results & monadic combinators
 
--- TODO: use syntactic conventions and capitalize error, result, etc.
-
-inductive error where
-   | assertionFailure: error
-   | integerOverflow: error
-   | arrayOutOfBounds: error
-   | maximumSizeExceeded: error
-   | panic: error
+inductive Error where
+   | assertionFailure: Error
+   | integerOverflow: Error
+   | arrayOutOfBounds: Error
+   | maximumSizeExceeded: Error
+   | panic: Error
 deriving Repr, BEq
 
-open error
+open Error
 
-inductive result (α : Type u) where
-  | ret (v: α): result α
-  | fail (e: error): result α 
+inductive Result (α : Type u) where
+  | ret (v: α): Result α
+  | fail (e: Error): Result α
 deriving Repr, BEq
 
-open result
+open Result
 
 /- HELPERS -/
 
--- TODO: is there automated syntax for these discriminators?
-def is_ret {α: Type} (r: result α): Bool :=
+def ret? {α: Type} (r: Result α): Bool :=
   match r with
-  | result.ret _ => true
-  | result.fail _ => false
+  | Result.ret _ => true
+  | Result.fail _ => false
 
-def massert (b:Bool) : result Unit :=
+def massert (b:Bool) : Result Unit :=
   if b then .ret () else fail assertionFailure
 
-def eval_global {α: Type} (x: result α) (_: is_ret x): α :=
+def eval_global {α: Type} (x: Result α) (_: ret? x): α :=
   match x with
-  | result.fail _ => by contradiction
-  | result.ret x => x
+  | Result.fail _ => by contradiction
+  | Result.ret x => x
 
 /- DO-DSL SUPPORT -/
 
-def bind (x: result α) (f: α -> result β) : result β :=
+def bind (x: Result α) (f: α -> Result β) : Result β :=
   match x with
   | ret v  => f v 
   | fail v => fail v
 
--- Allows using result in do-blocks
-instance : Bind result where
+-- Allows using Result in do-blocks
+instance : Bind Result where
   bind := bind
 
 -- Allows using return x in do-blocks
-instance : Pure result where
+instance : Pure Result where
   pure := fun x => ret x
 
 /- CUSTOM-DSL SUPPORT -/
 
--- Let-binding the result of a monadic operation is oftentimes not sufficient,
+-- Let-binding the Result of a monadic operation is oftentimes not sufficient,
 -- because we may need a hypothesis for equational reasoning in the scope. We
 -- rely on subtype, and a custom let-binding operator, in effect recreating our
 -- own variant of the do-dsl
 
-def result.attach : (o : result α) → result { x : α // o = ret x }
+def Result.attach {α: Type} (o : Result α): Result { x : α // o = ret x } :=
+  match o with
   | .ret x => .ret ⟨x, rfl⟩
-  | .fail e   => .fail e
+  | .fail e => .fail e
 
-macro "let" h:ident " : " e:term " <-- " f:term : doElem =>
-  `(doElem| let ⟨$e, $h⟩ ← result.attach $f)
+macro "let" e:term " ⟵ " f:term : doElem =>
+  `(doElem| let ⟨$e, h⟩ ← Result.attach $f)
 
--- Silly example of the kind of reasoning that this notation enables
+-- TODO: any way to factorize both definitions?
+macro "let" e:term " <-- " f:term : doElem =>
+  `(doElem| let ⟨$e, h⟩ ← Result.attach $f)
+
+-- We call the hypothesis `h`, in effect making it unavailable to the user
+-- (because too much shadowing). But in practice, once can use the French single
+-- quote notation (input with f< and f>), where `‹ h ›` finds a suitable
+-- hypothesis in the context, this is equivalent to `have x: h := by assumption in x`
 #eval do
-  let h: y <-- .ret (0: Nat)
-  let _: y = 0 := by cases h; decide
+  let y <-- .ret (0: Nat)
+  let _: y = 0 := by cases ‹ ret 0 = ret y › ; decide
   let r: { x: Nat // x = 0 } := ⟨ y, by assumption ⟩
   .ret r
 
@@ -84,36 +89,27 @@ macro "let" h:ident " : " e:term " <-- " f:term : doElem =>
 -- MACHINE INTEGERS --
 ----------------------
 
--- NOTE: we reuse the USize type from prelude.lean, because at least we know
--- it's defined in an idiomatic style that is going to make proofs easy (and
--- indeed, several proofs here are much shortened compared to Aymeric's earlier
--- attempt.) This is not stricto sensu the *correct* thing to do, because one
--- can query at run-time the value of USize, which we do *not* want to do (we
--- don't know what target we'll run on!), but when the day comes, we'll just
--- define our own USize.
--- ANOTHER NOTE: there is USize.sub but it has wraparound semantics, which is
--- not something we want to define (I think), so we use our own monadic sub (but
--- is it in line with the Rust behavior?)
-
--- TODO: I am somewhat under the impression that subtraction is defined as a
--- total function over nats...? the hypothesis in the if condition is not used
--- in the then-branch which confuses me quite a bit
-
--- TODO: add a refinement for the result (just like vec_push_back below) that
--- explains that the toNat of the result (in the case of success) is the sub of
--- the toNat of the arguments (i.e. intrinsic specification)
--- ... do we want intrinsic specifications for the builtins? that might require
--- some careful type annotations in the monadic notation for clients, but may
--- give us more "for free"
+-- NOTE: we reuse the fixed-width integer types from prelude.lean: UInt8, ...,
+-- USize. They are generally defined in an idiomatic style, except that there is
+-- not a single type class to rule them all (more on that below). The absence of
+-- type class is intentional, and allows the Lean compiler to efficiently map
+-- them to machine integers during compilation.
+
+-- USize is designed properly: you cannot reduce `getNumBits` using the
+-- simplifier, meaning that proofs do not depend on the compile-time value of
+-- USize.size. (Lean assumes 32 or 64-bit platforms, and Rust doesn't really
+-- support, at least officially, 16-bit microcontrollers, so this seems like a
+-- fine design decision for now.)
 
 -- Note from Chris Bailey: "If there's more than one salient property of your
 -- definition then the subtyping strategy might get messy, and the property part
 -- of a subtype is less discoverable by the simplifier or tactics like
--- library_search." Try to settle this with a Lean expert on what is the most
--- productive way to go about this?
+-- library_search." So, we will not add refinements on the return values of the
+-- operations defined on Primitives, but will rather rely on custom lemmas to
+-- invert on possible return values of the primitive operations.
 
--- One needs to perform a little bit of reasoning in order to successfully
--- inject constants into USize, so we provide a general-purpose macro
+-- Machine integer constants, done via `ofNatCore`, which requires a proof that
+-- the `Nat` fits within the desired integer type. We provide a custom tactic.
 
 syntax "intlit" : tactic
 
@@ -129,12 +125,21 @@ macro_rules
 -- Also works for other integer types (at the expense of a needless disjunction)
 #eval UInt32.ofNatCore 0 (by intlit)
 
+-- The machine integer operations (e.g. sub) are always total, which is not what
+-- we want. We therefore define "checked" variants, below. Note that we add a
+-- tiny bit of complexity for the USize variant: we first check whether the
+-- result is < 2^32; if it is, we can compute the definition, rather than
+-- returning a term that is computationally stuck (the comparison to USize.size
+-- cannot reduce at compile-time, per the remark about regarding `getNumBits`).
+-- This is useful for the various #asserts that we want to reduce at
+-- type-checking time.
+
 -- Further thoughts: look at what has been done here:
 -- https://github.com/leanprover-community/mathlib4/blob/master/Mathlib/Data/Fin/Basic.lean
 -- and
 -- https://github.com/leanprover-community/mathlib4/blob/master/Mathlib/Data/UInt.lean
 -- which both contain a fair amount of reasoning already!
-def USize.checked_sub (n: USize) (m: USize): result USize :=
+def USize.checked_sub (n: USize) (m: USize): Result USize :=
   -- NOTE: the test USize.toNat n - m >= 0 seems to always succeed?
   if n >= m then
     let n' := USize.toNat n
@@ -150,18 +155,19 @@ def USize.checked_sub (n: USize) (m: USize): result USize :=
   else
     fail integerOverflow
 
-def USize.checked_add (n: USize) (m: USize): result USize :=
-  if h: n.val.val + m.val.val <= 4294967295 then
-    .ret ⟨ n.val.val + m.val.val, by
-      have h': 4294967295 < USize.size := by intlit
-      apply Nat.lt_of_le_of_lt h h'
-    ⟩
-  else if h: n.val + m.val < USize.size then
+@[simp]
+theorem usize_fits (n: Nat) (h: n <= 4294967295): n < USize.size :=
+  match USize.size, usize_size_eq with
+  | _, Or.inl rfl => Nat.lt_of_le_of_lt h (by decide)
+  | _, Or.inr rfl => Nat.lt_of_le_of_lt h (by decide)
+
+def USize.checked_add (n: USize) (m: USize): Result USize :=
+  if h: n.val + m.val < USize.size then
     .ret ⟨ n.val + m.val, h ⟩
   else
     .fail integerOverflow
 
-def USize.checked_rem (n: USize) (m: USize): result USize :=
+def USize.checked_rem (n: USize) (m: USize): Result USize :=
   if h: m > 0 then
     .ret ⟨ n.val % m.val, by
       have h1: ↑m.val < USize.size := m.val.isLt
@@ -171,18 +177,13 @@ def USize.checked_rem (n: USize) (m: USize): result USize :=
   else
     .fail integerOverflow
 
-def USize.checked_mul (n: USize) (m: USize): result USize :=
-    if h: n.val.val * m.val.val <= 4294967295 then
-    .ret ⟨ n.val.val * m.val.val, by
-      have h': 4294967295 < USize.size := by intlit
-      apply Nat.lt_of_le_of_lt h h'
-    ⟩
-  else if h: n.val * m.val < USize.size then
+def USize.checked_mul (n: USize) (m: USize): Result USize :=
+  if h: n.val * m.val < USize.size then
     .ret ⟨ n.val * m.val, h ⟩
   else
     .fail integerOverflow
 
-def USize.checked_div (n: USize) (m: USize): result USize :=
+def USize.checked_div (n: USize) (m: USize): Result USize :=
   if m > 0 then
     .ret ⟨ n.val / m.val, by
       have h1: ↑n.val < USize.size := n.val.isLt
@@ -192,6 +193,19 @@ def USize.checked_div (n: USize) (m: USize): result USize :=
   else
     .fail integerOverflow
 
+-- Test behavior...
+#eval assert! USize.checked_sub 10 20 == fail integerOverflow; 0
+
+#eval USize.checked_sub 20 10
+-- NOTE: compare with concrete behavior here, which I do not think we want
+#eval USize.sub 0 1
+#eval UInt8.add 255 255
+
+-- We now define a type class that subsumes the various machine integer types, so
+-- as to write a concise definition for scalar_cast, rather than exhaustively
+-- enumerating all of the possible pairs. We remark that Rust has sane semantics
+-- and fails if a cast operation would involve a truncation or modulo.
+
 class MachineInteger (t: Type) where
   size: Nat
   val: t -> Fin size
@@ -209,30 +223,24 @@ run_cmd
     end $typeName
   ))
 
-def scalar_cast { src: Type } (dst: Type) [ MachineInteger src ] [ MachineInteger dst ] (x: src): result dst :=
+-- Aeneas only instantiates the destination type (`src` is implicit). We rely on
+-- Lean to infer `src`.
+
+def scalar_cast { src: Type } (dst: Type) [ MachineInteger src ] [ MachineInteger dst ] (x: src): Result dst :=
   if h: MachineInteger.val x < MachineInteger.size dst then
     .ret (MachineInteger.ofNatCore (MachineInteger.val x).val h)
   else
     .fail integerOverflow
 
-
--- Test behavior...
-#eval assert! USize.checked_sub 10 20 == fail integerOverflow; 0
-
-#eval USize.checked_sub 20 10
--- NOTE: compare with concrete behavior here, which I do not think we want
-#eval USize.sub 0 1
-#eval UInt8.add 255 255
-
 -------------
 -- VECTORS --
 -------------
 
 -- Note: unlike F*, Lean seems to use strict upper bounds (e.g. USize.size)
 -- rather than maximum values (usize_max).
-def vec (α : Type u) := { l : List α // List.length l < USize.size }
+def Vec (α : Type u) := { l : List α // List.length l < USize.size }
 
-def vec_new (α : Type u): vec α := ⟨ [], by {
+def vec_new (α : Type u): Vec α := ⟨ [], by {
   match USize.size, usize_size_eq with
   | _, Or.inl rfl => simp
   | _, Or.inr rfl => simp
@@ -240,20 +248,20 @@ def vec_new (α : Type u): vec α := ⟨ [], by {
 
 #check vec_new
 
-def vec_len (α : Type u) (v : vec α) : USize :=
+def vec_len (α : Type u) (v : Vec α) : USize :=
   let ⟨ v, l ⟩ := v
   USize.ofNatCore (List.length v) l
 
 #eval vec_len Nat (vec_new Nat)
  
-def vec_push_fwd (α : Type u) (_ : vec α) (_ : α) : Unit := ()
+def vec_push_fwd (α : Type u) (_ : Vec α) (_ : α) : Unit := ()
 
 -- NOTE: old version trying to use a subtype notation, but probably better to
--- leave result elimination to auxiliary lemmas with suitable preconditions
+-- leave Result elimination to auxiliary lemmas with suitable preconditions
 -- TODO: I originally wrote `List.length v.val < USize.size - 1`; how can one
 -- make the proof work in that case? Probably need to import tactics from
 -- mathlib to deal with inequalities... would love to see an example.
-def vec_push_back_old (α : Type u) (v : vec α) (x : α) : { res: result (vec α) //
+def vec_push_back_old (α : Type u) (v : Vec α) (x : α) : { res: Result (Vec α) //
   match res with | fail _ => True | ret v' => List.length v'.val = List.length v.val + 1}
   :=
   if h : List.length v.val + 1 < USize.size then
@@ -272,12 +280,12 @@ def vec_push_back_old (α : Type u) (v : vec α) (x : α) : { res: result (vec �
   -- annotate `x`, which relieves us of having to write `.val` on the right-hand
   -- side of the monadic let.
   let v := vec_new Nat
-  let x: vec Nat ← (vec_push_back_old Nat v 1: result (vec Nat)) -- WHY do we need the type annotation here?
+  let x: Vec Nat ← (vec_push_back_old Nat v 1: Result (Vec Nat)) -- WHY do we need the type annotation here?
   -- TODO: strengthen post-condition above and do a demo to show that we can
   -- safely eliminate the `fail` case
   return (vec_len Nat x)
 
-def vec_push_back (α : Type u) (v : vec α) (x : α) : result (vec α)
+def vec_push_back (α : Type u) (v : Vec α) (x : α) : Result (Vec α)
   :=
   if h : List.length v.val + 1 <= 4294967295 then
     return ⟨ List.concat v.val x,
@@ -295,13 +303,13 @@ def vec_push_back (α : Type u) (v : vec α) (x : α) : result (vec α)
   else
     fail maximumSizeExceeded
 
-def vec_insert_fwd (α : Type u) (v: vec α) (i: USize) (_: α): result Unit :=
+def vec_insert_fwd (α : Type u) (v: Vec α) (i: USize) (_: α): Result Unit :=
   if i.val < List.length v.val then
     .ret ()
   else
     .fail arrayOutOfBounds
 
-def vec_insert_back (α : Type u) (v: vec α) (i: USize) (x: α): result (vec α) :=
+def vec_insert_back (α : Type u) (v: Vec α) (i: USize) (x: α): Result (Vec α) :=
   if i.val < List.length v.val then
     .ret ⟨ List.set v.val i.val x, by
       have h: List.length v.val < USize.size := v.property
@@ -311,25 +319,25 @@ def vec_insert_back (α : Type u) (v: vec α) (i: USize) (x: α): result (vec α
   else
     .fail arrayOutOfBounds
 
-def vec_index_fwd (α : Type u) (v: vec α) (i: USize): result α :=
+def vec_index_fwd (α : Type u) (v: Vec α) (i: USize): Result α :=
   if h: i.val < List.length v.val then
     .ret (List.get v.val ⟨i.val, h⟩)
   else
     .fail arrayOutOfBounds
 
-def vec_index_back (α : Type u) (v: vec α) (i: USize) (_: α): result Unit :=
+def vec_index_back (α : Type u) (v: Vec α) (i: USize) (_: α): Result Unit :=
   if i.val < List.length v.val then
     .ret ()
   else
     .fail arrayOutOfBounds
 
-def vec_index_mut_fwd (α : Type u) (v: vec α) (i: USize): result α :=
+def vec_index_mut_fwd (α : Type u) (v: Vec α) (i: USize): Result α :=
   if h: i.val < List.length v.val then
     .ret (List.get v.val ⟨i.val, h⟩)
   else
     .fail arrayOutOfBounds
 
-def vec_index_mut_back (α : Type u) (v: vec α) (i: USize) (x: α): result (vec α) :=
+def vec_index_mut_back (α : Type u) (v: Vec α) (i: USize) (x: α): Result (Vec α) :=
   if i.val < List.length v.val then
     .ret ⟨ List.set v.val i.val x, by
       have h: List.length v.val < USize.size := v.property
@@ -349,6 +357,10 @@ def mem_replace_fwd (a : Type) (x : a) (_ : a) : a :=
 def mem_replace_back (a : Type) (_ : a) (y : a) : a :=
   y
 
+/-- Aeneas-translated function -- useful to reduce non-recursive definitions.
+ Use with `simp [ aeneas ]` -/
+register_simp_attr aeneas
+
 --------------------
 -- ASSERT COMMAND --
 --------------------
@@ -358,16 +370,23 @@ open Lean Elab Command Term Meta
 syntax (name := assert) "#assert" term: command
 
 @[command_elab assert]
+unsafe
 def assertImpl : CommandElab := fun (_stx: Syntax) => do
-  logInfo "Reducing and asserting: "
-  logInfo _stx[1]
   runTermElabM (fun _ => do
-    let e ← Term.elabTerm _stx[1] none
-    logInfo (Expr.dbgToString e)
-    -- How to evaluate the term and compare the result to true?
+    let r ← evalTerm Bool (mkConst ``Bool) _stx[1]
+    if not r then
+      logInfo "Assertion failed for: "
+      logInfo _stx[1]
+      logError "Expression reduced to false"
     pure ())
-  -- logInfo (Expr.dbgToString (``true))
-  -- throwError "TODO: assert"
 
 #eval 2 == 2
 #assert (2 == 2)
+
+-------------------
+-- SANITY CHECKS --
+-------------------
+
+-- TODO: add more once we have signed integers
+
+#assert (USize.checked_rem 1 2 == .ret 1)
diff --git a/tests/lean/hashmap/Hashmap/Clauses/Template.lean b/tests/lean/hashmap/Hashmap/Clauses/Template.lean
index 2bc92153..b767a0eb 100644
--- a/tests/lean/hashmap/Hashmap/Clauses/Template.lean
+++ b/tests/lean/hashmap/Hashmap/Clauses/Template.lean
@@ -5,7 +5,7 @@ import Hashmap.Types
 
 /- [hashmap::HashMap::{0}::allocate_slots]: termination measure -/
 @[simp]
-def hash_map_allocate_slots_loop_terminates (T : Type) (slots : vec (list_t T))
+def hash_map_allocate_slots_loop_terminates (T : Type) (slots : Vec (list_t T))
   (n : USize) :=
   (slots, n)
 
@@ -16,7 +16,7 @@ macro_rules
 
 /- [hashmap::HashMap::{0}::clear]: termination measure -/
 @[simp]
-def hash_map_clear_loop_terminates (T : Type) (slots : vec (list_t T))
+def hash_map_clear_loop_terminates (T : Type) (slots : Vec (list_t T))
   (i : USize) :=
   (slots, i)
 
@@ -52,7 +52,7 @@ macro_rules
 /- [hashmap::HashMap::{0}::move_elements]: termination measure -/
 @[simp]
 def hash_map_move_elements_loop_terminates (T : Type) (ntable : hash_map_t T)
-  (slots : vec (list_t T)) (i : USize) :=
+  (slots : Vec (list_t T)) (i : USize) :=
   (ntable, slots, i)
 
 syntax "hash_map_move_elements_loop_decreases" term+ : tactic
diff --git a/tests/lean/hashmap/Hashmap/Funs.lean b/tests/lean/hashmap/Hashmap/Funs.lean
index 3bdd9dd2..0b3708fa 100644
--- a/tests/lean/hashmap/Hashmap/Funs.lean
+++ b/tests/lean/hashmap/Hashmap/Funs.lean
@@ -5,42 +5,42 @@ import Hashmap.Types
 import Hashmap.Clauses.Clauses
 
 /- [hashmap::hash_key] -/
-def hash_key_fwd (k : USize) : result USize :=
-  result.ret k
+def hash_key_fwd (k : USize) : Result USize :=
+  Result.ret k
 
 /- [hashmap::HashMap::{0}::allocate_slots] -/
 def hash_map_allocate_slots_loop_fwd
-  (T : Type) (slots : vec (list_t T)) (n : USize) :
-  (result (vec (list_t T)))
+  (T : Type) (slots : Vec (list_t T)) (n : USize) :
+  (Result (Vec (list_t T)))
   :=
-  if n > (USize.ofNatCore 0 (by intlit))
+  if h: n > (USize.ofNatCore 0 (by intlit))
   then
     do
-      let slots0 <- vec_push_back (list_t T) slots list_t.ListNil
-      let n0 <- USize.checked_sub n (USize.ofNatCore 1 (by intlit))
+      let slots0 ← vec_push_back (list_t T) slots list_t.ListNil
+      let n0 ← USize.checked_sub n (USize.ofNatCore 1 (by intlit))
       hash_map_allocate_slots_loop_fwd T slots0 n0
-  else result.ret slots
+  else Result.ret slots
 termination_by hash_map_allocate_slots_loop_fwd slots n =>
   hash_map_allocate_slots_loop_terminates T slots n
 decreasing_by hash_map_allocate_slots_loop_decreases slots n
 
 /- [hashmap::HashMap::{0}::allocate_slots] -/
 def hash_map_allocate_slots_fwd
-  (T : Type) (slots : vec (list_t T)) (n : USize) : result (vec (list_t T)) :=
+  (T : Type) (slots : Vec (list_t T)) (n : USize) : Result (Vec (list_t T)) :=
   hash_map_allocate_slots_loop_fwd T slots n
 
 /- [hashmap::HashMap::{0}::new_with_capacity] -/
 def hash_map_new_with_capacity_fwd
   (T : Type) (capacity : USize) (max_load_dividend : USize)
   (max_load_divisor : USize) :
-  result (hash_map_t T)
+  Result (hash_map_t T)
   :=
   do
     let v := vec_new (list_t T)
-    let slots <- hash_map_allocate_slots_fwd T v capacity
-    let i <- USize.checked_mul capacity max_load_dividend
-    let i0 <- USize.checked_div i max_load_divisor
-    result.ret
+    let slots ← hash_map_allocate_slots_fwd T v capacity
+    let i ← USize.checked_mul capacity max_load_dividend
+    let i0 ← USize.checked_div i max_load_divisor
+    Result.ret
       {
         hash_map_num_entries := (USize.ofNatCore 0 (by intlit)),
         hash_map_max_load_factor := (max_load_dividend, max_load_divisor),
@@ -49,35 +49,35 @@ def hash_map_new_with_capacity_fwd
       }
 
 /- [hashmap::HashMap::{0}::new] -/
-def hash_map_new_fwd (T : Type) : result (hash_map_t T) :=
+def hash_map_new_fwd (T : Type) : Result (hash_map_t T) :=
   hash_map_new_with_capacity_fwd T (USize.ofNatCore 32 (by intlit))
     (USize.ofNatCore 4 (by intlit)) (USize.ofNatCore 5 (by intlit))
 
 /- [hashmap::HashMap::{0}::clear] -/
 def hash_map_clear_loop_fwd_back
-  (T : Type) (slots : vec (list_t T)) (i : USize) :
-  (result (vec (list_t T)))
+  (T : Type) (slots : Vec (list_t T)) (i : USize) :
+  (Result (Vec (list_t T)))
   :=
   let i0 := vec_len (list_t T) slots
-  if i < i0
+  if h: i < i0
   then
     do
-      let i1 <- USize.checked_add i (USize.ofNatCore 1 (by intlit))
-      let slots0 <- vec_index_mut_back (list_t T) slots i list_t.ListNil
+      let i1 ← USize.checked_add i (USize.ofNatCore 1 (by intlit))
+      let slots0 ← vec_index_mut_back (list_t T) slots i list_t.ListNil
       hash_map_clear_loop_fwd_back T slots0 i1
-  else result.ret slots
+  else Result.ret slots
 termination_by hash_map_clear_loop_fwd_back slots i =>
   hash_map_clear_loop_terminates T slots i
 decreasing_by hash_map_clear_loop_decreases slots i
 
 /- [hashmap::HashMap::{0}::clear] -/
 def hash_map_clear_fwd_back
-  (T : Type) (self : hash_map_t T) : result (hash_map_t T) :=
+  (T : Type) (self : hash_map_t T) : Result (hash_map_t T) :=
   do
-    let v <-
+    let v ←
       hash_map_clear_loop_fwd_back T self.hash_map_slots
         (USize.ofNatCore 0 (by intlit))
-    result.ret
+    Result.ret
       {
         hash_map_num_entries := (USize.ofNatCore 0 (by intlit)),
         hash_map_max_load_factor := self.hash_map_max_load_factor,
@@ -86,69 +86,69 @@ def hash_map_clear_fwd_back
       }
 
 /- [hashmap::HashMap::{0}::len] -/
-def hash_map_len_fwd (T : Type) (self : hash_map_t T) : result USize :=
-  result.ret self.hash_map_num_entries
+def hash_map_len_fwd (T : Type) (self : hash_map_t T) : Result USize :=
+  Result.ret self.hash_map_num_entries
 
 /- [hashmap::HashMap::{0}::insert_in_list] -/
 def hash_map_insert_in_list_loop_fwd
-  (T : Type) (key : USize) (value : T) (ls : list_t T) : (result Bool) :=
-  match ls with
+  (T : Type) (key : USize) (value : T) (ls : list_t T) : (Result Bool) :=
+  match h: ls with
   | list_t.ListCons ckey cvalue tl =>
-    if ckey = key
-    then result.ret false
+    if h: ckey = key
+    then Result.ret false
     else hash_map_insert_in_list_loop_fwd T key value tl
-  | list_t.ListNil => result.ret true
+  | list_t.ListNil => Result.ret true
 termination_by hash_map_insert_in_list_loop_fwd key value ls =>
   hash_map_insert_in_list_loop_terminates T key value ls
 decreasing_by hash_map_insert_in_list_loop_decreases key value ls
 
 /- [hashmap::HashMap::{0}::insert_in_list] -/
 def hash_map_insert_in_list_fwd
-  (T : Type) (key : USize) (value : T) (ls : list_t T) : result Bool :=
+  (T : Type) (key : USize) (value : T) (ls : list_t T) : Result Bool :=
   hash_map_insert_in_list_loop_fwd T key value ls
 
 /- [hashmap::HashMap::{0}::insert_in_list] -/
 def hash_map_insert_in_list_loop_back
-  (T : Type) (key : USize) (value : T) (ls : list_t T) : (result (list_t T)) :=
-  match ls with
+  (T : Type) (key : USize) (value : T) (ls : list_t T) : (Result (list_t T)) :=
+  match h: ls with
   | list_t.ListCons ckey cvalue tl =>
-    if ckey = key
-    then result.ret (list_t.ListCons ckey value tl)
+    if h: ckey = key
+    then Result.ret (list_t.ListCons ckey value tl)
     else
       do
-        let tl0 <- hash_map_insert_in_list_loop_back T key value tl
-        result.ret (list_t.ListCons ckey cvalue tl0)
+        let tl0 ← hash_map_insert_in_list_loop_back T key value tl
+        Result.ret (list_t.ListCons ckey cvalue tl0)
   | list_t.ListNil =>
     let l := list_t.ListNil
-    result.ret (list_t.ListCons key value l)
+    Result.ret (list_t.ListCons key value l)
 termination_by hash_map_insert_in_list_loop_back key value ls =>
   hash_map_insert_in_list_loop_terminates T key value ls
 decreasing_by hash_map_insert_in_list_loop_decreases key value ls
 
 /- [hashmap::HashMap::{0}::insert_in_list] -/
 def hash_map_insert_in_list_back
-  (T : Type) (key : USize) (value : T) (ls : list_t T) : result (list_t T) :=
+  (T : Type) (key : USize) (value : T) (ls : list_t T) : Result (list_t T) :=
   hash_map_insert_in_list_loop_back T key value ls
 
 /- [hashmap::HashMap::{0}::insert_no_resize] -/
 def hash_map_insert_no_resize_fwd_back
   (T : Type) (self : hash_map_t T) (key : USize) (value : T) :
-  result (hash_map_t T)
+  Result (hash_map_t T)
   :=
   do
-    let hash <- hash_key_fwd key
+    let hash ← hash_key_fwd key
     let i := vec_len (list_t T) self.hash_map_slots
-    let hash_mod <- USize.checked_rem hash i
-    let l <- vec_index_mut_fwd (list_t T) self.hash_map_slots hash_mod
-    let inserted <- hash_map_insert_in_list_fwd T key value l
-    if inserted
+    let hash_mod ← USize.checked_rem hash i
+    let l ← vec_index_mut_fwd (list_t T) self.hash_map_slots hash_mod
+    let inserted ← hash_map_insert_in_list_fwd T key value l
+    if h: inserted
     then
       do
-        let i0 <- USize.checked_add self.hash_map_num_entries
+        let i0 ← USize.checked_add self.hash_map_num_entries
           (USize.ofNatCore 1 (by intlit))
-        let l0 <- hash_map_insert_in_list_back T key value l
-        let v <- vec_index_mut_back (list_t T) self.hash_map_slots hash_mod l0
-        result.ret
+        let l0 ← hash_map_insert_in_list_back T key value l
+        let v ← vec_index_mut_back (list_t T) self.hash_map_slots hash_mod l0
+        Result.ret
           {
             hash_map_num_entries := i0,
             hash_map_max_load_factor := self.hash_map_max_load_factor,
@@ -157,9 +157,9 @@ def hash_map_insert_no_resize_fwd_back
           }
     else
       do
-        let l0 <- hash_map_insert_in_list_back T key value l
-        let v <- vec_index_mut_back (list_t T) self.hash_map_slots hash_mod l0
-        result.ret
+        let l0 ← hash_map_insert_in_list_back T key value l
+        let v ← vec_index_mut_back (list_t T) self.hash_map_slots hash_mod l0
+        Result.ret
           {
             hash_map_num_entries := self.hash_map_num_entries,
             hash_map_max_load_factor := self.hash_map_max_load_factor,
@@ -168,76 +168,76 @@ def hash_map_insert_no_resize_fwd_back
           }
 
 /- [core::num::u32::{9}::MAX] -/
-def core_num_u32_max_body : result UInt32 :=
-  result.ret (UInt32.ofNatCore 4294967295 (by intlit))
+def core_num_u32_max_body : Result UInt32 :=
+  Result.ret (UInt32.ofNatCore 4294967295 (by intlit))
 def core_num_u32_max_c : UInt32 := eval_global core_num_u32_max_body (by simp)
 
 /- [hashmap::HashMap::{0}::move_elements_from_list] -/
 def hash_map_move_elements_from_list_loop_fwd_back
   (T : Type) (ntable : hash_map_t T) (ls : list_t T) :
-  (result (hash_map_t T))
+  (Result (hash_map_t T))
   :=
-  match ls with
+  match h: ls with
   | list_t.ListCons k v tl =>
     do
-      let ntable0 <- hash_map_insert_no_resize_fwd_back T ntable k v
+      let ntable0 ← hash_map_insert_no_resize_fwd_back T ntable k v
       hash_map_move_elements_from_list_loop_fwd_back T ntable0 tl
-  | list_t.ListNil => result.ret ntable
+  | list_t.ListNil => Result.ret ntable
 termination_by hash_map_move_elements_from_list_loop_fwd_back ntable ls =>
   hash_map_move_elements_from_list_loop_terminates T ntable ls
 decreasing_by hash_map_move_elements_from_list_loop_decreases ntable ls
 
 /- [hashmap::HashMap::{0}::move_elements_from_list] -/
 def hash_map_move_elements_from_list_fwd_back
-  (T : Type) (ntable : hash_map_t T) (ls : list_t T) : result (hash_map_t T) :=
+  (T : Type) (ntable : hash_map_t T) (ls : list_t T) : Result (hash_map_t T) :=
   hash_map_move_elements_from_list_loop_fwd_back T ntable ls
 
 /- [hashmap::HashMap::{0}::move_elements] -/
 def hash_map_move_elements_loop_fwd_back
-  (T : Type) (ntable : hash_map_t T) (slots : vec (list_t T)) (i : USize) :
-  (result ((hash_map_t T) × (vec (list_t T))))
+  (T : Type) (ntable : hash_map_t T) (slots : Vec (list_t T)) (i : USize) :
+  (Result ((hash_map_t T) × (Vec (list_t T))))
   :=
   let i0 := vec_len (list_t T) slots
-  if i < i0
+  if h: i < i0
   then
     do
-      let l <- vec_index_mut_fwd (list_t T) slots i
+      let l ← vec_index_mut_fwd (list_t T) slots i
       let ls := mem_replace_fwd (list_t T) l list_t.ListNil
-      let ntable0 <- hash_map_move_elements_from_list_fwd_back T ntable ls
-      let i1 <- USize.checked_add i (USize.ofNatCore 1 (by intlit))
+      let ntable0 ← hash_map_move_elements_from_list_fwd_back T ntable ls
+      let i1 ← USize.checked_add i (USize.ofNatCore 1 (by intlit))
       let l0 := mem_replace_back (list_t T) l list_t.ListNil
-      let slots0 <- vec_index_mut_back (list_t T) slots i l0
+      let slots0 ← vec_index_mut_back (list_t T) slots i l0
       hash_map_move_elements_loop_fwd_back T ntable0 slots0 i1
-  else result.ret (ntable, slots)
+  else Result.ret (ntable, slots)
 termination_by hash_map_move_elements_loop_fwd_back ntable slots i =>
   hash_map_move_elements_loop_terminates T ntable slots i
 decreasing_by hash_map_move_elements_loop_decreases ntable slots i
 
 /- [hashmap::HashMap::{0}::move_elements] -/
 def hash_map_move_elements_fwd_back
-  (T : Type) (ntable : hash_map_t T) (slots : vec (list_t T)) (i : USize) :
-  result ((hash_map_t T) × (vec (list_t T)))
+  (T : Type) (ntable : hash_map_t T) (slots : Vec (list_t T)) (i : USize) :
+  Result ((hash_map_t T) × (Vec (list_t T)))
   :=
   hash_map_move_elements_loop_fwd_back T ntable slots i
 
 /- [hashmap::HashMap::{0}::try_resize] -/
 def hash_map_try_resize_fwd_back
-  (T : Type) (self : hash_map_t T) : result (hash_map_t T) :=
+  (T : Type) (self : hash_map_t T) : Result (hash_map_t T) :=
   do
-    let max_usize <- scalar_cast USize core_num_u32_max_c
+    let max_usize ← scalar_cast USize core_num_u32_max_c
     let capacity := vec_len (list_t T) self.hash_map_slots
-    let n1 <- USize.checked_div max_usize (USize.ofNatCore 2 (by intlit))
+    let n1 ← USize.checked_div max_usize (USize.ofNatCore 2 (by intlit))
     let (i, i0) := self.hash_map_max_load_factor
-    let i1 <- USize.checked_div n1 i
-    if capacity <= i1
+    let i1 ← USize.checked_div n1 i
+    if h: capacity <= i1
     then
       do
-        let i2 <- USize.checked_mul capacity (USize.ofNatCore 2 (by intlit))
-        let ntable <- hash_map_new_with_capacity_fwd T i2 i i0
-        let (ntable0, _) <-
+        let i2 ← USize.checked_mul capacity (USize.ofNatCore 2 (by intlit))
+        let ntable ← hash_map_new_with_capacity_fwd T i2 i i0
+        let (ntable0, _) ←
           hash_map_move_elements_fwd_back T ntable self.hash_map_slots
             (USize.ofNatCore 0 (by intlit))
-        result.ret
+        Result.ret
           {
             hash_map_num_entries := self.hash_map_num_entries,
             hash_map_max_load_factor := (i, i0),
@@ -245,7 +245,7 @@ def hash_map_try_resize_fwd_back
             hash_map_slots := ntable0.hash_map_slots
           }
     else
-      result.ret
+      Result.ret
         {
           hash_map_num_entries := self.hash_map_num_entries,
           hash_map_max_load_factor := (i, i0),
@@ -256,133 +256,133 @@ def hash_map_try_resize_fwd_back
 /- [hashmap::HashMap::{0}::insert] -/
 def hash_map_insert_fwd_back
   (T : Type) (self : hash_map_t T) (key : USize) (value : T) :
-  result (hash_map_t T)
+  Result (hash_map_t T)
   :=
   do
-    let self0 <- hash_map_insert_no_resize_fwd_back T self key value
-    let i <- hash_map_len_fwd T self0
-    if i > self0.hash_map_max_load
+    let self0 ← hash_map_insert_no_resize_fwd_back T self key value
+    let i ← hash_map_len_fwd T self0
+    if h: i > self0.hash_map_max_load
     then hash_map_try_resize_fwd_back T self0
-    else result.ret self0
+    else Result.ret self0
 
 /- [hashmap::HashMap::{0}::contains_key_in_list] -/
 def hash_map_contains_key_in_list_loop_fwd
-  (T : Type) (key : USize) (ls : list_t T) : (result Bool) :=
-  match ls with
+  (T : Type) (key : USize) (ls : list_t T) : (Result Bool) :=
+  match h: ls with
   | list_t.ListCons ckey t tl =>
-    if ckey = key
-    then result.ret true
+    if h: ckey = key
+    then Result.ret true
     else hash_map_contains_key_in_list_loop_fwd T key tl
-  | list_t.ListNil => result.ret false
+  | list_t.ListNil => Result.ret false
 termination_by hash_map_contains_key_in_list_loop_fwd key ls =>
   hash_map_contains_key_in_list_loop_terminates T key ls
 decreasing_by hash_map_contains_key_in_list_loop_decreases key ls
 
 /- [hashmap::HashMap::{0}::contains_key_in_list] -/
 def hash_map_contains_key_in_list_fwd
-  (T : Type) (key : USize) (ls : list_t T) : result Bool :=
+  (T : Type) (key : USize) (ls : list_t T) : Result Bool :=
   hash_map_contains_key_in_list_loop_fwd T key ls
 
 /- [hashmap::HashMap::{0}::contains_key] -/
 def hash_map_contains_key_fwd
-  (T : Type) (self : hash_map_t T) (key : USize) : result Bool :=
+  (T : Type) (self : hash_map_t T) (key : USize) : Result Bool :=
   do
-    let hash <- hash_key_fwd key
+    let hash ← hash_key_fwd key
     let i := vec_len (list_t T) self.hash_map_slots
-    let hash_mod <- USize.checked_rem hash i
-    let l <- vec_index_fwd (list_t T) self.hash_map_slots hash_mod
+    let hash_mod ← USize.checked_rem hash i
+    let l ← vec_index_fwd (list_t T) self.hash_map_slots hash_mod
     hash_map_contains_key_in_list_fwd T key l
 
 /- [hashmap::HashMap::{0}::get_in_list] -/
 def hash_map_get_in_list_loop_fwd
-  (T : Type) (key : USize) (ls : list_t T) : (result T) :=
-  match ls with
+  (T : Type) (key : USize) (ls : list_t T) : (Result T) :=
+  match h: ls with
   | list_t.ListCons ckey cvalue tl =>
-    if ckey = key
-    then result.ret cvalue
+    if h: ckey = key
+    then Result.ret cvalue
     else hash_map_get_in_list_loop_fwd T key tl
-  | list_t.ListNil => result.fail error.panic
+  | list_t.ListNil => Result.fail Error.panic
 termination_by hash_map_get_in_list_loop_fwd key ls =>
   hash_map_get_in_list_loop_terminates T key ls
 decreasing_by hash_map_get_in_list_loop_decreases key ls
 
 /- [hashmap::HashMap::{0}::get_in_list] -/
 def hash_map_get_in_list_fwd
-  (T : Type) (key : USize) (ls : list_t T) : result T :=
+  (T : Type) (key : USize) (ls : list_t T) : Result T :=
   hash_map_get_in_list_loop_fwd T key ls
 
 /- [hashmap::HashMap::{0}::get] -/
 def hash_map_get_fwd
-  (T : Type) (self : hash_map_t T) (key : USize) : result T :=
+  (T : Type) (self : hash_map_t T) (key : USize) : Result T :=
   do
-    let hash <- hash_key_fwd key
+    let hash ← hash_key_fwd key
     let i := vec_len (list_t T) self.hash_map_slots
-    let hash_mod <- USize.checked_rem hash i
-    let l <- vec_index_fwd (list_t T) self.hash_map_slots hash_mod
+    let hash_mod ← USize.checked_rem hash i
+    let l ← vec_index_fwd (list_t T) self.hash_map_slots hash_mod
     hash_map_get_in_list_fwd T key l
 
 /- [hashmap::HashMap::{0}::get_mut_in_list] -/
 def hash_map_get_mut_in_list_loop_fwd
-  (T : Type) (ls : list_t T) (key : USize) : (result T) :=
-  match ls with
+  (T : Type) (ls : list_t T) (key : USize) : (Result T) :=
+  match h: ls with
   | list_t.ListCons ckey cvalue tl =>
-    if ckey = key
-    then result.ret cvalue
+    if h: ckey = key
+    then Result.ret cvalue
     else hash_map_get_mut_in_list_loop_fwd T tl key
-  | list_t.ListNil => result.fail error.panic
+  | list_t.ListNil => Result.fail Error.panic
 termination_by hash_map_get_mut_in_list_loop_fwd ls key =>
   hash_map_get_mut_in_list_loop_terminates T ls key
 decreasing_by hash_map_get_mut_in_list_loop_decreases ls key
 
 /- [hashmap::HashMap::{0}::get_mut_in_list] -/
 def hash_map_get_mut_in_list_fwd
-  (T : Type) (ls : list_t T) (key : USize) : result T :=
+  (T : Type) (ls : list_t T) (key : USize) : Result T :=
   hash_map_get_mut_in_list_loop_fwd T ls key
 
 /- [hashmap::HashMap::{0}::get_mut_in_list] -/
 def hash_map_get_mut_in_list_loop_back
-  (T : Type) (ls : list_t T) (key : USize) (ret0 : T) : (result (list_t T)) :=
-  match ls with
+  (T : Type) (ls : list_t T) (key : USize) (ret0 : T) : (Result (list_t T)) :=
+  match h: ls with
   | list_t.ListCons ckey cvalue tl =>
-    if ckey = key
-    then result.ret (list_t.ListCons ckey ret0 tl)
+    if h: ckey = key
+    then Result.ret (list_t.ListCons ckey ret0 tl)
     else
       do
-        let tl0 <- hash_map_get_mut_in_list_loop_back T tl key ret0
-        result.ret (list_t.ListCons ckey cvalue tl0)
-  | list_t.ListNil => result.fail error.panic
+        let tl0 ← hash_map_get_mut_in_list_loop_back T tl key ret0
+        Result.ret (list_t.ListCons ckey cvalue tl0)
+  | list_t.ListNil => Result.fail Error.panic
 termination_by hash_map_get_mut_in_list_loop_back ls key ret0 =>
   hash_map_get_mut_in_list_loop_terminates T ls key
 decreasing_by hash_map_get_mut_in_list_loop_decreases ls key
 
 /- [hashmap::HashMap::{0}::get_mut_in_list] -/
 def hash_map_get_mut_in_list_back
-  (T : Type) (ls : list_t T) (key : USize) (ret0 : T) : result (list_t T) :=
+  (T : Type) (ls : list_t T) (key : USize) (ret0 : T) : Result (list_t T) :=
   hash_map_get_mut_in_list_loop_back T ls key ret0
 
 /- [hashmap::HashMap::{0}::get_mut] -/
 def hash_map_get_mut_fwd
-  (T : Type) (self : hash_map_t T) (key : USize) : result T :=
+  (T : Type) (self : hash_map_t T) (key : USize) : Result T :=
   do
-    let hash <- hash_key_fwd key
+    let hash ← hash_key_fwd key
     let i := vec_len (list_t T) self.hash_map_slots
-    let hash_mod <- USize.checked_rem hash i
-    let l <- vec_index_mut_fwd (list_t T) self.hash_map_slots hash_mod
+    let hash_mod ← USize.checked_rem hash i
+    let l ← vec_index_mut_fwd (list_t T) self.hash_map_slots hash_mod
     hash_map_get_mut_in_list_fwd T l key
 
 /- [hashmap::HashMap::{0}::get_mut] -/
 def hash_map_get_mut_back
   (T : Type) (self : hash_map_t T) (key : USize) (ret0 : T) :
-  result (hash_map_t T)
+  Result (hash_map_t T)
   :=
   do
-    let hash <- hash_key_fwd key
+    let hash ← hash_key_fwd key
     let i := vec_len (list_t T) self.hash_map_slots
-    let hash_mod <- USize.checked_rem hash i
-    let l <- vec_index_mut_fwd (list_t T) self.hash_map_slots hash_mod
-    let l0 <- hash_map_get_mut_in_list_back T l key ret0
-    let v <- vec_index_mut_back (list_t T) self.hash_map_slots hash_mod l0
-    result.ret
+    let hash_mod ← USize.checked_rem hash i
+    let l ← vec_index_mut_fwd (list_t T) self.hash_map_slots hash_mod
+    let l0 ← hash_map_get_mut_in_list_back T l key ret0
+    let v ← vec_index_mut_back (list_t T) self.hash_map_slots hash_mod l0
+    Result.ret
       {
         hash_map_num_entries := self.hash_map_num_entries,
         hash_map_max_load_factor := self.hash_map_max_load_factor,
@@ -392,85 +392,85 @@ def hash_map_get_mut_back
 
 /- [hashmap::HashMap::{0}::remove_from_list] -/
 def hash_map_remove_from_list_loop_fwd
-  (T : Type) (key : USize) (ls : list_t T) : (result (Option T)) :=
-  match ls with
+  (T : Type) (key : USize) (ls : list_t T) : (Result (Option T)) :=
+  match h: ls with
   | list_t.ListCons ckey t tl =>
-    if ckey = key
+    if h: ckey = key
     then
       let mv_ls :=
         mem_replace_fwd (list_t T) (list_t.ListCons ckey t tl) list_t.ListNil
-      match mv_ls with
-      | list_t.ListCons i cvalue tl0 => result.ret (Option.some cvalue)
-      | list_t.ListNil => result.fail error.panic
+      match h: mv_ls with
+      | list_t.ListCons i cvalue tl0 => Result.ret (Option.some cvalue)
+      | list_t.ListNil => Result.fail Error.panic
     else hash_map_remove_from_list_loop_fwd T key tl
-  | list_t.ListNil => result.ret Option.none
+  | list_t.ListNil => Result.ret Option.none
 termination_by hash_map_remove_from_list_loop_fwd key ls =>
   hash_map_remove_from_list_loop_terminates T key ls
 decreasing_by hash_map_remove_from_list_loop_decreases key ls
 
 /- [hashmap::HashMap::{0}::remove_from_list] -/
 def hash_map_remove_from_list_fwd
-  (T : Type) (key : USize) (ls : list_t T) : result (Option T) :=
+  (T : Type) (key : USize) (ls : list_t T) : Result (Option T) :=
   hash_map_remove_from_list_loop_fwd T key ls
 
 /- [hashmap::HashMap::{0}::remove_from_list] -/
 def hash_map_remove_from_list_loop_back
-  (T : Type) (key : USize) (ls : list_t T) : (result (list_t T)) :=
-  match ls with
+  (T : Type) (key : USize) (ls : list_t T) : (Result (list_t T)) :=
+  match h: ls with
   | list_t.ListCons ckey t tl =>
-    if ckey = key
+    if h: ckey = key
     then
       let mv_ls :=
         mem_replace_fwd (list_t T) (list_t.ListCons ckey t tl) list_t.ListNil
-      match mv_ls with
-      | list_t.ListCons i cvalue tl0 => result.ret tl0
-      | list_t.ListNil => result.fail error.panic
+      match h: mv_ls with
+      | list_t.ListCons i cvalue tl0 => Result.ret tl0
+      | list_t.ListNil => Result.fail Error.panic
     else
       do
-        let tl0 <- hash_map_remove_from_list_loop_back T key tl
-        result.ret (list_t.ListCons ckey t tl0)
-  | list_t.ListNil => result.ret list_t.ListNil
+        let tl0 ← hash_map_remove_from_list_loop_back T key tl
+        Result.ret (list_t.ListCons ckey t tl0)
+  | list_t.ListNil => Result.ret list_t.ListNil
 termination_by hash_map_remove_from_list_loop_back key ls =>
   hash_map_remove_from_list_loop_terminates T key ls
 decreasing_by hash_map_remove_from_list_loop_decreases key ls
 
 /- [hashmap::HashMap::{0}::remove_from_list] -/
 def hash_map_remove_from_list_back
-  (T : Type) (key : USize) (ls : list_t T) : result (list_t T) :=
+  (T : Type) (key : USize) (ls : list_t T) : Result (list_t T) :=
   hash_map_remove_from_list_loop_back T key ls
 
 /- [hashmap::HashMap::{0}::remove] -/
 def hash_map_remove_fwd
-  (T : Type) (self : hash_map_t T) (key : USize) : result (Option T) :=
+  (T : Type) (self : hash_map_t T) (key : USize) : Result (Option T) :=
   do
-    let hash <- hash_key_fwd key
+    let hash ← hash_key_fwd key
     let i := vec_len (list_t T) self.hash_map_slots
-    let hash_mod <- USize.checked_rem hash i
-    let l <- vec_index_mut_fwd (list_t T) self.hash_map_slots hash_mod
-    let x <- hash_map_remove_from_list_fwd T key l
-    match x with
-    | Option.none => result.ret Option.none
+    let hash_mod ← USize.checked_rem hash i
+    let l ← vec_index_mut_fwd (list_t T) self.hash_map_slots hash_mod
+    let x ← hash_map_remove_from_list_fwd T key l
+    match h: x with
+    | Option.none => Result.ret Option.none
     | Option.some x0 =>
       do
-        let _ <- USize.checked_sub self.hash_map_num_entries
+        let _ ← USize.checked_sub self.hash_map_num_entries
           (USize.ofNatCore 1 (by intlit))
-        result.ret (Option.some x0)
+        Result.ret (Option.some x0)
 
 /- [hashmap::HashMap::{0}::remove] -/
 def hash_map_remove_back
-  (T : Type) (self : hash_map_t T) (key : USize) : result (hash_map_t T) :=
+  (T : Type) (self : hash_map_t T) (key : USize) : Result (hash_map_t T) :=
   do
-    let hash <- hash_key_fwd key
+    let hash ← hash_key_fwd key
     let i := vec_len (list_t T) self.hash_map_slots
-    let hash_mod <- USize.checked_rem hash i
-    let l <- vec_index_mut_fwd (list_t T) self.hash_map_slots hash_mod
-    let x <- hash_map_remove_from_list_fwd T key l
-    match x with
+    let hash_mod ← USize.checked_rem hash i
+    let l ← vec_index_mut_fwd (list_t T) self.hash_map_slots hash_mod
+    let x ← hash_map_remove_from_list_fwd T key l
+    match h: x with
     | Option.none =>
       do
-        let l0 <- hash_map_remove_from_list_back T key l
-        let v <- vec_index_mut_back (list_t T) self.hash_map_slots hash_mod l0
-        result.ret
+        let l0 ← hash_map_remove_from_list_back T key l
+        let v ← vec_index_mut_back (list_t T) self.hash_map_slots hash_mod l0
+        Result.ret
           {
             hash_map_num_entries := self.hash_map_num_entries,
             hash_map_max_load_factor := self.hash_map_max_load_factor,
@@ -479,11 +479,11 @@ def hash_map_remove_back
           }
     | Option.some x0 =>
       do
-        let i0 <- USize.checked_sub self.hash_map_num_entries
+        let i0 ← USize.checked_sub self.hash_map_num_entries
           (USize.ofNatCore 1 (by intlit))
-        let l0 <- hash_map_remove_from_list_back T key l
-        let v <- vec_index_mut_back (list_t T) self.hash_map_slots hash_mod l0
-        result.ret
+        let l0 ← hash_map_remove_from_list_back T key l
+        let v ← vec_index_mut_back (list_t T) self.hash_map_slots hash_mod l0
+        Result.ret
           {
             hash_map_num_entries := i0,
             hash_map_max_load_factor := self.hash_map_max_load_factor,
@@ -492,67 +492,67 @@ def hash_map_remove_back
           }
 
 /- [hashmap::test1] -/
-def test1_fwd : result Unit :=
+def test1_fwd : Result Unit :=
   do
-    let hm <- hash_map_new_fwd UInt64
-    let hm0 <-
+    let hm ← hash_map_new_fwd UInt64
+    let hm0 ←
       hash_map_insert_fwd_back UInt64 hm (USize.ofNatCore 0 (by intlit))
         (UInt64.ofNatCore 42 (by intlit))
-    let hm1 <-
+    let hm1 ←
       hash_map_insert_fwd_back UInt64 hm0 (USize.ofNatCore 128 (by intlit))
         (UInt64.ofNatCore 18 (by intlit))
-    let hm2 <-
+    let hm2 ←
       hash_map_insert_fwd_back UInt64 hm1 (USize.ofNatCore 1024 (by intlit))
         (UInt64.ofNatCore 138 (by intlit))
-    let hm3 <-
+    let hm3 ←
       hash_map_insert_fwd_back UInt64 hm2 (USize.ofNatCore 1056 (by intlit))
         (UInt64.ofNatCore 256 (by intlit))
-    let i <- hash_map_get_fwd UInt64 hm3 (USize.ofNatCore 128 (by intlit))
-    if not (i = (UInt64.ofNatCore 18 (by intlit)))
-    then result.fail error.panic
+    let i ← hash_map_get_fwd UInt64 hm3 (USize.ofNatCore 128 (by intlit))
+    if h: not (i = (UInt64.ofNatCore 18 (by intlit)))
+    then Result.fail Error.panic
     else
       do
-        let hm4 <-
+        let hm4 ←
           hash_map_get_mut_back UInt64 hm3 (USize.ofNatCore 1024 (by intlit))
             (UInt64.ofNatCore 56 (by intlit))
-        let i0 <-
+        let i0 ←
           hash_map_get_fwd UInt64 hm4 (USize.ofNatCore 1024 (by intlit))
-        if not (i0 = (UInt64.ofNatCore 56 (by intlit)))
-        then result.fail error.panic
+        if h: not (i0 = (UInt64.ofNatCore 56 (by intlit)))
+        then Result.fail Error.panic
         else
           do
-            let x <-
+            let x ←
               hash_map_remove_fwd UInt64 hm4 (USize.ofNatCore 1024 (by intlit))
-            match x with
-            | Option.none => result.fail error.panic
+            match h: x with
+            | Option.none => Result.fail Error.panic
             | Option.some x0 =>
-              if not (x0 = (UInt64.ofNatCore 56 (by intlit)))
-              then result.fail error.panic
+              if h: not (x0 = (UInt64.ofNatCore 56 (by intlit)))
+              then Result.fail Error.panic
               else
                 do
-                  let hm5 <-
+                  let hm5 ←
                     hash_map_remove_back UInt64 hm4
                       (USize.ofNatCore 1024 (by intlit))
-                  let i1 <-
+                  let i1 ←
                     hash_map_get_fwd UInt64 hm5 (USize.ofNatCore 0 (by intlit))
-                  if not (i1 = (UInt64.ofNatCore 42 (by intlit)))
-                  then result.fail error.panic
+                  if h: not (i1 = (UInt64.ofNatCore 42 (by intlit)))
+                  then Result.fail Error.panic
                   else
                     do
-                      let i2 <-
+                      let i2 ←
                         hash_map_get_fwd UInt64 hm5
                           (USize.ofNatCore 128 (by intlit))
-                      if not (i2 = (UInt64.ofNatCore 18 (by intlit)))
-                      then result.fail error.panic
+                      if h: not (i2 = (UInt64.ofNatCore 18 (by intlit)))
+                      then Result.fail Error.panic
                       else
                         do
-                          let i3 <-
+                          let i3 ←
                             hash_map_get_fwd UInt64 hm5
                               (USize.ofNatCore 1056 (by intlit))
-                          if not (i3 = (UInt64.ofNatCore 256 (by intlit)))
-                          then result.fail error.panic
-                          else result.ret ()
+                          if h: not (i3 = (UInt64.ofNatCore 256 (by intlit)))
+                          then Result.fail Error.panic
+                          else Result.ret ()
 
 /- Unit test for [hashmap::test1] -/
-#assert (test1_fwd = .ret ())
+#assert (test1_fwd == .ret ())
 
diff --git a/tests/lean/hashmap/Hashmap/Types.lean b/tests/lean/hashmap/Hashmap/Types.lean
index 9f08bd60..750bf6de 100644
--- a/tests/lean/hashmap/Hashmap/Types.lean
+++ b/tests/lean/hashmap/Hashmap/Types.lean
@@ -13,6 +13,6 @@ structure hash_map_t (T : Type) where
   hash_map_num_entries : USize
   hash_map_max_load_factor : (USize × USize)
   hash_map_max_load : USize
-  hash_map_slots : vec (list_t T)
+  hash_map_slots : Vec (list_t T)
 
 
diff --git a/tests/lean/hashmap_on_disk/Base/Primitives.lean b/tests/lean/hashmap_on_disk/Base/Primitives.lean
index e230dcab..5b64e908 100644
--- a/tests/lean/hashmap_on_disk/Base/Primitives.lean
+++ b/tests/lean/hashmap_on_disk/Base/Primitives.lean
@@ -21,7 +21,7 @@ open Error
 
 inductive Result (α : Type u) where
   | ret (v: α): Result α
-  | fail (e: Error): Result α 
+  | fail (e: Error): Result α
 deriving Repr, BEq
 
 open Result
diff --git a/tests/lean/hashmap_on_disk/HashmapMain/Clauses/Template.lean b/tests/lean/hashmap_on_disk/HashmapMain/Clauses/Template.lean
index a514c58a..e64419a6 100644
--- a/tests/lean/hashmap_on_disk/HashmapMain/Clauses/Template.lean
+++ b/tests/lean/hashmap_on_disk/HashmapMain/Clauses/Template.lean
@@ -15,17 +15,16 @@ macro_rules
 | `(tactic| hashmap_hash_map_allocate_slots_loop_decreases $slots $n) =>
   `(tactic| sorry)
 
-/- [hashmap_main::hashmap::HashMap::{0}::clear_slots]: termination measure -/
+/- [hashmap_main::hashmap::HashMap::{0}::clear]: termination measure -/
 @[simp]
-def hashmap_hash_map_clear_slots_loop_terminates (T : Type)
+def hashmap_hash_map_clear_loop_terminates (T : Type)
   (slots : Vec (hashmap_list_t T)) (i : USize) :=
   (slots, i)
 
-syntax "hashmap_hash_map_clear_slots_loop_decreases" term+ : tactic
+syntax "hashmap_hash_map_clear_loop_decreases" term+ : tactic
 
 macro_rules
-| `(tactic| hashmap_hash_map_clear_slots_loop_decreases $slots $i) =>
-  `(tactic| sorry)
+| `(tactic| hashmap_hash_map_clear_loop_decreases $slots $i) =>`(tactic| sorry)
 
 /- [hashmap_main::hashmap::HashMap::{0}::insert_in_list]: termination measure -/
 @[simp]
diff --git a/tests/lean/hashmap_on_disk/HashmapMain/Funs.lean b/tests/lean/hashmap_on_disk/HashmapMain/Funs.lean
index 392c8816..c85d14f4 100644
--- a/tests/lean/hashmap_on_disk/HashmapMain/Funs.lean
+++ b/tests/lean/hashmap_on_disk/HashmapMain/Funs.lean
@@ -60,8 +60,8 @@ def hashmap_hash_map_new_fwd (T : Type) : Result (hashmap_hash_map_t T) :=
   hashmap_hash_map_new_with_capacity_fwd T (USize.ofNatCore 32 (by intlit))
     (USize.ofNatCore 4 (by intlit)) (USize.ofNatCore 5 (by intlit))
 
-/- [hashmap_main::hashmap::HashMap::{0}::clear_slots] -/
-def hashmap_hash_map_clear_slots_loop_fwd_back
+/- [hashmap_main::hashmap::HashMap::{0}::clear] -/
+def hashmap_hash_map_clear_loop_fwd_back
   (T : Type) (slots : Vec (hashmap_list_t T)) (i : USize) :
   (Result (Vec (hashmap_list_t T)))
   :=
@@ -73,26 +73,19 @@ def hashmap_hash_map_clear_slots_loop_fwd_back
       let slots0 ←
         vec_index_mut_back (hashmap_list_t T) slots i
           hashmap_list_t.HashmapListNil
-      hashmap_hash_map_clear_slots_loop_fwd_back T slots0 i1
+      hashmap_hash_map_clear_loop_fwd_back T slots0 i1
   else Result.ret slots
-termination_by hashmap_hash_map_clear_slots_loop_fwd_back slots i =>
-  hashmap_hash_map_clear_slots_loop_terminates T slots i
-decreasing_by hashmap_hash_map_clear_slots_loop_decreases slots i
-
-/- [hashmap_main::hashmap::HashMap::{0}::clear_slots] -/
-def hashmap_hash_map_clear_slots_fwd_back
-  (T : Type) (slots : Vec (hashmap_list_t T)) :
-  Result (Vec (hashmap_list_t T))
-  :=
-  hashmap_hash_map_clear_slots_loop_fwd_back T slots
-    (USize.ofNatCore 0 (by intlit))
+termination_by hashmap_hash_map_clear_loop_fwd_back slots i =>
+  hashmap_hash_map_clear_loop_terminates T slots i
+decreasing_by hashmap_hash_map_clear_loop_decreases slots i
 
 /- [hashmap_main::hashmap::HashMap::{0}::clear] -/
 def hashmap_hash_map_clear_fwd_back
   (T : Type) (self : hashmap_hash_map_t T) : Result (hashmap_hash_map_t T) :=
   do
     let v ←
-      hashmap_hash_map_clear_slots_fwd_back T self.hashmap_hash_map_slots
+      hashmap_hash_map_clear_loop_fwd_back T self.hashmap_hash_map_slots
+        (USize.ofNatCore 0 (by intlit))
     Result.ret
       {
         hashmap_hash_map_num_entries := (USize.ofNatCore 0 (by intlit)),
diff --git a/tests/lean/misc/constants/Base/Primitives.lean b/tests/lean/misc/constants/Base/Primitives.lean
index 79958d94..5b64e908 100644
--- a/tests/lean/misc/constants/Base/Primitives.lean
+++ b/tests/lean/misc/constants/Base/Primitives.lean
@@ -9,74 +9,79 @@ import Mathlib.Tactic.RunCmd
 
 -- Results & monadic combinators
 
--- TODO: use syntactic conventions and capitalize error, result, etc.
-
-inductive error where
-   | assertionFailure: error
-   | integerOverflow: error
-   | arrayOutOfBounds: error
-   | maximumSizeExceeded: error
-   | panic: error
+inductive Error where
+   | assertionFailure: Error
+   | integerOverflow: Error
+   | arrayOutOfBounds: Error
+   | maximumSizeExceeded: Error
+   | panic: Error
 deriving Repr, BEq
 
-open error
+open Error
 
-inductive result (α : Type u) where
-  | ret (v: α): result α
-  | fail (e: error): result α 
+inductive Result (α : Type u) where
+  | ret (v: α): Result α
+  | fail (e: Error): Result α
 deriving Repr, BEq
 
-open result
+open Result
 
 /- HELPERS -/
 
--- TODO: is there automated syntax for these discriminators?
-def is_ret {α: Type} (r: result α): Bool :=
+def ret? {α: Type} (r: Result α): Bool :=
   match r with
-  | result.ret _ => true
-  | result.fail _ => false
+  | Result.ret _ => true
+  | Result.fail _ => false
 
-def massert (b:Bool) : result Unit :=
+def massert (b:Bool) : Result Unit :=
   if b then .ret () else fail assertionFailure
 
-def eval_global {α: Type} (x: result α) (_: is_ret x): α :=
+def eval_global {α: Type} (x: Result α) (_: ret? x): α :=
   match x with
-  | result.fail _ => by contradiction
-  | result.ret x => x
+  | Result.fail _ => by contradiction
+  | Result.ret x => x
 
 /- DO-DSL SUPPORT -/
 
-def bind (x: result α) (f: α -> result β) : result β :=
+def bind (x: Result α) (f: α -> Result β) : Result β :=
   match x with
   | ret v  => f v 
   | fail v => fail v
 
--- Allows using result in do-blocks
-instance : Bind result where
+-- Allows using Result in do-blocks
+instance : Bind Result where
   bind := bind
 
 -- Allows using return x in do-blocks
-instance : Pure result where
+instance : Pure Result where
   pure := fun x => ret x
 
 /- CUSTOM-DSL SUPPORT -/
 
--- Let-binding the result of a monadic operation is oftentimes not sufficient,
+-- Let-binding the Result of a monadic operation is oftentimes not sufficient,
 -- because we may need a hypothesis for equational reasoning in the scope. We
 -- rely on subtype, and a custom let-binding operator, in effect recreating our
 -- own variant of the do-dsl
 
-def result.attach : (o : result α) → result { x : α // o = ret x }
+def Result.attach {α: Type} (o : Result α): Result { x : α // o = ret x } :=
+  match o with
   | .ret x => .ret ⟨x, rfl⟩
-  | .fail e   => .fail e
+  | .fail e => .fail e
 
-macro "let" h:ident " : " e:term " <-- " f:term : doElem =>
-  `(doElem| let ⟨$e, $h⟩ ← result.attach $f)
+macro "let" e:term " ⟵ " f:term : doElem =>
+  `(doElem| let ⟨$e, h⟩ ← Result.attach $f)
 
--- Silly example of the kind of reasoning that this notation enables
+-- TODO: any way to factorize both definitions?
+macro "let" e:term " <-- " f:term : doElem =>
+  `(doElem| let ⟨$e, h⟩ ← Result.attach $f)
+
+-- We call the hypothesis `h`, in effect making it unavailable to the user
+-- (because too much shadowing). But in practice, once can use the French single
+-- quote notation (input with f< and f>), where `‹ h ›` finds a suitable
+-- hypothesis in the context, this is equivalent to `have x: h := by assumption in x`
 #eval do
-  let h: y <-- .ret (0: Nat)
-  let _: y = 0 := by cases h; decide
+  let y <-- .ret (0: Nat)
+  let _: y = 0 := by cases ‹ ret 0 = ret y › ; decide
   let r: { x: Nat // x = 0 } := ⟨ y, by assumption ⟩
   .ret r
 
@@ -84,36 +89,27 @@ macro "let" h:ident " : " e:term " <-- " f:term : doElem =>
 -- MACHINE INTEGERS --
 ----------------------
 
--- NOTE: we reuse the USize type from prelude.lean, because at least we know
--- it's defined in an idiomatic style that is going to make proofs easy (and
--- indeed, several proofs here are much shortened compared to Aymeric's earlier
--- attempt.) This is not stricto sensu the *correct* thing to do, because one
--- can query at run-time the value of USize, which we do *not* want to do (we
--- don't know what target we'll run on!), but when the day comes, we'll just
--- define our own USize.
--- ANOTHER NOTE: there is USize.sub but it has wraparound semantics, which is
--- not something we want to define (I think), so we use our own monadic sub (but
--- is it in line with the Rust behavior?)
-
--- TODO: I am somewhat under the impression that subtraction is defined as a
--- total function over nats...? the hypothesis in the if condition is not used
--- in the then-branch which confuses me quite a bit
-
--- TODO: add a refinement for the result (just like vec_push_back below) that
--- explains that the toNat of the result (in the case of success) is the sub of
--- the toNat of the arguments (i.e. intrinsic specification)
--- ... do we want intrinsic specifications for the builtins? that might require
--- some careful type annotations in the monadic notation for clients, but may
--- give us more "for free"
+-- NOTE: we reuse the fixed-width integer types from prelude.lean: UInt8, ...,
+-- USize. They are generally defined in an idiomatic style, except that there is
+-- not a single type class to rule them all (more on that below). The absence of
+-- type class is intentional, and allows the Lean compiler to efficiently map
+-- them to machine integers during compilation.
+
+-- USize is designed properly: you cannot reduce `getNumBits` using the
+-- simplifier, meaning that proofs do not depend on the compile-time value of
+-- USize.size. (Lean assumes 32 or 64-bit platforms, and Rust doesn't really
+-- support, at least officially, 16-bit microcontrollers, so this seems like a
+-- fine design decision for now.)
 
 -- Note from Chris Bailey: "If there's more than one salient property of your
 -- definition then the subtyping strategy might get messy, and the property part
 -- of a subtype is less discoverable by the simplifier or tactics like
--- library_search." Try to settle this with a Lean expert on what is the most
--- productive way to go about this?
+-- library_search." So, we will not add refinements on the return values of the
+-- operations defined on Primitives, but will rather rely on custom lemmas to
+-- invert on possible return values of the primitive operations.
 
--- One needs to perform a little bit of reasoning in order to successfully
--- inject constants into USize, so we provide a general-purpose macro
+-- Machine integer constants, done via `ofNatCore`, which requires a proof that
+-- the `Nat` fits within the desired integer type. We provide a custom tactic.
 
 syntax "intlit" : tactic
 
@@ -129,12 +125,21 @@ macro_rules
 -- Also works for other integer types (at the expense of a needless disjunction)
 #eval UInt32.ofNatCore 0 (by intlit)
 
+-- The machine integer operations (e.g. sub) are always total, which is not what
+-- we want. We therefore define "checked" variants, below. Note that we add a
+-- tiny bit of complexity for the USize variant: we first check whether the
+-- result is < 2^32; if it is, we can compute the definition, rather than
+-- returning a term that is computationally stuck (the comparison to USize.size
+-- cannot reduce at compile-time, per the remark about regarding `getNumBits`).
+-- This is useful for the various #asserts that we want to reduce at
+-- type-checking time.
+
 -- Further thoughts: look at what has been done here:
 -- https://github.com/leanprover-community/mathlib4/blob/master/Mathlib/Data/Fin/Basic.lean
 -- and
 -- https://github.com/leanprover-community/mathlib4/blob/master/Mathlib/Data/UInt.lean
 -- which both contain a fair amount of reasoning already!
-def USize.checked_sub (n: USize) (m: USize): result USize :=
+def USize.checked_sub (n: USize) (m: USize): Result USize :=
   -- NOTE: the test USize.toNat n - m >= 0 seems to always succeed?
   if n >= m then
     let n' := USize.toNat n
@@ -150,18 +155,19 @@ def USize.checked_sub (n: USize) (m: USize): result USize :=
   else
     fail integerOverflow
 
-def USize.checked_add (n: USize) (m: USize): result USize :=
-  if h: n.val.val + m.val.val <= 4294967295 then
-    .ret ⟨ n.val.val + m.val.val, by
-      have h': 4294967295 < USize.size := by intlit
-      apply Nat.lt_of_le_of_lt h h'
-    ⟩
-  else if h: n.val + m.val < USize.size then
+@[simp]
+theorem usize_fits (n: Nat) (h: n <= 4294967295): n < USize.size :=
+  match USize.size, usize_size_eq with
+  | _, Or.inl rfl => Nat.lt_of_le_of_lt h (by decide)
+  | _, Or.inr rfl => Nat.lt_of_le_of_lt h (by decide)
+
+def USize.checked_add (n: USize) (m: USize): Result USize :=
+  if h: n.val + m.val < USize.size then
     .ret ⟨ n.val + m.val, h ⟩
   else
     .fail integerOverflow
 
-def USize.checked_rem (n: USize) (m: USize): result USize :=
+def USize.checked_rem (n: USize) (m: USize): Result USize :=
   if h: m > 0 then
     .ret ⟨ n.val % m.val, by
       have h1: ↑m.val < USize.size := m.val.isLt
@@ -171,18 +177,13 @@ def USize.checked_rem (n: USize) (m: USize): result USize :=
   else
     .fail integerOverflow
 
-def USize.checked_mul (n: USize) (m: USize): result USize :=
-    if h: n.val.val * m.val.val <= 4294967295 then
-    .ret ⟨ n.val.val * m.val.val, by
-      have h': 4294967295 < USize.size := by intlit
-      apply Nat.lt_of_le_of_lt h h'
-    ⟩
-  else if h: n.val * m.val < USize.size then
+def USize.checked_mul (n: USize) (m: USize): Result USize :=
+  if h: n.val * m.val < USize.size then
     .ret ⟨ n.val * m.val, h ⟩
   else
     .fail integerOverflow
 
-def USize.checked_div (n: USize) (m: USize): result USize :=
+def USize.checked_div (n: USize) (m: USize): Result USize :=
   if m > 0 then
     .ret ⟨ n.val / m.val, by
       have h1: ↑n.val < USize.size := n.val.isLt
@@ -192,6 +193,19 @@ def USize.checked_div (n: USize) (m: USize): result USize :=
   else
     .fail integerOverflow
 
+-- Test behavior...
+#eval assert! USize.checked_sub 10 20 == fail integerOverflow; 0
+
+#eval USize.checked_sub 20 10
+-- NOTE: compare with concrete behavior here, which I do not think we want
+#eval USize.sub 0 1
+#eval UInt8.add 255 255
+
+-- We now define a type class that subsumes the various machine integer types, so
+-- as to write a concise definition for scalar_cast, rather than exhaustively
+-- enumerating all of the possible pairs. We remark that Rust has sane semantics
+-- and fails if a cast operation would involve a truncation or modulo.
+
 class MachineInteger (t: Type) where
   size: Nat
   val: t -> Fin size
@@ -209,30 +223,24 @@ run_cmd
     end $typeName
   ))
 
-def scalar_cast { src: Type } (dst: Type) [ MachineInteger src ] [ MachineInteger dst ] (x: src): result dst :=
+-- Aeneas only instantiates the destination type (`src` is implicit). We rely on
+-- Lean to infer `src`.
+
+def scalar_cast { src: Type } (dst: Type) [ MachineInteger src ] [ MachineInteger dst ] (x: src): Result dst :=
   if h: MachineInteger.val x < MachineInteger.size dst then
     .ret (MachineInteger.ofNatCore (MachineInteger.val x).val h)
   else
     .fail integerOverflow
 
-
--- Test behavior...
-#eval assert! USize.checked_sub 10 20 == fail integerOverflow; 0
-
-#eval USize.checked_sub 20 10
--- NOTE: compare with concrete behavior here, which I do not think we want
-#eval USize.sub 0 1
-#eval UInt8.add 255 255
-
 -------------
 -- VECTORS --
 -------------
 
 -- Note: unlike F*, Lean seems to use strict upper bounds (e.g. USize.size)
 -- rather than maximum values (usize_max).
-def vec (α : Type u) := { l : List α // List.length l < USize.size }
+def Vec (α : Type u) := { l : List α // List.length l < USize.size }
 
-def vec_new (α : Type u): vec α := ⟨ [], by {
+def vec_new (α : Type u): Vec α := ⟨ [], by {
   match USize.size, usize_size_eq with
   | _, Or.inl rfl => simp
   | _, Or.inr rfl => simp
@@ -240,20 +248,20 @@ def vec_new (α : Type u): vec α := ⟨ [], by {
 
 #check vec_new
 
-def vec_len (α : Type u) (v : vec α) : USize :=
+def vec_len (α : Type u) (v : Vec α) : USize :=
   let ⟨ v, l ⟩ := v
   USize.ofNatCore (List.length v) l
 
 #eval vec_len Nat (vec_new Nat)
  
-def vec_push_fwd (α : Type u) (_ : vec α) (_ : α) : Unit := ()
+def vec_push_fwd (α : Type u) (_ : Vec α) (_ : α) : Unit := ()
 
 -- NOTE: old version trying to use a subtype notation, but probably better to
--- leave result elimination to auxiliary lemmas with suitable preconditions
+-- leave Result elimination to auxiliary lemmas with suitable preconditions
 -- TODO: I originally wrote `List.length v.val < USize.size - 1`; how can one
 -- make the proof work in that case? Probably need to import tactics from
 -- mathlib to deal with inequalities... would love to see an example.
-def vec_push_back_old (α : Type u) (v : vec α) (x : α) : { res: result (vec α) //
+def vec_push_back_old (α : Type u) (v : Vec α) (x : α) : { res: Result (Vec α) //
   match res with | fail _ => True | ret v' => List.length v'.val = List.length v.val + 1}
   :=
   if h : List.length v.val + 1 < USize.size then
@@ -272,12 +280,12 @@ def vec_push_back_old (α : Type u) (v : vec α) (x : α) : { res: result (vec �
   -- annotate `x`, which relieves us of having to write `.val` on the right-hand
   -- side of the monadic let.
   let v := vec_new Nat
-  let x: vec Nat ← (vec_push_back_old Nat v 1: result (vec Nat)) -- WHY do we need the type annotation here?
+  let x: Vec Nat ← (vec_push_back_old Nat v 1: Result (Vec Nat)) -- WHY do we need the type annotation here?
   -- TODO: strengthen post-condition above and do a demo to show that we can
   -- safely eliminate the `fail` case
   return (vec_len Nat x)
 
-def vec_push_back (α : Type u) (v : vec α) (x : α) : result (vec α)
+def vec_push_back (α : Type u) (v : Vec α) (x : α) : Result (Vec α)
   :=
   if h : List.length v.val + 1 <= 4294967295 then
     return ⟨ List.concat v.val x,
@@ -295,13 +303,13 @@ def vec_push_back (α : Type u) (v : vec α) (x : α) : result (vec α)
   else
     fail maximumSizeExceeded
 
-def vec_insert_fwd (α : Type u) (v: vec α) (i: USize) (_: α): result Unit :=
+def vec_insert_fwd (α : Type u) (v: Vec α) (i: USize) (_: α): Result Unit :=
   if i.val < List.length v.val then
     .ret ()
   else
     .fail arrayOutOfBounds
 
-def vec_insert_back (α : Type u) (v: vec α) (i: USize) (x: α): result (vec α) :=
+def vec_insert_back (α : Type u) (v: Vec α) (i: USize) (x: α): Result (Vec α) :=
   if i.val < List.length v.val then
     .ret ⟨ List.set v.val i.val x, by
       have h: List.length v.val < USize.size := v.property
@@ -311,25 +319,25 @@ def vec_insert_back (α : Type u) (v: vec α) (i: USize) (x: α): result (vec α
   else
     .fail arrayOutOfBounds
 
-def vec_index_fwd (α : Type u) (v: vec α) (i: USize): result α :=
+def vec_index_fwd (α : Type u) (v: Vec α) (i: USize): Result α :=
   if h: i.val < List.length v.val then
     .ret (List.get v.val ⟨i.val, h⟩)
   else
     .fail arrayOutOfBounds
 
-def vec_index_back (α : Type u) (v: vec α) (i: USize) (_: α): result Unit :=
+def vec_index_back (α : Type u) (v: Vec α) (i: USize) (_: α): Result Unit :=
   if i.val < List.length v.val then
     .ret ()
   else
     .fail arrayOutOfBounds
 
-def vec_index_mut_fwd (α : Type u) (v: vec α) (i: USize): result α :=
+def vec_index_mut_fwd (α : Type u) (v: Vec α) (i: USize): Result α :=
   if h: i.val < List.length v.val then
     .ret (List.get v.val ⟨i.val, h⟩)
   else
     .fail arrayOutOfBounds
 
-def vec_index_mut_back (α : Type u) (v: vec α) (i: USize) (x: α): result (vec α) :=
+def vec_index_mut_back (α : Type u) (v: Vec α) (i: USize) (x: α): Result (Vec α) :=
   if i.val < List.length v.val then
     .ret ⟨ List.set v.val i.val x, by
       have h: List.length v.val < USize.size := v.property
@@ -349,6 +357,10 @@ def mem_replace_fwd (a : Type) (x : a) (_ : a) : a :=
 def mem_replace_back (a : Type) (_ : a) (y : a) : a :=
   y
 
+/-- Aeneas-translated function -- useful to reduce non-recursive definitions.
+ Use with `simp [ aeneas ]` -/
+register_simp_attr aeneas
+
 --------------------
 -- ASSERT COMMAND --
 --------------------
@@ -358,16 +370,23 @@ open Lean Elab Command Term Meta
 syntax (name := assert) "#assert" term: command
 
 @[command_elab assert]
+unsafe
 def assertImpl : CommandElab := fun (_stx: Syntax) => do
-  logInfo "Reducing and asserting: "
-  logInfo _stx[1]
   runTermElabM (fun _ => do
-    let e ← Term.elabTerm _stx[1] none
-    logInfo (Expr.dbgToString e)
-    -- How to evaluate the term and compare the result to true?
+    let r ← evalTerm Bool (mkConst ``Bool) _stx[1]
+    if not r then
+      logInfo "Assertion failed for: "
+      logInfo _stx[1]
+      logError "Expression reduced to false"
     pure ())
-  -- logInfo (Expr.dbgToString (``true))
-  -- throwError "TODO: assert"
 
 #eval 2 == 2
 #assert (2 == 2)
+
+-------------------
+-- SANITY CHECKS --
+-------------------
+
+-- TODO: add more once we have signed integers
+
+#assert (USize.checked_rem 1 2 == .ret 1)
diff --git a/tests/lean/misc/constants/Constants.lean b/tests/lean/misc/constants/Constants.lean
index 9ef9ca44..57f6e403 100644
--- a/tests/lean/misc/constants/Constants.lean
+++ b/tests/lean/misc/constants/Constants.lean
@@ -5,64 +5,64 @@ import Base.Primitives
 structure OpaqueDefs where
   
   /- [constants::X0] -/
-  def x0_body : result UInt32 := result.ret (UInt32.ofNatCore 0 (by intlit))
+  def x0_body : Result UInt32 := Result.ret (UInt32.ofNatCore 0 (by intlit))
   def x0_c : UInt32 := eval_global x0_body (by simp)
   
   /- [core::num::u32::{9}::MAX] -/
-  def core_num_u32_max_body : result UInt32 :=
-    result.ret (UInt32.ofNatCore 4294967295 (by intlit))
+  def core_num_u32_max_body : Result UInt32 :=
+    Result.ret (UInt32.ofNatCore 4294967295 (by intlit))
   def core_num_u32_max_c : UInt32 :=
     eval_global core_num_u32_max_body (by simp)
   
   /- [constants::X1] -/
-  def x1_body : result UInt32 := result.ret core_num_u32_max_c
+  def x1_body : Result UInt32 := Result.ret core_num_u32_max_c
   def x1_c : UInt32 := eval_global x1_body (by simp)
   
   /- [constants::X2] -/
-  def x2_body : result UInt32 := result.ret (UInt32.ofNatCore 3 (by intlit))
+  def x2_body : Result UInt32 := Result.ret (UInt32.ofNatCore 3 (by intlit))
   def x2_c : UInt32 := eval_global x2_body (by simp)
   
   /- [constants::incr] -/
-  def incr_fwd (n : UInt32) : result UInt32 :=
+  def incr_fwd (n : UInt32) : Result UInt32 :=
     UInt32.checked_add n (UInt32.ofNatCore 1 (by intlit))
   
   /- [constants::X3] -/
-  def x3_body : result UInt32 := incr_fwd (UInt32.ofNatCore 32 (by intlit))
+  def x3_body : Result UInt32 := incr_fwd (UInt32.ofNatCore 32 (by intlit))
   def x3_c : UInt32 := eval_global x3_body (by simp)
   
   /- [constants::mk_pair0] -/
-  def mk_pair0_fwd (x : UInt32) (y : UInt32) : result (UInt32 × UInt32) :=
-    result.ret (x, y)
+  def mk_pair0_fwd (x : UInt32) (y : UInt32) : Result (UInt32 × UInt32) :=
+    Result.ret (x, y)
   
   /- [constants::Pair] -/
   structure pair_t (T1 T2 : Type) where  pair_x : T1 pair_y : T2 
   
   /- [constants::mk_pair1] -/
-  def mk_pair1_fwd (x : UInt32) (y : UInt32) : result (pair_t UInt32 UInt32) :=
-    result.ret { pair_x := x, pair_y := y }
+  def mk_pair1_fwd (x : UInt32) (y : UInt32) : Result (pair_t UInt32 UInt32) :=
+    Result.ret { pair_x := x, pair_y := y }
   
   /- [constants::P0] -/
-  def p0_body : result (UInt32 × UInt32) :=
+  def p0_body : Result (UInt32 × UInt32) :=
     mk_pair0_fwd (UInt32.ofNatCore 0 (by intlit))
       (UInt32.ofNatCore 1 (by intlit))
   def p0_c : (UInt32 × UInt32) := eval_global p0_body (by simp)
   
   /- [constants::P1] -/
-  def p1_body : result (pair_t UInt32 UInt32) :=
+  def p1_body : Result (pair_t UInt32 UInt32) :=
     mk_pair1_fwd (UInt32.ofNatCore 0 (by intlit))
       (UInt32.ofNatCore 1 (by intlit))
   def p1_c : pair_t UInt32 UInt32 := eval_global p1_body (by simp)
   
   /- [constants::P2] -/
-  def p2_body : result (UInt32 × UInt32) :=
-    result.ret
+  def p2_body : Result (UInt32 × UInt32) :=
+    Result.ret
     ((UInt32.ofNatCore 0 (by intlit)),
     (UInt32.ofNatCore 1 (by intlit)))
   def p2_c : (UInt32 × UInt32) := eval_global p2_body (by simp)
   
   /- [constants::P3] -/
-  def p3_body : result (pair_t UInt32 UInt32) :=
-    result.ret
+  def p3_body : Result (pair_t UInt32 UInt32) :=
+    Result.ret
     {
       pair_x := (UInt32.ofNatCore 0 (by intlit)),
       pair_y := (UInt32.ofNatCore 1 (by intlit))
@@ -73,68 +73,68 @@ structure OpaqueDefs where
   structure wrap_t (T : Type) where  wrap_val : T 
   
   /- [constants::Wrap::{0}::new] -/
-  def wrap_new_fwd (T : Type) (val : T) : result (wrap_t T) :=
-    result.ret { wrap_val := val }
+  def wrap_new_fwd (T : Type) (val : T) : Result (wrap_t T) :=
+    Result.ret { wrap_val := val }
   
   /- [constants::Y] -/
-  def y_body : result (wrap_t Int32) :=
+  def y_body : Result (wrap_t Int32) :=
     wrap_new_fwd Int32 (Int32.ofNatCore 2 (by intlit))
   def y_c : wrap_t Int32 := eval_global y_body (by simp)
   
   /- [constants::unwrap_y] -/
-  def unwrap_y_fwd : result Int32 :=
-    result.ret y_c.wrap_val
+  def unwrap_y_fwd : Result Int32 :=
+    Result.ret y_c.wrap_val
   
   /- [constants::YVAL] -/
-  def yval_body : result Int32 := unwrap_y_fwd
+  def yval_body : Result Int32 := unwrap_y_fwd
   def yval_c : Int32 := eval_global yval_body (by simp)
   
   /- [constants::get_z1::Z1] -/
-  def get_z1_z1_body : result Int32 :=
-    result.ret (Int32.ofNatCore 3 (by intlit))
+  def get_z1_z1_body : Result Int32 :=
+    Result.ret (Int32.ofNatCore 3 (by intlit))
   def get_z1_z1_c : Int32 := eval_global get_z1_z1_body (by simp)
   
   /- [constants::get_z1] -/
-  def get_z1_fwd : result Int32 :=
-    result.ret get_z1_z1_c
+  def get_z1_fwd : Result Int32 :=
+    Result.ret get_z1_z1_c
   
   /- [constants::add] -/
-  def add_fwd (a : Int32) (b : Int32) : result Int32 :=
+  def add_fwd (a : Int32) (b : Int32) : Result Int32 :=
     Int32.checked_add a b
   
   /- [constants::Q1] -/
-  def q1_body : result Int32 := result.ret (Int32.ofNatCore 5 (by intlit))
+  def q1_body : Result Int32 := Result.ret (Int32.ofNatCore 5 (by intlit))
   def q1_c : Int32 := eval_global q1_body (by simp)
   
   /- [constants::Q2] -/
-  def q2_body : result Int32 := result.ret q1_c
+  def q2_body : Result Int32 := Result.ret q1_c
   def q2_c : Int32 := eval_global q2_body (by simp)
   
   /- [constants::Q3] -/
-  def q3_body : result Int32 := add_fwd q2_c (Int32.ofNatCore 3 (by intlit))
+  def q3_body : Result Int32 := add_fwd q2_c (Int32.ofNatCore 3 (by intlit))
   def q3_c : Int32 := eval_global q3_body (by simp)
   
   /- [constants::get_z2] -/
-  def get_z2_fwd : result Int32 :=
+  def get_z2_fwd : Result Int32 :=
     do
-      let i <- get_z1_fwd
-      let i0 <- add_fwd i q3_c
+      let i ← get_z1_fwd
+      let i0 ← add_fwd i q3_c
       add_fwd q1_c i0
   
   /- [constants::S1] -/
-  def s1_body : result UInt32 := result.ret (UInt32.ofNatCore 6 (by intlit))
+  def s1_body : Result UInt32 := Result.ret (UInt32.ofNatCore 6 (by intlit))
   def s1_c : UInt32 := eval_global s1_body (by simp)
   
   /- [constants::S2] -/
-  def s2_body : result UInt32 := incr_fwd s1_c
+  def s2_body : Result UInt32 := incr_fwd s1_c
   def s2_c : UInt32 := eval_global s2_body (by simp)
   
   /- [constants::S3] -/
-  def s3_body : result (pair_t UInt32 UInt32) := result.ret p3_c
+  def s3_body : Result (pair_t UInt32 UInt32) := Result.ret p3_c
   def s3_c : pair_t UInt32 UInt32 := eval_global s3_body (by simp)
   
   /- [constants::S4] -/
-  def s4_body : result (pair_t UInt32 UInt32) :=
+  def s4_body : Result (pair_t UInt32 UInt32) :=
     mk_pair1_fwd (UInt32.ofNatCore 7 (by intlit))
       (UInt32.ofNatCore 8 (by intlit))
   def s4_c : pair_t UInt32 UInt32 := eval_global s4_body (by simp)
diff --git a/tests/lean/misc/external/Base/Primitives.lean b/tests/lean/misc/external/Base/Primitives.lean
index 79958d94..5b64e908 100644
--- a/tests/lean/misc/external/Base/Primitives.lean
+++ b/tests/lean/misc/external/Base/Primitives.lean
@@ -9,74 +9,79 @@ import Mathlib.Tactic.RunCmd
 
 -- Results & monadic combinators
 
--- TODO: use syntactic conventions and capitalize error, result, etc.
-
-inductive error where
-   | assertionFailure: error
-   | integerOverflow: error
-   | arrayOutOfBounds: error
-   | maximumSizeExceeded: error
-   | panic: error
+inductive Error where
+   | assertionFailure: Error
+   | integerOverflow: Error
+   | arrayOutOfBounds: Error
+   | maximumSizeExceeded: Error
+   | panic: Error
 deriving Repr, BEq
 
-open error
+open Error
 
-inductive result (α : Type u) where
-  | ret (v: α): result α
-  | fail (e: error): result α 
+inductive Result (α : Type u) where
+  | ret (v: α): Result α
+  | fail (e: Error): Result α
 deriving Repr, BEq
 
-open result
+open Result
 
 /- HELPERS -/
 
--- TODO: is there automated syntax for these discriminators?
-def is_ret {α: Type} (r: result α): Bool :=
+def ret? {α: Type} (r: Result α): Bool :=
   match r with
-  | result.ret _ => true
-  | result.fail _ => false
+  | Result.ret _ => true
+  | Result.fail _ => false
 
-def massert (b:Bool) : result Unit :=
+def massert (b:Bool) : Result Unit :=
   if b then .ret () else fail assertionFailure
 
-def eval_global {α: Type} (x: result α) (_: is_ret x): α :=
+def eval_global {α: Type} (x: Result α) (_: ret? x): α :=
   match x with
-  | result.fail _ => by contradiction
-  | result.ret x => x
+  | Result.fail _ => by contradiction
+  | Result.ret x => x
 
 /- DO-DSL SUPPORT -/
 
-def bind (x: result α) (f: α -> result β) : result β :=
+def bind (x: Result α) (f: α -> Result β) : Result β :=
   match x with
   | ret v  => f v 
   | fail v => fail v
 
--- Allows using result in do-blocks
-instance : Bind result where
+-- Allows using Result in do-blocks
+instance : Bind Result where
   bind := bind
 
 -- Allows using return x in do-blocks
-instance : Pure result where
+instance : Pure Result where
   pure := fun x => ret x
 
 /- CUSTOM-DSL SUPPORT -/
 
--- Let-binding the result of a monadic operation is oftentimes not sufficient,
+-- Let-binding the Result of a monadic operation is oftentimes not sufficient,
 -- because we may need a hypothesis for equational reasoning in the scope. We
 -- rely on subtype, and a custom let-binding operator, in effect recreating our
 -- own variant of the do-dsl
 
-def result.attach : (o : result α) → result { x : α // o = ret x }
+def Result.attach {α: Type} (o : Result α): Result { x : α // o = ret x } :=
+  match o with
   | .ret x => .ret ⟨x, rfl⟩
-  | .fail e   => .fail e
+  | .fail e => .fail e
 
-macro "let" h:ident " : " e:term " <-- " f:term : doElem =>
-  `(doElem| let ⟨$e, $h⟩ ← result.attach $f)
+macro "let" e:term " ⟵ " f:term : doElem =>
+  `(doElem| let ⟨$e, h⟩ ← Result.attach $f)
 
--- Silly example of the kind of reasoning that this notation enables
+-- TODO: any way to factorize both definitions?
+macro "let" e:term " <-- " f:term : doElem =>
+  `(doElem| let ⟨$e, h⟩ ← Result.attach $f)
+
+-- We call the hypothesis `h`, in effect making it unavailable to the user
+-- (because too much shadowing). But in practice, once can use the French single
+-- quote notation (input with f< and f>), where `‹ h ›` finds a suitable
+-- hypothesis in the context, this is equivalent to `have x: h := by assumption in x`
 #eval do
-  let h: y <-- .ret (0: Nat)
-  let _: y = 0 := by cases h; decide
+  let y <-- .ret (0: Nat)
+  let _: y = 0 := by cases ‹ ret 0 = ret y › ; decide
   let r: { x: Nat // x = 0 } := ⟨ y, by assumption ⟩
   .ret r
 
@@ -84,36 +89,27 @@ macro "let" h:ident " : " e:term " <-- " f:term : doElem =>
 -- MACHINE INTEGERS --
 ----------------------
 
--- NOTE: we reuse the USize type from prelude.lean, because at least we know
--- it's defined in an idiomatic style that is going to make proofs easy (and
--- indeed, several proofs here are much shortened compared to Aymeric's earlier
--- attempt.) This is not stricto sensu the *correct* thing to do, because one
--- can query at run-time the value of USize, which we do *not* want to do (we
--- don't know what target we'll run on!), but when the day comes, we'll just
--- define our own USize.
--- ANOTHER NOTE: there is USize.sub but it has wraparound semantics, which is
--- not something we want to define (I think), so we use our own monadic sub (but
--- is it in line with the Rust behavior?)
-
--- TODO: I am somewhat under the impression that subtraction is defined as a
--- total function over nats...? the hypothesis in the if condition is not used
--- in the then-branch which confuses me quite a bit
-
--- TODO: add a refinement for the result (just like vec_push_back below) that
--- explains that the toNat of the result (in the case of success) is the sub of
--- the toNat of the arguments (i.e. intrinsic specification)
--- ... do we want intrinsic specifications for the builtins? that might require
--- some careful type annotations in the monadic notation for clients, but may
--- give us more "for free"
+-- NOTE: we reuse the fixed-width integer types from prelude.lean: UInt8, ...,
+-- USize. They are generally defined in an idiomatic style, except that there is
+-- not a single type class to rule them all (more on that below). The absence of
+-- type class is intentional, and allows the Lean compiler to efficiently map
+-- them to machine integers during compilation.
+
+-- USize is designed properly: you cannot reduce `getNumBits` using the
+-- simplifier, meaning that proofs do not depend on the compile-time value of
+-- USize.size. (Lean assumes 32 or 64-bit platforms, and Rust doesn't really
+-- support, at least officially, 16-bit microcontrollers, so this seems like a
+-- fine design decision for now.)
 
 -- Note from Chris Bailey: "If there's more than one salient property of your
 -- definition then the subtyping strategy might get messy, and the property part
 -- of a subtype is less discoverable by the simplifier or tactics like
--- library_search." Try to settle this with a Lean expert on what is the most
--- productive way to go about this?
+-- library_search." So, we will not add refinements on the return values of the
+-- operations defined on Primitives, but will rather rely on custom lemmas to
+-- invert on possible return values of the primitive operations.
 
--- One needs to perform a little bit of reasoning in order to successfully
--- inject constants into USize, so we provide a general-purpose macro
+-- Machine integer constants, done via `ofNatCore`, which requires a proof that
+-- the `Nat` fits within the desired integer type. We provide a custom tactic.
 
 syntax "intlit" : tactic
 
@@ -129,12 +125,21 @@ macro_rules
 -- Also works for other integer types (at the expense of a needless disjunction)
 #eval UInt32.ofNatCore 0 (by intlit)
 
+-- The machine integer operations (e.g. sub) are always total, which is not what
+-- we want. We therefore define "checked" variants, below. Note that we add a
+-- tiny bit of complexity for the USize variant: we first check whether the
+-- result is < 2^32; if it is, we can compute the definition, rather than
+-- returning a term that is computationally stuck (the comparison to USize.size
+-- cannot reduce at compile-time, per the remark about regarding `getNumBits`).
+-- This is useful for the various #asserts that we want to reduce at
+-- type-checking time.
+
 -- Further thoughts: look at what has been done here:
 -- https://github.com/leanprover-community/mathlib4/blob/master/Mathlib/Data/Fin/Basic.lean
 -- and
 -- https://github.com/leanprover-community/mathlib4/blob/master/Mathlib/Data/UInt.lean
 -- which both contain a fair amount of reasoning already!
-def USize.checked_sub (n: USize) (m: USize): result USize :=
+def USize.checked_sub (n: USize) (m: USize): Result USize :=
   -- NOTE: the test USize.toNat n - m >= 0 seems to always succeed?
   if n >= m then
     let n' := USize.toNat n
@@ -150,18 +155,19 @@ def USize.checked_sub (n: USize) (m: USize): result USize :=
   else
     fail integerOverflow
 
-def USize.checked_add (n: USize) (m: USize): result USize :=
-  if h: n.val.val + m.val.val <= 4294967295 then
-    .ret ⟨ n.val.val + m.val.val, by
-      have h': 4294967295 < USize.size := by intlit
-      apply Nat.lt_of_le_of_lt h h'
-    ⟩
-  else if h: n.val + m.val < USize.size then
+@[simp]
+theorem usize_fits (n: Nat) (h: n <= 4294967295): n < USize.size :=
+  match USize.size, usize_size_eq with
+  | _, Or.inl rfl => Nat.lt_of_le_of_lt h (by decide)
+  | _, Or.inr rfl => Nat.lt_of_le_of_lt h (by decide)
+
+def USize.checked_add (n: USize) (m: USize): Result USize :=
+  if h: n.val + m.val < USize.size then
     .ret ⟨ n.val + m.val, h ⟩
   else
     .fail integerOverflow
 
-def USize.checked_rem (n: USize) (m: USize): result USize :=
+def USize.checked_rem (n: USize) (m: USize): Result USize :=
   if h: m > 0 then
     .ret ⟨ n.val % m.val, by
       have h1: ↑m.val < USize.size := m.val.isLt
@@ -171,18 +177,13 @@ def USize.checked_rem (n: USize) (m: USize): result USize :=
   else
     .fail integerOverflow
 
-def USize.checked_mul (n: USize) (m: USize): result USize :=
-    if h: n.val.val * m.val.val <= 4294967295 then
-    .ret ⟨ n.val.val * m.val.val, by
-      have h': 4294967295 < USize.size := by intlit
-      apply Nat.lt_of_le_of_lt h h'
-    ⟩
-  else if h: n.val * m.val < USize.size then
+def USize.checked_mul (n: USize) (m: USize): Result USize :=
+  if h: n.val * m.val < USize.size then
     .ret ⟨ n.val * m.val, h ⟩
   else
     .fail integerOverflow
 
-def USize.checked_div (n: USize) (m: USize): result USize :=
+def USize.checked_div (n: USize) (m: USize): Result USize :=
   if m > 0 then
     .ret ⟨ n.val / m.val, by
       have h1: ↑n.val < USize.size := n.val.isLt
@@ -192,6 +193,19 @@ def USize.checked_div (n: USize) (m: USize): result USize :=
   else
     .fail integerOverflow
 
+-- Test behavior...
+#eval assert! USize.checked_sub 10 20 == fail integerOverflow; 0
+
+#eval USize.checked_sub 20 10
+-- NOTE: compare with concrete behavior here, which I do not think we want
+#eval USize.sub 0 1
+#eval UInt8.add 255 255
+
+-- We now define a type class that subsumes the various machine integer types, so
+-- as to write a concise definition for scalar_cast, rather than exhaustively
+-- enumerating all of the possible pairs. We remark that Rust has sane semantics
+-- and fails if a cast operation would involve a truncation or modulo.
+
 class MachineInteger (t: Type) where
   size: Nat
   val: t -> Fin size
@@ -209,30 +223,24 @@ run_cmd
     end $typeName
   ))
 
-def scalar_cast { src: Type } (dst: Type) [ MachineInteger src ] [ MachineInteger dst ] (x: src): result dst :=
+-- Aeneas only instantiates the destination type (`src` is implicit). We rely on
+-- Lean to infer `src`.
+
+def scalar_cast { src: Type } (dst: Type) [ MachineInteger src ] [ MachineInteger dst ] (x: src): Result dst :=
   if h: MachineInteger.val x < MachineInteger.size dst then
     .ret (MachineInteger.ofNatCore (MachineInteger.val x).val h)
   else
     .fail integerOverflow
 
-
--- Test behavior...
-#eval assert! USize.checked_sub 10 20 == fail integerOverflow; 0
-
-#eval USize.checked_sub 20 10
--- NOTE: compare with concrete behavior here, which I do not think we want
-#eval USize.sub 0 1
-#eval UInt8.add 255 255
-
 -------------
 -- VECTORS --
 -------------
 
 -- Note: unlike F*, Lean seems to use strict upper bounds (e.g. USize.size)
 -- rather than maximum values (usize_max).
-def vec (α : Type u) := { l : List α // List.length l < USize.size }
+def Vec (α : Type u) := { l : List α // List.length l < USize.size }
 
-def vec_new (α : Type u): vec α := ⟨ [], by {
+def vec_new (α : Type u): Vec α := ⟨ [], by {
   match USize.size, usize_size_eq with
   | _, Or.inl rfl => simp
   | _, Or.inr rfl => simp
@@ -240,20 +248,20 @@ def vec_new (α : Type u): vec α := ⟨ [], by {
 
 #check vec_new
 
-def vec_len (α : Type u) (v : vec α) : USize :=
+def vec_len (α : Type u) (v : Vec α) : USize :=
   let ⟨ v, l ⟩ := v
   USize.ofNatCore (List.length v) l
 
 #eval vec_len Nat (vec_new Nat)
  
-def vec_push_fwd (α : Type u) (_ : vec α) (_ : α) : Unit := ()
+def vec_push_fwd (α : Type u) (_ : Vec α) (_ : α) : Unit := ()
 
 -- NOTE: old version trying to use a subtype notation, but probably better to
--- leave result elimination to auxiliary lemmas with suitable preconditions
+-- leave Result elimination to auxiliary lemmas with suitable preconditions
 -- TODO: I originally wrote `List.length v.val < USize.size - 1`; how can one
 -- make the proof work in that case? Probably need to import tactics from
 -- mathlib to deal with inequalities... would love to see an example.
-def vec_push_back_old (α : Type u) (v : vec α) (x : α) : { res: result (vec α) //
+def vec_push_back_old (α : Type u) (v : Vec α) (x : α) : { res: Result (Vec α) //
   match res with | fail _ => True | ret v' => List.length v'.val = List.length v.val + 1}
   :=
   if h : List.length v.val + 1 < USize.size then
@@ -272,12 +280,12 @@ def vec_push_back_old (α : Type u) (v : vec α) (x : α) : { res: result (vec �
   -- annotate `x`, which relieves us of having to write `.val` on the right-hand
   -- side of the monadic let.
   let v := vec_new Nat
-  let x: vec Nat ← (vec_push_back_old Nat v 1: result (vec Nat)) -- WHY do we need the type annotation here?
+  let x: Vec Nat ← (vec_push_back_old Nat v 1: Result (Vec Nat)) -- WHY do we need the type annotation here?
   -- TODO: strengthen post-condition above and do a demo to show that we can
   -- safely eliminate the `fail` case
   return (vec_len Nat x)
 
-def vec_push_back (α : Type u) (v : vec α) (x : α) : result (vec α)
+def vec_push_back (α : Type u) (v : Vec α) (x : α) : Result (Vec α)
   :=
   if h : List.length v.val + 1 <= 4294967295 then
     return ⟨ List.concat v.val x,
@@ -295,13 +303,13 @@ def vec_push_back (α : Type u) (v : vec α) (x : α) : result (vec α)
   else
     fail maximumSizeExceeded
 
-def vec_insert_fwd (α : Type u) (v: vec α) (i: USize) (_: α): result Unit :=
+def vec_insert_fwd (α : Type u) (v: Vec α) (i: USize) (_: α): Result Unit :=
   if i.val < List.length v.val then
     .ret ()
   else
     .fail arrayOutOfBounds
 
-def vec_insert_back (α : Type u) (v: vec α) (i: USize) (x: α): result (vec α) :=
+def vec_insert_back (α : Type u) (v: Vec α) (i: USize) (x: α): Result (Vec α) :=
   if i.val < List.length v.val then
     .ret ⟨ List.set v.val i.val x, by
       have h: List.length v.val < USize.size := v.property
@@ -311,25 +319,25 @@ def vec_insert_back (α : Type u) (v: vec α) (i: USize) (x: α): result (vec α
   else
     .fail arrayOutOfBounds
 
-def vec_index_fwd (α : Type u) (v: vec α) (i: USize): result α :=
+def vec_index_fwd (α : Type u) (v: Vec α) (i: USize): Result α :=
   if h: i.val < List.length v.val then
     .ret (List.get v.val ⟨i.val, h⟩)
   else
     .fail arrayOutOfBounds
 
-def vec_index_back (α : Type u) (v: vec α) (i: USize) (_: α): result Unit :=
+def vec_index_back (α : Type u) (v: Vec α) (i: USize) (_: α): Result Unit :=
   if i.val < List.length v.val then
     .ret ()
   else
     .fail arrayOutOfBounds
 
-def vec_index_mut_fwd (α : Type u) (v: vec α) (i: USize): result α :=
+def vec_index_mut_fwd (α : Type u) (v: Vec α) (i: USize): Result α :=
   if h: i.val < List.length v.val then
     .ret (List.get v.val ⟨i.val, h⟩)
   else
     .fail arrayOutOfBounds
 
-def vec_index_mut_back (α : Type u) (v: vec α) (i: USize) (x: α): result (vec α) :=
+def vec_index_mut_back (α : Type u) (v: Vec α) (i: USize) (x: α): Result (Vec α) :=
   if i.val < List.length v.val then
     .ret ⟨ List.set v.val i.val x, by
       have h: List.length v.val < USize.size := v.property
@@ -349,6 +357,10 @@ def mem_replace_fwd (a : Type) (x : a) (_ : a) : a :=
 def mem_replace_back (a : Type) (_ : a) (y : a) : a :=
   y
 
+/-- Aeneas-translated function -- useful to reduce non-recursive definitions.
+ Use with `simp [ aeneas ]` -/
+register_simp_attr aeneas
+
 --------------------
 -- ASSERT COMMAND --
 --------------------
@@ -358,16 +370,23 @@ open Lean Elab Command Term Meta
 syntax (name := assert) "#assert" term: command
 
 @[command_elab assert]
+unsafe
 def assertImpl : CommandElab := fun (_stx: Syntax) => do
-  logInfo "Reducing and asserting: "
-  logInfo _stx[1]
   runTermElabM (fun _ => do
-    let e ← Term.elabTerm _stx[1] none
-    logInfo (Expr.dbgToString e)
-    -- How to evaluate the term and compare the result to true?
+    let r ← evalTerm Bool (mkConst ``Bool) _stx[1]
+    if not r then
+      logInfo "Assertion failed for: "
+      logInfo _stx[1]
+      logError "Expression reduced to false"
     pure ())
-  -- logInfo (Expr.dbgToString (``true))
-  -- throwError "TODO: assert"
 
 #eval 2 == 2
 #assert (2 == 2)
+
+-------------------
+-- SANITY CHECKS --
+-------------------
+
+-- TODO: add more once we have signed integers
+
+#assert (USize.checked_rem 1 2 == .ret 1)
diff --git a/tests/lean/misc/external/External/Funs.lean b/tests/lean/misc/external/External/Funs.lean
index bb1e296d..4e1f36a1 100644
--- a/tests/lean/misc/external/External/Funs.lean
+++ b/tests/lean/misc/external/External/Funs.lean
@@ -8,86 +8,86 @@ section variable (opaque_defs: OpaqueDefs)
 
 /- [external::swap] -/
 def swap_fwd
-  (T : Type) (x : T) (y : T) (st : state) : result (state × Unit) :=
+  (T : Type) (x : T) (y : T) (st : State) : Result (State × Unit) :=
   do
-    let (st0, _) <- opaque_defs.core_mem_swap_fwd T x y st
-    let (st1, _) <- opaque_defs.core_mem_swap_back0 T x y st st0
-    let (st2, _) <- opaque_defs.core_mem_swap_back1 T x y st st1
-    result.ret (st2, ())
+    let (st0, _) ← opaque_defs.core_mem_swap_fwd T x y st
+    let (st1, _) ← opaque_defs.core_mem_swap_back0 T x y st st0
+    let (st2, _) ← opaque_defs.core_mem_swap_back1 T x y st st1
+    Result.ret (st2, ())
 
 /- [external::swap] -/
 def swap_back
-  (T : Type) (x : T) (y : T) (st : state) (st0 : state) :
-  result (state × (T × T))
+  (T : Type) (x : T) (y : T) (st : State) (st0 : State) :
+  Result (State × (T × T))
   :=
   do
-    let (st1, _) <- opaque_defs.core_mem_swap_fwd T x y st
-    let (st2, x0) <- opaque_defs.core_mem_swap_back0 T x y st st1
-    let (_, y0) <- opaque_defs.core_mem_swap_back1 T x y st st2
-    result.ret (st0, (x0, y0))
+    let (st1, _) ← opaque_defs.core_mem_swap_fwd T x y st
+    let (st2, x0) ← opaque_defs.core_mem_swap_back0 T x y st st1
+    let (_, y0) ← opaque_defs.core_mem_swap_back1 T x y st st2
+    Result.ret (st0, (x0, y0))
 
 /- [external::test_new_non_zero_u32] -/
 def test_new_non_zero_u32_fwd
-  (x : UInt32) (st : state) :
-  result (state × core_num_nonzero_non_zero_u32_t)
+  (x : UInt32) (st : State) :
+  Result (State × core_num_nonzero_non_zero_u32_t)
   :=
   do
-    let (st0, opt) <- opaque_defs.core_num_nonzero_non_zero_u32_new_fwd x st
+    let (st0, opt) ← opaque_defs.core_num_nonzero_non_zero_u32_new_fwd x st
     opaque_defs.core_option_option_unwrap_fwd core_num_nonzero_non_zero_u32_t
       opt st0
 
 /- [external::test_vec] -/
-def test_vec_fwd : result Unit :=
+def test_vec_fwd : Result Unit :=
   do
     let v := vec_new UInt32
-    let _ <- vec_push_back UInt32 v (UInt32.ofNatCore 0 (by intlit))
-    result.ret ()
+    let _ ← vec_push_back UInt32 v (UInt32.ofNatCore 0 (by intlit))
+    Result.ret ()
 
 /- Unit test for [external::test_vec] -/
-#assert (test_vec_fwd = .ret ())
+#assert (test_vec_fwd == .ret ())
 
 /- [external::custom_swap] -/
 def custom_swap_fwd
-  (T : Type) (x : T) (y : T) (st : state) : result (state × T) :=
+  (T : Type) (x : T) (y : T) (st : State) : Result (State × T) :=
   do
-    let (st0, _) <- opaque_defs.core_mem_swap_fwd T x y st
-    let (st1, x0) <- opaque_defs.core_mem_swap_back0 T x y st st0
-    let (st2, _) <- opaque_defs.core_mem_swap_back1 T x y st st1
-    result.ret (st2, x0)
+    let (st0, _) ← opaque_defs.core_mem_swap_fwd T x y st
+    let (st1, x0) ← opaque_defs.core_mem_swap_back0 T x y st st0
+    let (st2, _) ← opaque_defs.core_mem_swap_back1 T x y st st1
+    Result.ret (st2, x0)
 
 /- [external::custom_swap] -/
 def custom_swap_back
-  (T : Type) (x : T) (y : T) (st : state) (ret0 : T) (st0 : state) :
-  result (state × (T × T))
+  (T : Type) (x : T) (y : T) (st : State) (ret0 : T) (st0 : State) :
+  Result (State × (T × T))
   :=
   do
-    let (st1, _) <- opaque_defs.core_mem_swap_fwd T x y st
-    let (st2, _) <- opaque_defs.core_mem_swap_back0 T x y st st1
-    let (_, y0) <- opaque_defs.core_mem_swap_back1 T x y st st2
-    result.ret (st0, (ret0, y0))
+    let (st1, _) ← opaque_defs.core_mem_swap_fwd T x y st
+    let (st2, _) ← opaque_defs.core_mem_swap_back0 T x y st st1
+    let (_, y0) ← opaque_defs.core_mem_swap_back1 T x y st st2
+    Result.ret (st0, (ret0, y0))
 
 /- [external::test_custom_swap] -/
 def test_custom_swap_fwd
-  (x : UInt32) (y : UInt32) (st : state) : result (state × Unit) :=
+  (x : UInt32) (y : UInt32) (st : State) : Result (State × Unit) :=
   do
-    let (st0, _) <- custom_swap_fwd UInt32 x y st
-    result.ret (st0, ())
+    let (st0, _) ← custom_swap_fwd UInt32 x y st
+    Result.ret (st0, ())
 
 /- [external::test_custom_swap] -/
 def test_custom_swap_back
-  (x : UInt32) (y : UInt32) (st : state) (st0 : state) :
-  result (state × (UInt32 × UInt32))
+  (x : UInt32) (y : UInt32) (st : State) (st0 : State) :
+  Result (State × (UInt32 × UInt32))
   :=
   custom_swap_back UInt32 x y st (UInt32.ofNatCore 1 (by intlit)) st0
 
 /- [external::test_swap_non_zero] -/
 def test_swap_non_zero_fwd
-  (x : UInt32) (st : state) : result (state × UInt32) :=
+  (x : UInt32) (st : State) : Result (State × UInt32) :=
   do
-    let (st0, _) <- swap_fwd UInt32 x (UInt32.ofNatCore 0 (by intlit)) st
-    let (st1, (x0, _)) <-
+    let (st0, _) ← swap_fwd UInt32 x (UInt32.ofNatCore 0 (by intlit)) st
+    let (st1, (x0, _)) ←
       swap_back UInt32 x (UInt32.ofNatCore 0 (by intlit)) st st0
-    if x0 = (UInt32.ofNatCore 0 (by intlit))
-    then result.fail error.panic
-    else result.ret (st1, x0)
+    if h: x0 = (UInt32.ofNatCore 0 (by intlit))
+    then Result.fail Error.panic
+    else Result.ret (st1, x0)
 
diff --git a/tests/lean/misc/external/External/Opaque.lean b/tests/lean/misc/external/External/Opaque.lean
index 40ccc313..d3582de3 100644
--- a/tests/lean/misc/external/External/Opaque.lean
+++ b/tests/lean/misc/external/External/Opaque.lean
@@ -6,23 +6,23 @@ import External.Types
 structure OpaqueDefs where
   
   /- [core::mem::swap] -/
-  core_mem_swap_fwd (T : Type) : T -> T -> state -> result (state × Unit)
+  core_mem_swap_fwd (T : Type) : T -> T -> State -> Result (State × Unit)
   
   /- [core::mem::swap] -/
   core_mem_swap_back0
-    (T : Type) : T -> T -> state -> state -> result (state × T)
+    (T : Type) : T -> T -> State -> State -> Result (State × T)
   
   /- [core::mem::swap] -/
   core_mem_swap_back1
-    (T : Type) : T -> T -> state -> state -> result (state × T)
+    (T : Type) : T -> T -> State -> State -> Result (State × T)
   
   /- [core::num::nonzero::NonZeroU32::{14}::new] -/
   core_num_nonzero_non_zero_u32_new_fwd
     :
-    UInt32 -> state -> result (state × (Option
+    UInt32 -> State -> Result (State × (Option
       core_num_nonzero_non_zero_u32_t))
   
   /- [core::option::Option::{0}::unwrap] -/
   core_option_option_unwrap_fwd
-    (T : Type) : Option T -> state -> result (state × T)
+    (T : Type) : Option T -> State -> Result (State × T)
   
diff --git a/tests/lean/misc/external/External/Types.lean b/tests/lean/misc/external/External/Types.lean
index b6fa292b..386832f4 100644
--- a/tests/lean/misc/external/External/Types.lean
+++ b/tests/lean/misc/external/External/Types.lean
@@ -4,5 +4,5 @@ import Base.Primitives
 
 /- [core::num::nonzero::NonZeroU32] -/
 axiom core_num_nonzero_non_zero_u32_t : Type
-/- The state type used in the state-error monad -/ axiom state : Type
+/- The state type used in the state-error monad -/ axiom State : Type
 
diff --git a/tests/lean/misc/loops/Base/Primitives.lean b/tests/lean/misc/loops/Base/Primitives.lean
index 79958d94..5b64e908 100644
--- a/tests/lean/misc/loops/Base/Primitives.lean
+++ b/tests/lean/misc/loops/Base/Primitives.lean
@@ -9,74 +9,79 @@ import Mathlib.Tactic.RunCmd
 
 -- Results & monadic combinators
 
--- TODO: use syntactic conventions and capitalize error, result, etc.
-
-inductive error where
-   | assertionFailure: error
-   | integerOverflow: error
-   | arrayOutOfBounds: error
-   | maximumSizeExceeded: error
-   | panic: error
+inductive Error where
+   | assertionFailure: Error
+   | integerOverflow: Error
+   | arrayOutOfBounds: Error
+   | maximumSizeExceeded: Error
+   | panic: Error
 deriving Repr, BEq
 
-open error
+open Error
 
-inductive result (α : Type u) where
-  | ret (v: α): result α
-  | fail (e: error): result α 
+inductive Result (α : Type u) where
+  | ret (v: α): Result α
+  | fail (e: Error): Result α
 deriving Repr, BEq
 
-open result
+open Result
 
 /- HELPERS -/
 
--- TODO: is there automated syntax for these discriminators?
-def is_ret {α: Type} (r: result α): Bool :=
+def ret? {α: Type} (r: Result α): Bool :=
   match r with
-  | result.ret _ => true
-  | result.fail _ => false
+  | Result.ret _ => true
+  | Result.fail _ => false
 
-def massert (b:Bool) : result Unit :=
+def massert (b:Bool) : Result Unit :=
   if b then .ret () else fail assertionFailure
 
-def eval_global {α: Type} (x: result α) (_: is_ret x): α :=
+def eval_global {α: Type} (x: Result α) (_: ret? x): α :=
   match x with
-  | result.fail _ => by contradiction
-  | result.ret x => x
+  | Result.fail _ => by contradiction
+  | Result.ret x => x
 
 /- DO-DSL SUPPORT -/
 
-def bind (x: result α) (f: α -> result β) : result β :=
+def bind (x: Result α) (f: α -> Result β) : Result β :=
   match x with
   | ret v  => f v 
   | fail v => fail v
 
--- Allows using result in do-blocks
-instance : Bind result where
+-- Allows using Result in do-blocks
+instance : Bind Result where
   bind := bind
 
 -- Allows using return x in do-blocks
-instance : Pure result where
+instance : Pure Result where
   pure := fun x => ret x
 
 /- CUSTOM-DSL SUPPORT -/
 
--- Let-binding the result of a monadic operation is oftentimes not sufficient,
+-- Let-binding the Result of a monadic operation is oftentimes not sufficient,
 -- because we may need a hypothesis for equational reasoning in the scope. We
 -- rely on subtype, and a custom let-binding operator, in effect recreating our
 -- own variant of the do-dsl
 
-def result.attach : (o : result α) → result { x : α // o = ret x }
+def Result.attach {α: Type} (o : Result α): Result { x : α // o = ret x } :=
+  match o with
   | .ret x => .ret ⟨x, rfl⟩
-  | .fail e   => .fail e
+  | .fail e => .fail e
 
-macro "let" h:ident " : " e:term " <-- " f:term : doElem =>
-  `(doElem| let ⟨$e, $h⟩ ← result.attach $f)
+macro "let" e:term " ⟵ " f:term : doElem =>
+  `(doElem| let ⟨$e, h⟩ ← Result.attach $f)
 
--- Silly example of the kind of reasoning that this notation enables
+-- TODO: any way to factorize both definitions?
+macro "let" e:term " <-- " f:term : doElem =>
+  `(doElem| let ⟨$e, h⟩ ← Result.attach $f)
+
+-- We call the hypothesis `h`, in effect making it unavailable to the user
+-- (because too much shadowing). But in practice, once can use the French single
+-- quote notation (input with f< and f>), where `‹ h ›` finds a suitable
+-- hypothesis in the context, this is equivalent to `have x: h := by assumption in x`
 #eval do
-  let h: y <-- .ret (0: Nat)
-  let _: y = 0 := by cases h; decide
+  let y <-- .ret (0: Nat)
+  let _: y = 0 := by cases ‹ ret 0 = ret y › ; decide
   let r: { x: Nat // x = 0 } := ⟨ y, by assumption ⟩
   .ret r
 
@@ -84,36 +89,27 @@ macro "let" h:ident " : " e:term " <-- " f:term : doElem =>
 -- MACHINE INTEGERS --
 ----------------------
 
--- NOTE: we reuse the USize type from prelude.lean, because at least we know
--- it's defined in an idiomatic style that is going to make proofs easy (and
--- indeed, several proofs here are much shortened compared to Aymeric's earlier
--- attempt.) This is not stricto sensu the *correct* thing to do, because one
--- can query at run-time the value of USize, which we do *not* want to do (we
--- don't know what target we'll run on!), but when the day comes, we'll just
--- define our own USize.
--- ANOTHER NOTE: there is USize.sub but it has wraparound semantics, which is
--- not something we want to define (I think), so we use our own monadic sub (but
--- is it in line with the Rust behavior?)
-
--- TODO: I am somewhat under the impression that subtraction is defined as a
--- total function over nats...? the hypothesis in the if condition is not used
--- in the then-branch which confuses me quite a bit
-
--- TODO: add a refinement for the result (just like vec_push_back below) that
--- explains that the toNat of the result (in the case of success) is the sub of
--- the toNat of the arguments (i.e. intrinsic specification)
--- ... do we want intrinsic specifications for the builtins? that might require
--- some careful type annotations in the monadic notation for clients, but may
--- give us more "for free"
+-- NOTE: we reuse the fixed-width integer types from prelude.lean: UInt8, ...,
+-- USize. They are generally defined in an idiomatic style, except that there is
+-- not a single type class to rule them all (more on that below). The absence of
+-- type class is intentional, and allows the Lean compiler to efficiently map
+-- them to machine integers during compilation.
+
+-- USize is designed properly: you cannot reduce `getNumBits` using the
+-- simplifier, meaning that proofs do not depend on the compile-time value of
+-- USize.size. (Lean assumes 32 or 64-bit platforms, and Rust doesn't really
+-- support, at least officially, 16-bit microcontrollers, so this seems like a
+-- fine design decision for now.)
 
 -- Note from Chris Bailey: "If there's more than one salient property of your
 -- definition then the subtyping strategy might get messy, and the property part
 -- of a subtype is less discoverable by the simplifier or tactics like
--- library_search." Try to settle this with a Lean expert on what is the most
--- productive way to go about this?
+-- library_search." So, we will not add refinements on the return values of the
+-- operations defined on Primitives, but will rather rely on custom lemmas to
+-- invert on possible return values of the primitive operations.
 
--- One needs to perform a little bit of reasoning in order to successfully
--- inject constants into USize, so we provide a general-purpose macro
+-- Machine integer constants, done via `ofNatCore`, which requires a proof that
+-- the `Nat` fits within the desired integer type. We provide a custom tactic.
 
 syntax "intlit" : tactic
 
@@ -129,12 +125,21 @@ macro_rules
 -- Also works for other integer types (at the expense of a needless disjunction)
 #eval UInt32.ofNatCore 0 (by intlit)
 
+-- The machine integer operations (e.g. sub) are always total, which is not what
+-- we want. We therefore define "checked" variants, below. Note that we add a
+-- tiny bit of complexity for the USize variant: we first check whether the
+-- result is < 2^32; if it is, we can compute the definition, rather than
+-- returning a term that is computationally stuck (the comparison to USize.size
+-- cannot reduce at compile-time, per the remark about regarding `getNumBits`).
+-- This is useful for the various #asserts that we want to reduce at
+-- type-checking time.
+
 -- Further thoughts: look at what has been done here:
 -- https://github.com/leanprover-community/mathlib4/blob/master/Mathlib/Data/Fin/Basic.lean
 -- and
 -- https://github.com/leanprover-community/mathlib4/blob/master/Mathlib/Data/UInt.lean
 -- which both contain a fair amount of reasoning already!
-def USize.checked_sub (n: USize) (m: USize): result USize :=
+def USize.checked_sub (n: USize) (m: USize): Result USize :=
   -- NOTE: the test USize.toNat n - m >= 0 seems to always succeed?
   if n >= m then
     let n' := USize.toNat n
@@ -150,18 +155,19 @@ def USize.checked_sub (n: USize) (m: USize): result USize :=
   else
     fail integerOverflow
 
-def USize.checked_add (n: USize) (m: USize): result USize :=
-  if h: n.val.val + m.val.val <= 4294967295 then
-    .ret ⟨ n.val.val + m.val.val, by
-      have h': 4294967295 < USize.size := by intlit
-      apply Nat.lt_of_le_of_lt h h'
-    ⟩
-  else if h: n.val + m.val < USize.size then
+@[simp]
+theorem usize_fits (n: Nat) (h: n <= 4294967295): n < USize.size :=
+  match USize.size, usize_size_eq with
+  | _, Or.inl rfl => Nat.lt_of_le_of_lt h (by decide)
+  | _, Or.inr rfl => Nat.lt_of_le_of_lt h (by decide)
+
+def USize.checked_add (n: USize) (m: USize): Result USize :=
+  if h: n.val + m.val < USize.size then
     .ret ⟨ n.val + m.val, h ⟩
   else
     .fail integerOverflow
 
-def USize.checked_rem (n: USize) (m: USize): result USize :=
+def USize.checked_rem (n: USize) (m: USize): Result USize :=
   if h: m > 0 then
     .ret ⟨ n.val % m.val, by
       have h1: ↑m.val < USize.size := m.val.isLt
@@ -171,18 +177,13 @@ def USize.checked_rem (n: USize) (m: USize): result USize :=
   else
     .fail integerOverflow
 
-def USize.checked_mul (n: USize) (m: USize): result USize :=
-    if h: n.val.val * m.val.val <= 4294967295 then
-    .ret ⟨ n.val.val * m.val.val, by
-      have h': 4294967295 < USize.size := by intlit
-      apply Nat.lt_of_le_of_lt h h'
-    ⟩
-  else if h: n.val * m.val < USize.size then
+def USize.checked_mul (n: USize) (m: USize): Result USize :=
+  if h: n.val * m.val < USize.size then
     .ret ⟨ n.val * m.val, h ⟩
   else
     .fail integerOverflow
 
-def USize.checked_div (n: USize) (m: USize): result USize :=
+def USize.checked_div (n: USize) (m: USize): Result USize :=
   if m > 0 then
     .ret ⟨ n.val / m.val, by
       have h1: ↑n.val < USize.size := n.val.isLt
@@ -192,6 +193,19 @@ def USize.checked_div (n: USize) (m: USize): result USize :=
   else
     .fail integerOverflow
 
+-- Test behavior...
+#eval assert! USize.checked_sub 10 20 == fail integerOverflow; 0
+
+#eval USize.checked_sub 20 10
+-- NOTE: compare with concrete behavior here, which I do not think we want
+#eval USize.sub 0 1
+#eval UInt8.add 255 255
+
+-- We now define a type class that subsumes the various machine integer types, so
+-- as to write a concise definition for scalar_cast, rather than exhaustively
+-- enumerating all of the possible pairs. We remark that Rust has sane semantics
+-- and fails if a cast operation would involve a truncation or modulo.
+
 class MachineInteger (t: Type) where
   size: Nat
   val: t -> Fin size
@@ -209,30 +223,24 @@ run_cmd
     end $typeName
   ))
 
-def scalar_cast { src: Type } (dst: Type) [ MachineInteger src ] [ MachineInteger dst ] (x: src): result dst :=
+-- Aeneas only instantiates the destination type (`src` is implicit). We rely on
+-- Lean to infer `src`.
+
+def scalar_cast { src: Type } (dst: Type) [ MachineInteger src ] [ MachineInteger dst ] (x: src): Result dst :=
   if h: MachineInteger.val x < MachineInteger.size dst then
     .ret (MachineInteger.ofNatCore (MachineInteger.val x).val h)
   else
     .fail integerOverflow
 
-
--- Test behavior...
-#eval assert! USize.checked_sub 10 20 == fail integerOverflow; 0
-
-#eval USize.checked_sub 20 10
--- NOTE: compare with concrete behavior here, which I do not think we want
-#eval USize.sub 0 1
-#eval UInt8.add 255 255
-
 -------------
 -- VECTORS --
 -------------
 
 -- Note: unlike F*, Lean seems to use strict upper bounds (e.g. USize.size)
 -- rather than maximum values (usize_max).
-def vec (α : Type u) := { l : List α // List.length l < USize.size }
+def Vec (α : Type u) := { l : List α // List.length l < USize.size }
 
-def vec_new (α : Type u): vec α := ⟨ [], by {
+def vec_new (α : Type u): Vec α := ⟨ [], by {
   match USize.size, usize_size_eq with
   | _, Or.inl rfl => simp
   | _, Or.inr rfl => simp
@@ -240,20 +248,20 @@ def vec_new (α : Type u): vec α := ⟨ [], by {
 
 #check vec_new
 
-def vec_len (α : Type u) (v : vec α) : USize :=
+def vec_len (α : Type u) (v : Vec α) : USize :=
   let ⟨ v, l ⟩ := v
   USize.ofNatCore (List.length v) l
 
 #eval vec_len Nat (vec_new Nat)
  
-def vec_push_fwd (α : Type u) (_ : vec α) (_ : α) : Unit := ()
+def vec_push_fwd (α : Type u) (_ : Vec α) (_ : α) : Unit := ()
 
 -- NOTE: old version trying to use a subtype notation, but probably better to
--- leave result elimination to auxiliary lemmas with suitable preconditions
+-- leave Result elimination to auxiliary lemmas with suitable preconditions
 -- TODO: I originally wrote `List.length v.val < USize.size - 1`; how can one
 -- make the proof work in that case? Probably need to import tactics from
 -- mathlib to deal with inequalities... would love to see an example.
-def vec_push_back_old (α : Type u) (v : vec α) (x : α) : { res: result (vec α) //
+def vec_push_back_old (α : Type u) (v : Vec α) (x : α) : { res: Result (Vec α) //
   match res with | fail _ => True | ret v' => List.length v'.val = List.length v.val + 1}
   :=
   if h : List.length v.val + 1 < USize.size then
@@ -272,12 +280,12 @@ def vec_push_back_old (α : Type u) (v : vec α) (x : α) : { res: result (vec �
   -- annotate `x`, which relieves us of having to write `.val` on the right-hand
   -- side of the monadic let.
   let v := vec_new Nat
-  let x: vec Nat ← (vec_push_back_old Nat v 1: result (vec Nat)) -- WHY do we need the type annotation here?
+  let x: Vec Nat ← (vec_push_back_old Nat v 1: Result (Vec Nat)) -- WHY do we need the type annotation here?
   -- TODO: strengthen post-condition above and do a demo to show that we can
   -- safely eliminate the `fail` case
   return (vec_len Nat x)
 
-def vec_push_back (α : Type u) (v : vec α) (x : α) : result (vec α)
+def vec_push_back (α : Type u) (v : Vec α) (x : α) : Result (Vec α)
   :=
   if h : List.length v.val + 1 <= 4294967295 then
     return ⟨ List.concat v.val x,
@@ -295,13 +303,13 @@ def vec_push_back (α : Type u) (v : vec α) (x : α) : result (vec α)
   else
     fail maximumSizeExceeded
 
-def vec_insert_fwd (α : Type u) (v: vec α) (i: USize) (_: α): result Unit :=
+def vec_insert_fwd (α : Type u) (v: Vec α) (i: USize) (_: α): Result Unit :=
   if i.val < List.length v.val then
     .ret ()
   else
     .fail arrayOutOfBounds
 
-def vec_insert_back (α : Type u) (v: vec α) (i: USize) (x: α): result (vec α) :=
+def vec_insert_back (α : Type u) (v: Vec α) (i: USize) (x: α): Result (Vec α) :=
   if i.val < List.length v.val then
     .ret ⟨ List.set v.val i.val x, by
       have h: List.length v.val < USize.size := v.property
@@ -311,25 +319,25 @@ def vec_insert_back (α : Type u) (v: vec α) (i: USize) (x: α): result (vec α
   else
     .fail arrayOutOfBounds
 
-def vec_index_fwd (α : Type u) (v: vec α) (i: USize): result α :=
+def vec_index_fwd (α : Type u) (v: Vec α) (i: USize): Result α :=
   if h: i.val < List.length v.val then
     .ret (List.get v.val ⟨i.val, h⟩)
   else
     .fail arrayOutOfBounds
 
-def vec_index_back (α : Type u) (v: vec α) (i: USize) (_: α): result Unit :=
+def vec_index_back (α : Type u) (v: Vec α) (i: USize) (_: α): Result Unit :=
   if i.val < List.length v.val then
     .ret ()
   else
     .fail arrayOutOfBounds
 
-def vec_index_mut_fwd (α : Type u) (v: vec α) (i: USize): result α :=
+def vec_index_mut_fwd (α : Type u) (v: Vec α) (i: USize): Result α :=
   if h: i.val < List.length v.val then
     .ret (List.get v.val ⟨i.val, h⟩)
   else
     .fail arrayOutOfBounds
 
-def vec_index_mut_back (α : Type u) (v: vec α) (i: USize) (x: α): result (vec α) :=
+def vec_index_mut_back (α : Type u) (v: Vec α) (i: USize) (x: α): Result (Vec α) :=
   if i.val < List.length v.val then
     .ret ⟨ List.set v.val i.val x, by
       have h: List.length v.val < USize.size := v.property
@@ -349,6 +357,10 @@ def mem_replace_fwd (a : Type) (x : a) (_ : a) : a :=
 def mem_replace_back (a : Type) (_ : a) (y : a) : a :=
   y
 
+/-- Aeneas-translated function -- useful to reduce non-recursive definitions.
+ Use with `simp [ aeneas ]` -/
+register_simp_attr aeneas
+
 --------------------
 -- ASSERT COMMAND --
 --------------------
@@ -358,16 +370,23 @@ open Lean Elab Command Term Meta
 syntax (name := assert) "#assert" term: command
 
 @[command_elab assert]
+unsafe
 def assertImpl : CommandElab := fun (_stx: Syntax) => do
-  logInfo "Reducing and asserting: "
-  logInfo _stx[1]
   runTermElabM (fun _ => do
-    let e ← Term.elabTerm _stx[1] none
-    logInfo (Expr.dbgToString e)
-    -- How to evaluate the term and compare the result to true?
+    let r ← evalTerm Bool (mkConst ``Bool) _stx[1]
+    if not r then
+      logInfo "Assertion failed for: "
+      logInfo _stx[1]
+      logError "Expression reduced to false"
     pure ())
-  -- logInfo (Expr.dbgToString (``true))
-  -- throwError "TODO: assert"
 
 #eval 2 == 2
 #assert (2 == 2)
+
+-------------------
+-- SANITY CHECKS --
+-------------------
+
+-- TODO: add more once we have signed integers
+
+#assert (USize.checked_rem 1 2 == .ret 1)
diff --git a/tests/lean/misc/loops/Loops/Clauses/Template.lean b/tests/lean/misc/loops/Loops/Clauses/Template.lean
index 3c0f2f7c..d74f71e1 100644
--- a/tests/lean/misc/loops/Loops/Clauses/Template.lean
+++ b/tests/lean/misc/loops/Loops/Clauses/Template.lean
@@ -35,7 +35,7 @@ macro_rules
 | `(tactic| sum_with_shared_borrows_loop_decreases $max $i $s) =>`(tactic| sorry)
 
 /- [loops::clear]: termination measure -/
-@[simp] def clear_loop_terminates (v : vec UInt32) (i : USize) := (v, i)
+@[simp] def clear_loop_terminates (v : Vec UInt32) (i : USize) := (v, i)
 
 syntax "clear_loop_decreases" term+ : tactic
 
diff --git a/tests/lean/misc/loops/Loops/Funs.lean b/tests/lean/misc/loops/Loops/Funs.lean
index 55f0c87d..5a81ebff 100644
--- a/tests/lean/misc/loops/Loops/Funs.lean
+++ b/tests/lean/misc/loops/Loops/Funs.lean
@@ -5,30 +5,30 @@ import Loops.Types
 import Loops.Clauses.Clauses
 
 /- [loops::sum] -/
-def sum_loop_fwd (max : UInt32) (i : UInt32) (s : UInt32) : (result UInt32) :=
-  if i < max
+def sum_loop_fwd (max : UInt32) (i : UInt32) (s : UInt32) : (Result UInt32) :=
+  if h: i < max
   then
     do
-      let s0 <- UInt32.checked_add s i
-      let i0 <- UInt32.checked_add i (UInt32.ofNatCore 1 (by intlit))
+      let s0 ← UInt32.checked_add s i
+      let i0 ← UInt32.checked_add i (UInt32.ofNatCore 1 (by intlit))
       sum_loop_fwd max i0 s0
   else UInt32.checked_mul s (UInt32.ofNatCore 2 (by intlit))
 termination_by sum_loop_fwd max i s => sum_loop_terminates max i s
 decreasing_by sum_loop_decreases max i s
 
 /- [loops::sum] -/
-def sum_fwd (max : UInt32) : result UInt32 :=
+def sum_fwd (max : UInt32) : Result UInt32 :=
   sum_loop_fwd max (UInt32.ofNatCore 0 (by intlit))
     (UInt32.ofNatCore 0 (by intlit))
 
 /- [loops::sum_with_mut_borrows] -/
 def sum_with_mut_borrows_loop_fwd
-  (max : UInt32) (mi : UInt32) (ms : UInt32) : (result UInt32) :=
-  if mi < max
+  (max : UInt32) (mi : UInt32) (ms : UInt32) : (Result UInt32) :=
+  if h: mi < max
   then
     do
-      let ms0 <- UInt32.checked_add ms mi
-      let mi0 <- UInt32.checked_add mi (UInt32.ofNatCore 1 (by intlit))
+      let ms0 ← UInt32.checked_add ms mi
+      let mi0 ← UInt32.checked_add mi (UInt32.ofNatCore 1 (by intlit))
       sum_with_mut_borrows_loop_fwd max mi0 ms0
   else UInt32.checked_mul ms (UInt32.ofNatCore 2 (by intlit))
 termination_by sum_with_mut_borrows_loop_fwd max mi ms =>
@@ -36,18 +36,18 @@ termination_by sum_with_mut_borrows_loop_fwd max mi ms =>
 decreasing_by sum_with_mut_borrows_loop_decreases max mi ms
 
 /- [loops::sum_with_mut_borrows] -/
-def sum_with_mut_borrows_fwd (max : UInt32) : result UInt32 :=
+def sum_with_mut_borrows_fwd (max : UInt32) : Result UInt32 :=
   sum_with_mut_borrows_loop_fwd max (UInt32.ofNatCore 0 (by intlit))
     (UInt32.ofNatCore 0 (by intlit))
 
 /- [loops::sum_with_shared_borrows] -/
 def sum_with_shared_borrows_loop_fwd
-  (max : UInt32) (i : UInt32) (s : UInt32) : (result UInt32) :=
-  if i < max
+  (max : UInt32) (i : UInt32) (s : UInt32) : (Result UInt32) :=
+  if h: i < max
   then
     do
-      let i0 <- UInt32.checked_add i (UInt32.ofNatCore 1 (by intlit))
-      let s0 <- UInt32.checked_add s i0
+      let i0 ← UInt32.checked_add i (UInt32.ofNatCore 1 (by intlit))
+      let s0 ← UInt32.checked_add s i0
       sum_with_shared_borrows_loop_fwd max i0 s0
   else UInt32.checked_mul s (UInt32.ofNatCore 2 (by intlit))
 termination_by sum_with_shared_borrows_loop_fwd max i s =>
@@ -55,271 +55,272 @@ termination_by sum_with_shared_borrows_loop_fwd max i s =>
 decreasing_by sum_with_shared_borrows_loop_decreases max i s
 
 /- [loops::sum_with_shared_borrows] -/
-def sum_with_shared_borrows_fwd (max : UInt32) : result UInt32 :=
+def sum_with_shared_borrows_fwd (max : UInt32) : Result UInt32 :=
   sum_with_shared_borrows_loop_fwd max (UInt32.ofNatCore 0 (by intlit))
     (UInt32.ofNatCore 0 (by intlit))
 
 /- [loops::clear] -/
-def clear_loop_fwd_back (v : vec UInt32) (i : USize) : (result (vec UInt32)) :=
+def clear_loop_fwd_back (v : Vec UInt32) (i : USize) : (Result (Vec UInt32)) :=
   let i0 := vec_len UInt32 v
-  if i < i0
+  if h: i < i0
   then
     do
-      let i1 <- USize.checked_add i (USize.ofNatCore 1 (by intlit))
-      let v0 <- vec_index_mut_back UInt32 v i (UInt32.ofNatCore 0 (by intlit))
+      let i1 ← USize.checked_add i (USize.ofNatCore 1 (by intlit))
+      let v0 ← vec_index_mut_back UInt32 v i (UInt32.ofNatCore 0 (by intlit))
       clear_loop_fwd_back v0 i1
-  else result.ret v
+  else Result.ret v
 termination_by clear_loop_fwd_back v i => clear_loop_terminates v i
 decreasing_by clear_loop_decreases v i
 
 /- [loops::clear] -/
-def clear_fwd_back (v : vec UInt32) : result (vec UInt32) :=
+def clear_fwd_back (v : Vec UInt32) : Result (Vec UInt32) :=
   clear_loop_fwd_back v (USize.ofNatCore 0 (by intlit))
 
 /- [loops::list_mem] -/
-def list_mem_loop_fwd (x : UInt32) (ls : list_t UInt32) : (result Bool) :=
-  match ls with
+def list_mem_loop_fwd (x : UInt32) (ls : list_t UInt32) : (Result Bool) :=
+  match h: ls with
   | list_t.ListCons y tl =>
-    if y = x
-    then result.ret true
+    if h: y = x
+    then Result.ret true
     else list_mem_loop_fwd x tl
-  | list_t.ListNil => result.ret false
+  | list_t.ListNil => Result.ret false
 termination_by list_mem_loop_fwd x ls => list_mem_loop_terminates x ls
 decreasing_by list_mem_loop_decreases x ls
 
 /- [loops::list_mem] -/
-def list_mem_fwd (x : UInt32) (ls : list_t UInt32) : result Bool :=
+def list_mem_fwd (x : UInt32) (ls : list_t UInt32) : Result Bool :=
   list_mem_loop_fwd x ls
 
 /- [loops::list_nth_mut_loop] -/
 def list_nth_mut_loop_loop_fwd
-  (T : Type) (ls : list_t T) (i : UInt32) : (result T) :=
-  match ls with
+  (T : Type) (ls : list_t T) (i : UInt32) : (Result T) :=
+  match h: ls with
   | list_t.ListCons x tl =>
-    if i = (UInt32.ofNatCore 0 (by intlit))
-    then result.ret x
+    if h: i = (UInt32.ofNatCore 0 (by intlit))
+    then Result.ret x
     else
       do
-        let i0 <- UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+        let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
         list_nth_mut_loop_loop_fwd T tl i0
-  | list_t.ListNil => result.fail error.panic
+  | list_t.ListNil => Result.fail Error.panic
 termination_by list_nth_mut_loop_loop_fwd ls i =>
   list_nth_mut_loop_loop_terminates T ls i
 decreasing_by list_nth_mut_loop_loop_decreases ls i
 
 /- [loops::list_nth_mut_loop] -/
-def list_nth_mut_loop_fwd (T : Type) (ls : list_t T) (i : UInt32) : result T :=
+def list_nth_mut_loop_fwd (T : Type) (ls : list_t T) (i : UInt32) : Result T :=
   list_nth_mut_loop_loop_fwd T ls i
 
 /- [loops::list_nth_mut_loop] -/
 def list_nth_mut_loop_loop_back
-  (T : Type) (ls : list_t T) (i : UInt32) (ret0 : T) : (result (list_t T)) :=
-  match ls with
+  (T : Type) (ls : list_t T) (i : UInt32) (ret0 : T) : (Result (list_t T)) :=
+  match h: ls with
   | list_t.ListCons x tl =>
-    if i = (UInt32.ofNatCore 0 (by intlit))
-    then result.ret (list_t.ListCons ret0 tl)
+    if h: i = (UInt32.ofNatCore 0 (by intlit))
+    then Result.ret (list_t.ListCons ret0 tl)
     else
       do
-        let i0 <- UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
-        let tl0 <- list_nth_mut_loop_loop_back T tl i0 ret0
-        result.ret (list_t.ListCons x tl0)
-  | list_t.ListNil => result.fail error.panic
+        let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+        let tl0 ← list_nth_mut_loop_loop_back T tl i0 ret0
+        Result.ret (list_t.ListCons x tl0)
+  | list_t.ListNil => Result.fail Error.panic
 termination_by list_nth_mut_loop_loop_back ls i ret0 =>
   list_nth_mut_loop_loop_terminates T ls i
 decreasing_by list_nth_mut_loop_loop_decreases ls i
 
 /- [loops::list_nth_mut_loop] -/
 def list_nth_mut_loop_back
-  (T : Type) (ls : list_t T) (i : UInt32) (ret0 : T) : result (list_t T) :=
+  (T : Type) (ls : list_t T) (i : UInt32) (ret0 : T) : Result (list_t T) :=
   list_nth_mut_loop_loop_back T ls i ret0
 
 /- [loops::list_nth_shared_loop] -/
 def list_nth_shared_loop_loop_fwd
-  (T : Type) (ls : list_t T) (i : UInt32) : (result T) :=
-  match ls with
+  (T : Type) (ls : list_t T) (i : UInt32) : (Result T) :=
+  match h: ls with
   | list_t.ListCons x tl =>
-    if i = (UInt32.ofNatCore 0 (by intlit))
-    then result.ret x
+    if h: i = (UInt32.ofNatCore 0 (by intlit))
+    then Result.ret x
     else
       do
-        let i0 <- UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+        let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
         list_nth_shared_loop_loop_fwd T tl i0
-  | list_t.ListNil => result.fail error.panic
+  | list_t.ListNil => Result.fail Error.panic
 termination_by list_nth_shared_loop_loop_fwd ls i =>
   list_nth_shared_loop_loop_terminates T ls i
 decreasing_by list_nth_shared_loop_loop_decreases ls i
 
 /- [loops::list_nth_shared_loop] -/
 def list_nth_shared_loop_fwd
-  (T : Type) (ls : list_t T) (i : UInt32) : result T :=
+  (T : Type) (ls : list_t T) (i : UInt32) : Result T :=
   list_nth_shared_loop_loop_fwd T ls i
 
 /- [loops::get_elem_mut] -/
-def get_elem_mut_loop_fwd (x : USize) (ls : list_t USize) : (result USize) :=
-  match ls with
+def get_elem_mut_loop_fwd (x : USize) (ls : list_t USize) : (Result USize) :=
+  match h: ls with
   | list_t.ListCons y tl =>
-    if y = x
-    then result.ret y
+    if h: y = x
+    then Result.ret y
     else get_elem_mut_loop_fwd x tl
-  | list_t.ListNil => result.fail error.panic
+  | list_t.ListNil => Result.fail Error.panic
 termination_by get_elem_mut_loop_fwd x ls => get_elem_mut_loop_terminates x ls
 decreasing_by get_elem_mut_loop_decreases x ls
 
 /- [loops::get_elem_mut] -/
-def get_elem_mut_fwd (slots : vec (list_t USize)) (x : USize) : result USize :=
+def get_elem_mut_fwd (slots : Vec (list_t USize)) (x : USize) : Result USize :=
   do
-    let l <-
+    let l ←
       vec_index_mut_fwd (list_t USize) slots (USize.ofNatCore 0 (by intlit))
     get_elem_mut_loop_fwd x l
 
 /- [loops::get_elem_mut] -/
 def get_elem_mut_loop_back
-  (x : USize) (ls : list_t USize) (ret0 : USize) : (result (list_t USize)) :=
-  match ls with
+  (x : USize) (ls : list_t USize) (ret0 : USize) : (Result (list_t USize)) :=
+  match h: ls with
   | list_t.ListCons y tl =>
-    if y = x
-    then result.ret (list_t.ListCons ret0 tl)
+    if h: y = x
+    then Result.ret (list_t.ListCons ret0 tl)
     else
       do
-        let tl0 <- get_elem_mut_loop_back x tl ret0
-        result.ret (list_t.ListCons y tl0)
-  | list_t.ListNil => result.fail error.panic
+        let tl0 ← get_elem_mut_loop_back x tl ret0
+        Result.ret (list_t.ListCons y tl0)
+  | list_t.ListNil => Result.fail Error.panic
 termination_by get_elem_mut_loop_back x ls ret0 =>
   get_elem_mut_loop_terminates x ls
 decreasing_by get_elem_mut_loop_decreases x ls
 
 /- [loops::get_elem_mut] -/
 def get_elem_mut_back
-  (slots : vec (list_t USize)) (x : USize) (ret0 : USize) :
-  result (vec (list_t USize))
+  (slots : Vec (list_t USize)) (x : USize) (ret0 : USize) :
+  Result (Vec (list_t USize))
   :=
   do
-    let l <-
+    let l ←
       vec_index_mut_fwd (list_t USize) slots (USize.ofNatCore 0 (by intlit))
-    let l0 <- get_elem_mut_loop_back x l ret0
+    let l0 ← get_elem_mut_loop_back x l ret0
     vec_index_mut_back (list_t USize) slots (USize.ofNatCore 0 (by intlit)) l0
 
 /- [loops::get_elem_shared] -/
 def get_elem_shared_loop_fwd
-  (x : USize) (ls : list_t USize) : (result USize) :=
-  match ls with
+  (x : USize) (ls : list_t USize) : (Result USize) :=
+  match h: ls with
   | list_t.ListCons y tl =>
-    if y = x
-    then result.ret y
+    if h: y = x
+    then Result.ret y
     else get_elem_shared_loop_fwd x tl
-  | list_t.ListNil => result.fail error.panic
+  | list_t.ListNil => Result.fail Error.panic
 termination_by get_elem_shared_loop_fwd x ls =>
   get_elem_shared_loop_terminates x ls
 decreasing_by get_elem_shared_loop_decreases x ls
 
 /- [loops::get_elem_shared] -/
 def get_elem_shared_fwd
-  (slots : vec (list_t USize)) (x : USize) : result USize :=
+  (slots : Vec (list_t USize)) (x : USize) : Result USize :=
   do
-    let l <- vec_index_fwd (list_t USize) slots (USize.ofNatCore 0 (by intlit))
+    let l ←
+      vec_index_fwd (list_t USize) slots (USize.ofNatCore 0 (by intlit))
     get_elem_shared_loop_fwd x l
 
 /- [loops::id_mut] -/
-def id_mut_fwd (T : Type) (ls : list_t T) : result (list_t T) :=
-  result.ret ls
+def id_mut_fwd (T : Type) (ls : list_t T) : Result (list_t T) :=
+  Result.ret ls
 
 /- [loops::id_mut] -/
 def id_mut_back
-  (T : Type) (ls : list_t T) (ret0 : list_t T) : result (list_t T) :=
-  result.ret ret0
+  (T : Type) (ls : list_t T) (ret0 : list_t T) : Result (list_t T) :=
+  Result.ret ret0
 
 /- [loops::id_shared] -/
-def id_shared_fwd (T : Type) (ls : list_t T) : result (list_t T) :=
-  result.ret ls
+def id_shared_fwd (T : Type) (ls : list_t T) : Result (list_t T) :=
+  Result.ret ls
 
 /- [loops::list_nth_mut_loop_with_id] -/
 def list_nth_mut_loop_with_id_loop_fwd
-  (T : Type) (i : UInt32) (ls : list_t T) : (result T) :=
-  match ls with
+  (T : Type) (i : UInt32) (ls : list_t T) : (Result T) :=
+  match h: ls with
   | list_t.ListCons x tl =>
-    if i = (UInt32.ofNatCore 0 (by intlit))
-    then result.ret x
+    if h: i = (UInt32.ofNatCore 0 (by intlit))
+    then Result.ret x
     else
       do
-        let i0 <- UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+        let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
         list_nth_mut_loop_with_id_loop_fwd T i0 tl
-  | list_t.ListNil => result.fail error.panic
+  | list_t.ListNil => Result.fail Error.panic
 termination_by list_nth_mut_loop_with_id_loop_fwd i ls =>
   list_nth_mut_loop_with_id_loop_terminates T i ls
 decreasing_by list_nth_mut_loop_with_id_loop_decreases i ls
 
 /- [loops::list_nth_mut_loop_with_id] -/
 def list_nth_mut_loop_with_id_fwd
-  (T : Type) (ls : list_t T) (i : UInt32) : result T :=
+  (T : Type) (ls : list_t T) (i : UInt32) : Result T :=
   do
-    let ls0 <- id_mut_fwd T ls
+    let ls0 ← id_mut_fwd T ls
     list_nth_mut_loop_with_id_loop_fwd T i ls0
 
 /- [loops::list_nth_mut_loop_with_id] -/
 def list_nth_mut_loop_with_id_loop_back
-  (T : Type) (i : UInt32) (ls : list_t T) (ret0 : T) : (result (list_t T)) :=
-  match ls with
+  (T : Type) (i : UInt32) (ls : list_t T) (ret0 : T) : (Result (list_t T)) :=
+  match h: ls with
   | list_t.ListCons x tl =>
-    if i = (UInt32.ofNatCore 0 (by intlit))
-    then result.ret (list_t.ListCons ret0 tl)
+    if h: i = (UInt32.ofNatCore 0 (by intlit))
+    then Result.ret (list_t.ListCons ret0 tl)
     else
       do
-        let i0 <- UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
-        let tl0 <- list_nth_mut_loop_with_id_loop_back T i0 tl ret0
-        result.ret (list_t.ListCons x tl0)
-  | list_t.ListNil => result.fail error.panic
+        let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+        let tl0 ← list_nth_mut_loop_with_id_loop_back T i0 tl ret0
+        Result.ret (list_t.ListCons x tl0)
+  | list_t.ListNil => Result.fail Error.panic
 termination_by list_nth_mut_loop_with_id_loop_back i ls ret0 =>
   list_nth_mut_loop_with_id_loop_terminates T i ls
 decreasing_by list_nth_mut_loop_with_id_loop_decreases i ls
 
 /- [loops::list_nth_mut_loop_with_id] -/
 def list_nth_mut_loop_with_id_back
-  (T : Type) (ls : list_t T) (i : UInt32) (ret0 : T) : result (list_t T) :=
+  (T : Type) (ls : list_t T) (i : UInt32) (ret0 : T) : Result (list_t T) :=
   do
-    let ls0 <- id_mut_fwd T ls
-    let l <- list_nth_mut_loop_with_id_loop_back T i ls0 ret0
+    let ls0 ← id_mut_fwd T ls
+    let l ← list_nth_mut_loop_with_id_loop_back T i ls0 ret0
     id_mut_back T ls l
 
 /- [loops::list_nth_shared_loop_with_id] -/
 def list_nth_shared_loop_with_id_loop_fwd
-  (T : Type) (i : UInt32) (ls : list_t T) : (result T) :=
-  match ls with
+  (T : Type) (i : UInt32) (ls : list_t T) : (Result T) :=
+  match h: ls with
   | list_t.ListCons x tl =>
-    if i = (UInt32.ofNatCore 0 (by intlit))
-    then result.ret x
+    if h: i = (UInt32.ofNatCore 0 (by intlit))
+    then Result.ret x
     else
       do
-        let i0 <- UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+        let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
         list_nth_shared_loop_with_id_loop_fwd T i0 tl
-  | list_t.ListNil => result.fail error.panic
+  | list_t.ListNil => Result.fail Error.panic
 termination_by list_nth_shared_loop_with_id_loop_fwd i ls =>
   list_nth_shared_loop_with_id_loop_terminates T i ls
 decreasing_by list_nth_shared_loop_with_id_loop_decreases i ls
 
 /- [loops::list_nth_shared_loop_with_id] -/
 def list_nth_shared_loop_with_id_fwd
-  (T : Type) (ls : list_t T) (i : UInt32) : result T :=
+  (T : Type) (ls : list_t T) (i : UInt32) : Result T :=
   do
-    let ls0 <- id_shared_fwd T ls
+    let ls0 ← id_shared_fwd T ls
     list_nth_shared_loop_with_id_loop_fwd T i ls0
 
 /- [loops::list_nth_mut_loop_pair] -/
 def list_nth_mut_loop_pair_loop_fwd
   (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : UInt32) :
-  (result (T × T))
+  (Result (T × T))
   :=
-  match ls0 with
+  match h: ls0 with
   | list_t.ListCons x0 tl0 =>
-    match ls1 with
+    match h: ls1 with
     | list_t.ListCons x1 tl1 =>
-      if i = (UInt32.ofNatCore 0 (by intlit))
-      then result.ret (x0, x1)
+      if h: i = (UInt32.ofNatCore 0 (by intlit))
+      then Result.ret (x0, x1)
       else
         do
-          let i0 <- UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+          let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
           list_nth_mut_loop_pair_loop_fwd T tl0 tl1 i0
-    | list_t.ListNil => result.fail error.panic
-  | list_t.ListNil => result.fail error.panic
+    | list_t.ListNil => Result.fail Error.panic
+  | list_t.ListNil => Result.fail Error.panic
 termination_by list_nth_mut_loop_pair_loop_fwd ls0 ls1 i =>
   list_nth_mut_loop_pair_loop_terminates T ls0 ls1 i
 decreasing_by list_nth_mut_loop_pair_loop_decreases ls0 ls1 i
@@ -327,28 +328,28 @@ decreasing_by list_nth_mut_loop_pair_loop_decreases ls0 ls1 i
 /- [loops::list_nth_mut_loop_pair] -/
 def list_nth_mut_loop_pair_fwd
   (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : UInt32) :
-  result (T × T)
+  Result (T × T)
   :=
   list_nth_mut_loop_pair_loop_fwd T ls0 ls1 i
 
 /- [loops::list_nth_mut_loop_pair] -/
 def list_nth_mut_loop_pair_loop_back'a
   (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : UInt32) (ret0 : T) :
-  (result (list_t T))
+  (Result (list_t T))
   :=
-  match ls0 with
+  match h: ls0 with
   | list_t.ListCons x0 tl0 =>
-    match ls1 with
+    match h: ls1 with
     | list_t.ListCons x1 tl1 =>
-      if i = (UInt32.ofNatCore 0 (by intlit))
-      then result.ret (list_t.ListCons ret0 tl0)
+      if h: i = (UInt32.ofNatCore 0 (by intlit))
+      then Result.ret (list_t.ListCons ret0 tl0)
       else
         do
-          let i0 <- UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
-          let tl00 <- list_nth_mut_loop_pair_loop_back'a T tl0 tl1 i0 ret0
-          result.ret (list_t.ListCons x0 tl00)
-    | list_t.ListNil => result.fail error.panic
-  | list_t.ListNil => result.fail error.panic
+          let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+          let tl00 ← list_nth_mut_loop_pair_loop_back'a T tl0 tl1 i0 ret0
+          Result.ret (list_t.ListCons x0 tl00)
+    | list_t.ListNil => Result.fail Error.panic
+  | list_t.ListNil => Result.fail Error.panic
 termination_by list_nth_mut_loop_pair_loop_back'a ls0 ls1 i ret0 =>
   list_nth_mut_loop_pair_loop_terminates T ls0 ls1 i
 decreasing_by list_nth_mut_loop_pair_loop_decreases ls0 ls1 i
@@ -356,28 +357,28 @@ decreasing_by list_nth_mut_loop_pair_loop_decreases ls0 ls1 i
 /- [loops::list_nth_mut_loop_pair] -/
 def list_nth_mut_loop_pair_back'a
   (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : UInt32) (ret0 : T) :
-  result (list_t T)
+  Result (list_t T)
   :=
   list_nth_mut_loop_pair_loop_back'a T ls0 ls1 i ret0
 
 /- [loops::list_nth_mut_loop_pair] -/
 def list_nth_mut_loop_pair_loop_back'b
   (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : UInt32) (ret0 : T) :
-  (result (list_t T))
+  (Result (list_t T))
   :=
-  match ls0 with
+  match h: ls0 with
   | list_t.ListCons x0 tl0 =>
-    match ls1 with
+    match h: ls1 with
     | list_t.ListCons x1 tl1 =>
-      if i = (UInt32.ofNatCore 0 (by intlit))
-      then result.ret (list_t.ListCons ret0 tl1)
+      if h: i = (UInt32.ofNatCore 0 (by intlit))
+      then Result.ret (list_t.ListCons ret0 tl1)
       else
         do
-          let i0 <- UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
-          let tl10 <- list_nth_mut_loop_pair_loop_back'b T tl0 tl1 i0 ret0
-          result.ret (list_t.ListCons x1 tl10)
-    | list_t.ListNil => result.fail error.panic
-  | list_t.ListNil => result.fail error.panic
+          let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+          let tl10 ← list_nth_mut_loop_pair_loop_back'b T tl0 tl1 i0 ret0
+          Result.ret (list_t.ListCons x1 tl10)
+    | list_t.ListNil => Result.fail Error.panic
+  | list_t.ListNil => Result.fail Error.panic
 termination_by list_nth_mut_loop_pair_loop_back'b ls0 ls1 i ret0 =>
   list_nth_mut_loop_pair_loop_terminates T ls0 ls1 i
 decreasing_by list_nth_mut_loop_pair_loop_decreases ls0 ls1 i
@@ -385,27 +386,27 @@ decreasing_by list_nth_mut_loop_pair_loop_decreases ls0 ls1 i
 /- [loops::list_nth_mut_loop_pair] -/
 def list_nth_mut_loop_pair_back'b
   (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : UInt32) (ret0 : T) :
-  result (list_t T)
+  Result (list_t T)
   :=
   list_nth_mut_loop_pair_loop_back'b T ls0 ls1 i ret0
 
 /- [loops::list_nth_shared_loop_pair] -/
 def list_nth_shared_loop_pair_loop_fwd
   (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : UInt32) :
-  (result (T × T))
+  (Result (T × T))
   :=
-  match ls0 with
+  match h: ls0 with
   | list_t.ListCons x0 tl0 =>
-    match ls1 with
+    match h: ls1 with
     | list_t.ListCons x1 tl1 =>
-      if i = (UInt32.ofNatCore 0 (by intlit))
-      then result.ret (x0, x1)
+      if h: i = (UInt32.ofNatCore 0 (by intlit))
+      then Result.ret (x0, x1)
       else
         do
-          let i0 <- UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+          let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
           list_nth_shared_loop_pair_loop_fwd T tl0 tl1 i0
-    | list_t.ListNil => result.fail error.panic
-  | list_t.ListNil => result.fail error.panic
+    | list_t.ListNil => Result.fail Error.panic
+  | list_t.ListNil => Result.fail Error.panic
 termination_by list_nth_shared_loop_pair_loop_fwd ls0 ls1 i =>
   list_nth_shared_loop_pair_loop_terminates T ls0 ls1 i
 decreasing_by list_nth_shared_loop_pair_loop_decreases ls0 ls1 i
@@ -413,27 +414,27 @@ decreasing_by list_nth_shared_loop_pair_loop_decreases ls0 ls1 i
 /- [loops::list_nth_shared_loop_pair] -/
 def list_nth_shared_loop_pair_fwd
   (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : UInt32) :
-  result (T × T)
+  Result (T × T)
   :=
   list_nth_shared_loop_pair_loop_fwd T ls0 ls1 i
 
 /- [loops::list_nth_mut_loop_pair_merge] -/
 def list_nth_mut_loop_pair_merge_loop_fwd
   (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : UInt32) :
-  (result (T × T))
+  (Result (T × T))
   :=
-  match ls0 with
+  match h: ls0 with
   | list_t.ListCons x0 tl0 =>
-    match ls1 with
+    match h: ls1 with
     | list_t.ListCons x1 tl1 =>
-      if i = (UInt32.ofNatCore 0 (by intlit))
-      then result.ret (x0, x1)
+      if h: i = (UInt32.ofNatCore 0 (by intlit))
+      then Result.ret (x0, x1)
       else
         do
-          let i0 <- UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+          let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
           list_nth_mut_loop_pair_merge_loop_fwd T tl0 tl1 i0
-    | list_t.ListNil => result.fail error.panic
-  | list_t.ListNil => result.fail error.panic
+    | list_t.ListNil => Result.fail Error.panic
+  | list_t.ListNil => Result.fail Error.panic
 termination_by list_nth_mut_loop_pair_merge_loop_fwd ls0 ls1 i =>
   list_nth_mut_loop_pair_merge_loop_terminates T ls0 ls1 i
 decreasing_by list_nth_mut_loop_pair_merge_loop_decreases ls0 ls1 i
@@ -441,31 +442,31 @@ decreasing_by list_nth_mut_loop_pair_merge_loop_decreases ls0 ls1 i
 /- [loops::list_nth_mut_loop_pair_merge] -/
 def list_nth_mut_loop_pair_merge_fwd
   (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : UInt32) :
-  result (T × T)
+  Result (T × T)
   :=
   list_nth_mut_loop_pair_merge_loop_fwd T ls0 ls1 i
 
 /- [loops::list_nth_mut_loop_pair_merge] -/
 def list_nth_mut_loop_pair_merge_loop_back
   (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : UInt32) (ret0 : (T × T)) :
-  (result ((list_t T) × (list_t T)))
+  (Result ((list_t T) × (list_t T)))
   :=
-  match ls0 with
+  match h: ls0 with
   | list_t.ListCons x0 tl0 =>
-    match ls1 with
+    match h: ls1 with
     | list_t.ListCons x1 tl1 =>
-      if i = (UInt32.ofNatCore 0 (by intlit))
+      if h: i = (UInt32.ofNatCore 0 (by intlit))
       then
         let (t, t0) := ret0
-        result.ret (list_t.ListCons t tl0, list_t.ListCons t0 tl1)
+        Result.ret (list_t.ListCons t tl0, list_t.ListCons t0 tl1)
       else
         do
-          let i0 <- UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
-          let (tl00, tl10) <-
+          let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+          let (tl00, tl10) ←
             list_nth_mut_loop_pair_merge_loop_back T tl0 tl1 i0 ret0
-          result.ret (list_t.ListCons x0 tl00, list_t.ListCons x1 tl10)
-    | list_t.ListNil => result.fail error.panic
-  | list_t.ListNil => result.fail error.panic
+          Result.ret (list_t.ListCons x0 tl00, list_t.ListCons x1 tl10)
+    | list_t.ListNil => Result.fail Error.panic
+  | list_t.ListNil => Result.fail Error.panic
 termination_by list_nth_mut_loop_pair_merge_loop_back ls0 ls1 i ret0 =>
   list_nth_mut_loop_pair_merge_loop_terminates T ls0 ls1 i
 decreasing_by list_nth_mut_loop_pair_merge_loop_decreases ls0 ls1 i
@@ -473,27 +474,27 @@ decreasing_by list_nth_mut_loop_pair_merge_loop_decreases ls0 ls1 i
 /- [loops::list_nth_mut_loop_pair_merge] -/
 def list_nth_mut_loop_pair_merge_back
   (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : UInt32) (ret0 : (T × T)) :
-  result ((list_t T) × (list_t T))
+  Result ((list_t T) × (list_t T))
   :=
   list_nth_mut_loop_pair_merge_loop_back T ls0 ls1 i ret0
 
 /- [loops::list_nth_shared_loop_pair_merge] -/
 def list_nth_shared_loop_pair_merge_loop_fwd
   (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : UInt32) :
-  (result (T × T))
+  (Result (T × T))
   :=
-  match ls0 with
+  match h: ls0 with
   | list_t.ListCons x0 tl0 =>
-    match ls1 with
+    match h: ls1 with
     | list_t.ListCons x1 tl1 =>
-      if i = (UInt32.ofNatCore 0 (by intlit))
-      then result.ret (x0, x1)
+      if h: i = (UInt32.ofNatCore 0 (by intlit))
+      then Result.ret (x0, x1)
       else
         do
-          let i0 <- UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+          let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
           list_nth_shared_loop_pair_merge_loop_fwd T tl0 tl1 i0
-    | list_t.ListNil => result.fail error.panic
-  | list_t.ListNil => result.fail error.panic
+    | list_t.ListNil => Result.fail Error.panic
+  | list_t.ListNil => Result.fail Error.panic
 termination_by list_nth_shared_loop_pair_merge_loop_fwd ls0 ls1 i =>
   list_nth_shared_loop_pair_merge_loop_terminates T ls0 ls1 i
 decreasing_by list_nth_shared_loop_pair_merge_loop_decreases ls0 ls1 i
@@ -501,27 +502,27 @@ decreasing_by list_nth_shared_loop_pair_merge_loop_decreases ls0 ls1 i
 /- [loops::list_nth_shared_loop_pair_merge] -/
 def list_nth_shared_loop_pair_merge_fwd
   (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : UInt32) :
-  result (T × T)
+  Result (T × T)
   :=
   list_nth_shared_loop_pair_merge_loop_fwd T ls0 ls1 i
 
 /- [loops::list_nth_mut_shared_loop_pair] -/
 def list_nth_mut_shared_loop_pair_loop_fwd
   (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : UInt32) :
-  (result (T × T))
+  (Result (T × T))
   :=
-  match ls0 with
+  match h: ls0 with
   | list_t.ListCons x0 tl0 =>
-    match ls1 with
+    match h: ls1 with
     | list_t.ListCons x1 tl1 =>
-      if i = (UInt32.ofNatCore 0 (by intlit))
-      then result.ret (x0, x1)
+      if h: i = (UInt32.ofNatCore 0 (by intlit))
+      then Result.ret (x0, x1)
       else
         do
-          let i0 <- UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+          let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
           list_nth_mut_shared_loop_pair_loop_fwd T tl0 tl1 i0
-    | list_t.ListNil => result.fail error.panic
-  | list_t.ListNil => result.fail error.panic
+    | list_t.ListNil => Result.fail Error.panic
+  | list_t.ListNil => Result.fail Error.panic
 termination_by list_nth_mut_shared_loop_pair_loop_fwd ls0 ls1 i =>
   list_nth_mut_shared_loop_pair_loop_terminates T ls0 ls1 i
 decreasing_by list_nth_mut_shared_loop_pair_loop_decreases ls0 ls1 i
@@ -529,28 +530,29 @@ decreasing_by list_nth_mut_shared_loop_pair_loop_decreases ls0 ls1 i
 /- [loops::list_nth_mut_shared_loop_pair] -/
 def list_nth_mut_shared_loop_pair_fwd
   (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : UInt32) :
-  result (T × T)
+  Result (T × T)
   :=
   list_nth_mut_shared_loop_pair_loop_fwd T ls0 ls1 i
 
 /- [loops::list_nth_mut_shared_loop_pair] -/
 def list_nth_mut_shared_loop_pair_loop_back
   (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : UInt32) (ret0 : T) :
-  (result (list_t T))
+  (Result (list_t T))
   :=
-  match ls0 with
+  match h: ls0 with
   | list_t.ListCons x0 tl0 =>
-    match ls1 with
+    match h: ls1 with
     | list_t.ListCons x1 tl1 =>
-      if i = (UInt32.ofNatCore 0 (by intlit))
-      then result.ret (list_t.ListCons ret0 tl0)
+      if h: i = (UInt32.ofNatCore 0 (by intlit))
+      then Result.ret (list_t.ListCons ret0 tl0)
       else
         do
-          let i0 <- UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
-          let tl00 <- list_nth_mut_shared_loop_pair_loop_back T tl0 tl1 i0 ret0
-          result.ret (list_t.ListCons x0 tl00)
-    | list_t.ListNil => result.fail error.panic
-  | list_t.ListNil => result.fail error.panic
+          let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+          let tl00 ←
+            list_nth_mut_shared_loop_pair_loop_back T tl0 tl1 i0 ret0
+          Result.ret (list_t.ListCons x0 tl00)
+    | list_t.ListNil => Result.fail Error.panic
+  | list_t.ListNil => Result.fail Error.panic
 termination_by list_nth_mut_shared_loop_pair_loop_back ls0 ls1 i ret0 =>
   list_nth_mut_shared_loop_pair_loop_terminates T ls0 ls1 i
 decreasing_by list_nth_mut_shared_loop_pair_loop_decreases ls0 ls1 i
@@ -558,27 +560,27 @@ decreasing_by list_nth_mut_shared_loop_pair_loop_decreases ls0 ls1 i
 /- [loops::list_nth_mut_shared_loop_pair] -/
 def list_nth_mut_shared_loop_pair_back
   (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : UInt32) (ret0 : T) :
-  result (list_t T)
+  Result (list_t T)
   :=
   list_nth_mut_shared_loop_pair_loop_back T ls0 ls1 i ret0
 
 /- [loops::list_nth_mut_shared_loop_pair_merge] -/
 def list_nth_mut_shared_loop_pair_merge_loop_fwd
   (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : UInt32) :
-  (result (T × T))
+  (Result (T × T))
   :=
-  match ls0 with
+  match h: ls0 with
   | list_t.ListCons x0 tl0 =>
-    match ls1 with
+    match h: ls1 with
     | list_t.ListCons x1 tl1 =>
-      if i = (UInt32.ofNatCore 0 (by intlit))
-      then result.ret (x0, x1)
+      if h: i = (UInt32.ofNatCore 0 (by intlit))
+      then Result.ret (x0, x1)
       else
         do
-          let i0 <- UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+          let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
           list_nth_mut_shared_loop_pair_merge_loop_fwd T tl0 tl1 i0
-    | list_t.ListNil => result.fail error.panic
-  | list_t.ListNil => result.fail error.panic
+    | list_t.ListNil => Result.fail Error.panic
+  | list_t.ListNil => Result.fail Error.panic
 termination_by list_nth_mut_shared_loop_pair_merge_loop_fwd ls0 ls1 i =>
   list_nth_mut_shared_loop_pair_merge_loop_terminates T ls0 ls1 i
 decreasing_by list_nth_mut_shared_loop_pair_merge_loop_decreases ls0 ls1 i
@@ -586,29 +588,29 @@ decreasing_by list_nth_mut_shared_loop_pair_merge_loop_decreases ls0 ls1 i
 /- [loops::list_nth_mut_shared_loop_pair_merge] -/
 def list_nth_mut_shared_loop_pair_merge_fwd
   (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : UInt32) :
-  result (T × T)
+  Result (T × T)
   :=
   list_nth_mut_shared_loop_pair_merge_loop_fwd T ls0 ls1 i
 
 /- [loops::list_nth_mut_shared_loop_pair_merge] -/
 def list_nth_mut_shared_loop_pair_merge_loop_back
   (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : UInt32) (ret0 : T) :
-  (result (list_t T))
+  (Result (list_t T))
   :=
-  match ls0 with
+  match h: ls0 with
   | list_t.ListCons x0 tl0 =>
-    match ls1 with
+    match h: ls1 with
     | list_t.ListCons x1 tl1 =>
-      if i = (UInt32.ofNatCore 0 (by intlit))
-      then result.ret (list_t.ListCons ret0 tl0)
+      if h: i = (UInt32.ofNatCore 0 (by intlit))
+      then Result.ret (list_t.ListCons ret0 tl0)
       else
         do
-          let i0 <- UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
-          let tl00 <-
+          let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+          let tl00 ←
             list_nth_mut_shared_loop_pair_merge_loop_back T tl0 tl1 i0 ret0
-          result.ret (list_t.ListCons x0 tl00)
-    | list_t.ListNil => result.fail error.panic
-  | list_t.ListNil => result.fail error.panic
+          Result.ret (list_t.ListCons x0 tl00)
+    | list_t.ListNil => Result.fail Error.panic
+  | list_t.ListNil => Result.fail Error.panic
 termination_by list_nth_mut_shared_loop_pair_merge_loop_back ls0 ls1 i ret0 =>
   list_nth_mut_shared_loop_pair_merge_loop_terminates T ls0 ls1 i
 decreasing_by list_nth_mut_shared_loop_pair_merge_loop_decreases ls0 ls1 i
@@ -616,27 +618,27 @@ decreasing_by list_nth_mut_shared_loop_pair_merge_loop_decreases ls0 ls1 i
 /- [loops::list_nth_mut_shared_loop_pair_merge] -/
 def list_nth_mut_shared_loop_pair_merge_back
   (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : UInt32) (ret0 : T) :
-  result (list_t T)
+  Result (list_t T)
   :=
   list_nth_mut_shared_loop_pair_merge_loop_back T ls0 ls1 i ret0
 
 /- [loops::list_nth_shared_mut_loop_pair] -/
 def list_nth_shared_mut_loop_pair_loop_fwd
   (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : UInt32) :
-  (result (T × T))
+  (Result (T × T))
   :=
-  match ls0 with
+  match h: ls0 with
   | list_t.ListCons x0 tl0 =>
-    match ls1 with
+    match h: ls1 with
     | list_t.ListCons x1 tl1 =>
-      if i = (UInt32.ofNatCore 0 (by intlit))
-      then result.ret (x0, x1)
+      if h: i = (UInt32.ofNatCore 0 (by intlit))
+      then Result.ret (x0, x1)
       else
         do
-          let i0 <- UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+          let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
           list_nth_shared_mut_loop_pair_loop_fwd T tl0 tl1 i0
-    | list_t.ListNil => result.fail error.panic
-  | list_t.ListNil => result.fail error.panic
+    | list_t.ListNil => Result.fail Error.panic
+  | list_t.ListNil => Result.fail Error.panic
 termination_by list_nth_shared_mut_loop_pair_loop_fwd ls0 ls1 i =>
   list_nth_shared_mut_loop_pair_loop_terminates T ls0 ls1 i
 decreasing_by list_nth_shared_mut_loop_pair_loop_decreases ls0 ls1 i
@@ -644,28 +646,29 @@ decreasing_by list_nth_shared_mut_loop_pair_loop_decreases ls0 ls1 i
 /- [loops::list_nth_shared_mut_loop_pair] -/
 def list_nth_shared_mut_loop_pair_fwd
   (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : UInt32) :
-  result (T × T)
+  Result (T × T)
   :=
   list_nth_shared_mut_loop_pair_loop_fwd T ls0 ls1 i
 
 /- [loops::list_nth_shared_mut_loop_pair] -/
 def list_nth_shared_mut_loop_pair_loop_back
   (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : UInt32) (ret0 : T) :
-  (result (list_t T))
+  (Result (list_t T))
   :=
-  match ls0 with
+  match h: ls0 with
   | list_t.ListCons x0 tl0 =>
-    match ls1 with
+    match h: ls1 with
     | list_t.ListCons x1 tl1 =>
-      if i = (UInt32.ofNatCore 0 (by intlit))
-      then result.ret (list_t.ListCons ret0 tl1)
+      if h: i = (UInt32.ofNatCore 0 (by intlit))
+      then Result.ret (list_t.ListCons ret0 tl1)
       else
         do
-          let i0 <- UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
-          let tl10 <- list_nth_shared_mut_loop_pair_loop_back T tl0 tl1 i0 ret0
-          result.ret (list_t.ListCons x1 tl10)
-    | list_t.ListNil => result.fail error.panic
-  | list_t.ListNil => result.fail error.panic
+          let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+          let tl10 ←
+            list_nth_shared_mut_loop_pair_loop_back T tl0 tl1 i0 ret0
+          Result.ret (list_t.ListCons x1 tl10)
+    | list_t.ListNil => Result.fail Error.panic
+  | list_t.ListNil => Result.fail Error.panic
 termination_by list_nth_shared_mut_loop_pair_loop_back ls0 ls1 i ret0 =>
   list_nth_shared_mut_loop_pair_loop_terminates T ls0 ls1 i
 decreasing_by list_nth_shared_mut_loop_pair_loop_decreases ls0 ls1 i
@@ -673,27 +676,27 @@ decreasing_by list_nth_shared_mut_loop_pair_loop_decreases ls0 ls1 i
 /- [loops::list_nth_shared_mut_loop_pair] -/
 def list_nth_shared_mut_loop_pair_back
   (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : UInt32) (ret0 : T) :
-  result (list_t T)
+  Result (list_t T)
   :=
   list_nth_shared_mut_loop_pair_loop_back T ls0 ls1 i ret0
 
 /- [loops::list_nth_shared_mut_loop_pair_merge] -/
 def list_nth_shared_mut_loop_pair_merge_loop_fwd
   (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : UInt32) :
-  (result (T × T))
+  (Result (T × T))
   :=
-  match ls0 with
+  match h: ls0 with
   | list_t.ListCons x0 tl0 =>
-    match ls1 with
+    match h: ls1 with
     | list_t.ListCons x1 tl1 =>
-      if i = (UInt32.ofNatCore 0 (by intlit))
-      then result.ret (x0, x1)
+      if h: i = (UInt32.ofNatCore 0 (by intlit))
+      then Result.ret (x0, x1)
       else
         do
-          let i0 <- UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+          let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
           list_nth_shared_mut_loop_pair_merge_loop_fwd T tl0 tl1 i0
-    | list_t.ListNil => result.fail error.panic
-  | list_t.ListNil => result.fail error.panic
+    | list_t.ListNil => Result.fail Error.panic
+  | list_t.ListNil => Result.fail Error.panic
 termination_by list_nth_shared_mut_loop_pair_merge_loop_fwd ls0 ls1 i =>
   list_nth_shared_mut_loop_pair_merge_loop_terminates T ls0 ls1 i
 decreasing_by list_nth_shared_mut_loop_pair_merge_loop_decreases ls0 ls1 i
@@ -701,29 +704,29 @@ decreasing_by list_nth_shared_mut_loop_pair_merge_loop_decreases ls0 ls1 i
 /- [loops::list_nth_shared_mut_loop_pair_merge] -/
 def list_nth_shared_mut_loop_pair_merge_fwd
   (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : UInt32) :
-  result (T × T)
+  Result (T × T)
   :=
   list_nth_shared_mut_loop_pair_merge_loop_fwd T ls0 ls1 i
 
 /- [loops::list_nth_shared_mut_loop_pair_merge] -/
 def list_nth_shared_mut_loop_pair_merge_loop_back
   (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : UInt32) (ret0 : T) :
-  (result (list_t T))
+  (Result (list_t T))
   :=
-  match ls0 with
+  match h: ls0 with
   | list_t.ListCons x0 tl0 =>
-    match ls1 with
+    match h: ls1 with
     | list_t.ListCons x1 tl1 =>
-      if i = (UInt32.ofNatCore 0 (by intlit))
-      then result.ret (list_t.ListCons ret0 tl1)
+      if h: i = (UInt32.ofNatCore 0 (by intlit))
+      then Result.ret (list_t.ListCons ret0 tl1)
       else
         do
-          let i0 <- UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
-          let tl10 <-
+          let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+          let tl10 ←
             list_nth_shared_mut_loop_pair_merge_loop_back T tl0 tl1 i0 ret0
-          result.ret (list_t.ListCons x1 tl10)
-    | list_t.ListNil => result.fail error.panic
-  | list_t.ListNil => result.fail error.panic
+          Result.ret (list_t.ListCons x1 tl10)
+    | list_t.ListNil => Result.fail Error.panic
+  | list_t.ListNil => Result.fail Error.panic
 termination_by list_nth_shared_mut_loop_pair_merge_loop_back ls0 ls1 i ret0 =>
   list_nth_shared_mut_loop_pair_merge_loop_terminates T ls0 ls1 i
 decreasing_by list_nth_shared_mut_loop_pair_merge_loop_decreases ls0 ls1 i
@@ -731,7 +734,7 @@ decreasing_by list_nth_shared_mut_loop_pair_merge_loop_decreases ls0 ls1 i
 /- [loops::list_nth_shared_mut_loop_pair_merge] -/
 def list_nth_shared_mut_loop_pair_merge_back
   (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : UInt32) (ret0 : T) :
-  result (list_t T)
+  Result (list_t T)
   :=
   list_nth_shared_mut_loop_pair_merge_loop_back T ls0 ls1 i ret0
 
diff --git a/tests/lean/misc/no_nested_borrows/Base/Primitives.lean b/tests/lean/misc/no_nested_borrows/Base/Primitives.lean
index 79958d94..5b64e908 100644
--- a/tests/lean/misc/no_nested_borrows/Base/Primitives.lean
+++ b/tests/lean/misc/no_nested_borrows/Base/Primitives.lean
@@ -9,74 +9,79 @@ import Mathlib.Tactic.RunCmd
 
 -- Results & monadic combinators
 
--- TODO: use syntactic conventions and capitalize error, result, etc.
-
-inductive error where
-   | assertionFailure: error
-   | integerOverflow: error
-   | arrayOutOfBounds: error
-   | maximumSizeExceeded: error
-   | panic: error
+inductive Error where
+   | assertionFailure: Error
+   | integerOverflow: Error
+   | arrayOutOfBounds: Error
+   | maximumSizeExceeded: Error
+   | panic: Error
 deriving Repr, BEq
 
-open error
+open Error
 
-inductive result (α : Type u) where
-  | ret (v: α): result α
-  | fail (e: error): result α 
+inductive Result (α : Type u) where
+  | ret (v: α): Result α
+  | fail (e: Error): Result α
 deriving Repr, BEq
 
-open result
+open Result
 
 /- HELPERS -/
 
--- TODO: is there automated syntax for these discriminators?
-def is_ret {α: Type} (r: result α): Bool :=
+def ret? {α: Type} (r: Result α): Bool :=
   match r with
-  | result.ret _ => true
-  | result.fail _ => false
+  | Result.ret _ => true
+  | Result.fail _ => false
 
-def massert (b:Bool) : result Unit :=
+def massert (b:Bool) : Result Unit :=
   if b then .ret () else fail assertionFailure
 
-def eval_global {α: Type} (x: result α) (_: is_ret x): α :=
+def eval_global {α: Type} (x: Result α) (_: ret? x): α :=
   match x with
-  | result.fail _ => by contradiction
-  | result.ret x => x
+  | Result.fail _ => by contradiction
+  | Result.ret x => x
 
 /- DO-DSL SUPPORT -/
 
-def bind (x: result α) (f: α -> result β) : result β :=
+def bind (x: Result α) (f: α -> Result β) : Result β :=
   match x with
   | ret v  => f v 
   | fail v => fail v
 
--- Allows using result in do-blocks
-instance : Bind result where
+-- Allows using Result in do-blocks
+instance : Bind Result where
   bind := bind
 
 -- Allows using return x in do-blocks
-instance : Pure result where
+instance : Pure Result where
   pure := fun x => ret x
 
 /- CUSTOM-DSL SUPPORT -/
 
--- Let-binding the result of a monadic operation is oftentimes not sufficient,
+-- Let-binding the Result of a monadic operation is oftentimes not sufficient,
 -- because we may need a hypothesis for equational reasoning in the scope. We
 -- rely on subtype, and a custom let-binding operator, in effect recreating our
 -- own variant of the do-dsl
 
-def result.attach : (o : result α) → result { x : α // o = ret x }
+def Result.attach {α: Type} (o : Result α): Result { x : α // o = ret x } :=
+  match o with
   | .ret x => .ret ⟨x, rfl⟩
-  | .fail e   => .fail e
+  | .fail e => .fail e
 
-macro "let" h:ident " : " e:term " <-- " f:term : doElem =>
-  `(doElem| let ⟨$e, $h⟩ ← result.attach $f)
+macro "let" e:term " ⟵ " f:term : doElem =>
+  `(doElem| let ⟨$e, h⟩ ← Result.attach $f)
 
--- Silly example of the kind of reasoning that this notation enables
+-- TODO: any way to factorize both definitions?
+macro "let" e:term " <-- " f:term : doElem =>
+  `(doElem| let ⟨$e, h⟩ ← Result.attach $f)
+
+-- We call the hypothesis `h`, in effect making it unavailable to the user
+-- (because too much shadowing). But in practice, once can use the French single
+-- quote notation (input with f< and f>), where `‹ h ›` finds a suitable
+-- hypothesis in the context, this is equivalent to `have x: h := by assumption in x`
 #eval do
-  let h: y <-- .ret (0: Nat)
-  let _: y = 0 := by cases h; decide
+  let y <-- .ret (0: Nat)
+  let _: y = 0 := by cases ‹ ret 0 = ret y › ; decide
   let r: { x: Nat // x = 0 } := ⟨ y, by assumption ⟩
   .ret r
 
@@ -84,36 +89,27 @@ macro "let" h:ident " : " e:term " <-- " f:term : doElem =>
 -- MACHINE INTEGERS --
 ----------------------
 
--- NOTE: we reuse the USize type from prelude.lean, because at least we know
--- it's defined in an idiomatic style that is going to make proofs easy (and
--- indeed, several proofs here are much shortened compared to Aymeric's earlier
--- attempt.) This is not stricto sensu the *correct* thing to do, because one
--- can query at run-time the value of USize, which we do *not* want to do (we
--- don't know what target we'll run on!), but when the day comes, we'll just
--- define our own USize.
--- ANOTHER NOTE: there is USize.sub but it has wraparound semantics, which is
--- not something we want to define (I think), so we use our own monadic sub (but
--- is it in line with the Rust behavior?)
-
--- TODO: I am somewhat under the impression that subtraction is defined as a
--- total function over nats...? the hypothesis in the if condition is not used
--- in the then-branch which confuses me quite a bit
-
--- TODO: add a refinement for the result (just like vec_push_back below) that
--- explains that the toNat of the result (in the case of success) is the sub of
--- the toNat of the arguments (i.e. intrinsic specification)
--- ... do we want intrinsic specifications for the builtins? that might require
--- some careful type annotations in the monadic notation for clients, but may
--- give us more "for free"
+-- NOTE: we reuse the fixed-width integer types from prelude.lean: UInt8, ...,
+-- USize. They are generally defined in an idiomatic style, except that there is
+-- not a single type class to rule them all (more on that below). The absence of
+-- type class is intentional, and allows the Lean compiler to efficiently map
+-- them to machine integers during compilation.
+
+-- USize is designed properly: you cannot reduce `getNumBits` using the
+-- simplifier, meaning that proofs do not depend on the compile-time value of
+-- USize.size. (Lean assumes 32 or 64-bit platforms, and Rust doesn't really
+-- support, at least officially, 16-bit microcontrollers, so this seems like a
+-- fine design decision for now.)
 
 -- Note from Chris Bailey: "If there's more than one salient property of your
 -- definition then the subtyping strategy might get messy, and the property part
 -- of a subtype is less discoverable by the simplifier or tactics like
--- library_search." Try to settle this with a Lean expert on what is the most
--- productive way to go about this?
+-- library_search." So, we will not add refinements on the return values of the
+-- operations defined on Primitives, but will rather rely on custom lemmas to
+-- invert on possible return values of the primitive operations.
 
--- One needs to perform a little bit of reasoning in order to successfully
--- inject constants into USize, so we provide a general-purpose macro
+-- Machine integer constants, done via `ofNatCore`, which requires a proof that
+-- the `Nat` fits within the desired integer type. We provide a custom tactic.
 
 syntax "intlit" : tactic
 
@@ -129,12 +125,21 @@ macro_rules
 -- Also works for other integer types (at the expense of a needless disjunction)
 #eval UInt32.ofNatCore 0 (by intlit)
 
+-- The machine integer operations (e.g. sub) are always total, which is not what
+-- we want. We therefore define "checked" variants, below. Note that we add a
+-- tiny bit of complexity for the USize variant: we first check whether the
+-- result is < 2^32; if it is, we can compute the definition, rather than
+-- returning a term that is computationally stuck (the comparison to USize.size
+-- cannot reduce at compile-time, per the remark about regarding `getNumBits`).
+-- This is useful for the various #asserts that we want to reduce at
+-- type-checking time.
+
 -- Further thoughts: look at what has been done here:
 -- https://github.com/leanprover-community/mathlib4/blob/master/Mathlib/Data/Fin/Basic.lean
 -- and
 -- https://github.com/leanprover-community/mathlib4/blob/master/Mathlib/Data/UInt.lean
 -- which both contain a fair amount of reasoning already!
-def USize.checked_sub (n: USize) (m: USize): result USize :=
+def USize.checked_sub (n: USize) (m: USize): Result USize :=
   -- NOTE: the test USize.toNat n - m >= 0 seems to always succeed?
   if n >= m then
     let n' := USize.toNat n
@@ -150,18 +155,19 @@ def USize.checked_sub (n: USize) (m: USize): result USize :=
   else
     fail integerOverflow
 
-def USize.checked_add (n: USize) (m: USize): result USize :=
-  if h: n.val.val + m.val.val <= 4294967295 then
-    .ret ⟨ n.val.val + m.val.val, by
-      have h': 4294967295 < USize.size := by intlit
-      apply Nat.lt_of_le_of_lt h h'
-    ⟩
-  else if h: n.val + m.val < USize.size then
+@[simp]
+theorem usize_fits (n: Nat) (h: n <= 4294967295): n < USize.size :=
+  match USize.size, usize_size_eq with
+  | _, Or.inl rfl => Nat.lt_of_le_of_lt h (by decide)
+  | _, Or.inr rfl => Nat.lt_of_le_of_lt h (by decide)
+
+def USize.checked_add (n: USize) (m: USize): Result USize :=
+  if h: n.val + m.val < USize.size then
     .ret ⟨ n.val + m.val, h ⟩
   else
     .fail integerOverflow
 
-def USize.checked_rem (n: USize) (m: USize): result USize :=
+def USize.checked_rem (n: USize) (m: USize): Result USize :=
   if h: m > 0 then
     .ret ⟨ n.val % m.val, by
       have h1: ↑m.val < USize.size := m.val.isLt
@@ -171,18 +177,13 @@ def USize.checked_rem (n: USize) (m: USize): result USize :=
   else
     .fail integerOverflow
 
-def USize.checked_mul (n: USize) (m: USize): result USize :=
-    if h: n.val.val * m.val.val <= 4294967295 then
-    .ret ⟨ n.val.val * m.val.val, by
-      have h': 4294967295 < USize.size := by intlit
-      apply Nat.lt_of_le_of_lt h h'
-    ⟩
-  else if h: n.val * m.val < USize.size then
+def USize.checked_mul (n: USize) (m: USize): Result USize :=
+  if h: n.val * m.val < USize.size then
     .ret ⟨ n.val * m.val, h ⟩
   else
     .fail integerOverflow
 
-def USize.checked_div (n: USize) (m: USize): result USize :=
+def USize.checked_div (n: USize) (m: USize): Result USize :=
   if m > 0 then
     .ret ⟨ n.val / m.val, by
       have h1: ↑n.val < USize.size := n.val.isLt
@@ -192,6 +193,19 @@ def USize.checked_div (n: USize) (m: USize): result USize :=
   else
     .fail integerOverflow
 
+-- Test behavior...
+#eval assert! USize.checked_sub 10 20 == fail integerOverflow; 0
+
+#eval USize.checked_sub 20 10
+-- NOTE: compare with concrete behavior here, which I do not think we want
+#eval USize.sub 0 1
+#eval UInt8.add 255 255
+
+-- We now define a type class that subsumes the various machine integer types, so
+-- as to write a concise definition for scalar_cast, rather than exhaustively
+-- enumerating all of the possible pairs. We remark that Rust has sane semantics
+-- and fails if a cast operation would involve a truncation or modulo.
+
 class MachineInteger (t: Type) where
   size: Nat
   val: t -> Fin size
@@ -209,30 +223,24 @@ run_cmd
     end $typeName
   ))
 
-def scalar_cast { src: Type } (dst: Type) [ MachineInteger src ] [ MachineInteger dst ] (x: src): result dst :=
+-- Aeneas only instantiates the destination type (`src` is implicit). We rely on
+-- Lean to infer `src`.
+
+def scalar_cast { src: Type } (dst: Type) [ MachineInteger src ] [ MachineInteger dst ] (x: src): Result dst :=
   if h: MachineInteger.val x < MachineInteger.size dst then
     .ret (MachineInteger.ofNatCore (MachineInteger.val x).val h)
   else
     .fail integerOverflow
 
-
--- Test behavior...
-#eval assert! USize.checked_sub 10 20 == fail integerOverflow; 0
-
-#eval USize.checked_sub 20 10
--- NOTE: compare with concrete behavior here, which I do not think we want
-#eval USize.sub 0 1
-#eval UInt8.add 255 255
-
 -------------
 -- VECTORS --
 -------------
 
 -- Note: unlike F*, Lean seems to use strict upper bounds (e.g. USize.size)
 -- rather than maximum values (usize_max).
-def vec (α : Type u) := { l : List α // List.length l < USize.size }
+def Vec (α : Type u) := { l : List α // List.length l < USize.size }
 
-def vec_new (α : Type u): vec α := ⟨ [], by {
+def vec_new (α : Type u): Vec α := ⟨ [], by {
   match USize.size, usize_size_eq with
   | _, Or.inl rfl => simp
   | _, Or.inr rfl => simp
@@ -240,20 +248,20 @@ def vec_new (α : Type u): vec α := ⟨ [], by {
 
 #check vec_new
 
-def vec_len (α : Type u) (v : vec α) : USize :=
+def vec_len (α : Type u) (v : Vec α) : USize :=
   let ⟨ v, l ⟩ := v
   USize.ofNatCore (List.length v) l
 
 #eval vec_len Nat (vec_new Nat)
  
-def vec_push_fwd (α : Type u) (_ : vec α) (_ : α) : Unit := ()
+def vec_push_fwd (α : Type u) (_ : Vec α) (_ : α) : Unit := ()
 
 -- NOTE: old version trying to use a subtype notation, but probably better to
--- leave result elimination to auxiliary lemmas with suitable preconditions
+-- leave Result elimination to auxiliary lemmas with suitable preconditions
 -- TODO: I originally wrote `List.length v.val < USize.size - 1`; how can one
 -- make the proof work in that case? Probably need to import tactics from
 -- mathlib to deal with inequalities... would love to see an example.
-def vec_push_back_old (α : Type u) (v : vec α) (x : α) : { res: result (vec α) //
+def vec_push_back_old (α : Type u) (v : Vec α) (x : α) : { res: Result (Vec α) //
   match res with | fail _ => True | ret v' => List.length v'.val = List.length v.val + 1}
   :=
   if h : List.length v.val + 1 < USize.size then
@@ -272,12 +280,12 @@ def vec_push_back_old (α : Type u) (v : vec α) (x : α) : { res: result (vec �
   -- annotate `x`, which relieves us of having to write `.val` on the right-hand
   -- side of the monadic let.
   let v := vec_new Nat
-  let x: vec Nat ← (vec_push_back_old Nat v 1: result (vec Nat)) -- WHY do we need the type annotation here?
+  let x: Vec Nat ← (vec_push_back_old Nat v 1: Result (Vec Nat)) -- WHY do we need the type annotation here?
   -- TODO: strengthen post-condition above and do a demo to show that we can
   -- safely eliminate the `fail` case
   return (vec_len Nat x)
 
-def vec_push_back (α : Type u) (v : vec α) (x : α) : result (vec α)
+def vec_push_back (α : Type u) (v : Vec α) (x : α) : Result (Vec α)
   :=
   if h : List.length v.val + 1 <= 4294967295 then
     return ⟨ List.concat v.val x,
@@ -295,13 +303,13 @@ def vec_push_back (α : Type u) (v : vec α) (x : α) : result (vec α)
   else
     fail maximumSizeExceeded
 
-def vec_insert_fwd (α : Type u) (v: vec α) (i: USize) (_: α): result Unit :=
+def vec_insert_fwd (α : Type u) (v: Vec α) (i: USize) (_: α): Result Unit :=
   if i.val < List.length v.val then
     .ret ()
   else
     .fail arrayOutOfBounds
 
-def vec_insert_back (α : Type u) (v: vec α) (i: USize) (x: α): result (vec α) :=
+def vec_insert_back (α : Type u) (v: Vec α) (i: USize) (x: α): Result (Vec α) :=
   if i.val < List.length v.val then
     .ret ⟨ List.set v.val i.val x, by
       have h: List.length v.val < USize.size := v.property
@@ -311,25 +319,25 @@ def vec_insert_back (α : Type u) (v: vec α) (i: USize) (x: α): result (vec α
   else
     .fail arrayOutOfBounds
 
-def vec_index_fwd (α : Type u) (v: vec α) (i: USize): result α :=
+def vec_index_fwd (α : Type u) (v: Vec α) (i: USize): Result α :=
   if h: i.val < List.length v.val then
     .ret (List.get v.val ⟨i.val, h⟩)
   else
     .fail arrayOutOfBounds
 
-def vec_index_back (α : Type u) (v: vec α) (i: USize) (_: α): result Unit :=
+def vec_index_back (α : Type u) (v: Vec α) (i: USize) (_: α): Result Unit :=
   if i.val < List.length v.val then
     .ret ()
   else
     .fail arrayOutOfBounds
 
-def vec_index_mut_fwd (α : Type u) (v: vec α) (i: USize): result α :=
+def vec_index_mut_fwd (α : Type u) (v: Vec α) (i: USize): Result α :=
   if h: i.val < List.length v.val then
     .ret (List.get v.val ⟨i.val, h⟩)
   else
     .fail arrayOutOfBounds
 
-def vec_index_mut_back (α : Type u) (v: vec α) (i: USize) (x: α): result (vec α) :=
+def vec_index_mut_back (α : Type u) (v: Vec α) (i: USize) (x: α): Result (Vec α) :=
   if i.val < List.length v.val then
     .ret ⟨ List.set v.val i.val x, by
       have h: List.length v.val < USize.size := v.property
@@ -349,6 +357,10 @@ def mem_replace_fwd (a : Type) (x : a) (_ : a) : a :=
 def mem_replace_back (a : Type) (_ : a) (y : a) : a :=
   y
 
+/-- Aeneas-translated function -- useful to reduce non-recursive definitions.
+ Use with `simp [ aeneas ]` -/
+register_simp_attr aeneas
+
 --------------------
 -- ASSERT COMMAND --
 --------------------
@@ -358,16 +370,23 @@ open Lean Elab Command Term Meta
 syntax (name := assert) "#assert" term: command
 
 @[command_elab assert]
+unsafe
 def assertImpl : CommandElab := fun (_stx: Syntax) => do
-  logInfo "Reducing and asserting: "
-  logInfo _stx[1]
   runTermElabM (fun _ => do
-    let e ← Term.elabTerm _stx[1] none
-    logInfo (Expr.dbgToString e)
-    -- How to evaluate the term and compare the result to true?
+    let r ← evalTerm Bool (mkConst ``Bool) _stx[1]
+    if not r then
+      logInfo "Assertion failed for: "
+      logInfo _stx[1]
+      logError "Expression reduced to false"
     pure ())
-  -- logInfo (Expr.dbgToString (``true))
-  -- throwError "TODO: assert"
 
 #eval 2 == 2
 #assert (2 == 2)
+
+-------------------
+-- SANITY CHECKS --
+-------------------
+
+-- TODO: add more once we have signed integers
+
+#assert (USize.checked_rem 1 2 == .ret 1)
diff --git a/tests/lean/misc/no_nested_borrows/NoNestedBorrows.lean b/tests/lean/misc/no_nested_borrows/NoNestedBorrows.lean
index 608aabc1..a20ee9fd 100644
--- a/tests/lean/misc/no_nested_borrows/NoNestedBorrows.lean
+++ b/tests/lean/misc/no_nested_borrows/NoNestedBorrows.lean
@@ -30,232 +30,234 @@ structure OpaqueDefs where
   | SumRight : T2 -> sum_t T1 T2
   
   /- [no_nested_borrows::neg_test] -/
-  def neg_test_fwd (x : Int32) : result Int32 :=
+  def neg_test_fwd (x : Int32) : Result Int32 :=
     Int32.checked_neg x
   
   /- [no_nested_borrows::add_test] -/
-  def add_test_fwd (x : UInt32) (y : UInt32) : result UInt32 :=
+  def add_test_fwd (x : UInt32) (y : UInt32) : Result UInt32 :=
     UInt32.checked_add x y
   
   /- [no_nested_borrows::subs_test] -/
-  def subs_test_fwd (x : UInt32) (y : UInt32) : result UInt32 :=
+  def subs_test_fwd (x : UInt32) (y : UInt32) : Result UInt32 :=
     UInt32.checked_sub x y
   
   /- [no_nested_borrows::div_test] -/
-  def div_test_fwd (x : UInt32) (y : UInt32) : result UInt32 :=
+  def div_test_fwd (x : UInt32) (y : UInt32) : Result UInt32 :=
     UInt32.checked_div x y
   
   /- [no_nested_borrows::div_test1] -/
-  def div_test1_fwd (x : UInt32) : result UInt32 :=
+  def div_test1_fwd (x : UInt32) : Result UInt32 :=
     UInt32.checked_div x (UInt32.ofNatCore 2 (by intlit))
   
   /- [no_nested_borrows::rem_test] -/
-  def rem_test_fwd (x : UInt32) (y : UInt32) : result UInt32 :=
+  def rem_test_fwd (x : UInt32) (y : UInt32) : Result UInt32 :=
     UInt32.checked_rem x y
   
   /- [no_nested_borrows::cast_test] -/
-  def cast_test_fwd (x : UInt32) : result Int32 :=
+  def cast_test_fwd (x : UInt32) : Result Int32 :=
     scalar_cast Int32 x
   
   /- [no_nested_borrows::test2] -/
-  def test2_fwd : result Unit :=
+  def test2_fwd : Result Unit :=
     do
-      let _ <- UInt32.checked_add (UInt32.ofNatCore 23 (by intlit))
+      let _ ← UInt32.checked_add (UInt32.ofNatCore 23 (by intlit))
         (UInt32.ofNatCore 44 (by intlit))
-      result.ret ()
+      Result.ret ()
   
   /- Unit test for [no_nested_borrows::test2] -/
-  #assert (test2_fwd = .ret ())
+  #assert (test2_fwd == .ret ())
   
   /- [no_nested_borrows::get_max] -/
-  def get_max_fwd (x : UInt32) (y : UInt32) : result UInt32 :=
-    if x >= y
-    then result.ret x
-    else result.ret y
+  def get_max_fwd (x : UInt32) (y : UInt32) : Result UInt32 :=
+    if h: x >= y
+    then Result.ret x
+    else Result.ret y
   
   /- [no_nested_borrows::test3] -/
-  def test3_fwd : result Unit :=
+  def test3_fwd : Result Unit :=
     do
-      let x <-
+      let x ←
         get_max_fwd (UInt32.ofNatCore 4 (by intlit))
           (UInt32.ofNatCore 3 (by intlit))
-      let y <-
+      let y ←
         get_max_fwd (UInt32.ofNatCore 10 (by intlit))
           (UInt32.ofNatCore 11 (by intlit))
-      let z <- UInt32.checked_add x y
-      if not (z = (UInt32.ofNatCore 15 (by intlit)))
-      then result.fail error.panic
-      else result.ret ()
+      let z ← UInt32.checked_add x y
+      if h: not (z = (UInt32.ofNatCore 15 (by intlit)))
+      then Result.fail Error.panic
+      else Result.ret ()
   
   /- Unit test for [no_nested_borrows::test3] -/
-  #assert (test3_fwd = .ret ())
+  #assert (test3_fwd == .ret ())
   
   /- [no_nested_borrows::test_neg1] -/
-  def test_neg1_fwd : result Unit :=
+  def test_neg1_fwd : Result Unit :=
     do
-      let y <- Int32.checked_neg (Int32.ofNatCore 3 (by intlit))
-      if not (y = (Int32.ofNatCore -3 (by intlit)))
-      then result.fail error.panic
-      else result.ret ()
+      let y ← Int32.checked_neg (Int32.ofNatCore 3 (by intlit))
+      if h: not (y = (Int32.ofNatCore -3 (by intlit)))
+      then Result.fail Error.panic
+      else Result.ret ()
   
   /- Unit test for [no_nested_borrows::test_neg1] -/
-  #assert (test_neg1_fwd = .ret ())
+  #assert (test_neg1_fwd == .ret ())
   
   /- [no_nested_borrows::refs_test1] -/
-  def refs_test1_fwd : result Unit :=
-    if not ((Int32.ofNatCore 1 (by intlit)) = (Int32.ofNatCore 1 (by intlit)))
-    then result.fail error.panic
-    else result.ret ()
+  def refs_test1_fwd : Result Unit :=
+    if h: not ((Int32.ofNatCore 1 (by intlit)) =
+      (Int32.ofNatCore 1 (by intlit)))
+    then Result.fail Error.panic
+    else Result.ret ()
   
   /- Unit test for [no_nested_borrows::refs_test1] -/
-  #assert (refs_test1_fwd = .ret ())
+  #assert (refs_test1_fwd == .ret ())
   
   /- [no_nested_borrows::refs_test2] -/
-  def refs_test2_fwd : result Unit :=
-    if not ((Int32.ofNatCore 2 (by intlit)) = (Int32.ofNatCore 2 (by intlit)))
-    then result.fail error.panic
+  def refs_test2_fwd : Result Unit :=
+    if h: not ((Int32.ofNatCore 2 (by intlit)) =
+      (Int32.ofNatCore 2 (by intlit)))
+    then Result.fail Error.panic
     else
-      if not ((Int32.ofNatCore 0 (by intlit)) =
+      if h: not ((Int32.ofNatCore 0 (by intlit)) =
         (Int32.ofNatCore 0 (by intlit)))
-      then result.fail error.panic
+      then Result.fail Error.panic
       else
-        if not ((Int32.ofNatCore 2 (by intlit)) =
+        if h: not ((Int32.ofNatCore 2 (by intlit)) =
           (Int32.ofNatCore 2 (by intlit)))
-        then result.fail error.panic
+        then Result.fail Error.panic
         else
-          if not ((Int32.ofNatCore 2 (by intlit)) =
+          if h: not ((Int32.ofNatCore 2 (by intlit)) =
             (Int32.ofNatCore 2 (by intlit)))
-          then result.fail error.panic
-          else result.ret ()
+          then Result.fail Error.panic
+          else Result.ret ()
   
   /- Unit test for [no_nested_borrows::refs_test2] -/
-  #assert (refs_test2_fwd = .ret ())
+  #assert (refs_test2_fwd == .ret ())
   
   /- [no_nested_borrows::test_list1] -/
-  def test_list1_fwd : result Unit :=
-    result.ret ()
+  def test_list1_fwd : Result Unit :=
+    Result.ret ()
   
   /- Unit test for [no_nested_borrows::test_list1] -/
-  #assert (test_list1_fwd = .ret ())
+  #assert (test_list1_fwd == .ret ())
   
   /- [no_nested_borrows::test_box1] -/
-  def test_box1_fwd : result Unit :=
+  def test_box1_fwd : Result Unit :=
     let b := (Int32.ofNatCore 1 (by intlit))
     let x := b
-    if not (x = (Int32.ofNatCore 1 (by intlit)))
-    then result.fail error.panic
-    else result.ret ()
+    if h: not (x = (Int32.ofNatCore 1 (by intlit)))
+    then Result.fail Error.panic
+    else Result.ret ()
   
   /- Unit test for [no_nested_borrows::test_box1] -/
-  #assert (test_box1_fwd = .ret ())
+  #assert (test_box1_fwd == .ret ())
   
   /- [no_nested_borrows::copy_int] -/
-  def copy_int_fwd (x : Int32) : result Int32 :=
-    result.ret x
+  def copy_int_fwd (x : Int32) : Result Int32 :=
+    Result.ret x
   
   /- [no_nested_borrows::test_unreachable] -/
-  def test_unreachable_fwd (b : Bool) : result Unit :=
-    if b
-    then result.fail error.panic
-    else result.ret ()
+  def test_unreachable_fwd (b : Bool) : Result Unit :=
+    if h: b
+    then Result.fail Error.panic
+    else Result.ret ()
   
   /- [no_nested_borrows::test_panic] -/
-  def test_panic_fwd (b : Bool) : result Unit :=
-    if b
-    then result.fail error.panic
-    else result.ret ()
+  def test_panic_fwd (b : Bool) : Result Unit :=
+    if h: b
+    then Result.fail Error.panic
+    else Result.ret ()
   
   /- [no_nested_borrows::test_copy_int] -/
-  def test_copy_int_fwd : result Unit :=
+  def test_copy_int_fwd : Result Unit :=
     do
-      let y <- copy_int_fwd (Int32.ofNatCore 0 (by intlit))
-      if not ((Int32.ofNatCore 0 (by intlit)) = y)
-      then result.fail error.panic
-      else result.ret ()
+      let y ← copy_int_fwd (Int32.ofNatCore 0 (by intlit))
+      if h: not ((Int32.ofNatCore 0 (by intlit)) = y)
+      then Result.fail Error.panic
+      else Result.ret ()
   
   /- Unit test for [no_nested_borrows::test_copy_int] -/
-  #assert (test_copy_int_fwd = .ret ())
+  #assert (test_copy_int_fwd == .ret ())
   
   /- [no_nested_borrows::is_cons] -/
-  def is_cons_fwd (T : Type) (l : list_t T) : result Bool :=
-    match l with
-    | list_t.ListCons t l0 => result.ret true
-    | list_t.ListNil => result.ret false
+  def is_cons_fwd (T : Type) (l : list_t T) : Result Bool :=
+    match h: l with
+    | list_t.ListCons t l0 => Result.ret true
+    | list_t.ListNil => Result.ret false
   
   /- [no_nested_borrows::test_is_cons] -/
-  def test_is_cons_fwd : result Unit :=
+  def test_is_cons_fwd : Result Unit :=
     do
       let l := list_t.ListNil
-      let b <-
+      let b ←
         is_cons_fwd Int32 (list_t.ListCons (Int32.ofNatCore 0 (by intlit)) l)
-      if not b
-      then result.fail error.panic
-      else result.ret ()
+      if h: not b
+      then Result.fail Error.panic
+      else Result.ret ()
   
   /- Unit test for [no_nested_borrows::test_is_cons] -/
-  #assert (test_is_cons_fwd = .ret ())
+  #assert (test_is_cons_fwd == .ret ())
   
   /- [no_nested_borrows::split_list] -/
-  def split_list_fwd (T : Type) (l : list_t T) : result (T × (list_t T)) :=
-    match l with
-    | list_t.ListCons hd tl => result.ret (hd, tl)
-    | list_t.ListNil => result.fail error.panic
+  def split_list_fwd (T : Type) (l : list_t T) : Result (T × (list_t T)) :=
+    match h: l with
+    | list_t.ListCons hd tl => Result.ret (hd, tl)
+    | list_t.ListNil => Result.fail Error.panic
   
   /- [no_nested_borrows::test_split_list] -/
-  def test_split_list_fwd : result Unit :=
+  def test_split_list_fwd : Result Unit :=
     do
       let l := list_t.ListNil
-      let p <-
+      let p ←
         split_list_fwd Int32 (list_t.ListCons (Int32.ofNatCore 0 (by intlit))
           l)
       let (hd, _) := p
-      if not (hd = (Int32.ofNatCore 0 (by intlit)))
-      then result.fail error.panic
-      else result.ret ()
+      if h: not (hd = (Int32.ofNatCore 0 (by intlit)))
+      then Result.fail Error.panic
+      else Result.ret ()
   
   /- Unit test for [no_nested_borrows::test_split_list] -/
-  #assert (test_split_list_fwd = .ret ())
+  #assert (test_split_list_fwd == .ret ())
   
   /- [no_nested_borrows::choose] -/
-  def choose_fwd (T : Type) (b : Bool) (x : T) (y : T) : result T :=
-    if b
-    then result.ret x
-    else result.ret y
+  def choose_fwd (T : Type) (b : Bool) (x : T) (y : T) : Result T :=
+    if h: b
+    then Result.ret x
+    else Result.ret y
   
   /- [no_nested_borrows::choose] -/
   def choose_back
-    (T : Type) (b : Bool) (x : T) (y : T) (ret0 : T) : result (T × T) :=
-    if b
-    then result.ret (ret0, y)
-    else result.ret (x, ret0)
+    (T : Type) (b : Bool) (x : T) (y : T) (ret0 : T) : Result (T × T) :=
+    if h: b
+    then Result.ret (ret0, y)
+    else Result.ret (x, ret0)
   
   /- [no_nested_borrows::choose_test] -/
-  def choose_test_fwd : result Unit :=
+  def choose_test_fwd : Result Unit :=
     do
-      let z <-
+      let z ←
         choose_fwd Int32 true (Int32.ofNatCore 0 (by intlit))
           (Int32.ofNatCore 0 (by intlit))
-      let z0 <- Int32.checked_add z (Int32.ofNatCore 1 (by intlit))
-      if not (z0 = (Int32.ofNatCore 1 (by intlit)))
-      then result.fail error.panic
+      let z0 ← Int32.checked_add z (Int32.ofNatCore 1 (by intlit))
+      if h: not (z0 = (Int32.ofNatCore 1 (by intlit)))
+      then Result.fail Error.panic
       else
         do
-          let (x, y) <-
+          let (x, y) ←
             choose_back Int32 true (Int32.ofNatCore 0 (by intlit))
               (Int32.ofNatCore 0 (by intlit)) z0
-          if not (x = (Int32.ofNatCore 1 (by intlit)))
-          then result.fail error.panic
+          if h: not (x = (Int32.ofNatCore 1 (by intlit)))
+          then Result.fail Error.panic
           else
-            if not (y = (Int32.ofNatCore 0 (by intlit)))
-            then result.fail error.panic
-            else result.ret ()
+            if h: not (y = (Int32.ofNatCore 0 (by intlit)))
+            then Result.fail Error.panic
+            else Result.ret ()
   
   /- Unit test for [no_nested_borrows::choose_test] -/
-  #assert (choose_test_fwd = .ret ())
+  #assert (choose_test_fwd == .ret ())
   
   /- [no_nested_borrows::test_char] -/
-  def test_char_fwd : result Char :=
-    result.ret 'a'
+  def test_char_fwd : Result Char :=
+    Result.ret 'a'
   
   /- [no_nested_borrows::NodeElem] -/
   mutual inductive node_elem_t (T : Type) :=
@@ -268,179 +270,179 @@ structure OpaqueDefs where
   | TreeNode : T -> node_elem_t T -> tree_t T -> tree_t T
   
   /- [no_nested_borrows::list_length] -/
-  def list_length_fwd (T : Type) (l : list_t T) : result UInt32 :=
-    match l with
+  def list_length_fwd (T : Type) (l : list_t T) : Result UInt32 :=
+    match h: l with
     | list_t.ListCons t l1 =>
       do
-        let i <- list_length_fwd T l1
+        let i ← list_length_fwd T l1
         UInt32.checked_add (UInt32.ofNatCore 1 (by intlit)) i
-    | list_t.ListNil => result.ret (UInt32.ofNatCore 0 (by intlit))
+    | list_t.ListNil => Result.ret (UInt32.ofNatCore 0 (by intlit))
   
   /- [no_nested_borrows::list_nth_shared] -/
-  def list_nth_shared_fwd (T : Type) (l : list_t T) (i : UInt32) : result T :=
-    match l with
+  def list_nth_shared_fwd (T : Type) (l : list_t T) (i : UInt32) : Result T :=
+    match h: l with
     | list_t.ListCons x tl =>
-      if i = (UInt32.ofNatCore 0 (by intlit))
-      then result.ret x
+      if h: i = (UInt32.ofNatCore 0 (by intlit))
+      then Result.ret x
       else
         do
-          let i0 <- UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+          let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
           list_nth_shared_fwd T tl i0
-    | list_t.ListNil => result.fail error.panic
+    | list_t.ListNil => Result.fail Error.panic
   
   /- [no_nested_borrows::list_nth_mut] -/
-  def list_nth_mut_fwd (T : Type) (l : list_t T) (i : UInt32) : result T :=
-    match l with
+  def list_nth_mut_fwd (T : Type) (l : list_t T) (i : UInt32) : Result T :=
+    match h: l with
     | list_t.ListCons x tl =>
-      if i = (UInt32.ofNatCore 0 (by intlit))
-      then result.ret x
+      if h: i = (UInt32.ofNatCore 0 (by intlit))
+      then Result.ret x
       else
         do
-          let i0 <- UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+          let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
           list_nth_mut_fwd T tl i0
-    | list_t.ListNil => result.fail error.panic
+    | list_t.ListNil => Result.fail Error.panic
   
   /- [no_nested_borrows::list_nth_mut] -/
   def list_nth_mut_back
-    (T : Type) (l : list_t T) (i : UInt32) (ret0 : T) : result (list_t T) :=
-    match l with
+    (T : Type) (l : list_t T) (i : UInt32) (ret0 : T) : Result (list_t T) :=
+    match h: l with
     | list_t.ListCons x tl =>
-      if i = (UInt32.ofNatCore 0 (by intlit))
-      then result.ret (list_t.ListCons ret0 tl)
+      if h: i = (UInt32.ofNatCore 0 (by intlit))
+      then Result.ret (list_t.ListCons ret0 tl)
       else
         do
-          let i0 <- UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
-          let tl0 <- list_nth_mut_back T tl i0 ret0
-          result.ret (list_t.ListCons x tl0)
-    | list_t.ListNil => result.fail error.panic
+          let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+          let tl0 ← list_nth_mut_back T tl i0 ret0
+          Result.ret (list_t.ListCons x tl0)
+    | list_t.ListNil => Result.fail Error.panic
   
   /- [no_nested_borrows::list_rev_aux] -/
   def list_rev_aux_fwd
-    (T : Type) (li : list_t T) (lo : list_t T) : result (list_t T) :=
-    match li with
+    (T : Type) (li : list_t T) (lo : list_t T) : Result (list_t T) :=
+    match h: li with
     | list_t.ListCons hd tl => list_rev_aux_fwd T tl (list_t.ListCons hd lo)
-    | list_t.ListNil => result.ret lo
+    | list_t.ListNil => Result.ret lo
   
   /- [no_nested_borrows::list_rev] -/
-  def list_rev_fwd_back (T : Type) (l : list_t T) : result (list_t T) :=
+  def list_rev_fwd_back (T : Type) (l : list_t T) : Result (list_t T) :=
     let li := mem_replace_fwd (list_t T) l list_t.ListNil
     list_rev_aux_fwd T li list_t.ListNil
   
   /- [no_nested_borrows::test_list_functions] -/
-  def test_list_functions_fwd : result Unit :=
+  def test_list_functions_fwd : Result Unit :=
     do
       let l := list_t.ListNil
       let l0 := list_t.ListCons (Int32.ofNatCore 2 (by intlit)) l
       let l1 := list_t.ListCons (Int32.ofNatCore 1 (by intlit)) l0
-      let i <-
+      let i ←
         list_length_fwd Int32 (list_t.ListCons (Int32.ofNatCore 0 (by intlit))
           l1)
-      if not (i = (UInt32.ofNatCore 3 (by intlit)))
-      then result.fail error.panic
+      if h: not (i = (UInt32.ofNatCore 3 (by intlit)))
+      then Result.fail Error.panic
       else
         do
-          let i0 <-
+          let i0 ←
             list_nth_shared_fwd Int32 (list_t.ListCons
               (Int32.ofNatCore 0 (by intlit)) l1)
               (UInt32.ofNatCore 0 (by intlit))
-          if not (i0 = (Int32.ofNatCore 0 (by intlit)))
-          then result.fail error.panic
+          if h: not (i0 = (Int32.ofNatCore 0 (by intlit)))
+          then Result.fail Error.panic
           else
             do
-              let i1 <-
+              let i1 ←
                 list_nth_shared_fwd Int32 (list_t.ListCons
                   (Int32.ofNatCore 0 (by intlit)) l1)
                   (UInt32.ofNatCore 1 (by intlit))
-              if not (i1 = (Int32.ofNatCore 1 (by intlit)))
-              then result.fail error.panic
+              if h: not (i1 = (Int32.ofNatCore 1 (by intlit)))
+              then Result.fail Error.panic
               else
                 do
-                  let i2 <-
+                  let i2 ←
                     list_nth_shared_fwd Int32 (list_t.ListCons
                       (Int32.ofNatCore 0 (by intlit)) l1)
                       (UInt32.ofNatCore 2 (by intlit))
-                  if not (i2 = (Int32.ofNatCore 2 (by intlit)))
-                  then result.fail error.panic
+                  if h: not (i2 = (Int32.ofNatCore 2 (by intlit)))
+                  then Result.fail Error.panic
                   else
                     do
-                      let ls <-
+                      let ls ←
                         list_nth_mut_back Int32 (list_t.ListCons
                           (Int32.ofNatCore 0 (by intlit)) l1)
                           (UInt32.ofNatCore 1 (by intlit))
                           (Int32.ofNatCore 3 (by intlit))
-                      let i3 <-
+                      let i3 ←
                         list_nth_shared_fwd Int32 ls
                           (UInt32.ofNatCore 0 (by intlit))
-                      if not (i3 = (Int32.ofNatCore 0 (by intlit)))
-                      then result.fail error.panic
+                      if h: not (i3 = (Int32.ofNatCore 0 (by intlit)))
+                      then Result.fail Error.panic
                       else
                         do
-                          let i4 <-
+                          let i4 ←
                             list_nth_shared_fwd Int32 ls
                               (UInt32.ofNatCore 1 (by intlit))
-                          if not (i4 = (Int32.ofNatCore 3 (by intlit)))
-                          then result.fail error.panic
+                          if h: not (i4 = (Int32.ofNatCore 3 (by intlit)))
+                          then Result.fail Error.panic
                           else
                             do
-                              let i5 <-
+                              let i5 ←
                                 list_nth_shared_fwd Int32 ls
                                   (UInt32.ofNatCore 2 (by intlit))
-                              if not (i5 = (Int32.ofNatCore 2 (by intlit)))
-                              then result.fail error.panic
-                              else result.ret ()
+                              if h: not (i5 = (Int32.ofNatCore 2 (by intlit)))
+                              then Result.fail Error.panic
+                              else Result.ret ()
   
   /- Unit test for [no_nested_borrows::test_list_functions] -/
-  #assert (test_list_functions_fwd = .ret ())
+  #assert (test_list_functions_fwd == .ret ())
   
   /- [no_nested_borrows::id_mut_pair1] -/
-  def id_mut_pair1_fwd (T1 T2 : Type) (x : T1) (y : T2) : result (T1 × T2) :=
-    result.ret (x, y)
+  def id_mut_pair1_fwd (T1 T2 : Type) (x : T1) (y : T2) : Result (T1 × T2) :=
+    Result.ret (x, y)
   
   /- [no_nested_borrows::id_mut_pair1] -/
   def id_mut_pair1_back
-    (T1 T2 : Type) (x : T1) (y : T2) (ret0 : (T1 × T2)) : result (T1 × T2) :=
+    (T1 T2 : Type) (x : T1) (y : T2) (ret0 : (T1 × T2)) : Result (T1 × T2) :=
     let (t, t0) := ret0
-    result.ret (t, t0)
+    Result.ret (t, t0)
   
   /- [no_nested_borrows::id_mut_pair2] -/
-  def id_mut_pair2_fwd (T1 T2 : Type) (p : (T1 × T2)) : result (T1 × T2) :=
+  def id_mut_pair2_fwd (T1 T2 : Type) (p : (T1 × T2)) : Result (T1 × T2) :=
     let (t, t0) := p
-    result.ret (t, t0)
+    Result.ret (t, t0)
   
   /- [no_nested_borrows::id_mut_pair2] -/
   def id_mut_pair2_back
-    (T1 T2 : Type) (p : (T1 × T2)) (ret0 : (T1 × T2)) : result (T1 × T2) :=
+    (T1 T2 : Type) (p : (T1 × T2)) (ret0 : (T1 × T2)) : Result (T1 × T2) :=
     let (t, t0) := ret0
-    result.ret (t, t0)
+    Result.ret (t, t0)
   
   /- [no_nested_borrows::id_mut_pair3] -/
-  def id_mut_pair3_fwd (T1 T2 : Type) (x : T1) (y : T2) : result (T1 × T2) :=
-    result.ret (x, y)
+  def id_mut_pair3_fwd (T1 T2 : Type) (x : T1) (y : T2) : Result (T1 × T2) :=
+    Result.ret (x, y)
   
   /- [no_nested_borrows::id_mut_pair3] -/
   def id_mut_pair3_back'a
-    (T1 T2 : Type) (x : T1) (y : T2) (ret0 : T1) : result T1 :=
-    result.ret ret0
+    (T1 T2 : Type) (x : T1) (y : T2) (ret0 : T1) : Result T1 :=
+    Result.ret ret0
   
   /- [no_nested_borrows::id_mut_pair3] -/
   def id_mut_pair3_back'b
-    (T1 T2 : Type) (x : T1) (y : T2) (ret0 : T2) : result T2 :=
-    result.ret ret0
+    (T1 T2 : Type) (x : T1) (y : T2) (ret0 : T2) : Result T2 :=
+    Result.ret ret0
   
   /- [no_nested_borrows::id_mut_pair4] -/
-  def id_mut_pair4_fwd (T1 T2 : Type) (p : (T1 × T2)) : result (T1 × T2) :=
+  def id_mut_pair4_fwd (T1 T2 : Type) (p : (T1 × T2)) : Result (T1 × T2) :=
     let (t, t0) := p
-    result.ret (t, t0)
+    Result.ret (t, t0)
   
   /- [no_nested_borrows::id_mut_pair4] -/
   def id_mut_pair4_back'a
-    (T1 T2 : Type) (p : (T1 × T2)) (ret0 : T1) : result T1 :=
-    result.ret ret0
+    (T1 T2 : Type) (p : (T1 × T2)) (ret0 : T1) : Result T1 :=
+    Result.ret ret0
   
   /- [no_nested_borrows::id_mut_pair4] -/
   def id_mut_pair4_back'b
-    (T1 T2 : Type) (p : (T1 × T2)) (ret0 : T2) : result T2 :=
-    result.ret ret0
+    (T1 T2 : Type) (p : (T1 × T2)) (ret0 : T2) : Result T2 :=
+    Result.ret ret0
   
   /- [no_nested_borrows::StructWithTuple] -/
   structure struct_with_tuple_t (T1 T2 : Type) where
@@ -449,24 +451,24 @@ structure OpaqueDefs where
   
   
   /- [no_nested_borrows::new_tuple1] -/
-  def new_tuple1_fwd : result (struct_with_tuple_t UInt32 UInt32) :=
-    result.ret
+  def new_tuple1_fwd : Result (struct_with_tuple_t UInt32 UInt32) :=
+    Result.ret
     {
       struct_with_tuple_p := ((UInt32.ofNatCore 1 (by intlit)),
                                (UInt32.ofNatCore 2 (by intlit)))
     }
   
   /- [no_nested_borrows::new_tuple2] -/
-  def new_tuple2_fwd : result (struct_with_tuple_t Int16 Int16) :=
-    result.ret
+  def new_tuple2_fwd : Result (struct_with_tuple_t Int16 Int16) :=
+    Result.ret
     {
       struct_with_tuple_p := ((Int16.ofNatCore 1 (by intlit)),
                                (Int16.ofNatCore 2 (by intlit)))
     }
   
   /- [no_nested_borrows::new_tuple3] -/
-  def new_tuple3_fwd : result (struct_with_tuple_t UInt64 Int64) :=
-    result.ret
+  def new_tuple3_fwd : Result (struct_with_tuple_t UInt64 Int64) :=
+    Result.ret
     {
       struct_with_tuple_p := ((UInt64.ofNatCore 1 (by intlit)),
                                (Int64.ofNatCore 2 (by intlit)))
@@ -479,8 +481,8 @@ structure OpaqueDefs where
   
   
   /- [no_nested_borrows::new_pair1] -/
-  def new_pair1_fwd : result (struct_with_pair_t UInt32 UInt32) :=
-    result.ret
+  def new_pair1_fwd : Result (struct_with_pair_t UInt32 UInt32) :=
+    Result.ret
     {
       struct_with_pair_p := {
                               pair_x := (UInt32.ofNatCore 1 (by intlit)),
@@ -489,66 +491,66 @@ structure OpaqueDefs where
     }
   
   /- [no_nested_borrows::test_constants] -/
-  def test_constants_fwd : result Unit :=
+  def test_constants_fwd : Result Unit :=
     do
-      let swt <- new_tuple1_fwd
+      let swt ← new_tuple1_fwd
       let (i, _) := swt.struct_with_tuple_p
-      if not (i = (UInt32.ofNatCore 1 (by intlit)))
-      then result.fail error.panic
+      if h: not (i = (UInt32.ofNatCore 1 (by intlit)))
+      then Result.fail Error.panic
       else
         do
-          let swt0 <- new_tuple2_fwd
+          let swt0 ← new_tuple2_fwd
           let (i0, _) := swt0.struct_with_tuple_p
-          if not (i0 = (Int16.ofNatCore 1 (by intlit)))
-          then result.fail error.panic
+          if h: not (i0 = (Int16.ofNatCore 1 (by intlit)))
+          then Result.fail Error.panic
           else
             do
-              let swt1 <- new_tuple3_fwd
+              let swt1 ← new_tuple3_fwd
               let (i1, _) := swt1.struct_with_tuple_p
-              if not (i1 = (UInt64.ofNatCore 1 (by intlit)))
-              then result.fail error.panic
+              if h: not (i1 = (UInt64.ofNatCore 1 (by intlit)))
+              then Result.fail Error.panic
               else
                 do
-                  let swp <- new_pair1_fwd
-                  if not (swp.struct_with_pair_p.pair_x =
+                  let swp ← new_pair1_fwd
+                  if h: not (swp.struct_with_pair_p.pair_x =
                     (UInt32.ofNatCore 1 (by intlit)))
-                  then result.fail error.panic
-                  else result.ret ()
+                  then Result.fail Error.panic
+                  else Result.ret ()
   
   /- Unit test for [no_nested_borrows::test_constants] -/
-  #assert (test_constants_fwd = .ret ())
+  #assert (test_constants_fwd == .ret ())
   
   /- [no_nested_borrows::test_weird_borrows1] -/
-  def test_weird_borrows1_fwd : result Unit :=
-    result.ret ()
+  def test_weird_borrows1_fwd : Result Unit :=
+    Result.ret ()
   
   /- Unit test for [no_nested_borrows::test_weird_borrows1] -/
-  #assert (test_weird_borrows1_fwd = .ret ())
+  #assert (test_weird_borrows1_fwd == .ret ())
   
   /- [no_nested_borrows::test_mem_replace] -/
-  def test_mem_replace_fwd_back (px : UInt32) : result UInt32 :=
+  def test_mem_replace_fwd_back (px : UInt32) : Result UInt32 :=
     let y := mem_replace_fwd UInt32 px (UInt32.ofNatCore 1 (by intlit))
-    if not (y = (UInt32.ofNatCore 0 (by intlit)))
-    then result.fail error.panic
-    else result.ret (UInt32.ofNatCore 2 (by intlit))
+    if h: not (y = (UInt32.ofNatCore 0 (by intlit)))
+    then Result.fail Error.panic
+    else Result.ret (UInt32.ofNatCore 2 (by intlit))
   
   /- [no_nested_borrows::test_shared_borrow_bool1] -/
-  def test_shared_borrow_bool1_fwd (b : Bool) : result UInt32 :=
-    if b
-    then result.ret (UInt32.ofNatCore 0 (by intlit))
-    else result.ret (UInt32.ofNatCore 1 (by intlit))
+  def test_shared_borrow_bool1_fwd (b : Bool) : Result UInt32 :=
+    if h: b
+    then Result.ret (UInt32.ofNatCore 0 (by intlit))
+    else Result.ret (UInt32.ofNatCore 1 (by intlit))
   
   /- [no_nested_borrows::test_shared_borrow_bool2] -/
-  def test_shared_borrow_bool2_fwd : result UInt32 :=
-    result.ret (UInt32.ofNatCore 0 (by intlit))
+  def test_shared_borrow_bool2_fwd : Result UInt32 :=
+    Result.ret (UInt32.ofNatCore 0 (by intlit))
   
   /- [no_nested_borrows::test_shared_borrow_enum1] -/
-  def test_shared_borrow_enum1_fwd (l : list_t UInt32) : result UInt32 :=
-    match l with
-    | list_t.ListCons i l0 => result.ret (UInt32.ofNatCore 1 (by intlit))
-    | list_t.ListNil => result.ret (UInt32.ofNatCore 0 (by intlit))
+  def test_shared_borrow_enum1_fwd (l : list_t UInt32) : Result UInt32 :=
+    match h: l with
+    | list_t.ListCons i l0 => Result.ret (UInt32.ofNatCore 1 (by intlit))
+    | list_t.ListNil => Result.ret (UInt32.ofNatCore 0 (by intlit))
   
   /- [no_nested_borrows::test_shared_borrow_enum2] -/
-  def test_shared_borrow_enum2_fwd : result UInt32 :=
-    result.ret (UInt32.ofNatCore 0 (by intlit))
+  def test_shared_borrow_enum2_fwd : Result UInt32 :=
+    Result.ret (UInt32.ofNatCore 0 (by intlit))
   
diff --git a/tests/lean/misc/paper/Base/Primitives.lean b/tests/lean/misc/paper/Base/Primitives.lean
index 79958d94..5b64e908 100644
--- a/tests/lean/misc/paper/Base/Primitives.lean
+++ b/tests/lean/misc/paper/Base/Primitives.lean
@@ -9,74 +9,79 @@ import Mathlib.Tactic.RunCmd
 
 -- Results & monadic combinators
 
--- TODO: use syntactic conventions and capitalize error, result, etc.
-
-inductive error where
-   | assertionFailure: error
-   | integerOverflow: error
-   | arrayOutOfBounds: error
-   | maximumSizeExceeded: error
-   | panic: error
+inductive Error where
+   | assertionFailure: Error
+   | integerOverflow: Error
+   | arrayOutOfBounds: Error
+   | maximumSizeExceeded: Error
+   | panic: Error
 deriving Repr, BEq
 
-open error
+open Error
 
-inductive result (α : Type u) where
-  | ret (v: α): result α
-  | fail (e: error): result α 
+inductive Result (α : Type u) where
+  | ret (v: α): Result α
+  | fail (e: Error): Result α
 deriving Repr, BEq
 
-open result
+open Result
 
 /- HELPERS -/
 
--- TODO: is there automated syntax for these discriminators?
-def is_ret {α: Type} (r: result α): Bool :=
+def ret? {α: Type} (r: Result α): Bool :=
   match r with
-  | result.ret _ => true
-  | result.fail _ => false
+  | Result.ret _ => true
+  | Result.fail _ => false
 
-def massert (b:Bool) : result Unit :=
+def massert (b:Bool) : Result Unit :=
   if b then .ret () else fail assertionFailure
 
-def eval_global {α: Type} (x: result α) (_: is_ret x): α :=
+def eval_global {α: Type} (x: Result α) (_: ret? x): α :=
   match x with
-  | result.fail _ => by contradiction
-  | result.ret x => x
+  | Result.fail _ => by contradiction
+  | Result.ret x => x
 
 /- DO-DSL SUPPORT -/
 
-def bind (x: result α) (f: α -> result β) : result β :=
+def bind (x: Result α) (f: α -> Result β) : Result β :=
   match x with
   | ret v  => f v 
   | fail v => fail v
 
--- Allows using result in do-blocks
-instance : Bind result where
+-- Allows using Result in do-blocks
+instance : Bind Result where
   bind := bind
 
 -- Allows using return x in do-blocks
-instance : Pure result where
+instance : Pure Result where
   pure := fun x => ret x
 
 /- CUSTOM-DSL SUPPORT -/
 
--- Let-binding the result of a monadic operation is oftentimes not sufficient,
+-- Let-binding the Result of a monadic operation is oftentimes not sufficient,
 -- because we may need a hypothesis for equational reasoning in the scope. We
 -- rely on subtype, and a custom let-binding operator, in effect recreating our
 -- own variant of the do-dsl
 
-def result.attach : (o : result α) → result { x : α // o = ret x }
+def Result.attach {α: Type} (o : Result α): Result { x : α // o = ret x } :=
+  match o with
   | .ret x => .ret ⟨x, rfl⟩
-  | .fail e   => .fail e
+  | .fail e => .fail e
 
-macro "let" h:ident " : " e:term " <-- " f:term : doElem =>
-  `(doElem| let ⟨$e, $h⟩ ← result.attach $f)
+macro "let" e:term " ⟵ " f:term : doElem =>
+  `(doElem| let ⟨$e, h⟩ ← Result.attach $f)
 
--- Silly example of the kind of reasoning that this notation enables
+-- TODO: any way to factorize both definitions?
+macro "let" e:term " <-- " f:term : doElem =>
+  `(doElem| let ⟨$e, h⟩ ← Result.attach $f)
+
+-- We call the hypothesis `h`, in effect making it unavailable to the user
+-- (because too much shadowing). But in practice, once can use the French single
+-- quote notation (input with f< and f>), where `‹ h ›` finds a suitable
+-- hypothesis in the context, this is equivalent to `have x: h := by assumption in x`
 #eval do
-  let h: y <-- .ret (0: Nat)
-  let _: y = 0 := by cases h; decide
+  let y <-- .ret (0: Nat)
+  let _: y = 0 := by cases ‹ ret 0 = ret y › ; decide
   let r: { x: Nat // x = 0 } := ⟨ y, by assumption ⟩
   .ret r
 
@@ -84,36 +89,27 @@ macro "let" h:ident " : " e:term " <-- " f:term : doElem =>
 -- MACHINE INTEGERS --
 ----------------------
 
--- NOTE: we reuse the USize type from prelude.lean, because at least we know
--- it's defined in an idiomatic style that is going to make proofs easy (and
--- indeed, several proofs here are much shortened compared to Aymeric's earlier
--- attempt.) This is not stricto sensu the *correct* thing to do, because one
--- can query at run-time the value of USize, which we do *not* want to do (we
--- don't know what target we'll run on!), but when the day comes, we'll just
--- define our own USize.
--- ANOTHER NOTE: there is USize.sub but it has wraparound semantics, which is
--- not something we want to define (I think), so we use our own monadic sub (but
--- is it in line with the Rust behavior?)
-
--- TODO: I am somewhat under the impression that subtraction is defined as a
--- total function over nats...? the hypothesis in the if condition is not used
--- in the then-branch which confuses me quite a bit
-
--- TODO: add a refinement for the result (just like vec_push_back below) that
--- explains that the toNat of the result (in the case of success) is the sub of
--- the toNat of the arguments (i.e. intrinsic specification)
--- ... do we want intrinsic specifications for the builtins? that might require
--- some careful type annotations in the monadic notation for clients, but may
--- give us more "for free"
+-- NOTE: we reuse the fixed-width integer types from prelude.lean: UInt8, ...,
+-- USize. They are generally defined in an idiomatic style, except that there is
+-- not a single type class to rule them all (more on that below). The absence of
+-- type class is intentional, and allows the Lean compiler to efficiently map
+-- them to machine integers during compilation.
+
+-- USize is designed properly: you cannot reduce `getNumBits` using the
+-- simplifier, meaning that proofs do not depend on the compile-time value of
+-- USize.size. (Lean assumes 32 or 64-bit platforms, and Rust doesn't really
+-- support, at least officially, 16-bit microcontrollers, so this seems like a
+-- fine design decision for now.)
 
 -- Note from Chris Bailey: "If there's more than one salient property of your
 -- definition then the subtyping strategy might get messy, and the property part
 -- of a subtype is less discoverable by the simplifier or tactics like
--- library_search." Try to settle this with a Lean expert on what is the most
--- productive way to go about this?
+-- library_search." So, we will not add refinements on the return values of the
+-- operations defined on Primitives, but will rather rely on custom lemmas to
+-- invert on possible return values of the primitive operations.
 
--- One needs to perform a little bit of reasoning in order to successfully
--- inject constants into USize, so we provide a general-purpose macro
+-- Machine integer constants, done via `ofNatCore`, which requires a proof that
+-- the `Nat` fits within the desired integer type. We provide a custom tactic.
 
 syntax "intlit" : tactic
 
@@ -129,12 +125,21 @@ macro_rules
 -- Also works for other integer types (at the expense of a needless disjunction)
 #eval UInt32.ofNatCore 0 (by intlit)
 
+-- The machine integer operations (e.g. sub) are always total, which is not what
+-- we want. We therefore define "checked" variants, below. Note that we add a
+-- tiny bit of complexity for the USize variant: we first check whether the
+-- result is < 2^32; if it is, we can compute the definition, rather than
+-- returning a term that is computationally stuck (the comparison to USize.size
+-- cannot reduce at compile-time, per the remark about regarding `getNumBits`).
+-- This is useful for the various #asserts that we want to reduce at
+-- type-checking time.
+
 -- Further thoughts: look at what has been done here:
 -- https://github.com/leanprover-community/mathlib4/blob/master/Mathlib/Data/Fin/Basic.lean
 -- and
 -- https://github.com/leanprover-community/mathlib4/blob/master/Mathlib/Data/UInt.lean
 -- which both contain a fair amount of reasoning already!
-def USize.checked_sub (n: USize) (m: USize): result USize :=
+def USize.checked_sub (n: USize) (m: USize): Result USize :=
   -- NOTE: the test USize.toNat n - m >= 0 seems to always succeed?
   if n >= m then
     let n' := USize.toNat n
@@ -150,18 +155,19 @@ def USize.checked_sub (n: USize) (m: USize): result USize :=
   else
     fail integerOverflow
 
-def USize.checked_add (n: USize) (m: USize): result USize :=
-  if h: n.val.val + m.val.val <= 4294967295 then
-    .ret ⟨ n.val.val + m.val.val, by
-      have h': 4294967295 < USize.size := by intlit
-      apply Nat.lt_of_le_of_lt h h'
-    ⟩
-  else if h: n.val + m.val < USize.size then
+@[simp]
+theorem usize_fits (n: Nat) (h: n <= 4294967295): n < USize.size :=
+  match USize.size, usize_size_eq with
+  | _, Or.inl rfl => Nat.lt_of_le_of_lt h (by decide)
+  | _, Or.inr rfl => Nat.lt_of_le_of_lt h (by decide)
+
+def USize.checked_add (n: USize) (m: USize): Result USize :=
+  if h: n.val + m.val < USize.size then
     .ret ⟨ n.val + m.val, h ⟩
   else
     .fail integerOverflow
 
-def USize.checked_rem (n: USize) (m: USize): result USize :=
+def USize.checked_rem (n: USize) (m: USize): Result USize :=
   if h: m > 0 then
     .ret ⟨ n.val % m.val, by
       have h1: ↑m.val < USize.size := m.val.isLt
@@ -171,18 +177,13 @@ def USize.checked_rem (n: USize) (m: USize): result USize :=
   else
     .fail integerOverflow
 
-def USize.checked_mul (n: USize) (m: USize): result USize :=
-    if h: n.val.val * m.val.val <= 4294967295 then
-    .ret ⟨ n.val.val * m.val.val, by
-      have h': 4294967295 < USize.size := by intlit
-      apply Nat.lt_of_le_of_lt h h'
-    ⟩
-  else if h: n.val * m.val < USize.size then
+def USize.checked_mul (n: USize) (m: USize): Result USize :=
+  if h: n.val * m.val < USize.size then
     .ret ⟨ n.val * m.val, h ⟩
   else
     .fail integerOverflow
 
-def USize.checked_div (n: USize) (m: USize): result USize :=
+def USize.checked_div (n: USize) (m: USize): Result USize :=
   if m > 0 then
     .ret ⟨ n.val / m.val, by
       have h1: ↑n.val < USize.size := n.val.isLt
@@ -192,6 +193,19 @@ def USize.checked_div (n: USize) (m: USize): result USize :=
   else
     .fail integerOverflow
 
+-- Test behavior...
+#eval assert! USize.checked_sub 10 20 == fail integerOverflow; 0
+
+#eval USize.checked_sub 20 10
+-- NOTE: compare with concrete behavior here, which I do not think we want
+#eval USize.sub 0 1
+#eval UInt8.add 255 255
+
+-- We now define a type class that subsumes the various machine integer types, so
+-- as to write a concise definition for scalar_cast, rather than exhaustively
+-- enumerating all of the possible pairs. We remark that Rust has sane semantics
+-- and fails if a cast operation would involve a truncation or modulo.
+
 class MachineInteger (t: Type) where
   size: Nat
   val: t -> Fin size
@@ -209,30 +223,24 @@ run_cmd
     end $typeName
   ))
 
-def scalar_cast { src: Type } (dst: Type) [ MachineInteger src ] [ MachineInteger dst ] (x: src): result dst :=
+-- Aeneas only instantiates the destination type (`src` is implicit). We rely on
+-- Lean to infer `src`.
+
+def scalar_cast { src: Type } (dst: Type) [ MachineInteger src ] [ MachineInteger dst ] (x: src): Result dst :=
   if h: MachineInteger.val x < MachineInteger.size dst then
     .ret (MachineInteger.ofNatCore (MachineInteger.val x).val h)
   else
     .fail integerOverflow
 
-
--- Test behavior...
-#eval assert! USize.checked_sub 10 20 == fail integerOverflow; 0
-
-#eval USize.checked_sub 20 10
--- NOTE: compare with concrete behavior here, which I do not think we want
-#eval USize.sub 0 1
-#eval UInt8.add 255 255
-
 -------------
 -- VECTORS --
 -------------
 
 -- Note: unlike F*, Lean seems to use strict upper bounds (e.g. USize.size)
 -- rather than maximum values (usize_max).
-def vec (α : Type u) := { l : List α // List.length l < USize.size }
+def Vec (α : Type u) := { l : List α // List.length l < USize.size }
 
-def vec_new (α : Type u): vec α := ⟨ [], by {
+def vec_new (α : Type u): Vec α := ⟨ [], by {
   match USize.size, usize_size_eq with
   | _, Or.inl rfl => simp
   | _, Or.inr rfl => simp
@@ -240,20 +248,20 @@ def vec_new (α : Type u): vec α := ⟨ [], by {
 
 #check vec_new
 
-def vec_len (α : Type u) (v : vec α) : USize :=
+def vec_len (α : Type u) (v : Vec α) : USize :=
   let ⟨ v, l ⟩ := v
   USize.ofNatCore (List.length v) l
 
 #eval vec_len Nat (vec_new Nat)
  
-def vec_push_fwd (α : Type u) (_ : vec α) (_ : α) : Unit := ()
+def vec_push_fwd (α : Type u) (_ : Vec α) (_ : α) : Unit := ()
 
 -- NOTE: old version trying to use a subtype notation, but probably better to
--- leave result elimination to auxiliary lemmas with suitable preconditions
+-- leave Result elimination to auxiliary lemmas with suitable preconditions
 -- TODO: I originally wrote `List.length v.val < USize.size - 1`; how can one
 -- make the proof work in that case? Probably need to import tactics from
 -- mathlib to deal with inequalities... would love to see an example.
-def vec_push_back_old (α : Type u) (v : vec α) (x : α) : { res: result (vec α) //
+def vec_push_back_old (α : Type u) (v : Vec α) (x : α) : { res: Result (Vec α) //
   match res with | fail _ => True | ret v' => List.length v'.val = List.length v.val + 1}
   :=
   if h : List.length v.val + 1 < USize.size then
@@ -272,12 +280,12 @@ def vec_push_back_old (α : Type u) (v : vec α) (x : α) : { res: result (vec �
   -- annotate `x`, which relieves us of having to write `.val` on the right-hand
   -- side of the monadic let.
   let v := vec_new Nat
-  let x: vec Nat ← (vec_push_back_old Nat v 1: result (vec Nat)) -- WHY do we need the type annotation here?
+  let x: Vec Nat ← (vec_push_back_old Nat v 1: Result (Vec Nat)) -- WHY do we need the type annotation here?
   -- TODO: strengthen post-condition above and do a demo to show that we can
   -- safely eliminate the `fail` case
   return (vec_len Nat x)
 
-def vec_push_back (α : Type u) (v : vec α) (x : α) : result (vec α)
+def vec_push_back (α : Type u) (v : Vec α) (x : α) : Result (Vec α)
   :=
   if h : List.length v.val + 1 <= 4294967295 then
     return ⟨ List.concat v.val x,
@@ -295,13 +303,13 @@ def vec_push_back (α : Type u) (v : vec α) (x : α) : result (vec α)
   else
     fail maximumSizeExceeded
 
-def vec_insert_fwd (α : Type u) (v: vec α) (i: USize) (_: α): result Unit :=
+def vec_insert_fwd (α : Type u) (v: Vec α) (i: USize) (_: α): Result Unit :=
   if i.val < List.length v.val then
     .ret ()
   else
     .fail arrayOutOfBounds
 
-def vec_insert_back (α : Type u) (v: vec α) (i: USize) (x: α): result (vec α) :=
+def vec_insert_back (α : Type u) (v: Vec α) (i: USize) (x: α): Result (Vec α) :=
   if i.val < List.length v.val then
     .ret ⟨ List.set v.val i.val x, by
       have h: List.length v.val < USize.size := v.property
@@ -311,25 +319,25 @@ def vec_insert_back (α : Type u) (v: vec α) (i: USize) (x: α): result (vec α
   else
     .fail arrayOutOfBounds
 
-def vec_index_fwd (α : Type u) (v: vec α) (i: USize): result α :=
+def vec_index_fwd (α : Type u) (v: Vec α) (i: USize): Result α :=
   if h: i.val < List.length v.val then
     .ret (List.get v.val ⟨i.val, h⟩)
   else
     .fail arrayOutOfBounds
 
-def vec_index_back (α : Type u) (v: vec α) (i: USize) (_: α): result Unit :=
+def vec_index_back (α : Type u) (v: Vec α) (i: USize) (_: α): Result Unit :=
   if i.val < List.length v.val then
     .ret ()
   else
     .fail arrayOutOfBounds
 
-def vec_index_mut_fwd (α : Type u) (v: vec α) (i: USize): result α :=
+def vec_index_mut_fwd (α : Type u) (v: Vec α) (i: USize): Result α :=
   if h: i.val < List.length v.val then
     .ret (List.get v.val ⟨i.val, h⟩)
   else
     .fail arrayOutOfBounds
 
-def vec_index_mut_back (α : Type u) (v: vec α) (i: USize) (x: α): result (vec α) :=
+def vec_index_mut_back (α : Type u) (v: Vec α) (i: USize) (x: α): Result (Vec α) :=
   if i.val < List.length v.val then
     .ret ⟨ List.set v.val i.val x, by
       have h: List.length v.val < USize.size := v.property
@@ -349,6 +357,10 @@ def mem_replace_fwd (a : Type) (x : a) (_ : a) : a :=
 def mem_replace_back (a : Type) (_ : a) (y : a) : a :=
   y
 
+/-- Aeneas-translated function -- useful to reduce non-recursive definitions.
+ Use with `simp [ aeneas ]` -/
+register_simp_attr aeneas
+
 --------------------
 -- ASSERT COMMAND --
 --------------------
@@ -358,16 +370,23 @@ open Lean Elab Command Term Meta
 syntax (name := assert) "#assert" term: command
 
 @[command_elab assert]
+unsafe
 def assertImpl : CommandElab := fun (_stx: Syntax) => do
-  logInfo "Reducing and asserting: "
-  logInfo _stx[1]
   runTermElabM (fun _ => do
-    let e ← Term.elabTerm _stx[1] none
-    logInfo (Expr.dbgToString e)
-    -- How to evaluate the term and compare the result to true?
+    let r ← evalTerm Bool (mkConst ``Bool) _stx[1]
+    if not r then
+      logInfo "Assertion failed for: "
+      logInfo _stx[1]
+      logError "Expression reduced to false"
     pure ())
-  -- logInfo (Expr.dbgToString (``true))
-  -- throwError "TODO: assert"
 
 #eval 2 == 2
 #assert (2 == 2)
+
+-------------------
+-- SANITY CHECKS --
+-------------------
+
+-- TODO: add more once we have signed integers
+
+#assert (USize.checked_rem 1 2 == .ret 1)
diff --git a/tests/lean/misc/paper/Paper.lean b/tests/lean/misc/paper/Paper.lean
index adcd1eae..4faf36ee 100644
--- a/tests/lean/misc/paper/Paper.lean
+++ b/tests/lean/misc/paper/Paper.lean
@@ -5,56 +5,56 @@ import Base.Primitives
 structure OpaqueDefs where
   
   /- [paper::ref_incr] -/
-  def ref_incr_fwd_back (x : Int32) : result Int32 :=
+  def ref_incr_fwd_back (x : Int32) : Result Int32 :=
     Int32.checked_add x (Int32.ofNatCore 1 (by intlit))
   
   /- [paper::test_incr] -/
-  def test_incr_fwd : result Unit :=
+  def test_incr_fwd : Result Unit :=
     do
-      let x <- ref_incr_fwd_back (Int32.ofNatCore 0 (by intlit))
-      if not (x = (Int32.ofNatCore 1 (by intlit)))
-      then result.fail error.panic
-      else result.ret ()
+      let x ← ref_incr_fwd_back (Int32.ofNatCore 0 (by intlit))
+      if h: not (x = (Int32.ofNatCore 1 (by intlit)))
+      then Result.fail Error.panic
+      else Result.ret ()
   
   /- Unit test for [paper::test_incr] -/
-  #assert (test_incr_fwd = .ret ())
+  #assert (test_incr_fwd == .ret ())
   
   /- [paper::choose] -/
-  def choose_fwd (T : Type) (b : Bool) (x : T) (y : T) : result T :=
-    if b
-    then result.ret x
-    else result.ret y
+  def choose_fwd (T : Type) (b : Bool) (x : T) (y : T) : Result T :=
+    if h: b
+    then Result.ret x
+    else Result.ret y
   
   /- [paper::choose] -/
   def choose_back
-    (T : Type) (b : Bool) (x : T) (y : T) (ret0 : T) : result (T × T) :=
-    if b
-    then result.ret (ret0, y)
-    else result.ret (x, ret0)
+    (T : Type) (b : Bool) (x : T) (y : T) (ret0 : T) : Result (T × T) :=
+    if h: b
+    then Result.ret (ret0, y)
+    else Result.ret (x, ret0)
   
   /- [paper::test_choose] -/
-  def test_choose_fwd : result Unit :=
+  def test_choose_fwd : Result Unit :=
     do
-      let z <-
+      let z ←
         choose_fwd Int32 true (Int32.ofNatCore 0 (by intlit))
           (Int32.ofNatCore 0 (by intlit))
-      let z0 <- Int32.checked_add z (Int32.ofNatCore 1 (by intlit))
-      if not (z0 = (Int32.ofNatCore 1 (by intlit)))
-      then result.fail error.panic
+      let z0 ← Int32.checked_add z (Int32.ofNatCore 1 (by intlit))
+      if h: not (z0 = (Int32.ofNatCore 1 (by intlit)))
+      then Result.fail Error.panic
       else
         do
-          let (x, y) <-
+          let (x, y) ←
             choose_back Int32 true (Int32.ofNatCore 0 (by intlit))
               (Int32.ofNatCore 0 (by intlit)) z0
-          if not (x = (Int32.ofNatCore 1 (by intlit)))
-          then result.fail error.panic
+          if h: not (x = (Int32.ofNatCore 1 (by intlit)))
+          then Result.fail Error.panic
           else
-            if not (y = (Int32.ofNatCore 0 (by intlit)))
-            then result.fail error.panic
-            else result.ret ()
+            if h: not (y = (Int32.ofNatCore 0 (by intlit)))
+            then Result.fail Error.panic
+            else Result.ret ()
   
   /- Unit test for [paper::test_choose] -/
-  #assert (test_choose_fwd = .ret ())
+  #assert (test_choose_fwd == .ret ())
   
   /- [paper::List] -/
   inductive list_t (T : Type) :=
@@ -62,67 +62,67 @@ structure OpaqueDefs where
   | ListNil : list_t T
   
   /- [paper::list_nth_mut] -/
-  def list_nth_mut_fwd (T : Type) (l : list_t T) (i : UInt32) : result T :=
-    match l with
+  def list_nth_mut_fwd (T : Type) (l : list_t T) (i : UInt32) : Result T :=
+    match h: l with
     | list_t.ListCons x tl =>
-      if i = (UInt32.ofNatCore 0 (by intlit))
-      then result.ret x
+      if h: i = (UInt32.ofNatCore 0 (by intlit))
+      then Result.ret x
       else
         do
-          let i0 <- UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+          let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
           list_nth_mut_fwd T tl i0
-    | list_t.ListNil => result.fail error.panic
+    | list_t.ListNil => Result.fail Error.panic
   
   /- [paper::list_nth_mut] -/
   def list_nth_mut_back
-    (T : Type) (l : list_t T) (i : UInt32) (ret0 : T) : result (list_t T) :=
-    match l with
+    (T : Type) (l : list_t T) (i : UInt32) (ret0 : T) : Result (list_t T) :=
+    match h: l with
     | list_t.ListCons x tl =>
-      if i = (UInt32.ofNatCore 0 (by intlit))
-      then result.ret (list_t.ListCons ret0 tl)
+      if h: i = (UInt32.ofNatCore 0 (by intlit))
+      then Result.ret (list_t.ListCons ret0 tl)
       else
         do
-          let i0 <- UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
-          let tl0 <- list_nth_mut_back T tl i0 ret0
-          result.ret (list_t.ListCons x tl0)
-    | list_t.ListNil => result.fail error.panic
+          let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+          let tl0 ← list_nth_mut_back T tl i0 ret0
+          Result.ret (list_t.ListCons x tl0)
+    | list_t.ListNil => Result.fail Error.panic
   
   /- [paper::sum] -/
-  def sum_fwd (l : list_t Int32) : result Int32 :=
-    match l with
+  def sum_fwd (l : list_t Int32) : Result Int32 :=
+    match h: l with
     | list_t.ListCons x tl => do
-                                let i <- sum_fwd tl
+                                let i ← sum_fwd tl
                                 Int32.checked_add x i
-    | list_t.ListNil => result.ret (Int32.ofNatCore 0 (by intlit))
+    | list_t.ListNil => Result.ret (Int32.ofNatCore 0 (by intlit))
   
   /- [paper::test_nth] -/
-  def test_nth_fwd : result Unit :=
+  def test_nth_fwd : Result Unit :=
     do
       let l := list_t.ListNil
       let l0 := list_t.ListCons (Int32.ofNatCore 3 (by intlit)) l
       let l1 := list_t.ListCons (Int32.ofNatCore 2 (by intlit)) l0
-      let x <-
+      let x ←
         list_nth_mut_fwd Int32 (list_t.ListCons (Int32.ofNatCore 1 (by intlit))
           l1) (UInt32.ofNatCore 2 (by intlit))
-      let x0 <- Int32.checked_add x (Int32.ofNatCore 1 (by intlit))
-      let l2 <-
+      let x0 ← Int32.checked_add x (Int32.ofNatCore 1 (by intlit))
+      let l2 ←
         list_nth_mut_back Int32 (list_t.ListCons
           (Int32.ofNatCore 1 (by intlit)) l1) (UInt32.ofNatCore 2 (by intlit))
           x0
-      let i <- sum_fwd l2
-      if not (i = (Int32.ofNatCore 7 (by intlit)))
-      then result.fail error.panic
-      else result.ret ()
+      let i ← sum_fwd l2
+      if h: not (i = (Int32.ofNatCore 7 (by intlit)))
+      then Result.fail Error.panic
+      else Result.ret ()
   
   /- Unit test for [paper::test_nth] -/
-  #assert (test_nth_fwd = .ret ())
+  #assert (test_nth_fwd == .ret ())
   
   /- [paper::call_choose] -/
-  def call_choose_fwd (p : (UInt32 × UInt32)) : result UInt32 :=
+  def call_choose_fwd (p : (UInt32 × UInt32)) : Result UInt32 :=
     do
       let (px, py) := p
-      let pz <- choose_fwd UInt32 true px py
-      let pz0 <- UInt32.checked_add pz (UInt32.ofNatCore 1 (by intlit))
-      let (px0, _) <- choose_back UInt32 true px py pz0
-      result.ret px0
+      let pz ← choose_fwd UInt32 true px py
+      let pz0 ← UInt32.checked_add pz (UInt32.ofNatCore 1 (by intlit))
+      let (px0, _) ← choose_back UInt32 true px py pz0
+      Result.ret px0