16 files changed, 3397 insertions, 2362 deletions
diff --git a/tests/lean/misc-constants/Base/Primitives.lean b/tests/lean/misc-constants/Base/Primitives.lean
index 5b64e908..034f41b2 100644
--- a/tests/lean/misc-constants/Base/Primitives.lean
+++ b/tests/lean/misc-constants/Base/Primitives.lean
@@ -3,6 +3,28 @@ import Lean.Meta.Tactic.Simp
 import Init.Data.List.Basic
 import Mathlib.Tactic.RunCmd
 
+--------------------
+-- ASSERT COMMAND --
+--------------------
+
+open Lean Elab Command Term Meta
+
+syntax (name := assert) "#assert" term: command
+
+@[command_elab assert]
+unsafe
+def assertImpl : CommandElab := fun (_stx: Syntax) => do
+  runTermElabM (fun _ => do
+    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 ())
+
+#eval 2 == 2
+#assert (2 == 2)
+
 -------------
 -- PRELUDE --
 -------------
@@ -12,6 +34,7 @@ import Mathlib.Tactic.RunCmd
 inductive Error where
    | assertionFailure: Error
    | integerOverflow: Error
+   | divisionByZero: Error
    | arrayOutOfBounds: Error
    | maximumSizeExceeded: Error
    | panic: Error
@@ -89,17 +112,13 @@ macro "let" e:term " <-- " f:term : doElem =>
 -- MACHINE INTEGERS --
 ----------------------
 
--- 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.
+-- We redefine our machine integers types.
 
--- 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.)
+-- For Isize/Usize, we reuse `getNumBits` from `USize`. 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
@@ -111,236 +130,435 @@ macro "let" e:term " <-- " f:term : doElem =>
 -- 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
-
-macro_rules
-  | `(tactic| intlit) => `(tactic|
-    match USize.size, usize_size_eq with
-    | _, Or.inl rfl => decide
-    | _, Or.inr rfl => decide)
-
--- This is how the macro is expected to be used
-#eval USize.ofNatCore 0 (by intlit)
-
--- 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`).
+open System.Platform.getNumBits
+
+-- TODO: is there a way of only importing System.Platform.getNumBits?
+--
+@[simp] def size_num_bits : Nat := (System.Platform.getNumBits ()).val
+
+-- Remark: Lean seems to use < for the comparisons with the upper bounds by convention.
+-- We keep the F* convention for now.
+@[simp] def Isize.min : Int := - (HPow.hPow 2 (size_num_bits - 1))
+@[simp] def Isize.max : Int := (HPow.hPow 2 (size_num_bits - 1)) - 1
+@[simp] def I8.min    : Int := - (HPow.hPow 2 7)
+@[simp] def I8.max    : Int := HPow.hPow 2 7 - 1
+@[simp] def I16.min   : Int := - (HPow.hPow 2 15)
+@[simp] def I16.max   : Int := HPow.hPow 2 15 - 1
+@[simp] def I32.min   : Int := -(HPow.hPow 2 31)
+@[simp] def I32.max   : Int := HPow.hPow 2 31 - 1
+@[simp] def I64.min   : Int := -(HPow.hPow 2 63)
+@[simp] def I64.max   : Int := HPow.hPow 2 63 - 1
+@[simp] def I128.min  : Int := -(HPow.hPow 2 127)
+@[simp] def I128.max  : Int := HPow.hPow 2 127 - 1
+@[simp] def Usize.min : Int := 0
+@[simp] def Usize.max : Int := HPow.hPow 2 size_num_bits - 1
+@[simp] def U8.min    : Int := 0
+@[simp] def U8.max    : Int := HPow.hPow 2 8 - 1
+@[simp] def U16.min   : Int := 0
+@[simp] def U16.max   : Int := HPow.hPow 2 16 - 1
+@[simp] def U32.min   : Int := 0
+@[simp] def U32.max   : Int := HPow.hPow 2 32 - 1
+@[simp] def U64.min   : Int := 0
+@[simp] def U64.max   : Int := HPow.hPow 2 64 - 1
+@[simp] def U128.min  : Int := 0
+@[simp] def U128.max  : Int := HPow.hPow 2 128 - 1
+
+#assert (I8.min   == -128)
+#assert (I8.max   == 127)
+#assert (I16.min  == -32768)
+#assert (I16.max  == 32767)
+#assert (I32.min  == -2147483648)
+#assert (I32.max  == 2147483647)
+#assert (I64.min  == -9223372036854775808)
+#assert (I64.max  == 9223372036854775807)
+#assert (I128.min == -170141183460469231731687303715884105728)
+#assert (I128.max == 170141183460469231731687303715884105727)
+#assert (U8.min   == 0)
+#assert (U8.max   == 255)
+#assert (U16.min  == 0)
+#assert (U16.max  == 65535)
+#assert (U32.min  == 0)
+#assert (U32.max  == 4294967295)
+#assert (U64.min  == 0)
+#assert (U64.max  == 18446744073709551615)
+#assert (U128.min == 0)
+#assert (U128.max == 340282366920938463463374607431768211455)
+
+inductive ScalarTy :=
+| Isize
+| I8
+| I16
+| I32
+| I64
+| I128
+| Usize
+| U8
+| U16
+| U32
+| U64
+| U128
+
+def Scalar.min (ty : ScalarTy) : Int :=
+  match ty with
+  | .Isize => Isize.min
+  | .I8    => I8.min
+  | .I16   => I16.min
+  | .I32   => I32.min
+  | .I64   => I64.min
+  | .I128  => I128.min
+  | .Usize => Usize.min
+  | .U8    => U8.min
+  | .U16   => U16.min
+  | .U32   => U32.min
+  | .U64   => U64.min
+  | .U128  => U128.min
+
+def Scalar.max (ty : ScalarTy) : Int :=
+  match ty with
+  | .Isize => Isize.max
+  | .I8    => I8.max
+  | .I16   => I16.max
+  | .I32   => I32.max
+  | .I64   => I64.max
+  | .I128  => I128.max
+  | .Usize => Usize.max
+  | .U8    => U8.max
+  | .U16   => U16.max
+  | .U32   => U32.max
+  | .U64   => U64.max
+  | .U128  => U128.max
+
+-- "Conservative" bounds
+-- We use those because we can't compare to the isize bounds (which can't
+-- reduce at compile-time). Whenever we perform an arithmetic operation like
+-- addition we need to check that the result is in bounds: we first compare
+-- to the conservative bounds, which reduce, then compare to the real bounds.
 -- This is useful for the various #asserts that we want to reduce at
 -- type-checking time.
+def Scalar.cMin (ty : ScalarTy) : Int :=
+  match ty with
+  | .Isize => I32.min
+  | _ => Scalar.min ty
+
+def Scalar.cMax (ty : ScalarTy) : Int :=
+  match ty with
+  | .Isize => I32.max
+  | .Usize => U32.max
+  | _ => Scalar.max ty
+
+theorem Scalar.cMin_bound ty : Scalar.min ty <= Scalar.cMin ty := by sorry
+theorem Scalar.cMax_bound ty : Scalar.min ty <= Scalar.cMin ty := by sorry
+
+structure Scalar (ty : ScalarTy) where
+  val : Int
+  hmin : Scalar.min ty <= val
+  hmax : val <= Scalar.max ty
+
+theorem Scalar.bound_suffices (ty : ScalarTy) (x : Int) :
+  Scalar.cMin ty <= x && x <= Scalar.cMax ty ->
+  (decide (Scalar.min ty ≤ x) && decide (x ≤ Scalar.max ty)) = true
+  := by sorry
+
+def Scalar.ofIntCore {ty : ScalarTy} (x : Int)
+  (hmin : Scalar.min ty <= x) (hmax : x <= Scalar.max ty) : Scalar ty :=
+  { val := x, hmin := hmin, hmax := hmax }
+
+def Scalar.ofInt {ty : ScalarTy} (x : Int)
+  (h : Scalar.min ty <= x && x <= Scalar.max ty) : Scalar ty :=
+  let hmin: Scalar.min ty <= x := by sorry
+  let hmax: x <= Scalar.max ty := by sorry
+  Scalar.ofIntCore x hmin hmax
 
 -- 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 :=
-  -- NOTE: the test USize.toNat n - m >= 0 seems to always succeed?
-  if n >= m then
-    let n' := USize.toNat n
-    let m' := USize.toNat n
-    let r := USize.ofNatCore (n' - m') (by
-      have h: n' - m' <= n' := by
-        apply Nat.sub_le_of_le_add
-        case h => rewrite [ Nat.add_comm ]; apply Nat.le_add_left
-      apply Nat.lt_of_le_of_lt h
-      apply n.val.isLt
-    )
-    return r
-  else
-    fail integerOverflow
-
-@[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 :=
-  if h: m > 0 then
-    .ret ⟨ n.val % m.val, by
-      have h1: ↑m.val < USize.size := m.val.isLt
-      have h2: n.val.val % m.val.val < m.val.val := @Nat.mod_lt n.val m.val h
-      apply Nat.lt_trans h2 h1
-    ⟩
-  else
-    .fail integerOverflow
+def Scalar.tryMk (ty : ScalarTy) (x : Int) : Result (Scalar ty) :=
+  -- TODO: write this with only one if then else
+  if hmin_cons: Scalar.cMin ty <= x || Scalar.min ty <= x then
+    if hmax_cons: x <= Scalar.cMax ty || x <= Scalar.max ty then
+      let hmin: Scalar.min ty <= x := by sorry
+      let hmax: x <= Scalar.max ty := by sorry
+      return Scalar.ofIntCore x hmin hmax
+    else fail integerOverflow
+  else fail integerOverflow
+
+def Scalar.neg {ty : ScalarTy} (x : Scalar ty) : Result (Scalar ty) := Scalar.tryMk ty (- x.val)
+
+def Scalar.div {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+  if y.val != 0 then Scalar.tryMk ty (x.val / y.val) else fail divisionByZero
+
+-- Checking that the % operation in Lean computes the same as the remainder operation in Rust
+#assert 1 % 2 = (1:Int)
+#assert (-1) % 2 = -1
+#assert 1 % (-2) = 1
+#assert (-1) % (-2) = -1
+
+def Scalar.rem {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+  if y.val != 0 then Scalar.tryMk ty (x.val % y.val) else fail divisionByZero
+
+def Scalar.add {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+  Scalar.tryMk ty (x.val + y.val)
+
+def Scalar.sub {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+  Scalar.tryMk ty (x.val - y.val)
+
+def Scalar.mul {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+  Scalar.tryMk ty (x.val * y.val)
+
+-- TODO: instances of +, -, * etc. for scalars
+
+-- Cast an integer from a [src_ty] to a [tgt_ty]
+-- TODO: check the semantics of casts in Rust
+def Scalar.cast {src_ty : ScalarTy} (tgt_ty : ScalarTy) (x : Scalar src_ty) : Result (Scalar tgt_ty) :=
+  Scalar.tryMk tgt_ty x.val
+
+-- The scalar types
+-- We declare the definitions as reducible so that Lean can unfold them (useful
+-- for type class resolution for instance).
+@[reducible] def Isize := Scalar .Isize
+@[reducible] def I8    := Scalar .I8
+@[reducible] def I16   := Scalar .I16
+@[reducible] def I32   := Scalar .I32
+@[reducible] def I64   := Scalar .I64
+@[reducible] def I128  := Scalar .I128
+@[reducible] def Usize := Scalar .Usize
+@[reducible] def U8    := Scalar .U8
+@[reducible] def U16   := Scalar .U16
+@[reducible] def U32   := Scalar .U32
+@[reducible] def U64   := Scalar .U64
+@[reducible] def U128  := Scalar .U128
+
+-- TODO: below: not sure this is the best way.
+-- Should we rather overload operations like +, -, etc.?
+-- Also, it is possible to automate the generation of those definitions
+-- with macros (but would it be a good idea? It would be less easy to
+-- read the file, which is not supposed to change a lot)
+
+-- Negation
+
+/--
+Remark: there is no heterogeneous negation in the Lean prelude: we thus introduce
+one here.
+
+The notation typeclass for heterogeneous addition.
+This enables the notation `- a : β` where `a : α`.
+-/
+class HNeg (α : Type u) (β : outParam (Type v)) where
+  /-- `- a` computes the negation of `a`.
+  The meaning of this notation is type-dependent. -/
+  hNeg : α → β
+
+prefix:75  "-"   => HNeg.hNeg
+
+instance : HNeg Isize (Result Isize) where hNeg x := Scalar.neg x
+instance : HNeg I8 (Result I8) where hNeg x := Scalar.neg x
+instance : HNeg I16 (Result I16) where hNeg x := Scalar.neg x
+instance : HNeg I32 (Result I32) where hNeg x := Scalar.neg x
+instance : HNeg I64 (Result I64) where hNeg x := Scalar.neg x
+instance : HNeg I128 (Result I128) where hNeg x := Scalar.neg x
+
+-- Addition
+instance {ty} : HAdd (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+  hAdd x y := Scalar.add x y
+
+-- Substraction
+instance {ty} : HSub (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+  hSub x y := Scalar.sub x y
+
+-- Multiplication
+instance {ty} : HMul (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+  hMul x y := Scalar.mul x y
+
+-- Division
+instance {ty} : HDiv (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+  hDiv x y := Scalar.div x y
+
+-- Remainder
+instance {ty} : HMod (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+  hMod x y := Scalar.rem x y
+
+-- ofIntCore
+-- TODO: typeclass?
+def Isize.ofIntCore := @Scalar.ofIntCore .Isize
+def I8.ofIntCore    := @Scalar.ofIntCore .I8
+def I16.ofIntCore   := @Scalar.ofIntCore .I16
+def I32.ofIntCore   := @Scalar.ofIntCore .I32
+def I64.ofIntCore   := @Scalar.ofIntCore .I64
+def I128.ofIntCore  := @Scalar.ofIntCore .I128
+def Usize.ofIntCore := @Scalar.ofIntCore .Usize
+def U8.ofIntCore    := @Scalar.ofIntCore .U8
+def U16.ofIntCore   := @Scalar.ofIntCore .U16
+def U32.ofIntCore   := @Scalar.ofIntCore .U32
+def U64.ofIntCore   := @Scalar.ofIntCore .U64
+def U128.ofIntCore  := @Scalar.ofIntCore .U128
+
+--  ofInt
+-- TODO: typeclass?
+def Isize.ofInt := @Scalar.ofInt .Isize
+def I8.ofInt    := @Scalar.ofInt .I8
+def I16.ofInt   := @Scalar.ofInt .I16
+def I32.ofInt   := @Scalar.ofInt .I32
+def I64.ofInt   := @Scalar.ofInt .I64
+def I128.ofInt  := @Scalar.ofInt .I128
+def Usize.ofInt := @Scalar.ofInt .Usize
+def U8.ofInt    := @Scalar.ofInt .U8
+def U16.ofInt   := @Scalar.ofInt .U16
+def U32.ofInt   := @Scalar.ofInt .U32
+def U64.ofInt   := @Scalar.ofInt .U64
+def U128.ofInt  := @Scalar.ofInt .U128
+
+-- Comparisons
+instance {ty} : LT (Scalar ty) where
+  lt a b := LT.lt a.val b.val
+
+instance {ty} : LE (Scalar ty) where le a b := LE.le a.val b.val
+
+instance Scalar.decLt {ty} (a b : Scalar ty) : Decidable (LT.lt a b) := Int.decLt ..
+instance Scalar.decLe {ty} (a b : Scalar ty) : Decidable (LE.le a b) := Int.decLe ..
+
+theorem Scalar.eq_of_val_eq {ty} : ∀ {i j : Scalar ty}, Eq i.val j.val → Eq i j
+  | ⟨_, _, _⟩, ⟨_, _, _⟩, rfl => rfl
+
+theorem Scalar.val_eq_of_eq {ty} {i j : Scalar ty} (h : Eq i j) : Eq i.val j.val :=
+  h ▸ rfl
+
+theorem Scalar.ne_of_val_ne {ty} {i j : Scalar ty} (h : Not (Eq i.val j.val)) : Not (Eq i j) :=
+  fun h' => absurd (val_eq_of_eq h') h
+
+instance (ty : ScalarTy) : DecidableEq (Scalar ty) :=
+  fun i j =>
+    match decEq i.val j.val with
+    | isTrue h  => isTrue (Scalar.eq_of_val_eq h)
+    | isFalse h => isFalse (Scalar.ne_of_val_ne h)
+
+def Scalar.toInt {ty} (n : Scalar ty) : Int := n.val
+
+-- Tactic to prove that integers are in bounds
+syntax "intlit" : tactic
 
-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 :=
-  if m > 0 then
-    .ret ⟨ n.val / m.val, by
-      have h1: ↑n.val < USize.size := n.val.isLt
-      have h2: n.val.val / m.val.val <= n.val.val := @Nat.div_le_self n.val m.val
-      apply Nat.lt_of_le_of_lt h2 h1
-    ⟩
-  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
-  ofNatCore: (n:Nat) -> LT.lt n size -> t
-
-set_option hygiene false in
-run_cmd
-  for typeName in [`UInt8, `UInt16, `UInt32, `UInt64, `USize].map Lean.mkIdent do
-  Lean.Elab.Command.elabCommand (← `(
-    namespace $typeName
-    instance: MachineInteger $typeName where
-      size := size
-      val := val
-      ofNatCore := ofNatCore
-    end $typeName
-  ))
-
--- 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
+macro_rules
+  | `(tactic| intlit) => `(tactic| apply Scalar.bound_suffices ; decide)
+
+-- -- 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
+--   ofNatCore: (n:Nat) -> LT.lt n size -> t
+
+-- set_option hygiene false in
+-- run_cmd
+--   for typeName in [`UInt8, `UInt16, `UInt32, `UInt64, `USize].map Lean.mkIdent do
+--   Lean.Elab.Command.elabCommand (← `(
+--     namespace $typeName
+--     instance: MachineInteger $typeName where
+--       size := size
+--       val := val
+--       ofNatCore := ofNatCore
+--     end $typeName
+--   ))
+
+-- -- 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
 
 -------------
 -- 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_new (α : Type u): Vec α := ⟨ [], by {
-  match USize.size, usize_size_eq with
-  | _, Or.inl rfl => simp
-  | _, Or.inr rfl => simp
-  } ⟩
+def Vec (α : Type u) := { l : List α // List.length l <= Usize.max }
 
-#check vec_new
+def vec_new (α : Type u): Vec α := ⟨ [], by sorry ⟩
 
-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)
+  Usize.ofIntCore (List.length v) (by sorry) l
  
 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
--- 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) //
-  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
-    ⟨ return ⟨List.concat v.val x,
-      by
-        rw [List.length_concat]
-        assumption
-     ⟩, by simp ⟩
-  else
-    ⟨ fail maximumSizeExceeded, by simp ⟩
-
-#eval do
-  -- NOTE: the // notation is syntactic sugar for Subtype, a refinement with
-  -- fields val and property. However, Lean's elaborator can automatically
-  -- select the `val` field if the context provides a type annotation. We
-  -- 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?
-  -- 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)
   :=
-  if h : List.length v.val + 1 <= 4294967295 then
-    return ⟨ List.concat v.val x,
-      by
-        rw [List.length_concat]
-        have h': 4294967295 < USize.size := by intlit
-        apply Nat.lt_of_le_of_lt h h'
-    ⟩
-  else if h: List.length v.val + 1 < USize.size then
-    return ⟨List.concat v.val x,
-      by
-        rw [List.length_concat]
-        assumption
-     ⟩
+  if h : List.length v.val <= U32.max || List.length v.val <= Usize.max then
+    return ⟨ List.concat v.val x, by sorry ⟩
   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
+    -- TODO: maybe we should redefine a list library which uses integers
+    -- (instead of natural numbers)
+    let i : Nat :=
+      match i.val with
+      | .ofNat n => n
+      | .negSucc n => by sorry -- TODO: we can't get here
+    let isLt: i < USize.size := by sorry
+    let i : Fin USize.size := { val := i, isLt := isLt }
     .ret ⟨ List.set v.val i.val x, by
-      have h: List.length v.val < USize.size := v.property
+      have h: List.length v.val <= Usize.max := v.property
       rewrite [ List.length_set v.val i.val x ]
       assumption
     ⟩
   else
     .fail arrayOutOfBounds
 
-def vec_index_fwd (α : Type u) (v: Vec α) (i: USize): Result α :=
-  if h: i.val < List.length v.val then
+def vec_index_fwd (α : Type u) (v: Vec α) (i: Usize): Result α :=
+  if i.val < List.length v.val then
+    let i : Nat :=
+      match i.val with
+      | .ofNat n => n
+      | .negSucc n => by sorry -- TODO: we can't get here
+    let isLt: i < USize.size := by sorry
+    let i : Fin USize.size := { val := i, isLt := isLt }
+    let h: i < List.length v.val := by sorry
     .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 :=
-  if h: i.val < List.length v.val then
+def vec_index_mut_fwd (α : Type u) (v: Vec α) (i: Usize): Result α :=
+  if i.val < List.length v.val then
+    let i : Nat :=
+      match i.val with
+      | .ofNat n => n
+      | .negSucc n => by sorry -- TODO: we can't get here
+    let isLt: i < USize.size := by sorry
+    let i : Fin USize.size := { val := i, isLt := isLt }
+    let h: i < List.length v.val := by sorry
     .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
+    let i : Nat :=
+      match i.val with
+      | .ofNat n => n
+      | .negSucc n => by sorry -- TODO: we can't get here
+    let isLt: i < USize.size := by sorry
+    let i : Fin USize.size := { val := i, isLt := isLt }
     .ret ⟨ List.set v.val i.val x, by
-      have h: List.length v.val < USize.size := v.property
+      have h: List.length v.val <= Usize.max := v.property
       rewrite [ List.length_set v.val i.val x ]
       assumption
     ⟩
@@ -360,33 +578,3 @@ def mem_replace_back (a : Type) (_ : a) (y : a) : a :=
 /-- Aeneas-translated function -- useful to reduce non-recursive definitions.
  Use with `simp [ aeneas ]` -/
 register_simp_attr aeneas
-
---------------------
--- ASSERT COMMAND --
---------------------
-
-open Lean Elab Command Term Meta
-
-syntax (name := assert) "#assert" term: command
-
-@[command_elab assert]
-unsafe
-def assertImpl : CommandElab := fun (_stx: Syntax) => do
-  runTermElabM (fun _ => do
-    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 ())
-
-#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 937a15e5..8306ed85 100644
--- a/tests/lean/misc-constants/Constants.lean
+++ b/tests/lean/misc-constants/Constants.lean
@@ -2,143 +2,130 @@
 -- [constants]
 import Base.Primitives
 
-structure OpaqueDefs where
-  
-  /- [constants::X0] -/
-  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_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_c : UInt32 := eval_global x1_body (by simp)
-  
-  /- [constants::X2] -/
-  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 :=
-    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_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)
-  
-  /- [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 }
-  
-  /- [constants::P0] -/
-  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) :=
-    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
-    ((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
-    {
-      pair_x := (UInt32.ofNatCore 0 (by intlit)),
-      pair_y := (UInt32.ofNatCore 1 (by intlit))
-    }
-  def p3_c : pair_t UInt32 UInt32 := eval_global p3_body (by simp)
-  
-  /- [constants::Wrap] -/
-  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 }
-  
-  /- [constants::Y] -/
-  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
-  
-  /- [constants::YVAL] -/
-  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_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
-  
-  /- [constants::add] -/
-  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_c : Int32 := eval_global q1_body (by simp)
-  
-  /- [constants::Q2] -/
-  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_c : Int32 := eval_global q3_body (by simp)
-  
-  /- [constants::get_z2] -/
-  def get_z2_fwd : Result Int32 :=
-    do
-      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_c : UInt32 := eval_global s1_body (by simp)
-  
-  /- [constants::S2] -/
-  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_c : pair_t UInt32 UInt32 := eval_global s3_body (by simp)
-  
-  /- [constants::S4] -/
-  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)
-  
+/- [constants::X0] -/
+def x0_body : Result U32 := Result.ret (U32.ofInt 0 (by intlit))
+def x0_c : U32 := eval_global x0_body (by simp)
+
+/- [core::num::u32::{9}::MAX] -/
+def core_num_u32_max_body : Result U32 :=
+  Result.ret (U32.ofInt 4294967295 (by intlit))
+def core_num_u32_max_c : U32 := eval_global core_num_u32_max_body (by simp)
+
+/- [constants::X1] -/
+def x1_body : Result U32 := Result.ret core_num_u32_max_c
+def x1_c : U32 := eval_global x1_body (by simp)
+
+/- [constants::X2] -/
+def x2_body : Result U32 := Result.ret (U32.ofInt 3 (by intlit))
+def x2_c : U32 := eval_global x2_body (by simp)
+
+/- [constants::incr] -/
+def incr_fwd (n : U32) : Result U32 :=
+  n + (U32.ofInt 1 (by intlit))
+
+/- [constants::X3] -/
+def x3_body : Result U32 := incr_fwd (U32.ofInt 32 (by intlit))
+def x3_c : U32 := eval_global x3_body (by simp)
+
+/- [constants::mk_pair0] -/
+def mk_pair0_fwd (x : U32) (y : U32) : Result (U32 × U32) :=
+  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 : U32) (y : U32) : Result (pair_t U32 U32) :=
+  Result.ret { pair_x := x, pair_y := y }
+
+/- [constants::P0] -/
+def p0_body : Result (U32 × U32) :=
+  mk_pair0_fwd (U32.ofInt 0 (by intlit)) (U32.ofInt 1 (by intlit))
+def p0_c : (U32 × U32) := eval_global p0_body (by simp)
+
+/- [constants::P1] -/
+def p1_body : Result (pair_t U32 U32) :=
+  mk_pair1_fwd (U32.ofInt 0 (by intlit)) (U32.ofInt 1 (by intlit))
+def p1_c : pair_t U32 U32 := eval_global p1_body (by simp)
+
+/- [constants::P2] -/
+def p2_body : Result (U32 × U32) :=
+  Result.ret ((U32.ofInt 0 (by intlit)), (U32.ofInt 1 (by intlit)))
+def p2_c : (U32 × U32) := eval_global p2_body (by simp)
+
+/- [constants::P3] -/
+def p3_body : Result (pair_t U32 U32) :=
+  Result.ret
+  { pair_x := (U32.ofInt 0 (by intlit)), pair_y := (U32.ofInt 1 (by intlit)) }
+def p3_c : pair_t U32 U32 := eval_global p3_body (by simp)
+
+/- [constants::Wrap] -/
+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 }
+
+/- [constants::Y] -/
+def y_body : Result (wrap_t I32) := wrap_new_fwd I32 (I32.ofInt 2 (by intlit))
+def y_c : wrap_t I32 := eval_global y_body (by simp)
+
+/- [constants::unwrap_y] -/
+def unwrap_y_fwd : Result I32 :=
+  Result.ret y_c.wrap_val
+
+/- [constants::YVAL] -/
+def yval_body : Result I32 := unwrap_y_fwd
+def yval_c : I32 := eval_global yval_body (by simp)
+
+/- [constants::get_z1::Z1] -/
+def get_z1_z1_body : Result I32 := Result.ret (I32.ofInt 3 (by intlit))
+def get_z1_z1_c : I32 := eval_global get_z1_z1_body (by simp)
+
+/- [constants::get_z1] -/
+def get_z1_fwd : Result I32 :=
+  Result.ret get_z1_z1_c
+
+/- [constants::add] -/
+def add_fwd (a : I32) (b : I32) : Result I32 :=
+  a + b
+
+/- [constants::Q1] -/
+def q1_body : Result I32 := Result.ret (I32.ofInt 5 (by intlit))
+def q1_c : I32 := eval_global q1_body (by simp)
+
+/- [constants::Q2] -/
+def q2_body : Result I32 := Result.ret q1_c
+def q2_c : I32 := eval_global q2_body (by simp)
+
+/- [constants::Q3] -/
+def q3_body : Result I32 := add_fwd q2_c (I32.ofInt 3 (by intlit))
+def q3_c : I32 := eval_global q3_body (by simp)
+
+/- [constants::get_z2] -/
+def get_z2_fwd : Result I32 :=
+  do
+    let i ← get_z1_fwd
+    let i0 ← add_fwd i q3_c
+    add_fwd q1_c i0
+
+/- [constants::S1] -/
+def s1_body : Result U32 := Result.ret (U32.ofInt 6 (by intlit))
+def s1_c : U32 := eval_global s1_body (by simp)
+
+/- [constants::S2] -/
+def s2_body : Result U32 := incr_fwd s1_c
+def s2_c : U32 := eval_global s2_body (by simp)
+
+/- [constants::S3] -/
+def s3_body : Result (pair_t U32 U32) := Result.ret p3_c
+def s3_c : pair_t U32 U32 := eval_global s3_body (by simp)
+
+/- [constants::S4] -/
+def s4_body : Result (pair_t U32 U32) :=
+  mk_pair1_fwd (U32.ofInt 7 (by intlit)) (U32.ofInt 8 (by intlit))
+def s4_c : pair_t U32 U32 := 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 5b64e908..034f41b2 100644
--- a/tests/lean/misc-external/Base/Primitives.lean
+++ b/tests/lean/misc-external/Base/Primitives.lean
@@ -3,6 +3,28 @@ import Lean.Meta.Tactic.Simp
 import Init.Data.List.Basic
 import Mathlib.Tactic.RunCmd
 
+--------------------
+-- ASSERT COMMAND --
+--------------------
+
+open Lean Elab Command Term Meta
+
+syntax (name := assert) "#assert" term: command
+
+@[command_elab assert]
+unsafe
+def assertImpl : CommandElab := fun (_stx: Syntax) => do
+  runTermElabM (fun _ => do
+    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 ())
+
+#eval 2 == 2
+#assert (2 == 2)
+
 -------------
 -- PRELUDE --
 -------------
@@ -12,6 +34,7 @@ import Mathlib.Tactic.RunCmd
 inductive Error where
    | assertionFailure: Error
    | integerOverflow: Error
+   | divisionByZero: Error
    | arrayOutOfBounds: Error
    | maximumSizeExceeded: Error
    | panic: Error
@@ -89,17 +112,13 @@ macro "let" e:term " <-- " f:term : doElem =>
 -- MACHINE INTEGERS --
 ----------------------
 
--- 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.
+-- We redefine our machine integers types.
 
--- 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.)
+-- For Isize/Usize, we reuse `getNumBits` from `USize`. 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
@@ -111,236 +130,435 @@ macro "let" e:term " <-- " f:term : doElem =>
 -- 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
-
-macro_rules
-  | `(tactic| intlit) => `(tactic|
-    match USize.size, usize_size_eq with
-    | _, Or.inl rfl => decide
-    | _, Or.inr rfl => decide)
-
--- This is how the macro is expected to be used
-#eval USize.ofNatCore 0 (by intlit)
-
--- 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`).
+open System.Platform.getNumBits
+
+-- TODO: is there a way of only importing System.Platform.getNumBits?
+--
+@[simp] def size_num_bits : Nat := (System.Platform.getNumBits ()).val
+
+-- Remark: Lean seems to use < for the comparisons with the upper bounds by convention.
+-- We keep the F* convention for now.
+@[simp] def Isize.min : Int := - (HPow.hPow 2 (size_num_bits - 1))
+@[simp] def Isize.max : Int := (HPow.hPow 2 (size_num_bits - 1)) - 1
+@[simp] def I8.min    : Int := - (HPow.hPow 2 7)
+@[simp] def I8.max    : Int := HPow.hPow 2 7 - 1
+@[simp] def I16.min   : Int := - (HPow.hPow 2 15)
+@[simp] def I16.max   : Int := HPow.hPow 2 15 - 1
+@[simp] def I32.min   : Int := -(HPow.hPow 2 31)
+@[simp] def I32.max   : Int := HPow.hPow 2 31 - 1
+@[simp] def I64.min   : Int := -(HPow.hPow 2 63)
+@[simp] def I64.max   : Int := HPow.hPow 2 63 - 1
+@[simp] def I128.min  : Int := -(HPow.hPow 2 127)
+@[simp] def I128.max  : Int := HPow.hPow 2 127 - 1
+@[simp] def Usize.min : Int := 0
+@[simp] def Usize.max : Int := HPow.hPow 2 size_num_bits - 1
+@[simp] def U8.min    : Int := 0
+@[simp] def U8.max    : Int := HPow.hPow 2 8 - 1
+@[simp] def U16.min   : Int := 0
+@[simp] def U16.max   : Int := HPow.hPow 2 16 - 1
+@[simp] def U32.min   : Int := 0
+@[simp] def U32.max   : Int := HPow.hPow 2 32 - 1
+@[simp] def U64.min   : Int := 0
+@[simp] def U64.max   : Int := HPow.hPow 2 64 - 1
+@[simp] def U128.min  : Int := 0
+@[simp] def U128.max  : Int := HPow.hPow 2 128 - 1
+
+#assert (I8.min   == -128)
+#assert (I8.max   == 127)
+#assert (I16.min  == -32768)
+#assert (I16.max  == 32767)
+#assert (I32.min  == -2147483648)
+#assert (I32.max  == 2147483647)
+#assert (I64.min  == -9223372036854775808)
+#assert (I64.max  == 9223372036854775807)
+#assert (I128.min == -170141183460469231731687303715884105728)
+#assert (I128.max == 170141183460469231731687303715884105727)
+#assert (U8.min   == 0)
+#assert (U8.max   == 255)
+#assert (U16.min  == 0)
+#assert (U16.max  == 65535)
+#assert (U32.min  == 0)
+#assert (U32.max  == 4294967295)
+#assert (U64.min  == 0)
+#assert (U64.max  == 18446744073709551615)
+#assert (U128.min == 0)
+#assert (U128.max == 340282366920938463463374607431768211455)
+
+inductive ScalarTy :=
+| Isize
+| I8
+| I16
+| I32
+| I64
+| I128
+| Usize
+| U8
+| U16
+| U32
+| U64
+| U128
+
+def Scalar.min (ty : ScalarTy) : Int :=
+  match ty with
+  | .Isize => Isize.min
+  | .I8    => I8.min
+  | .I16   => I16.min
+  | .I32   => I32.min
+  | .I64   => I64.min
+  | .I128  => I128.min
+  | .Usize => Usize.min
+  | .U8    => U8.min
+  | .U16   => U16.min
+  | .U32   => U32.min
+  | .U64   => U64.min
+  | .U128  => U128.min
+
+def Scalar.max (ty : ScalarTy) : Int :=
+  match ty with
+  | .Isize => Isize.max
+  | .I8    => I8.max
+  | .I16   => I16.max
+  | .I32   => I32.max
+  | .I64   => I64.max
+  | .I128  => I128.max
+  | .Usize => Usize.max
+  | .U8    => U8.max
+  | .U16   => U16.max
+  | .U32   => U32.max
+  | .U64   => U64.max
+  | .U128  => U128.max
+
+-- "Conservative" bounds
+-- We use those because we can't compare to the isize bounds (which can't
+-- reduce at compile-time). Whenever we perform an arithmetic operation like
+-- addition we need to check that the result is in bounds: we first compare
+-- to the conservative bounds, which reduce, then compare to the real bounds.
 -- This is useful for the various #asserts that we want to reduce at
 -- type-checking time.
+def Scalar.cMin (ty : ScalarTy) : Int :=
+  match ty with
+  | .Isize => I32.min
+  | _ => Scalar.min ty
+
+def Scalar.cMax (ty : ScalarTy) : Int :=
+  match ty with
+  | .Isize => I32.max
+  | .Usize => U32.max
+  | _ => Scalar.max ty
+
+theorem Scalar.cMin_bound ty : Scalar.min ty <= Scalar.cMin ty := by sorry
+theorem Scalar.cMax_bound ty : Scalar.min ty <= Scalar.cMin ty := by sorry
+
+structure Scalar (ty : ScalarTy) where
+  val : Int
+  hmin : Scalar.min ty <= val
+  hmax : val <= Scalar.max ty
+
+theorem Scalar.bound_suffices (ty : ScalarTy) (x : Int) :
+  Scalar.cMin ty <= x && x <= Scalar.cMax ty ->
+  (decide (Scalar.min ty ≤ x) && decide (x ≤ Scalar.max ty)) = true
+  := by sorry
+
+def Scalar.ofIntCore {ty : ScalarTy} (x : Int)
+  (hmin : Scalar.min ty <= x) (hmax : x <= Scalar.max ty) : Scalar ty :=
+  { val := x, hmin := hmin, hmax := hmax }
+
+def Scalar.ofInt {ty : ScalarTy} (x : Int)
+  (h : Scalar.min ty <= x && x <= Scalar.max ty) : Scalar ty :=
+  let hmin: Scalar.min ty <= x := by sorry
+  let hmax: x <= Scalar.max ty := by sorry
+  Scalar.ofIntCore x hmin hmax
 
 -- 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 :=
-  -- NOTE: the test USize.toNat n - m >= 0 seems to always succeed?
-  if n >= m then
-    let n' := USize.toNat n
-    let m' := USize.toNat n
-    let r := USize.ofNatCore (n' - m') (by
-      have h: n' - m' <= n' := by
-        apply Nat.sub_le_of_le_add
-        case h => rewrite [ Nat.add_comm ]; apply Nat.le_add_left
-      apply Nat.lt_of_le_of_lt h
-      apply n.val.isLt
-    )
-    return r
-  else
-    fail integerOverflow
-
-@[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 :=
-  if h: m > 0 then
-    .ret ⟨ n.val % m.val, by
-      have h1: ↑m.val < USize.size := m.val.isLt
-      have h2: n.val.val % m.val.val < m.val.val := @Nat.mod_lt n.val m.val h
-      apply Nat.lt_trans h2 h1
-    ⟩
-  else
-    .fail integerOverflow
+def Scalar.tryMk (ty : ScalarTy) (x : Int) : Result (Scalar ty) :=
+  -- TODO: write this with only one if then else
+  if hmin_cons: Scalar.cMin ty <= x || Scalar.min ty <= x then
+    if hmax_cons: x <= Scalar.cMax ty || x <= Scalar.max ty then
+      let hmin: Scalar.min ty <= x := by sorry
+      let hmax: x <= Scalar.max ty := by sorry
+      return Scalar.ofIntCore x hmin hmax
+    else fail integerOverflow
+  else fail integerOverflow
+
+def Scalar.neg {ty : ScalarTy} (x : Scalar ty) : Result (Scalar ty) := Scalar.tryMk ty (- x.val)
+
+def Scalar.div {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+  if y.val != 0 then Scalar.tryMk ty (x.val / y.val) else fail divisionByZero
+
+-- Checking that the % operation in Lean computes the same as the remainder operation in Rust
+#assert 1 % 2 = (1:Int)
+#assert (-1) % 2 = -1
+#assert 1 % (-2) = 1
+#assert (-1) % (-2) = -1
+
+def Scalar.rem {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+  if y.val != 0 then Scalar.tryMk ty (x.val % y.val) else fail divisionByZero
+
+def Scalar.add {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+  Scalar.tryMk ty (x.val + y.val)
+
+def Scalar.sub {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+  Scalar.tryMk ty (x.val - y.val)
+
+def Scalar.mul {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+  Scalar.tryMk ty (x.val * y.val)
+
+-- TODO: instances of +, -, * etc. for scalars
+
+-- Cast an integer from a [src_ty] to a [tgt_ty]
+-- TODO: check the semantics of casts in Rust
+def Scalar.cast {src_ty : ScalarTy} (tgt_ty : ScalarTy) (x : Scalar src_ty) : Result (Scalar tgt_ty) :=
+  Scalar.tryMk tgt_ty x.val
+
+-- The scalar types
+-- We declare the definitions as reducible so that Lean can unfold them (useful
+-- for type class resolution for instance).
+@[reducible] def Isize := Scalar .Isize
+@[reducible] def I8    := Scalar .I8
+@[reducible] def I16   := Scalar .I16
+@[reducible] def I32   := Scalar .I32
+@[reducible] def I64   := Scalar .I64
+@[reducible] def I128  := Scalar .I128
+@[reducible] def Usize := Scalar .Usize
+@[reducible] def U8    := Scalar .U8
+@[reducible] def U16   := Scalar .U16
+@[reducible] def U32   := Scalar .U32
+@[reducible] def U64   := Scalar .U64
+@[reducible] def U128  := Scalar .U128
+
+-- TODO: below: not sure this is the best way.
+-- Should we rather overload operations like +, -, etc.?
+-- Also, it is possible to automate the generation of those definitions
+-- with macros (but would it be a good idea? It would be less easy to
+-- read the file, which is not supposed to change a lot)
+
+-- Negation
+
+/--
+Remark: there is no heterogeneous negation in the Lean prelude: we thus introduce
+one here.
+
+The notation typeclass for heterogeneous addition.
+This enables the notation `- a : β` where `a : α`.
+-/
+class HNeg (α : Type u) (β : outParam (Type v)) where
+  /-- `- a` computes the negation of `a`.
+  The meaning of this notation is type-dependent. -/
+  hNeg : α → β
+
+prefix:75  "-"   => HNeg.hNeg
+
+instance : HNeg Isize (Result Isize) where hNeg x := Scalar.neg x
+instance : HNeg I8 (Result I8) where hNeg x := Scalar.neg x
+instance : HNeg I16 (Result I16) where hNeg x := Scalar.neg x
+instance : HNeg I32 (Result I32) where hNeg x := Scalar.neg x
+instance : HNeg I64 (Result I64) where hNeg x := Scalar.neg x
+instance : HNeg I128 (Result I128) where hNeg x := Scalar.neg x
+
+-- Addition
+instance {ty} : HAdd (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+  hAdd x y := Scalar.add x y
+
+-- Substraction
+instance {ty} : HSub (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+  hSub x y := Scalar.sub x y
+
+-- Multiplication
+instance {ty} : HMul (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+  hMul x y := Scalar.mul x y
+
+-- Division
+instance {ty} : HDiv (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+  hDiv x y := Scalar.div x y
+
+-- Remainder
+instance {ty} : HMod (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+  hMod x y := Scalar.rem x y
+
+-- ofIntCore
+-- TODO: typeclass?
+def Isize.ofIntCore := @Scalar.ofIntCore .Isize
+def I8.ofIntCore    := @Scalar.ofIntCore .I8
+def I16.ofIntCore   := @Scalar.ofIntCore .I16
+def I32.ofIntCore   := @Scalar.ofIntCore .I32
+def I64.ofIntCore   := @Scalar.ofIntCore .I64
+def I128.ofIntCore  := @Scalar.ofIntCore .I128
+def Usize.ofIntCore := @Scalar.ofIntCore .Usize
+def U8.ofIntCore    := @Scalar.ofIntCore .U8
+def U16.ofIntCore   := @Scalar.ofIntCore .U16
+def U32.ofIntCore   := @Scalar.ofIntCore .U32
+def U64.ofIntCore   := @Scalar.ofIntCore .U64
+def U128.ofIntCore  := @Scalar.ofIntCore .U128
+
+--  ofInt
+-- TODO: typeclass?
+def Isize.ofInt := @Scalar.ofInt .Isize
+def I8.ofInt    := @Scalar.ofInt .I8
+def I16.ofInt   := @Scalar.ofInt .I16
+def I32.ofInt   := @Scalar.ofInt .I32
+def I64.ofInt   := @Scalar.ofInt .I64
+def I128.ofInt  := @Scalar.ofInt .I128
+def Usize.ofInt := @Scalar.ofInt .Usize
+def U8.ofInt    := @Scalar.ofInt .U8
+def U16.ofInt   := @Scalar.ofInt .U16
+def U32.ofInt   := @Scalar.ofInt .U32
+def U64.ofInt   := @Scalar.ofInt .U64
+def U128.ofInt  := @Scalar.ofInt .U128
+
+-- Comparisons
+instance {ty} : LT (Scalar ty) where
+  lt a b := LT.lt a.val b.val
+
+instance {ty} : LE (Scalar ty) where le a b := LE.le a.val b.val
+
+instance Scalar.decLt {ty} (a b : Scalar ty) : Decidable (LT.lt a b) := Int.decLt ..
+instance Scalar.decLe {ty} (a b : Scalar ty) : Decidable (LE.le a b) := Int.decLe ..
+
+theorem Scalar.eq_of_val_eq {ty} : ∀ {i j : Scalar ty}, Eq i.val j.val → Eq i j
+  | ⟨_, _, _⟩, ⟨_, _, _⟩, rfl => rfl
+
+theorem Scalar.val_eq_of_eq {ty} {i j : Scalar ty} (h : Eq i j) : Eq i.val j.val :=
+  h ▸ rfl
+
+theorem Scalar.ne_of_val_ne {ty} {i j : Scalar ty} (h : Not (Eq i.val j.val)) : Not (Eq i j) :=
+  fun h' => absurd (val_eq_of_eq h') h
+
+instance (ty : ScalarTy) : DecidableEq (Scalar ty) :=
+  fun i j =>
+    match decEq i.val j.val with
+    | isTrue h  => isTrue (Scalar.eq_of_val_eq h)
+    | isFalse h => isFalse (Scalar.ne_of_val_ne h)
+
+def Scalar.toInt {ty} (n : Scalar ty) : Int := n.val
+
+-- Tactic to prove that integers are in bounds
+syntax "intlit" : tactic
 
-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 :=
-  if m > 0 then
-    .ret ⟨ n.val / m.val, by
-      have h1: ↑n.val < USize.size := n.val.isLt
-      have h2: n.val.val / m.val.val <= n.val.val := @Nat.div_le_self n.val m.val
-      apply Nat.lt_of_le_of_lt h2 h1
-    ⟩
-  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
-  ofNatCore: (n:Nat) -> LT.lt n size -> t
-
-set_option hygiene false in
-run_cmd
-  for typeName in [`UInt8, `UInt16, `UInt32, `UInt64, `USize].map Lean.mkIdent do
-  Lean.Elab.Command.elabCommand (← `(
-    namespace $typeName
-    instance: MachineInteger $typeName where
-      size := size
-      val := val
-      ofNatCore := ofNatCore
-    end $typeName
-  ))
-
--- 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
+macro_rules
+  | `(tactic| intlit) => `(tactic| apply Scalar.bound_suffices ; decide)
+
+-- -- 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
+--   ofNatCore: (n:Nat) -> LT.lt n size -> t
+
+-- set_option hygiene false in
+-- run_cmd
+--   for typeName in [`UInt8, `UInt16, `UInt32, `UInt64, `USize].map Lean.mkIdent do
+--   Lean.Elab.Command.elabCommand (← `(
+--     namespace $typeName
+--     instance: MachineInteger $typeName where
+--       size := size
+--       val := val
+--       ofNatCore := ofNatCore
+--     end $typeName
+--   ))
+
+-- -- 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
 
 -------------
 -- 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_new (α : Type u): Vec α := ⟨ [], by {
-  match USize.size, usize_size_eq with
-  | _, Or.inl rfl => simp
-  | _, Or.inr rfl => simp
-  } ⟩
+def Vec (α : Type u) := { l : List α // List.length l <= Usize.max }
 
-#check vec_new
+def vec_new (α : Type u): Vec α := ⟨ [], by sorry ⟩
 
-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)
+  Usize.ofIntCore (List.length v) (by sorry) l
  
 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
--- 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) //
-  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
-    ⟨ return ⟨List.concat v.val x,
-      by
-        rw [List.length_concat]
-        assumption
-     ⟩, by simp ⟩
-  else
-    ⟨ fail maximumSizeExceeded, by simp ⟩
-
-#eval do
-  -- NOTE: the // notation is syntactic sugar for Subtype, a refinement with
-  -- fields val and property. However, Lean's elaborator can automatically
-  -- select the `val` field if the context provides a type annotation. We
-  -- 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?
-  -- 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)
   :=
-  if h : List.length v.val + 1 <= 4294967295 then
-    return ⟨ List.concat v.val x,
-      by
-        rw [List.length_concat]
-        have h': 4294967295 < USize.size := by intlit
-        apply Nat.lt_of_le_of_lt h h'
-    ⟩
-  else if h: List.length v.val + 1 < USize.size then
-    return ⟨List.concat v.val x,
-      by
-        rw [List.length_concat]
-        assumption
-     ⟩
+  if h : List.length v.val <= U32.max || List.length v.val <= Usize.max then
+    return ⟨ List.concat v.val x, by sorry ⟩
   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
+    -- TODO: maybe we should redefine a list library which uses integers
+    -- (instead of natural numbers)
+    let i : Nat :=
+      match i.val with
+      | .ofNat n => n
+      | .negSucc n => by sorry -- TODO: we can't get here
+    let isLt: i < USize.size := by sorry
+    let i : Fin USize.size := { val := i, isLt := isLt }
     .ret ⟨ List.set v.val i.val x, by
-      have h: List.length v.val < USize.size := v.property
+      have h: List.length v.val <= Usize.max := v.property
       rewrite [ List.length_set v.val i.val x ]
       assumption
     ⟩
   else
     .fail arrayOutOfBounds
 
-def vec_index_fwd (α : Type u) (v: Vec α) (i: USize): Result α :=
-  if h: i.val < List.length v.val then
+def vec_index_fwd (α : Type u) (v: Vec α) (i: Usize): Result α :=
+  if i.val < List.length v.val then
+    let i : Nat :=
+      match i.val with
+      | .ofNat n => n
+      | .negSucc n => by sorry -- TODO: we can't get here
+    let isLt: i < USize.size := by sorry
+    let i : Fin USize.size := { val := i, isLt := isLt }
+    let h: i < List.length v.val := by sorry
     .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 :=
-  if h: i.val < List.length v.val then
+def vec_index_mut_fwd (α : Type u) (v: Vec α) (i: Usize): Result α :=
+  if i.val < List.length v.val then
+    let i : Nat :=
+      match i.val with
+      | .ofNat n => n
+      | .negSucc n => by sorry -- TODO: we can't get here
+    let isLt: i < USize.size := by sorry
+    let i : Fin USize.size := { val := i, isLt := isLt }
+    let h: i < List.length v.val := by sorry
     .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
+    let i : Nat :=
+      match i.val with
+      | .ofNat n => n
+      | .negSucc n => by sorry -- TODO: we can't get here
+    let isLt: i < USize.size := by sorry
+    let i : Fin USize.size := { val := i, isLt := isLt }
     .ret ⟨ List.set v.val i.val x, by
-      have h: List.length v.val < USize.size := v.property
+      have h: List.length v.val <= Usize.max := v.property
       rewrite [ List.length_set v.val i.val x ]
       assumption
     ⟩
@@ -360,33 +578,3 @@ def mem_replace_back (a : Type) (_ : a) (y : a) : a :=
 /-- Aeneas-translated function -- useful to reduce non-recursive definitions.
  Use with `simp [ aeneas ]` -/
 register_simp_attr aeneas
-
---------------------
--- ASSERT COMMAND --
---------------------
-
-open Lean Elab Command Term Meta
-
-syntax (name := assert) "#assert" term: command
-
-@[command_elab assert]
-unsafe
-def assertImpl : CommandElab := fun (_stx: Syntax) => do
-  runTermElabM (fun _ => do
-    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 ())
-
-#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/ExternalFuns.lean b/tests/lean/misc-external/External/ExternalFuns.lean
new file mode 100644
index 00000000..6bd4f4a9
--- /dev/null
+++ b/tests/lean/misc-external/External/ExternalFuns.lean
@@ -0,0 +1,5 @@
+import Base.Primitives
+import External.Types
+import External.Opaque
+
+def opaque_defs : OpaqueDefs := sorry
diff --git a/tests/lean/misc-external/External/Funs.lean b/tests/lean/misc-external/External/Funs.lean
index 4e1f36a1..eeb83989 100644
--- a/tests/lean/misc-external/External/Funs.lean
+++ b/tests/lean/misc-external/External/Funs.lean
@@ -2,9 +2,7 @@
 -- [external]: function definitions
 import Base.Primitives
 import External.Types
-import External.Opaque
-
-section variable (opaque_defs: OpaqueDefs)
+import External.ExternalFuns
 
 /- [external::swap] -/
 def swap_fwd
@@ -28,9 +26,7 @@ def swap_back
 
 /- [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 : U32) (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
     opaque_defs.core_option_option_unwrap_fwd core_num_nonzero_non_zero_u32_t
@@ -39,13 +35,10 @@ def test_new_non_zero_u32_fwd
 /- [external::test_vec] -/
 def test_vec_fwd : Result Unit :=
   do
-    let v := vec_new UInt32
-    let _ ← vec_push_back UInt32 v (UInt32.ofNatCore 0 (by intlit))
+    let v := vec_new U32
+    let _ ← vec_push_back U32 v (U32.ofInt 0 (by intlit))
     Result.ret ()
 
-/- Unit test for [external::test_vec] -/
-#assert (test_vec_fwd == .ret ())
-
 /- [external::custom_swap] -/
 def custom_swap_fwd
   (T : Type) (x : T) (y : T) (st : State) : Result (State × T) :=
@@ -68,26 +61,24 @@ def custom_swap_back
 
 /- [external::test_custom_swap] -/
 def test_custom_swap_fwd
-  (x : UInt32) (y : UInt32) (st : State) : Result (State × Unit) :=
+  (x : U32) (y : U32) (st : State) : Result (State × Unit) :=
   do
-    let (st0, _) ← custom_swap_fwd UInt32 x y st
+    let (st0, _) ← custom_swap_fwd U32 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 : U32) (y : U32) (st : State) (st0 : State) :
+  Result (State × (U32 × U32))
   :=
-  custom_swap_back UInt32 x y st (UInt32.ofNatCore 1 (by intlit)) st0
+  custom_swap_back U32 x y st (U32.ofInt 1 (by intlit)) st0
 
 /- [external::test_swap_non_zero] -/
-def test_swap_non_zero_fwd
-  (x : UInt32) (st : State) : Result (State × UInt32) :=
+def test_swap_non_zero_fwd (x : U32) (st : State) : Result (State × U32) :=
   do
-    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 h: x0 = (UInt32.ofNatCore 0 (by intlit))
+    let (st0, _) ← swap_fwd U32 x (U32.ofInt 0 (by intlit)) st
+    let (st1, (x0, _)) ← swap_back U32 x (U32.ofInt 0 (by intlit)) st st0
+    if h: x0 = (U32.ofInt 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 d3582de3..d641912b 100644
--- a/tests/lean/misc-external/External/Opaque.lean
+++ b/tests/lean/misc-external/External/Opaque.lean
@@ -19,8 +19,7 @@ structure OpaqueDefs where
   /- [core::num::nonzero::NonZeroU32::{14}::new] -/
   core_num_nonzero_non_zero_u32_new_fwd
     :
-    UInt32 -> State -> Result (State × (Option
-      core_num_nonzero_non_zero_u32_t))
+    U32 -> State -> Result (State × (Option core_num_nonzero_non_zero_u32_t))
   
   /- [core::option::Option::{0}::unwrap] -/
   core_option_option_unwrap_fwd
diff --git a/tests/lean/misc-loops/Base/Primitives.lean b/tests/lean/misc-loops/Base/Primitives.lean
index 5b64e908..034f41b2 100644
--- a/tests/lean/misc-loops/Base/Primitives.lean
+++ b/tests/lean/misc-loops/Base/Primitives.lean
@@ -3,6 +3,28 @@ import Lean.Meta.Tactic.Simp
 import Init.Data.List.Basic
 import Mathlib.Tactic.RunCmd
 
+--------------------
+-- ASSERT COMMAND --
+--------------------
+
+open Lean Elab Command Term Meta
+
+syntax (name := assert) "#assert" term: command
+
+@[command_elab assert]
+unsafe
+def assertImpl : CommandElab := fun (_stx: Syntax) => do
+  runTermElabM (fun _ => do
+    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 ())
+
+#eval 2 == 2
+#assert (2 == 2)
+
 -------------
 -- PRELUDE --
 -------------
@@ -12,6 +34,7 @@ import Mathlib.Tactic.RunCmd
 inductive Error where
    | assertionFailure: Error
    | integerOverflow: Error
+   | divisionByZero: Error
    | arrayOutOfBounds: Error
    | maximumSizeExceeded: Error
    | panic: Error
@@ -89,17 +112,13 @@ macro "let" e:term " <-- " f:term : doElem =>
 -- MACHINE INTEGERS --
 ----------------------
 
--- 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.
+-- We redefine our machine integers types.
 
--- 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.)
+-- For Isize/Usize, we reuse `getNumBits` from `USize`. 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
@@ -111,236 +130,435 @@ macro "let" e:term " <-- " f:term : doElem =>
 -- 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
-
-macro_rules
-  | `(tactic| intlit) => `(tactic|
-    match USize.size, usize_size_eq with
-    | _, Or.inl rfl => decide
-    | _, Or.inr rfl => decide)
-
--- This is how the macro is expected to be used
-#eval USize.ofNatCore 0 (by intlit)
-
--- 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`).
+open System.Platform.getNumBits
+
+-- TODO: is there a way of only importing System.Platform.getNumBits?
+--
+@[simp] def size_num_bits : Nat := (System.Platform.getNumBits ()).val
+
+-- Remark: Lean seems to use < for the comparisons with the upper bounds by convention.
+-- We keep the F* convention for now.
+@[simp] def Isize.min : Int := - (HPow.hPow 2 (size_num_bits - 1))
+@[simp] def Isize.max : Int := (HPow.hPow 2 (size_num_bits - 1)) - 1
+@[simp] def I8.min    : Int := - (HPow.hPow 2 7)
+@[simp] def I8.max    : Int := HPow.hPow 2 7 - 1
+@[simp] def I16.min   : Int := - (HPow.hPow 2 15)
+@[simp] def I16.max   : Int := HPow.hPow 2 15 - 1
+@[simp] def I32.min   : Int := -(HPow.hPow 2 31)
+@[simp] def I32.max   : Int := HPow.hPow 2 31 - 1
+@[simp] def I64.min   : Int := -(HPow.hPow 2 63)
+@[simp] def I64.max   : Int := HPow.hPow 2 63 - 1
+@[simp] def I128.min  : Int := -(HPow.hPow 2 127)
+@[simp] def I128.max  : Int := HPow.hPow 2 127 - 1
+@[simp] def Usize.min : Int := 0
+@[simp] def Usize.max : Int := HPow.hPow 2 size_num_bits - 1
+@[simp] def U8.min    : Int := 0
+@[simp] def U8.max    : Int := HPow.hPow 2 8 - 1
+@[simp] def U16.min   : Int := 0
+@[simp] def U16.max   : Int := HPow.hPow 2 16 - 1
+@[simp] def U32.min   : Int := 0
+@[simp] def U32.max   : Int := HPow.hPow 2 32 - 1
+@[simp] def U64.min   : Int := 0
+@[simp] def U64.max   : Int := HPow.hPow 2 64 - 1
+@[simp] def U128.min  : Int := 0
+@[simp] def U128.max  : Int := HPow.hPow 2 128 - 1
+
+#assert (I8.min   == -128)
+#assert (I8.max   == 127)
+#assert (I16.min  == -32768)
+#assert (I16.max  == 32767)
+#assert (I32.min  == -2147483648)
+#assert (I32.max  == 2147483647)
+#assert (I64.min  == -9223372036854775808)
+#assert (I64.max  == 9223372036854775807)
+#assert (I128.min == -170141183460469231731687303715884105728)
+#assert (I128.max == 170141183460469231731687303715884105727)
+#assert (U8.min   == 0)
+#assert (U8.max   == 255)
+#assert (U16.min  == 0)
+#assert (U16.max  == 65535)
+#assert (U32.min  == 0)
+#assert (U32.max  == 4294967295)
+#assert (U64.min  == 0)
+#assert (U64.max  == 18446744073709551615)
+#assert (U128.min == 0)
+#assert (U128.max == 340282366920938463463374607431768211455)
+
+inductive ScalarTy :=
+| Isize
+| I8
+| I16
+| I32
+| I64
+| I128
+| Usize
+| U8
+| U16
+| U32
+| U64
+| U128
+
+def Scalar.min (ty : ScalarTy) : Int :=
+  match ty with
+  | .Isize => Isize.min
+  | .I8    => I8.min
+  | .I16   => I16.min
+  | .I32   => I32.min
+  | .I64   => I64.min
+  | .I128  => I128.min
+  | .Usize => Usize.min
+  | .U8    => U8.min
+  | .U16   => U16.min
+  | .U32   => U32.min
+  | .U64   => U64.min
+  | .U128  => U128.min
+
+def Scalar.max (ty : ScalarTy) : Int :=
+  match ty with
+  | .Isize => Isize.max
+  | .I8    => I8.max
+  | .I16   => I16.max
+  | .I32   => I32.max
+  | .I64   => I64.max
+  | .I128  => I128.max
+  | .Usize => Usize.max
+  | .U8    => U8.max
+  | .U16   => U16.max
+  | .U32   => U32.max
+  | .U64   => U64.max
+  | .U128  => U128.max
+
+-- "Conservative" bounds
+-- We use those because we can't compare to the isize bounds (which can't
+-- reduce at compile-time). Whenever we perform an arithmetic operation like
+-- addition we need to check that the result is in bounds: we first compare
+-- to the conservative bounds, which reduce, then compare to the real bounds.
 -- This is useful for the various #asserts that we want to reduce at
 -- type-checking time.
+def Scalar.cMin (ty : ScalarTy) : Int :=
+  match ty with
+  | .Isize => I32.min
+  | _ => Scalar.min ty
+
+def Scalar.cMax (ty : ScalarTy) : Int :=
+  match ty with
+  | .Isize => I32.max
+  | .Usize => U32.max
+  | _ => Scalar.max ty
+
+theorem Scalar.cMin_bound ty : Scalar.min ty <= Scalar.cMin ty := by sorry
+theorem Scalar.cMax_bound ty : Scalar.min ty <= Scalar.cMin ty := by sorry
+
+structure Scalar (ty : ScalarTy) where
+  val : Int
+  hmin : Scalar.min ty <= val
+  hmax : val <= Scalar.max ty
+
+theorem Scalar.bound_suffices (ty : ScalarTy) (x : Int) :
+  Scalar.cMin ty <= x && x <= Scalar.cMax ty ->
+  (decide (Scalar.min ty ≤ x) && decide (x ≤ Scalar.max ty)) = true
+  := by sorry
+
+def Scalar.ofIntCore {ty : ScalarTy} (x : Int)
+  (hmin : Scalar.min ty <= x) (hmax : x <= Scalar.max ty) : Scalar ty :=
+  { val := x, hmin := hmin, hmax := hmax }
+
+def Scalar.ofInt {ty : ScalarTy} (x : Int)
+  (h : Scalar.min ty <= x && x <= Scalar.max ty) : Scalar ty :=
+  let hmin: Scalar.min ty <= x := by sorry
+  let hmax: x <= Scalar.max ty := by sorry
+  Scalar.ofIntCore x hmin hmax
 
 -- 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 :=
-  -- NOTE: the test USize.toNat n - m >= 0 seems to always succeed?
-  if n >= m then
-    let n' := USize.toNat n
-    let m' := USize.toNat n
-    let r := USize.ofNatCore (n' - m') (by
-      have h: n' - m' <= n' := by
-        apply Nat.sub_le_of_le_add
-        case h => rewrite [ Nat.add_comm ]; apply Nat.le_add_left
-      apply Nat.lt_of_le_of_lt h
-      apply n.val.isLt
-    )
-    return r
-  else
-    fail integerOverflow
-
-@[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 :=
-  if h: m > 0 then
-    .ret ⟨ n.val % m.val, by
-      have h1: ↑m.val < USize.size := m.val.isLt
-      have h2: n.val.val % m.val.val < m.val.val := @Nat.mod_lt n.val m.val h
-      apply Nat.lt_trans h2 h1
-    ⟩
-  else
-    .fail integerOverflow
+def Scalar.tryMk (ty : ScalarTy) (x : Int) : Result (Scalar ty) :=
+  -- TODO: write this with only one if then else
+  if hmin_cons: Scalar.cMin ty <= x || Scalar.min ty <= x then
+    if hmax_cons: x <= Scalar.cMax ty || x <= Scalar.max ty then
+      let hmin: Scalar.min ty <= x := by sorry
+      let hmax: x <= Scalar.max ty := by sorry
+      return Scalar.ofIntCore x hmin hmax
+    else fail integerOverflow
+  else fail integerOverflow
+
+def Scalar.neg {ty : ScalarTy} (x : Scalar ty) : Result (Scalar ty) := Scalar.tryMk ty (- x.val)
+
+def Scalar.div {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+  if y.val != 0 then Scalar.tryMk ty (x.val / y.val) else fail divisionByZero
+
+-- Checking that the % operation in Lean computes the same as the remainder operation in Rust
+#assert 1 % 2 = (1:Int)
+#assert (-1) % 2 = -1
+#assert 1 % (-2) = 1
+#assert (-1) % (-2) = -1
+
+def Scalar.rem {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+  if y.val != 0 then Scalar.tryMk ty (x.val % y.val) else fail divisionByZero
+
+def Scalar.add {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+  Scalar.tryMk ty (x.val + y.val)
+
+def Scalar.sub {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+  Scalar.tryMk ty (x.val - y.val)
+
+def Scalar.mul {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+  Scalar.tryMk ty (x.val * y.val)
+
+-- TODO: instances of +, -, * etc. for scalars
+
+-- Cast an integer from a [src_ty] to a [tgt_ty]
+-- TODO: check the semantics of casts in Rust
+def Scalar.cast {src_ty : ScalarTy} (tgt_ty : ScalarTy) (x : Scalar src_ty) : Result (Scalar tgt_ty) :=
+  Scalar.tryMk tgt_ty x.val
+
+-- The scalar types
+-- We declare the definitions as reducible so that Lean can unfold them (useful
+-- for type class resolution for instance).
+@[reducible] def Isize := Scalar .Isize
+@[reducible] def I8    := Scalar .I8
+@[reducible] def I16   := Scalar .I16
+@[reducible] def I32   := Scalar .I32
+@[reducible] def I64   := Scalar .I64
+@[reducible] def I128  := Scalar .I128
+@[reducible] def Usize := Scalar .Usize
+@[reducible] def U8    := Scalar .U8
+@[reducible] def U16   := Scalar .U16
+@[reducible] def U32   := Scalar .U32
+@[reducible] def U64   := Scalar .U64
+@[reducible] def U128  := Scalar .U128
+
+-- TODO: below: not sure this is the best way.
+-- Should we rather overload operations like +, -, etc.?
+-- Also, it is possible to automate the generation of those definitions
+-- with macros (but would it be a good idea? It would be less easy to
+-- read the file, which is not supposed to change a lot)
+
+-- Negation
+
+/--
+Remark: there is no heterogeneous negation in the Lean prelude: we thus introduce
+one here.
+
+The notation typeclass for heterogeneous addition.
+This enables the notation `- a : β` where `a : α`.
+-/
+class HNeg (α : Type u) (β : outParam (Type v)) where
+  /-- `- a` computes the negation of `a`.
+  The meaning of this notation is type-dependent. -/
+  hNeg : α → β
+
+prefix:75  "-"   => HNeg.hNeg
+
+instance : HNeg Isize (Result Isize) where hNeg x := Scalar.neg x
+instance : HNeg I8 (Result I8) where hNeg x := Scalar.neg x
+instance : HNeg I16 (Result I16) where hNeg x := Scalar.neg x
+instance : HNeg I32 (Result I32) where hNeg x := Scalar.neg x
+instance : HNeg I64 (Result I64) where hNeg x := Scalar.neg x
+instance : HNeg I128 (Result I128) where hNeg x := Scalar.neg x
+
+-- Addition
+instance {ty} : HAdd (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+  hAdd x y := Scalar.add x y
+
+-- Substraction
+instance {ty} : HSub (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+  hSub x y := Scalar.sub x y
+
+-- Multiplication
+instance {ty} : HMul (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+  hMul x y := Scalar.mul x y
+
+-- Division
+instance {ty} : HDiv (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+  hDiv x y := Scalar.div x y
+
+-- Remainder
+instance {ty} : HMod (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+  hMod x y := Scalar.rem x y
+
+-- ofIntCore
+-- TODO: typeclass?
+def Isize.ofIntCore := @Scalar.ofIntCore .Isize
+def I8.ofIntCore    := @Scalar.ofIntCore .I8
+def I16.ofIntCore   := @Scalar.ofIntCore .I16
+def I32.ofIntCore   := @Scalar.ofIntCore .I32
+def I64.ofIntCore   := @Scalar.ofIntCore .I64
+def I128.ofIntCore  := @Scalar.ofIntCore .I128
+def Usize.ofIntCore := @Scalar.ofIntCore .Usize
+def U8.ofIntCore    := @Scalar.ofIntCore .U8
+def U16.ofIntCore   := @Scalar.ofIntCore .U16
+def U32.ofIntCore   := @Scalar.ofIntCore .U32
+def U64.ofIntCore   := @Scalar.ofIntCore .U64
+def U128.ofIntCore  := @Scalar.ofIntCore .U128
+
+--  ofInt
+-- TODO: typeclass?
+def Isize.ofInt := @Scalar.ofInt .Isize
+def I8.ofInt    := @Scalar.ofInt .I8
+def I16.ofInt   := @Scalar.ofInt .I16
+def I32.ofInt   := @Scalar.ofInt .I32
+def I64.ofInt   := @Scalar.ofInt .I64
+def I128.ofInt  := @Scalar.ofInt .I128
+def Usize.ofInt := @Scalar.ofInt .Usize
+def U8.ofInt    := @Scalar.ofInt .U8
+def U16.ofInt   := @Scalar.ofInt .U16
+def U32.ofInt   := @Scalar.ofInt .U32
+def U64.ofInt   := @Scalar.ofInt .U64
+def U128.ofInt  := @Scalar.ofInt .U128
+
+-- Comparisons
+instance {ty} : LT (Scalar ty) where
+  lt a b := LT.lt a.val b.val
+
+instance {ty} : LE (Scalar ty) where le a b := LE.le a.val b.val
+
+instance Scalar.decLt {ty} (a b : Scalar ty) : Decidable (LT.lt a b) := Int.decLt ..
+instance Scalar.decLe {ty} (a b : Scalar ty) : Decidable (LE.le a b) := Int.decLe ..
+
+theorem Scalar.eq_of_val_eq {ty} : ∀ {i j : Scalar ty}, Eq i.val j.val → Eq i j
+  | ⟨_, _, _⟩, ⟨_, _, _⟩, rfl => rfl
+
+theorem Scalar.val_eq_of_eq {ty} {i j : Scalar ty} (h : Eq i j) : Eq i.val j.val :=
+  h ▸ rfl
+
+theorem Scalar.ne_of_val_ne {ty} {i j : Scalar ty} (h : Not (Eq i.val j.val)) : Not (Eq i j) :=
+  fun h' => absurd (val_eq_of_eq h') h
+
+instance (ty : ScalarTy) : DecidableEq (Scalar ty) :=
+  fun i j =>
+    match decEq i.val j.val with
+    | isTrue h  => isTrue (Scalar.eq_of_val_eq h)
+    | isFalse h => isFalse (Scalar.ne_of_val_ne h)
+
+def Scalar.toInt {ty} (n : Scalar ty) : Int := n.val
+
+-- Tactic to prove that integers are in bounds
+syntax "intlit" : tactic
 
-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 :=
-  if m > 0 then
-    .ret ⟨ n.val / m.val, by
-      have h1: ↑n.val < USize.size := n.val.isLt
-      have h2: n.val.val / m.val.val <= n.val.val := @Nat.div_le_self n.val m.val
-      apply Nat.lt_of_le_of_lt h2 h1
-    ⟩
-  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
-  ofNatCore: (n:Nat) -> LT.lt n size -> t
-
-set_option hygiene false in
-run_cmd
-  for typeName in [`UInt8, `UInt16, `UInt32, `UInt64, `USize].map Lean.mkIdent do
-  Lean.Elab.Command.elabCommand (← `(
-    namespace $typeName
-    instance: MachineInteger $typeName where
-      size := size
-      val := val
-      ofNatCore := ofNatCore
-    end $typeName
-  ))
-
--- 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
+macro_rules
+  | `(tactic| intlit) => `(tactic| apply Scalar.bound_suffices ; decide)
+
+-- -- 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
+--   ofNatCore: (n:Nat) -> LT.lt n size -> t
+
+-- set_option hygiene false in
+-- run_cmd
+--   for typeName in [`UInt8, `UInt16, `UInt32, `UInt64, `USize].map Lean.mkIdent do
+--   Lean.Elab.Command.elabCommand (← `(
+--     namespace $typeName
+--     instance: MachineInteger $typeName where
+--       size := size
+--       val := val
+--       ofNatCore := ofNatCore
+--     end $typeName
+--   ))
+
+-- -- 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
 
 -------------
 -- 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_new (α : Type u): Vec α := ⟨ [], by {
-  match USize.size, usize_size_eq with
-  | _, Or.inl rfl => simp
-  | _, Or.inr rfl => simp
-  } ⟩
+def Vec (α : Type u) := { l : List α // List.length l <= Usize.max }
 
-#check vec_new
+def vec_new (α : Type u): Vec α := ⟨ [], by sorry ⟩
 
-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)
+  Usize.ofIntCore (List.length v) (by sorry) l
  
 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
--- 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) //
-  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
-    ⟨ return ⟨List.concat v.val x,
-      by
-        rw [List.length_concat]
-        assumption
-     ⟩, by simp ⟩
-  else
-    ⟨ fail maximumSizeExceeded, by simp ⟩
-
-#eval do
-  -- NOTE: the // notation is syntactic sugar for Subtype, a refinement with
-  -- fields val and property. However, Lean's elaborator can automatically
-  -- select the `val` field if the context provides a type annotation. We
-  -- 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?
-  -- 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)
   :=
-  if h : List.length v.val + 1 <= 4294967295 then
-    return ⟨ List.concat v.val x,
-      by
-        rw [List.length_concat]
-        have h': 4294967295 < USize.size := by intlit
-        apply Nat.lt_of_le_of_lt h h'
-    ⟩
-  else if h: List.length v.val + 1 < USize.size then
-    return ⟨List.concat v.val x,
-      by
-        rw [List.length_concat]
-        assumption
-     ⟩
+  if h : List.length v.val <= U32.max || List.length v.val <= Usize.max then
+    return ⟨ List.concat v.val x, by sorry ⟩
   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
+    -- TODO: maybe we should redefine a list library which uses integers
+    -- (instead of natural numbers)
+    let i : Nat :=
+      match i.val with
+      | .ofNat n => n
+      | .negSucc n => by sorry -- TODO: we can't get here
+    let isLt: i < USize.size := by sorry
+    let i : Fin USize.size := { val := i, isLt := isLt }
     .ret ⟨ List.set v.val i.val x, by
-      have h: List.length v.val < USize.size := v.property
+      have h: List.length v.val <= Usize.max := v.property
       rewrite [ List.length_set v.val i.val x ]
       assumption
     ⟩
   else
     .fail arrayOutOfBounds
 
-def vec_index_fwd (α : Type u) (v: Vec α) (i: USize): Result α :=
-  if h: i.val < List.length v.val then
+def vec_index_fwd (α : Type u) (v: Vec α) (i: Usize): Result α :=
+  if i.val < List.length v.val then
+    let i : Nat :=
+      match i.val with
+      | .ofNat n => n
+      | .negSucc n => by sorry -- TODO: we can't get here
+    let isLt: i < USize.size := by sorry
+    let i : Fin USize.size := { val := i, isLt := isLt }
+    let h: i < List.length v.val := by sorry
     .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 :=
-  if h: i.val < List.length v.val then
+def vec_index_mut_fwd (α : Type u) (v: Vec α) (i: Usize): Result α :=
+  if i.val < List.length v.val then
+    let i : Nat :=
+      match i.val with
+      | .ofNat n => n
+      | .negSucc n => by sorry -- TODO: we can't get here
+    let isLt: i < USize.size := by sorry
+    let i : Fin USize.size := { val := i, isLt := isLt }
+    let h: i < List.length v.val := by sorry
     .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
+    let i : Nat :=
+      match i.val with
+      | .ofNat n => n
+      | .negSucc n => by sorry -- TODO: we can't get here
+    let isLt: i < USize.size := by sorry
+    let i : Fin USize.size := { val := i, isLt := isLt }
     .ret ⟨ List.set v.val i.val x, by
-      have h: List.length v.val < USize.size := v.property
+      have h: List.length v.val <= Usize.max := v.property
       rewrite [ List.length_set v.val i.val x ]
       assumption
     ⟩
@@ -360,33 +578,3 @@ def mem_replace_back (a : Type) (_ : a) (y : a) : a :=
 /-- Aeneas-translated function -- useful to reduce non-recursive definitions.
  Use with `simp [ aeneas ]` -/
 register_simp_attr aeneas
-
---------------------
--- ASSERT COMMAND --
---------------------
-
-open Lean Elab Command Term Meta
-
-syntax (name := assert) "#assert" term: command
-
-@[command_elab assert]
-unsafe
-def assertImpl : CommandElab := fun (_stx: Syntax) => do
-  runTermElabM (fun _ => do
-    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 ())
-
-#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/Clauses.lean b/tests/lean/misc-loops/Loops/Clauses/Clauses.lean
index 5ddb65ca..89a7ce34 100644
--- a/tests/lean/misc-loops/Loops/Clauses/Clauses.lean
+++ b/tests/lean/misc-loops/Loops/Clauses/Clauses.lean
@@ -4,7 +4,7 @@ import Loops.Types
 
 /- [loops::sum]: termination measure -/
 @[simp]
-def sum_loop_terminates (max : UInt32) (i : UInt32) (s : UInt32) := (max, i, s)
+def sum_loop_terminates (max : U32) (i : U32) (s : U32) := (max, i, s)
 
 syntax "sum_loop_decreases" term+ : tactic
 
@@ -13,8 +13,7 @@ macro_rules
 
 /- [loops::sum_with_mut_borrows]: termination measure -/
 @[simp]
-def sum_with_mut_borrows_loop_terminates (max : UInt32) (mi : UInt32)
-  (ms : UInt32) :=
+def sum_with_mut_borrows_loop_terminates (max : U32) (mi : U32) (ms : U32) :=
   (max, mi, ms)
 
 syntax "sum_with_mut_borrows_loop_decreases" term+ : tactic
@@ -24,8 +23,7 @@ macro_rules
 
 /- [loops::sum_with_shared_borrows]: termination measure -/
 @[simp]
-def sum_with_shared_borrows_loop_terminates (max : UInt32) (i : UInt32)
-  (s : UInt32) :=
+def sum_with_shared_borrows_loop_terminates (max : U32) (i : U32) (s : U32) :=
   (max, i, s)
 
 syntax "sum_with_shared_borrows_loop_decreases" term+ : tactic
@@ -34,7 +32,7 @@ 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 U32) (i : Usize) := (v, i)
 
 syntax "clear_loop_decreases" term+ : tactic
 
@@ -43,7 +41,7 @@ macro_rules
 
 /- [loops::list_mem]: termination measure -/
 @[simp]
-def list_mem_loop_terminates (x : UInt32) (ls : list_t UInt32) := (x, ls)
+def list_mem_loop_terminates (x : U32) (ls : list_t U32) := (x, ls)
 
 syntax "list_mem_loop_decreases" term+ : tactic
 
@@ -52,8 +50,7 @@ macro_rules
 
 /- [loops::list_nth_mut_loop]: termination measure -/
 @[simp]
-def list_nth_mut_loop_loop_terminates (T : Type) (ls : list_t T) (i : UInt32)
-  :=
+def list_nth_mut_loop_loop_terminates (T : Type) (ls : list_t T) (i : U32) :=
   (ls, i)
 
 syntax "list_nth_mut_loop_loop_decreases" term+ : tactic
@@ -63,8 +60,7 @@ macro_rules
 
 /- [loops::list_nth_shared_loop]: termination measure -/
 @[simp]
-def list_nth_shared_loop_loop_terminates (T : Type) (ls : list_t T)
-  (i : UInt32) :=
+def list_nth_shared_loop_loop_terminates (T : Type) (ls : list_t T) (i : U32) :=
   (ls, i)
 
 syntax "list_nth_shared_loop_loop_decreases" term+ : tactic
@@ -74,7 +70,7 @@ macro_rules
 
 /- [loops::get_elem_mut]: termination measure -/
 @[simp]
-def get_elem_mut_loop_terminates (x : USize) (ls : list_t USize) := (x, ls)
+def get_elem_mut_loop_terminates (x : Usize) (ls : list_t Usize) := (x, ls)
 
 syntax "get_elem_mut_loop_decreases" term+ : tactic
 
@@ -83,7 +79,7 @@ macro_rules
 
 /- [loops::get_elem_shared]: termination measure -/
 @[simp]
-def get_elem_shared_loop_terminates (x : USize) (ls : list_t USize) := (x, ls)
+def get_elem_shared_loop_terminates (x : Usize) (ls : list_t Usize) := (x, ls)
 
 syntax "get_elem_shared_loop_decreases" term+ : tactic
 
@@ -92,7 +88,7 @@ macro_rules
 
 /- [loops::list_nth_mut_loop_with_id]: termination measure -/
 @[simp]
-def list_nth_mut_loop_with_id_loop_terminates (T : Type) (i : UInt32)
+def list_nth_mut_loop_with_id_loop_terminates (T : Type) (i : U32)
   (ls : list_t T) :=
   (i, ls)
 
@@ -103,7 +99,7 @@ macro_rules
 
 /- [loops::list_nth_shared_loop_with_id]: termination measure -/
 @[simp]
-def list_nth_shared_loop_with_id_loop_terminates (T : Type) (i : UInt32)
+def list_nth_shared_loop_with_id_loop_terminates (T : Type) (i : U32)
   (ls : list_t T) :=
   (i, ls)
 
@@ -115,7 +111,7 @@ macro_rules
 /- [loops::list_nth_mut_loop_pair]: termination measure -/
 @[simp]
 def list_nth_mut_loop_pair_loop_terminates (T : Type) (ls0 : list_t T)
-  (ls1 : list_t T) (i : UInt32) :=
+  (ls1 : list_t T) (i : U32) :=
   (ls0, ls1, i)
 
 syntax "list_nth_mut_loop_pair_loop_decreases" term+ : tactic
@@ -126,7 +122,7 @@ macro_rules
 /- [loops::list_nth_shared_loop_pair]: termination measure -/
 @[simp]
 def list_nth_shared_loop_pair_loop_terminates (T : Type) (ls0 : list_t T)
-  (ls1 : list_t T) (i : UInt32) :=
+  (ls1 : list_t T) (i : U32) :=
   (ls0, ls1, i)
 
 syntax "list_nth_shared_loop_pair_loop_decreases" term+ : tactic
@@ -138,7 +134,7 @@ macro_rules
 /- [loops::list_nth_mut_loop_pair_merge]: termination measure -/
 @[simp]
 def list_nth_mut_loop_pair_merge_loop_terminates (T : Type) (ls0 : list_t T)
-  (ls1 : list_t T) (i : UInt32) :=
+  (ls1 : list_t T) (i : U32) :=
   (ls0, ls1, i)
 
 syntax "list_nth_mut_loop_pair_merge_loop_decreases" term+ : tactic
@@ -150,7 +146,7 @@ macro_rules
 /- [loops::list_nth_shared_loop_pair_merge]: termination measure -/
 @[simp]
 def list_nth_shared_loop_pair_merge_loop_terminates (T : Type) (ls0 : list_t T)
-  (ls1 : list_t T) (i : UInt32) :=
+  (ls1 : list_t T) (i : U32) :=
   (ls0, ls1, i)
 
 syntax "list_nth_shared_loop_pair_merge_loop_decreases" term+ : tactic
@@ -162,7 +158,7 @@ macro_rules
 /- [loops::list_nth_mut_shared_loop_pair]: termination measure -/
 @[simp]
 def list_nth_mut_shared_loop_pair_loop_terminates (T : Type) (ls0 : list_t T)
-  (ls1 : list_t T) (i : UInt32) :=
+  (ls1 : list_t T) (i : U32) :=
   (ls0, ls1, i)
 
 syntax "list_nth_mut_shared_loop_pair_loop_decreases" term+ : tactic
@@ -174,7 +170,7 @@ macro_rules
 /- [loops::list_nth_mut_shared_loop_pair_merge]: termination measure -/
 @[simp]
 def list_nth_mut_shared_loop_pair_merge_loop_terminates (T : Type)
-  (ls0 : list_t T) (ls1 : list_t T) (i : UInt32) :=
+  (ls0 : list_t T) (ls1 : list_t T) (i : U32) :=
   (ls0, ls1, i)
 
 syntax "list_nth_mut_shared_loop_pair_merge_loop_decreases" term+ : tactic
@@ -186,7 +182,7 @@ macro_rules
 /- [loops::list_nth_shared_mut_loop_pair]: termination measure -/
 @[simp]
 def list_nth_shared_mut_loop_pair_loop_terminates (T : Type) (ls0 : list_t T)
-  (ls1 : list_t T) (i : UInt32) :=
+  (ls1 : list_t T) (i : U32) :=
   (ls0, ls1, i)
 
 syntax "list_nth_shared_mut_loop_pair_loop_decreases" term+ : tactic
@@ -198,7 +194,7 @@ macro_rules
 /- [loops::list_nth_shared_mut_loop_pair_merge]: termination measure -/
 @[simp]
 def list_nth_shared_mut_loop_pair_merge_loop_terminates (T : Type)
-  (ls0 : list_t T) (ls1 : list_t T) (i : UInt32) :=
+  (ls0 : list_t T) (ls1 : list_t T) (i : U32) :=
   (ls0, ls1, i)
 
 syntax "list_nth_shared_mut_loop_pair_merge_loop_decreases" term+ : tactic
diff --git a/tests/lean/misc-loops/Loops/Clauses/Template.lean b/tests/lean/misc-loops/Loops/Clauses/Template.lean
index d1e72d65..2e28a6c0 100644
--- a/tests/lean/misc-loops/Loops/Clauses/Template.lean
+++ b/tests/lean/misc-loops/Loops/Clauses/Template.lean
@@ -4,8 +4,7 @@ import Base.Primitives
 import Loops.Types
 
 /- [loops::sum]: termination measure -/
-@[simp]
-def sum_loop_terminates (max : UInt32) (i : UInt32) (s : UInt32) := (max, i, s)
+@[simp] def sum_loop_terminates (max : U32) (i : U32) (s : U32) := (max, i, s)
 
 /- [loops::sum]: decreases_by tactic -/
 syntax "sum_loop_decreases" term+ : tactic
@@ -14,8 +13,7 @@ macro_rules
 
 /- [loops::sum_with_mut_borrows]: termination measure -/
 @[simp]
-def sum_with_mut_borrows_loop_terminates (max : UInt32) (mi : UInt32)
-  (ms : UInt32) :=
+def sum_with_mut_borrows_loop_terminates (max : U32) (mi : U32) (ms : U32) :=
   (max, mi, ms)
 
 /- [loops::sum_with_mut_borrows]: decreases_by tactic -/
@@ -25,8 +23,7 @@ macro_rules
 
 /- [loops::sum_with_shared_borrows]: termination measure -/
 @[simp]
-def sum_with_shared_borrows_loop_terminates (max : UInt32) (i : UInt32)
-  (s : UInt32) :=
+def sum_with_shared_borrows_loop_terminates (max : U32) (i : U32) (s : U32) :=
   (max, i, s)
 
 /- [loops::sum_with_shared_borrows]: decreases_by tactic -/
@@ -35,7 +32,7 @@ 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 U32) (i : Usize) := (v, i)
 
 /- [loops::clear]: decreases_by tactic -/
 syntax "clear_loop_decreases" term+ : tactic
@@ -43,8 +40,7 @@ macro_rules
 | `(tactic| clear_loop_decreases $v $i) =>`(tactic| sorry)
 
 /- [loops::list_mem]: termination measure -/
-@[simp]
-def list_mem_loop_terminates (x : UInt32) (ls : list_t UInt32) := (x, ls)
+@[simp] def list_mem_loop_terminates (x : U32) (ls : list_t U32) := (x, ls)
 
 /- [loops::list_mem]: decreases_by tactic -/
 syntax "list_mem_loop_decreases" term+ : tactic
@@ -53,8 +49,7 @@ macro_rules
 
 /- [loops::list_nth_mut_loop]: termination measure -/
 @[simp]
-def list_nth_mut_loop_loop_terminates (T : Type) (ls : list_t T) (i : UInt32)
-  :=
+def list_nth_mut_loop_loop_terminates (T : Type) (ls : list_t T) (i : U32) :=
   (ls, i)
 
 /- [loops::list_nth_mut_loop]: decreases_by tactic -/
@@ -64,8 +59,8 @@ macro_rules
 
 /- [loops::list_nth_shared_loop]: termination measure -/
 @[simp]
-def list_nth_shared_loop_loop_terminates (T : Type) (ls : list_t T)
-  (i : UInt32) :=
+def list_nth_shared_loop_loop_terminates (T : Type) (ls : list_t T) (i : U32)
+  :=
   (ls, i)
 
 /- [loops::list_nth_shared_loop]: decreases_by tactic -/
@@ -75,7 +70,7 @@ macro_rules
 
 /- [loops::get_elem_mut]: termination measure -/
 @[simp]
-def get_elem_mut_loop_terminates (x : USize) (ls : list_t USize) := (x, ls)
+def get_elem_mut_loop_terminates (x : Usize) (ls : list_t Usize) := (x, ls)
 
 /- [loops::get_elem_mut]: decreases_by tactic -/
 syntax "get_elem_mut_loop_decreases" term+ : tactic
@@ -84,7 +79,7 @@ macro_rules
 
 /- [loops::get_elem_shared]: termination measure -/
 @[simp]
-def get_elem_shared_loop_terminates (x : USize) (ls : list_t USize) := (x, ls)
+def get_elem_shared_loop_terminates (x : Usize) (ls : list_t Usize) := (x, ls)
 
 /- [loops::get_elem_shared]: decreases_by tactic -/
 syntax "get_elem_shared_loop_decreases" term+ : tactic
@@ -93,7 +88,7 @@ macro_rules
 
 /- [loops::list_nth_mut_loop_with_id]: termination measure -/
 @[simp]
-def list_nth_mut_loop_with_id_loop_terminates (T : Type) (i : UInt32)
+def list_nth_mut_loop_with_id_loop_terminates (T : Type) (i : U32)
   (ls : list_t T) :=
   (i, ls)
 
@@ -104,7 +99,7 @@ macro_rules
 
 /- [loops::list_nth_shared_loop_with_id]: termination measure -/
 @[simp]
-def list_nth_shared_loop_with_id_loop_terminates (T : Type) (i : UInt32)
+def list_nth_shared_loop_with_id_loop_terminates (T : Type) (i : U32)
   (ls : list_t T) :=
   (i, ls)
 
@@ -116,7 +111,7 @@ macro_rules
 /- [loops::list_nth_mut_loop_pair]: termination measure -/
 @[simp]
 def list_nth_mut_loop_pair_loop_terminates (T : Type) (ls0 : list_t T)
-  (ls1 : list_t T) (i : UInt32) :=
+  (ls1 : list_t T) (i : U32) :=
   (ls0, ls1, i)
 
 /- [loops::list_nth_mut_loop_pair]: decreases_by tactic -/
@@ -127,7 +122,7 @@ macro_rules
 /- [loops::list_nth_shared_loop_pair]: termination measure -/
 @[simp]
 def list_nth_shared_loop_pair_loop_terminates (T : Type) (ls0 : list_t T)
-  (ls1 : list_t T) (i : UInt32) :=
+  (ls1 : list_t T) (i : U32) :=
   (ls0, ls1, i)
 
 /- [loops::list_nth_shared_loop_pair]: decreases_by tactic -/
@@ -139,7 +134,7 @@ macro_rules
 /- [loops::list_nth_mut_loop_pair_merge]: termination measure -/
 @[simp]
 def list_nth_mut_loop_pair_merge_loop_terminates (T : Type) (ls0 : list_t T)
-  (ls1 : list_t T) (i : UInt32) :=
+  (ls1 : list_t T) (i : U32) :=
   (ls0, ls1, i)
 
 /- [loops::list_nth_mut_loop_pair_merge]: decreases_by tactic -/
@@ -151,7 +146,7 @@ macro_rules
 /- [loops::list_nth_shared_loop_pair_merge]: termination measure -/
 @[simp]
 def list_nth_shared_loop_pair_merge_loop_terminates (T : Type) (ls0 : list_t T)
-  (ls1 : list_t T) (i : UInt32) :=
+  (ls1 : list_t T) (i : U32) :=
   (ls0, ls1, i)
 
 /- [loops::list_nth_shared_loop_pair_merge]: decreases_by tactic -/
@@ -163,7 +158,7 @@ macro_rules
 /- [loops::list_nth_mut_shared_loop_pair]: termination measure -/
 @[simp]
 def list_nth_mut_shared_loop_pair_loop_terminates (T : Type) (ls0 : list_t T)
-  (ls1 : list_t T) (i : UInt32) :=
+  (ls1 : list_t T) (i : U32) :=
   (ls0, ls1, i)
 
 /- [loops::list_nth_mut_shared_loop_pair]: decreases_by tactic -/
@@ -175,7 +170,7 @@ macro_rules
 /- [loops::list_nth_mut_shared_loop_pair_merge]: termination measure -/
 @[simp]
 def list_nth_mut_shared_loop_pair_merge_loop_terminates (T : Type)
-  (ls0 : list_t T) (ls1 : list_t T) (i : UInt32) :=
+  (ls0 : list_t T) (ls1 : list_t T) (i : U32) :=
   (ls0, ls1, i)
 
 /- [loops::list_nth_mut_shared_loop_pair_merge]: decreases_by tactic -/
@@ -187,7 +182,7 @@ macro_rules
 /- [loops::list_nth_shared_mut_loop_pair]: termination measure -/
 @[simp]
 def list_nth_shared_mut_loop_pair_loop_terminates (T : Type) (ls0 : list_t T)
-  (ls1 : list_t T) (i : UInt32) :=
+  (ls1 : list_t T) (i : U32) :=
   (ls0, ls1, i)
 
 /- [loops::list_nth_shared_mut_loop_pair]: decreases_by tactic -/
@@ -199,7 +194,7 @@ macro_rules
 /- [loops::list_nth_shared_mut_loop_pair_merge]: termination measure -/
 @[simp]
 def list_nth_shared_mut_loop_pair_merge_loop_terminates (T : Type)
-  (ls0 : list_t T) (ls1 : list_t T) (i : UInt32) :=
+  (ls0 : list_t T) (ls1 : list_t T) (i : U32) :=
   (ls0, ls1, i)
 
 /- [loops::list_nth_shared_mut_loop_pair_merge]: decreases_by tactic -/
diff --git a/tests/lean/misc-loops/Loops/Funs.lean b/tests/lean/misc-loops/Loops/Funs.lean
index f79a27a9..fd8d62d7 100644
--- a/tests/lean/misc-loops/Loops/Funs.lean
+++ b/tests/lean/misc-loops/Loops/Funs.lean
@@ -5,79 +5,78 @@ import Loops.Types
 import Loops.Clauses.Clauses
 
 /- [loops::sum] -/
-def sum_loop_fwd (max : UInt32) (i : UInt32) (s : UInt32) : (Result UInt32) :=
+def sum_loop_fwd (max : U32) (i : U32) (s : U32) : (Result U32) :=
   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 ← s + i
+      let i0 ← i + (U32.ofInt 1 (by intlit))
       sum_loop_fwd max i0 s0
-  else UInt32.checked_mul s (UInt32.ofNatCore 2 (by intlit))
+  else s * (U32.ofInt 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 :=
-  sum_loop_fwd max (UInt32.ofNatCore 0 (by intlit))
-    (UInt32.ofNatCore 0 (by intlit))
+def sum_fwd (max : U32) : Result U32 :=
+  sum_loop_fwd max (U32.ofInt 0 (by intlit)) (U32.ofInt 0 (by intlit))
 
 /- [loops::sum_with_mut_borrows] -/
 def sum_with_mut_borrows_loop_fwd
-  (max : UInt32) (mi : UInt32) (ms : UInt32) : (Result UInt32) :=
+  (max : U32) (mi : U32) (ms : U32) : (Result U32) :=
   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 ← ms + mi
+      let mi0 ← mi + (U32.ofInt 1 (by intlit))
       sum_with_mut_borrows_loop_fwd max mi0 ms0
-  else UInt32.checked_mul ms (UInt32.ofNatCore 2 (by intlit))
+  else ms * (U32.ofInt 2 (by intlit))
 termination_by sum_with_mut_borrows_loop_fwd max mi ms =>
   sum_with_mut_borrows_loop_terminates 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 :=
-  sum_with_mut_borrows_loop_fwd max (UInt32.ofNatCore 0 (by intlit))
-    (UInt32.ofNatCore 0 (by intlit))
+def sum_with_mut_borrows_fwd (max : U32) : Result U32 :=
+  sum_with_mut_borrows_loop_fwd max (U32.ofInt 0 (by intlit))
+    (U32.ofInt 0 (by intlit))
 
 /- [loops::sum_with_shared_borrows] -/
 def sum_with_shared_borrows_loop_fwd
-  (max : UInt32) (i : UInt32) (s : UInt32) : (Result UInt32) :=
+  (max : U32) (i : U32) (s : U32) : (Result U32) :=
   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 ← i + (U32.ofInt 1 (by intlit))
+      let s0 ← s + i0
       sum_with_shared_borrows_loop_fwd max i0 s0
-  else UInt32.checked_mul s (UInt32.ofNatCore 2 (by intlit))
+  else s * (U32.ofInt 2 (by intlit))
 termination_by sum_with_shared_borrows_loop_fwd max i s =>
   sum_with_shared_borrows_loop_terminates 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 :=
-  sum_with_shared_borrows_loop_fwd max (UInt32.ofNatCore 0 (by intlit))
-    (UInt32.ofNatCore 0 (by intlit))
+def sum_with_shared_borrows_fwd (max : U32) : Result U32 :=
+  sum_with_shared_borrows_loop_fwd max (U32.ofInt 0 (by intlit))
+    (U32.ofInt 0 (by intlit))
 
 /- [loops::clear] -/
-def clear_loop_fwd_back (v : Vec UInt32) (i : USize) : (Result (Vec UInt32)) :=
-  let i0 := vec_len UInt32 v
+def clear_loop_fwd_back (v : Vec U32) (i : Usize) : (Result (Vec U32)) :=
+  let i0 := vec_len U32 v
   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 ← i + (Usize.ofInt 1 (by intlit))
+      let v0 ← vec_index_mut_back U32 v i (U32.ofInt 0 (by intlit))
       clear_loop_fwd_back v0 i1
   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) :=
-  clear_loop_fwd_back v (USize.ofNatCore 0 (by intlit))
+def clear_fwd_back (v : Vec U32) : Result (Vec U32) :=
+  clear_loop_fwd_back v (Usize.ofInt 0 (by intlit))
 
 /- [loops::list_mem] -/
-def list_mem_loop_fwd (x : UInt32) (ls : list_t UInt32) : (Result Bool) :=
+def list_mem_loop_fwd (x : U32) (ls : list_t U32) : (Result Bool) :=
   match h: ls with
   | list_t.Cons y tl =>
     if h: y = x
@@ -88,19 +87,19 @@ 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 : U32) (ls : list_t U32) : 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) :=
+  (T : Type) (ls : list_t T) (i : U32) : (Result T) :=
   match h: ls with
   | list_t.Cons x tl =>
-    if h: i = (UInt32.ofNatCore 0 (by intlit))
+    if h: i = (U32.ofInt 0 (by intlit))
     then Result.ret x
     else
       do
-        let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+        let i0 ← i - (U32.ofInt 1 (by intlit))
         list_nth_mut_loop_loop_fwd T tl i0
   | list_t.Nil => Result.fail Error.panic
 termination_by list_nth_mut_loop_loop_fwd ls i =>
@@ -108,19 +107,19 @@ termination_by list_nth_mut_loop_loop_fwd 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 : U32) : 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)) :=
+  (T : Type) (ls : list_t T) (i : U32) (ret0 : T) : (Result (list_t T)) :=
   match h: ls with
   | list_t.Cons x tl =>
-    if h: i = (UInt32.ofNatCore 0 (by intlit))
+    if h: i = (U32.ofInt 0 (by intlit))
     then Result.ret (list_t.Cons ret0 tl)
     else
       do
-        let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+        let i0 ← i - (U32.ofInt 1 (by intlit))
         let tl0 ← list_nth_mut_loop_loop_back T tl i0 ret0
         Result.ret (list_t.Cons x tl0)
   | list_t.Nil => Result.fail Error.panic
@@ -130,19 +129,19 @@ 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 : U32) (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) :=
+  (T : Type) (ls : list_t T) (i : U32) : (Result T) :=
   match h: ls with
   | list_t.Cons x tl =>
-    if h: i = (UInt32.ofNatCore 0 (by intlit))
+    if h: i = (U32.ofInt 0 (by intlit))
     then Result.ret x
     else
       do
-        let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+        let i0 ← i - (U32.ofInt 1 (by intlit))
         list_nth_shared_loop_loop_fwd T tl i0
   | list_t.Nil => Result.fail Error.panic
 termination_by list_nth_shared_loop_loop_fwd ls i =>
@@ -150,12 +149,11 @@ termination_by list_nth_shared_loop_loop_fwd 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 :=
+def list_nth_shared_loop_fwd (T : Type) (ls : list_t T) (i : U32) : 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) :=
+def get_elem_mut_loop_fwd (x : Usize) (ls : list_t Usize) : (Result Usize) :=
   match h: ls with
   | list_t.Cons y tl =>
     if h: y = x
@@ -166,15 +164,15 @@ 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 ←
-      vec_index_mut_fwd (list_t USize) slots (USize.ofNatCore 0 (by intlit))
+      vec_index_mut_fwd (list_t Usize) slots (Usize.ofInt 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)) :=
+  (x : Usize) (ls : list_t Usize) (ret0 : Usize) : (Result (list_t Usize)) :=
   match h: ls with
   | list_t.Cons y tl =>
     if h: y = x
@@ -190,18 +188,18 @@ 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 ←
-      vec_index_mut_fwd (list_t USize) slots (USize.ofNatCore 0 (by intlit))
+      vec_index_mut_fwd (list_t Usize) slots (Usize.ofInt 0 (by intlit))
     let l0 ← get_elem_mut_loop_back x l ret0
-    vec_index_mut_back (list_t USize) slots (USize.ofNatCore 0 (by intlit)) l0
+    vec_index_mut_back (list_t Usize) slots (Usize.ofInt 0 (by intlit)) l0
 
 /- [loops::get_elem_shared] -/
 def get_elem_shared_loop_fwd
-  (x : USize) (ls : list_t USize) : (Result USize) :=
+  (x : Usize) (ls : list_t Usize) : (Result Usize) :=
   match h: ls with
   | list_t.Cons y tl =>
     if h: y = x
@@ -214,10 +212,9 @@ 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.ofInt 0 (by intlit))
     get_elem_shared_loop_fwd x l
 
 /- [loops::id_mut] -/
@@ -235,14 +232,14 @@ def id_shared_fwd (T : Type) (ls : list_t T) : Result (list_t T) :=
 
 /- [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) :=
+  (T : Type) (i : U32) (ls : list_t T) : (Result T) :=
   match h: ls with
   | list_t.Cons x tl =>
-    if h: i = (UInt32.ofNatCore 0 (by intlit))
+    if h: i = (U32.ofInt 0 (by intlit))
     then Result.ret x
     else
       do
-        let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+        let i0 ← i - (U32.ofInt 1 (by intlit))
         list_nth_mut_loop_with_id_loop_fwd T i0 tl
   | list_t.Nil => Result.fail Error.panic
 termination_by list_nth_mut_loop_with_id_loop_fwd i ls =>
@@ -251,21 +248,21 @@ 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 : U32) : Result T :=
   do
     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)) :=
+  (T : Type) (i : U32) (ls : list_t T) (ret0 : T) : (Result (list_t T)) :=
   match h: ls with
   | list_t.Cons x tl =>
-    if h: i = (UInt32.ofNatCore 0 (by intlit))
+    if h: i = (U32.ofInt 0 (by intlit))
     then Result.ret (list_t.Cons ret0 tl)
     else
       do
-        let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+        let i0 ← i - (U32.ofInt 1 (by intlit))
         let tl0 ← list_nth_mut_loop_with_id_loop_back T i0 tl ret0
         Result.ret (list_t.Cons x tl0)
   | list_t.Nil => Result.fail Error.panic
@@ -275,7 +272,7 @@ 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 : U32) (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
@@ -283,14 +280,14 @@ def list_nth_mut_loop_with_id_back
 
 /- [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) :=
+  (T : Type) (i : U32) (ls : list_t T) : (Result T) :=
   match h: ls with
   | list_t.Cons x tl =>
-    if h: i = (UInt32.ofNatCore 0 (by intlit))
+    if h: i = (U32.ofInt 0 (by intlit))
     then Result.ret x
     else
       do
-        let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+        let i0 ← i - (U32.ofInt 1 (by intlit))
         list_nth_shared_loop_with_id_loop_fwd T i0 tl
   | list_t.Nil => Result.fail Error.panic
 termination_by list_nth_shared_loop_with_id_loop_fwd i ls =>
@@ -299,25 +296,23 @@ 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 : U32) : Result T :=
   do
     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))
-  :=
+  (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : U32) : (Result (T × T)) :=
   match h: ls0 with
   | list_t.Cons x0 tl0 =>
     match h: ls1 with
     | list_t.Cons x1 tl1 =>
-      if h: i = (UInt32.ofNatCore 0 (by intlit))
+      if h: i = (U32.ofInt 0 (by intlit))
       then Result.ret (x0, x1)
       else
         do
-          let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+          let i0 ← i - (U32.ofInt 1 (by intlit))
           list_nth_mut_loop_pair_loop_fwd T tl0 tl1 i0
     | list_t.Nil => Result.fail Error.panic
   | list_t.Nil => Result.fail Error.panic
@@ -327,25 +322,23 @@ 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)
-  :=
+  (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : U32) : 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) :
+  (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : U32) (ret0 : T) :
   (Result (list_t T))
   :=
   match h: ls0 with
   | list_t.Cons x0 tl0 =>
     match h: ls1 with
     | list_t.Cons x1 tl1 =>
-      if h: i = (UInt32.ofNatCore 0 (by intlit))
+      if h: i = (U32.ofInt 0 (by intlit))
       then Result.ret (list_t.Cons ret0 tl0)
       else
         do
-          let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+          let i0 ← i - (U32.ofInt 1 (by intlit))
           let tl00 ← list_nth_mut_loop_pair_loop_back'a T tl0 tl1 i0 ret0
           Result.ret (list_t.Cons x0 tl00)
     | list_t.Nil => Result.fail Error.panic
@@ -356,25 +349,25 @@ 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) :
+  (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : U32) (ret0 : 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) :
+  (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : U32) (ret0 : T) :
   (Result (list_t T))
   :=
   match h: ls0 with
   | list_t.Cons x0 tl0 =>
     match h: ls1 with
     | list_t.Cons x1 tl1 =>
-      if h: i = (UInt32.ofNatCore 0 (by intlit))
+      if h: i = (U32.ofInt 0 (by intlit))
       then Result.ret (list_t.Cons ret0 tl1)
       else
         do
-          let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+          let i0 ← i - (U32.ofInt 1 (by intlit))
           let tl10 ← list_nth_mut_loop_pair_loop_back'b T tl0 tl1 i0 ret0
           Result.ret (list_t.Cons x1 tl10)
     | list_t.Nil => Result.fail Error.panic
@@ -385,25 +378,23 @@ 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) :
+  (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : U32) (ret0 : 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))
-  :=
+  (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : U32) : (Result (T × T)) :=
   match h: ls0 with
   | list_t.Cons x0 tl0 =>
     match h: ls1 with
     | list_t.Cons x1 tl1 =>
-      if h: i = (UInt32.ofNatCore 0 (by intlit))
+      if h: i = (U32.ofInt 0 (by intlit))
       then Result.ret (x0, x1)
       else
         do
-          let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+          let i0 ← i - (U32.ofInt 1 (by intlit))
           list_nth_shared_loop_pair_loop_fwd T tl0 tl1 i0
     | list_t.Nil => Result.fail Error.panic
   | list_t.Nil => Result.fail Error.panic
@@ -413,25 +404,21 @@ 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)
-  :=
+  (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : U32) : 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))
-  :=
+  (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : U32) : (Result (T × T)) :=
   match h: ls0 with
   | list_t.Cons x0 tl0 =>
     match h: ls1 with
     | list_t.Cons x1 tl1 =>
-      if h: i = (UInt32.ofNatCore 0 (by intlit))
+      if h: i = (U32.ofInt 0 (by intlit))
       then Result.ret (x0, x1)
       else
         do
-          let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+          let i0 ← i - (U32.ofInt 1 (by intlit))
           list_nth_mut_loop_pair_merge_loop_fwd T tl0 tl1 i0
     | list_t.Nil => Result.fail Error.panic
   | list_t.Nil => Result.fail Error.panic
@@ -441,27 +428,25 @@ 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)
-  :=
+  (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : U32) : 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)) :
+  (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : U32) (ret0 : (T × T)) :
   (Result ((list_t T) × (list_t T)))
   :=
   match h: ls0 with
   | list_t.Cons x0 tl0 =>
     match h: ls1 with
     | list_t.Cons x1 tl1 =>
-      if h: i = (UInt32.ofNatCore 0 (by intlit))
+      if h: i = (U32.ofInt 0 (by intlit))
       then
         let (t, t0) := ret0
         Result.ret (list_t.Cons t tl0, list_t.Cons t0 tl1)
       else
         do
-          let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+          let i0 ← i - (U32.ofInt 1 (by intlit))
           let (tl00, tl10) ←
             list_nth_mut_loop_pair_merge_loop_back T tl0 tl1 i0 ret0
           Result.ret (list_t.Cons x0 tl00, list_t.Cons x1 tl10)
@@ -473,25 +458,23 @@ 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)) :
+  (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : U32) (ret0 : (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))
-  :=
+  (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : U32) : (Result (T × T)) :=
   match h: ls0 with
   | list_t.Cons x0 tl0 =>
     match h: ls1 with
     | list_t.Cons x1 tl1 =>
-      if h: i = (UInt32.ofNatCore 0 (by intlit))
+      if h: i = (U32.ofInt 0 (by intlit))
       then Result.ret (x0, x1)
       else
         do
-          let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+          let i0 ← i - (U32.ofInt 1 (by intlit))
           list_nth_shared_loop_pair_merge_loop_fwd T tl0 tl1 i0
     | list_t.Nil => Result.fail Error.panic
   | list_t.Nil => Result.fail Error.panic
@@ -501,25 +484,21 @@ 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)
-  :=
+  (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : U32) : 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))
-  :=
+  (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : U32) : (Result (T × T)) :=
   match h: ls0 with
   | list_t.Cons x0 tl0 =>
     match h: ls1 with
     | list_t.Cons x1 tl1 =>
-      if h: i = (UInt32.ofNatCore 0 (by intlit))
+      if h: i = (U32.ofInt 0 (by intlit))
       then Result.ret (x0, x1)
       else
         do
-          let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+          let i0 ← i - (U32.ofInt 1 (by intlit))
           list_nth_mut_shared_loop_pair_loop_fwd T tl0 tl1 i0
     | list_t.Nil => Result.fail Error.panic
   | list_t.Nil => Result.fail Error.panic
@@ -529,25 +508,23 @@ 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)
-  :=
+  (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : U32) : 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) :
+  (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : U32) (ret0 : T) :
   (Result (list_t T))
   :=
   match h: ls0 with
   | list_t.Cons x0 tl0 =>
     match h: ls1 with
     | list_t.Cons x1 tl1 =>
-      if h: i = (UInt32.ofNatCore 0 (by intlit))
+      if h: i = (U32.ofInt 0 (by intlit))
       then Result.ret (list_t.Cons ret0 tl0)
       else
         do
-          let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+          let i0 ← i - (U32.ofInt 1 (by intlit))
           let tl00 ←
             list_nth_mut_shared_loop_pair_loop_back T tl0 tl1 i0 ret0
           Result.ret (list_t.Cons x0 tl00)
@@ -559,25 +536,23 @@ 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) :
+  (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : U32) (ret0 : 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))
-  :=
+  (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : U32) : (Result (T × T)) :=
   match h: ls0 with
   | list_t.Cons x0 tl0 =>
     match h: ls1 with
     | list_t.Cons x1 tl1 =>
-      if h: i = (UInt32.ofNatCore 0 (by intlit))
+      if h: i = (U32.ofInt 0 (by intlit))
       then Result.ret (x0, x1)
       else
         do
-          let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+          let i0 ← i - (U32.ofInt 1 (by intlit))
           list_nth_mut_shared_loop_pair_merge_loop_fwd T tl0 tl1 i0
     | list_t.Nil => Result.fail Error.panic
   | list_t.Nil => Result.fail Error.panic
@@ -587,25 +562,23 @@ 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)
-  :=
+  (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : U32) : 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) :
+  (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : U32) (ret0 : T) :
   (Result (list_t T))
   :=
   match h: ls0 with
   | list_t.Cons x0 tl0 =>
     match h: ls1 with
     | list_t.Cons x1 tl1 =>
-      if h: i = (UInt32.ofNatCore 0 (by intlit))
+      if h: i = (U32.ofInt 0 (by intlit))
       then Result.ret (list_t.Cons ret0 tl0)
       else
         do
-          let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+          let i0 ← i - (U32.ofInt 1 (by intlit))
           let tl00 ←
             list_nth_mut_shared_loop_pair_merge_loop_back T tl0 tl1 i0 ret0
           Result.ret (list_t.Cons x0 tl00)
@@ -617,25 +590,23 @@ 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) :
+  (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : U32) (ret0 : 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))
-  :=
+  (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : U32) : (Result (T × T)) :=
   match h: ls0 with
   | list_t.Cons x0 tl0 =>
     match h: ls1 with
     | list_t.Cons x1 tl1 =>
-      if h: i = (UInt32.ofNatCore 0 (by intlit))
+      if h: i = (U32.ofInt 0 (by intlit))
       then Result.ret (x0, x1)
       else
         do
-          let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+          let i0 ← i - (U32.ofInt 1 (by intlit))
           list_nth_shared_mut_loop_pair_loop_fwd T tl0 tl1 i0
     | list_t.Nil => Result.fail Error.panic
   | list_t.Nil => Result.fail Error.panic
@@ -645,25 +616,23 @@ 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)
-  :=
+  (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : U32) : 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) :
+  (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : U32) (ret0 : T) :
   (Result (list_t T))
   :=
   match h: ls0 with
   | list_t.Cons x0 tl0 =>
     match h: ls1 with
     | list_t.Cons x1 tl1 =>
-      if h: i = (UInt32.ofNatCore 0 (by intlit))
+      if h: i = (U32.ofInt 0 (by intlit))
       then Result.ret (list_t.Cons ret0 tl1)
       else
         do
-          let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+          let i0 ← i - (U32.ofInt 1 (by intlit))
           let tl10 ←
             list_nth_shared_mut_loop_pair_loop_back T tl0 tl1 i0 ret0
           Result.ret (list_t.Cons x1 tl10)
@@ -675,25 +644,23 @@ 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) :
+  (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : U32) (ret0 : 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))
-  :=
+  (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : U32) : (Result (T × T)) :=
   match h: ls0 with
   | list_t.Cons x0 tl0 =>
     match h: ls1 with
     | list_t.Cons x1 tl1 =>
-      if h: i = (UInt32.ofNatCore 0 (by intlit))
+      if h: i = (U32.ofInt 0 (by intlit))
       then Result.ret (x0, x1)
       else
         do
-          let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+          let i0 ← i - (U32.ofInt 1 (by intlit))
           list_nth_shared_mut_loop_pair_merge_loop_fwd T tl0 tl1 i0
     | list_t.Nil => Result.fail Error.panic
   | list_t.Nil => Result.fail Error.panic
@@ -703,25 +670,23 @@ 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)
-  :=
+  (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : U32) : 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) :
+  (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : U32) (ret0 : T) :
   (Result (list_t T))
   :=
   match h: ls0 with
   | list_t.Cons x0 tl0 =>
     match h: ls1 with
     | list_t.Cons x1 tl1 =>
-      if h: i = (UInt32.ofNatCore 0 (by intlit))
+      if h: i = (U32.ofInt 0 (by intlit))
       then Result.ret (list_t.Cons ret0 tl1)
       else
         do
-          let i0 ← UInt32.checked_sub i (UInt32.ofNatCore 1 (by intlit))
+          let i0 ← i - (U32.ofInt 1 (by intlit))
           let tl10 ←
             list_nth_shared_mut_loop_pair_merge_loop_back T tl0 tl1 i0 ret0
           Result.ret (list_t.Cons x1 tl10)
@@ -733,7 +698,7 @@ 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) :
+  (T : Type) (ls0 : list_t T) (ls1 : list_t T) (i : U32) (ret0 : 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 5b64e908..034f41b2 100644
--- a/tests/lean/misc-no_nested_borrows/Base/Primitives.lean
+++ b/tests/lean/misc-no_nested_borrows/Base/Primitives.lean
@@ -3,6 +3,28 @@ import Lean.Meta.Tactic.Simp
 import Init.Data.List.Basic
 import Mathlib.Tactic.RunCmd
 
+--------------------
+-- ASSERT COMMAND --
+--------------------
+
+open Lean Elab Command Term Meta
+
+syntax (name := assert) "#assert" term: command
+
+@[command_elab assert]
+unsafe
+def assertImpl : CommandElab := fun (_stx: Syntax) => do
+  runTermElabM (fun _ => do
+    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 ())
+
+#eval 2 == 2
+#assert (2 == 2)
+
 -------------
 -- PRELUDE --
 -------------
@@ -12,6 +34,7 @@ import Mathlib.Tactic.RunCmd
 inductive Error where
    | assertionFailure: Error
    | integerOverflow: Error
+   | divisionByZero: Error
    | arrayOutOfBounds: Error
    | maximumSizeExceeded: Error
    | panic: Error
@@ -89,17 +112,13 @@ macro "let" e:term " <-- " f:term : doElem =>
 -- MACHINE INTEGERS --
 ----------------------
 
--- 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.
+-- We redefine our machine integers types.
 
--- 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.)
+-- For Isize/Usize, we reuse `getNumBits` from `USize`. 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
@@ -111,236 +130,435 @@ macro "let" e:term " <-- " f:term : doElem =>
 -- 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
-
-macro_rules
-  | `(tactic| intlit) => `(tactic|
-    match USize.size, usize_size_eq with
-    | _, Or.inl rfl => decide
-    | _, Or.inr rfl => decide)
-
--- This is how the macro is expected to be used
-#eval USize.ofNatCore 0 (by intlit)
-
--- 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`).
+open System.Platform.getNumBits
+
+-- TODO: is there a way of only importing System.Platform.getNumBits?
+--
+@[simp] def size_num_bits : Nat := (System.Platform.getNumBits ()).val
+
+-- Remark: Lean seems to use < for the comparisons with the upper bounds by convention.
+-- We keep the F* convention for now.
+@[simp] def Isize.min : Int := - (HPow.hPow 2 (size_num_bits - 1))
+@[simp] def Isize.max : Int := (HPow.hPow 2 (size_num_bits - 1)) - 1
+@[simp] def I8.min    : Int := - (HPow.hPow 2 7)
+@[simp] def I8.max    : Int := HPow.hPow 2 7 - 1
+@[simp] def I16.min   : Int := - (HPow.hPow 2 15)
+@[simp] def I16.max   : Int := HPow.hPow 2 15 - 1
+@[simp] def I32.min   : Int := -(HPow.hPow 2 31)
+@[simp] def I32.max   : Int := HPow.hPow 2 31 - 1
+@[simp] def I64.min   : Int := -(HPow.hPow 2 63)
+@[simp] def I64.max   : Int := HPow.hPow 2 63 - 1
+@[simp] def I128.min  : Int := -(HPow.hPow 2 127)
+@[simp] def I128.max  : Int := HPow.hPow 2 127 - 1
+@[simp] def Usize.min : Int := 0
+@[simp] def Usize.max : Int := HPow.hPow 2 size_num_bits - 1
+@[simp] def U8.min    : Int := 0
+@[simp] def U8.max    : Int := HPow.hPow 2 8 - 1
+@[simp] def U16.min   : Int := 0
+@[simp] def U16.max   : Int := HPow.hPow 2 16 - 1
+@[simp] def U32.min   : Int := 0
+@[simp] def U32.max   : Int := HPow.hPow 2 32 - 1
+@[simp] def U64.min   : Int := 0
+@[simp] def U64.max   : Int := HPow.hPow 2 64 - 1
+@[simp] def U128.min  : Int := 0
+@[simp] def U128.max  : Int := HPow.hPow 2 128 - 1
+
+#assert (I8.min   == -128)
+#assert (I8.max   == 127)
+#assert (I16.min  == -32768)
+#assert (I16.max  == 32767)
+#assert (I32.min  == -2147483648)
+#assert (I32.max  == 2147483647)
+#assert (I64.min  == -9223372036854775808)
+#assert (I64.max  == 9223372036854775807)
+#assert (I128.min == -170141183460469231731687303715884105728)
+#assert (I128.max == 170141183460469231731687303715884105727)
+#assert (U8.min   == 0)
+#assert (U8.max   == 255)
+#assert (U16.min  == 0)
+#assert (U16.max  == 65535)
+#assert (U32.min  == 0)
+#assert (U32.max  == 4294967295)
+#assert (U64.min  == 0)
+#assert (U64.max  == 18446744073709551615)
+#assert (U128.min == 0)
+#assert (U128.max == 340282366920938463463374607431768211455)
+
+inductive ScalarTy :=
+| Isize
+| I8
+| I16
+| I32
+| I64
+| I128
+| Usize
+| U8
+| U16
+| U32
+| U64
+| U128
+
+def Scalar.min (ty : ScalarTy) : Int :=
+  match ty with
+  | .Isize => Isize.min
+  | .I8    => I8.min
+  | .I16   => I16.min
+  | .I32   => I32.min
+  | .I64   => I64.min
+  | .I128  => I128.min
+  | .Usize => Usize.min
+  | .U8    => U8.min
+  | .U16   => U16.min
+  | .U32   => U32.min
+  | .U64   => U64.min
+  | .U128  => U128.min
+
+def Scalar.max (ty : ScalarTy) : Int :=
+  match ty with
+  | .Isize => Isize.max
+  | .I8    => I8.max
+  | .I16   => I16.max
+  | .I32   => I32.max
+  | .I64   => I64.max
+  | .I128  => I128.max
+  | .Usize => Usize.max
+  | .U8    => U8.max
+  | .U16   => U16.max
+  | .U32   => U32.max
+  | .U64   => U64.max
+  | .U128  => U128.max
+
+-- "Conservative" bounds
+-- We use those because we can't compare to the isize bounds (which can't
+-- reduce at compile-time). Whenever we perform an arithmetic operation like
+-- addition we need to check that the result is in bounds: we first compare
+-- to the conservative bounds, which reduce, then compare to the real bounds.
 -- This is useful for the various #asserts that we want to reduce at
 -- type-checking time.
+def Scalar.cMin (ty : ScalarTy) : Int :=
+  match ty with
+  | .Isize => I32.min
+  | _ => Scalar.min ty
+
+def Scalar.cMax (ty : ScalarTy) : Int :=
+  match ty with
+  | .Isize => I32.max
+  | .Usize => U32.max
+  | _ => Scalar.max ty
+
+theorem Scalar.cMin_bound ty : Scalar.min ty <= Scalar.cMin ty := by sorry
+theorem Scalar.cMax_bound ty : Scalar.min ty <= Scalar.cMin ty := by sorry
+
+structure Scalar (ty : ScalarTy) where
+  val : Int
+  hmin : Scalar.min ty <= val
+  hmax : val <= Scalar.max ty
+
+theorem Scalar.bound_suffices (ty : ScalarTy) (x : Int) :
+  Scalar.cMin ty <= x && x <= Scalar.cMax ty ->
+  (decide (Scalar.min ty ≤ x) && decide (x ≤ Scalar.max ty)) = true
+  := by sorry
+
+def Scalar.ofIntCore {ty : ScalarTy} (x : Int)
+  (hmin : Scalar.min ty <= x) (hmax : x <= Scalar.max ty) : Scalar ty :=
+  { val := x, hmin := hmin, hmax := hmax }
+
+def Scalar.ofInt {ty : ScalarTy} (x : Int)
+  (h : Scalar.min ty <= x && x <= Scalar.max ty) : Scalar ty :=
+  let hmin: Scalar.min ty <= x := by sorry
+  let hmax: x <= Scalar.max ty := by sorry
+  Scalar.ofIntCore x hmin hmax
 
 -- 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 :=
-  -- NOTE: the test USize.toNat n - m >= 0 seems to always succeed?
-  if n >= m then
-    let n' := USize.toNat n
-    let m' := USize.toNat n
-    let r := USize.ofNatCore (n' - m') (by
-      have h: n' - m' <= n' := by
-        apply Nat.sub_le_of_le_add
-        case h => rewrite [ Nat.add_comm ]; apply Nat.le_add_left
-      apply Nat.lt_of_le_of_lt h
-      apply n.val.isLt
-    )
-    return r
-  else
-    fail integerOverflow
-
-@[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 :=
-  if h: m > 0 then
-    .ret ⟨ n.val % m.val, by
-      have h1: ↑m.val < USize.size := m.val.isLt
-      have h2: n.val.val % m.val.val < m.val.val := @Nat.mod_lt n.val m.val h
-      apply Nat.lt_trans h2 h1
-    ⟩
-  else
-    .fail integerOverflow
+def Scalar.tryMk (ty : ScalarTy) (x : Int) : Result (Scalar ty) :=
+  -- TODO: write this with only one if then else
+  if hmin_cons: Scalar.cMin ty <= x || Scalar.min ty <= x then
+    if hmax_cons: x <= Scalar.cMax ty || x <= Scalar.max ty then
+      let hmin: Scalar.min ty <= x := by sorry
+      let hmax: x <= Scalar.max ty := by sorry
+      return Scalar.ofIntCore x hmin hmax
+    else fail integerOverflow
+  else fail integerOverflow
+
+def Scalar.neg {ty : ScalarTy} (x : Scalar ty) : Result (Scalar ty) := Scalar.tryMk ty (- x.val)
+
+def Scalar.div {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+  if y.val != 0 then Scalar.tryMk ty (x.val / y.val) else fail divisionByZero
+
+-- Checking that the % operation in Lean computes the same as the remainder operation in Rust
+#assert 1 % 2 = (1:Int)
+#assert (-1) % 2 = -1
+#assert 1 % (-2) = 1
+#assert (-1) % (-2) = -1
+
+def Scalar.rem {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+  if y.val != 0 then Scalar.tryMk ty (x.val % y.val) else fail divisionByZero
+
+def Scalar.add {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+  Scalar.tryMk ty (x.val + y.val)
+
+def Scalar.sub {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+  Scalar.tryMk ty (x.val - y.val)
+
+def Scalar.mul {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+  Scalar.tryMk ty (x.val * y.val)
+
+-- TODO: instances of +, -, * etc. for scalars
+
+-- Cast an integer from a [src_ty] to a [tgt_ty]
+-- TODO: check the semantics of casts in Rust
+def Scalar.cast {src_ty : ScalarTy} (tgt_ty : ScalarTy) (x : Scalar src_ty) : Result (Scalar tgt_ty) :=
+  Scalar.tryMk tgt_ty x.val
+
+-- The scalar types
+-- We declare the definitions as reducible so that Lean can unfold them (useful
+-- for type class resolution for instance).
+@[reducible] def Isize := Scalar .Isize
+@[reducible] def I8    := Scalar .I8
+@[reducible] def I16   := Scalar .I16
+@[reducible] def I32   := Scalar .I32
+@[reducible] def I64   := Scalar .I64
+@[reducible] def I128  := Scalar .I128
+@[reducible] def Usize := Scalar .Usize
+@[reducible] def U8    := Scalar .U8
+@[reducible] def U16   := Scalar .U16
+@[reducible] def U32   := Scalar .U32
+@[reducible] def U64   := Scalar .U64
+@[reducible] def U128  := Scalar .U128
+
+-- TODO: below: not sure this is the best way.
+-- Should we rather overload operations like +, -, etc.?
+-- Also, it is possible to automate the generation of those definitions
+-- with macros (but would it be a good idea? It would be less easy to
+-- read the file, which is not supposed to change a lot)
+
+-- Negation
+
+/--
+Remark: there is no heterogeneous negation in the Lean prelude: we thus introduce
+one here.
+
+The notation typeclass for heterogeneous addition.
+This enables the notation `- a : β` where `a : α`.
+-/
+class HNeg (α : Type u) (β : outParam (Type v)) where
+  /-- `- a` computes the negation of `a`.
+  The meaning of this notation is type-dependent. -/
+  hNeg : α → β
+
+prefix:75  "-"   => HNeg.hNeg
+
+instance : HNeg Isize (Result Isize) where hNeg x := Scalar.neg x
+instance : HNeg I8 (Result I8) where hNeg x := Scalar.neg x
+instance : HNeg I16 (Result I16) where hNeg x := Scalar.neg x
+instance : HNeg I32 (Result I32) where hNeg x := Scalar.neg x
+instance : HNeg I64 (Result I64) where hNeg x := Scalar.neg x
+instance : HNeg I128 (Result I128) where hNeg x := Scalar.neg x
+
+-- Addition
+instance {ty} : HAdd (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+  hAdd x y := Scalar.add x y
+
+-- Substraction
+instance {ty} : HSub (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+  hSub x y := Scalar.sub x y
+
+-- Multiplication
+instance {ty} : HMul (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+  hMul x y := Scalar.mul x y
+
+-- Division
+instance {ty} : HDiv (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+  hDiv x y := Scalar.div x y
+
+-- Remainder
+instance {ty} : HMod (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+  hMod x y := Scalar.rem x y
+
+-- ofIntCore
+-- TODO: typeclass?
+def Isize.ofIntCore := @Scalar.ofIntCore .Isize
+def I8.ofIntCore    := @Scalar.ofIntCore .I8
+def I16.ofIntCore   := @Scalar.ofIntCore .I16
+def I32.ofIntCore   := @Scalar.ofIntCore .I32
+def I64.ofIntCore   := @Scalar.ofIntCore .I64
+def I128.ofIntCore  := @Scalar.ofIntCore .I128
+def Usize.ofIntCore := @Scalar.ofIntCore .Usize
+def U8.ofIntCore    := @Scalar.ofIntCore .U8
+def U16.ofIntCore   := @Scalar.ofIntCore .U16
+def U32.ofIntCore   := @Scalar.ofIntCore .U32
+def U64.ofIntCore   := @Scalar.ofIntCore .U64
+def U128.ofIntCore  := @Scalar.ofIntCore .U128
+
+--  ofInt
+-- TODO: typeclass?
+def Isize.ofInt := @Scalar.ofInt .Isize
+def I8.ofInt    := @Scalar.ofInt .I8
+def I16.ofInt   := @Scalar.ofInt .I16
+def I32.ofInt   := @Scalar.ofInt .I32
+def I64.ofInt   := @Scalar.ofInt .I64
+def I128.ofInt  := @Scalar.ofInt .I128
+def Usize.ofInt := @Scalar.ofInt .Usize
+def U8.ofInt    := @Scalar.ofInt .U8
+def U16.ofInt   := @Scalar.ofInt .U16
+def U32.ofInt   := @Scalar.ofInt .U32
+def U64.ofInt   := @Scalar.ofInt .U64
+def U128.ofInt  := @Scalar.ofInt .U128
+
+-- Comparisons
+instance {ty} : LT (Scalar ty) where
+  lt a b := LT.lt a.val b.val
+
+instance {ty} : LE (Scalar ty) where le a b := LE.le a.val b.val
+
+instance Scalar.decLt {ty} (a b : Scalar ty) : Decidable (LT.lt a b) := Int.decLt ..
+instance Scalar.decLe {ty} (a b : Scalar ty) : Decidable (LE.le a b) := Int.decLe ..
+
+theorem Scalar.eq_of_val_eq {ty} : ∀ {i j : Scalar ty}, Eq i.val j.val → Eq i j
+  | ⟨_, _, _⟩, ⟨_, _, _⟩, rfl => rfl
+
+theorem Scalar.val_eq_of_eq {ty} {i j : Scalar ty} (h : Eq i j) : Eq i.val j.val :=
+  h ▸ rfl
+
+theorem Scalar.ne_of_val_ne {ty} {i j : Scalar ty} (h : Not (Eq i.val j.val)) : Not (Eq i j) :=
+  fun h' => absurd (val_eq_of_eq h') h
+
+instance (ty : ScalarTy) : DecidableEq (Scalar ty) :=
+  fun i j =>
+    match decEq i.val j.val with
+    | isTrue h  => isTrue (Scalar.eq_of_val_eq h)
+    | isFalse h => isFalse (Scalar.ne_of_val_ne h)
+
+def Scalar.toInt {ty} (n : Scalar ty) : Int := n.val
+
+-- Tactic to prove that integers are in bounds
+syntax "intlit" : tactic
 
-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 :=
-  if m > 0 then
-    .ret ⟨ n.val / m.val, by
-      have h1: ↑n.val < USize.size := n.val.isLt
-      have h2: n.val.val / m.val.val <= n.val.val := @Nat.div_le_self n.val m.val
-      apply Nat.lt_of_le_of_lt h2 h1
-    ⟩
-  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
-  ofNatCore: (n:Nat) -> LT.lt n size -> t
-
-set_option hygiene false in
-run_cmd
-  for typeName in [`UInt8, `UInt16, `UInt32, `UInt64, `USize].map Lean.mkIdent do
-  Lean.Elab.Command.elabCommand (← `(
-    namespace $typeName
-    instance: MachineInteger $typeName where
-      size := size
-      val := val
-      ofNatCore := ofNatCore
-    end $typeName
-  ))
-
--- 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
+macro_rules
+  | `(tactic| intlit) => `(tactic| apply Scalar.bound_suffices ; decide)
+
+-- -- 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
+--   ofNatCore: (n:Nat) -> LT.lt n size -> t
+
+-- set_option hygiene false in
+-- run_cmd
+--   for typeName in [`UInt8, `UInt16, `UInt32, `UInt64, `USize].map Lean.mkIdent do
+--   Lean.Elab.Command.elabCommand (← `(
+--     namespace $typeName
+--     instance: MachineInteger $typeName where
+--       size := size
+--       val := val
+--       ofNatCore := ofNatCore
+--     end $typeName
+--   ))
+
+-- -- 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
 
 -------------
 -- 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_new (α : Type u): Vec α := ⟨ [], by {
-  match USize.size, usize_size_eq with
-  | _, Or.inl rfl => simp
-  | _, Or.inr rfl => simp
-  } ⟩
+def Vec (α : Type u) := { l : List α // List.length l <= Usize.max }
 
-#check vec_new
+def vec_new (α : Type u): Vec α := ⟨ [], by sorry ⟩
 
-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)
+  Usize.ofIntCore (List.length v) (by sorry) l
  
 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
--- 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) //
-  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
-    ⟨ return ⟨List.concat v.val x,
-      by
-        rw [List.length_concat]
-        assumption
-     ⟩, by simp ⟩
-  else
-    ⟨ fail maximumSizeExceeded, by simp ⟩
-
-#eval do
-  -- NOTE: the // notation is syntactic sugar for Subtype, a refinement with
-  -- fields val and property. However, Lean's elaborator can automatically
-  -- select the `val` field if the context provides a type annotation. We
-  -- 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?
-  -- 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)
   :=
-  if h : List.length v.val + 1 <= 4294967295 then
-    return ⟨ List.concat v.val x,
-      by
-        rw [List.length_concat]
-        have h': 4294967295 < USize.size := by intlit
-        apply Nat.lt_of_le_of_lt h h'
-    ⟩
-  else if h: List.length v.val + 1 < USize.size then
-    return ⟨List.concat v.val x,
-      by
-        rw [List.length_concat]
-        assumption
-     ⟩
+  if h : List.length v.val <= U32.max || List.length v.val <= Usize.max then
+    return ⟨ List.concat v.val x, by sorry ⟩
   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
+    -- TODO: maybe we should redefine a list library which uses integers
+    -- (instead of natural numbers)
+    let i : Nat :=
+      match i.val with
+      | .ofNat n => n
+      | .negSucc n => by sorry -- TODO: we can't get here
+    let isLt: i < USize.size := by sorry
+    let i : Fin USize.size := { val := i, isLt := isLt }
     .ret ⟨ List.set v.val i.val x, by
-      have h: List.length v.val < USize.size := v.property
+      have h: List.length v.val <= Usize.max := v.property
       rewrite [ List.length_set v.val i.val x ]
       assumption
     ⟩
   else
     .fail arrayOutOfBounds
 
-def vec_index_fwd (α : Type u) (v: Vec α) (i: USize): Result α :=
-  if h: i.val < List.length v.val then
+def vec_index_fwd (α : Type u) (v: Vec α) (i: Usize): Result α :=
+  if i.val < List.length v.val then
+    let i : Nat :=
+      match i.val with
+      | .ofNat n => n
+      | .negSucc n => by sorry -- TODO: we can't get here
+    let isLt: i < USize.size := by sorry
+    let i : Fin USize.size := { val := i, isLt := isLt }
+    let h: i < List.length v.val := by sorry
     .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 :=
-  if h: i.val < List.length v.val then
+def vec_index_mut_fwd (α : Type u) (v: Vec α) (i: Usize): Result α :=
+  if i.val < List.length v.val then
+    let i : Nat :=
+      match i.val with
+      | .ofNat n => n
+      | .negSucc n => by sorry -- TODO: we can't get here
+    let isLt: i < USize.size := by sorry
+    let i : Fin USize.size := { val := i, isLt := isLt }
+    let h: i < List.length v.val := by sorry
     .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
+    let i : Nat :=
+      match i.val with
+      | .ofNat n => n
+      | .negSucc n => by sorry -- TODO: we can't get here
+    let isLt: i < USize.size := by sorry
+    let i : Fin USize.size := { val := i, isLt := isLt }
     .ret ⟨ List.set v.val i.val x, by
-      have h: List.length v.val < USize.size := v.property
+      have h: List.length v.val <= Usize.max := v.property
       rewrite [ List.length_set v.val i.val x ]
       assumption
     ⟩
@@ -360,33 +578,3 @@ def mem_replace_back (a : Type) (_ : a) (y : a) : a :=
 /-- Aeneas-translated function -- useful to reduce non-recursive definitions.
  Use with `simp [ aeneas ]` -/
 register_simp_attr aeneas
-
---------------------
--- ASSERT COMMAND --
---------------------
-
-open Lean Elab Command Term Meta
-
-syntax (name := assert) "#assert" term: command
-
-@[command_elab assert]
-unsafe
-def assertImpl : CommandElab := fun (_stx: Syntax) => do
-  runTermElabM (fun _ => do
-    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 ())
-
-#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 e2697385..a73848de 100644
--- a/tests/lean/misc-no_nested_borrows/NoNestedBorrows.lean
+++ b/tests/lean/misc-no_nested_borrows/NoNestedBorrows.lean
@@ -2,556 +2,534 @@
 -- [no_nested_borrows]
 import Base.Primitives
 
-structure OpaqueDefs where
-  
-  /- [no_nested_borrows::Pair] -/
-  structure pair_t (T1 T2 : Type) where
-    pair_x : T1
-    pair_y : T2
-  
-  /- [no_nested_borrows::List] -/
-  inductive list_t (T : Type) :=
-  | Cons : T -> list_t T -> list_t T
-  | Nil : list_t T
-  
-  /- [no_nested_borrows::One] -/
-  inductive one_t (T1 : Type) :=
-  | One : T1 -> one_t T1
-  
-  /- [no_nested_borrows::EmptyEnum] -/
-  inductive empty_enum_t :=
-  | Empty : empty_enum_t
-  
-  /- [no_nested_borrows::Enum] -/
-  inductive enum_t :=
-  | Variant1 : enum_t
-  | Variant2 : enum_t
-  
-  /- [no_nested_borrows::EmptyStruct] -/
-  structure empty_struct_t where
-  
-  /- [no_nested_borrows::Sum] -/
-  inductive sum_t (T1 T2 : Type) :=
-  | Left : T1 -> sum_t T1 T2
-  | Right : T2 -> sum_t T1 T2
-  
-  /- [no_nested_borrows::neg_test] -/
-  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 :=
-    UInt32.checked_add x y
-  
-  /- [no_nested_borrows::subs_test] -/
-  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 :=
-    UInt32.checked_div x y
-  
-  /- [no_nested_borrows::div_test1] -/
-  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 :=
-    UInt32.checked_rem x y
-  
-  /- [no_nested_borrows::cast_test] -/
-  def cast_test_fwd (x : UInt32) : Result Int32 :=
-    scalar_cast Int32 x
-  
-  /- [no_nested_borrows::test2] -/
-  def test2_fwd : Result Unit :=
-    do
-      let _ ← UInt32.checked_add (UInt32.ofNatCore 23 (by intlit))
-        (UInt32.ofNatCore 44 (by intlit))
-      Result.ret ()
-  
-  /- Unit test for [no_nested_borrows::test2] -/
-  #assert (test2_fwd == .ret ())
-  
-  /- [no_nested_borrows::get_max] -/
-  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 :=
-    do
-      let x ←
-        get_max_fwd (UInt32.ofNatCore 4 (by intlit))
-          (UInt32.ofNatCore 3 (by intlit))
-      let y ←
-        get_max_fwd (UInt32.ofNatCore 10 (by intlit))
-          (UInt32.ofNatCore 11 (by intlit))
-      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 ())
-  
-  /- [no_nested_borrows::test_neg1] -/
-  def test_neg1_fwd : Result Unit :=
-    do
-      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 ())
-  
-  /- [no_nested_borrows::refs_test1] -/
-  def refs_test1_fwd : Result Unit :=
-    if h: not ((Int32.ofNatCore 1 (by intlit)) =
-      (Int32.ofNatCore 1 (by intlit)))
+/- [no_nested_borrows::Pair] -/
+structure pair_t (T1 T2 : Type) where
+  pair_x : T1
+  pair_y : T2
+
+/- [no_nested_borrows::List] -/
+inductive list_t (T : Type) :=
+| Cons : T -> list_t T -> list_t T
+| Nil : list_t T
+
+/- [no_nested_borrows::One] -/
+inductive one_t (T1 : Type) :=
+| One : T1 -> one_t T1
+
+/- [no_nested_borrows::EmptyEnum] -/
+inductive empty_enum_t :=
+| Empty : empty_enum_t
+
+/- [no_nested_borrows::Enum] -/
+inductive enum_t :=
+| Variant1 : enum_t
+| Variant2 : enum_t
+
+/- [no_nested_borrows::EmptyStruct] -/
+structure empty_struct_t where
+
+/- [no_nested_borrows::Sum] -/
+inductive sum_t (T1 T2 : Type) :=
+| Left : T1 -> sum_t T1 T2
+| Right : T2 -> sum_t T1 T2
+
+/- [no_nested_borrows::neg_test] -/
+def neg_test_fwd (x : I32) : Result I32 :=
+  - x
+
+/- [no_nested_borrows::add_test] -/
+def add_test_fwd (x : U32) (y : U32) : Result U32 :=
+  x + y
+
+/- [no_nested_borrows::subs_test] -/
+def subs_test_fwd (x : U32) (y : U32) : Result U32 :=
+  x - y
+
+/- [no_nested_borrows::div_test] -/
+def div_test_fwd (x : U32) (y : U32) : Result U32 :=
+  x / y
+
+/- [no_nested_borrows::div_test1] -/
+def div_test1_fwd (x : U32) : Result U32 :=
+  x / (U32.ofInt 2 (by intlit))
+
+/- [no_nested_borrows::rem_test] -/
+def rem_test_fwd (x : U32) (y : U32) : Result U32 :=
+  x % y
+
+/- [no_nested_borrows::cast_test] -/
+def cast_test_fwd (x : U32) : Result I32 :=
+  Scalar.cast .I32 x
+
+/- [no_nested_borrows::test2] -/
+def test2_fwd : Result Unit :=
+  do
+    let _ ← (U32.ofInt 23 (by intlit)) + (U32.ofInt 44 (by intlit))
+    Result.ret ()
+
+/- Unit test for [no_nested_borrows::test2] -/
+#assert (test2_fwd == .ret ())
+
+/- [no_nested_borrows::get_max] -/
+def get_max_fwd (x : U32) (y : U32) : Result U32 :=
+  if h: x >= y
+  then Result.ret x
+  else Result.ret y
+
+/- [no_nested_borrows::test3] -/
+def test3_fwd : Result Unit :=
+  do
+    let x ← get_max_fwd (U32.ofInt 4 (by intlit)) (U32.ofInt 3 (by intlit))
+    let y ← get_max_fwd (U32.ofInt 10 (by intlit)) (U32.ofInt 11 (by intlit))
+    let z ← x + y
+    if h: not (z = (U32.ofInt 15 (by intlit)))
     then Result.fail Error.panic
     else Result.ret ()
-  
-  /- Unit test for [no_nested_borrows::refs_test1] -/
-  #assert (refs_test1_fwd == .ret ())
-  
-  /- [no_nested_borrows::refs_test2] -/
-  def refs_test2_fwd : Result Unit :=
-    if h: not ((Int32.ofNatCore 2 (by intlit)) =
-      (Int32.ofNatCore 2 (by intlit)))
+
+/- Unit test for [no_nested_borrows::test3] -/
+#assert (test3_fwd == .ret ())
+
+/- [no_nested_borrows::test_neg1] -/
+def test_neg1_fwd : Result Unit :=
+  do
+    let y ← - (I32.ofInt 3 (by intlit))
+    if h: not (y = (I32.ofInt (-(3:Int)) (by intlit)))
+    then Result.fail Error.panic
+    else Result.ret ()
+
+/- Unit test for [no_nested_borrows::test_neg1] -/
+#assert (test_neg1_fwd == .ret ())
+
+/- [no_nested_borrows::refs_test1] -/
+def refs_test1_fwd : Result Unit :=
+  if h: not ((I32.ofInt 1 (by intlit)) = (I32.ofInt 1 (by intlit)))
+  then Result.fail Error.panic
+  else Result.ret ()
+
+/- Unit test for [no_nested_borrows::refs_test1] -/
+#assert (refs_test1_fwd == .ret ())
+
+/- [no_nested_borrows::refs_test2] -/
+def refs_test2_fwd : Result Unit :=
+  if h: not ((I32.ofInt 2 (by intlit)) = (I32.ofInt 2 (by intlit)))
+  then Result.fail Error.panic
+  else
+    if h: not ((I32.ofInt 0 (by intlit)) = (I32.ofInt 0 (by intlit)))
     then Result.fail Error.panic
     else
-      if h: not ((Int32.ofNatCore 0 (by intlit)) =
-        (Int32.ofNatCore 0 (by intlit)))
+      if h: not ((I32.ofInt 2 (by intlit)) = (I32.ofInt 2 (by intlit)))
       then Result.fail Error.panic
       else
-        if h: not ((Int32.ofNatCore 2 (by intlit)) =
-          (Int32.ofNatCore 2 (by intlit)))
+        if h: not ((I32.ofInt 2 (by intlit)) = (I32.ofInt 2 (by intlit)))
         then Result.fail Error.panic
-        else
-          if h: not ((Int32.ofNatCore 2 (by intlit)) =
-            (Int32.ofNatCore 2 (by intlit)))
-          then Result.fail Error.panic
-          else Result.ret ()
-  
-  /- Unit test for [no_nested_borrows::refs_test2] -/
-  #assert (refs_test2_fwd == .ret ())
-  
-  /- [no_nested_borrows::test_list1] -/
-  def test_list1_fwd : Result Unit :=
-    Result.ret ()
-  
-  /- Unit test for [no_nested_borrows::test_list1] -/
-  #assert (test_list1_fwd == .ret ())
-  
-  /- [no_nested_borrows::test_box1] -/
-  def test_box1_fwd : Result Unit :=
-    let b := (Int32.ofNatCore 1 (by intlit))
-    let x := b
-    if h: not (x = (Int32.ofNatCore 1 (by intlit)))
+        else Result.ret ()
+
+/- Unit test for [no_nested_borrows::refs_test2] -/
+#assert (refs_test2_fwd == .ret ())
+
+/- [no_nested_borrows::test_list1] -/
+def test_list1_fwd : Result Unit :=
+  Result.ret ()
+
+/- Unit test for [no_nested_borrows::test_list1] -/
+#assert (test_list1_fwd == .ret ())
+
+/- [no_nested_borrows::test_box1] -/
+def test_box1_fwd : Result Unit :=
+  let b := (I32.ofInt 1 (by intlit))
+  let x := b
+  if h: not (x = (I32.ofInt 1 (by intlit)))
+  then Result.fail Error.panic
+  else Result.ret ()
+
+/- Unit test for [no_nested_borrows::test_box1] -/
+#assert (test_box1_fwd == .ret ())
+
+/- [no_nested_borrows::copy_int] -/
+def copy_int_fwd (x : I32) : Result I32 :=
+  Result.ret x
+
+/- [no_nested_borrows::test_unreachable] -/
+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 h: b
+  then Result.fail Error.panic
+  else Result.ret ()
+
+/- [no_nested_borrows::test_copy_int] -/
+def test_copy_int_fwd : Result Unit :=
+  do
+    let y ← copy_int_fwd (I32.ofInt 0 (by intlit))
+    if h: not ((I32.ofInt 0 (by intlit)) = y)
     then Result.fail Error.panic
     else Result.ret ()
-  
-  /- Unit test for [no_nested_borrows::test_box1] -/
-  #assert (test_box1_fwd == .ret ())
-  
-  /- [no_nested_borrows::copy_int] -/
-  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 h: b
+
+/- Unit test for [no_nested_borrows::test_copy_int] -/
+#assert (test_copy_int_fwd == .ret ())
+
+/- [no_nested_borrows::is_cons] -/
+def is_cons_fwd (T : Type) (l : list_t T) : Result Bool :=
+  match h: l with
+  | list_t.Cons t l0 => Result.ret true
+  | list_t.Nil => Result.ret false
+
+/- [no_nested_borrows::test_is_cons] -/
+def test_is_cons_fwd : Result Unit :=
+  do
+    let l := list_t.Nil
+    let b ← is_cons_fwd I32 (list_t.Cons (I32.ofInt 0 (by intlit)) l)
+    if h: not b
     then Result.fail Error.panic
     else Result.ret ()
-  
-  /- [no_nested_borrows::test_panic] -/
-  def test_panic_fwd (b : Bool) : Result Unit :=
-    if h: b
+
+/- Unit test for [no_nested_borrows::test_is_cons] -/
+#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 h: l with
+  | list_t.Cons hd tl => Result.ret (hd, tl)
+  | list_t.Nil => Result.fail Error.panic
+
+/- [no_nested_borrows::test_split_list] -/
+def test_split_list_fwd : Result Unit :=
+  do
+    let l := list_t.Nil
+    let p ← split_list_fwd I32 (list_t.Cons (I32.ofInt 0 (by intlit)) l)
+    let (hd, _) := p
+    if h: not (hd = (I32.ofInt 0 (by intlit)))
     then Result.fail Error.panic
     else Result.ret ()
-  
-  /- [no_nested_borrows::test_copy_int] -/
-  def test_copy_int_fwd : Result Unit :=
-    do
-      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 ())
-  
-  /- [no_nested_borrows::is_cons] -/
-  def is_cons_fwd (T : Type) (l : list_t T) : Result Bool :=
-    match h: l with
-    | list_t.Cons t l0 => Result.ret true
-    | list_t.Nil => Result.ret false
-  
-  /- [no_nested_borrows::test_is_cons] -/
-  def test_is_cons_fwd : Result Unit :=
-    do
-      let l := list_t.Nil
-      let b ←
-        is_cons_fwd Int32 (list_t.Cons (Int32.ofNatCore 0 (by intlit)) l)
-      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 ())
-  
-  /- [no_nested_borrows::split_list] -/
-  def split_list_fwd (T : Type) (l : list_t T) : Result (T × (list_t T)) :=
-    match h: l with
-    | list_t.Cons hd tl => Result.ret (hd, tl)
-    | list_t.Nil => Result.fail Error.panic
-  
-  /- [no_nested_borrows::test_split_list] -/
-  def test_split_list_fwd : Result Unit :=
-    do
-      let l := list_t.Nil
-      let p ←
-        split_list_fwd Int32 (list_t.Cons (Int32.ofNatCore 0 (by intlit)) l)
-      let (hd, _) := p
-      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 ())
-  
-  /- [no_nested_borrows::choose] -/
-  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 h: b
-    then Result.ret (ret0, y)
-    else Result.ret (x, ret0)
-  
-  /- [no_nested_borrows::choose_test] -/
-  def choose_test_fwd : Result Unit :=
-    do
-      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 h: not (z0 = (Int32.ofNatCore 1 (by intlit)))
-      then Result.fail Error.panic
-      else
-        do
-          let (x, y) ←
-            choose_back Int32 true (Int32.ofNatCore 0 (by intlit))
-              (Int32.ofNatCore 0 (by intlit)) z0
-          if h: not (x = (Int32.ofNatCore 1 (by intlit)))
-          then Result.fail Error.panic
-          else
-            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 ())
-  
-  /- [no_nested_borrows::test_char] -/
-  def test_char_fwd : Result Char :=
-    Result.ret 'a'
-  
-  /- [no_nested_borrows::NodeElem] -/
-  mutual inductive node_elem_t (T : Type) :=
-  | Cons : tree_t T -> node_elem_t T -> node_elem_t T
-  | Nil : node_elem_t T
-  
-  /- [no_nested_borrows::Tree] -/
-  inductive tree_t (T : Type) :=
-  | Leaf : T -> tree_t T
-  | Node : 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 h: l with
-    | list_t.Cons t l1 =>
+
+/- Unit test for [no_nested_borrows::test_split_list] -/
+#assert (test_split_list_fwd == .ret ())
+
+/- [no_nested_borrows::choose] -/
+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 h: b
+  then Result.ret (ret0, y)
+  else Result.ret (x, ret0)
+
+/- [no_nested_borrows::choose_test] -/
+def choose_test_fwd : Result Unit :=
+  do
+    let z ←
+      choose_fwd I32 true (I32.ofInt 0 (by intlit)) (I32.ofInt 0 (by intlit))
+    let z0 ← z + (I32.ofInt 1 (by intlit))
+    if h: not (z0 = (I32.ofInt 1 (by intlit)))
+    then Result.fail Error.panic
+    else
       do
-        let i ← list_length_fwd T l1
-        UInt32.checked_add (UInt32.ofNatCore 1 (by intlit)) i
-    | list_t.Nil => 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 h: l with
-    | list_t.Cons x tl =>
-      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))
-          list_nth_shared_fwd T tl i0
-    | list_t.Nil => 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 h: l with
-    | list_t.Cons x tl =>
-      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))
-          list_nth_mut_fwd T tl i0
-    | list_t.Nil => 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 h: l with
-    | list_t.Cons x tl =>
-      if h: i = (UInt32.ofNatCore 0 (by intlit))
-      then Result.ret (list_t.Cons 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.Cons x tl0)
-    | list_t.Nil => 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 h: li with
-    | list_t.Cons hd tl => list_rev_aux_fwd T tl (list_t.Cons hd lo)
-    | list_t.Nil => Result.ret lo
-  
-  /- [no_nested_borrows::list_rev] -/
-  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.Nil
-    list_rev_aux_fwd T li list_t.Nil
-  
-  /- [no_nested_borrows::test_list_functions] -/
-  def test_list_functions_fwd : Result Unit :=
-    do
-      let l := list_t.Nil
-      let l0 := list_t.Cons (Int32.ofNatCore 2 (by intlit)) l
-      let l1 := list_t.Cons (Int32.ofNatCore 1 (by intlit)) l0
-      let i ←
-        list_length_fwd Int32 (list_t.Cons (Int32.ofNatCore 0 (by intlit)) l1)
-      if h: not (i = (UInt32.ofNatCore 3 (by intlit)))
-      then Result.fail Error.panic
-      else
-        do
-          let i0 ←
-            list_nth_shared_fwd Int32 (list_t.Cons
-              (Int32.ofNatCore 0 (by intlit)) l1)
-              (UInt32.ofNatCore 0 (by intlit))
-          if h: not (i0 = (Int32.ofNatCore 0 (by intlit)))
+        let (x, y) ←
+          choose_back I32 true (I32.ofInt 0 (by intlit))
+            (I32.ofInt 0 (by intlit)) z0
+        if h: not (x = (I32.ofInt 1 (by intlit)))
+        then Result.fail Error.panic
+        else
+          if h: not (y = (I32.ofInt 0 (by intlit)))
           then Result.fail Error.panic
-          else
-            do
-              let i1 ←
-                list_nth_shared_fwd Int32 (list_t.Cons
-                  (Int32.ofNatCore 0 (by intlit)) l1)
-                  (UInt32.ofNatCore 1 (by intlit))
-              if h: not (i1 = (Int32.ofNatCore 1 (by intlit)))
-              then Result.fail Error.panic
-              else
-                do
-                  let i2 ←
-                    list_nth_shared_fwd Int32 (list_t.Cons
-                      (Int32.ofNatCore 0 (by intlit)) l1)
-                      (UInt32.ofNatCore 2 (by intlit))
-                  if h: not (i2 = (Int32.ofNatCore 2 (by intlit)))
-                  then Result.fail Error.panic
-                  else
-                    do
-                      let ls ←
-                        list_nth_mut_back Int32 (list_t.Cons
-                          (Int32.ofNatCore 0 (by intlit)) l1)
-                          (UInt32.ofNatCore 1 (by intlit))
-                          (Int32.ofNatCore 3 (by intlit))
-                      let i3 ←
-                        list_nth_shared_fwd Int32 ls
-                          (UInt32.ofNatCore 0 (by intlit))
-                      if h: not (i3 = (Int32.ofNatCore 0 (by intlit)))
-                      then Result.fail Error.panic
-                      else
-                        do
-                          let i4 ←
-                            list_nth_shared_fwd Int32 ls
-                              (UInt32.ofNatCore 1 (by intlit))
-                          if h: not (i4 = (Int32.ofNatCore 3 (by intlit)))
-                          then Result.fail Error.panic
-                          else
-                            do
-                              let i5 ←
-                                list_nth_shared_fwd Int32 ls
-                                  (UInt32.ofNatCore 2 (by intlit))
-                              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 ())
-  
-  /- [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)
-  
-  /- [no_nested_borrows::id_mut_pair1] -/
-  def id_mut_pair1_back
-    (T1 T2 : Type) (x : T1) (y : T2) (ret0 : (T1 × T2)) : Result (T1 × T2) :=
-    let (t, t0) := ret0
-    Result.ret (t, t0)
-  
-  /- [no_nested_borrows::id_mut_pair2] -/
-  def id_mut_pair2_fwd (T1 T2 : Type) (p : (T1 × T2)) : Result (T1 × T2) :=
-    let (t, t0) := p
-    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) :=
-    let (t, t0) := ret0
-    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)
-  
-  /- [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
-  
-  /- [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
-  
-  /- [no_nested_borrows::id_mut_pair4] -/
-  def id_mut_pair4_fwd (T1 T2 : Type) (p : (T1 × T2)) : Result (T1 × T2) :=
-    let (t, t0) := p
-    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
-  
-  /- [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
-  
-  /- [no_nested_borrows::StructWithTuple] -/
-  structure struct_with_tuple_t (T1 T2 : Type) where
-    struct_with_tuple_p : (T1 × T2)
-  
-  /- [no_nested_borrows::new_tuple1] -/
-  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
-    {
-      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
-    {
-      struct_with_tuple_p :=
-        ((UInt64.ofNatCore 1 (by intlit)), (Int64.ofNatCore 2 (by intlit)))
-    }
-  
-  /- [no_nested_borrows::StructWithPair] -/
-  structure struct_with_pair_t (T1 T2 : Type) where
-    struct_with_pair_p : pair_t T1 T2
-  
-  /- [no_nested_borrows::new_pair1] -/
-  def new_pair1_fwd : Result (struct_with_pair_t UInt32 UInt32) :=
-    Result.ret
-    {
-      struct_with_pair_p :=
-        {
-          pair_x := (UInt32.ofNatCore 1 (by intlit)),
-          pair_y := (UInt32.ofNatCore 2 (by intlit))
-        }
-    }
-  
-  /- [no_nested_borrows::test_constants] -/
-  def test_constants_fwd : Result Unit :=
+          else Result.ret ()
+
+/- Unit test for [no_nested_borrows::choose_test] -/
+#assert (choose_test_fwd == .ret ())
+
+/- [no_nested_borrows::test_char] -/
+def test_char_fwd : Result Char :=
+  Result.ret 'a'
+
+/- [no_nested_borrows::NodeElem] -/
+mutual inductive node_elem_t (T : Type) :=
+| Cons : tree_t T -> node_elem_t T -> node_elem_t T
+| Nil : node_elem_t T
+
+/- [no_nested_borrows::Tree] -/
+inductive tree_t (T : Type) :=
+| Leaf : T -> tree_t T
+| Node : T -> node_elem_t T -> tree_t T -> tree_t T
+end
+
+/- [no_nested_borrows::list_length] -/
+def list_length_fwd (T : Type) (l : list_t T) : Result U32 :=
+  match h: l with
+  | list_t.Cons t l1 =>
     do
-      let swt ← new_tuple1_fwd
-      let (i, _) := swt.struct_with_tuple_p
-      if h: not (i = (UInt32.ofNatCore 1 (by intlit)))
-      then Result.fail Error.panic
-      else
-        do
-          let swt0 ← new_tuple2_fwd
-          let (i0, _) := swt0.struct_with_tuple_p
-          if h: not (i0 = (Int16.ofNatCore 1 (by intlit)))
-          then Result.fail Error.panic
-          else
-            do
-              let swt1 ← new_tuple3_fwd
-              let (i1, _) := swt1.struct_with_tuple_p
-              if h: not (i1 = (UInt64.ofNatCore 1 (by intlit)))
-              then Result.fail Error.panic
-              else
-                do
-                  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 ()
-  
-  /- Unit test for [no_nested_borrows::test_constants] -/
-  #assert (test_constants_fwd == .ret ())
-  
-  /- [no_nested_borrows::test_weird_borrows1] -/
-  def test_weird_borrows1_fwd : Result Unit :=
-    Result.ret ()
-  
-  /- Unit test for [no_nested_borrows::test_weird_borrows1] -/
-  #assert (test_weird_borrows1_fwd == .ret ())
-  
-  /- [no_nested_borrows::test_mem_replace] -/
-  def test_mem_replace_fwd_back (px : UInt32) : Result UInt32 :=
-    let y := mem_replace_fwd UInt32 px (UInt32.ofNatCore 1 (by intlit))
-    if h: not (y = (UInt32.ofNatCore 0 (by intlit)))
+      let i ← list_length_fwd T l1
+      (U32.ofInt 1 (by intlit)) + i
+  | list_t.Nil => Result.ret (U32.ofInt 0 (by intlit))
+
+/- [no_nested_borrows::list_nth_shared] -/
+def list_nth_shared_fwd (T : Type) (l : list_t T) (i : U32) : Result T :=
+  match h: l with
+  | list_t.Cons x tl =>
+    if h: i = (U32.ofInt 0 (by intlit))
+    then Result.ret x
+    else
+      do
+        let i0 ← i - (U32.ofInt 1 (by intlit))
+        list_nth_shared_fwd T tl i0
+  | list_t.Nil => Result.fail Error.panic
+
+/- [no_nested_borrows::list_nth_mut] -/
+def list_nth_mut_fwd (T : Type) (l : list_t T) (i : U32) : Result T :=
+  match h: l with
+  | list_t.Cons x tl =>
+    if h: i = (U32.ofInt 0 (by intlit))
+    then Result.ret x
+    else do
+           let i0 ← i - (U32.ofInt 1 (by intlit))
+           list_nth_mut_fwd T tl i0
+  | list_t.Nil => Result.fail Error.panic
+
+/- [no_nested_borrows::list_nth_mut] -/
+def list_nth_mut_back
+  (T : Type) (l : list_t T) (i : U32) (ret0 : T) : Result (list_t T) :=
+  match h: l with
+  | list_t.Cons x tl =>
+    if h: i = (U32.ofInt 0 (by intlit))
+    then Result.ret (list_t.Cons ret0 tl)
+    else
+      do
+        let i0 ← i - (U32.ofInt 1 (by intlit))
+        let tl0 ← list_nth_mut_back T tl i0 ret0
+        Result.ret (list_t.Cons x tl0)
+  | list_t.Nil => 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 h: li with
+  | list_t.Cons hd tl => list_rev_aux_fwd T tl (list_t.Cons hd lo)
+  | list_t.Nil => Result.ret lo
+
+/- [no_nested_borrows::list_rev] -/
+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.Nil
+  list_rev_aux_fwd T li list_t.Nil
+
+/- [no_nested_borrows::test_list_functions] -/
+def test_list_functions_fwd : Result Unit :=
+  do
+    let l := list_t.Nil
+    let l0 := list_t.Cons (I32.ofInt 2 (by intlit)) l
+    let l1 := list_t.Cons (I32.ofInt 1 (by intlit)) l0
+    let i ← list_length_fwd I32 (list_t.Cons (I32.ofInt 0 (by intlit)) l1)
+    if h: not (i = (U32.ofInt 3 (by intlit)))
+    then Result.fail Error.panic
+    else
+      do
+        let i0 ←
+          list_nth_shared_fwd I32 (list_t.Cons (I32.ofInt 0 (by intlit)) l1)
+            (U32.ofInt 0 (by intlit))
+        if h: not (i0 = (I32.ofInt 0 (by intlit)))
+        then Result.fail Error.panic
+        else
+          do
+            let i1 ←
+              list_nth_shared_fwd I32 (list_t.Cons (I32.ofInt 0 (by intlit))
+                l1) (U32.ofInt 1 (by intlit))
+            if h: not (i1 = (I32.ofInt 1 (by intlit)))
+            then Result.fail Error.panic
+            else
+              do
+                let i2 ←
+                  list_nth_shared_fwd I32 (list_t.Cons
+                    (I32.ofInt 0 (by intlit)) l1) (U32.ofInt 2 (by intlit))
+                if h: not (i2 = (I32.ofInt 2 (by intlit)))
+                then Result.fail Error.panic
+                else
+                  do
+                    let ls ←
+                      list_nth_mut_back I32 (list_t.Cons
+                        (I32.ofInt 0 (by intlit)) l1) (U32.ofInt 1 (by intlit))
+                        (I32.ofInt 3 (by intlit))
+                    let i3 ←
+                      list_nth_shared_fwd I32 ls (U32.ofInt 0 (by intlit))
+                    if h: not (i3 = (I32.ofInt 0 (by intlit)))
+                    then Result.fail Error.panic
+                    else
+                      do
+                        let i4 ←
+                          list_nth_shared_fwd I32 ls (U32.ofInt 1 (by intlit))
+                        if h: not (i4 = (I32.ofInt 3 (by intlit)))
+                        then Result.fail Error.panic
+                        else
+                          do
+                            let i5 ←
+                              list_nth_shared_fwd I32 ls
+                                (U32.ofInt 2 (by intlit))
+                            if h: not (i5 = (I32.ofInt 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 ())
+
+/- [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)
+
+/- [no_nested_borrows::id_mut_pair1] -/
+def id_mut_pair1_back
+  (T1 T2 : Type) (x : T1) (y : T2) (ret0 : (T1 × T2)) : Result (T1 × T2) :=
+  let (t, t0) := ret0
+  Result.ret (t, t0)
+
+/- [no_nested_borrows::id_mut_pair2] -/
+def id_mut_pair2_fwd (T1 T2 : Type) (p : (T1 × T2)) : Result (T1 × T2) :=
+  let (t, t0) := p
+  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) :=
+  let (t, t0) := ret0
+  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)
+
+/- [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
+
+/- [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
+
+/- [no_nested_borrows::id_mut_pair4] -/
+def id_mut_pair4_fwd (T1 T2 : Type) (p : (T1 × T2)) : Result (T1 × T2) :=
+  let (t, t0) := p
+  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
+
+/- [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
+
+/- [no_nested_borrows::StructWithTuple] -/
+structure struct_with_tuple_t (T1 T2 : Type) where
+  struct_with_tuple_p : (T1 × T2)
+
+/- [no_nested_borrows::new_tuple1] -/
+def new_tuple1_fwd : Result (struct_with_tuple_t U32 U32) :=
+  Result.ret
+  {
+    struct_with_tuple_p :=
+      ((U32.ofInt 1 (by intlit)), (U32.ofInt 2 (by intlit)))
+  }
+
+/- [no_nested_borrows::new_tuple2] -/
+def new_tuple2_fwd : Result (struct_with_tuple_t I16 I16) :=
+  Result.ret
+  {
+    struct_with_tuple_p :=
+      ((I16.ofInt 1 (by intlit)), (I16.ofInt 2 (by intlit)))
+  }
+
+/- [no_nested_borrows::new_tuple3] -/
+def new_tuple3_fwd : Result (struct_with_tuple_t U64 I64) :=
+  Result.ret
+  {
+    struct_with_tuple_p :=
+      ((U64.ofInt 1 (by intlit)), (I64.ofInt 2 (by intlit)))
+  }
+
+/- [no_nested_borrows::StructWithPair] -/
+structure struct_with_pair_t (T1 T2 : Type) where
+  struct_with_pair_p : pair_t T1 T2
+
+/- [no_nested_borrows::new_pair1] -/
+def new_pair1_fwd : Result (struct_with_pair_t U32 U32) :=
+  Result.ret
+  {
+    struct_with_pair_p :=
+      {
+        pair_x := (U32.ofInt 1 (by intlit)),
+        pair_y := (U32.ofInt 2 (by intlit))
+      }
+  }
+
+/- [no_nested_borrows::test_constants] -/
+def test_constants_fwd : Result Unit :=
+  do
+    let swt ← new_tuple1_fwd
+    let (i, _) := swt.struct_with_tuple_p
+    if h: not (i = (U32.ofInt 1 (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 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))
-  
-  /- [no_nested_borrows::test_shared_borrow_enum1] -/
-  def test_shared_borrow_enum1_fwd (l : list_t UInt32) : Result UInt32 :=
-    match h: l with
-    | list_t.Cons i l0 => Result.ret (UInt32.ofNatCore 1 (by intlit))
-    | list_t.Nil => 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))
-  
+    else
+      do
+        let swt0 ← new_tuple2_fwd
+        let (i0, _) := swt0.struct_with_tuple_p
+        if h: not (i0 = (I16.ofInt 1 (by intlit)))
+        then Result.fail Error.panic
+        else
+          do
+            let swt1 ← new_tuple3_fwd
+            let (i1, _) := swt1.struct_with_tuple_p
+            if h: not (i1 = (U64.ofInt 1 (by intlit)))
+            then Result.fail Error.panic
+            else
+              do
+                let swp ← new_pair1_fwd
+                if h: not (swp.struct_with_pair_p.pair_x =
+                  (U32.ofInt 1 (by intlit)))
+                then Result.fail Error.panic
+                else Result.ret ()
+
+/- Unit test for [no_nested_borrows::test_constants] -/
+#assert (test_constants_fwd == .ret ())
+
+/- [no_nested_borrows::test_weird_borrows1] -/
+def test_weird_borrows1_fwd : Result Unit :=
+  Result.ret ()
+
+/- Unit test for [no_nested_borrows::test_weird_borrows1] -/
+#assert (test_weird_borrows1_fwd == .ret ())
+
+/- [no_nested_borrows::test_mem_replace] -/
+def test_mem_replace_fwd_back (px : U32) : Result U32 :=
+  let y := mem_replace_fwd U32 px (U32.ofInt 1 (by intlit))
+  if h: not (y = (U32.ofInt 0 (by intlit)))
+  then Result.fail Error.panic
+  else Result.ret (U32.ofInt 2 (by intlit))
+
+/- [no_nested_borrows::test_shared_borrow_bool1] -/
+def test_shared_borrow_bool1_fwd (b : Bool) : Result U32 :=
+  if h: b
+  then Result.ret (U32.ofInt 0 (by intlit))
+  else Result.ret (U32.ofInt 1 (by intlit))
+
+/- [no_nested_borrows::test_shared_borrow_bool2] -/
+def test_shared_borrow_bool2_fwd : Result U32 :=
+  Result.ret (U32.ofInt 0 (by intlit))
+
+/- [no_nested_borrows::test_shared_borrow_enum1] -/
+def test_shared_borrow_enum1_fwd (l : list_t U32) : Result U32 :=
+  match h: l with
+  | list_t.Cons i l0 => Result.ret (U32.ofInt 1 (by intlit))
+  | list_t.Nil => Result.ret (U32.ofInt 0 (by intlit))
+
+/- [no_nested_borrows::test_shared_borrow_enum2] -/
+def test_shared_borrow_enum2_fwd : Result U32 :=
+  Result.ret (U32.ofInt 0 (by intlit))
+
diff --git a/tests/lean/misc-paper/Base/Primitives.lean b/tests/lean/misc-paper/Base/Primitives.lean
index 5b64e908..034f41b2 100644
--- a/tests/lean/misc-paper/Base/Primitives.lean
+++ b/tests/lean/misc-paper/Base/Primitives.lean
@@ -3,6 +3,28 @@ import Lean.Meta.Tactic.Simp
 import Init.Data.List.Basic
 import Mathlib.Tactic.RunCmd
 
+--------------------
+-- ASSERT COMMAND --
+--------------------
+
+open Lean Elab Command Term Meta
+
+syntax (name := assert) "#assert" term: command
+
+@[command_elab assert]
+unsafe
+def assertImpl : CommandElab := fun (_stx: Syntax) => do
+  runTermElabM (fun _ => do
+    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 ())
+
+#eval 2 == 2
+#assert (2 == 2)
+
 -------------
 -- PRELUDE --
 -------------
@@ -12,6 +34,7 @@ import Mathlib.Tactic.RunCmd
 inductive Error where
    | assertionFailure: Error
    | integerOverflow: Error
+   | divisionByZero: Error
    | arrayOutOfBounds: Error
    | maximumSizeExceeded: Error
    | panic: Error
@@ -89,17 +112,13 @@ macro "let" e:term " <-- " f:term : doElem =>
 -- MACHINE INTEGERS --
 ----------------------
 
--- 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.
+-- We redefine our machine integers types.
 
--- 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.)
+-- For Isize/Usize, we reuse `getNumBits` from `USize`. 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
@@ -111,236 +130,435 @@ macro "let" e:term " <-- " f:term : doElem =>
 -- 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
-
-macro_rules
-  | `(tactic| intlit) => `(tactic|
-    match USize.size, usize_size_eq with
-    | _, Or.inl rfl => decide
-    | _, Or.inr rfl => decide)
-
--- This is how the macro is expected to be used
-#eval USize.ofNatCore 0 (by intlit)
-
--- 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`).
+open System.Platform.getNumBits
+
+-- TODO: is there a way of only importing System.Platform.getNumBits?
+--
+@[simp] def size_num_bits : Nat := (System.Platform.getNumBits ()).val
+
+-- Remark: Lean seems to use < for the comparisons with the upper bounds by convention.
+-- We keep the F* convention for now.
+@[simp] def Isize.min : Int := - (HPow.hPow 2 (size_num_bits - 1))
+@[simp] def Isize.max : Int := (HPow.hPow 2 (size_num_bits - 1)) - 1
+@[simp] def I8.min    : Int := - (HPow.hPow 2 7)
+@[simp] def I8.max    : Int := HPow.hPow 2 7 - 1
+@[simp] def I16.min   : Int := - (HPow.hPow 2 15)
+@[simp] def I16.max   : Int := HPow.hPow 2 15 - 1
+@[simp] def I32.min   : Int := -(HPow.hPow 2 31)
+@[simp] def I32.max   : Int := HPow.hPow 2 31 - 1
+@[simp] def I64.min   : Int := -(HPow.hPow 2 63)
+@[simp] def I64.max   : Int := HPow.hPow 2 63 - 1
+@[simp] def I128.min  : Int := -(HPow.hPow 2 127)
+@[simp] def I128.max  : Int := HPow.hPow 2 127 - 1
+@[simp] def Usize.min : Int := 0
+@[simp] def Usize.max : Int := HPow.hPow 2 size_num_bits - 1
+@[simp] def U8.min    : Int := 0
+@[simp] def U8.max    : Int := HPow.hPow 2 8 - 1
+@[simp] def U16.min   : Int := 0
+@[simp] def U16.max   : Int := HPow.hPow 2 16 - 1
+@[simp] def U32.min   : Int := 0
+@[simp] def U32.max   : Int := HPow.hPow 2 32 - 1
+@[simp] def U64.min   : Int := 0
+@[simp] def U64.max   : Int := HPow.hPow 2 64 - 1
+@[simp] def U128.min  : Int := 0
+@[simp] def U128.max  : Int := HPow.hPow 2 128 - 1
+
+#assert (I8.min   == -128)
+#assert (I8.max   == 127)
+#assert (I16.min  == -32768)
+#assert (I16.max  == 32767)
+#assert (I32.min  == -2147483648)
+#assert (I32.max  == 2147483647)
+#assert (I64.min  == -9223372036854775808)
+#assert (I64.max  == 9223372036854775807)
+#assert (I128.min == -170141183460469231731687303715884105728)
+#assert (I128.max == 170141183460469231731687303715884105727)
+#assert (U8.min   == 0)
+#assert (U8.max   == 255)
+#assert (U16.min  == 0)
+#assert (U16.max  == 65535)
+#assert (U32.min  == 0)
+#assert (U32.max  == 4294967295)
+#assert (U64.min  == 0)
+#assert (U64.max  == 18446744073709551615)
+#assert (U128.min == 0)
+#assert (U128.max == 340282366920938463463374607431768211455)
+
+inductive ScalarTy :=
+| Isize
+| I8
+| I16
+| I32
+| I64
+| I128
+| Usize
+| U8
+| U16
+| U32
+| U64
+| U128
+
+def Scalar.min (ty : ScalarTy) : Int :=
+  match ty with
+  | .Isize => Isize.min
+  | .I8    => I8.min
+  | .I16   => I16.min
+  | .I32   => I32.min
+  | .I64   => I64.min
+  | .I128  => I128.min
+  | .Usize => Usize.min
+  | .U8    => U8.min
+  | .U16   => U16.min
+  | .U32   => U32.min
+  | .U64   => U64.min
+  | .U128  => U128.min
+
+def Scalar.max (ty : ScalarTy) : Int :=
+  match ty with
+  | .Isize => Isize.max
+  | .I8    => I8.max
+  | .I16   => I16.max
+  | .I32   => I32.max
+  | .I64   => I64.max
+  | .I128  => I128.max
+  | .Usize => Usize.max
+  | .U8    => U8.max
+  | .U16   => U16.max
+  | .U32   => U32.max
+  | .U64   => U64.max
+  | .U128  => U128.max
+
+-- "Conservative" bounds
+-- We use those because we can't compare to the isize bounds (which can't
+-- reduce at compile-time). Whenever we perform an arithmetic operation like
+-- addition we need to check that the result is in bounds: we first compare
+-- to the conservative bounds, which reduce, then compare to the real bounds.
 -- This is useful for the various #asserts that we want to reduce at
 -- type-checking time.
+def Scalar.cMin (ty : ScalarTy) : Int :=
+  match ty with
+  | .Isize => I32.min
+  | _ => Scalar.min ty
+
+def Scalar.cMax (ty : ScalarTy) : Int :=
+  match ty with
+  | .Isize => I32.max
+  | .Usize => U32.max
+  | _ => Scalar.max ty
+
+theorem Scalar.cMin_bound ty : Scalar.min ty <= Scalar.cMin ty := by sorry
+theorem Scalar.cMax_bound ty : Scalar.min ty <= Scalar.cMin ty := by sorry
+
+structure Scalar (ty : ScalarTy) where
+  val : Int
+  hmin : Scalar.min ty <= val
+  hmax : val <= Scalar.max ty
+
+theorem Scalar.bound_suffices (ty : ScalarTy) (x : Int) :
+  Scalar.cMin ty <= x && x <= Scalar.cMax ty ->
+  (decide (Scalar.min ty ≤ x) && decide (x ≤ Scalar.max ty)) = true
+  := by sorry
+
+def Scalar.ofIntCore {ty : ScalarTy} (x : Int)
+  (hmin : Scalar.min ty <= x) (hmax : x <= Scalar.max ty) : Scalar ty :=
+  { val := x, hmin := hmin, hmax := hmax }
+
+def Scalar.ofInt {ty : ScalarTy} (x : Int)
+  (h : Scalar.min ty <= x && x <= Scalar.max ty) : Scalar ty :=
+  let hmin: Scalar.min ty <= x := by sorry
+  let hmax: x <= Scalar.max ty := by sorry
+  Scalar.ofIntCore x hmin hmax
 
 -- 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 :=
-  -- NOTE: the test USize.toNat n - m >= 0 seems to always succeed?
-  if n >= m then
-    let n' := USize.toNat n
-    let m' := USize.toNat n
-    let r := USize.ofNatCore (n' - m') (by
-      have h: n' - m' <= n' := by
-        apply Nat.sub_le_of_le_add
-        case h => rewrite [ Nat.add_comm ]; apply Nat.le_add_left
-      apply Nat.lt_of_le_of_lt h
-      apply n.val.isLt
-    )
-    return r
-  else
-    fail integerOverflow
-
-@[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 :=
-  if h: m > 0 then
-    .ret ⟨ n.val % m.val, by
-      have h1: ↑m.val < USize.size := m.val.isLt
-      have h2: n.val.val % m.val.val < m.val.val := @Nat.mod_lt n.val m.val h
-      apply Nat.lt_trans h2 h1
-    ⟩
-  else
-    .fail integerOverflow
+def Scalar.tryMk (ty : ScalarTy) (x : Int) : Result (Scalar ty) :=
+  -- TODO: write this with only one if then else
+  if hmin_cons: Scalar.cMin ty <= x || Scalar.min ty <= x then
+    if hmax_cons: x <= Scalar.cMax ty || x <= Scalar.max ty then
+      let hmin: Scalar.min ty <= x := by sorry
+      let hmax: x <= Scalar.max ty := by sorry
+      return Scalar.ofIntCore x hmin hmax
+    else fail integerOverflow
+  else fail integerOverflow
+
+def Scalar.neg {ty : ScalarTy} (x : Scalar ty) : Result (Scalar ty) := Scalar.tryMk ty (- x.val)
+
+def Scalar.div {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+  if y.val != 0 then Scalar.tryMk ty (x.val / y.val) else fail divisionByZero
+
+-- Checking that the % operation in Lean computes the same as the remainder operation in Rust
+#assert 1 % 2 = (1:Int)
+#assert (-1) % 2 = -1
+#assert 1 % (-2) = 1
+#assert (-1) % (-2) = -1
+
+def Scalar.rem {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+  if y.val != 0 then Scalar.tryMk ty (x.val % y.val) else fail divisionByZero
+
+def Scalar.add {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+  Scalar.tryMk ty (x.val + y.val)
+
+def Scalar.sub {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+  Scalar.tryMk ty (x.val - y.val)
+
+def Scalar.mul {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+  Scalar.tryMk ty (x.val * y.val)
+
+-- TODO: instances of +, -, * etc. for scalars
+
+-- Cast an integer from a [src_ty] to a [tgt_ty]
+-- TODO: check the semantics of casts in Rust
+def Scalar.cast {src_ty : ScalarTy} (tgt_ty : ScalarTy) (x : Scalar src_ty) : Result (Scalar tgt_ty) :=
+  Scalar.tryMk tgt_ty x.val
+
+-- The scalar types
+-- We declare the definitions as reducible so that Lean can unfold them (useful
+-- for type class resolution for instance).
+@[reducible] def Isize := Scalar .Isize
+@[reducible] def I8    := Scalar .I8
+@[reducible] def I16   := Scalar .I16
+@[reducible] def I32   := Scalar .I32
+@[reducible] def I64   := Scalar .I64
+@[reducible] def I128  := Scalar .I128
+@[reducible] def Usize := Scalar .Usize
+@[reducible] def U8    := Scalar .U8
+@[reducible] def U16   := Scalar .U16
+@[reducible] def U32   := Scalar .U32
+@[reducible] def U64   := Scalar .U64
+@[reducible] def U128  := Scalar .U128
+
+-- TODO: below: not sure this is the best way.
+-- Should we rather overload operations like +, -, etc.?
+-- Also, it is possible to automate the generation of those definitions
+-- with macros (but would it be a good idea? It would be less easy to
+-- read the file, which is not supposed to change a lot)
+
+-- Negation
+
+/--
+Remark: there is no heterogeneous negation in the Lean prelude: we thus introduce
+one here.
+
+The notation typeclass for heterogeneous addition.
+This enables the notation `- a : β` where `a : α`.
+-/
+class HNeg (α : Type u) (β : outParam (Type v)) where
+  /-- `- a` computes the negation of `a`.
+  The meaning of this notation is type-dependent. -/
+  hNeg : α → β
+
+prefix:75  "-"   => HNeg.hNeg
+
+instance : HNeg Isize (Result Isize) where hNeg x := Scalar.neg x
+instance : HNeg I8 (Result I8) where hNeg x := Scalar.neg x
+instance : HNeg I16 (Result I16) where hNeg x := Scalar.neg x
+instance : HNeg I32 (Result I32) where hNeg x := Scalar.neg x
+instance : HNeg I64 (Result I64) where hNeg x := Scalar.neg x
+instance : HNeg I128 (Result I128) where hNeg x := Scalar.neg x
+
+-- Addition
+instance {ty} : HAdd (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+  hAdd x y := Scalar.add x y
+
+-- Substraction
+instance {ty} : HSub (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+  hSub x y := Scalar.sub x y
+
+-- Multiplication
+instance {ty} : HMul (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+  hMul x y := Scalar.mul x y
+
+-- Division
+instance {ty} : HDiv (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+  hDiv x y := Scalar.div x y
+
+-- Remainder
+instance {ty} : HMod (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+  hMod x y := Scalar.rem x y
+
+-- ofIntCore
+-- TODO: typeclass?
+def Isize.ofIntCore := @Scalar.ofIntCore .Isize
+def I8.ofIntCore    := @Scalar.ofIntCore .I8
+def I16.ofIntCore   := @Scalar.ofIntCore .I16
+def I32.ofIntCore   := @Scalar.ofIntCore .I32
+def I64.ofIntCore   := @Scalar.ofIntCore .I64
+def I128.ofIntCore  := @Scalar.ofIntCore .I128
+def Usize.ofIntCore := @Scalar.ofIntCore .Usize
+def U8.ofIntCore    := @Scalar.ofIntCore .U8
+def U16.ofIntCore   := @Scalar.ofIntCore .U16
+def U32.ofIntCore   := @Scalar.ofIntCore .U32
+def U64.ofIntCore   := @Scalar.ofIntCore .U64
+def U128.ofIntCore  := @Scalar.ofIntCore .U128
+
+--  ofInt
+-- TODO: typeclass?
+def Isize.ofInt := @Scalar.ofInt .Isize
+def I8.ofInt    := @Scalar.ofInt .I8
+def I16.ofInt   := @Scalar.ofInt .I16
+def I32.ofInt   := @Scalar.ofInt .I32
+def I64.ofInt   := @Scalar.ofInt .I64
+def I128.ofInt  := @Scalar.ofInt .I128
+def Usize.ofInt := @Scalar.ofInt .Usize
+def U8.ofInt    := @Scalar.ofInt .U8
+def U16.ofInt   := @Scalar.ofInt .U16
+def U32.ofInt   := @Scalar.ofInt .U32
+def U64.ofInt   := @Scalar.ofInt .U64
+def U128.ofInt  := @Scalar.ofInt .U128
+
+-- Comparisons
+instance {ty} : LT (Scalar ty) where
+  lt a b := LT.lt a.val b.val
+
+instance {ty} : LE (Scalar ty) where le a b := LE.le a.val b.val
+
+instance Scalar.decLt {ty} (a b : Scalar ty) : Decidable (LT.lt a b) := Int.decLt ..
+instance Scalar.decLe {ty} (a b : Scalar ty) : Decidable (LE.le a b) := Int.decLe ..
+
+theorem Scalar.eq_of_val_eq {ty} : ∀ {i j : Scalar ty}, Eq i.val j.val → Eq i j
+  | ⟨_, _, _⟩, ⟨_, _, _⟩, rfl => rfl
+
+theorem Scalar.val_eq_of_eq {ty} {i j : Scalar ty} (h : Eq i j) : Eq i.val j.val :=
+  h ▸ rfl
+
+theorem Scalar.ne_of_val_ne {ty} {i j : Scalar ty} (h : Not (Eq i.val j.val)) : Not (Eq i j) :=
+  fun h' => absurd (val_eq_of_eq h') h
+
+instance (ty : ScalarTy) : DecidableEq (Scalar ty) :=
+  fun i j =>
+    match decEq i.val j.val with
+    | isTrue h  => isTrue (Scalar.eq_of_val_eq h)
+    | isFalse h => isFalse (Scalar.ne_of_val_ne h)
+
+def Scalar.toInt {ty} (n : Scalar ty) : Int := n.val
+
+-- Tactic to prove that integers are in bounds
+syntax "intlit" : tactic
 
-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 :=
-  if m > 0 then
-    .ret ⟨ n.val / m.val, by
-      have h1: ↑n.val < USize.size := n.val.isLt
-      have h2: n.val.val / m.val.val <= n.val.val := @Nat.div_le_self n.val m.val
-      apply Nat.lt_of_le_of_lt h2 h1
-    ⟩
-  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
-  ofNatCore: (n:Nat) -> LT.lt n size -> t
-
-set_option hygiene false in
-run_cmd
-  for typeName in [`UInt8, `UInt16, `UInt32, `UInt64, `USize].map Lean.mkIdent do
-  Lean.Elab.Command.elabCommand (← `(
-    namespace $typeName
-    instance: MachineInteger $typeName where
-      size := size
-      val := val
-      ofNatCore := ofNatCore
-    end $typeName
-  ))
-
--- 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
+macro_rules
+  | `(tactic| intlit) => `(tactic| apply Scalar.bound_suffices ; decide)
+
+-- -- 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
+--   ofNatCore: (n:Nat) -> LT.lt n size -> t
+
+-- set_option hygiene false in
+-- run_cmd
+--   for typeName in [`UInt8, `UInt16, `UInt32, `UInt64, `USize].map Lean.mkIdent do
+--   Lean.Elab.Command.elabCommand (← `(
+--     namespace $typeName
+--     instance: MachineInteger $typeName where
+--       size := size
+--       val := val
+--       ofNatCore := ofNatCore
+--     end $typeName
+--   ))
+
+-- -- 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
 
 -------------
 -- 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_new (α : Type u): Vec α := ⟨ [], by {
-  match USize.size, usize_size_eq with
-  | _, Or.inl rfl => simp
-  | _, Or.inr rfl => simp
-  } ⟩
+def Vec (α : Type u) := { l : List α // List.length l <= Usize.max }
 
-#check vec_new
+def vec_new (α : Type u): Vec α := ⟨ [], by sorry ⟩
 
-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)
+  Usize.ofIntCore (List.length v) (by sorry) l
  
 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
--- 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) //
-  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
-    ⟨ return ⟨List.concat v.val x,
-      by
-        rw [List.length_concat]
-        assumption
-     ⟩, by simp ⟩
-  else
-    ⟨ fail maximumSizeExceeded, by simp ⟩
-
-#eval do
-  -- NOTE: the // notation is syntactic sugar for Subtype, a refinement with
-  -- fields val and property. However, Lean's elaborator can automatically
-  -- select the `val` field if the context provides a type annotation. We
-  -- 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?
-  -- 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)
   :=
-  if h : List.length v.val + 1 <= 4294967295 then
-    return ⟨ List.concat v.val x,
-      by
-        rw [List.length_concat]
-        have h': 4294967295 < USize.size := by intlit
-        apply Nat.lt_of_le_of_lt h h'
-    ⟩
-  else if h: List.length v.val + 1 < USize.size then
-    return ⟨List.concat v.val x,
-      by
-        rw [List.length_concat]
-        assumption
-     ⟩
+  if h : List.length v.val <= U32.max || List.length v.val <= Usize.max then
+    return ⟨ List.concat v.val x, by sorry ⟩
   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
+    -- TODO: maybe we should redefine a list library which uses integers
+    -- (instead of natural numbers)
+    let i : Nat :=
+      match i.val with
+      | .ofNat n => n
+      | .negSucc n => by sorry -- TODO: we can't get here
+    let isLt: i < USize.size := by sorry
+    let i : Fin USize.size := { val := i, isLt := isLt }
     .ret ⟨ List.set v.val i.val x, by
-      have h: List.length v.val < USize.size := v.property
+      have h: List.length v.val <= Usize.max := v.property
       rewrite [ List.length_set v.val i.val x ]
       assumption
     ⟩
   else
     .fail arrayOutOfBounds
 
-def vec_index_fwd (α : Type u) (v: Vec α) (i: USize): Result α :=
-  if h: i.val < List.length v.val then
+def vec_index_fwd (α : Type u) (v: Vec α) (i: Usize): Result α :=
+  if i.val < List.length v.val then
+    let i : Nat :=
+      match i.val with
+      | .ofNat n => n
+      | .negSucc n => by sorry -- TODO: we can't get here
+    let isLt: i < USize.size := by sorry
+    let i : Fin USize.size := { val := i, isLt := isLt }
+    let h: i < List.length v.val := by sorry
     .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 :=
-  if h: i.val < List.length v.val then
+def vec_index_mut_fwd (α : Type u) (v: Vec α) (i: Usize): Result α :=
+  if i.val < List.length v.val then
+    let i : Nat :=
+      match i.val with
+      | .ofNat n => n
+      | .negSucc n => by sorry -- TODO: we can't get here
+    let isLt: i < USize.size := by sorry
+    let i : Fin USize.size := { val := i, isLt := isLt }
+    let h: i < List.length v.val := by sorry
     .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
+    let i : Nat :=
+      match i.val with
+      | .ofNat n => n
+      | .negSucc n => by sorry -- TODO: we can't get here
+    let isLt: i < USize.size := by sorry
+    let i : Fin USize.size := { val := i, isLt := isLt }
     .ret ⟨ List.set v.val i.val x, by
-      have h: List.length v.val < USize.size := v.property
+      have h: List.length v.val <= Usize.max := v.property
       rewrite [ List.length_set v.val i.val x ]
       assumption
     ⟩
@@ -360,33 +578,3 @@ def mem_replace_back (a : Type) (_ : a) (y : a) : a :=
 /-- Aeneas-translated function -- useful to reduce non-recursive definitions.
  Use with `simp [ aeneas ]` -/
 register_simp_attr aeneas
-
---------------------
--- ASSERT COMMAND --
---------------------
-
-open Lean Elab Command Term Meta
-
-syntax (name := assert) "#assert" term: command
-
-@[command_elab assert]
-unsafe
-def assertImpl : CommandElab := fun (_stx: Syntax) => do
-  runTermElabM (fun _ => do
-    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 ())
-
-#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 05fde52c..0b16fb8e 100644
--- a/tests/lean/misc-paper/Paper.lean
+++ b/tests/lean/misc-paper/Paper.lean
@@ -2,126 +2,122 @@
 -- [paper]
 import Base.Primitives
 
-structure OpaqueDefs where
-  
-  /- [paper::ref_incr] -/
-  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 :=
-    do
-      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 ())
-  
-  /- [paper::choose] -/
-  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 h: b
-    then Result.ret (ret0, y)
-    else Result.ret (x, ret0)
-  
-  /- [paper::test_choose] -/
-  def test_choose_fwd : Result Unit :=
-    do
-      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 h: not (z0 = (Int32.ofNatCore 1 (by intlit)))
-      then Result.fail Error.panic
-      else
-        do
-          let (x, y) ←
-            choose_back Int32 true (Int32.ofNatCore 0 (by intlit))
-              (Int32.ofNatCore 0 (by intlit)) z0
-          if h: not (x = (Int32.ofNatCore 1 (by intlit)))
+/- [paper::ref_incr] -/
+def ref_incr_fwd_back (x : I32) : Result I32 :=
+  x + (I32.ofInt 1 (by intlit))
+
+/- [paper::test_incr] -/
+def test_incr_fwd : Result Unit :=
+  do
+    let x ← ref_incr_fwd_back (I32.ofInt 0 (by intlit))
+    if h: not (x = (I32.ofInt 1 (by intlit)))
+    then Result.fail Error.panic
+    else Result.ret ()
+
+/- Unit test for [paper::test_incr] -/
+#assert (test_incr_fwd == .ret ())
+
+/- [paper::choose] -/
+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 h: b
+  then Result.ret (ret0, y)
+  else Result.ret (x, ret0)
+
+/- [paper::test_choose] -/
+def test_choose_fwd : Result Unit :=
+  do
+    let z ←
+      choose_fwd I32 true (I32.ofInt 0 (by intlit)) (I32.ofInt 0 (by intlit))
+    let z0 ← z + (I32.ofInt 1 (by intlit))
+    if h: not (z0 = (I32.ofInt 1 (by intlit)))
+    then Result.fail Error.panic
+    else
+      do
+        let (x, y) ←
+          choose_back I32 true (I32.ofInt 0 (by intlit))
+            (I32.ofInt 0 (by intlit)) z0
+        if h: not (x = (I32.ofInt 1 (by intlit)))
+        then Result.fail Error.panic
+        else
+          if h: not (y = (I32.ofInt 0 (by intlit)))
           then Result.fail Error.panic
-          else
-            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 ())
-  
-  /- [paper::List] -/
-  inductive list_t (T : Type) :=
-  | Cons : T -> list_t T -> list_t T
-  | Nil : list_t T
-  
-  /- [paper::list_nth_mut] -/
-  def list_nth_mut_fwd (T : Type) (l : list_t T) (i : UInt32) : Result T :=
-    match h: l with
-    | list_t.Cons x tl =>
-      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))
-          list_nth_mut_fwd T tl i0
-    | list_t.Nil => 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 h: l with
-    | list_t.Cons x tl =>
-      if h: i = (UInt32.ofNatCore 0 (by intlit))
-      then Result.ret (list_t.Cons 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.Cons x tl0)
-    | list_t.Nil => Result.fail Error.panic
-  
-  /- [paper::sum] -/
-  def sum_fwd (l : list_t Int32) : Result Int32 :=
-    match h: l with
-    | list_t.Cons x tl => do
-                            let i ← sum_fwd tl
-                            Int32.checked_add x i
-    | list_t.Nil => Result.ret (Int32.ofNatCore 0 (by intlit))
-  
-  /- [paper::test_nth] -/
-  def test_nth_fwd : Result Unit :=
-    do
-      let l := list_t.Nil
-      let l0 := list_t.Cons (Int32.ofNatCore 3 (by intlit)) l
-      let l1 := list_t.Cons (Int32.ofNatCore 2 (by intlit)) l0
-      let x ←
-        list_nth_mut_fwd Int32 (list_t.Cons (Int32.ofNatCore 1 (by intlit)) l1)
-          (UInt32.ofNatCore 2 (by intlit))
-      let x0 ← Int32.checked_add x (Int32.ofNatCore 1 (by intlit))
-      let l2 ←
-        list_nth_mut_back Int32 (list_t.Cons (Int32.ofNatCore 1 (by intlit))
-          l1) (UInt32.ofNatCore 2 (by intlit)) x0
-      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 ())
-  
-  /- [paper::call_choose] -/
-  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
-  
+          else Result.ret ()
+
+/- Unit test for [paper::test_choose] -/
+#assert (test_choose_fwd == .ret ())
+
+/- [paper::List] -/
+inductive list_t (T : Type) :=
+| Cons : T -> list_t T -> list_t T
+| Nil : list_t T
+
+/- [paper::list_nth_mut] -/
+def list_nth_mut_fwd (T : Type) (l : list_t T) (i : U32) : Result T :=
+  match h: l with
+  | list_t.Cons x tl =>
+    if h: i = (U32.ofInt 0 (by intlit))
+    then Result.ret x
+    else do
+           let i0 ← i - (U32.ofInt 1 (by intlit))
+           list_nth_mut_fwd T tl i0
+  | list_t.Nil => Result.fail Error.panic
+
+/- [paper::list_nth_mut] -/
+def list_nth_mut_back
+  (T : Type) (l : list_t T) (i : U32) (ret0 : T) : Result (list_t T) :=
+  match h: l with
+  | list_t.Cons x tl =>
+    if h: i = (U32.ofInt 0 (by intlit))
+    then Result.ret (list_t.Cons ret0 tl)
+    else
+      do
+        let i0 ← i - (U32.ofInt 1 (by intlit))
+        let tl0 ← list_nth_mut_back T tl i0 ret0
+        Result.ret (list_t.Cons x tl0)
+  | list_t.Nil => Result.fail Error.panic
+
+/- [paper::sum] -/
+def sum_fwd (l : list_t I32) : Result I32 :=
+  match h: l with
+  | list_t.Cons x tl => do
+                          let i ← sum_fwd tl
+                          x + i
+  | list_t.Nil => Result.ret (I32.ofInt 0 (by intlit))
+
+/- [paper::test_nth] -/
+def test_nth_fwd : Result Unit :=
+  do
+    let l := list_t.Nil
+    let l0 := list_t.Cons (I32.ofInt 3 (by intlit)) l
+    let l1 := list_t.Cons (I32.ofInt 2 (by intlit)) l0
+    let x ←
+      list_nth_mut_fwd I32 (list_t.Cons (I32.ofInt 1 (by intlit)) l1)
+        (U32.ofInt 2 (by intlit))
+    let x0 ← x + (I32.ofInt 1 (by intlit))
+    let l2 ←
+      list_nth_mut_back I32 (list_t.Cons (I32.ofInt 1 (by intlit)) l1)
+        (U32.ofInt 2 (by intlit)) x0
+    let i ← sum_fwd l2
+    if h: not (i = (I32.ofInt 7 (by intlit)))
+    then Result.fail Error.panic
+    else Result.ret ()
+
+/- Unit test for [paper::test_nth] -/
+#assert (test_nth_fwd == .ret ())
+
+/- [paper::call_choose] -/
+def call_choose_fwd (p : (U32 × U32)) : Result U32 :=
+  do
+    let (px, py) := p
+    let pz ← choose_fwd U32 true px py
+    let pz0 ← pz + (U32.ofInt 1 (by intlit))
+    let (px0, _) ← choose_back U32 true px py pz0
+    Result.ret px0
+
diff --git a/tests/lean/misc-polonius_list/Base/Primitives.lean b/tests/lean/misc-polonius_list/Base/Primitives.lean
index 5b64e908..034f41b2 100644
--- a/tests/lean/misc-polonius_list/Base/Primitives.lean
+++ b/tests/lean/misc-polonius_list/Base/Primitives.lean
@@ -3,6 +3,28 @@ import Lean.Meta.Tactic.Simp
 import Init.Data.List.Basic
 import Mathlib.Tactic.RunCmd
 
+--------------------
+-- ASSERT COMMAND --
+--------------------
+
+open Lean Elab Command Term Meta
+
+syntax (name := assert) "#assert" term: command
+
+@[command_elab assert]
+unsafe
+def assertImpl : CommandElab := fun (_stx: Syntax) => do
+  runTermElabM (fun _ => do
+    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 ())
+
+#eval 2 == 2
+#assert (2 == 2)
+
 -------------
 -- PRELUDE --
 -------------
@@ -12,6 +34,7 @@ import Mathlib.Tactic.RunCmd
 inductive Error where
    | assertionFailure: Error
    | integerOverflow: Error
+   | divisionByZero: Error
    | arrayOutOfBounds: Error
    | maximumSizeExceeded: Error
    | panic: Error
@@ -89,17 +112,13 @@ macro "let" e:term " <-- " f:term : doElem =>
 -- MACHINE INTEGERS --
 ----------------------
 
--- 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.
+-- We redefine our machine integers types.
 
--- 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.)
+-- For Isize/Usize, we reuse `getNumBits` from `USize`. 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
@@ -111,236 +130,435 @@ macro "let" e:term " <-- " f:term : doElem =>
 -- 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
-
-macro_rules
-  | `(tactic| intlit) => `(tactic|
-    match USize.size, usize_size_eq with
-    | _, Or.inl rfl => decide
-    | _, Or.inr rfl => decide)
-
--- This is how the macro is expected to be used
-#eval USize.ofNatCore 0 (by intlit)
-
--- 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`).
+open System.Platform.getNumBits
+
+-- TODO: is there a way of only importing System.Platform.getNumBits?
+--
+@[simp] def size_num_bits : Nat := (System.Platform.getNumBits ()).val
+
+-- Remark: Lean seems to use < for the comparisons with the upper bounds by convention.
+-- We keep the F* convention for now.
+@[simp] def Isize.min : Int := - (HPow.hPow 2 (size_num_bits - 1))
+@[simp] def Isize.max : Int := (HPow.hPow 2 (size_num_bits - 1)) - 1
+@[simp] def I8.min    : Int := - (HPow.hPow 2 7)
+@[simp] def I8.max    : Int := HPow.hPow 2 7 - 1
+@[simp] def I16.min   : Int := - (HPow.hPow 2 15)
+@[simp] def I16.max   : Int := HPow.hPow 2 15 - 1
+@[simp] def I32.min   : Int := -(HPow.hPow 2 31)
+@[simp] def I32.max   : Int := HPow.hPow 2 31 - 1
+@[simp] def I64.min   : Int := -(HPow.hPow 2 63)
+@[simp] def I64.max   : Int := HPow.hPow 2 63 - 1
+@[simp] def I128.min  : Int := -(HPow.hPow 2 127)
+@[simp] def I128.max  : Int := HPow.hPow 2 127 - 1
+@[simp] def Usize.min : Int := 0
+@[simp] def Usize.max : Int := HPow.hPow 2 size_num_bits - 1
+@[simp] def U8.min    : Int := 0
+@[simp] def U8.max    : Int := HPow.hPow 2 8 - 1
+@[simp] def U16.min   : Int := 0
+@[simp] def U16.max   : Int := HPow.hPow 2 16 - 1
+@[simp] def U32.min   : Int := 0
+@[simp] def U32.max   : Int := HPow.hPow 2 32 - 1
+@[simp] def U64.min   : Int := 0
+@[simp] def U64.max   : Int := HPow.hPow 2 64 - 1
+@[simp] def U128.min  : Int := 0
+@[simp] def U128.max  : Int := HPow.hPow 2 128 - 1
+
+#assert (I8.min   == -128)
+#assert (I8.max   == 127)
+#assert (I16.min  == -32768)
+#assert (I16.max  == 32767)
+#assert (I32.min  == -2147483648)
+#assert (I32.max  == 2147483647)
+#assert (I64.min  == -9223372036854775808)
+#assert (I64.max  == 9223372036854775807)
+#assert (I128.min == -170141183460469231731687303715884105728)
+#assert (I128.max == 170141183460469231731687303715884105727)
+#assert (U8.min   == 0)
+#assert (U8.max   == 255)
+#assert (U16.min  == 0)
+#assert (U16.max  == 65535)
+#assert (U32.min  == 0)
+#assert (U32.max  == 4294967295)
+#assert (U64.min  == 0)
+#assert (U64.max  == 18446744073709551615)
+#assert (U128.min == 0)
+#assert (U128.max == 340282366920938463463374607431768211455)
+
+inductive ScalarTy :=
+| Isize
+| I8
+| I16
+| I32
+| I64
+| I128
+| Usize
+| U8
+| U16
+| U32
+| U64
+| U128
+
+def Scalar.min (ty : ScalarTy) : Int :=
+  match ty with
+  | .Isize => Isize.min
+  | .I8    => I8.min
+  | .I16   => I16.min
+  | .I32   => I32.min
+  | .I64   => I64.min
+  | .I128  => I128.min
+  | .Usize => Usize.min
+  | .U8    => U8.min
+  | .U16   => U16.min
+  | .U32   => U32.min
+  | .U64   => U64.min
+  | .U128  => U128.min
+
+def Scalar.max (ty : ScalarTy) : Int :=
+  match ty with
+  | .Isize => Isize.max
+  | .I8    => I8.max
+  | .I16   => I16.max
+  | .I32   => I32.max
+  | .I64   => I64.max
+  | .I128  => I128.max
+  | .Usize => Usize.max
+  | .U8    => U8.max
+  | .U16   => U16.max
+  | .U32   => U32.max
+  | .U64   => U64.max
+  | .U128  => U128.max
+
+-- "Conservative" bounds
+-- We use those because we can't compare to the isize bounds (which can't
+-- reduce at compile-time). Whenever we perform an arithmetic operation like
+-- addition we need to check that the result is in bounds: we first compare
+-- to the conservative bounds, which reduce, then compare to the real bounds.
 -- This is useful for the various #asserts that we want to reduce at
 -- type-checking time.
+def Scalar.cMin (ty : ScalarTy) : Int :=
+  match ty with
+  | .Isize => I32.min
+  | _ => Scalar.min ty
+
+def Scalar.cMax (ty : ScalarTy) : Int :=
+  match ty with
+  | .Isize => I32.max
+  | .Usize => U32.max
+  | _ => Scalar.max ty
+
+theorem Scalar.cMin_bound ty : Scalar.min ty <= Scalar.cMin ty := by sorry
+theorem Scalar.cMax_bound ty : Scalar.min ty <= Scalar.cMin ty := by sorry
+
+structure Scalar (ty : ScalarTy) where
+  val : Int
+  hmin : Scalar.min ty <= val
+  hmax : val <= Scalar.max ty
+
+theorem Scalar.bound_suffices (ty : ScalarTy) (x : Int) :
+  Scalar.cMin ty <= x && x <= Scalar.cMax ty ->
+  (decide (Scalar.min ty ≤ x) && decide (x ≤ Scalar.max ty)) = true
+  := by sorry
+
+def Scalar.ofIntCore {ty : ScalarTy} (x : Int)
+  (hmin : Scalar.min ty <= x) (hmax : x <= Scalar.max ty) : Scalar ty :=
+  { val := x, hmin := hmin, hmax := hmax }
+
+def Scalar.ofInt {ty : ScalarTy} (x : Int)
+  (h : Scalar.min ty <= x && x <= Scalar.max ty) : Scalar ty :=
+  let hmin: Scalar.min ty <= x := by sorry
+  let hmax: x <= Scalar.max ty := by sorry
+  Scalar.ofIntCore x hmin hmax
 
 -- 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 :=
-  -- NOTE: the test USize.toNat n - m >= 0 seems to always succeed?
-  if n >= m then
-    let n' := USize.toNat n
-    let m' := USize.toNat n
-    let r := USize.ofNatCore (n' - m') (by
-      have h: n' - m' <= n' := by
-        apply Nat.sub_le_of_le_add
-        case h => rewrite [ Nat.add_comm ]; apply Nat.le_add_left
-      apply Nat.lt_of_le_of_lt h
-      apply n.val.isLt
-    )
-    return r
-  else
-    fail integerOverflow
-
-@[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 :=
-  if h: m > 0 then
-    .ret ⟨ n.val % m.val, by
-      have h1: ↑m.val < USize.size := m.val.isLt
-      have h2: n.val.val % m.val.val < m.val.val := @Nat.mod_lt n.val m.val h
-      apply Nat.lt_trans h2 h1
-    ⟩
-  else
-    .fail integerOverflow
+def Scalar.tryMk (ty : ScalarTy) (x : Int) : Result (Scalar ty) :=
+  -- TODO: write this with only one if then else
+  if hmin_cons: Scalar.cMin ty <= x || Scalar.min ty <= x then
+    if hmax_cons: x <= Scalar.cMax ty || x <= Scalar.max ty then
+      let hmin: Scalar.min ty <= x := by sorry
+      let hmax: x <= Scalar.max ty := by sorry
+      return Scalar.ofIntCore x hmin hmax
+    else fail integerOverflow
+  else fail integerOverflow
+
+def Scalar.neg {ty : ScalarTy} (x : Scalar ty) : Result (Scalar ty) := Scalar.tryMk ty (- x.val)
+
+def Scalar.div {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+  if y.val != 0 then Scalar.tryMk ty (x.val / y.val) else fail divisionByZero
+
+-- Checking that the % operation in Lean computes the same as the remainder operation in Rust
+#assert 1 % 2 = (1:Int)
+#assert (-1) % 2 = -1
+#assert 1 % (-2) = 1
+#assert (-1) % (-2) = -1
+
+def Scalar.rem {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+  if y.val != 0 then Scalar.tryMk ty (x.val % y.val) else fail divisionByZero
+
+def Scalar.add {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+  Scalar.tryMk ty (x.val + y.val)
+
+def Scalar.sub {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+  Scalar.tryMk ty (x.val - y.val)
+
+def Scalar.mul {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+  Scalar.tryMk ty (x.val * y.val)
+
+-- TODO: instances of +, -, * etc. for scalars
+
+-- Cast an integer from a [src_ty] to a [tgt_ty]
+-- TODO: check the semantics of casts in Rust
+def Scalar.cast {src_ty : ScalarTy} (tgt_ty : ScalarTy) (x : Scalar src_ty) : Result (Scalar tgt_ty) :=
+  Scalar.tryMk tgt_ty x.val
+
+-- The scalar types
+-- We declare the definitions as reducible so that Lean can unfold them (useful
+-- for type class resolution for instance).
+@[reducible] def Isize := Scalar .Isize
+@[reducible] def I8    := Scalar .I8
+@[reducible] def I16   := Scalar .I16
+@[reducible] def I32   := Scalar .I32
+@[reducible] def I64   := Scalar .I64
+@[reducible] def I128  := Scalar .I128
+@[reducible] def Usize := Scalar .Usize
+@[reducible] def U8    := Scalar .U8
+@[reducible] def U16   := Scalar .U16
+@[reducible] def U32   := Scalar .U32
+@[reducible] def U64   := Scalar .U64
+@[reducible] def U128  := Scalar .U128
+
+-- TODO: below: not sure this is the best way.
+-- Should we rather overload operations like +, -, etc.?
+-- Also, it is possible to automate the generation of those definitions
+-- with macros (but would it be a good idea? It would be less easy to
+-- read the file, which is not supposed to change a lot)
+
+-- Negation
+
+/--
+Remark: there is no heterogeneous negation in the Lean prelude: we thus introduce
+one here.
+
+The notation typeclass for heterogeneous addition.
+This enables the notation `- a : β` where `a : α`.
+-/
+class HNeg (α : Type u) (β : outParam (Type v)) where
+  /-- `- a` computes the negation of `a`.
+  The meaning of this notation is type-dependent. -/
+  hNeg : α → β
+
+prefix:75  "-"   => HNeg.hNeg
+
+instance : HNeg Isize (Result Isize) where hNeg x := Scalar.neg x
+instance : HNeg I8 (Result I8) where hNeg x := Scalar.neg x
+instance : HNeg I16 (Result I16) where hNeg x := Scalar.neg x
+instance : HNeg I32 (Result I32) where hNeg x := Scalar.neg x
+instance : HNeg I64 (Result I64) where hNeg x := Scalar.neg x
+instance : HNeg I128 (Result I128) where hNeg x := Scalar.neg x
+
+-- Addition
+instance {ty} : HAdd (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+  hAdd x y := Scalar.add x y
+
+-- Substraction
+instance {ty} : HSub (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+  hSub x y := Scalar.sub x y
+
+-- Multiplication
+instance {ty} : HMul (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+  hMul x y := Scalar.mul x y
+
+-- Division
+instance {ty} : HDiv (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+  hDiv x y := Scalar.div x y
+
+-- Remainder
+instance {ty} : HMod (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+  hMod x y := Scalar.rem x y
+
+-- ofIntCore
+-- TODO: typeclass?
+def Isize.ofIntCore := @Scalar.ofIntCore .Isize
+def I8.ofIntCore    := @Scalar.ofIntCore .I8
+def I16.ofIntCore   := @Scalar.ofIntCore .I16
+def I32.ofIntCore   := @Scalar.ofIntCore .I32
+def I64.ofIntCore   := @Scalar.ofIntCore .I64
+def I128.ofIntCore  := @Scalar.ofIntCore .I128
+def Usize.ofIntCore := @Scalar.ofIntCore .Usize
+def U8.ofIntCore    := @Scalar.ofIntCore .U8
+def U16.ofIntCore   := @Scalar.ofIntCore .U16
+def U32.ofIntCore   := @Scalar.ofIntCore .U32
+def U64.ofIntCore   := @Scalar.ofIntCore .U64
+def U128.ofIntCore  := @Scalar.ofIntCore .U128
+
+--  ofInt
+-- TODO: typeclass?
+def Isize.ofInt := @Scalar.ofInt .Isize
+def I8.ofInt    := @Scalar.ofInt .I8
+def I16.ofInt   := @Scalar.ofInt .I16
+def I32.ofInt   := @Scalar.ofInt .I32
+def I64.ofInt   := @Scalar.ofInt .I64
+def I128.ofInt  := @Scalar.ofInt .I128
+def Usize.ofInt := @Scalar.ofInt .Usize
+def U8.ofInt    := @Scalar.ofInt .U8
+def U16.ofInt   := @Scalar.ofInt .U16
+def U32.ofInt   := @Scalar.ofInt .U32
+def U64.ofInt   := @Scalar.ofInt .U64
+def U128.ofInt  := @Scalar.ofInt .U128
+
+-- Comparisons
+instance {ty} : LT (Scalar ty) where
+  lt a b := LT.lt a.val b.val
+
+instance {ty} : LE (Scalar ty) where le a b := LE.le a.val b.val
+
+instance Scalar.decLt {ty} (a b : Scalar ty) : Decidable (LT.lt a b) := Int.decLt ..
+instance Scalar.decLe {ty} (a b : Scalar ty) : Decidable (LE.le a b) := Int.decLe ..
+
+theorem Scalar.eq_of_val_eq {ty} : ∀ {i j : Scalar ty}, Eq i.val j.val → Eq i j
+  | ⟨_, _, _⟩, ⟨_, _, _⟩, rfl => rfl
+
+theorem Scalar.val_eq_of_eq {ty} {i j : Scalar ty} (h : Eq i j) : Eq i.val j.val :=
+  h ▸ rfl
+
+theorem Scalar.ne_of_val_ne {ty} {i j : Scalar ty} (h : Not (Eq i.val j.val)) : Not (Eq i j) :=
+  fun h' => absurd (val_eq_of_eq h') h
+
+instance (ty : ScalarTy) : DecidableEq (Scalar ty) :=
+  fun i j =>
+    match decEq i.val j.val with
+    | isTrue h  => isTrue (Scalar.eq_of_val_eq h)
+    | isFalse h => isFalse (Scalar.ne_of_val_ne h)
+
+def Scalar.toInt {ty} (n : Scalar ty) : Int := n.val
+
+-- Tactic to prove that integers are in bounds
+syntax "intlit" : tactic
 
-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 :=
-  if m > 0 then
-    .ret ⟨ n.val / m.val, by
-      have h1: ↑n.val < USize.size := n.val.isLt
-      have h2: n.val.val / m.val.val <= n.val.val := @Nat.div_le_self n.val m.val
-      apply Nat.lt_of_le_of_lt h2 h1
-    ⟩
-  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
-  ofNatCore: (n:Nat) -> LT.lt n size -> t
-
-set_option hygiene false in
-run_cmd
-  for typeName in [`UInt8, `UInt16, `UInt32, `UInt64, `USize].map Lean.mkIdent do
-  Lean.Elab.Command.elabCommand (← `(
-    namespace $typeName
-    instance: MachineInteger $typeName where
-      size := size
-      val := val
-      ofNatCore := ofNatCore
-    end $typeName
-  ))
-
--- 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
+macro_rules
+  | `(tactic| intlit) => `(tactic| apply Scalar.bound_suffices ; decide)
+
+-- -- 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
+--   ofNatCore: (n:Nat) -> LT.lt n size -> t
+
+-- set_option hygiene false in
+-- run_cmd
+--   for typeName in [`UInt8, `UInt16, `UInt32, `UInt64, `USize].map Lean.mkIdent do
+--   Lean.Elab.Command.elabCommand (← `(
+--     namespace $typeName
+--     instance: MachineInteger $typeName where
+--       size := size
+--       val := val
+--       ofNatCore := ofNatCore
+--     end $typeName
+--   ))
+
+-- -- 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
 
 -------------
 -- 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_new (α : Type u): Vec α := ⟨ [], by {
-  match USize.size, usize_size_eq with
-  | _, Or.inl rfl => simp
-  | _, Or.inr rfl => simp
-  } ⟩
+def Vec (α : Type u) := { l : List α // List.length l <= Usize.max }
 
-#check vec_new
+def vec_new (α : Type u): Vec α := ⟨ [], by sorry ⟩
 
-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)
+  Usize.ofIntCore (List.length v) (by sorry) l
  
 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
--- 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) //
-  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
-    ⟨ return ⟨List.concat v.val x,
-      by
-        rw [List.length_concat]
-        assumption
-     ⟩, by simp ⟩
-  else
-    ⟨ fail maximumSizeExceeded, by simp ⟩
-
-#eval do
-  -- NOTE: the // notation is syntactic sugar for Subtype, a refinement with
-  -- fields val and property. However, Lean's elaborator can automatically
-  -- select the `val` field if the context provides a type annotation. We
-  -- 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?
-  -- 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)
   :=
-  if h : List.length v.val + 1 <= 4294967295 then
-    return ⟨ List.concat v.val x,
-      by
-        rw [List.length_concat]
-        have h': 4294967295 < USize.size := by intlit
-        apply Nat.lt_of_le_of_lt h h'
-    ⟩
-  else if h: List.length v.val + 1 < USize.size then
-    return ⟨List.concat v.val x,
-      by
-        rw [List.length_concat]
-        assumption
-     ⟩
+  if h : List.length v.val <= U32.max || List.length v.val <= Usize.max then
+    return ⟨ List.concat v.val x, by sorry ⟩
   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
+    -- TODO: maybe we should redefine a list library which uses integers
+    -- (instead of natural numbers)
+    let i : Nat :=
+      match i.val with
+      | .ofNat n => n
+      | .negSucc n => by sorry -- TODO: we can't get here
+    let isLt: i < USize.size := by sorry
+    let i : Fin USize.size := { val := i, isLt := isLt }
     .ret ⟨ List.set v.val i.val x, by
-      have h: List.length v.val < USize.size := v.property
+      have h: List.length v.val <= Usize.max := v.property
       rewrite [ List.length_set v.val i.val x ]
       assumption
     ⟩
   else
     .fail arrayOutOfBounds
 
-def vec_index_fwd (α : Type u) (v: Vec α) (i: USize): Result α :=
-  if h: i.val < List.length v.val then
+def vec_index_fwd (α : Type u) (v: Vec α) (i: Usize): Result α :=
+  if i.val < List.length v.val then
+    let i : Nat :=
+      match i.val with
+      | .ofNat n => n
+      | .negSucc n => by sorry -- TODO: we can't get here
+    let isLt: i < USize.size := by sorry
+    let i : Fin USize.size := { val := i, isLt := isLt }
+    let h: i < List.length v.val := by sorry
     .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 :=
-  if h: i.val < List.length v.val then
+def vec_index_mut_fwd (α : Type u) (v: Vec α) (i: Usize): Result α :=
+  if i.val < List.length v.val then
+    let i : Nat :=
+      match i.val with
+      | .ofNat n => n
+      | .negSucc n => by sorry -- TODO: we can't get here
+    let isLt: i < USize.size := by sorry
+    let i : Fin USize.size := { val := i, isLt := isLt }
+    let h: i < List.length v.val := by sorry
     .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
+    let i : Nat :=
+      match i.val with
+      | .ofNat n => n
+      | .negSucc n => by sorry -- TODO: we can't get here
+    let isLt: i < USize.size := by sorry
+    let i : Fin USize.size := { val := i, isLt := isLt }
     .ret ⟨ List.set v.val i.val x, by
-      have h: List.length v.val < USize.size := v.property
+      have h: List.length v.val <= Usize.max := v.property
       rewrite [ List.length_set v.val i.val x ]
       assumption
     ⟩
@@ -360,33 +578,3 @@ def mem_replace_back (a : Type) (_ : a) (y : a) : a :=
 /-- Aeneas-translated function -- useful to reduce non-recursive definitions.
  Use with `simp [ aeneas ]` -/
 register_simp_attr aeneas
-
---------------------
--- ASSERT COMMAND --
---------------------
-
-open Lean Elab Command Term Meta
-
-syntax (name := assert) "#assert" term: command
-
-@[command_elab assert]
-unsafe
-def assertImpl : CommandElab := fun (_stx: Syntax) => do
-  runTermElabM (fun _ => do
-    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 ())
-
-#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-polonius_list/PoloniusList.lean b/tests/lean/misc-polonius_list/PoloniusList.lean
index a3bbfd0a..79696996 100644
--- a/tests/lean/misc-polonius_list/PoloniusList.lean
+++ b/tests/lean/misc-polonius_list/PoloniusList.lean
@@ -2,35 +2,30 @@
 -- [polonius_list]
 import Base.Primitives
 
-structure OpaqueDefs where
-  
-  /- [polonius_list::List] -/
-  inductive list_t (T : Type) :=
-  | Cons : T -> list_t T -> list_t T
-  | Nil : list_t T
-  
-  /- [polonius_list::get_list_at_x] -/
-  def get_list_at_x_fwd
-    (ls : list_t UInt32) (x : UInt32) : Result (list_t UInt32) :=
-    match h: ls with
-    | list_t.Cons hd tl =>
-      if h: hd = x
-      then Result.ret (list_t.Cons hd tl)
-      else get_list_at_x_fwd tl x
-    | list_t.Nil => Result.ret list_t.Nil
-  
-  /- [polonius_list::get_list_at_x] -/
-  def get_list_at_x_back
-    (ls : list_t UInt32) (x : UInt32) (ret0 : list_t UInt32) :
-    Result (list_t UInt32)
-    :=
-    match h: ls with
-    | list_t.Cons hd tl =>
-      if h: hd = x
-      then Result.ret ret0
-      else
-        do
-          let tl0 ← get_list_at_x_back tl x ret0
-          Result.ret (list_t.Cons hd tl0)
-    | list_t.Nil => Result.ret ret0
-  
+/- [polonius_list::List] -/
+inductive list_t (T : Type) :=
+| Cons : T -> list_t T -> list_t T
+| Nil : list_t T
+
+/- [polonius_list::get_list_at_x] -/
+def get_list_at_x_fwd (ls : list_t U32) (x : U32) : Result (list_t U32) :=
+  match h: ls with
+  | list_t.Cons hd tl =>
+    if h: hd = x
+    then Result.ret (list_t.Cons hd tl)
+    else get_list_at_x_fwd tl x
+  | list_t.Nil => Result.ret list_t.Nil
+
+/- [polonius_list::get_list_at_x] -/
+def get_list_at_x_back
+  (ls : list_t U32) (x : U32) (ret0 : list_t U32) : Result (list_t U32) :=
+  match h: ls with
+  | list_t.Cons hd tl =>
+    if h: hd = x
+    then Result.ret ret0
+    else
+      do
+        let tl0 ← get_list_at_x_back tl x ret0
+        Result.ret (list_t.Cons hd tl0)
+  | list_t.Nil => Result.ret ret0
+